[{"data":1,"prerenderedAt":71840},["ShallowReactive",2],{"expert-notes-index":3},[4,575,41214,67231],{"id":5,"title":6,"authors":7,"body":9,"breadcrumb":517,"builders":522,"byline":533,"challenge":535,"courseAuthor":535,"courseLead":535,"dek":536,"description":537,"draft":538,"extension":539,"eyebrow":540,"finish":535,"fork":535,"hero":535,"heroAlt":535,"heroCta":535,"heroImage":535,"kind":541,"lessonCount":535,"meta":542,"navigation":104,"newsItems":535,"next":535,"ogImage":543,"order":535,"outcomes":535,"path":544,"publishDate":545,"readingTime":546,"related":547,"relatedProjects":548,"seo":568,"stem":570,"tags":571,"track":535,"trackName":535,"__hash__":574},"blog\u002Fblog\u002Fexpert-notes\u002F2-qubit-state-visualization.md","2-qubit states, visualized 8 ways",[8],"onri-jay-benally",{"type":10,"value":11,"toc":505},"minimark",[12,16,30,35,38,369,372,376,379,386,389,395,399,402,408,412,415,421,425,428,434,438,441,445,448,454,458,461,467,471,474,480,486,490,501],[13,14,15],"p",{},"A 2-qubit state is a four-dimensional complex vector. That's easy to write down and hard to picture, and entanglement makes it harder: the most interesting thing about the state is precisely the part you can't see by looking at each qubit alone. This walkthrough takes one small circuit and renders its state eight different ways, because each visualization makes a different property visible.",[13,17,18,19,24,25,29],{},"Everything below comes from Onri's runnable notebook, shared openly on ",[20,21,23],"a",{"href":22},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002F2_Qubit_State_Visualization_in_Python.ipynb","GitHub",". The code and figures are his, republished here with permission. You can run the whole thing end to end in ",[20,26,28],{"href":27},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002F2_Qubit_State_Visualization_in_Python.ipynb","Google Colab",".",[31,32,34],"h2",{"id":33},"one-circuit-two-states","One circuit, two states",[13,36,37],{},"The experiment is a controlled comparison. First, the baseline: the product state |00⟩, where each qubit has its own well-defined state and nothing is correlated. Then the same eight visualizations run again on an entangled state, built with a Hadamard, a CNOT, and a couple of single-qubit rotations:",[39,40,43],"code-block",{"name":41,"tag":42},"2_qubit_state_visualization.ipynb","Python",[44,45,50],"pre",{"className":46,"code":47,"language":48,"meta":49,"style":49},"language-python shiki shiki-themes one-dark-pro","import numpy as np\nfrom qiskit import QuantumCircuit\nfrom qiskit.quantum_info import Statevector, DensityMatrix\n\n# Create a circuit with 2 qubits\nqc = QuantumCircuit(2)\nqc.h(0) # Hadamard on q0\nqc.cx(0, 1) # CNOT with q0 as control, q1 as target\nqc.ry(np.pi \u002F 4, 1) # Y-rotation of pi\u002F4 on q1\nqc.z(1) # Phase flip (Z gate) on q1\n\nfinal_statevector = Statevector(qc)\nfinal_density_matrix = DensityMatrix(qc)\n\n# Generate all 8 plots for the final state\nplot_bloch_multivector_viz(final_density_matrix, \"Final Entangled State\")\nplot_q_sphere_viz(final_density_matrix, \"Final Entangled State\")\nplot_steering_ellipsoid(final_density_matrix, \"Steering Ellipsoid for Final State\")\nvisualize_schmidt_decomposition(final_statevector, \"Schmidt Decomposition for Final State\")\nexplain_toroidal_geometry()\nplot_correlation_invariants(final_density_matrix, \"Final Entangled State\")\nplot_wigner_function(final_density_matrix, \"Wigner Functions for Final State\")\nplot_density_matrix(final_density_matrix, \"Density Matrix for Final State\")\n","python","",[51,52,53,72,86,99,106,113,137,157,181,208,225,230,244,257,262,268,283,295,308,322,331,343,356],"code",{"__ignoreMap":49},[54,55,58,62,66,69],"span",{"class":56,"line":57},"line",1,[54,59,61],{"class":60},"seHd6","import",[54,63,65],{"class":64},"sn6KH"," numpy ",[54,67,68],{"class":60},"as",[54,70,71],{"class":64}," np\n",[54,73,75,78,81,83],{"class":56,"line":74},2,[54,76,77],{"class":60},"from",[54,79,80],{"class":64}," qiskit ",[54,82,61],{"class":60},[54,84,85],{"class":64}," QuantumCircuit\n",[54,87,89,91,94,96],{"class":56,"line":88},3,[54,90,77],{"class":60},[54,92,93],{"class":64}," qiskit.quantum_info ",[54,95,61],{"class":60},[54,97,98],{"class":64}," Statevector, DensityMatrix\n",[54,100,102],{"class":56,"line":101},4,[54,103,105],{"emptyLinePlaceholder":104},true,"\n",[54,107,109],{"class":56,"line":108},5,[54,110,112],{"class":111},"sV9Aq","# Create a circuit with 2 qubits\n",[54,114,116,119,123,127,130,134],{"class":56,"line":115},6,[54,117,118],{"class":64},"qc ",[54,120,122],{"class":121},"sjrmR","=",[54,124,126],{"class":125},"sVbv2"," QuantumCircuit",[54,128,129],{"class":64},"(",[54,131,133],{"class":132},"sVC51","2",[54,135,136],{"class":64},")\n",[54,138,140,143,146,148,151,154],{"class":56,"line":139},7,[54,141,142],{"class":64},"qc.",[54,144,145],{"class":125},"h",[54,147,129],{"class":64},[54,149,150],{"class":132},"0",[54,152,153],{"class":64},") ",[54,155,156],{"class":111},"# Hadamard on q0\n",[54,158,160,162,165,167,169,172,175,178],{"class":56,"line":159},8,[54,161,142],{"class":64},[54,163,164],{"class":125},"cx",[54,166,129],{"class":64},[54,168,150],{"class":132},[54,170,171],{"class":64},", ",[54,173,174],{"class":132},"1",[54,176,177],{"class":64},") ",[54,179,180],{"class":111},"# CNOT with q0 as control, q1 as target\n",[54,182,184,186,189,192,195,198,200,202,205],{"class":56,"line":183},9,[54,185,142],{"class":64},[54,187,188],{"class":125},"ry",[54,190,191],{"class":64},"(np.pi ",[54,193,194],{"class":121},"\u002F",[54,196,197],{"class":132}," 4",[54,199,171],{"class":64},[54,201,174],{"class":132},[54,203,204],{"class":64},") ",[54,206,207],{"class":111},"# Y-rotation of pi\u002F4 on q1\n",[54,209,211,213,216,218,220,222],{"class":56,"line":210},10,[54,212,142],{"class":64},[54,214,215],{"class":125},"z",[54,217,129],{"class":64},[54,219,174],{"class":132},[54,221,153],{"class":64},[54,223,224],{"class":111},"# Phase flip (Z gate) on q1\n",[54,226,228],{"class":56,"line":227},11,[54,229,105],{"emptyLinePlaceholder":104},[54,231,233,236,238,241],{"class":56,"line":232},12,[54,234,235],{"class":64},"final_statevector ",[54,237,122],{"class":121},[54,239,240],{"class":125}," Statevector",[54,242,243],{"class":64},"(qc)\n",[54,245,247,250,252,255],{"class":56,"line":246},13,[54,248,249],{"class":64},"final_density_matrix ",[54,251,122],{"class":121},[54,253,254],{"class":125}," DensityMatrix",[54,256,243],{"class":64},[54,258,260],{"class":56,"line":259},14,[54,261,105],{"emptyLinePlaceholder":104},[54,263,265],{"class":56,"line":264},15,[54,266,267],{"class":111},"# Generate all 8 plots for the final state\n",[54,269,271,274,277,281],{"class":56,"line":270},16,[54,272,273],{"class":125},"plot_bloch_multivector_viz",[54,275,276],{"class":64},"(final_density_matrix, ",[54,278,280],{"class":279},"subq3","\"Final Entangled State\"",[54,282,136],{"class":64},[54,284,286,289,291,293],{"class":56,"line":285},17,[54,287,288],{"class":125},"plot_q_sphere_viz",[54,290,276],{"class":64},[54,292,280],{"class":279},[54,294,136],{"class":64},[54,296,298,301,303,306],{"class":56,"line":297},18,[54,299,300],{"class":125},"plot_steering_ellipsoid",[54,302,276],{"class":64},[54,304,305],{"class":279},"\"Steering Ellipsoid for Final State\"",[54,307,136],{"class":64},[54,309,311,314,317,320],{"class":56,"line":310},19,[54,312,313],{"class":125},"visualize_schmidt_decomposition",[54,315,316],{"class":64},"(final_statevector, ",[54,318,319],{"class":279},"\"Schmidt Decomposition for Final State\"",[54,321,136],{"class":64},[54,323,325,328],{"class":56,"line":324},20,[54,326,327],{"class":125},"explain_toroidal_geometry",[54,329,330],{"class":64},"()\n",[54,332,334,337,339,341],{"class":56,"line":333},21,[54,335,336],{"class":125},"plot_correlation_invariants",[54,338,276],{"class":64},[54,340,280],{"class":279},[54,342,136],{"class":64},[54,344,346,349,351,354],{"class":56,"line":345},22,[54,347,348],{"class":125},"plot_wigner_function",[54,350,276],{"class":64},[54,352,353],{"class":279},"\"Wigner Functions for Final State\"",[54,355,136],{"class":64},[54,357,359,362,364,367],{"class":56,"line":358},23,[54,360,361],{"class":125},"plot_density_matrix",[54,363,276],{"class":64},[54,365,366],{"class":279},"\"Density Matrix for Final State\"",[54,368,136],{"class":64},[13,370,371],{},"Every method below shows the entangled state; where the contrast with |00⟩ is the whole point, both are shown.",[31,373,375],{"id":374},"_1-two-bloch-spheres-one-giveaway","1. Two Bloch spheres, one giveaway",[13,377,378],{},"The Bloch multivector shows the state of each qubit on its own Bloch sphere. For |00⟩, both vectors reach the surface: each qubit is in a pure, definite state.",[380,381],"blog-figure",{"alt":382,"caption":383,"no":384,"src":385},"Two Bloch spheres for the initial state |00>, each with a vector pointing to the north pole","The product state |00⟩: each qubit's vector reaches the surface of its own sphere.","Fig. 1","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fbloch-initial.webp",[13,387,388],{},"For the entangled state, the vectors retreat inside the sphere. Each qubit on its own looks like a mixed state. The information isn't in either qubit, it's in the correlation between them. The shrunken vector is the visual signature of entanglement.",[380,390],{"alt":391,"caption":392,"no":393,"src":394},"Two Bloch spheres for the entangled state, each vector noticeably inside the sphere","The entangled state: both vectors pull inside the sphere; each qubit alone is mixed.","Fig. 2","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fbloch-entangled.webp",[31,396,398],{"id":397},"_2-the-q-sphere","2. The Q-sphere",[13,400,401],{},"The Q-sphere maps state amplitudes and phases onto a single sphere: the size of each point is the magnitude of the amplitude for a basis state, and its color is the phase. Where the Bloch view loses the correlations, the Q-sphere shows the full 2-qubit state at once.",[380,403],{"alt":404,"caption":405,"no":406,"src":407},"Q-sphere of the entangled state showing amplitude points with phase coloring","The entangled state on the Q-sphere: amplitude as point size, phase as color.","Fig. 3","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fqsphere-entangled.webp",[31,409,411],{"id":410},"_3-the-steering-ellipsoid","3. The steering ellipsoid",[13,413,414],{},"The steering ellipsoid shows the set of states qubit B can be \"steered\" into by measuring qubit A. If it collapses to a single point, there is no steering at all; a non-trivial ellipsoid means the state is entangled. For |00⟩ the notebook can't even draw one: the ellipsoid degenerates to a point, which is its own kind of answer.",[380,416],{"alt":417,"caption":418,"no":419,"src":420},"Quantum steering ellipsoid for the entangled state","The entangled state's steering ellipsoid: measuring qubit A can steer qubit B anywhere on this surface.","Fig. 4","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fsteering-ellipsoid-entangled.webp",[31,422,424],{"id":423},"_4-schmidt-decomposition","4. Schmidt decomposition",[13,426,427],{},"Any pure 2-qubit state can be written as |ψ⟩ = λ₀|u₀⟩|v₀⟩ + λ₁|u₁⟩|v₁⟩. The Schmidt coefficients λ₀ and λ₁ quantify entanglement directly: a product state has one coefficient equal to 1 and the other 0, while an entangled state splits the weight between both terms.",[380,429],{"alt":430,"caption":431,"no":432,"src":433},"Bar chart of the Schmidt coefficients for the entangled state","Both Schmidt coefficients are non-zero: the weight is shared, so the state is entangled.","Fig. 5","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fschmidt-entangled.webp",[31,435,437],{"id":436},"_5-toroidal-geometry","5. Toroidal geometry",[13,439,440],{},"Not everything has a standard plot. The toroidal view is a highly abstract method that maps entanglement invariants onto the surface of a torus. Powerful for classifying entanglement, but beyond standard plotting libraries. The notebook explains the concept and moves on; it's included because knowing a representation exists is half of finding a use for it.",[31,442,444],{"id":443},"_6-correlation-invariants","6. Correlation invariants",[13,446,447],{},"This plot shows the invariants of the state under local unitary operations: the eigenvalues of TᵀT, where T is the correlation matrix. These numbers don't change when either qubit is rotated on its own, so they capture the nature and strength of the correlation itself: the part of the state that no local operation can create or destroy.",[380,449],{"alt":450,"caption":451,"no":452,"src":453},"Bar plot of local correlation invariants for the entangled state","Local invariants of the entangled state: what remains when you ignore everything each qubit does alone.","Fig. 6","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fcorrelation-invariants-entangled.webp",[31,455,457],{"id":456},"_7-the-wigner-function","7. The Wigner function",[13,459,460],{},"A phase-space representation of each qubit's reduced state. Negative values, shown in red, are a key signature of non-classicality. No classical probability distribution can go below zero.",[380,462],{"alt":463,"caption":464,"no":465,"src":466},"Wigner function plots for the entangled state with negative regions shown in red","Wigner functions for the entangled state: the red, negative regions have no classical counterpart.","Fig. 7","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fwigner-entangled.webp",[31,468,470],{"id":469},"_8-the-density-matrix","8. The density matrix",[13,472,473],{},"The workhorse. Real parts in blue, imaginary parts in red; the diagonal holds the populations and the off-diagonal elements indicate coherence and entanglement. For |00⟩, a single bar stands alone.",[380,475],{"alt":476,"caption":477,"no":478,"src":479},"Density matrix of the initial state |00>, a single bar at the 00 position","The density matrix of |00⟩: one population, no coherences.","Fig. 8","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fdensity-initial.webp",[380,481],{"alt":482,"caption":483,"no":484,"src":485},"Density matrix of the entangled state with multiple bars including off-diagonal elements","The entangled state: off-diagonal structure appears, the coherences that carry the correlations.","Fig. 9","\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fdensity-entangled.webp",[31,487,489],{"id":488},"run-it-yourself","Run it yourself",[13,491,492,493,496,497,500],{},"Every figure above comes out of one script you can run top to bottom: ",[20,494,495],{"href":22},"open the notebook on GitHub"," or ",[20,498,499],{"href":27},"run it directly in Colab",". Change the circuit in Step 5, run again, and watch all eight views shift together. That's where the intuition comes from.",[502,503,504],"style",{},"html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":49,"searchDepth":74,"depth":74,"links":506},[507,508,509,510,511,512,513,514,515,516],{"id":33,"depth":74,"text":34},{"id":374,"depth":74,"text":375},{"id":397,"depth":74,"text":398},{"id":410,"depth":74,"text":411},{"id":423,"depth":74,"text":424},{"id":436,"depth":74,"text":437},{"id":443,"depth":74,"text":444},{"id":456,"depth":74,"text":457},{"id":469,"depth":74,"text":470},{"id":488,"depth":74,"text":489},[518,519,520,521],"Qollab","Blog","Expert notes","2-qubit states",[523],{"username":8,"name":524,"role":525,"bio":526,"links":527},"Onri Jay Benally","Quantum Hardware Engineer, IBM · University of Minnesota","A quantum hardware engineer at IBM and graduate researcher at the Nano Magnetism & Quantum Spintronics Lab, University of Minnesota-Twin Cities. Onri shares runnable notebooks that make hardware-level quantum concepts visible.",[528,531],{"label":529,"href":530},"GitHub ↗","https:\u002F\u002Fgithub.com\u002FOJB-Quantum",{"label":532,"href":22},"Original notebook ↗",{"username":8,"name":524,"role":534},"Quantum Hardware Engineer, IBM",null,"One entangled state, eight ways to see it. A runnable notebook that turns the same 2-qubit circuit into Bloch spheres, steering ellipsoids, Wigner functions, and more.","One entangled state, eight ways to see it. Onri Jay Benally's runnable notebook renders a 2-qubit state as Bloch spheres, Q-spheres, steering ellipsoids, Wigner functions, and more.",false,"md","Deep dive · Visualization","article",{},"\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fbloch-entangled.png","\u002Fblog\u002Fexpert-notes\u002F2-qubit-state-visualization","2026-07-15","6 min read",[],[549,556,563],{"username":550,"project":551,"title":552,"category":553,"thumb":554,"to":555},"q-inho","qave","QAVE","Visualization","\u002F_content\u002Fimages\u002Fqave\u002Fthumbnail.webp","\u002Fexplore\u002Fqave",{"username":557,"project":558,"title":559,"category":560,"thumb":561,"to":562},"AmberPincar","quantum-garden","Quantum Garden","Art","\u002F_content\u002Fimages\u002Fquantum-garden\u002Fthumbnail.webp","\u002Fexplore\u002Fquantum-garden",{"username":564,"project":565,"title":566,"category":560,"thumb":567},"xinyi","quantum-butterfly-field","Quantum Butterfly Field","\u002F_content\u002Fimages\u002Fquantum-butterfly-field\u002Fthumbnail.webp",{"description":569,"title":6},"One entangled state, eight ways to see it. Onri Jay Benally's runnable notebook renders a 2-qubit state as Bloch spheres, Q-spheres, steering ellipsoids, and more.","blog\u002Fexpert-notes\u002F2-qubit-state-visualization",[572,573,48],"visualization","quantum","hbpIuxS-SqH0GmOc1n1ds2B-L8OsTlp8DNHQ4vd3Ylw",{"id":576,"title":577,"authors":578,"body":580,"breadcrumb":41190,"builders":41192,"byline":41200,"challenge":535,"courseAuthor":535,"courseLead":535,"dek":535,"description":41202,"draft":538,"extension":539,"eyebrow":41203,"finish":535,"fork":535,"hero":535,"heroAlt":535,"heroCta":535,"heroImage":535,"kind":541,"lessonCount":535,"meta":41204,"navigation":104,"newsItems":535,"next":535,"ogImage":535,"order":535,"outcomes":535,"path":41205,"publishDate":545,"readingTime":41206,"related":41207,"relatedProjects":535,"seo":41208,"stem":41209,"tags":41210,"track":535,"trackName":535,"__hash__":41213},"blog\u002Fblog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization.md","Krylov Quantum Diagonalization of Lattice Hamiltonians",[579],"mirko-amico",{"type":10,"value":581,"toc":41161},[582,593,598,613,617,620,648,652,655,673,677,2064,2068,2073,2337,2648,3315,3635,3638,3642,3946,4207,4478,4682,5461,5698,5701,5939,6261,6268,7215,7220,7706,8084,8091,8095,8193,8529,8535,8538,8586,8590,8672,8738,8742,8746,8854,8978,8984,8988,8994,10265,11094,11644,11647,12635,13027,13274,13684,13690,13693,13696,13763,13769,13773,13777,13780,13786,14430,17197,17405,18624,19858,19903,19907,20455,20887,20890,20894,20984,20987,21112,21115,21173,21398,21401,21456,21462,21465,21469,21472,21579,21583,21586,21975,21978,22193,22199,22203,22671,22675,22730,23060,23066,23158,23531,23535,23560,23564,23604,23694,23725,24051,24064,24127,24466,24478,24542,24714,24798,24891,24897,24900,24917,24923,25172,25176,25179,25553,25679,26275,26307,26772,26854,26860,26926,27643,27746,28009,28263,28594,28777,29119,29859,30173,31788,31820,33316,33793,34487,35270,35604,35931,35993,36002,36006,36205,36563,36566,37094,37097,37397,37453,37717,38210,38469,39411,40415,40855,41110,41114,41123,41128,41133,41138,41142,41145,41151,41153,41158],[13,583,584],{},[585,586,587,588,592],"em",{},"Republished with the author's permission from ",[20,589,591],{"href":590},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fen\u002Ftutorials\u002Fkrylov-quantum-diagonalization","the original tutorial",". The text, code, and figures below are Mirko's, as published, lightly reformatted for this page.",[13,594,595],{},[585,596,597],{},"Usage estimate: 20 minutes on a Heron r2 (NOTE: This is an estimate only. Your runtime might vary.)",[44,599,601],{"className":46,"code":600,"language":48,"meta":49,"style":49},"# This cell is hidden from users – it disables some lint rules\n# ruff: noqa: E402 E722 F601\n",[51,602,603,608],{"__ignoreMap":49},[54,604,605],{"class":56,"line":57},[54,606,607],{"class":111},"# This cell is hidden from users – it disables some lint rules\n",[54,609,610],{"class":56,"line":74},[54,611,612],{"class":111},"# ruff: noqa: E402 E722 F601\n",[31,614,616],{"id":615},"background","Background",[13,618,619],{},"This tutorial demonstrates how to implement the Krylov Quantum Diagonalization Algorithm (KQD) within the context of Qiskit patterns. You will first learn about the theory behind the algorithm and then see a demonstration of its execution on a QPU.",[13,621,622,623,627,628,632,633,637,638,642,643,647],{},"Across disciplines, we're interested in learning ground state properties of quantum systems. Examples include understanding the fundamental nature of particles and forces, predicting and understanding the behavior of complex materials and understanding bio-chemical interactions and reactions. Because of the exponential growth of the Hilbert space and the correlation that arise in entangled systems, classical algorithm struggle to solve this problem for quantum systems of increasing size. At one end of the spectrum is the existing approach that takes advantage of the quantum hardware focus on variational quantum methods (for example, ",[20,624,626],{"href":625},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Ftutorials\u002Fspin-chain-vqe","variational quantum eigensolver","). These techniques face challenges with current devices because of the high number of function calls required in the optimization process, which adds a large overhead in resources once advanced error mitigation techniques are introduced, thus limiting their efficacy to small systems. At the other end of the spectrum, there are fault-tolerant quantum methods with performance guarantees (for example, ",[20,629,631],{"href":630},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fquant-ph\u002F0604193","quantum phase estimation","), which require deep circuits that can be executed only on a fault-tolerant device. For these reasons, we introduce here a quantum algorithm based on subspace methods (as described in this ",[20,634,636],{"href":635},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2312.00178","review paper","), the Krylov quantum diagonalization (KQD) algorithm. This algorithm performs well at large scale ",[20,639,641],{"href":640},"#references","[1]"," on existing quantum hardware, shares similar ",[20,644,646],{"href":645},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2110.07492","performance guarantees"," as phase estimation, is compatible with advanced error mitigation techniques, and could provide results that are classically inaccessible.",[31,649,651],{"id":650},"requirements","Requirements",[13,653,654],{},"Before starting this tutorial, be sure you have the following installed:",[656,657,658,666],"ul",{},[659,660,661,662,665],"li",{},"Qiskit SDK v2.0 or later, with ",[20,663,572],{"href":664},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fvisualization"," support",[659,667,668,669,672],{},"Qiskit Runtime v0.22 or later ( ",[51,670,671],{},"pip install qiskit-ibm-runtime"," )",[31,674,676],{"id":675},"setup","Setup",[44,678,680],{"className":46,"code":679,"language":48,"meta":49,"style":49},"import numpy as np\nimport scipy as sp\nimport matplotlib.pylab as plt\nfrom typing import Union, List\nimport itertools as it\nimport copy\nfrom sympy import Matrix\nimport warnings\n\nwarnings.filterwarnings(\"ignore\")\n\nfrom qiskit.quantum_info import SparsePauliOp, Pauli, StabilizerState\nfrom qiskit.circuit import Parameter, IfElseOp\nfrom qiskit import QuantumCircuit, QuantumRegister\nfrom qiskit.circuit.library import PauliEvolutionGate\nfrom qiskit.synthesis import LieTrotter\nfrom qiskit.transpiler import Target, CouplingMap\nfrom qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager\n\n\nfrom qiskit_ibm_runtime import (\n QiskitRuntimeService,\n EstimatorV2 as Estimator,\n)\n\n\ndef solve_regularized_gen_eig(\n h: np.ndarray,\n s: np.ndarray,\n threshold: float,\n k: int = 1,\n return_dimn: bool = False,\n) -> Union[float, List[float]]:\n \"\"\"\n Method for solving the generalized eigenvalue problem with regularization\n\n Args:\n h (numpy.ndarray):\n The effective representation of the matrix in the Krylov subspace\n s (numpy.ndarray):\n The matrix of overlaps between vectors of the Krylov subspace\n threshold (float):\n Cut-off value for the eigenvalue of s\n k (int):\n Number of eigenvalues to return\n return_dimn (bool):\n Whether to return the size of the regularized subspace\n\n Returns:\n lowest k-eigenvalue(s) that are the solution of the regularized generalized eigenvalue problem\n\n\n \"\"\"\n s_vals, s_vecs = sp.linalg.eigh(s)\n s_vecs = s_vecs.T\n good_vecs = np.array(\n [vec for val, vec in zip(s_vals, s_vecs) if val > threshold]\n )\n h_reg = good_vecs.conj() @ h @ good_vecs.T\n s_reg = good_vecs.conj() @ s @ good_vecs.T\n if k == 1:\n if return_dimn:\n return sp.linalg.eigh(h_reg, s_reg)[0][0], len(good_vecs)\n else:\n return sp.linalg.eigh(h_reg, s_reg)[0][0]\n else:\n if return_dimn:\n return sp.linalg.eigh(h_reg, s_reg)[0][:k], len(good_vecs)\n else:\n return sp.linalg.eigh(h_reg, s_reg)[0][:k]\n\n\ndef single_particle_gs(H_op, n_qubits):\n \"\"\"\n Find the ground state of the single particle(excitation) sector\n \"\"\"\n H_x = []\n for p, coeff in H_op.to_list():\n H_x.append(set([i for i, v in enumerate(Pauli(p).x) if v]))\n\n H_z = []\n for p, coeff in H_op.to_list():\n H_z.append(set([i for i, v in enumerate(Pauli(p).z) if v]))\n\n H_c = H_op.coeffs\n\n print(\"n_sys_qubits\", n_qubits)\n\n n_exc = 1\n sub_dimn = int(sp.special.comb(n_qubits + 1, n_exc))\n print(\"n_exc\", n_exc, \", subspace dimension\", sub_dimn)\n\n few_particle_H = np.zeros((sub_dimn, sub_dimn), dtype=complex)\n\n sparse_vecs = [\n set(vec) for vec in it.combinations(range(n_qubits + 1), r=n_exc)\n ] # list all of the possible sets of n_exc indices of 1s in n_exc-particle states\n\n m = 0\n for i, i_set in enumerate(sparse_vecs):\n for j, j_set in enumerate(sparse_vecs):\n m += 1\n\n if len(i_set.symmetric_difference(j_set)) \u003C= 2:\n for p_x, p_z, coeff in zip(H_x, H_z, H_c):\n if i_set.symmetric_difference(j_set) == p_x:\n sgn = ((-1j) ** len(p_x.intersection(p_z))) * (\n (-1) ** len(i_set.intersection(p_z))\n )\n else:\n sgn = 0\n\n few_particle_H[i, j] += sgn * coeff\n\n gs_en = min(np.linalg.eigvalsh(few_particle_H))\n print(\"single particle ground state energy: \", gs_en)\n return gs_en\n",[51,681,682,692,704,716,728,740,747,759,766,770,785,789,800,812,823,835,847,859,871,875,879,891,896,906,911,916,921,933,943,951,966,985,1003,1019,1025,1031,1036,1042,1048,1054,1060,1066,1072,1078,1084,1090,1096,1102,1107,1113,1119,1124,1129,1134,1151,1162,1178,1211,1217,1245,1268,1285,1294,1323,1331,1351,1359,1366,1386,1393,1409,1414,1419,1440,1445,1451,1456,1467,1487,1527,1532,1542,1557,1590,1595,1606,1611,1625,1630,1641,1669,1688,1693,1718,1723,1734,1778,1787,1792,1803,1818,1833,1844,1849,1875,1891,1910,1950,1973,1979,1987,1996,2001,2017,2022,2042,2055],{"__ignoreMap":49},[54,683,684,686,688,690],{"class":56,"line":57},[54,685,61],{"class":60},[54,687,65],{"class":64},[54,689,68],{"class":60},[54,691,71],{"class":64},[54,693,694,696,699,701],{"class":56,"line":74},[54,695,61],{"class":60},[54,697,698],{"class":64}," scipy ",[54,700,68],{"class":60},[54,702,703],{"class":64}," sp\n",[54,705,706,708,711,713],{"class":56,"line":88},[54,707,61],{"class":60},[54,709,710],{"class":64}," matplotlib.pylab ",[54,712,68],{"class":60},[54,714,715],{"class":64}," plt\n",[54,717,718,720,723,725],{"class":56,"line":101},[54,719,77],{"class":60},[54,721,722],{"class":64}," typing ",[54,724,61],{"class":60},[54,726,727],{"class":64}," Union, List\n",[54,729,730,732,735,737],{"class":56,"line":108},[54,731,61],{"class":60},[54,733,734],{"class":64}," itertools ",[54,736,68],{"class":60},[54,738,739],{"class":64}," it\n",[54,741,742,744],{"class":56,"line":115},[54,743,61],{"class":60},[54,745,746],{"class":64}," copy\n",[54,748,749,751,754,756],{"class":56,"line":139},[54,750,77],{"class":60},[54,752,753],{"class":64}," sympy ",[54,755,61],{"class":60},[54,757,758],{"class":64}," Matrix\n",[54,760,761,763],{"class":56,"line":159},[54,762,61],{"class":60},[54,764,765],{"class":64}," warnings\n",[54,767,768],{"class":56,"line":183},[54,769,105],{"emptyLinePlaceholder":104},[54,771,772,775,778,780,783],{"class":56,"line":210},[54,773,774],{"class":64},"warnings.",[54,776,777],{"class":125},"filterwarnings",[54,779,129],{"class":64},[54,781,782],{"class":279},"\"ignore\"",[54,784,136],{"class":64},[54,786,787],{"class":56,"line":227},[54,788,105],{"emptyLinePlaceholder":104},[54,790,791,793,795,797],{"class":56,"line":232},[54,792,77],{"class":60},[54,794,93],{"class":64},[54,796,61],{"class":60},[54,798,799],{"class":64}," SparsePauliOp, Pauli, StabilizerState\n",[54,801,802,804,807,809],{"class":56,"line":246},[54,803,77],{"class":60},[54,805,806],{"class":64}," qiskit.circuit ",[54,808,61],{"class":60},[54,810,811],{"class":64}," Parameter, IfElseOp\n",[54,813,814,816,818,820],{"class":56,"line":259},[54,815,77],{"class":60},[54,817,80],{"class":64},[54,819,61],{"class":60},[54,821,822],{"class":64}," QuantumCircuit, QuantumRegister\n",[54,824,825,827,830,832],{"class":56,"line":264},[54,826,77],{"class":60},[54,828,829],{"class":64}," qiskit.circuit.library ",[54,831,61],{"class":60},[54,833,834],{"class":64}," PauliEvolutionGate\n",[54,836,837,839,842,844],{"class":56,"line":270},[54,838,77],{"class":60},[54,840,841],{"class":64}," qiskit.synthesis ",[54,843,61],{"class":60},[54,845,846],{"class":64}," LieTrotter\n",[54,848,849,851,854,856],{"class":56,"line":285},[54,850,77],{"class":60},[54,852,853],{"class":64}," qiskit.transpiler ",[54,855,61],{"class":60},[54,857,858],{"class":64}," Target, CouplingMap\n",[54,860,861,863,866,868],{"class":56,"line":297},[54,862,77],{"class":60},[54,864,865],{"class":64}," qiskit.transpiler.preset_passmanagers ",[54,867,61],{"class":60},[54,869,870],{"class":64}," generate_preset_pass_manager\n",[54,872,873],{"class":56,"line":310},[54,874,105],{"emptyLinePlaceholder":104},[54,876,877],{"class":56,"line":324},[54,878,105],{"emptyLinePlaceholder":104},[54,880,881,883,886,888],{"class":56,"line":333},[54,882,77],{"class":60},[54,884,885],{"class":64}," qiskit_ibm_runtime ",[54,887,61],{"class":60},[54,889,890],{"class":64}," (\n",[54,892,893],{"class":56,"line":345},[54,894,895],{"class":64}," QiskitRuntimeService,\n",[54,897,898,901,903],{"class":56,"line":358},[54,899,900],{"class":64}," EstimatorV2 ",[54,902,68],{"class":60},[54,904,905],{"class":64}," Estimator,\n",[54,907,909],{"class":56,"line":908},24,[54,910,136],{"class":64},[54,912,914],{"class":56,"line":913},25,[54,915,105],{"emptyLinePlaceholder":104},[54,917,919],{"class":56,"line":918},26,[54,920,105],{"emptyLinePlaceholder":104},[54,922,924,927,930],{"class":56,"line":923},27,[54,925,926],{"class":60},"def",[54,928,929],{"class":125}," solve_regularized_gen_eig",[54,931,932],{"class":64},"(\n",[54,934,936,940],{"class":56,"line":935},28,[54,937,939],{"class":938},"sb9H8"," h",[54,941,942],{"class":64},": np.ndarray,\n",[54,944,946,949],{"class":56,"line":945},29,[54,947,948],{"class":938}," s",[54,950,942],{"class":64},[54,952,954,957,960,963],{"class":56,"line":953},30,[54,955,956],{"class":938}," threshold",[54,958,959],{"class":64},": ",[54,961,962],{"class":121},"float",[54,964,965],{"class":64},",\n",[54,967,969,972,974,977,980,983],{"class":56,"line":968},31,[54,970,971],{"class":938}," k",[54,973,959],{"class":64},[54,975,976],{"class":121},"int",[54,978,979],{"class":121}," =",[54,981,982],{"class":132}," 1",[54,984,965],{"class":64},[54,986,988,991,993,996,998,1001],{"class":56,"line":987},32,[54,989,990],{"class":938}," return_dimn",[54,992,959],{"class":64},[54,994,995],{"class":121},"bool",[54,997,979],{"class":121},[54,999,1000],{"class":132}," False",[54,1002,965],{"class":64},[54,1004,1006,1009,1011,1014,1016],{"class":56,"line":1005},33,[54,1007,1008],{"class":64},") -> Union[",[54,1010,962],{"class":121},[54,1012,1013],{"class":64},", List[",[54,1015,962],{"class":121},[54,1017,1018],{"class":64},"]]:\n",[54,1020,1022],{"class":56,"line":1021},34,[54,1023,1024],{"class":279}," \"\"\"\n",[54,1026,1028],{"class":56,"line":1027},35,[54,1029,1030],{"class":279}," Method for solving the generalized eigenvalue problem with regularization\n",[54,1032,1034],{"class":56,"line":1033},36,[54,1035,105],{"emptyLinePlaceholder":104},[54,1037,1039],{"class":56,"line":1038},37,[54,1040,1041],{"class":279}," Args:\n",[54,1043,1045],{"class":56,"line":1044},38,[54,1046,1047],{"class":279}," h (numpy.ndarray):\n",[54,1049,1051],{"class":56,"line":1050},39,[54,1052,1053],{"class":279}," The effective representation of the matrix in the Krylov subspace\n",[54,1055,1057],{"class":56,"line":1056},40,[54,1058,1059],{"class":279}," s (numpy.ndarray):\n",[54,1061,1063],{"class":56,"line":1062},41,[54,1064,1065],{"class":279}," The matrix of overlaps between vectors of the Krylov subspace\n",[54,1067,1069],{"class":56,"line":1068},42,[54,1070,1071],{"class":279}," threshold (float):\n",[54,1073,1075],{"class":56,"line":1074},43,[54,1076,1077],{"class":279}," Cut-off value for the eigenvalue of s\n",[54,1079,1081],{"class":56,"line":1080},44,[54,1082,1083],{"class":279}," k (int):\n",[54,1085,1087],{"class":56,"line":1086},45,[54,1088,1089],{"class":279}," Number of eigenvalues to return\n",[54,1091,1093],{"class":56,"line":1092},46,[54,1094,1095],{"class":279}," return_dimn (bool):\n",[54,1097,1099],{"class":56,"line":1098},47,[54,1100,1101],{"class":279}," Whether to return the size of the regularized subspace\n",[54,1103,1105],{"class":56,"line":1104},48,[54,1106,105],{"emptyLinePlaceholder":104},[54,1108,1110],{"class":56,"line":1109},49,[54,1111,1112],{"class":279}," Returns:\n",[54,1114,1116],{"class":56,"line":1115},50,[54,1117,1118],{"class":279}," lowest k-eigenvalue(s) that are the solution of the regularized generalized eigenvalue problem\n",[54,1120,1122],{"class":56,"line":1121},51,[54,1123,105],{"emptyLinePlaceholder":104},[54,1125,1127],{"class":56,"line":1126},52,[54,1128,105],{"emptyLinePlaceholder":104},[54,1130,1132],{"class":56,"line":1131},53,[54,1133,1024],{"class":279},[54,1135,1137,1140,1142,1145,1148],{"class":56,"line":1136},54,[54,1138,1139],{"class":64}," s_vals, s_vecs ",[54,1141,122],{"class":121},[54,1143,1144],{"class":64}," sp.linalg.",[54,1146,1147],{"class":125},"eigh",[54,1149,1150],{"class":64},"(s)\n",[54,1152,1154,1157,1159],{"class":56,"line":1153},55,[54,1155,1156],{"class":64}," s_vecs ",[54,1158,122],{"class":121},[54,1160,1161],{"class":64}," s_vecs.T\n",[54,1163,1165,1168,1170,1173,1176],{"class":56,"line":1164},56,[54,1166,1167],{"class":64}," good_vecs ",[54,1169,122],{"class":121},[54,1171,1172],{"class":64}," np.",[54,1174,1175],{"class":125},"array",[54,1177,932],{"class":64},[54,1179,1181,1184,1187,1190,1193,1196,1199,1202,1205,1208],{"class":56,"line":1180},57,[54,1182,1183],{"class":64}," [vec ",[54,1185,1186],{"class":60},"for",[54,1188,1189],{"class":64}," val, vec ",[54,1191,1192],{"class":60},"in",[54,1194,1195],{"class":121}," zip",[54,1197,1198],{"class":64},"(s_vals, s_vecs) ",[54,1200,1201],{"class":60},"if",[54,1203,1204],{"class":64}," val ",[54,1206,1207],{"class":121},">",[54,1209,1210],{"class":64}," threshold]\n",[54,1212,1214],{"class":56,"line":1213},58,[54,1215,1216],{"class":64}," )\n",[54,1218,1220,1223,1225,1228,1231,1234,1237,1240,1242],{"class":56,"line":1219},59,[54,1221,1222],{"class":64}," h_reg ",[54,1224,122],{"class":121},[54,1226,1227],{"class":64}," good_vecs.",[54,1229,1230],{"class":125},"conj",[54,1232,1233],{"class":64},"() ",[54,1235,1236],{"class":121},"@",[54,1238,1239],{"class":64}," h ",[54,1241,1236],{"class":121},[54,1243,1244],{"class":64}," good_vecs.T\n",[54,1246,1248,1251,1253,1255,1257,1259,1261,1264,1266],{"class":56,"line":1247},60,[54,1249,1250],{"class":64}," s_reg ",[54,1252,122],{"class":121},[54,1254,1227],{"class":64},[54,1256,1230],{"class":125},[54,1258,1233],{"class":64},[54,1260,1236],{"class":121},[54,1262,1263],{"class":64}," s ",[54,1265,1236],{"class":121},[54,1267,1244],{"class":64},[54,1269,1271,1274,1277,1280,1282],{"class":56,"line":1270},61,[54,1272,1273],{"class":60}," if",[54,1275,1276],{"class":64}," k ",[54,1278,1279],{"class":121},"==",[54,1281,982],{"class":132},[54,1283,1284],{"class":64},":\n",[54,1286,1288,1291],{"class":56,"line":1287},62,[54,1289,1290],{"class":60}," if",[54,1292,1293],{"class":64}," return_dimn:\n",[54,1295,1297,1300,1302,1304,1307,1309,1312,1314,1317,1320],{"class":56,"line":1296},63,[54,1298,1299],{"class":60}," return",[54,1301,1144],{"class":64},[54,1303,1147],{"class":125},[54,1305,1306],{"class":64},"(h_reg, s_reg)[",[54,1308,150],{"class":132},[54,1310,1311],{"class":64},"][",[54,1313,150],{"class":132},[54,1315,1316],{"class":64},"], ",[54,1318,1319],{"class":121},"len",[54,1321,1322],{"class":64},"(good_vecs)\n",[54,1324,1326,1329],{"class":56,"line":1325},64,[54,1327,1328],{"class":60}," else",[54,1330,1284],{"class":64},[54,1332,1334,1336,1338,1340,1342,1344,1346,1348],{"class":56,"line":1333},65,[54,1335,1299],{"class":60},[54,1337,1144],{"class":64},[54,1339,1147],{"class":125},[54,1341,1306],{"class":64},[54,1343,150],{"class":132},[54,1345,1311],{"class":64},[54,1347,150],{"class":132},[54,1349,1350],{"class":64},"]\n",[54,1352,1354,1357],{"class":56,"line":1353},66,[54,1355,1356],{"class":60}," else",[54,1358,1284],{"class":64},[54,1360,1362,1364],{"class":56,"line":1361},67,[54,1363,1290],{"class":60},[54,1365,1293],{"class":64},[54,1367,1369,1371,1373,1375,1377,1379,1382,1384],{"class":56,"line":1368},68,[54,1370,1299],{"class":60},[54,1372,1144],{"class":64},[54,1374,1147],{"class":125},[54,1376,1306],{"class":64},[54,1378,150],{"class":132},[54,1380,1381],{"class":64},"][:k], ",[54,1383,1319],{"class":121},[54,1385,1322],{"class":64},[54,1387,1389,1391],{"class":56,"line":1388},69,[54,1390,1328],{"class":60},[54,1392,1284],{"class":64},[54,1394,1396,1398,1400,1402,1404,1406],{"class":56,"line":1395},70,[54,1397,1299],{"class":60},[54,1399,1144],{"class":64},[54,1401,1147],{"class":125},[54,1403,1306],{"class":64},[54,1405,150],{"class":132},[54,1407,1408],{"class":64},"][:k]\n",[54,1410,1412],{"class":56,"line":1411},71,[54,1413,105],{"emptyLinePlaceholder":104},[54,1415,1417],{"class":56,"line":1416},72,[54,1418,105],{"emptyLinePlaceholder":104},[54,1420,1422,1424,1427,1429,1432,1434,1437],{"class":56,"line":1421},73,[54,1423,926],{"class":60},[54,1425,1426],{"class":125}," single_particle_gs",[54,1428,129],{"class":64},[54,1430,1431],{"class":938},"H_op",[54,1433,171],{"class":64},[54,1435,1436],{"class":938},"n_qubits",[54,1438,1439],{"class":64},"):\n",[54,1441,1443],{"class":56,"line":1442},74,[54,1444,1024],{"class":279},[54,1446,1448],{"class":56,"line":1447},75,[54,1449,1450],{"class":279}," Find the ground state of the single particle(excitation) sector\n",[54,1452,1454],{"class":56,"line":1453},76,[54,1455,1024],{"class":279},[54,1457,1459,1462,1464],{"class":56,"line":1458},77,[54,1460,1461],{"class":64}," H_x ",[54,1463,122],{"class":121},[54,1465,1466],{"class":64}," []\n",[54,1468,1470,1473,1476,1478,1481,1484],{"class":56,"line":1469},78,[54,1471,1472],{"class":60}," for",[54,1474,1475],{"class":64}," p, coeff ",[54,1477,1192],{"class":60},[54,1479,1480],{"class":64}," H_op.",[54,1482,1483],{"class":125},"to_list",[54,1485,1486],{"class":64},"():\n",[54,1488,1490,1493,1496,1498,1501,1504,1506,1509,1511,1514,1516,1519,1522,1524],{"class":56,"line":1489},79,[54,1491,1492],{"class":64}," H_x.",[54,1494,1495],{"class":125},"append",[54,1497,129],{"class":64},[54,1499,1500],{"class":121},"set",[54,1502,1503],{"class":64},"([i ",[54,1505,1186],{"class":60},[54,1507,1508],{"class":64}," i, v ",[54,1510,1192],{"class":60},[54,1512,1513],{"class":121}," enumerate",[54,1515,129],{"class":64},[54,1517,1518],{"class":125},"Pauli",[54,1520,1521],{"class":64},"(p).x) ",[54,1523,1201],{"class":60},[54,1525,1526],{"class":64}," v]))\n",[54,1528,1530],{"class":56,"line":1529},80,[54,1531,105],{"emptyLinePlaceholder":104},[54,1533,1535,1538,1540],{"class":56,"line":1534},81,[54,1536,1537],{"class":64}," H_z ",[54,1539,122],{"class":121},[54,1541,1466],{"class":64},[54,1543,1545,1547,1549,1551,1553,1555],{"class":56,"line":1544},82,[54,1546,1472],{"class":60},[54,1548,1475],{"class":64},[54,1550,1192],{"class":60},[54,1552,1480],{"class":64},[54,1554,1483],{"class":125},[54,1556,1486],{"class":64},[54,1558,1560,1563,1565,1567,1569,1571,1573,1575,1577,1579,1581,1583,1586,1588],{"class":56,"line":1559},83,[54,1561,1562],{"class":64}," H_z.",[54,1564,1495],{"class":125},[54,1566,129],{"class":64},[54,1568,1500],{"class":121},[54,1570,1503],{"class":64},[54,1572,1186],{"class":60},[54,1574,1508],{"class":64},[54,1576,1192],{"class":60},[54,1578,1513],{"class":121},[54,1580,129],{"class":64},[54,1582,1518],{"class":125},[54,1584,1585],{"class":64},"(p).z) ",[54,1587,1201],{"class":60},[54,1589,1526],{"class":64},[54,1591,1593],{"class":56,"line":1592},84,[54,1594,105],{"emptyLinePlaceholder":104},[54,1596,1598,1601,1603],{"class":56,"line":1597},85,[54,1599,1600],{"class":64}," H_c ",[54,1602,122],{"class":121},[54,1604,1605],{"class":64}," H_op.coeffs\n",[54,1607,1609],{"class":56,"line":1608},86,[54,1610,105],{"emptyLinePlaceholder":104},[54,1612,1614,1617,1619,1622],{"class":56,"line":1613},87,[54,1615,1616],{"class":121}," print",[54,1618,129],{"class":64},[54,1620,1621],{"class":279},"\"n_sys_qubits\"",[54,1623,1624],{"class":64},", n_qubits)\n",[54,1626,1628],{"class":56,"line":1627},88,[54,1629,105],{"emptyLinePlaceholder":104},[54,1631,1633,1636,1638],{"class":56,"line":1632},89,[54,1634,1635],{"class":64}," n_exc ",[54,1637,122],{"class":121},[54,1639,1640],{"class":132}," 1\n",[54,1642,1644,1647,1649,1652,1655,1658,1661,1664,1666],{"class":56,"line":1643},90,[54,1645,1646],{"class":64}," sub_dimn ",[54,1648,122],{"class":121},[54,1650,1651],{"class":121}," int",[54,1653,1654],{"class":64},"(sp.special.",[54,1656,1657],{"class":125},"comb",[54,1659,1660],{"class":64},"(n_qubits ",[54,1662,1663],{"class":121},"+",[54,1665,982],{"class":132},[54,1667,1668],{"class":64},", n_exc))\n",[54,1670,1672,1674,1676,1679,1682,1685],{"class":56,"line":1671},91,[54,1673,1616],{"class":121},[54,1675,129],{"class":64},[54,1677,1678],{"class":279},"\"n_exc\"",[54,1680,1681],{"class":64},", n_exc, ",[54,1683,1684],{"class":279},"\", subspace dimension\"",[54,1686,1687],{"class":64},", sub_dimn)\n",[54,1689,1691],{"class":56,"line":1690},92,[54,1692,105],{"emptyLinePlaceholder":104},[54,1694,1696,1699,1701,1703,1706,1709,1713,1716],{"class":56,"line":1695},93,[54,1697,1698],{"class":64}," few_particle_H ",[54,1700,122],{"class":121},[54,1702,1172],{"class":64},[54,1704,1705],{"class":125},"zeros",[54,1707,1708],{"class":64},"((sub_dimn, sub_dimn), ",[54,1710,1712],{"class":1711},"s_ZVi","dtype",[54,1714,1715],{"class":121},"=complex",[54,1717,136],{"class":64},[54,1719,1721],{"class":56,"line":1720},94,[54,1722,105],{"emptyLinePlaceholder":104},[54,1724,1726,1729,1731],{"class":56,"line":1725},95,[54,1727,1728],{"class":64}," sparse_vecs ",[54,1730,122],{"class":121},[54,1732,1733],{"class":64}," [\n",[54,1735,1737,1740,1743,1745,1748,1750,1753,1756,1758,1761,1763,1765,1767,1770,1773,1775],{"class":56,"line":1736},96,[54,1738,1739],{"class":121}," set",[54,1741,1742],{"class":64},"(vec) ",[54,1744,1186],{"class":60},[54,1746,1747],{"class":64}," vec ",[54,1749,1192],{"class":60},[54,1751,1752],{"class":64}," it.",[54,1754,1755],{"class":125},"combinations",[54,1757,129],{"class":64},[54,1759,1760],{"class":121},"range",[54,1762,1660],{"class":64},[54,1764,1663],{"class":121},[54,1766,982],{"class":132},[54,1768,1769],{"class":64},"), ",[54,1771,1772],{"class":1711},"r",[54,1774,122],{"class":121},[54,1776,1777],{"class":64},"n_exc)\n",[54,1779,1781,1784],{"class":56,"line":1780},97,[54,1782,1783],{"class":64}," ] ",[54,1785,1786],{"class":111},"# list all of the possible sets of n_exc indices of 1s in n_exc-particle states\n",[54,1788,1790],{"class":56,"line":1789},98,[54,1791,105],{"emptyLinePlaceholder":104},[54,1793,1795,1798,1800],{"class":56,"line":1794},99,[54,1796,1797],{"class":64}," m ",[54,1799,122],{"class":121},[54,1801,1802],{"class":132}," 0\n",[54,1804,1806,1808,1811,1813,1815],{"class":56,"line":1805},100,[54,1807,1472],{"class":60},[54,1809,1810],{"class":64}," i, i_set ",[54,1812,1192],{"class":60},[54,1814,1513],{"class":121},[54,1816,1817],{"class":64},"(sparse_vecs):\n",[54,1819,1821,1824,1827,1829,1831],{"class":56,"line":1820},101,[54,1822,1823],{"class":60}," for",[54,1825,1826],{"class":64}," j, j_set ",[54,1828,1192],{"class":60},[54,1830,1513],{"class":121},[54,1832,1817],{"class":64},[54,1834,1836,1839,1842],{"class":56,"line":1835},102,[54,1837,1838],{"class":64}," m ",[54,1840,1841],{"class":121},"+=",[54,1843,1640],{"class":132},[54,1845,1847],{"class":56,"line":1846},103,[54,1848,105],{"emptyLinePlaceholder":104},[54,1850,1852,1855,1858,1861,1864,1867,1870,1873],{"class":56,"line":1851},104,[54,1853,1854],{"class":60}," if",[54,1856,1857],{"class":121}," len",[54,1859,1860],{"class":64},"(i_set.",[54,1862,1863],{"class":125},"symmetric_difference",[54,1865,1866],{"class":64},"(j_set)) ",[54,1868,1869],{"class":121},"\u003C=",[54,1871,1872],{"class":132}," 2",[54,1874,1284],{"class":64},[54,1876,1878,1881,1884,1886,1888],{"class":56,"line":1877},105,[54,1879,1880],{"class":60}," for",[54,1882,1883],{"class":64}," p_x, p_z, coeff ",[54,1885,1192],{"class":60},[54,1887,1195],{"class":121},[54,1889,1890],{"class":64},"(H_x, H_z, H_c):\n",[54,1892,1894,1897,1900,1902,1905,1907],{"class":56,"line":1893},106,[54,1895,1896],{"class":60}," if",[54,1898,1899],{"class":64}," i_set.",[54,1901,1863],{"class":125},[54,1903,1904],{"class":64},"(j_set) ",[54,1906,1279],{"class":121},[54,1908,1909],{"class":64}," p_x:\n",[54,1911,1913,1916,1918,1921,1924,1926,1929,1931,1934,1936,1939,1942,1945,1948],{"class":56,"line":1912},107,[54,1914,1915],{"class":64}," sgn ",[54,1917,122],{"class":121},[54,1919,1920],{"class":64}," ((",[54,1922,1923],{"class":121},"-",[54,1925,174],{"class":132},[54,1927,1928],{"class":60},"j",[54,1930,204],{"class":64},[54,1932,1933],{"class":121},"**",[54,1935,1857],{"class":121},[54,1937,1938],{"class":64},"(p_x.",[54,1940,1941],{"class":125},"intersection",[54,1943,1944],{"class":64},"(p_z))) ",[54,1946,1947],{"class":121},"*",[54,1949,890],{"class":64},[54,1951,1953,1956,1958,1960,1962,1964,1966,1968,1970],{"class":56,"line":1952},108,[54,1954,1955],{"class":64}," (",[54,1957,1923],{"class":121},[54,1959,174],{"class":132},[54,1961,204],{"class":64},[54,1963,1933],{"class":121},[54,1965,1857],{"class":121},[54,1967,1860],{"class":64},[54,1969,1941],{"class":125},[54,1971,1972],{"class":64},"(p_z))\n",[54,1974,1976],{"class":56,"line":1975},109,[54,1977,1978],{"class":64}," )\n",[54,1980,1982,1985],{"class":56,"line":1981},110,[54,1983,1984],{"class":60}," else",[54,1986,1284],{"class":64},[54,1988,1990,1992,1994],{"class":56,"line":1989},111,[54,1991,1915],{"class":64},[54,1993,122],{"class":121},[54,1995,1802],{"class":132},[54,1997,1999],{"class":56,"line":1998},112,[54,2000,105],{"emptyLinePlaceholder":104},[54,2002,2004,2007,2009,2012,2014],{"class":56,"line":2003},113,[54,2005,2006],{"class":64}," few_particle_H[i, j] ",[54,2008,1841],{"class":121},[54,2010,2011],{"class":64}," sgn ",[54,2013,1947],{"class":121},[54,2015,2016],{"class":64}," coeff\n",[54,2018,2020],{"class":56,"line":2019},114,[54,2021,105],{"emptyLinePlaceholder":104},[54,2023,2025,2028,2030,2033,2036,2039],{"class":56,"line":2024},115,[54,2026,2027],{"class":64}," gs_en ",[54,2029,122],{"class":121},[54,2031,2032],{"class":121}," min",[54,2034,2035],{"class":64},"(np.linalg.",[54,2037,2038],{"class":125},"eigvalsh",[54,2040,2041],{"class":64},"(few_particle_H))\n",[54,2043,2045,2047,2049,2052],{"class":56,"line":2044},116,[54,2046,1616],{"class":121},[54,2048,129],{"class":64},[54,2050,2051],{"class":279},"\"single particle ground state energy: \"",[54,2053,2054],{"class":64},", gs_en)\n",[54,2056,2058,2061],{"class":56,"line":2057},117,[54,2059,2060],{"class":60}," return",[54,2062,2063],{"class":64}," gs_en\n",[31,2065,2067],{"id":2066},"step-1-map-classical-inputs-to-a-quantum-problem","Step 1: Map classical inputs to a quantum problem",[2069,2070,2072],"h3",{"id":2071},"the-krylov-space","The Krylov space",[13,2074,2075,2076,2169,2170,2199,2200,2229,2230,2289,2290,29],{},"The Krylov space ",[54,2077,2080,2109],{"className":2078},[2079],"katex",[54,2081,2084],{"className":2082},[2083],"katex-mathml",[2085,2086,2088],"math",{"xmlns":2087},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[2089,2090,2091,2104],"semantics",{},[2092,2093,2094],"mrow",{},[2095,2096,2097,2102],"msup",{},[2098,2099,2101],"mi",{"mathvariant":2100},"script","K",[2098,2103,1772],{},[2105,2106,2108],"annotation",{"encoding":2107},"application\u002Fx-tex","\\mathcal{K}^r",[54,2110,2114],{"className":2111,"ariaHidden":2113},[2112],"katex-html","true",[54,2115,2118,2123],{"className":2116},[2117],"base",[54,2119],{"className":2120,"style":2122},[2121],"strut","height:0.6833em;",[54,2124,2127,2132],{"className":2125},[2126],"mord",[54,2128,2101],{"className":2129,"style":2131},[2126,2130],"mathcal","margin-right:0.0144em;",[54,2133,2136],{"className":2134},[2135],"msupsub",[54,2137,2140],{"className":2138},[2139],"vlist-t",[54,2141,2144],{"className":2142},[2143],"vlist-r",[54,2145,2149],{"className":2146,"style":2148},[2147],"vlist","height:0.6644em;",[54,2150,2152,2157],{"style":2151},"top:-3.063em;margin-right:0.05em;",[54,2153],{"className":2154,"style":2156},[2155],"pstrut","height:2.7em;",[54,2158,2164],{"className":2159},[2160,2161,2162,2163],"sizing","reset-size6","size3","mtight",[54,2165,1772],{"className":2166,"style":2168},[2126,2167,2163],"mathnormal","margin-right:0.0278em;"," of order ",[54,2171,2173,2186],{"className":2172},[2079],[54,2174,2176],{"className":2175},[2083],[2085,2177,2178],{"xmlns":2087},[2089,2179,2180,2184],{},[2092,2181,2182],{},[2098,2183,1772],{},[2105,2185,1772],{"encoding":2107},[54,2187,2189],{"className":2188,"ariaHidden":2113},[2112],[54,2190,2192,2196],{"className":2191},[2117],[54,2193],{"className":2194,"style":2195},[2121],"height:0.4306em;",[54,2197,1772],{"className":2198,"style":2168},[2126,2167]," is the space spanned by vectors obtained by multiplying higher powers of a matrix ",[54,2201,2203,2217],{"className":2202},[2079],[54,2204,2206],{"className":2205},[2083],[2085,2207,2208],{"xmlns":2087},[2089,2209,2210,2215],{},[2092,2211,2212],{},[2098,2213,2214],{},"A",[2105,2216,2214],{"encoding":2107},[54,2218,2220],{"className":2219,"ariaHidden":2113},[2112],[54,2221,2223,2226],{"className":2222},[2117],[54,2224],{"className":2225,"style":2122},[2121],[54,2227,2214],{"className":2228},[2126,2167],", up to ",[54,2231,2233,2254],{"className":2232},[2079],[54,2234,2236],{"className":2235},[2083],[2085,2237,2238],{"xmlns":2087},[2089,2239,2240,2251],{},[2092,2241,2242,2244,2248],{},[2098,2243,1772],{},[2245,2246,2247],"mo",{},"−",[2249,2250,174],"mn",{},[2105,2252,2253],{"encoding":2107},"r-1",[54,2255,2257,2279],{"className":2256,"ariaHidden":2113},[2112],[54,2258,2260,2264,2267,2272,2276],{"className":2259},[2117],[54,2261],{"className":2262,"style":2263},[2121],"height:0.6667em;vertical-align:-0.0833em;",[54,2265,1772],{"className":2266,"style":2168},[2126,2167],[54,2268],{"className":2269,"style":2271},[2270],"mspace","margin-right:0.2222em;",[54,2273,2247],{"className":2274},[2275],"mbin",[54,2277],{"className":2278,"style":2271},[2270],[54,2280,2282,2286],{"className":2281},[2117],[54,2283],{"className":2284,"style":2285},[2121],"height:0.6444em;",[54,2287,174],{"className":2288},[2126],", with a reference vector ",[54,2291,2293,2316],{"className":2292},[2079],[54,2294,2296],{"className":2295},[2083],[2085,2297,2298],{"xmlns":2087},[2089,2299,2300,2313],{},[2092,2301,2302,2306,2309],{},[2098,2303,2305],{"mathvariant":2304},"normal","∣",[2098,2307,2308],{},"v",[2245,2310,2312],{"stretchy":2311},"false","⟩",[2105,2314,2315],{"encoding":2107},"\\vert v \\rangle",[54,2317,2319],{"className":2318,"ariaHidden":2113},[2112],[54,2320,2322,2326,2329,2333],{"className":2321},[2117],[54,2323],{"className":2324,"style":2325},[2121],"height:1em;vertical-align:-0.25em;",[54,2327,2305],{"className":2328},[2126],[54,2330,2308],{"className":2331,"style":2332},[2126,2167],"margin-right:0.0359em;",[54,2334,2312],{"className":2335},[2336],"mclose",[54,2338,2341],{"className":2339},[2340],"katex-display",[54,2342,2344,2432],{"className":2343},[2079],[54,2345,2347],{"className":2346},[2083],[2085,2348,2350],{"xmlns":2087,"display":2349},"block",[2089,2351,2352,2429],{},[2092,2353,2354,2360,2362],{},[2095,2355,2356,2358],{},[2098,2357,2101],{"mathvariant":2100},[2098,2359,1772],{},[2245,2361,122],{},[2092,2363,2364,2367,2369,2371,2373,2376,2378,2380,2382,2384,2386,2392,2394,2396,2398,2400,2402,2404,2406,2408,2420,2422,2424,2426],{},[2245,2365,2366],{"fence":2113},"{",[2098,2368,2305],{"mathvariant":2304},[2098,2370,2308],{},[2245,2372,2312],{"stretchy":2311},[2245,2374,2375],{"separator":2113},",",[2098,2377,2214],{},[2098,2379,2305],{"mathvariant":2304},[2098,2381,2308],{},[2245,2383,2312],{"stretchy":2311},[2245,2385,2375],{"separator":2113},[2095,2387,2388,2390],{},[2098,2389,2214],{},[2249,2391,133],{},[2098,2393,2305],{"mathvariant":2304},[2098,2395,2308],{},[2245,2397,2312],{"stretchy":2311},[2245,2399,2375],{"separator":2113},[2098,2401,29],{"mathvariant":2304},[2098,2403,29],{"mathvariant":2304},[2098,2405,29],{"mathvariant":2304},[2245,2407,2375],{"separator":2113},[2095,2409,2410,2412],{},[2098,2411,2214],{},[2092,2413,2414,2416,2418],{},[2098,2415,1772],{},[2245,2417,2247],{},[2249,2419,174],{},[2098,2421,2305],{"mathvariant":2304},[2098,2423,2308],{},[2245,2425,2312],{"stretchy":2311},[2245,2427,2428],{"fence":2113},"}",[2105,2430,2431],{"encoding":2107},"\\mathcal{K}^r = \\left\\{ \\vert v \\rangle, A \\vert v \\rangle, A^2 \\vert v \\rangle, ..., A^{r-1} \\vert v \\rangle \\right\\}",[54,2433,2435,2483],{"className":2434,"ariaHidden":2113},[2112],[54,2436,2438,2442,2472,2476,2480],{"className":2437},[2117],[54,2439],{"className":2440,"style":2441},[2121],"height:0.7144em;",[54,2443,2445,2448],{"className":2444},[2126],[54,2446,2101],{"className":2447,"style":2131},[2126,2130],[54,2449,2451],{"className":2450},[2135],[54,2452,2454],{"className":2453},[2139],[54,2455,2457],{"className":2456},[2143],[54,2458,2460],{"className":2459,"style":2441},[2147],[54,2461,2463,2466],{"style":2462},"top:-3.113em;margin-right:0.05em;",[54,2464],{"className":2465,"style":2156},[2155],[54,2467,2469],{"className":2468},[2160,2161,2162,2163],[54,2470,1772],{"className":2471,"style":2168},[2126,2167,2163],[54,2473],{"className":2474,"style":2475},[2270],"margin-right:0.2778em;",[54,2477,122],{"className":2478},[2479],"mrel",[54,2481],{"className":2482,"style":2475},[2270],[54,2484,2486,2490],{"className":2485},[2117],[54,2487],{"className":2488,"style":2489},[2121],"height:1.2141em;vertical-align:-0.35em;",[54,2491,2494,2505,2508,2511,2514,2518,2522,2525,2528,2531,2534,2537,2540,2570,2573,2576,2579,2582,2585,2589,2592,2595,2633,2636,2639,2642],{"className":2492},[2493],"minner",[54,2495,2500],{"className":2496,"style":2499},[2497,2498],"mopen","delimcenter","top:0em;",[54,2501,2366],{"className":2502},[2503,2504],"delimsizing","size1",[54,2506,2305],{"className":2507},[2126],[54,2509,2308],{"className":2510,"style":2332},[2126,2167],[54,2512,2312],{"className":2513},[2336],[54,2515,2375],{"className":2516},[2517],"mpunct",[54,2519],{"className":2520,"style":2521},[2270],"margin-right:0.1667em;",[54,2523,2214],{"className":2524},[2126,2167],[54,2526,2305],{"className":2527},[2126],[54,2529,2308],{"className":2530,"style":2332},[2126,2167],[54,2532,2312],{"className":2533},[2336],[54,2535,2375],{"className":2536},[2517],[54,2538],{"className":2539,"style":2521},[2270],[54,2541,2543,2546],{"className":2542},[2126],[54,2544,2214],{"className":2545},[2126,2167],[54,2547,2549],{"className":2548},[2135],[54,2550,2552],{"className":2551},[2139],[54,2553,2555],{"className":2554},[2143],[54,2556,2559],{"className":2557,"style":2558},[2147],"height:0.8641em;",[54,2560,2561,2564],{"style":2462},[54,2562],{"className":2563,"style":2156},[2155],[54,2565,2567],{"className":2566},[2160,2161,2162,2163],[54,2568,133],{"className":2569},[2126,2163],[54,2571,2305],{"className":2572},[2126],[54,2574,2308],{"className":2575,"style":2332},[2126,2167],[54,2577,2312],{"className":2578},[2336],[54,2580,2375],{"className":2581},[2517],[54,2583],{"className":2584,"style":2521},[2270],[54,2586,2588],{"className":2587},[2126],"...",[54,2590,2375],{"className":2591},[2517],[54,2593],{"className":2594,"style":2521},[2270],[54,2596,2598,2601],{"className":2597},[2126],[54,2599,2214],{"className":2600},[2126,2167],[54,2602,2604],{"className":2603},[2135],[54,2605,2607],{"className":2606},[2139],[54,2608,2610],{"className":2609},[2143],[54,2611,2613],{"className":2612,"style":2558},[2147],[54,2614,2615,2618],{"style":2462},[54,2616],{"className":2617,"style":2156},[2155],[54,2619,2621],{"className":2620},[2160,2161,2162,2163],[54,2622,2624,2627,2630],{"className":2623},[2126,2163],[54,2625,1772],{"className":2626,"style":2168},[2126,2167,2163],[54,2628,2247],{"className":2629},[2275,2163],[54,2631,174],{"className":2632},[2126,2163],[54,2634,2305],{"className":2635},[2126],[54,2637,2308],{"className":2638,"style":2332},[2126,2167],[54,2640,2312],{"className":2641},[2336],[54,2643,2645],{"className":2644,"style":2499},[2336,2498],[54,2646,2428],{"className":2647},[2503,2504],[13,2649,2650,2651,2679,2680,2710,2711,2791,2792,2820,2821,2928,2929,2999,3000,3028,3029,3130,3131,3134,3135,3163,3164,29],{},"If the matrix ",[54,2652,2654,2667],{"className":2653},[2079],[54,2655,2657],{"className":2656},[2083],[2085,2658,2659],{"xmlns":2087},[2089,2660,2661,2665],{},[2092,2662,2663],{},[2098,2664,2214],{},[2105,2666,2214],{"encoding":2107},[54,2668,2670],{"className":2669,"ariaHidden":2113},[2112],[54,2671,2673,2676],{"className":2672},[2117],[54,2674],{"className":2675,"style":2122},[2121],[54,2677,2214],{"className":2678},[2126,2167]," is the Hamiltonian ",[54,2681,2683,2697],{"className":2682},[2079],[54,2684,2686],{"className":2685},[2083],[2085,2687,2688],{"xmlns":2087},[2089,2689,2690,2695],{},[2092,2691,2692],{},[2098,2693,2694],{},"H",[2105,2696,2694],{"encoding":2107},[54,2698,2700],{"className":2699,"ariaHidden":2113},[2112],[54,2701,2703,2706],{"className":2702},[2117],[54,2704],{"className":2705,"style":2122},[2121],[54,2707,2694],{"className":2708,"style":2709},[2126,2167],"margin-right:0.0813em;",", we'll refer to the corresponding space as the power Krylov space ",[54,2712,2714,2734],{"className":2713},[2079],[54,2715,2717],{"className":2716},[2083],[2085,2718,2719],{"xmlns":2087},[2089,2720,2721,2731],{},[2092,2722,2723],{},[2724,2725,2726,2728],"msub",{},[2098,2727,2101],{"mathvariant":2100},[2098,2729,2730],{},"P",[2105,2732,2733],{"encoding":2107},"\\mathcal{K}_P",[54,2735,2737],{"className":2736,"ariaHidden":2113},[2112],[54,2738,2740,2744],{"className":2739},[2117],[54,2741],{"className":2742,"style":2743},[2121],"height:0.8333em;vertical-align:-0.15em;",[54,2745,2747,2750],{"className":2746},[2126],[54,2748,2101],{"className":2749,"style":2131},[2126,2130],[54,2751,2753],{"className":2752},[2135],[54,2754,2757,2782],{"className":2755},[2139,2756],"vlist-t2",[54,2758,2760,2777],{"className":2759},[2143],[54,2761,2764],{"className":2762,"style":2763},[2147],"height:0.3283em;",[54,2765,2767,2770],{"style":2766},"top:-2.55em;margin-left:-0.0144em;margin-right:0.05em;",[54,2768],{"className":2769,"style":2156},[2155],[54,2771,2773],{"className":2772},[2160,2161,2162,2163],[54,2774,2730],{"className":2775,"style":2776},[2126,2167,2163],"margin-right:0.1389em;",[54,2778,2781],{"className":2779},[2780],"vlist-s","​",[54,2783,2785],{"className":2784},[2143],[54,2786,2789],{"className":2787,"style":2788},[2147],"height:0.15em;",[54,2790],{},". In the case where ",[54,2793,2795,2808],{"className":2794},[2079],[54,2796,2798],{"className":2797},[2083],[2085,2799,2800],{"xmlns":2087},[2089,2801,2802,2806],{},[2092,2803,2804],{},[2098,2805,2214],{},[2105,2807,2214],{"encoding":2107},[54,2809,2811],{"className":2810,"ariaHidden":2113},[2112],[54,2812,2814,2817],{"className":2813},[2117],[54,2815],{"className":2816,"style":2122},[2121],[54,2818,2214],{"className":2819},[2126,2167]," is the time-evolution operator generated by the Hamiltonian ",[54,2822,2824,2858],{"className":2823},[2079],[54,2825,2827],{"className":2826},[2083],[2085,2828,2829],{"xmlns":2087},[2089,2830,2831,2855],{},[2092,2832,2833,2836,2838],{},[2098,2834,2835],{},"U",[2245,2837,122],{},[2095,2839,2840,2843],{},[2098,2841,2842],{},"e",[2092,2844,2845,2847,2850,2852],{},[2245,2846,2247],{},[2098,2848,2849],{},"i",[2098,2851,2694],{},[2098,2853,2854],{},"t",[2105,2856,2857],{"encoding":2107},"U=e^{-iHt}",[54,2859,2861,2880],{"className":2860,"ariaHidden":2113},[2112],[54,2862,2864,2867,2871,2874,2877],{"className":2863},[2117],[54,2865],{"className":2866,"style":2122},[2121],[54,2868,2835],{"className":2869,"style":2870},[2126,2167],"margin-right:0.109em;",[54,2872],{"className":2873,"style":2475},[2270],[54,2875,122],{"className":2876},[2479],[54,2878],{"className":2879,"style":2475},[2270],[54,2881,2883,2887],{"className":2882},[2117],[54,2884],{"className":2885,"style":2886},[2121],"height:0.8413em;",[54,2888,2890,2893],{"className":2889},[2126],[54,2891,2842],{"className":2892},[2126,2167],[54,2894,2896],{"className":2895},[2135],[54,2897,2899],{"className":2898},[2139],[54,2900,2902],{"className":2901},[2143],[54,2903,2905],{"className":2904,"style":2886},[2147],[54,2906,2907,2910],{"style":2151},[54,2908],{"className":2909,"style":2156},[2155],[54,2911,2913],{"className":2912},[2160,2161,2162,2163],[54,2914,2916,2919,2922,2925],{"className":2915},[2126,2163],[54,2917,2247],{"className":2918},[2126,2163],[54,2920,2849],{"className":2921},[2126,2167,2163],[54,2923,2694],{"className":2924,"style":2709},[2126,2167,2163],[54,2926,2854],{"className":2927},[2126,2167,2163],", we'll refer to the space as the unitary Krylov space ",[54,2930,2932,2950],{"className":2931},[2079],[54,2933,2935],{"className":2934},[2083],[2085,2936,2937],{"xmlns":2087},[2089,2938,2939,2947],{},[2092,2940,2941],{},[2724,2942,2943,2945],{},[2098,2944,2101],{"mathvariant":2100},[2098,2946,2835],{},[2105,2948,2949],{"encoding":2107},"\\mathcal{K}_U",[54,2951,2953],{"className":2952,"ariaHidden":2113},[2112],[54,2954,2956,2959],{"className":2955},[2117],[54,2957],{"className":2958,"style":2743},[2121],[54,2960,2962,2965],{"className":2961},[2126],[54,2963,2101],{"className":2964,"style":2131},[2126,2130],[54,2966,2968],{"className":2967},[2135],[54,2969,2971,2991],{"className":2970},[2139,2756],[54,2972,2974,2988],{"className":2973},[2143],[54,2975,2977],{"className":2976,"style":2763},[2147],[54,2978,2979,2982],{"style":2766},[54,2980],{"className":2981,"style":2156},[2155],[54,2983,2985],{"className":2984},[2160,2161,2162,2163],[54,2986,2835],{"className":2987,"style":2870},[2126,2167,2163],[54,2989,2781],{"className":2990},[2780],[54,2992,2994],{"className":2993},[2143],[54,2995,2997],{"className":2996,"style":2788},[2147],[54,2998],{},". The power Krylov subspace that we use classically cannot be generated directly on a quantum computer as ",[54,3001,3003,3016],{"className":3002},[2079],[54,3004,3006],{"className":3005},[2083],[2085,3007,3008],{"xmlns":2087},[2089,3009,3010,3014],{},[2092,3011,3012],{},[2098,3013,2694],{},[2105,3015,2694],{"encoding":2107},[54,3017,3019],{"className":3018,"ariaHidden":2113},[2112],[54,3020,3022,3025],{"className":3021},[2117],[54,3023],{"className":3024,"style":2122},[2121],[54,3026,2694],{"className":3027,"style":2709},[2126,2167]," is not a unitary operator. Instead, we can use the time-evolution operator ",[54,3030,3032,3062],{"className":3031},[2079],[54,3033,3035],{"className":3034},[2083],[2085,3036,3037],{"xmlns":2087},[2089,3038,3039,3059],{},[2092,3040,3041,3043,3045],{},[2098,3042,2835],{},[2245,3044,122],{},[2095,3046,3047,3049],{},[2098,3048,2842],{},[2092,3050,3051,3053,3055,3057],{},[2245,3052,2247],{},[2098,3054,2849],{},[2098,3056,2694],{},[2098,3058,2854],{},[2105,3060,3061],{"encoding":2107},"U = e^{-iHt}",[54,3063,3065,3083],{"className":3064,"ariaHidden":2113},[2112],[54,3066,3068,3071,3074,3077,3080],{"className":3067},[2117],[54,3069],{"className":3070,"style":2122},[2121],[54,3072,2835],{"className":3073,"style":2870},[2126,2167],[54,3075],{"className":3076,"style":2475},[2270],[54,3078,122],{"className":3079},[2479],[54,3081],{"className":3082,"style":2475},[2270],[54,3084,3086,3089],{"className":3085},[2117],[54,3087],{"className":3088,"style":2886},[2121],[54,3090,3092,3095],{"className":3091},[2126],[54,3093,2842],{"className":3094},[2126,2167],[54,3096,3098],{"className":3097},[2135],[54,3099,3101],{"className":3100},[2139],[54,3102,3104],{"className":3103},[2143],[54,3105,3107],{"className":3106,"style":2886},[2147],[54,3108,3109,3112],{"style":2151},[54,3110],{"className":3111,"style":2156},[2155],[54,3113,3115],{"className":3114},[2160,2161,2162,2163],[54,3116,3118,3121,3124,3127],{"className":3117},[2126,2163],[54,3119,2247],{"className":3120},[2126,2163],[54,3122,2849],{"className":3123},[2126,2167,2163],[54,3125,2694],{"className":3126,"style":2709},[2126,2167,2163],[54,3128,2854],{"className":3129},[2126,2167,2163]," which can be shown to give similar ",[20,3132,3133],{"href":645},"convergence guarantees"," as the power method. Powers of ",[54,3136,3138,3151],{"className":3137},[2079],[54,3139,3141],{"className":3140},[2083],[2085,3142,3143],{"xmlns":2087},[2089,3144,3145,3149],{},[2092,3146,3147],{},[2098,3148,2835],{},[2105,3150,2835],{"encoding":2107},[54,3152,3154],{"className":3153,"ariaHidden":2113},[2112],[54,3155,3157,3160],{"className":3156},[2117],[54,3158],{"className":3159,"style":2122},[2121],[54,3161,2835],{"className":3162,"style":2870},[2126,2167]," then become different time steps ",[54,3165,3167,3209],{"className":3166},[2079],[54,3168,3170],{"className":3169},[2083],[2085,3171,3172],{"xmlns":2087},[2089,3173,3174,3206],{},[2092,3175,3176,3183,3185],{},[2095,3177,3178,3180],{},[2098,3179,2835],{},[2098,3181,3182],{},"k",[2245,3184,122],{},[2095,3186,3187,3189],{},[2098,3188,2842],{},[2092,3190,3191,3193,3195,3197,3199,3201,3203],{},[2245,3192,2247],{},[2098,3194,2849],{},[2098,3196,2694],{},[2245,3198,129],{"stretchy":2311},[2098,3200,3182],{},[2098,3202,2854],{},[2245,3204,3205],{"stretchy":2311},")",[2105,3207,3208],{"encoding":2107},"U^k = e^{-iH(kt)}",[54,3210,3212,3258],{"className":3211,"ariaHidden":2113},[2112],[54,3213,3215,3219,3249,3252,3255],{"className":3214},[2117],[54,3216],{"className":3217,"style":3218},[2121],"height:0.8491em;",[54,3220,3222,3225],{"className":3221},[2126],[54,3223,2835],{"className":3224,"style":2870},[2126,2167],[54,3226,3228],{"className":3227},[2135],[54,3229,3231],{"className":3230},[2139],[54,3232,3234],{"className":3233},[2143],[54,3235,3237],{"className":3236,"style":3218},[2147],[54,3238,3239,3242],{"style":2151},[54,3240],{"className":3241,"style":2156},[2155],[54,3243,3245],{"className":3244},[2160,2161,2162,2163],[54,3246,3182],{"className":3247,"style":3248},[2126,2167,2163],"margin-right:0.0315em;",[54,3250],{"className":3251,"style":2475},[2270],[54,3253,122],{"className":3254},[2479],[54,3256],{"className":3257,"style":2475},[2270],[54,3259,3261,3265],{"className":3260},[2117],[54,3262],{"className":3263,"style":3264},[2121],"height:0.888em;",[54,3266,3268,3271],{"className":3267},[2126],[54,3269,2842],{"className":3270},[2126,2167],[54,3272,3274],{"className":3273},[2135],[54,3275,3277],{"className":3276},[2139],[54,3278,3280],{"className":3279},[2143],[54,3281,3283],{"className":3282,"style":3264},[2147],[54,3284,3285,3288],{"style":2151},[54,3286],{"className":3287,"style":2156},[2155],[54,3289,3291],{"className":3290},[2160,2161,2162,2163],[54,3292,3294,3297,3300,3303,3306,3309,3312],{"className":3293},[2126,2163],[54,3295,2247],{"className":3296},[2126,2163],[54,3298,2849],{"className":3299},[2126,2167,2163],[54,3301,2694],{"className":3302,"style":2709},[2126,2167,2163],[54,3304,129],{"className":3305},[2497,2163],[54,3307,3182],{"className":3308,"style":3248},[2126,2167,2163],[54,3310,2854],{"className":3311},[2126,2167,2163],[54,3313,3205],{"className":3314},[2336,2163],[54,3316,3318],{"className":3317},[2340],[54,3319,3321,3409],{"className":3320},[2079],[54,3322,3324],{"className":3323},[2083],[2085,3325,3326],{"xmlns":2087,"display":2349},[2089,3327,3328,3406],{},[2092,3329,3330,3339,3341],{},[3331,3332,3333,3335,3337],"msubsup",{},[2098,3334,2101],{"mathvariant":2100},[2098,3336,2835],{},[2098,3338,1772],{},[2245,3340,122],{},[2092,3342,3343,3345,3347,3350,3352,3354,3356,3358,3360,3362,3364,3370,3372,3374,3376,3378,3380,3382,3384,3386,3398,3400,3402,3404],{},[2245,3344,2366],{"fence":2113},[2098,3346,2305],{"mathvariant":2304},[2098,3348,3349],{},"ψ",[2245,3351,2312],{"stretchy":2311},[2245,3353,2375],{"separator":2113},[2098,3355,2835],{},[2098,3357,2305],{"mathvariant":2304},[2098,3359,3349],{},[2245,3361,2312],{"stretchy":2311},[2245,3363,2375],{"separator":2113},[2095,3365,3366,3368],{},[2098,3367,2835],{},[2249,3369,133],{},[2098,3371,2305],{"mathvariant":2304},[2098,3373,3349],{},[2245,3375,2312],{"stretchy":2311},[2245,3377,2375],{"separator":2113},[2098,3379,29],{"mathvariant":2304},[2098,3381,29],{"mathvariant":2304},[2098,3383,29],{"mathvariant":2304},[2245,3385,2375],{"separator":2113},[2095,3387,3388,3390],{},[2098,3389,2835],{},[2092,3391,3392,3394,3396],{},[2098,3393,1772],{},[2245,3395,2247],{},[2249,3397,174],{},[2098,3399,2305],{"mathvariant":2304},[2098,3401,3349],{},[2245,3403,2312],{"stretchy":2311},[2245,3405,2428],{"fence":2113},[2105,3407,3408],{"encoding":2107},"\\mathcal{K}_U^r = \\left\\{ \\vert \\psi \\rangle, U \\vert \\psi \\rangle, U^2 \\vert \\psi \\rangle, ..., U^{r-1} \\vert \\psi \\rangle \\right\\}",[54,3410,3412,3481],{"className":3411,"ariaHidden":2113},[2112],[54,3413,3415,3419,3472,3475,3478],{"className":3414},[2117],[54,3416],{"className":3417,"style":3418},[2121],"height:0.9614em;vertical-align:-0.247em;",[54,3420,3422,3425],{"className":3421},[2126],[54,3423,2101],{"className":3424,"style":2131},[2126,2130],[54,3426,3428],{"className":3427},[2135],[54,3429,3431,3463],{"className":3430},[2139,2756],[54,3432,3434,3460],{"className":3433},[2143],[54,3435,3437,3449],{"className":3436,"style":2441},[2147],[54,3438,3440,3443],{"style":3439},"top:-2.453em;margin-left:-0.0144em;margin-right:0.05em;",[54,3441],{"className":3442,"style":2156},[2155],[54,3444,3446],{"className":3445},[2160,2161,2162,2163],[54,3447,2835],{"className":3448,"style":2870},[2126,2167,2163],[54,3450,3451,3454],{"style":2462},[54,3452],{"className":3453,"style":2156},[2155],[54,3455,3457],{"className":3456},[2160,2161,2162,2163],[54,3458,1772],{"className":3459,"style":2168},[2126,2167,2163],[54,3461,2781],{"className":3462},[2780],[54,3464,3466],{"className":3465},[2143],[54,3467,3470],{"className":3468,"style":3469},[2147],"height:0.247em;",[54,3471],{},[54,3473],{"className":3474,"style":2475},[2270],[54,3476,122],{"className":3477},[2479],[54,3479],{"className":3480,"style":2475},[2270],[54,3482,3484,3487],{"className":3483},[2117],[54,3485],{"className":3486,"style":2489},[2121],[54,3488,3490,3496,3499,3502,3505,3508,3511,3514,3517,3520,3523,3526,3529,3558,3561,3564,3567,3570,3573,3576,3579,3582,3620,3623,3626,3629],{"className":3489},[2493],[54,3491,3493],{"className":3492,"style":2499},[2497,2498],[54,3494,2366],{"className":3495},[2503,2504],[54,3497,2305],{"className":3498},[2126],[54,3500,3349],{"className":3501,"style":2332},[2126,2167],[54,3503,2312],{"className":3504},[2336],[54,3506,2375],{"className":3507},[2517],[54,3509],{"className":3510,"style":2521},[2270],[54,3512,2835],{"className":3513,"style":2870},[2126,2167],[54,3515,2305],{"className":3516},[2126],[54,3518,3349],{"className":3519,"style":2332},[2126,2167],[54,3521,2312],{"className":3522},[2336],[54,3524,2375],{"className":3525},[2517],[54,3527],{"className":3528,"style":2521},[2270],[54,3530,3532,3535],{"className":3531},[2126],[54,3533,2835],{"className":3534,"style":2870},[2126,2167],[54,3536,3538],{"className":3537},[2135],[54,3539,3541],{"className":3540},[2139],[54,3542,3544],{"className":3543},[2143],[54,3545,3547],{"className":3546,"style":2558},[2147],[54,3548,3549,3552],{"style":2462},[54,3550],{"className":3551,"style":2156},[2155],[54,3553,3555],{"className":3554},[2160,2161,2162,2163],[54,3556,133],{"className":3557},[2126,2163],[54,3559,2305],{"className":3560},[2126],[54,3562,3349],{"className":3563,"style":2332},[2126,2167],[54,3565,2312],{"className":3566},[2336],[54,3568,2375],{"className":3569},[2517],[54,3571],{"className":3572,"style":2521},[2270],[54,3574,2588],{"className":3575},[2126],[54,3577,2375],{"className":3578},[2517],[54,3580],{"className":3581,"style":2521},[2270],[54,3583,3585,3588],{"className":3584},[2126],[54,3586,2835],{"className":3587,"style":2870},[2126,2167],[54,3589,3591],{"className":3590},[2135],[54,3592,3594],{"className":3593},[2139],[54,3595,3597],{"className":3596},[2143],[54,3598,3600],{"className":3599,"style":2558},[2147],[54,3601,3602,3605],{"style":2462},[54,3603],{"className":3604,"style":2156},[2155],[54,3606,3608],{"className":3607},[2160,2161,2162,2163],[54,3609,3611,3614,3617],{"className":3610},[2126,2163],[54,3612,1772],{"className":3613,"style":2168},[2126,2167,2163],[54,3615,2247],{"className":3616},[2275,2163],[54,3618,174],{"className":3619},[2126,2163],[54,3621,2305],{"className":3622},[2126],[54,3624,3349],{"className":3625,"style":2332},[2126,2167],[54,3627,2312],{"className":3628},[2336],[54,3630,3632],{"className":3631,"style":2499},[2336,2498],[54,3633,2428],{"className":3634},[2503,2504],[13,3636,3637],{},"See the Appendix for a detailed derivation of how the unitary Krylov space allows to represents low-energy eigenstates accurately.",[2069,3639,3641],{"id":3640},"krylov-quantum-diagonalization-algorithm","Krylov quantum diagonalization algorithm",[13,3643,3644,3645,3673,3674,3743,3744,3813,3814,3884,3885,3945],{},"Given an Hamiltonian ",[54,3646,3648,3661],{"className":3647},[2079],[54,3649,3651],{"className":3650},[2083],[2085,3652,3653],{"xmlns":2087},[2089,3654,3655,3659],{},[2092,3656,3657],{},[2098,3658,2694],{},[2105,3660,2694],{"encoding":2107},[54,3662,3664],{"className":3663,"ariaHidden":2113},[2112],[54,3665,3667,3670],{"className":3666},[2117],[54,3668],{"className":3669,"style":2122},[2121],[54,3671,2694],{"className":3672,"style":2709},[2126,2167]," that we wish to diagonalize, first we consider the corresponding unitary Krylov space ",[54,3675,3677,3694],{"className":3676},[2079],[54,3678,3680],{"className":3679},[2083],[2085,3681,3682],{"xmlns":2087},[2089,3683,3684,3692],{},[2092,3685,3686],{},[2724,3687,3688,3690],{},[2098,3689,2101],{"mathvariant":2100},[2098,3691,2835],{},[2105,3693,2949],{"encoding":2107},[54,3695,3697],{"className":3696,"ariaHidden":2113},[2112],[54,3698,3700,3703],{"className":3699},[2117],[54,3701],{"className":3702,"style":2743},[2121],[54,3704,3706,3709],{"className":3705},[2126],[54,3707,2101],{"className":3708,"style":2131},[2126,2130],[54,3710,3712],{"className":3711},[2135],[54,3713,3715,3735],{"className":3714},[2139,2756],[54,3716,3718,3732],{"className":3717},[2143],[54,3719,3721],{"className":3720,"style":2763},[2147],[54,3722,3723,3726],{"style":2766},[54,3724],{"className":3725,"style":2156},[2155],[54,3727,3729],{"className":3728},[2160,2161,2162,2163],[54,3730,2835],{"className":3731,"style":2870},[2126,2167,2163],[54,3733,2781],{"className":3734},[2780],[54,3736,3738],{"className":3737},[2143],[54,3739,3741],{"className":3740,"style":2788},[2147],[54,3742],{},". The goal is to find a compact representation of the Hamiltonian in ",[54,3745,3747,3764],{"className":3746},[2079],[54,3748,3750],{"className":3749},[2083],[2085,3751,3752],{"xmlns":2087},[2089,3753,3754,3762],{},[2092,3755,3756],{},[2724,3757,3758,3760],{},[2098,3759,2101],{"mathvariant":2100},[2098,3761,2835],{},[2105,3763,2949],{"encoding":2107},[54,3765,3767],{"className":3766,"ariaHidden":2113},[2112],[54,3768,3770,3773],{"className":3769},[2117],[54,3771],{"className":3772,"style":2743},[2121],[54,3774,3776,3779],{"className":3775},[2126],[54,3777,2101],{"className":3778,"style":2131},[2126,2130],[54,3780,3782],{"className":3781},[2135],[54,3783,3785,3805],{"className":3784},[2139,2756],[54,3786,3788,3802],{"className":3787},[2143],[54,3789,3791],{"className":3790,"style":2763},[2147],[54,3792,3793,3796],{"style":2766},[54,3794],{"className":3795,"style":2156},[2155],[54,3797,3799],{"className":3798},[2160,2161,2162,2163],[54,3800,2835],{"className":3801,"style":2870},[2126,2167,2163],[54,3803,2781],{"className":3804},[2780],[54,3806,3808],{"className":3807},[2143],[54,3809,3811],{"className":3810,"style":2788},[2147],[54,3812],{},", which we'll refer to as ",[54,3815,3817,3837],{"className":3816},[2079],[54,3818,3820],{"className":3819},[2083],[2085,3821,3822],{"xmlns":2087},[2089,3823,3824,3834],{},[2092,3825,3826],{},[3827,3828,3829,3831],"mover",{"accent":2113},[2098,3830,2694],{},[2245,3832,3833],{},"~",[2105,3835,3836],{"encoding":2107},"\\tilde{H}",[54,3838,3840],{"className":3839,"ariaHidden":2113},[2112],[54,3841,3843,3847],{"className":3842},[2117],[54,3844],{"className":3845,"style":3846},[2121],"height:0.9202em;",[54,3848,3851],{"className":3849},[2126,3850],"accent",[54,3852,3854],{"className":3853},[2139],[54,3855,3857],{"className":3856},[2143],[54,3858,3860,3870],{"className":3859,"style":3846},[2147],[54,3861,3863,3867],{"style":3862},"top:-3em;",[54,3864],{"className":3865,"style":3866},[2155],"height:3em;",[54,3868,2694],{"className":3869,"style":2709},[2126,2167],[54,3871,3873,3876],{"style":3872},"top:-3.6023em;",[54,3874],{"className":3875,"style":3866},[2155],[54,3877,3881],{"className":3878,"style":3880},[3879],"accent-body","left:-0.1944em;",[54,3882,3833],{"className":3883},[2126],". The matrix elements of ",[54,3886,3888,3905],{"className":3887},[2079],[54,3889,3891],{"className":3890},[2083],[2085,3892,3893],{"xmlns":2087},[2089,3894,3895,3903],{},[2092,3896,3897],{},[3827,3898,3899,3901],{"accent":2113},[2098,3900,2694],{},[2245,3902,3833],{},[2105,3904,3836],{"encoding":2107},[54,3906,3908],{"className":3907,"ariaHidden":2113},[2112],[54,3909,3911,3914],{"className":3910},[2117],[54,3912],{"className":3913,"style":3846},[2121],[54,3915,3917],{"className":3916},[2126,3850],[54,3918,3920],{"className":3919},[2139],[54,3921,3923],{"className":3922},[2143],[54,3924,3926,3934],{"className":3925,"style":3846},[2147],[54,3927,3928,3931],{"style":3862},[54,3929],{"className":3930,"style":3866},[2155],[54,3932,2694],{"className":3933,"style":2709},[2126,2167],[54,3935,3936,3939],{"style":3872},[54,3937],{"className":3938,"style":3866},[2155],[54,3940,3942],{"className":3941,"style":3880},[3879],[54,3943,3833],{"className":3944},[2126],", the projection of the Hamiltonian in the Krylov space, can be calculated by calculating the following expectation values",[54,3947,3949],{"className":3948},[2340],[54,3950,3952,4007],{"className":3951},[2079],[54,3953,3955],{"className":3954},[2083],[2085,3956,3957],{"xmlns":2087,"display":2349},[2089,3958,3959,4004],{},[2092,3960,3961,3977,3979,3982,3988,3990,3992,3994,4000,4002],{},[2724,3962,3963,3969],{},[3827,3964,3965,3967],{"accent":2113},[2098,3966,2694],{},[2245,3968,3833],{},[2092,3970,3971,3974],{},[2098,3972,3973],{},"m",[2098,3975,3976],{},"n",[2245,3978,122],{},[2245,3980,3981],{"stretchy":2311},"⟨",[2724,3983,3984,3986],{},[2098,3985,3349],{},[2098,3987,3973],{},[2098,3989,2305],{"mathvariant":2304},[2098,3991,2694],{},[2098,3993,2305],{"mathvariant":2304},[2724,3995,3996,3998],{},[2098,3997,3349],{},[2098,3999,3976],{},[2245,4001,2312],{"stretchy":2311},[2245,4003,122],{},[2105,4005,4006],{"encoding":2107},"\\tilde{H}_{mn} = \\langle \\psi_m \\vert H \\vert \\psi_n \\rangle =",[54,4008,4010,4099],{"className":4009,"ariaHidden":2113},[2112],[54,4011,4013,4017,4090,4093,4096],{"className":4012},[2117],[54,4014],{"className":4015,"style":4016},[2121],"height:1.0702em;vertical-align:-0.15em;",[54,4018,4020,4051],{"className":4019},[2126],[54,4021,4023],{"className":4022},[2126,3850],[54,4024,4026],{"className":4025},[2139],[54,4027,4029],{"className":4028},[2143],[54,4030,4032,4040],{"className":4031,"style":3846},[2147],[54,4033,4034,4037],{"style":3862},[54,4035],{"className":4036,"style":3866},[2155],[54,4038,2694],{"className":4039,"style":2709},[2126,2167],[54,4041,4042,4045],{"style":3872},[54,4043],{"className":4044,"style":3866},[2155],[54,4046,4048],{"className":4047,"style":3880},[3879],[54,4049,3833],{"className":4050},[2126],[54,4052,4054],{"className":4053},[2135],[54,4055,4057,4082],{"className":4056},[2139,2756],[54,4058,4060,4079],{"className":4059},[2143],[54,4061,4064],{"className":4062,"style":4063},[2147],"height:0.1514em;",[54,4065,4067,4070],{"style":4066},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[54,4068],{"className":4069,"style":2156},[2155],[54,4071,4073],{"className":4072},[2160,2161,2162,2163],[54,4074,4076],{"className":4075},[2126,2163],[54,4077,2249],{"className":4078},[2126,2167,2163],[54,4080,2781],{"className":4081},[2780],[54,4083,4085],{"className":4084},[2143],[54,4086,4088],{"className":4087,"style":2788},[2147],[54,4089],{},[54,4091],{"className":4092,"style":2475},[2270],[54,4094,122],{"className":4095},[2479],[54,4097],{"className":4098,"style":2475},[2270],[54,4100,4102,4105,4108,4149,4152,4155,4158,4198,4201,4204],{"className":4101},[2117],[54,4103],{"className":4104,"style":2325},[2121],[54,4106,3981],{"className":4107},[2497],[54,4109,4111,4114],{"className":4110},[2126],[54,4112,3349],{"className":4113,"style":2332},[2126,2167],[54,4115,4117],{"className":4116},[2135],[54,4118,4120,4141],{"className":4119},[2139,2756],[54,4121,4123,4138],{"className":4122},[2143],[54,4124,4126],{"className":4125,"style":4063},[2147],[54,4127,4129,4132],{"style":4128},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[54,4130],{"className":4131,"style":2156},[2155],[54,4133,4135],{"className":4134},[2160,2161,2162,2163],[54,4136,3973],{"className":4137},[2126,2167,2163],[54,4139,2781],{"className":4140},[2780],[54,4142,4144],{"className":4143},[2143],[54,4145,4147],{"className":4146,"style":2788},[2147],[54,4148],{},[54,4150,2305],{"className":4151},[2126],[54,4153,2694],{"className":4154,"style":2709},[2126,2167],[54,4156,2305],{"className":4157},[2126],[54,4159,4161,4164],{"className":4160},[2126],[54,4162,3349],{"className":4163,"style":2332},[2126,2167],[54,4165,4167],{"className":4166},[2135],[54,4168,4170,4190],{"className":4169},[2139,2756],[54,4171,4173,4187],{"className":4172},[2143],[54,4174,4176],{"className":4175,"style":4063},[2147],[54,4177,4178,4181],{"style":4128},[54,4179],{"className":4180,"style":2156},[2155],[54,4182,4184],{"className":4183},[2160,2161,2162,2163],[54,4185,3976],{"className":4186},[2126,2167,2163],[54,4188,2781],{"className":4189},[2780],[54,4191,4193],{"className":4192},[2143],[54,4194,4196],{"className":4195,"style":2788},[2147],[54,4197],{},[54,4199,2312],{"className":4200},[2336],[54,4202],{"className":4203,"style":2475},[2270],[54,4205,122],{"className":4206},[2479],[54,4208,4210],{"className":4209},[2340],[54,4211,4213,4275],{"className":4212},[2079],[54,4214,4216],{"className":4215},[2083],[2085,4217,4218],{"xmlns":2087,"display":2349},[2089,4219,4220,4272],{},[2092,4221,4222,4224,4226,4228,4230,4246,4248,4266,4268,4270],{},[2245,4223,122],{},[2245,4225,3981],{"stretchy":2311},[2098,4227,3349],{},[2098,4229,2305],{"mathvariant":2304},[2095,4231,4232,4234],{},[2098,4233,2842],{},[2092,4235,4236,4238,4240],{},[2098,4237,2849],{},[2098,4239,2694],{},[2724,4241,4242,4244],{},[2098,4243,2854],{},[2098,4245,3973],{},[2098,4247,2694],{},[2095,4249,4250,4252],{},[2098,4251,2842],{},[2092,4253,4254,4256,4258,4260],{},[2245,4255,2247],{},[2098,4257,2849],{},[2098,4259,2694],{},[2724,4261,4262,4264],{},[2098,4263,2854],{},[2098,4265,3976],{},[2098,4267,2305],{"mathvariant":2304},[2098,4269,3349],{},[2245,4271,2312],{"stretchy":2311},[2105,4273,4274],{"encoding":2107},"= \\langle \\psi \\vert e^{i H t_m} H e^{-i H t_n} \\vert \\psi \\rangle",[54,4276,4278,4291],{"className":4277,"ariaHidden":2113},[2112],[54,4279,4281,4285,4288],{"className":4280},[2117],[54,4282],{"className":4283,"style":4284},[2121],"height:0.3669em;",[54,4286,122],{"className":4287},[2479],[54,4289],{"className":4290,"style":2475},[2270],[54,4292,4294,4298,4301,4304,4307,4388,4391,4469,4472,4475],{"className":4293},[2117],[54,4295],{"className":4296,"style":4297},[2121],"height:1.1413em;vertical-align:-0.25em;",[54,4299,3981],{"className":4300},[2497],[54,4302,3349],{"className":4303,"style":2332},[2126,2167],[54,4305,2305],{"className":4306},[2126],[54,4308,4310,4313],{"className":4309},[2126],[54,4311,2842],{"className":4312},[2126,2167],[54,4314,4316],{"className":4315},[2135],[54,4317,4319],{"className":4318},[2139],[54,4320,4322],{"className":4321},[2143],[54,4323,4326],{"className":4324,"style":4325},[2147],"height:0.8913em;",[54,4327,4328,4331],{"style":2462},[54,4329],{"className":4330,"style":2156},[2155],[54,4332,4334],{"className":4333},[2160,2161,2162,2163],[54,4335,4337,4340,4343],{"className":4336},[2126,2163],[54,4338,2849],{"className":4339},[2126,2167,2163],[54,4341,2694],{"className":4342,"style":2709},[2126,2167,2163],[54,4344,4346,4349],{"className":4345},[2126,2163],[54,4347,2854],{"className":4348},[2126,2167,2163],[54,4350,4352],{"className":4351},[2135],[54,4353,4355,4379],{"className":4354},[2139,2756],[54,4356,4358,4376],{"className":4357},[2143],[54,4359,4362],{"className":4360,"style":4361},[2147],"height:0.1645em;",[54,4363,4365,4369],{"style":4364},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[54,4366],{"className":4367,"style":4368},[2155],"height:2.5em;",[54,4370,4373],{"className":4371},[2160,4372,2504,2163],"reset-size3",[54,4374,3973],{"className":4375},[2126,2167,2163],[54,4377,2781],{"className":4378},[2780],[54,4380,4382],{"className":4381},[2143],[54,4383,4386],{"className":4384,"style":4385},[2147],"height:0.143em;",[54,4387],{},[54,4389,2694],{"className":4390,"style":2709},[2126,2167],[54,4392,4394,4397],{"className":4393},[2126],[54,4395,2842],{"className":4396},[2126,2167],[54,4398,4400],{"className":4399},[2135],[54,4401,4403],{"className":4402},[2139],[54,4404,4406],{"className":4405},[2143],[54,4407,4409],{"className":4408,"style":4325},[2147],[54,4410,4411,4414],{"style":2462},[54,4412],{"className":4413,"style":2156},[2155],[54,4415,4417],{"className":4416},[2160,2161,2162,2163],[54,4418,4420,4423,4426,4429],{"className":4419},[2126,2163],[54,4421,2247],{"className":4422},[2126,2163],[54,4424,2849],{"className":4425},[2126,2167,2163],[54,4427,2694],{"className":4428,"style":2709},[2126,2167,2163],[54,4430,4432,4435],{"className":4431},[2126,2163],[54,4433,2854],{"className":4434},[2126,2167,2163],[54,4436,4438],{"className":4437},[2135],[54,4439,4441,4461],{"className":4440},[2139,2756],[54,4442,4444,4458],{"className":4443},[2143],[54,4445,4447],{"className":4446,"style":4361},[2147],[54,4448,4449,4452],{"style":4364},[54,4450],{"className":4451,"style":4368},[2155],[54,4453,4455],{"className":4454},[2160,4372,2504,2163],[54,4456,3976],{"className":4457},[2126,2167,2163],[54,4459,2781],{"className":4460},[2780],[54,4462,4464],{"className":4463},[2143],[54,4465,4467],{"className":4466,"style":4385},[2147],[54,4468],{},[54,4470,2305],{"className":4471},[2126],[54,4473,3349],{"className":4474,"style":2332},[2126,2167],[54,4476,2312],{"className":4477},[2336],[54,4479,4481],{"className":4480},[2340],[54,4482,4484,4547],{"className":4483},[2079],[54,4485,4487],{"className":4486},[2083],[2085,4488,4489],{"xmlns":2087,"display":2349},[2089,4490,4491,4544],{},[2092,4492,4493,4495,4497,4499,4501,4518,4520,4538,4540,4542],{},[2245,4494,122],{},[2245,4496,3981],{"stretchy":2311},[2098,4498,3349],{},[2098,4500,2305],{"mathvariant":2304},[2095,4502,4503,4505],{},[2098,4504,2842],{},[2092,4506,4507,4509,4511,4513,4516],{},[2098,4508,2849],{},[2098,4510,2694],{},[2098,4512,3973],{},[2098,4514,4515],{},"d",[2098,4517,2854],{},[2098,4519,2694],{},[2095,4521,4522,4524],{},[2098,4523,2842],{},[2092,4525,4526,4528,4530,4532,4534,4536],{},[2245,4527,2247],{},[2098,4529,2849],{},[2098,4531,2694],{},[2098,4533,3976],{},[2098,4535,4515],{},[2098,4537,2854],{},[2098,4539,2305],{"mathvariant":2304},[2098,4541,3349],{},[2245,4543,2312],{"stretchy":2311},[2105,4545,4546],{"encoding":2107},"= \\langle \\psi \\vert e^{i H m dt} H e^{-i H n dt} \\vert \\psi \\rangle",[54,4548,4550,4562],{"className":4549,"ariaHidden":2113},[2112],[54,4551,4553,4556,4559],{"className":4552},[2117],[54,4554],{"className":4555,"style":4284},[2121],[54,4557,122],{"className":4558},[2479],[54,4560],{"className":4561,"style":2475},[2270],[54,4563,4565,4569,4572,4575,4578,4623,4626,4673,4676,4679],{"className":4564},[2117],[54,4566],{"className":4567,"style":4568},[2121],"height:1.1491em;vertical-align:-0.25em;",[54,4570,3981],{"className":4571},[2497],[54,4573,3349],{"className":4574,"style":2332},[2126,2167],[54,4576,2305],{"className":4577},[2126],[54,4579,4581,4584],{"className":4580},[2126],[54,4582,2842],{"className":4583},[2126,2167],[54,4585,4587],{"className":4586},[2135],[54,4588,4590],{"className":4589},[2139],[54,4591,4593],{"className":4592},[2143],[54,4594,4597],{"className":4595,"style":4596},[2147],"height:0.8991em;",[54,4598,4599,4602],{"style":2462},[54,4600],{"className":4601,"style":2156},[2155],[54,4603,4605],{"className":4604},[2160,2161,2162,2163],[54,4606,4608,4611,4614,4617,4620],{"className":4607},[2126,2163],[54,4609,2849],{"className":4610},[2126,2167,2163],[54,4612,2694],{"className":4613,"style":2709},[2126,2167,2163],[54,4615,3973],{"className":4616},[2126,2167,2163],[54,4618,4515],{"className":4619},[2126,2167,2163],[54,4621,2854],{"className":4622},[2126,2167,2163],[54,4624,2694],{"className":4625,"style":2709},[2126,2167],[54,4627,4629,4632],{"className":4628},[2126],[54,4630,2842],{"className":4631},[2126,2167],[54,4633,4635],{"className":4634},[2135],[54,4636,4638],{"className":4637},[2139],[54,4639,4641],{"className":4640},[2143],[54,4642,4644],{"className":4643,"style":4596},[2147],[54,4645,4646,4649],{"style":2462},[54,4647],{"className":4648,"style":2156},[2155],[54,4650,4652],{"className":4651},[2160,2161,2162,2163],[54,4653,4655,4658,4661,4664,4667,4670],{"className":4654},[2126,2163],[54,4656,2247],{"className":4657},[2126,2163],[54,4659,2849],{"className":4660},[2126,2167,2163],[54,4662,2694],{"className":4663,"style":2709},[2126,2167,2163],[54,4665,3976],{"className":4666},[2126,2167,2163],[54,4668,4515],{"className":4669},[2126,2167,2163],[54,4671,2854],{"className":4672},[2126,2167,2163],[54,4674,2305],{"className":4675},[2126],[54,4677,3349],{"className":4678,"style":2332},[2126,2167],[54,4680,2312],{"className":4681},[2336],[13,4683,4684,4685,4894,4895,5000,5001,5035,5036,5105,5106,5134,5135,5187,5188,5216,5217,5274,5275,5335,5336,5396,5397],{},"Where ",[54,4686,4688,4736],{"className":4687},[2079],[54,4689,4691],{"className":4690},[2083],[2085,4692,4693],{"xmlns":2087},[2089,4694,4695,4733],{},[2092,4696,4697,4699,4705,4707,4709,4727,4729,4731],{},[2098,4698,2305],{"mathvariant":2304},[2724,4700,4701,4703],{},[2098,4702,3349],{},[2098,4704,3976],{},[2245,4706,2312],{"stretchy":2311},[2245,4708,122],{},[2095,4710,4711,4713],{},[2098,4712,2842],{},[2092,4714,4715,4717,4719,4721],{},[2245,4716,2247],{},[2098,4718,2849],{},[2098,4720,2694],{},[2724,4722,4723,4725],{},[2098,4724,2854],{},[2098,4726,3976],{},[2098,4728,2305],{"mathvariant":2304},[2098,4730,3349],{},[2245,4732,2312],{"stretchy":2311},[2105,4734,4735],{"encoding":2107},"\\vert \\psi_n \\rangle = e^{-i H t_n} \\vert \\psi \\rangle",[54,4737,4739,4800],{"className":4738,"ariaHidden":2113},[2112],[54,4740,4742,4745,4748,4788,4791,4794,4797],{"className":4741},[2117],[54,4743],{"className":4744,"style":2325},[2121],[54,4746,2305],{"className":4747},[2126],[54,4749,4751,4754],{"className":4750},[2126],[54,4752,3349],{"className":4753,"style":2332},[2126,2167],[54,4755,4757],{"className":4756},[2135],[54,4758,4760,4780],{"className":4759},[2139,2756],[54,4761,4763,4777],{"className":4762},[2143],[54,4764,4766],{"className":4765,"style":4063},[2147],[54,4767,4768,4771],{"style":4128},[54,4769],{"className":4770,"style":2156},[2155],[54,4772,4774],{"className":4773},[2160,2161,2162,2163],[54,4775,3976],{"className":4776},[2126,2167,2163],[54,4778,2781],{"className":4779},[2780],[54,4781,4783],{"className":4782},[2143],[54,4784,4786],{"className":4785,"style":2788},[2147],[54,4787],{},[54,4789,2312],{"className":4790},[2336],[54,4792],{"className":4793,"style":2475},[2270],[54,4795,122],{"className":4796},[2479],[54,4798],{"className":4799,"style":2475},[2270],[54,4801,4803,4807,4885,4888,4891],{"className":4802},[2117],[54,4804],{"className":4805,"style":4806},[2121],"height:1.0913em;vertical-align:-0.25em;",[54,4808,4810,4813],{"className":4809},[2126],[54,4811,2842],{"className":4812},[2126,2167],[54,4814,4816],{"className":4815},[2135],[54,4817,4819],{"className":4818},[2139],[54,4820,4822],{"className":4821},[2143],[54,4823,4825],{"className":4824,"style":2886},[2147],[54,4826,4827,4830],{"style":2151},[54,4828],{"className":4829,"style":2156},[2155],[54,4831,4833],{"className":4832},[2160,2161,2162,2163],[54,4834,4836,4839,4842,4845],{"className":4835},[2126,2163],[54,4837,2247],{"className":4838},[2126,2163],[54,4840,2849],{"className":4841},[2126,2167,2163],[54,4843,2694],{"className":4844,"style":2709},[2126,2167,2163],[54,4846,4848,4851],{"className":4847},[2126,2163],[54,4849,2854],{"className":4850},[2126,2167,2163],[54,4852,4854],{"className":4853},[2135],[54,4855,4857,4877],{"className":4856},[2139,2756],[54,4858,4860,4874],{"className":4859},[2143],[54,4861,4863],{"className":4862,"style":4361},[2147],[54,4864,4865,4868],{"style":4364},[54,4866],{"className":4867,"style":4368},[2155],[54,4869,4871],{"className":4870},[2160,4372,2504,2163],[54,4872,3976],{"className":4873},[2126,2167,2163],[54,4875,2781],{"className":4876},[2780],[54,4878,4880],{"className":4879},[2143],[54,4881,4883],{"className":4882,"style":4385},[2147],[54,4884],{},[54,4886,2305],{"className":4887},[2126],[54,4889,3349],{"className":4890,"style":2332},[2126,2167],[54,4892,2312],{"className":4893},[2336]," are the vectors of the unitary Krylov space and ",[54,4896,4898,4924],{"className":4897},[2079],[54,4899,4901],{"className":4900},[2083],[2085,4902,4903],{"xmlns":2087},[2089,4904,4905,4921],{},[2092,4906,4907,4913,4915,4917,4919],{},[2724,4908,4909,4911],{},[2098,4910,2854],{},[2098,4912,3976],{},[2245,4914,122],{},[2098,4916,3976],{},[2098,4918,4515],{},[2098,4920,2854],{},[2105,4922,4923],{"encoding":2107},"t_n = n dt",[54,4925,4927,4984],{"className":4926,"ariaHidden":2113},[2112],[54,4928,4930,4934,4975,4978,4981],{"className":4929},[2117],[54,4931],{"className":4932,"style":4933},[2121],"height:0.7651em;vertical-align:-0.15em;",[54,4935,4937,4940],{"className":4936},[2126],[54,4938,2854],{"className":4939},[2126,2167],[54,4941,4943],{"className":4942},[2135],[54,4944,4946,4967],{"className":4945},[2139,2756],[54,4947,4949,4964],{"className":4948},[2143],[54,4950,4952],{"className":4951,"style":4063},[2147],[54,4953,4955,4958],{"style":4954},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[54,4956],{"className":4957,"style":2156},[2155],[54,4959,4961],{"className":4960},[2160,2161,2162,2163],[54,4962,3976],{"className":4963},[2126,2167,2163],[54,4965,2781],{"className":4966},[2780],[54,4968,4970],{"className":4969},[2143],[54,4971,4973],{"className":4972,"style":2788},[2147],[54,4974],{},[54,4976],{"className":4977,"style":2475},[2270],[54,4979,122],{"className":4980},[2479],[54,4982],{"className":4983,"style":2475},[2270],[54,4985,4987,4991,4994,4997],{"className":4986},[2117],[54,4988],{"className":4989,"style":4990},[2121],"height:0.6944em;",[54,4992,3976],{"className":4993},[2126,2167],[54,4995,4515],{"className":4996},[2126,2167],[54,4998,2854],{"className":4999},[2126,2167]," are the multiples of the time step ",[54,5002,5004,5020],{"className":5003},[2079],[54,5005,5007],{"className":5006},[2083],[2085,5008,5009],{"xmlns":2087},[2089,5010,5011,5017],{},[2092,5012,5013,5015],{},[2098,5014,4515],{},[2098,5016,2854],{},[2105,5018,5019],{"encoding":2107},"dt",[54,5021,5023],{"className":5022,"ariaHidden":2113},[2112],[54,5024,5026,5029,5032],{"className":5025},[2117],[54,5027],{"className":5028,"style":4990},[2121],[54,5030,4515],{"className":5031},[2126,2167],[54,5033,2854],{"className":5034},[2126,2167]," chosen. On a quantum computer, the calculation of each matrix elements can be done with any algorithm which allows to obtain overlap between quantum states. This tutorial focuses on the Hadamard test. Given that the ",[54,5037,5039,5056],{"className":5038},[2079],[54,5040,5042],{"className":5041},[2083],[2085,5043,5044],{"xmlns":2087},[2089,5045,5046,5054],{},[2092,5047,5048],{},[2724,5049,5050,5052],{},[2098,5051,2101],{"mathvariant":2100},[2098,5053,2835],{},[2105,5055,2949],{"encoding":2107},[54,5057,5059],{"className":5058,"ariaHidden":2113},[2112],[54,5060,5062,5065],{"className":5061},[2117],[54,5063],{"className":5064,"style":2743},[2121],[54,5066,5068,5071],{"className":5067},[2126],[54,5069,2101],{"className":5070,"style":2131},[2126,2130],[54,5072,5074],{"className":5073},[2135],[54,5075,5077,5097],{"className":5076},[2139,2756],[54,5078,5080,5094],{"className":5079},[2143],[54,5081,5083],{"className":5082,"style":2763},[2147],[54,5084,5085,5088],{"style":2766},[54,5086],{"className":5087,"style":2156},[2155],[54,5089,5091],{"className":5090},[2160,2161,2162,2163],[54,5092,2835],{"className":5093,"style":2870},[2126,2167,2163],[54,5095,2781],{"className":5096},[2780],[54,5098,5100],{"className":5099},[2143],[54,5101,5103],{"className":5102,"style":2788},[2147],[54,5104],{}," has dimension ",[54,5107,5109,5122],{"className":5108},[2079],[54,5110,5112],{"className":5111},[2083],[2085,5113,5114],{"xmlns":2087},[2089,5115,5116,5120],{},[2092,5117,5118],{},[2098,5119,1772],{},[2105,5121,1772],{"encoding":2107},[54,5123,5125],{"className":5124,"ariaHidden":2113},[2112],[54,5126,5128,5131],{"className":5127},[2117],[54,5129],{"className":5130,"style":2195},[2121],[54,5132,1772],{"className":5133,"style":2168},[2126,2167],", the Hamiltonian projected into the subspace will have dimensions ",[54,5136,5138,5157],{"className":5137},[2079],[54,5139,5141],{"className":5140},[2083],[2085,5142,5143],{"xmlns":2087},[2089,5144,5145,5154],{},[2092,5146,5147,5149,5152],{},[2098,5148,1772],{},[2245,5150,5151],{},"×",[2098,5153,1772],{},[2105,5155,5156],{"encoding":2107},"r \\times r",[54,5158,5160,5178],{"className":5159,"ariaHidden":2113},[2112],[54,5161,5163,5166,5169,5172,5175],{"className":5162},[2117],[54,5164],{"className":5165,"style":2263},[2121],[54,5167,1772],{"className":5168,"style":2168},[2126,2167],[54,5170],{"className":5171,"style":2271},[2270],[54,5173,5151],{"className":5174},[2275],[54,5176],{"className":5177,"style":2271},[2270],[54,5179,5181,5184],{"className":5180},[2117],[54,5182],{"className":5183,"style":2195},[2121],[54,5185,1772],{"className":5186,"style":2168},[2126,2167],". With ",[54,5189,5191,5204],{"className":5190},[2079],[54,5192,5194],{"className":5193},[2083],[2085,5195,5196],{"xmlns":2087},[2089,5197,5198,5202],{},[2092,5199,5200],{},[2098,5201,1772],{},[2105,5203,1772],{"encoding":2107},[54,5205,5207],{"className":5206,"ariaHidden":2113},[2112],[54,5208,5210,5213],{"className":5209},[2117],[54,5211],{"className":5212,"style":2195},[2121],[54,5214,1772],{"className":5215,"style":2168},[2126,2167]," small enough (generally ",[54,5218,5220,5242],{"className":5219},[2079],[54,5221,5223],{"className":5222},[2083],[2085,5224,5225],{"xmlns":2087},[2089,5226,5227,5239],{},[2092,5228,5229,5231,5234,5236],{},[2098,5230,1772],{},[2245,5232,5233],{},"\u003C",[2245,5235,5233],{},[2249,5237,5238],{},"100",[2105,5240,5241],{"encoding":2107},"r\u003C\u003C100",[54,5243,5245,5265],{"className":5244,"ariaHidden":2113},[2112],[54,5246,5248,5252,5255,5258,5262],{"className":5247},[2117],[54,5249],{"className":5250,"style":5251},[2121],"height:0.5782em;vertical-align:-0.0391em;",[54,5253,1772],{"className":5254,"style":2168},[2126,2167],[54,5256],{"className":5257,"style":2475},[2270],[54,5259,5261],{"className":5260},[2479],"\u003C\u003C",[54,5263],{"className":5264,"style":2475},[2270],[54,5266,5268,5271],{"className":5267},[2117],[54,5269],{"className":5270,"style":2285},[2121],[54,5272,5238],{"className":5273},[2126]," is sufficient to obtain convergence of estimates of eigenenergies) we can then easily diagonalize the projected Hamiltonian ",[54,5276,5278,5295],{"className":5277},[2079],[54,5279,5281],{"className":5280},[2083],[2085,5282,5283],{"xmlns":2087},[2089,5284,5285,5293],{},[2092,5286,5287],{},[3827,5288,5289,5291],{"accent":2113},[2098,5290,2694],{},[2245,5292,3833],{},[2105,5294,3836],{"encoding":2107},[54,5296,5298],{"className":5297,"ariaHidden":2113},[2112],[54,5299,5301,5304],{"className":5300},[2117],[54,5302],{"className":5303,"style":3846},[2121],[54,5305,5307],{"className":5306},[2126,3850],[54,5308,5310],{"className":5309},[2139],[54,5311,5313],{"className":5312},[2143],[54,5314,5316,5324],{"className":5315,"style":3846},[2147],[54,5317,5318,5321],{"style":3862},[54,5319],{"className":5320,"style":3866},[2155],[54,5322,2694],{"className":5323,"style":2709},[2126,2167],[54,5325,5326,5329],{"style":3872},[54,5327],{"className":5328,"style":3866},[2155],[54,5330,5332],{"className":5331,"style":3880},[3879],[54,5333,3833],{"className":5334},[2126],". However, we cannot directly diagonalize ",[54,5337,5339,5356],{"className":5338},[2079],[54,5340,5342],{"className":5341},[2083],[2085,5343,5344],{"xmlns":2087},[2089,5345,5346,5354],{},[2092,5347,5348],{},[3827,5349,5350,5352],{"accent":2113},[2098,5351,2694],{},[2245,5353,3833],{},[2105,5355,3836],{"encoding":2107},[54,5357,5359],{"className":5358,"ariaHidden":2113},[2112],[54,5360,5362,5365],{"className":5361},[2117],[54,5363],{"className":5364,"style":3846},[2121],[54,5366,5368],{"className":5367},[2126,3850],[54,5369,5371],{"className":5370},[2139],[54,5372,5374],{"className":5373},[2143],[54,5375,5377,5385],{"className":5376,"style":3846},[2147],[54,5378,5379,5382],{"style":3862},[54,5380],{"className":5381,"style":3866},[2155],[54,5383,2694],{"className":5384,"style":2709},[2126,2167],[54,5386,5387,5390],{"style":3872},[54,5388],{"className":5389,"style":3866},[2155],[54,5391,5393],{"className":5392,"style":3880},[3879],[54,5394,3833],{"className":5395},[2126]," because of the non-orthogonality of the Krylov space vectors. We'll have to measure their overlaps and construct a matrix ",[54,5398,5400,5419],{"className":5399},[2079],[54,5401,5403],{"className":5402},[2083],[2085,5404,5405],{"xmlns":2087},[2089,5406,5407,5416],{},[2092,5408,5409],{},[3827,5410,5411,5414],{"accent":2113},[2098,5412,5413],{},"S",[2245,5415,3833],{},[2105,5417,5418],{"encoding":2107},"\\tilde{S}",[54,5420,5422],{"className":5421,"ariaHidden":2113},[2112],[54,5423,5425,5428],{"className":5424},[2117],[54,5426],{"className":5427,"style":3846},[2121],[54,5429,5431],{"className":5430},[2126,3850],[54,5432,5434],{"className":5433},[2139],[54,5435,5437],{"className":5436},[2143],[54,5438,5440,5449],{"className":5439,"style":3846},[2147],[54,5441,5442,5445],{"style":3862},[54,5443],{"className":5444,"style":3866},[2155],[54,5446,5413],{"className":5447,"style":5448},[2126,2167],"margin-right:0.0576em;",[54,5450,5451,5454],{"style":3872},[54,5452],{"className":5453,"style":3866},[2155],[54,5455,5458],{"className":5456,"style":5457},[3879],"left:-0.1667em;",[54,5459,3833],{"className":5460},[2126],[54,5462,5464],{"className":5463},[2340],[54,5465,5467,5513],{"className":5466},[2079],[54,5468,5470],{"className":5469},[2083],[2085,5471,5472],{"xmlns":2087,"display":2349},[2089,5473,5474,5510],{},[2092,5475,5476,5490,5492,5494,5500,5502,5508],{},[2724,5477,5478,5484],{},[3827,5479,5480,5482],{"accent":2113},[2098,5481,5413],{},[2245,5483,3833],{},[2092,5485,5486,5488],{},[2098,5487,3973],{},[2098,5489,3976],{},[2245,5491,122],{},[2245,5493,3981],{"stretchy":2311},[2724,5495,5496,5498],{},[2098,5497,3349],{},[2098,5499,3973],{},[2098,5501,2305],{"mathvariant":2304},[2724,5503,5504,5506],{},[2098,5505,3349],{},[2098,5507,3976],{},[2245,5509,2312],{"stretchy":2311},[2105,5511,5512],{"encoding":2107},"\\tilde{S}_{mn} = \\langle \\psi_m \\vert \\psi_n \\rangle",[54,5514,5516,5603],{"className":5515,"ariaHidden":2113},[2112],[54,5517,5519,5522,5594,5597,5600],{"className":5518},[2117],[54,5520],{"className":5521,"style":4016},[2121],[54,5523,5525,5556],{"className":5524},[2126],[54,5526,5528],{"className":5527},[2126,3850],[54,5529,5531],{"className":5530},[2139],[54,5532,5534],{"className":5533},[2143],[54,5535,5537,5545],{"className":5536,"style":3846},[2147],[54,5538,5539,5542],{"style":3862},[54,5540],{"className":5541,"style":3866},[2155],[54,5543,5413],{"className":5544,"style":5448},[2126,2167],[54,5546,5547,5550],{"style":3872},[54,5548],{"className":5549,"style":3866},[2155],[54,5551,5553],{"className":5552,"style":5457},[3879],[54,5554,3833],{"className":5555},[2126],[54,5557,5559],{"className":5558},[2135],[54,5560,5562,5586],{"className":5561},[2139,2756],[54,5563,5565,5583],{"className":5564},[2143],[54,5566,5568],{"className":5567,"style":4063},[2147],[54,5569,5571,5574],{"style":5570},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[54,5572],{"className":5573,"style":2156},[2155],[54,5575,5577],{"className":5576},[2160,2161,2162,2163],[54,5578,5580],{"className":5579},[2126,2163],[54,5581,2249],{"className":5582},[2126,2167,2163],[54,5584,2781],{"className":5585},[2780],[54,5587,5589],{"className":5588},[2143],[54,5590,5592],{"className":5591,"style":2788},[2147],[54,5593],{},[54,5595],{"className":5596,"style":2475},[2270],[54,5598,122],{"className":5599},[2479],[54,5601],{"className":5602,"style":2475},[2270],[54,5604,5606,5609,5612,5652,5655,5695],{"className":5605},[2117],[54,5607],{"className":5608,"style":2325},[2121],[54,5610,3981],{"className":5611},[2497],[54,5613,5615,5618],{"className":5614},[2126],[54,5616,3349],{"className":5617,"style":2332},[2126,2167],[54,5619,5621],{"className":5620},[2135],[54,5622,5624,5644],{"className":5623},[2139,2756],[54,5625,5627,5641],{"className":5626},[2143],[54,5628,5630],{"className":5629,"style":4063},[2147],[54,5631,5632,5635],{"style":4128},[54,5633],{"className":5634,"style":2156},[2155],[54,5636,5638],{"className":5637},[2160,2161,2162,2163],[54,5639,3973],{"className":5640},[2126,2167,2163],[54,5642,2781],{"className":5643},[2780],[54,5645,5647],{"className":5646},[2143],[54,5648,5650],{"className":5649,"style":2788},[2147],[54,5651],{},[54,5653,2305],{"className":5654},[2126],[54,5656,5658,5661],{"className":5657},[2126],[54,5659,3349],{"className":5660,"style":2332},[2126,2167],[54,5662,5664],{"className":5663},[2135],[54,5665,5667,5687],{"className":5666},[2139,2756],[54,5668,5670,5684],{"className":5669},[2143],[54,5671,5673],{"className":5672,"style":4063},[2147],[54,5674,5675,5678],{"style":4128},[54,5676],{"className":5677,"style":2156},[2155],[54,5679,5681],{"className":5680},[2160,2161,2162,2163],[54,5682,3976],{"className":5683},[2126,2167,2163],[54,5685,2781],{"className":5686},[2780],[54,5688,5690],{"className":5689},[2143],[54,5691,5693],{"className":5692,"style":2788},[2147],[54,5694],{},[54,5696,2312],{"className":5697},[2336],[13,5699,5700],{},"This allows us to solve the eigenvalue problem in a non-orthogonal space (also called generalized eigenvalue problem)",[54,5702,5704],{"className":5703},[2340],[54,5705,5707,5758],{"className":5706},[2079],[54,5708,5710],{"className":5709},[2083],[2085,5711,5712],{"xmlns":2087,"display":2349},[2089,5713,5714,5755],{},[2092,5715,5716,5722,5726,5734,5736,5739,5741,5747,5749],{},[3827,5717,5718,5720],{"accent":2113},[2098,5719,2694],{},[2245,5721,3833],{},[5723,5724,5725],"mtext",{}," ",[3827,5727,5728,5731],{"accent":2113},[2098,5729,5730],{},"c",[2245,5732,5733],{},"⃗",[2245,5735,122],{},[2098,5737,5738],{},"E",[5723,5740,5725],{},[3827,5742,5743,5745],{"accent":2113},[2098,5744,5413],{},[2245,5746,3833],{},[5723,5748,5725],{},[3827,5750,5751,5753],{"accent":2113},[2098,5752,5730],{},[2245,5754,5733],{},[2105,5756,5757],{"encoding":2107},"\\tilde{H} \\ \\vec{c} = E \\ \\tilde{S} \\ \\vec{c}",[54,5759,5761,5858],{"className":5760,"ariaHidden":2113},[2112],[54,5762,5764,5767,5798,5801,5849,5852,5855],{"className":5763},[2117],[54,5765],{"className":5766,"style":3846},[2121],[54,5768,5770],{"className":5769},[2126,3850],[54,5771,5773],{"className":5772},[2139],[54,5774,5776],{"className":5775},[2143],[54,5777,5779,5787],{"className":5778,"style":3846},[2147],[54,5780,5781,5784],{"style":3862},[54,5782],{"className":5783,"style":3866},[2155],[54,5785,2694],{"className":5786,"style":2709},[2126,2167],[54,5788,5789,5792],{"style":3872},[54,5790],{"className":5791,"style":3866},[2155],[54,5793,5795],{"className":5794,"style":3880},[3879],[54,5796,3833],{"className":5797},[2126],[54,5799,5725],{"className":5800},[2270],[54,5802,5804],{"className":5803},[2126,3850],[54,5805,5807],{"className":5806},[2139],[54,5808,5810],{"className":5809},[2143],[54,5811,5814,5822],{"className":5812,"style":5813},[2147],"height:0.714em;",[54,5815,5816,5819],{"style":3862},[54,5817],{"className":5818,"style":3866},[2155],[54,5820,5730],{"className":5821},[2126,2167],[54,5823,5824,5827],{"style":3862},[54,5825],{"className":5826,"style":3866},[2155],[54,5828,5831],{"className":5829,"style":5830},[3879],"left:-0.1799em;",[54,5832,5836],{"className":5833,"style":5835},[5834],"overlay","height:0.714em;width:0.471em;",[5837,5838,5845],"svg",{"xmlns":5839,"width":5840,"height":5841,"style":5842,"viewBox":5843,"preserveAspectRatio":5844},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.471em","0.714em","width:0.471em","0 0 471 714","xMinYMin",[5846,5847],"path",{"d":5848},"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z",[54,5850],{"className":5851,"style":2475},[2270],[54,5853,122],{"className":5854},[2479],[54,5856],{"className":5857,"style":2475},[2270],[54,5859,5861,5864,5867,5870,5901,5904],{"className":5860},[2117],[54,5862],{"className":5863,"style":3846},[2121],[54,5865,5738],{"className":5866,"style":5448},[2126,2167],[54,5868,5725],{"className":5869},[2270],[54,5871,5873],{"className":5872},[2126,3850],[54,5874,5876],{"className":5875},[2139],[54,5877,5879],{"className":5878},[2143],[54,5880,5882,5890],{"className":5881,"style":3846},[2147],[54,5883,5884,5887],{"style":3862},[54,5885],{"className":5886,"style":3866},[2155],[54,5888,5413],{"className":5889,"style":5448},[2126,2167],[54,5891,5892,5895],{"style":3872},[54,5893],{"className":5894,"style":3866},[2155],[54,5896,5898],{"className":5897,"style":5457},[3879],[54,5899,3833],{"className":5900},[2126],[54,5902,5725],{"className":5903},[2270],[54,5905,5907],{"className":5906},[2126,3850],[54,5908,5910],{"className":5909},[2139],[54,5911,5913],{"className":5912},[2143],[54,5914,5916,5924],{"className":5915,"style":5813},[2147],[54,5917,5918,5921],{"style":3862},[54,5919],{"className":5920,"style":3866},[2155],[54,5922,5730],{"className":5923},[2126,2167],[54,5925,5926,5929],{"style":3862},[54,5927],{"className":5928,"style":3866},[2155],[54,5930,5932],{"className":5931,"style":5830},[3879],[54,5933,5935],{"className":5934,"style":5835},[5834],[5837,5936,5937],{"xmlns":5839,"width":5840,"height":5841,"style":5842,"viewBox":5843,"preserveAspectRatio":5844},[5846,5938],{"d":5848},[13,5940,5941,5942,5970,5971,6031,6032,6060,6061,6126,6127,6191,6192,29],{},"One can then obtain estimates of the eigenvalues and eigenstates of ",[54,5943,5945,5958],{"className":5944},[2079],[54,5946,5948],{"className":5947},[2083],[2085,5949,5950],{"xmlns":2087},[2089,5951,5952,5956],{},[2092,5953,5954],{},[2098,5955,2694],{},[2105,5957,2694],{"encoding":2107},[54,5959,5961],{"className":5960,"ariaHidden":2113},[2112],[54,5962,5964,5967],{"className":5963},[2117],[54,5965],{"className":5966,"style":2122},[2121],[54,5968,2694],{"className":5969,"style":2709},[2126,2167]," by looking at the ones of ",[54,5972,5974,5991],{"className":5973},[2079],[54,5975,5977],{"className":5976},[2083],[2085,5978,5979],{"xmlns":2087},[2089,5980,5981,5989],{},[2092,5982,5983],{},[3827,5984,5985,5987],{"accent":2113},[2098,5986,2694],{},[2245,5988,3833],{},[2105,5990,3836],{"encoding":2107},[54,5992,5994],{"className":5993,"ariaHidden":2113},[2112],[54,5995,5997,6000],{"className":5996},[2117],[54,5998],{"className":5999,"style":3846},[2121],[54,6001,6003],{"className":6002},[2126,3850],[54,6004,6006],{"className":6005},[2139],[54,6007,6009],{"className":6008},[2143],[54,6010,6012,6020],{"className":6011,"style":3846},[2147],[54,6013,6014,6017],{"style":3862},[54,6015],{"className":6016,"style":3866},[2155],[54,6018,2694],{"className":6019,"style":2709},[2126,2167],[54,6021,6022,6025],{"style":3872},[54,6023],{"className":6024,"style":3866},[2155],[54,6026,6028],{"className":6027,"style":3880},[3879],[54,6029,3833],{"className":6030},[2126],". For example, the estimate of the ground state energy is obtained by taking the smallest eigenvalue ",[54,6033,6035,6048],{"className":6034},[2079],[54,6036,6038],{"className":6037},[2083],[2085,6039,6040],{"xmlns":2087},[2089,6041,6042,6046],{},[2092,6043,6044],{},[2098,6045,5730],{},[2105,6047,5730],{"encoding":2107},[54,6049,6051],{"className":6050,"ariaHidden":2113},[2112],[54,6052,6054,6057],{"className":6053},[2117],[54,6055],{"className":6056,"style":2195},[2121],[54,6058,5730],{"className":6059},[2126,2167]," and the ground state from the corresponding eigenvector ",[54,6062,6064,6082],{"className":6063},[2079],[54,6065,6067],{"className":6066},[2083],[2085,6068,6069],{"xmlns":2087},[2089,6070,6071,6079],{},[2092,6072,6073],{},[3827,6074,6075,6077],{"accent":2113},[2098,6076,5730],{},[2245,6078,5733],{},[2105,6080,6081],{"encoding":2107},"\\vec{c}",[54,6083,6085],{"className":6084,"ariaHidden":2113},[2112],[54,6086,6088,6091],{"className":6087},[2117],[54,6089],{"className":6090,"style":5813},[2121],[54,6092,6094],{"className":6093},[2126,3850],[54,6095,6097],{"className":6096},[2139],[54,6098,6100],{"className":6099},[2143],[54,6101,6103,6111],{"className":6102,"style":5813},[2147],[54,6104,6105,6108],{"style":3862},[54,6106],{"className":6107,"style":3866},[2155],[54,6109,5730],{"className":6110},[2126,2167],[54,6112,6113,6116],{"style":3862},[54,6114],{"className":6115,"style":3866},[2155],[54,6117,6119],{"className":6118,"style":5830},[3879],[54,6120,6122],{"className":6121,"style":5835},[5834],[5837,6123,6124],{"xmlns":5839,"width":5840,"height":5841,"style":5842,"viewBox":5843,"preserveAspectRatio":5844},[5846,6125],{"d":5848},". The coefficients in ",[54,6128,6130,6147],{"className":6129},[2079],[54,6131,6133],{"className":6132},[2083],[2085,6134,6135],{"xmlns":2087},[2089,6136,6137,6145],{},[2092,6138,6139],{},[3827,6140,6141,6143],{"accent":2113},[2098,6142,5730],{},[2245,6144,5733],{},[2105,6146,6081],{"encoding":2107},[54,6148,6150],{"className":6149,"ariaHidden":2113},[2112],[54,6151,6153,6156],{"className":6152},[2117],[54,6154],{"className":6155,"style":5813},[2121],[54,6157,6159],{"className":6158},[2126,3850],[54,6160,6162],{"className":6161},[2139],[54,6163,6165],{"className":6164},[2143],[54,6166,6168,6176],{"className":6167,"style":5813},[2147],[54,6169,6170,6173],{"style":3862},[54,6171],{"className":6172,"style":3866},[2155],[54,6174,5730],{"className":6175},[2126,2167],[54,6177,6178,6181],{"style":3862},[54,6179],{"className":6180,"style":3866},[2155],[54,6182,6184],{"className":6183,"style":5830},[3879],[54,6185,6187],{"className":6186,"style":5835},[5834],[5837,6188,6189],{"xmlns":5839,"width":5840,"height":5841,"style":5842,"viewBox":5843,"preserveAspectRatio":5844},[5846,6190],{"d":5848}," determines the contribution of the different vectors that span ",[54,6193,6195,6212],{"className":6194},[2079],[54,6196,6198],{"className":6197},[2083],[2085,6199,6200],{"xmlns":2087},[2089,6201,6202,6210],{},[2092,6203,6204],{},[2724,6205,6206,6208],{},[2098,6207,2101],{"mathvariant":2100},[2098,6209,2835],{},[2105,6211,2949],{"encoding":2107},[54,6213,6215],{"className":6214,"ariaHidden":2113},[2112],[54,6216,6218,6221],{"className":6217},[2117],[54,6219],{"className":6220,"style":2743},[2121],[54,6222,6224,6227],{"className":6223},[2126],[54,6225,2101],{"className":6226,"style":2131},[2126,2130],[54,6228,6230],{"className":6229},[2135],[54,6231,6233,6253],{"className":6232},[2139,2756],[54,6234,6236,6250],{"className":6235},[2143],[54,6237,6239],{"className":6238,"style":2763},[2147],[54,6240,6241,6244],{"style":2766},[54,6242],{"className":6243,"style":2156},[2155],[54,6245,6247],{"className":6246},[2160,2161,2162,2163],[54,6248,2835],{"className":6249,"style":2870},[2126,2167,2163],[54,6251,2781],{"className":6252},[2780],[54,6254,6256],{"className":6255},[2143],[54,6257,6259],{"className":6258,"style":2788},[2147],[54,6260],{},[13,6262,6263],{},[6264,6265],"img",{"alt":6266,"src":6267},"fig1.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-06.avif",[13,6269,6270,6271,6392,6393,171,6473,6554,6555,171,6642,6726,6727,6787,6788,6870,6871,6950,6951,7033,7034,7062,7063,7214],{},"The Figure shows a circuit representation of the modified Hadamard test, a method that is used to compute the overlap between different quantum states. For each matrix element ",[54,6272,6274,6302],{"className":6273},[2079],[54,6275,6277],{"className":6276},[2083],[2085,6278,6279],{"xmlns":2087},[2089,6280,6281,6299],{},[2092,6282,6283],{},[2724,6284,6285,6291],{},[3827,6286,6287,6289],{"accent":2113},[2098,6288,2694],{},[2245,6290,3833],{},[2092,6292,6293,6295,6297],{},[2098,6294,2849],{},[2245,6296,2375],{"separator":2113},[2098,6298,1928],{},[2105,6300,6301],{"encoding":2107},"\\tilde{H}_{i,j}",[54,6303,6305],{"className":6304,"ariaHidden":2113},[2112],[54,6306,6308,6312],{"className":6307},[2117],[54,6309],{"className":6310,"style":6311},[2121],"height:1.2063em;vertical-align:-0.2861em;",[54,6313,6315,6346],{"className":6314},[2126],[54,6316,6318],{"className":6317},[2126,3850],[54,6319,6321],{"className":6320},[2139],[54,6322,6324],{"className":6323},[2143],[54,6325,6327,6335],{"className":6326,"style":3846},[2147],[54,6328,6329,6332],{"style":3862},[54,6330],{"className":6331,"style":3866},[2155],[54,6333,2694],{"className":6334,"style":2709},[2126,2167],[54,6336,6337,6340],{"style":3872},[54,6338],{"className":6339,"style":3866},[2155],[54,6341,6343],{"className":6342,"style":3880},[3879],[54,6344,3833],{"className":6345},[2126],[54,6347,6349],{"className":6348},[2135],[54,6350,6352,6383],{"className":6351},[2139,2756],[54,6353,6355,6380],{"className":6354},[2143],[54,6356,6359],{"className":6357,"style":6358},[2147],"height:0.3117em;",[54,6360,6361,6364],{"style":4066},[54,6362],{"className":6363,"style":2156},[2155],[54,6365,6367],{"className":6366},[2160,2161,2162,2163],[54,6368,6370,6373,6376],{"className":6369},[2126,2163],[54,6371,2849],{"className":6372},[2126,2167,2163],[54,6374,2375],{"className":6375},[2517,2163],[54,6377,1928],{"className":6378,"style":6379},[2126,2167,2163],"margin-right:0.0572em;",[54,6381,2781],{"className":6382},[2780],[54,6384,6386],{"className":6385},[2143],[54,6387,6390],{"className":6388,"style":6389},[2147],"height:0.2861em;",[54,6391],{},", a Hadamard test between the state ",[54,6394,6396,6418],{"className":6395},[2079],[54,6397,6399],{"className":6398},[2083],[2085,6400,6401],{"xmlns":2087},[2089,6402,6403,6415],{},[2092,6404,6405,6407,6413],{},[2098,6406,2305],{"mathvariant":2304},[2724,6408,6409,6411],{},[2098,6410,3349],{},[2098,6412,2849],{},[2245,6414,2312],{"stretchy":2311},[2105,6416,6417],{"encoding":2107},"\\vert \\psi_i \\rangle",[54,6419,6421],{"className":6420,"ariaHidden":2113},[2112],[54,6422,6424,6427,6430,6470],{"className":6423},[2117],[54,6425],{"className":6426,"style":2325},[2121],[54,6428,2305],{"className":6429},[2126],[54,6431,6433,6436],{"className":6432},[2126],[54,6434,3349],{"className":6435,"style":2332},[2126,2167],[54,6437,6439],{"className":6438},[2135],[54,6440,6442,6462],{"className":6441},[2139,2756],[54,6443,6445,6459],{"className":6444},[2143],[54,6446,6448],{"className":6447,"style":6358},[2147],[54,6449,6450,6453],{"style":4128},[54,6451],{"className":6452,"style":2156},[2155],[54,6454,6456],{"className":6455},[2160,2161,2162,2163],[54,6457,2849],{"className":6458},[2126,2167,2163],[54,6460,2781],{"className":6461},[2780],[54,6463,6465],{"className":6464},[2143],[54,6466,6468],{"className":6467,"style":2788},[2147],[54,6469],{},[54,6471,2312],{"className":6472},[2336],[54,6474,6476,6498],{"className":6475},[2079],[54,6477,6479],{"className":6478},[2083],[2085,6480,6481],{"xmlns":2087},[2089,6482,6483,6495],{},[2092,6484,6485,6487,6493],{},[2098,6486,2305],{"mathvariant":2304},[2724,6488,6489,6491],{},[2098,6490,3349],{},[2098,6492,1928],{},[2245,6494,2312],{"stretchy":2311},[2105,6496,6497],{"encoding":2107},"\\vert \\psi_j \\rangle",[54,6499,6501],{"className":6500,"ariaHidden":2113},[2112],[54,6502,6504,6508,6511,6551],{"className":6503},[2117],[54,6505],{"className":6506,"style":6507},[2121],"height:1.0361em;vertical-align:-0.2861em;",[54,6509,2305],{"className":6510},[2126],[54,6512,6514,6517],{"className":6513},[2126],[54,6515,3349],{"className":6516,"style":2332},[2126,2167],[54,6518,6520],{"className":6519},[2135],[54,6521,6523,6543],{"className":6522},[2139,2756],[54,6524,6526,6540],{"className":6525},[2143],[54,6527,6529],{"className":6528,"style":6358},[2147],[54,6530,6531,6534],{"style":4128},[54,6532],{"className":6533,"style":2156},[2155],[54,6535,6537],{"className":6536},[2160,2161,2162,2163],[54,6538,1928],{"className":6539,"style":6379},[2126,2167,2163],[54,6541,2781],{"className":6542},[2780],[54,6544,6546],{"className":6545},[2143],[54,6547,6549],{"className":6548,"style":6389},[2147],[54,6550],{},[54,6552,2312],{"className":6553},[2336]," is carried out. This is highlighted in the figure by the color scheme for the matrix elements and the corresponding ",[54,6556,6558,6582],{"className":6557},[2079],[54,6559,6561],{"className":6560},[2083],[2085,6562,6563],{"xmlns":2087},[2089,6564,6565,6579],{},[2092,6566,6567,6570,6573],{},[5723,6568,6569],{},"Prep",[5723,6571,6572],{},"  ",[2724,6574,6575,6577],{},[2098,6576,3349],{},[2098,6578,2849],{},[2105,6580,6581],{"encoding":2107},"\\text{Prep} \\; \\psi_i",[54,6583,6585],{"className":6584,"ariaHidden":2113},[2112],[54,6586,6588,6592,6599,6602],{"className":6587},[2117],[54,6589],{"className":6590,"style":6591},[2121],"height:0.8889em;vertical-align:-0.1944em;",[54,6593,6596],{"className":6594},[2126,6595],"text",[54,6597,6569],{"className":6598},[2126],[54,6600],{"className":6601,"style":2475},[2270],[54,6603,6605,6608],{"className":6604},[2126],[54,6606,3349],{"className":6607,"style":2332},[2126,2167],[54,6609,6611],{"className":6610},[2135],[54,6612,6614,6634],{"className":6613},[2139,2756],[54,6615,6617,6631],{"className":6616},[2143],[54,6618,6620],{"className":6619,"style":6358},[2147],[54,6621,6622,6625],{"style":4128},[54,6623],{"className":6624,"style":2156},[2155],[54,6626,6628],{"className":6627},[2160,2161,2162,2163],[54,6629,2849],{"className":6630},[2126,2167,2163],[54,6632,2781],{"className":6633},[2780],[54,6635,6637],{"className":6636},[2143],[54,6638,6640],{"className":6639,"style":2788},[2147],[54,6641],{},[54,6643,6645,6667],{"className":6644},[2079],[54,6646,6648],{"className":6647},[2083],[2085,6649,6650],{"xmlns":2087},[2089,6651,6652,6664],{},[2092,6653,6654,6656,6658],{},[5723,6655,6569],{},[5723,6657,6572],{},[2724,6659,6660,6662],{},[2098,6661,3349],{},[2098,6663,1928],{},[2105,6665,6666],{"encoding":2107},"\\text{Prep} \\; \\psi_j",[54,6668,6670],{"className":6669,"ariaHidden":2113},[2112],[54,6671,6673,6677,6683,6686],{"className":6672},[2117],[54,6674],{"className":6675,"style":6676},[2121],"height:0.9805em;vertical-align:-0.2861em;",[54,6678,6680],{"className":6679},[2126,6595],[54,6681,6569],{"className":6682},[2126],[54,6684],{"className":6685,"style":2475},[2270],[54,6687,6689,6692],{"className":6688},[2126],[54,6690,3349],{"className":6691,"style":2332},[2126,2167],[54,6693,6695],{"className":6694},[2135],[54,6696,6698,6718],{"className":6697},[2139,2756],[54,6699,6701,6715],{"className":6700},[2143],[54,6702,6704],{"className":6703,"style":6358},[2147],[54,6705,6706,6709],{"style":4128},[54,6707],{"className":6708,"style":2156},[2155],[54,6710,6712],{"className":6711},[2160,2161,2162,2163],[54,6713,1928],{"className":6714,"style":6379},[2126,2167,2163],[54,6716,2781],{"className":6717},[2780],[54,6719,6721],{"className":6720},[2143],[54,6722,6724],{"className":6723,"style":6389},[2147],[54,6725],{}," operations. Thus, a set of Hadamard tests for all the possible combinations of Krylov space vectors is required to compute all the matrix elements of the projected Hamiltonian ",[54,6728,6730,6747],{"className":6729},[2079],[54,6731,6733],{"className":6732},[2083],[2085,6734,6735],{"xmlns":2087},[2089,6736,6737,6745],{},[2092,6738,6739],{},[3827,6740,6741,6743],{"accent":2113},[2098,6742,2694],{},[2245,6744,3833],{},[2105,6746,3836],{"encoding":2107},[54,6748,6750],{"className":6749,"ariaHidden":2113},[2112],[54,6751,6753,6756],{"className":6752},[2117],[54,6754],{"className":6755,"style":3846},[2121],[54,6757,6759],{"className":6758},[2126,3850],[54,6760,6762],{"className":6761},[2139],[54,6763,6765],{"className":6764},[2143],[54,6766,6768,6776],{"className":6767,"style":3846},[2147],[54,6769,6770,6773],{"style":3862},[54,6771],{"className":6772,"style":3866},[2155],[54,6774,2694],{"className":6775,"style":2709},[2126,2167],[54,6777,6778,6781],{"style":3872},[54,6779],{"className":6780,"style":3866},[2155],[54,6782,6784],{"className":6783,"style":3880},[3879],[54,6785,3833],{"className":6786},[2126],". The top wire in the Hadamard test circuit is an ancilla qubit which is measured either in the X or Y basis, its expectation value determines the value of the overlap between the states. The bottom wire represents all the qubits of the system Hamiltonian. The ",[54,6789,6791,6812],{"className":6790},[2079],[54,6792,6794],{"className":6793},[2083],[2085,6795,6796],{"xmlns":2087},[2089,6797,6798,6810],{},[2092,6799,6800,6802,6804],{},[5723,6801,6569],{},[5723,6803,6572],{},[2724,6805,6806,6808],{},[2098,6807,3349],{},[2098,6809,2849],{},[2105,6811,6581],{"encoding":2107},[54,6813,6815],{"className":6814,"ariaHidden":2113},[2112],[54,6816,6818,6821,6827,6830],{"className":6817},[2117],[54,6819],{"className":6820,"style":6591},[2121],[54,6822,6824],{"className":6823},[2126,6595],[54,6825,6569],{"className":6826},[2126],[54,6828],{"className":6829,"style":2475},[2270],[54,6831,6833,6836],{"className":6832},[2126],[54,6834,3349],{"className":6835,"style":2332},[2126,2167],[54,6837,6839],{"className":6838},[2135],[54,6840,6842,6862],{"className":6841},[2139,2756],[54,6843,6845,6859],{"className":6844},[2143],[54,6846,6848],{"className":6847,"style":6358},[2147],[54,6849,6850,6853],{"style":4128},[54,6851],{"className":6852,"style":2156},[2155],[54,6854,6856],{"className":6855},[2160,2161,2162,2163],[54,6857,2849],{"className":6858},[2126,2167,2163],[54,6860,2781],{"className":6861},[2780],[54,6863,6865],{"className":6864},[2143],[54,6866,6868],{"className":6867,"style":2788},[2147],[54,6869],{}," operation prepares the system qubit in the state ",[54,6872,6874,6895],{"className":6873},[2079],[54,6875,6877],{"className":6876},[2083],[2085,6878,6879],{"xmlns":2087},[2089,6880,6881,6893],{},[2092,6882,6883,6885,6891],{},[2098,6884,2305],{"mathvariant":2304},[2724,6886,6887,6889],{},[2098,6888,3349],{},[2098,6890,2849],{},[2245,6892,2312],{"stretchy":2311},[2105,6894,6417],{"encoding":2107},[54,6896,6898],{"className":6897,"ariaHidden":2113},[2112],[54,6899,6901,6904,6907,6947],{"className":6900},[2117],[54,6902],{"className":6903,"style":2325},[2121],[54,6905,2305],{"className":6906},[2126],[54,6908,6910,6913],{"className":6909},[2126],[54,6911,3349],{"className":6912,"style":2332},[2126,2167],[54,6914,6916],{"className":6915},[2135],[54,6917,6919,6939],{"className":6918},[2139,2756],[54,6920,6922,6936],{"className":6921},[2143],[54,6923,6925],{"className":6924,"style":6358},[2147],[54,6926,6927,6930],{"style":4128},[54,6928],{"className":6929,"style":2156},[2155],[54,6931,6933],{"className":6932},[2160,2161,2162,2163],[54,6934,2849],{"className":6935},[2126,2167,2163],[54,6937,2781],{"className":6938},[2780],[54,6940,6942],{"className":6941},[2143],[54,6943,6945],{"className":6944,"style":2788},[2147],[54,6946],{},[54,6948,2312],{"className":6949},[2336]," controlled by the state of the ancilla qubit (similarly for ",[54,6952,6954,6975],{"className":6953},[2079],[54,6955,6957],{"className":6956},[2083],[2085,6958,6959],{"xmlns":2087},[2089,6960,6961,6973],{},[2092,6962,6963,6965,6967],{},[5723,6964,6569],{},[5723,6966,6572],{},[2724,6968,6969,6971],{},[2098,6970,3349],{},[2098,6972,1928],{},[2105,6974,6666],{"encoding":2107},[54,6976,6978],{"className":6977,"ariaHidden":2113},[2112],[54,6979,6981,6984,6990,6993],{"className":6980},[2117],[54,6982],{"className":6983,"style":6676},[2121],[54,6985,6987],{"className":6986},[2126,6595],[54,6988,6569],{"className":6989},[2126],[54,6991],{"className":6992,"style":2475},[2270],[54,6994,6996,6999],{"className":6995},[2126],[54,6997,3349],{"className":6998,"style":2332},[2126,2167],[54,7000,7002],{"className":7001},[2135],[54,7003,7005,7025],{"className":7004},[2139,2756],[54,7006,7008,7022],{"className":7007},[2143],[54,7009,7011],{"className":7010,"style":6358},[2147],[54,7012,7013,7016],{"style":4128},[54,7014],{"className":7015,"style":2156},[2155],[54,7017,7019],{"className":7018},[2160,2161,2162,2163],[54,7020,1928],{"className":7021,"style":6379},[2126,2167,2163],[54,7023,2781],{"className":7024},[2780],[54,7026,7028],{"className":7027},[2143],[54,7029,7031],{"className":7030,"style":6389},[2147],[54,7032],{},") and the operation ",[54,7035,7037,7050],{"className":7036},[2079],[54,7038,7040],{"className":7039},[2083],[2085,7041,7042],{"xmlns":2087},[2089,7043,7044,7048],{},[2092,7045,7046],{},[2098,7047,2730],{},[2105,7049,2730],{"encoding":2107},[54,7051,7053],{"className":7052,"ariaHidden":2113},[2112],[54,7054,7056,7059],{"className":7055},[2117],[54,7057],{"className":7058,"style":2122},[2121],[54,7060,2730],{"className":7061,"style":2776},[2126,2167]," represents Pauli decomposition of the system Hamiltonian ",[54,7064,7066,7095],{"className":7065},[2079],[54,7067,7069],{"className":7068},[2083],[2085,7070,7071],{"xmlns":2087},[2089,7072,7073,7092],{},[2092,7074,7075,7077,7079,7086],{},[2098,7076,2694],{},[2245,7078,122],{},[2724,7080,7081,7084],{},[2245,7082,7083],{},"∑",[2098,7085,2849],{},[2724,7087,7088,7090],{},[2098,7089,2730],{},[2098,7091,2849],{},[2105,7093,7094],{"encoding":2107},"H = \\sum_i P_i",[54,7096,7098,7116],{"className":7097,"ariaHidden":2113},[2112],[54,7099,7101,7104,7107,7110,7113],{"className":7100},[2117],[54,7102],{"className":7103,"style":2122},[2121],[54,7105,2694],{"className":7106,"style":2709},[2126,2167],[54,7108],{"className":7109,"style":2475},[2270],[54,7111,122],{"className":7112},[2479],[54,7114],{"className":7115,"style":2475},[2270],[54,7117,7119,7123,7170,7173],{"className":7118},[2117],[54,7120],{"className":7121,"style":7122},[2121],"height:1.0497em;vertical-align:-0.2997em;",[54,7124,7127,7133],{"className":7125},[7126],"mop",[54,7128,7083],{"className":7129,"style":7132},[7126,7130,7131],"op-symbol","small-op","position:relative;top:0em;",[54,7134,7136],{"className":7135},[2135],[54,7137,7139,7161],{"className":7138},[2139,2756],[54,7140,7142,7158],{"className":7141},[2143],[54,7143,7146],{"className":7144,"style":7145},[2147],"height:0.162em;",[54,7147,7149,7152],{"style":7148},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[54,7150],{"className":7151,"style":2156},[2155],[54,7153,7155],{"className":7154},[2160,2161,2162,2163],[54,7156,2849],{"className":7157},[2126,2167,2163],[54,7159,2781],{"className":7160},[2780],[54,7162,7164],{"className":7163},[2143],[54,7165,7168],{"className":7166,"style":7167},[2147],"height:0.2997em;",[54,7169],{},[54,7171],{"className":7172,"style":2521},[2270],[54,7174,7176,7179],{"className":7175},[2126],[54,7177,2730],{"className":7178,"style":2776},[2126,2167],[54,7180,7182],{"className":7181},[2135],[54,7183,7185,7206],{"className":7184},[2139,2756],[54,7186,7188,7203],{"className":7187},[2143],[54,7189,7191],{"className":7190,"style":6358},[2147],[54,7192,7194,7197],{"style":7193},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[54,7195],{"className":7196,"style":2156},[2155],[54,7198,7200],{"className":7199},[2160,2161,2162,2163],[54,7201,2849],{"className":7202},[2126,2167,2163],[54,7204,2781],{"className":7205},[2780],[54,7207,7209],{"className":7208},[2143],[54,7210,7212],{"className":7211,"style":2788},[2147],[54,7213],{},". A more detailed derivation of the operations calculated by the Hadamard test is given below.",[7216,7217,7219],"h4",{"id":7218},"define-hamiltonian","Define Hamiltonian",[13,7221,7222,7223,7252,7253],{},"Let's consider the Heisenberg Hamiltonian for ",[54,7224,7226,7240],{"className":7225},[2079],[54,7227,7229],{"className":7228},[2083],[2085,7230,7231],{"xmlns":2087},[2089,7232,7233,7238],{},[2092,7234,7235],{},[2098,7236,7237],{},"N",[2105,7239,7237],{"encoding":2107},[54,7241,7243],{"className":7242,"ariaHidden":2113},[2112],[54,7244,7246,7249],{"className":7245},[2117],[54,7247],{"className":7248,"style":2122},[2121],[54,7250,7237],{"className":7251,"style":2870},[2126,2167]," qubits on a linear chain: ",[54,7254,7256,7332],{"className":7255},[2079],[54,7257,7259],{"className":7258},[2083],[2085,7260,7261],{"xmlns":2087},[2089,7262,7263,7329],{},[2092,7264,7265,7267,7269,7283,7290,7296,7298,7305,7311,7313,7316,7323],{},[2098,7266,2694],{},[2245,7268,122],{},[3331,7270,7271,7273,7281],{},[2245,7272,7083],{},[2092,7274,7275,7277,7279],{},[2098,7276,2849],{},[2245,7278,2375],{"separator":2113},[2098,7280,1928],{},[2098,7282,7237],{},[2724,7284,7285,7288],{},[2098,7286,7287],{},"X",[2098,7289,2849],{},[2724,7291,7292,7294],{},[2098,7293,7287],{},[2098,7295,1928],{},[2245,7297,1663],{},[2724,7299,7300,7303],{},[2098,7301,7302],{},"Y",[2098,7304,2849],{},[2724,7306,7307,7309],{},[2098,7308,7302],{},[2098,7310,1928],{},[2245,7312,2247],{},[2098,7314,7315],{},"J",[2724,7317,7318,7321],{},[2098,7319,7320],{},"Z",[2098,7322,2849],{},[2724,7324,7325,7327],{},[2098,7326,7320],{},[2098,7328,1928],{},[2105,7330,7331],{"encoding":2107},"H= \\sum_{i,j}^N X_i X_j + Y_i Y_j - J Z_i Z_j",[54,7333,7335,7353,7517,7614],{"className":7334,"ariaHidden":2113},[2112],[54,7336,7338,7341,7344,7347,7350],{"className":7337},[2117],[54,7339],{"className":7340,"style":2122},[2121],[54,7342,2694],{"className":7343,"style":2709},[2126,2167],[54,7345],{"className":7346,"style":2475},[2270],[54,7348,122],{"className":7349},[2479],[54,7351],{"className":7352,"style":2475},[2270],[54,7354,7356,7360,7423,7426,7468,7508,7511,7514],{"className":7355},[2117],[54,7357],{"className":7358,"style":7359},[2121],"height:1.417em;vertical-align:-0.4358em;",[54,7361,7363,7366],{"className":7362},[7126],[54,7364,7083],{"className":7365,"style":7132},[7126,7130,7131],[54,7367,7369],{"className":7368},[2135],[54,7370,7372,7414],{"className":7371},[2139,2756],[54,7373,7375,7411],{"className":7374},[2143],[54,7376,7379,7399],{"className":7377,"style":7378},[2147],"height:0.9812em;",[54,7380,7381,7384],{"style":7148},[54,7382],{"className":7383,"style":2156},[2155],[54,7385,7387],{"className":7386},[2160,2161,2162,2163],[54,7388,7390,7393,7396],{"className":7389},[2126,2163],[54,7391,2849],{"className":7392},[2126,2167,2163],[54,7394,2375],{"className":7395},[2517,2163],[54,7397,1928],{"className":7398,"style":6379},[2126,2167,2163],[54,7400,7402,7405],{"style":7401},"top:-3.2029em;margin-right:0.05em;",[54,7403],{"className":7404,"style":2156},[2155],[54,7406,7408],{"className":7407},[2160,2161,2162,2163],[54,7409,7237],{"className":7410,"style":2870},[2126,2167,2163],[54,7412,2781],{"className":7413},[2780],[54,7415,7417],{"className":7416},[2143],[54,7418,7421],{"className":7419,"style":7420},[2147],"height:0.4358em;",[54,7422],{},[54,7424],{"className":7425,"style":2521},[2270],[54,7427,7429,7433],{"className":7428},[2126],[54,7430,7287],{"className":7431,"style":7432},[2126,2167],"margin-right:0.0785em;",[54,7434,7436],{"className":7435},[2135],[54,7437,7439,7460],{"className":7438},[2139,2756],[54,7440,7442,7457],{"className":7441},[2143],[54,7443,7445],{"className":7444,"style":6358},[2147],[54,7446,7448,7451],{"style":7447},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[54,7449],{"className":7450,"style":2156},[2155],[54,7452,7454],{"className":7453},[2160,2161,2162,2163],[54,7455,2849],{"className":7456},[2126,2167,2163],[54,7458,2781],{"className":7459},[2780],[54,7461,7463],{"className":7462},[2143],[54,7464,7466],{"className":7465,"style":2788},[2147],[54,7467],{},[54,7469,7471,7474],{"className":7470},[2126],[54,7472,7287],{"className":7473,"style":7432},[2126,2167],[54,7475,7477],{"className":7476},[2135],[54,7478,7480,7500],{"className":7479},[2139,2756],[54,7481,7483,7497],{"className":7482},[2143],[54,7484,7486],{"className":7485,"style":6358},[2147],[54,7487,7488,7491],{"style":7447},[54,7489],{"className":7490,"style":2156},[2155],[54,7492,7494],{"className":7493},[2160,2161,2162,2163],[54,7495,1928],{"className":7496,"style":6379},[2126,2167,2163],[54,7498,2781],{"className":7499},[2780],[54,7501,7503],{"className":7502},[2143],[54,7504,7506],{"className":7505,"style":6389},[2147],[54,7507],{},[54,7509],{"className":7510,"style":2271},[2270],[54,7512,1663],{"className":7513},[2275],[54,7515],{"className":7516,"style":2271},[2270],[54,7518,7520,7524,7565,7605,7608,7611],{"className":7519},[2117],[54,7521],{"className":7522,"style":7523},[2121],"height:0.9694em;vertical-align:-0.2861em;",[54,7525,7527,7530],{"className":7526},[2126],[54,7528,7302],{"className":7529,"style":2271},[2126,2167],[54,7531,7533],{"className":7532},[2135],[54,7534,7536,7557],{"className":7535},[2139,2756],[54,7537,7539,7554],{"className":7538},[2143],[54,7540,7542],{"className":7541,"style":6358},[2147],[54,7543,7545,7548],{"style":7544},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[54,7546],{"className":7547,"style":2156},[2155],[54,7549,7551],{"className":7550},[2160,2161,2162,2163],[54,7552,2849],{"className":7553},[2126,2167,2163],[54,7555,2781],{"className":7556},[2780],[54,7558,7560],{"className":7559},[2143],[54,7561,7563],{"className":7562,"style":2788},[2147],[54,7564],{},[54,7566,7568,7571],{"className":7567},[2126],[54,7569,7302],{"className":7570,"style":2271},[2126,2167],[54,7572,7574],{"className":7573},[2135],[54,7575,7577,7597],{"className":7576},[2139,2756],[54,7578,7580,7594],{"className":7579},[2143],[54,7581,7583],{"className":7582,"style":6358},[2147],[54,7584,7585,7588],{"style":7544},[54,7586],{"className":7587,"style":2156},[2155],[54,7589,7591],{"className":7590},[2160,2161,2162,2163],[54,7592,1928],{"className":7593,"style":6379},[2126,2167,2163],[54,7595,2781],{"className":7596},[2780],[54,7598,7600],{"className":7599},[2143],[54,7601,7603],{"className":7602,"style":6389},[2147],[54,7604],{},[54,7606],{"className":7607,"style":2271},[2270],[54,7609,2247],{"className":7610},[2275],[54,7612],{"className":7613,"style":2271},[2270],[54,7615,7617,7620,7624,7666],{"className":7616},[2117],[54,7618],{"className":7619,"style":7523},[2121],[54,7621,7315],{"className":7622,"style":7623},[2126,2167],"margin-right:0.0962em;",[54,7625,7627,7631],{"className":7626},[2126],[54,7628,7320],{"className":7629,"style":7630},[2126,2167],"margin-right:0.0715em;",[54,7632,7634],{"className":7633},[2135],[54,7635,7637,7658],{"className":7636},[2139,2756],[54,7638,7640,7655],{"className":7639},[2143],[54,7641,7643],{"className":7642,"style":6358},[2147],[54,7644,7646,7649],{"style":7645},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[54,7647],{"className":7648,"style":2156},[2155],[54,7650,7652],{"className":7651},[2160,2161,2162,2163],[54,7653,2849],{"className":7654},[2126,2167,2163],[54,7656,2781],{"className":7657},[2780],[54,7659,7661],{"className":7660},[2143],[54,7662,7664],{"className":7663,"style":2788},[2147],[54,7665],{},[54,7667,7669,7672],{"className":7668},[2126],[54,7670,7320],{"className":7671,"style":7630},[2126,2167],[54,7673,7675],{"className":7674},[2135],[54,7676,7678,7698],{"className":7677},[2139,2756],[54,7679,7681,7695],{"className":7680},[2143],[54,7682,7684],{"className":7683,"style":6358},[2147],[54,7685,7686,7689],{"style":7645},[54,7687],{"className":7688,"style":2156},[2155],[54,7690,7692],{"className":7691},[2160,2161,2162,2163],[54,7693,1928],{"className":7694,"style":6379},[2126,2167,2163],[54,7696,2781],{"className":7697},[2780],[54,7699,7701],{"className":7700},[2143],[54,7702,7704],{"className":7703,"style":6389},[2147],[54,7705],{},[44,7707,7709],{"className":46,"code":7708,"language":48,"meta":49,"style":49},"# Define problem Hamiltonian.\nn_qubits = 30\nJ = 1 # coupling strength for ZZ interaction\n\n# Define the Hamiltonian:\nH_int = [[\"I\"] * n_qubits for _ in range(3 * (n_qubits - 1))]\nfor i in range(n_qubits - 1):\n H_int[i][i] = \"Z\"\n H_int[i][i + 1] = \"Z\"\nfor i in range(n_qubits - 1):\n H_int[n_qubits - 1 + i][i] = \"X\"\n H_int[n_qubits - 1 + i][i + 1] = \"X\"\nfor i in range(n_qubits - 1):\n H_int[2 * (n_qubits - 1) + i][i] = \"Y\"\n H_int[2 * (n_qubits - 1) + i][i + 1] = \"Y\"\nH_int = [\"\".join(term) for term in H_int]\nH_tot = [(term, J) if term.count(\"Z\") == 2 else (term, 1) for term in H_int]\n\n# Get operator\nH_op = SparsePauliOp.from_list(H_tot)\nprint(H_tot)\n",[51,7710,7711,7716,7726,7738,7742,7747,7796,7815,7825,7840,7858,7878,7901,7919,7945,7975,8005,8052,8056,8061,8077],{"__ignoreMap":49},[54,7712,7713],{"class":56,"line":57},[54,7714,7715],{"class":111},"# Define problem Hamiltonian.\n",[54,7717,7718,7721,7723],{"class":56,"line":74},[54,7719,7720],{"class":64},"n_qubits ",[54,7722,122],{"class":121},[54,7724,7725],{"class":132}," 30\n",[54,7727,7728,7731,7733,7735],{"class":56,"line":88},[54,7729,7730],{"class":64},"J ",[54,7732,122],{"class":121},[54,7734,982],{"class":132},[54,7736,7737],{"class":111}," # coupling strength for ZZ interaction\n",[54,7739,7740],{"class":56,"line":101},[54,7741,105],{"emptyLinePlaceholder":104},[54,7743,7744],{"class":56,"line":108},[54,7745,7746],{"class":111},"# Define the Hamiltonian:\n",[54,7748,7749,7752,7754,7757,7760,7763,7765,7768,7770,7773,7775,7778,7780,7783,7786,7789,7791,7793],{"class":56,"line":115},[54,7750,7751],{"class":64},"H_int ",[54,7753,122],{"class":121},[54,7755,7756],{"class":64}," [[",[54,7758,7759],{"class":279},"\"I\"",[54,7761,7762],{"class":64},"] ",[54,7764,1947],{"class":121},[54,7766,7767],{"class":64}," n_qubits ",[54,7769,1186],{"class":60},[54,7771,7772],{"class":64}," _ ",[54,7774,1192],{"class":60},[54,7776,7777],{"class":121}," range",[54,7779,129],{"class":64},[54,7781,7782],{"class":132},"3",[54,7784,7785],{"class":121}," *",[54,7787,7788],{"class":64}," (n_qubits ",[54,7790,1923],{"class":121},[54,7792,982],{"class":132},[54,7794,7795],{"class":64},"))]\n",[54,7797,7798,7800,7803,7805,7807,7809,7811,7813],{"class":56,"line":139},[54,7799,1186],{"class":60},[54,7801,7802],{"class":64}," i ",[54,7804,1192],{"class":60},[54,7806,7777],{"class":121},[54,7808,1660],{"class":64},[54,7810,1923],{"class":121},[54,7812,982],{"class":132},[54,7814,1439],{"class":64},[54,7816,7817,7820,7822],{"class":56,"line":159},[54,7818,7819],{"class":64}," H_int[i][i] ",[54,7821,122],{"class":121},[54,7823,7824],{"class":279}," \"Z\"\n",[54,7826,7827,7830,7832,7834,7836,7838],{"class":56,"line":183},[54,7828,7829],{"class":64}," H_int[i][i ",[54,7831,1663],{"class":121},[54,7833,982],{"class":132},[54,7835,7762],{"class":64},[54,7837,122],{"class":121},[54,7839,7824],{"class":279},[54,7841,7842,7844,7846,7848,7850,7852,7854,7856],{"class":56,"line":210},[54,7843,1186],{"class":60},[54,7845,7802],{"class":64},[54,7847,1192],{"class":60},[54,7849,7777],{"class":121},[54,7851,1660],{"class":64},[54,7853,1923],{"class":121},[54,7855,982],{"class":132},[54,7857,1439],{"class":64},[54,7859,7860,7863,7865,7867,7870,7873,7875],{"class":56,"line":227},[54,7861,7862],{"class":64}," H_int[n_qubits ",[54,7864,1923],{"class":121},[54,7866,982],{"class":132},[54,7868,7869],{"class":121}," +",[54,7871,7872],{"class":64}," i][i] ",[54,7874,122],{"class":121},[54,7876,7877],{"class":279}," \"X\"\n",[54,7879,7880,7882,7884,7886,7888,7891,7893,7895,7897,7899],{"class":56,"line":232},[54,7881,7862],{"class":64},[54,7883,1923],{"class":121},[54,7885,982],{"class":132},[54,7887,7869],{"class":121},[54,7889,7890],{"class":64}," i][i ",[54,7892,1663],{"class":121},[54,7894,982],{"class":132},[54,7896,7762],{"class":64},[54,7898,122],{"class":121},[54,7900,7877],{"class":279},[54,7902,7903,7905,7907,7909,7911,7913,7915,7917],{"class":56,"line":246},[54,7904,1186],{"class":60},[54,7906,7802],{"class":64},[54,7908,1192],{"class":60},[54,7910,7777],{"class":121},[54,7912,1660],{"class":64},[54,7914,1923],{"class":121},[54,7916,982],{"class":132},[54,7918,1439],{"class":64},[54,7920,7921,7924,7926,7928,7930,7932,7934,7936,7938,7940,7942],{"class":56,"line":259},[54,7922,7923],{"class":64}," H_int[",[54,7925,133],{"class":132},[54,7927,7785],{"class":121},[54,7929,7788],{"class":64},[54,7931,1923],{"class":121},[54,7933,982],{"class":132},[54,7935,204],{"class":64},[54,7937,1663],{"class":121},[54,7939,7872],{"class":64},[54,7941,122],{"class":121},[54,7943,7944],{"class":279}," \"Y\"\n",[54,7946,7947,7949,7951,7953,7955,7957,7959,7961,7963,7965,7967,7969,7971,7973],{"class":56,"line":264},[54,7948,7923],{"class":64},[54,7950,133],{"class":132},[54,7952,7785],{"class":121},[54,7954,7788],{"class":64},[54,7956,1923],{"class":121},[54,7958,982],{"class":132},[54,7960,204],{"class":64},[54,7962,1663],{"class":121},[54,7964,7890],{"class":64},[54,7966,1663],{"class":121},[54,7968,982],{"class":132},[54,7970,7762],{"class":64},[54,7972,122],{"class":121},[54,7974,7944],{"class":279},[54,7976,7977,7979,7981,7984,7987,7989,7992,7995,7997,8000,8002],{"class":56,"line":270},[54,7978,7751],{"class":64},[54,7980,122],{"class":121},[54,7982,7983],{"class":64}," [",[54,7985,7986],{"class":279},"\"\"",[54,7988,29],{"class":64},[54,7990,7991],{"class":125},"join",[54,7993,7994],{"class":64},"(term) ",[54,7996,1186],{"class":60},[54,7998,7999],{"class":64}," term ",[54,8001,1192],{"class":60},[54,8003,8004],{"class":64}," H_int]\n",[54,8006,8007,8010,8012,8015,8017,8020,8023,8025,8028,8030,8032,8034,8037,8040,8042,8044,8046,8048,8050],{"class":56,"line":285},[54,8008,8009],{"class":64},"H_tot ",[54,8011,122],{"class":121},[54,8013,8014],{"class":64}," [(term, J) ",[54,8016,1201],{"class":60},[54,8018,8019],{"class":64}," term.",[54,8021,8022],{"class":125},"count",[54,8024,129],{"class":64},[54,8026,8027],{"class":279},"\"Z\"",[54,8029,204],{"class":64},[54,8031,1279],{"class":121},[54,8033,1872],{"class":132},[54,8035,8036],{"class":60}," else",[54,8038,8039],{"class":64}," (term, ",[54,8041,174],{"class":132},[54,8043,204],{"class":64},[54,8045,1186],{"class":60},[54,8047,7999],{"class":64},[54,8049,1192],{"class":60},[54,8051,8004],{"class":64},[54,8053,8054],{"class":56,"line":297},[54,8055,105],{"emptyLinePlaceholder":104},[54,8057,8058],{"class":56,"line":310},[54,8059,8060],{"class":111},"# Get operator\n",[54,8062,8063,8066,8068,8071,8074],{"class":56,"line":324},[54,8064,8065],{"class":64},"H_op ",[54,8067,122],{"class":121},[54,8069,8070],{"class":64}," SparsePauliOp.",[54,8072,8073],{"class":125},"from_list",[54,8075,8076],{"class":64},"(H_tot)\n",[54,8078,8079,8082],{"class":56,"line":333},[54,8080,8081],{"class":121},"print",[54,8083,8076],{"class":64},[44,8085,8089],{"className":8086,"code":8088,"language":6595,"meta":49},[8087],"language-text","[('ZZIIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IZZIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIZZIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIZZIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIZZIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIZZIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIZZIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIZZIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIZZIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIZZIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIZZIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIZZIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIZZIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIZZIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIZZIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIZZIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIZZIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIZZIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIZZIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIZZIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIZZIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIZZIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIZZIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIZZIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIZZIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIZZIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIZZII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIZZI', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIIZZ', 1), ('XXIIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IXXIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIXXIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIXXIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIXXIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIXXIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIXXIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIXXIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIXXIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIXXIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIXXIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIXXIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIXXIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIXXIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIXXIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIXXIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIXXIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIXXIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIXXIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIXXIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIXXIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIXXIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIXXIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIXXIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIXXIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIXXIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIXXII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIXXI', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIIXX', 1), ('YYIIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IYYIIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIYYIIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIYYIIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIYYIIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIYYIIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIYYIIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIYYIIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIYYIIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIYYIIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIYYIIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIYYIIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIYYIIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIYYIIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIYYIIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIYYIIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIYYIIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIYYIIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIYYIIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIYYIIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIYYIIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIYYIIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIYYIIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIYYIIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIYYIIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIYYIII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIYYII', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIYYI', 1), ('IIIIIIIIIIIIIIIIIIIIIIIIIIIIYY', 1)]\n",[51,8090,8088],{"__ignoreMap":49},[7216,8092,8094],{"id":8093},"set-parameters-for-the-algorithm","Set parameters for the algorithm",[13,8096,8097,8098,8100,8101,8104,8105,8158,8159,8192],{},"We heuristically choose a value for the time-step ",[51,8099,5019],{}," (based on upper bounds on the Hamiltonian norm). Ref ",[20,8102,8103],{"href":640},"[2]"," showed that a sufficiently small timestep is ",[54,8106,8108,8135],{"className":8107},[2079],[54,8109,8111],{"className":8110},[2083],[2085,8112,8113],{"xmlns":2087},[2089,8114,8115,8132],{},[2092,8116,8117,8120,8122,8124,8126,8128,8130],{},[2098,8118,8119],{},"π",[2098,8121,194],{"mathvariant":2304},[2098,8123,2305],{"mathvariant":2304},[2098,8125,2305],{"mathvariant":2304},[2098,8127,2694],{},[2098,8129,2305],{"mathvariant":2304},[2098,8131,2305],{"mathvariant":2304},[2105,8133,8134],{"encoding":2107},"\\pi\u002F\\vert \\vert H \\vert \\vert",[54,8136,8138],{"className":8137,"ariaHidden":2113},[2112],[54,8139,8141,8144,8147,8151,8154],{"className":8140},[2117],[54,8142],{"className":8143,"style":2325},[2121],[54,8145,8119],{"className":8146,"style":2332},[2126,2167],[54,8148,8150],{"className":8149},[2126],"\u002F∣∣",[54,8152,2694],{"className":8153,"style":2709},[2126,2167],[54,8155,8157],{"className":8156},[2126],"∣∣",", and that it is preferable up to a point to underestimate this value rather than overestimate, since overestimating can allow contributions from high-energy states to corrupt even the optimal state in the Krylov space. On the other hand, choosing ",[54,8160,8162,8177],{"className":8161},[2079],[54,8163,8165],{"className":8164},[2083],[2085,8166,8167],{"xmlns":2087},[2089,8168,8169,8175],{},[2092,8170,8171,8173],{},[2098,8172,4515],{},[2098,8174,2854],{},[2105,8176,5019],{"encoding":2107},[54,8178,8180],{"className":8179,"ariaHidden":2113},[2112],[54,8181,8183,8186,8189],{"className":8182},[2117],[54,8184],{"className":8185,"style":4990},[2121],[54,8187,4515],{"className":8188},[2126,2167],[54,8190,2854],{"className":8191},[2126,2167]," to be too small leads to worse conditioning of the Krylov subspace, since the Krylov basis vectors differ less from timestep to timestep.",[44,8194,8196],{"className":46,"code":8195,"language":48,"meta":49,"style":49},"# Get Hamiltonian restricted to single-particle states\nsingle_particle_H = np.zeros((n_qubits, n_qubits))\nfor i in range(n_qubits):\n for j in range(i + 1):\n for p, coeff in H_op.to_list():\n p_x = Pauli(p).x\n p_z = Pauli(p).z\n if all(\n p_x[k] == ((i == k) + (j == k)) % 2 for k in range(n_qubits)\n ):\n sgn = (\n (-1j) ** sum(p_z[k] and p_x[k] for k in range(n_qubits))\n ) * ((-1) ** p_z[i])\n else:\n sgn = 0\n single_particle_H[i, j] += sgn * coeff\nfor i in range(n_qubits):\n for j in range(i + 1, n_qubits):\n single_particle_H[i, j] = np.conj(single_particle_H[j, i])\n\n# Set dt according to spectral norm\ndt = np.pi \u002F np.linalg.norm(single_particle_H, ord=2)\ndt\n",[51,8197,8198,8203,8217,8230,8250,8264,8277,8289,8298,8340,8345,8354,8392,8412,8419,8427,8440,8452,8471,8485,8489,8494,8524],{"__ignoreMap":49},[54,8199,8200],{"class":56,"line":57},[54,8201,8202],{"class":111},"# Get Hamiltonian restricted to single-particle states\n",[54,8204,8205,8208,8210,8212,8214],{"class":56,"line":74},[54,8206,8207],{"class":64},"single_particle_H ",[54,8209,122],{"class":121},[54,8211,1172],{"class":64},[54,8213,1705],{"class":125},[54,8215,8216],{"class":64},"((n_qubits, n_qubits))\n",[54,8218,8219,8221,8223,8225,8227],{"class":56,"line":88},[54,8220,1186],{"class":60},[54,8222,7802],{"class":64},[54,8224,1192],{"class":60},[54,8226,7777],{"class":121},[54,8228,8229],{"class":64},"(n_qubits):\n",[54,8231,8232,8234,8237,8239,8241,8244,8246,8248],{"class":56,"line":101},[54,8233,1472],{"class":60},[54,8235,8236],{"class":64}," j ",[54,8238,1192],{"class":60},[54,8240,7777],{"class":121},[54,8242,8243],{"class":64},"(i ",[54,8245,1663],{"class":121},[54,8247,982],{"class":132},[54,8249,1439],{"class":64},[54,8251,8252,8254,8256,8258,8260,8262],{"class":56,"line":108},[54,8253,1823],{"class":60},[54,8255,1475],{"class":64},[54,8257,1192],{"class":60},[54,8259,1480],{"class":64},[54,8261,1483],{"class":125},[54,8263,1486],{"class":64},[54,8265,8266,8269,8271,8274],{"class":56,"line":115},[54,8267,8268],{"class":64}," p_x ",[54,8270,122],{"class":121},[54,8272,8273],{"class":125}," Pauli",[54,8275,8276],{"class":64},"(p).x\n",[54,8278,8279,8282,8284,8286],{"class":56,"line":139},[54,8280,8281],{"class":64}," p_z ",[54,8283,122],{"class":121},[54,8285,8273],{"class":125},[54,8287,8288],{"class":64},"(p).z\n",[54,8290,8291,8293,8296],{"class":56,"line":159},[54,8292,1854],{"class":60},[54,8294,8295],{"class":121}," all",[54,8297,932],{"class":64},[54,8299,8300,8303,8305,8308,8310,8313,8315,8318,8320,8323,8326,8328,8331,8333,8335,8337],{"class":56,"line":183},[54,8301,8302],{"class":64}," p_x[k] ",[54,8304,1279],{"class":121},[54,8306,8307],{"class":64}," ((i ",[54,8309,1279],{"class":121},[54,8311,8312],{"class":64}," k) ",[54,8314,1663],{"class":121},[54,8316,8317],{"class":64}," (j ",[54,8319,1279],{"class":121},[54,8321,8322],{"class":64}," k)) ",[54,8324,8325],{"class":121},"%",[54,8327,1872],{"class":132},[54,8329,8330],{"class":60}," for",[54,8332,1276],{"class":64},[54,8334,1192],{"class":60},[54,8336,7777],{"class":121},[54,8338,8339],{"class":64},"(n_qubits)\n",[54,8341,8342],{"class":56,"line":210},[54,8343,8344],{"class":64}," ):\n",[54,8346,8347,8350,8352],{"class":56,"line":227},[54,8348,8349],{"class":64}," sgn ",[54,8351,122],{"class":121},[54,8353,890],{"class":64},[54,8355,8356,8359,8361,8363,8365,8367,8369,8372,8375,8378,8381,8383,8385,8387,8389],{"class":56,"line":232},[54,8357,8358],{"class":64}," (",[54,8360,1923],{"class":121},[54,8362,174],{"class":132},[54,8364,1928],{"class":60},[54,8366,204],{"class":64},[54,8368,1933],{"class":121},[54,8370,8371],{"class":121}," sum",[54,8373,8374],{"class":64},"(p_z[k] ",[54,8376,8377],{"class":60},"and",[54,8379,8380],{"class":64}," p_x[k] ",[54,8382,1186],{"class":60},[54,8384,1276],{"class":64},[54,8386,1192],{"class":60},[54,8388,7777],{"class":121},[54,8390,8391],{"class":64},"(n_qubits))\n",[54,8393,8394,8397,8399,8401,8403,8405,8407,8409],{"class":56,"line":246},[54,8395,8396],{"class":64}," ) ",[54,8398,1947],{"class":121},[54,8400,1920],{"class":64},[54,8402,1923],{"class":121},[54,8404,174],{"class":132},[54,8406,204],{"class":64},[54,8408,1933],{"class":121},[54,8410,8411],{"class":64}," p_z[i])\n",[54,8413,8414,8417],{"class":56,"line":259},[54,8415,8416],{"class":60}," else",[54,8418,1284],{"class":64},[54,8420,8421,8423,8425],{"class":56,"line":264},[54,8422,8349],{"class":64},[54,8424,122],{"class":121},[54,8426,1802],{"class":132},[54,8428,8429,8432,8434,8436,8438],{"class":56,"line":270},[54,8430,8431],{"class":64}," single_particle_H[i, j] ",[54,8433,1841],{"class":121},[54,8435,2011],{"class":64},[54,8437,1947],{"class":121},[54,8439,2016],{"class":64},[54,8441,8442,8444,8446,8448,8450],{"class":56,"line":285},[54,8443,1186],{"class":60},[54,8445,7802],{"class":64},[54,8447,1192],{"class":60},[54,8449,7777],{"class":121},[54,8451,8229],{"class":64},[54,8453,8454,8456,8458,8460,8462,8464,8466,8468],{"class":56,"line":297},[54,8455,1472],{"class":60},[54,8457,8236],{"class":64},[54,8459,1192],{"class":60},[54,8461,7777],{"class":121},[54,8463,8243],{"class":64},[54,8465,1663],{"class":121},[54,8467,982],{"class":132},[54,8469,8470],{"class":64},", n_qubits):\n",[54,8472,8473,8476,8478,8480,8482],{"class":56,"line":310},[54,8474,8475],{"class":64}," single_particle_H[i, j] ",[54,8477,122],{"class":121},[54,8479,1172],{"class":64},[54,8481,1230],{"class":125},[54,8483,8484],{"class":64},"(single_particle_H[j, i])\n",[54,8486,8487],{"class":56,"line":324},[54,8488,105],{"emptyLinePlaceholder":104},[54,8490,8491],{"class":56,"line":333},[54,8492,8493],{"class":111},"# Set dt according to spectral norm\n",[54,8495,8496,8499,8501,8504,8506,8509,8512,8515,8518,8520,8522],{"class":56,"line":345},[54,8497,8498],{"class":64},"dt ",[54,8500,122],{"class":121},[54,8502,8503],{"class":64}," np.pi ",[54,8505,194],{"class":121},[54,8507,8508],{"class":64}," np.linalg.",[54,8510,8511],{"class":125},"norm",[54,8513,8514],{"class":64},"(single_particle_H, ",[54,8516,8517],{"class":1711},"ord",[54,8519,122],{"class":121},[54,8521,133],{"class":132},[54,8523,136],{"class":64},[54,8525,8526],{"class":56,"line":358},[54,8527,8528],{"class":64},"dt\n",[44,8530,8533],{"className":8531,"code":8532,"language":6595,"meta":49},[8087],"np.float64(0.10833078115826875)\n",[51,8534,8532],{"__ignoreMap":49},[13,8536,8537],{},"And set other parameters of the algorithm. For the sake of this tutorial, we'll limit ourselves to using a Krylov space with only five dimensions, which is quite limiting.",[44,8539,8541],{"className":46,"code":8540,"language":48,"meta":49,"style":49},"# Set parameters for quantum Krylov algorithm\nkrylov_dim = 5 # size of Krylov subspace\nnum_trotter_steps = 6\ndt_circ = dt \u002F num_trotter_steps\n",[51,8542,8543,8548,8561,8571],{"__ignoreMap":49},[54,8544,8545],{"class":56,"line":57},[54,8546,8547],{"class":111},"# Set parameters for quantum Krylov algorithm\n",[54,8549,8550,8553,8555,8558],{"class":56,"line":74},[54,8551,8552],{"class":64},"krylov_dim ",[54,8554,122],{"class":121},[54,8556,8557],{"class":132}," 5",[54,8559,8560],{"class":111}," # size of Krylov subspace\n",[54,8562,8563,8566,8568],{"class":56,"line":88},[54,8564,8565],{"class":64},"num_trotter_steps ",[54,8567,122],{"class":121},[54,8569,8570],{"class":132}," 6\n",[54,8572,8573,8576,8578,8581,8583],{"class":56,"line":101},[54,8574,8575],{"class":64},"dt_circ ",[54,8577,122],{"class":121},[54,8579,8580],{"class":64}," dt ",[54,8582,194],{"class":121},[54,8584,8585],{"class":64}," num_trotter_steps\n",[7216,8587,8589],{"id":8588},"state-preparation","State preparation",[13,8591,8592,8593,8632,8633,8671],{},"Pick a reference state ",[54,8594,8596,8614],{"className":8595},[2079],[54,8597,8599],{"className":8598},[2083],[2085,8600,8601],{"xmlns":2087},[2089,8602,8603,8611],{},[2092,8604,8605,8607,8609],{},[2098,8606,2305],{"mathvariant":2304},[2098,8608,3349],{},[2245,8610,2312],{"stretchy":2311},[2105,8612,8613],{"encoding":2107},"\\vert \\psi \\rangle",[54,8615,8617],{"className":8616,"ariaHidden":2113},[2112],[54,8618,8620,8623,8626,8629],{"className":8619},[2117],[54,8621],{"className":8622,"style":2325},[2121],[54,8624,2305],{"className":8625},[2126],[54,8627,3349],{"className":8628,"style":2332},[2126,2167],[54,8630,2312],{"className":8631},[2336]," that has some overlap with the ground state. For this Hamiltonian, We use the a state with an excitation in the middle qubit ",[54,8634,8636,8655],{"className":8635},[2079],[54,8637,8639],{"className":8638},[2083],[2085,8640,8641],{"xmlns":2087},[2089,8642,8643,8652],{},[2092,8644,8645,8647,8650],{},[2098,8646,2305],{"mathvariant":2304},[2249,8648,8649],{},"00..010...00",[2245,8651,2312],{"stretchy":2311},[2105,8653,8654],{"encoding":2107},"\\vert 00..010...00 \\rangle",[54,8656,8658],{"className":8657,"ariaHidden":2113},[2112],[54,8659,8661,8664,8668],{"className":8660},[2117],[54,8662],{"className":8663,"style":2325},[2121],[54,8665,8667],{"className":8666},[2126],"∣00..010...00",[54,8669,2312],{"className":8670},[2336]," as our reference state.",[44,8673,8675],{"className":46,"code":8674,"language":48,"meta":49,"style":49},"qc_state_prep = QuantumCircuit(n_qubits)\nqc_state_prep.x(int(n_qubits \u002F 2) + 1)\nqc_state_prep.draw(\"mpl\", scale=0.5)\n",[51,8676,8677,8688,8714],{"__ignoreMap":49},[54,8678,8679,8682,8684,8686],{"class":56,"line":57},[54,8680,8681],{"class":64},"qc_state_prep ",[54,8683,122],{"class":121},[54,8685,126],{"class":125},[54,8687,8339],{"class":64},[54,8689,8690,8693,8696,8698,8700,8702,8704,8706,8708,8710,8712],{"class":56,"line":74},[54,8691,8692],{"class":64},"qc_state_prep.",[54,8694,8695],{"class":125},"x",[54,8697,129],{"class":64},[54,8699,976],{"class":121},[54,8701,1660],{"class":64},[54,8703,194],{"class":121},[54,8705,1872],{"class":132},[54,8707,204],{"class":64},[54,8709,1663],{"class":121},[54,8711,982],{"class":132},[54,8713,136],{"class":64},[54,8715,8716,8718,8721,8723,8726,8728,8731,8733,8736],{"class":56,"line":88},[54,8717,8692],{"class":64},[54,8719,8720],{"class":125},"draw",[54,8722,129],{"class":64},[54,8724,8725],{"class":279},"\"mpl\"",[54,8727,171],{"class":64},[54,8729,8730],{"class":1711},"scale",[54,8732,122],{"class":121},[54,8734,8735],{"class":132},"0.5",[54,8737,136],{"class":64},[380,8739],{"alt":8740,"src":8741},"Output of the previous code cell","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-01.avif",[7216,8743,8745],{"id":8744},"time-evolution","Time evolution",[13,8747,8748,8749,8849,8850,29],{},"We can realize the time-evolution operator generated by a given Hamiltonian: ",[54,8750,8752,8781],{"className":8751},[2079],[54,8753,8755],{"className":8754},[2083],[2085,8756,8757],{"xmlns":2087},[2089,8758,8759,8779],{},[2092,8760,8761,8763,8765],{},[2098,8762,2835],{},[2245,8764,122],{},[2095,8766,8767,8769],{},[2098,8768,2842],{},[2092,8770,8771,8773,8775,8777],{},[2245,8772,2247],{},[2098,8774,2849],{},[2098,8776,2694],{},[2098,8778,2854],{},[2105,8780,2857],{"encoding":2107},[54,8782,8784,8802],{"className":8783,"ariaHidden":2113},[2112],[54,8785,8787,8790,8793,8796,8799],{"className":8786},[2117],[54,8788],{"className":8789,"style":2122},[2121],[54,8791,2835],{"className":8792,"style":2870},[2126,2167],[54,8794],{"className":8795,"style":2475},[2270],[54,8797,122],{"className":8798},[2479],[54,8800],{"className":8801,"style":2475},[2270],[54,8803,8805,8808],{"className":8804},[2117],[54,8806],{"className":8807,"style":2886},[2121],[54,8809,8811,8814],{"className":8810},[2126],[54,8812,2842],{"className":8813},[2126,2167],[54,8815,8817],{"className":8816},[2135],[54,8818,8820],{"className":8819},[2139],[54,8821,8823],{"className":8822},[2143],[54,8824,8826],{"className":8825,"style":2886},[2147],[54,8827,8828,8831],{"style":2151},[54,8829],{"className":8830,"style":2156},[2155],[54,8832,8834],{"className":8833},[2160,2161,2162,2163],[54,8835,8837,8840,8843,8846],{"className":8836},[2126,2163],[54,8838,2247],{"className":8839},[2126,2163],[54,8841,2849],{"className":8842},[2126,2167,2163],[54,8844,2694],{"className":8845,"style":2709},[2126,2167,2163],[54,8847,2854],{"className":8848},[2126,2167,2163]," via the ",[20,8851,8853],{"href":8852},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fqiskit.synthesis.LieTrotter","Lie-Trotter approximation",[44,8855,8857],{"className":46,"code":8856,"language":48,"meta":49,"style":49},"t = Parameter(\"t\")\n\n## Create the time-evo op circuit\nevol_gate = PauliEvolutionGate(\n H_op, time=t, synthesis=LieTrotter(reps=num_trotter_steps)\n)\n\nqr = QuantumRegister(n_qubits)\nqc_evol = QuantumCircuit(qr)\nqc_evol.append(evol_gate, qargs=qr)\n",[51,8858,8859,8876,8880,8885,8897,8928,8932,8936,8948,8960],{"__ignoreMap":49},[54,8860,8861,8864,8866,8869,8871,8874],{"class":56,"line":57},[54,8862,8863],{"class":64},"t ",[54,8865,122],{"class":121},[54,8867,8868],{"class":125}," Parameter",[54,8870,129],{"class":64},[54,8872,8873],{"class":279},"\"t\"",[54,8875,136],{"class":64},[54,8877,8878],{"class":56,"line":74},[54,8879,105],{"emptyLinePlaceholder":104},[54,8881,8882],{"class":56,"line":88},[54,8883,8884],{"class":111},"## Create the time-evo op circuit\n",[54,8886,8887,8890,8892,8895],{"class":56,"line":101},[54,8888,8889],{"class":64},"evol_gate ",[54,8891,122],{"class":121},[54,8893,8894],{"class":125}," PauliEvolutionGate",[54,8896,932],{"class":64},[54,8898,8899,8902,8905,8907,8910,8913,8915,8918,8920,8923,8925],{"class":56,"line":108},[54,8900,8901],{"class":64}," H_op, ",[54,8903,8904],{"class":1711},"time",[54,8906,122],{"class":121},[54,8908,8909],{"class":64},"t, ",[54,8911,8912],{"class":1711},"synthesis",[54,8914,122],{"class":121},[54,8916,8917],{"class":125},"LieTrotter",[54,8919,129],{"class":64},[54,8921,8922],{"class":1711},"reps",[54,8924,122],{"class":121},[54,8926,8927],{"class":64},"num_trotter_steps)\n",[54,8929,8930],{"class":56,"line":115},[54,8931,136],{"class":64},[54,8933,8934],{"class":56,"line":139},[54,8935,105],{"emptyLinePlaceholder":104},[54,8937,8938,8941,8943,8946],{"class":56,"line":159},[54,8939,8940],{"class":64},"qr ",[54,8942,122],{"class":121},[54,8944,8945],{"class":125}," QuantumRegister",[54,8947,8339],{"class":64},[54,8949,8950,8953,8955,8957],{"class":56,"line":183},[54,8951,8952],{"class":64},"qc_evol ",[54,8954,122],{"class":121},[54,8956,126],{"class":125},[54,8958,8959],{"class":64},"(qr)\n",[54,8961,8962,8965,8967,8970,8973,8975],{"class":56,"line":210},[54,8963,8964],{"class":64},"qc_evol.",[54,8966,1495],{"class":125},[54,8968,8969],{"class":64},"(evol_gate, ",[54,8971,8972],{"class":1711},"qargs",[54,8974,122],{"class":121},[54,8976,8977],{"class":64},"qr)\n",[44,8979,8982],{"className":8980,"code":8981,"language":6595,"meta":49},[8087],"\u003Cqiskit.circuit.instructionset.InstructionSet at 0x11eef9be0>\n",[51,8983,8981],{"__ignoreMap":49},[7216,8985,8987],{"id":8986},"hadamard-test","Hadamard test",[13,8989,8990],{},[6264,8991],{"alt":8992,"src":8993},"fig2.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-07.avif",[54,8995,8997],{"className":8996},[2340],[54,8998,9000,9248],{"className":8999},[2079],[54,9001,9003],{"className":9002},[2083],[2085,9004,9005],{"xmlns":2087,"display":2349},[2089,9006,9007,9245],{},[9008,9009,9012],"mtable",{"rowspacing":9010,"columnspacing":9011},"0.16em","1em",[9013,9014,9015],"mtr",{},[9016,9017,9018],"mtd",{},[9019,9020,9021],"mstyle",{"scriptlevel":150,"displaystyle":2113},[2092,9022,9023,9025,9027,9029,9031,9033,9039,9041,9044,9046,9056,9059,9061,9063,9065,9067,9069,9071,9073,9075,9077,9079,9085,9087,9089,9091,9099,9101,9103,9105,9107,9109,9111,9117,9119,9121,9123,9125,9127,9133,9135,9137,9139,9141,9143,9151,9153,9155,9157,9159,9161,9163,9169,9171,9173,9175,9177,9179,9181,9187,9189,9191,9193,9195,9197,9205,9207,9209,9211,9213,9215,9221,9223,9225,9227,9229,9231,9233,9235,9241,9243],{},[2098,9024,2305],{"mathvariant":2304},[2249,9026,150],{},[2245,9028,2312],{"stretchy":2311},[2098,9030,2305],{"mathvariant":2304},[2249,9032,150],{},[2095,9034,9035,9037],{},[2245,9036,2312],{"stretchy":2311},[2098,9038,7237],{},[2270,9040],{"width":9011},[2245,9042,9043],{},"⟶",[2270,9045],{"width":9011},[9047,9048,9049,9051],"mfrac",{},[2249,9050,174],{},[9052,9053,9054],"msqrt",{},[2249,9055,133],{},[2245,9057,129],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},"1.8em",[2098,9060,2305],{"mathvariant":2304},[2249,9062,150],{},[2245,9064,2312],{"stretchy":2311},[2245,9066,1663],{},[2098,9068,2305],{"mathvariant":2304},[2249,9070,174],{},[2245,9072,2312],{"stretchy":2311},[2245,9074,3205],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2098,9076,2305],{"mathvariant":2304},[2249,9078,150],{},[2095,9080,9081,9083],{},[2245,9082,2312],{"stretchy":2311},[2098,9084,7237],{},[2270,9086],{"width":9011},[2245,9088,9043],{},[2270,9090],{"width":9011},[9047,9092,9093,9095],{},[2249,9094,174],{},[9052,9096,9097],{},[2249,9098,133],{},[2245,9100,129],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2098,9102,2305],{"mathvariant":2304},[2249,9104,150],{},[2245,9106,2312],{"stretchy":2311},[2098,9108,2305],{"mathvariant":2304},[2249,9110,150],{},[2095,9112,9113,9115],{},[2245,9114,2312],{"stretchy":2311},[2098,9116,7237],{},[2245,9118,1663],{},[2098,9120,2305],{"mathvariant":2304},[2249,9122,174],{},[2245,9124,2312],{"stretchy":2311},[2098,9126,2305],{"mathvariant":2304},[2724,9128,9129,9131],{},[2098,9130,3349],{},[2098,9132,2849],{},[2245,9134,2312],{"stretchy":2311},[2245,9136,3205],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2270,9138],{"width":9011},[2245,9140,9043],{},[2270,9142],{"width":9011},[9047,9144,9145,9147],{},[2249,9146,174],{},[9052,9148,9149],{},[2249,9150,133],{},[2245,9152,129],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2098,9154,2305],{"mathvariant":2304},[2249,9156,150],{},[2245,9158,2312],{"stretchy":2311},[2098,9160,2305],{"mathvariant":2304},[2249,9162,150],{},[2095,9164,9165,9167],{},[2245,9166,2312],{"stretchy":2311},[2098,9168,7237],{},[2245,9170,1663],{},[2098,9172,2305],{"mathvariant":2304},[2249,9174,174],{},[2245,9176,2312],{"stretchy":2311},[2098,9178,2730],{},[2098,9180,2305],{"mathvariant":2304},[2724,9182,9183,9185],{},[2098,9184,3349],{},[2098,9186,2849],{},[2245,9188,2312],{"stretchy":2311},[2245,9190,3205],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2270,9192],{"width":9011},[2245,9194,9043],{},[2270,9196],{"width":9011},[9047,9198,9199,9201],{},[2249,9200,174],{},[9052,9202,9203],{},[2249,9204,133],{},[2245,9206,129],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2098,9208,2305],{"mathvariant":2304},[2249,9210,150],{},[2245,9212,2312],{"stretchy":2311},[2098,9214,2305],{"mathvariant":2304},[2724,9216,9217,9219],{},[2098,9218,3349],{},[2098,9220,1928],{},[2245,9222,2312],{"stretchy":2311},[2245,9224,1663],{},[2098,9226,2305],{"mathvariant":2304},[2249,9228,174],{},[2245,9230,2312],{"stretchy":2311},[2098,9232,2730],{},[2098,9234,2305],{"mathvariant":2304},[2724,9236,9237,9239],{},[2098,9238,3349],{},[2098,9240,2849],{},[2245,9242,2312],{"stretchy":2311},[2245,9244,3205],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2105,9246,9247],{"encoding":2107},"\\begin{equation*}\n |0\\rangle|0\\rangle^N \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle + |1\\rangle \\Big)|0\\rangle^N \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle|0\\rangle^N+|1\\rangle |\\psi_i\\rangle\\Big) \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle |0\\rangle^N+|1\\rangle P |\\psi_i\\rangle\\Big) \\quad\\longrightarrow\\quad\\frac{1}{\\sqrt{2}}\\Big(|0\\rangle |\\psi_j\\rangle+|1\\rangle P|\\psi_i\\rangle\\Big)\n\\end{equation*}",[54,9249,9251],{"className":9250,"ariaHidden":2113},[2112],[54,9252,9254,9258],{"className":9253},[2117],[54,9255],{"className":9256,"style":9257},[2121],"height:2.2514em;vertical-align:-0.8757em;",[54,9259,9261],{"className":9260},[2126],[54,9262,9264],{"className":9263},[9008],[54,9265,9268],{"className":9266},[9267],"col-align-c",[54,9269,9271,10256],{"className":9270},[2139,2756],[54,9272,9274,10253],{"className":9273},[2143],[54,9275,9278],{"className":9276,"style":9277},[2147],"height:1.3757em;",[54,9279,9281,9285],{"style":9280},"top:-3.3757em;",[54,9282],{"className":9283,"style":9284},[2155],"height:3.3214em;",[54,9286,9288,9292,9295,9298,9327,9331,9334,9337,9340,9343,9470,9477,9480,9483,9486,9489,9492,9496,9499,9505,9508,9537,9540,9543,9546,9549,9552,9658,9664,9667,9670,9673,9702,9705,9708,9711,9714,9717,9720,9760,9763,9769,9772,9775,9778,9781,9784,9890,9896,9899,9902,9905,9934,9937,9940,9943,9946,9949,9952,9955,9995,9998,10004,10007,10010,10013,10016,10019,10125,10131,10134,10137,10140,10180,10183,10186,10189,10192,10195,10198,10201,10204,10244,10247],{"className":9287},[2126],[54,9289,9291],{"className":9290},[2126],"∣0",[54,9293,2312],{"className":9294},[2336],[54,9296,9291],{"className":9297},[2126],[54,9299,9301,9304],{"className":9300},[2336],[54,9302,2312],{"className":9303},[2336],[54,9305,9307],{"className":9306},[2135],[54,9308,9310],{"className":9309},[2139],[54,9311,9313],{"className":9312},[2143],[54,9314,9316],{"className":9315,"style":4325},[2147],[54,9317,9318,9321],{"style":2462},[54,9319],{"className":9320,"style":2156},[2155],[54,9322,9324],{"className":9323},[2160,2161,2162,2163],[54,9325,7237],{"className":9326,"style":2870},[2126,2167,2163],[54,9328],{"className":9329,"style":9330},[2270],"margin-right:1em;",[54,9332],{"className":9333,"style":2475},[2270],[54,9335,9043],{"className":9336},[2479],[54,9338],{"className":9339,"style":9330},[2270],[54,9341],{"className":9342,"style":2475},[2270],[54,9344,9346,9350,9467],{"className":9345},[2126],[54,9347],{"className":9348},[2497,9349],"nulldelimiter",[54,9351,9353],{"className":9352},[9047],[54,9354,9356,9458],{"className":9355},[2139,2756],[54,9357,9359,9455],{"className":9358},[2143],[54,9360,9363,9432,9443],{"className":9361,"style":9362},[2147],"height:1.3214em;",[54,9364,9366,9369],{"style":9365},"top:-2.2028em;",[54,9367],{"className":9368,"style":3866},[2155],[54,9370,9372],{"className":9371},[2126],[54,9373,9376],{"className":9374},[2126,9375],"sqrt",[54,9377,9379,9423],{"className":9378},[2139,2756],[54,9380,9382,9420],{"className":9381},[2143],[54,9383,9386,9400],{"className":9384,"style":9385},[2147],"height:0.9072em;",[54,9387,9390,9393],{"className":9388,"style":3862},[9389],"svg-align",[54,9391],{"className":9392,"style":3866},[2155],[54,9394,9397],{"className":9395,"style":9396},[2126],"padding-left:0.833em;",[54,9398,133],{"className":9399},[2126],[54,9401,9403,9406],{"style":9402},"top:-2.8672em;",[54,9404],{"className":9405,"style":3866},[2155],[54,9407,9411],{"className":9408,"style":9410},[9409],"hide-tail","min-width:0.853em;height:1.08em;",[5837,9412,9417],{"xmlns":5839,"width":9413,"height":9414,"viewBox":9415,"preserveAspectRatio":9416},"400em","1.08em","0 0 400000 1080","xMinYMin slice",[5846,9418],{"d":9419},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[54,9421,2781],{"className":9422},[2780],[54,9424,9426],{"className":9425},[2143],[54,9427,9430],{"className":9428,"style":9429},[2147],"height:0.1328em;",[54,9431],{},[54,9433,9435,9438],{"style":9434},"top:-3.23em;",[54,9436],{"className":9437,"style":3866},[2155],[54,9439],{"className":9440,"style":9442},[9441],"frac-line","border-bottom-width:0.04em;",[54,9444,9446,9449],{"style":9445},"top:-3.677em;",[54,9447],{"className":9448,"style":3866},[2155],[54,9450,9452],{"className":9451},[2126],[54,9453,174],{"className":9454},[2126],[54,9456,2781],{"className":9457},[2780],[54,9459,9461],{"className":9460},[2143],[54,9462,9465],{"className":9463,"style":9464},[2147],"height:0.93em;",[54,9466],{},[54,9468],{"className":9469},[2336,9349],[54,9471,9473],{"className":9472},[2126],[54,9474,129],{"className":9475},[2503,9476],"size2",[54,9478,9291],{"className":9479},[2126],[54,9481,2312],{"className":9482},[2336],[54,9484],{"className":9485,"style":2271},[2270],[54,9487,1663],{"className":9488},[2275],[54,9490],{"className":9491,"style":2271},[2270],[54,9493,9495],{"className":9494},[2126],"∣1",[54,9497,2312],{"className":9498},[2336],[54,9500,9502],{"className":9501},[2126],[54,9503,3205],{"className":9504},[2503,9476],[54,9506,9291],{"className":9507},[2126],[54,9509,9511,9514],{"className":9510},[2336],[54,9512,2312],{"className":9513},[2336],[54,9515,9517],{"className":9516},[2135],[54,9518,9520],{"className":9519},[2139],[54,9521,9523],{"className":9522},[2143],[54,9524,9526],{"className":9525,"style":4325},[2147],[54,9527,9528,9531],{"style":2462},[54,9529],{"className":9530,"style":2156},[2155],[54,9532,9534],{"className":9533},[2160,2161,2162,2163],[54,9535,7237],{"className":9536,"style":2870},[2126,2167,2163],[54,9538],{"className":9539,"style":9330},[2270],[54,9541],{"className":9542,"style":2475},[2270],[54,9544,9043],{"className":9545},[2479],[54,9547],{"className":9548,"style":9330},[2270],[54,9550],{"className":9551,"style":2475},[2270],[54,9553,9555,9558,9655],{"className":9554},[2126],[54,9556],{"className":9557},[2497,9349],[54,9559,9561],{"className":9560},[9047],[54,9562,9564,9647],{"className":9563},[2139,2756],[54,9565,9567,9644],{"className":9566},[2143],[54,9568,9570,9625,9633],{"className":9569,"style":9362},[2147],[54,9571,9572,9575],{"style":9365},[54,9573],{"className":9574,"style":3866},[2155],[54,9576,9578],{"className":9577},[2126],[54,9579,9581],{"className":9580},[2126,9375],[54,9582,9584,9617],{"className":9583},[2139,2756],[54,9585,9587,9614],{"className":9586},[2143],[54,9588,9590,9602],{"className":9589,"style":9385},[2147],[54,9591,9593,9596],{"className":9592,"style":3862},[9389],[54,9594],{"className":9595,"style":3866},[2155],[54,9597,9599],{"className":9598,"style":9396},[2126],[54,9600,133],{"className":9601},[2126],[54,9603,9604,9607],{"style":9402},[54,9605],{"className":9606,"style":3866},[2155],[54,9608,9610],{"className":9609,"style":9410},[9409],[5837,9611,9612],{"xmlns":5839,"width":9413,"height":9414,"viewBox":9415,"preserveAspectRatio":9416},[5846,9613],{"d":9419},[54,9615,2781],{"className":9616},[2780],[54,9618,9620],{"className":9619},[2143],[54,9621,9623],{"className":9622,"style":9429},[2147],[54,9624],{},[54,9626,9627,9630],{"style":9434},[54,9628],{"className":9629,"style":3866},[2155],[54,9631],{"className":9632,"style":9442},[9441],[54,9634,9635,9638],{"style":9445},[54,9636],{"className":9637,"style":3866},[2155],[54,9639,9641],{"className":9640},[2126],[54,9642,174],{"className":9643},[2126],[54,9645,2781],{"className":9646},[2780],[54,9648,9650],{"className":9649},[2143],[54,9651,9653],{"className":9652,"style":9464},[2147],[54,9654],{},[54,9656],{"className":9657},[2336,9349],[54,9659,9661],{"className":9660},[2126],[54,9662,129],{"className":9663},[2503,9476],[54,9665,9291],{"className":9666},[2126],[54,9668,2312],{"className":9669},[2336],[54,9671,9291],{"className":9672},[2126],[54,9674,9676,9679],{"className":9675},[2336],[54,9677,2312],{"className":9678},[2336],[54,9680,9682],{"className":9681},[2135],[54,9683,9685],{"className":9684},[2139],[54,9686,9688],{"className":9687},[2143],[54,9689,9691],{"className":9690,"style":4325},[2147],[54,9692,9693,9696],{"style":2462},[54,9694],{"className":9695,"style":2156},[2155],[54,9697,9699],{"className":9698},[2160,2161,2162,2163],[54,9700,7237],{"className":9701,"style":2870},[2126,2167,2163],[54,9703],{"className":9704,"style":2271},[2270],[54,9706,1663],{"className":9707},[2275],[54,9709],{"className":9710,"style":2271},[2270],[54,9712,9495],{"className":9713},[2126],[54,9715,2312],{"className":9716},[2336],[54,9718,2305],{"className":9719},[2126],[54,9721,9723,9726],{"className":9722},[2126],[54,9724,3349],{"className":9725,"style":2332},[2126,2167],[54,9727,9729],{"className":9728},[2135],[54,9730,9732,9752],{"className":9731},[2139,2756],[54,9733,9735,9749],{"className":9734},[2143],[54,9736,9738],{"className":9737,"style":6358},[2147],[54,9739,9740,9743],{"style":4128},[54,9741],{"className":9742,"style":2156},[2155],[54,9744,9746],{"className":9745},[2160,2161,2162,2163],[54,9747,2849],{"className":9748},[2126,2167,2163],[54,9750,2781],{"className":9751},[2780],[54,9753,9755],{"className":9754},[2143],[54,9756,9758],{"className":9757,"style":2788},[2147],[54,9759],{},[54,9761,2312],{"className":9762},[2336],[54,9764,9766],{"className":9765},[2126],[54,9767,3205],{"className":9768},[2503,9476],[54,9770],{"className":9771,"style":9330},[2270],[54,9773],{"className":9774,"style":2475},[2270],[54,9776,9043],{"className":9777},[2479],[54,9779],{"className":9780,"style":9330},[2270],[54,9782],{"className":9783,"style":2475},[2270],[54,9785,9787,9790,9887],{"className":9786},[2126],[54,9788],{"className":9789},[2497,9349],[54,9791,9793],{"className":9792},[9047],[54,9794,9796,9879],{"className":9795},[2139,2756],[54,9797,9799,9876],{"className":9798},[2143],[54,9800,9802,9857,9865],{"className":9801,"style":9362},[2147],[54,9803,9804,9807],{"style":9365},[54,9805],{"className":9806,"style":3866},[2155],[54,9808,9810],{"className":9809},[2126],[54,9811,9813],{"className":9812},[2126,9375],[54,9814,9816,9849],{"className":9815},[2139,2756],[54,9817,9819,9846],{"className":9818},[2143],[54,9820,9822,9834],{"className":9821,"style":9385},[2147],[54,9823,9825,9828],{"className":9824,"style":3862},[9389],[54,9826],{"className":9827,"style":3866},[2155],[54,9829,9831],{"className":9830,"style":9396},[2126],[54,9832,133],{"className":9833},[2126],[54,9835,9836,9839],{"style":9402},[54,9837],{"className":9838,"style":3866},[2155],[54,9840,9842],{"className":9841,"style":9410},[9409],[5837,9843,9844],{"xmlns":5839,"width":9413,"height":9414,"viewBox":9415,"preserveAspectRatio":9416},[5846,9845],{"d":9419},[54,9847,2781],{"className":9848},[2780],[54,9850,9852],{"className":9851},[2143],[54,9853,9855],{"className":9854,"style":9429},[2147],[54,9856],{},[54,9858,9859,9862],{"style":9434},[54,9860],{"className":9861,"style":3866},[2155],[54,9863],{"className":9864,"style":9442},[9441],[54,9866,9867,9870],{"style":9445},[54,9868],{"className":9869,"style":3866},[2155],[54,9871,9873],{"className":9872},[2126],[54,9874,174],{"className":9875},[2126],[54,9877,2781],{"className":9878},[2780],[54,9880,9882],{"className":9881},[2143],[54,9883,9885],{"className":9884,"style":9464},[2147],[54,9886],{},[54,9888],{"className":9889},[2336,9349],[54,9891,9893],{"className":9892},[2126],[54,9894,129],{"className":9895},[2503,9476],[54,9897,9291],{"className":9898},[2126],[54,9900,2312],{"className":9901},[2336],[54,9903,9291],{"className":9904},[2126],[54,9906,9908,9911],{"className":9907},[2336],[54,9909,2312],{"className":9910},[2336],[54,9912,9914],{"className":9913},[2135],[54,9915,9917],{"className":9916},[2139],[54,9918,9920],{"className":9919},[2143],[54,9921,9923],{"className":9922,"style":4325},[2147],[54,9924,9925,9928],{"style":2462},[54,9926],{"className":9927,"style":2156},[2155],[54,9929,9931],{"className":9930},[2160,2161,2162,2163],[54,9932,7237],{"className":9933,"style":2870},[2126,2167,2163],[54,9935],{"className":9936,"style":2271},[2270],[54,9938,1663],{"className":9939},[2275],[54,9941],{"className":9942,"style":2271},[2270],[54,9944,9495],{"className":9945},[2126],[54,9947,2312],{"className":9948},[2336],[54,9950,2730],{"className":9951,"style":2776},[2126,2167],[54,9953,2305],{"className":9954},[2126],[54,9956,9958,9961],{"className":9957},[2126],[54,9959,3349],{"className":9960,"style":2332},[2126,2167],[54,9962,9964],{"className":9963},[2135],[54,9965,9967,9987],{"className":9966},[2139,2756],[54,9968,9970,9984],{"className":9969},[2143],[54,9971,9973],{"className":9972,"style":6358},[2147],[54,9974,9975,9978],{"style":4128},[54,9976],{"className":9977,"style":2156},[2155],[54,9979,9981],{"className":9980},[2160,2161,2162,2163],[54,9982,2849],{"className":9983},[2126,2167,2163],[54,9985,2781],{"className":9986},[2780],[54,9988,9990],{"className":9989},[2143],[54,9991,9993],{"className":9992,"style":2788},[2147],[54,9994],{},[54,9996,2312],{"className":9997},[2336],[54,9999,10001],{"className":10000},[2126],[54,10002,3205],{"className":10003},[2503,9476],[54,10005],{"className":10006,"style":9330},[2270],[54,10008],{"className":10009,"style":2475},[2270],[54,10011,9043],{"className":10012},[2479],[54,10014],{"className":10015,"style":9330},[2270],[54,10017],{"className":10018,"style":2475},[2270],[54,10020,10022,10025,10122],{"className":10021},[2126],[54,10023],{"className":10024},[2497,9349],[54,10026,10028],{"className":10027},[9047],[54,10029,10031,10114],{"className":10030},[2139,2756],[54,10032,10034,10111],{"className":10033},[2143],[54,10035,10037,10092,10100],{"className":10036,"style":9362},[2147],[54,10038,10039,10042],{"style":9365},[54,10040],{"className":10041,"style":3866},[2155],[54,10043,10045],{"className":10044},[2126],[54,10046,10048],{"className":10047},[2126,9375],[54,10049,10051,10084],{"className":10050},[2139,2756],[54,10052,10054,10081],{"className":10053},[2143],[54,10055,10057,10069],{"className":10056,"style":9385},[2147],[54,10058,10060,10063],{"className":10059,"style":3862},[9389],[54,10061],{"className":10062,"style":3866},[2155],[54,10064,10066],{"className":10065,"style":9396},[2126],[54,10067,133],{"className":10068},[2126],[54,10070,10071,10074],{"style":9402},[54,10072],{"className":10073,"style":3866},[2155],[54,10075,10077],{"className":10076,"style":9410},[9409],[5837,10078,10079],{"xmlns":5839,"width":9413,"height":9414,"viewBox":9415,"preserveAspectRatio":9416},[5846,10080],{"d":9419},[54,10082,2781],{"className":10083},[2780],[54,10085,10087],{"className":10086},[2143],[54,10088,10090],{"className":10089,"style":9429},[2147],[54,10091],{},[54,10093,10094,10097],{"style":9434},[54,10095],{"className":10096,"style":3866},[2155],[54,10098],{"className":10099,"style":9442},[9441],[54,10101,10102,10105],{"style":9445},[54,10103],{"className":10104,"style":3866},[2155],[54,10106,10108],{"className":10107},[2126],[54,10109,174],{"className":10110},[2126],[54,10112,2781],{"className":10113},[2780],[54,10115,10117],{"className":10116},[2143],[54,10118,10120],{"className":10119,"style":9464},[2147],[54,10121],{},[54,10123],{"className":10124},[2336,9349],[54,10126,10128],{"className":10127},[2126],[54,10129,129],{"className":10130},[2503,9476],[54,10132,9291],{"className":10133},[2126],[54,10135,2312],{"className":10136},[2336],[54,10138,2305],{"className":10139},[2126],[54,10141,10143,10146],{"className":10142},[2126],[54,10144,3349],{"className":10145,"style":2332},[2126,2167],[54,10147,10149],{"className":10148},[2135],[54,10150,10152,10172],{"className":10151},[2139,2756],[54,10153,10155,10169],{"className":10154},[2143],[54,10156,10158],{"className":10157,"style":6358},[2147],[54,10159,10160,10163],{"style":4128},[54,10161],{"className":10162,"style":2156},[2155],[54,10164,10166],{"className":10165},[2160,2161,2162,2163],[54,10167,1928],{"className":10168,"style":6379},[2126,2167,2163],[54,10170,2781],{"className":10171},[2780],[54,10173,10175],{"className":10174},[2143],[54,10176,10178],{"className":10177,"style":6389},[2147],[54,10179],{},[54,10181,2312],{"className":10182},[2336],[54,10184],{"className":10185,"style":2271},[2270],[54,10187,1663],{"className":10188},[2275],[54,10190],{"className":10191,"style":2271},[2270],[54,10193,9495],{"className":10194},[2126],[54,10196,2312],{"className":10197},[2336],[54,10199,2730],{"className":10200,"style":2776},[2126,2167],[54,10202,2305],{"className":10203},[2126],[54,10205,10207,10210],{"className":10206},[2126],[54,10208,3349],{"className":10209,"style":2332},[2126,2167],[54,10211,10213],{"className":10212},[2135],[54,10214,10216,10236],{"className":10215},[2139,2756],[54,10217,10219,10233],{"className":10218},[2143],[54,10220,10222],{"className":10221,"style":6358},[2147],[54,10223,10224,10227],{"style":4128},[54,10225],{"className":10226,"style":2156},[2155],[54,10228,10230],{"className":10229},[2160,2161,2162,2163],[54,10231,2849],{"className":10232},[2126,2167,2163],[54,10234,2781],{"className":10235},[2780],[54,10237,10239],{"className":10238},[2143],[54,10240,10242],{"className":10241,"style":2788},[2147],[54,10243],{},[54,10245,2312],{"className":10246},[2336],[54,10248,10250],{"className":10249},[2126],[54,10251,3205],{"className":10252},[2503,9476],[54,10254,2781],{"className":10255},[2780],[54,10257,10259],{"className":10258},[2143],[54,10260,10263],{"className":10261,"style":10262},[2147],"height:0.8757em;",[54,10264],{},[13,10266,4684,10267,10295,10296,10355,10356,171,10438,10520,10521,171,10601,10681,10682,11036,11037,11065,11066,2588],{},[54,10268,10270,10283],{"className":10269},[2079],[54,10271,10273],{"className":10272},[2083],[2085,10274,10275],{"xmlns":2087},[2089,10276,10277,10281],{},[2092,10278,10279],{},[2098,10280,2730],{},[2105,10282,2730],{"encoding":2107},[54,10284,10286],{"className":10285,"ariaHidden":2113},[2112],[54,10287,10289,10292],{"className":10288},[2117],[54,10290],{"className":10291,"style":2122},[2121],[54,10293,2730],{"className":10294,"style":2776},[2126,2167]," is one of the terms in the decomposition of the Hamiltonian ",[54,10297,10299,10319],{"className":10298},[2079],[54,10300,10302],{"className":10301},[2083],[2085,10303,10304],{"xmlns":2087},[2089,10305,10306,10316],{},[2092,10307,10308,10310,10312,10314],{},[2098,10309,2694],{},[2245,10311,122],{},[2245,10313,7083],{},[2098,10315,2730],{},[2105,10317,10318],{"encoding":2107},"H=\\sum P",[54,10320,10322,10340],{"className":10321,"ariaHidden":2113},[2112],[54,10323,10325,10328,10331,10334,10337],{"className":10324},[2117],[54,10326],{"className":10327,"style":2122},[2121],[54,10329,2694],{"className":10330,"style":2709},[2126,2167],[54,10332],{"className":10333,"style":2475},[2270],[54,10335,122],{"className":10336},[2479],[54,10338],{"className":10339,"style":2475},[2270],[54,10341,10343,10346,10349,10352],{"className":10342},[2117],[54,10344],{"className":10345,"style":2325},[2121],[54,10347,7083],{"className":10348,"style":7132},[7126,7130,7131],[54,10350],{"className":10351,"style":2521},[2270],[54,10353,2730],{"className":10354,"style":2776},[2126,2167]," and ",[54,10357,10359,10380],{"className":10358},[2079],[54,10360,10362],{"className":10361},[2083],[2085,10363,10364],{"xmlns":2087},[2089,10365,10366,10378],{},[2092,10367,10368,10370,10372],{},[5723,10369,6569],{},[5723,10371,6572],{},[2724,10373,10374,10376],{},[2098,10375,3349],{},[2098,10377,2849],{},[2105,10379,6581],{"encoding":2107},[54,10381,10383],{"className":10382,"ariaHidden":2113},[2112],[54,10384,10386,10389,10395,10398],{"className":10385},[2117],[54,10387],{"className":10388,"style":6591},[2121],[54,10390,10392],{"className":10391},[2126,6595],[54,10393,6569],{"className":10394},[2126],[54,10396],{"className":10397,"style":2475},[2270],[54,10399,10401,10404],{"className":10400},[2126],[54,10402,3349],{"className":10403,"style":2332},[2126,2167],[54,10405,10407],{"className":10406},[2135],[54,10408,10410,10430],{"className":10409},[2139,2756],[54,10411,10413,10427],{"className":10412},[2143],[54,10414,10416],{"className":10415,"style":6358},[2147],[54,10417,10418,10421],{"style":4128},[54,10419],{"className":10420,"style":2156},[2155],[54,10422,10424],{"className":10423},[2160,2161,2162,2163],[54,10425,2849],{"className":10426},[2126,2167,2163],[54,10428,2781],{"className":10429},[2780],[54,10431,10433],{"className":10432},[2143],[54,10434,10436],{"className":10435,"style":2788},[2147],[54,10437],{},[54,10439,10441,10462],{"className":10440},[2079],[54,10442,10444],{"className":10443},[2083],[2085,10445,10446],{"xmlns":2087},[2089,10447,10448,10460],{},[2092,10449,10450,10452,10454],{},[5723,10451,6569],{},[5723,10453,6572],{},[2724,10455,10456,10458],{},[2098,10457,3349],{},[2098,10459,1928],{},[2105,10461,6666],{"encoding":2107},[54,10463,10465],{"className":10464,"ariaHidden":2113},[2112],[54,10466,10468,10471,10477,10480],{"className":10467},[2117],[54,10469],{"className":10470,"style":6676},[2121],[54,10472,10474],{"className":10473},[2126,6595],[54,10475,6569],{"className":10476},[2126],[54,10478],{"className":10479,"style":2475},[2270],[54,10481,10483,10486],{"className":10482},[2126],[54,10484,3349],{"className":10485,"style":2332},[2126,2167],[54,10487,10489],{"className":10488},[2135],[54,10490,10492,10512],{"className":10491},[2139,2756],[54,10493,10495,10509],{"className":10494},[2143],[54,10496,10498],{"className":10497,"style":6358},[2147],[54,10499,10500,10503],{"style":4128},[54,10501],{"className":10502,"style":2156},[2155],[54,10504,10506],{"className":10505},[2160,2161,2162,2163],[54,10507,1928],{"className":10508,"style":6379},[2126,2167,2163],[54,10510,2781],{"className":10511},[2780],[54,10513,10515],{"className":10514},[2143],[54,10516,10518],{"className":10517,"style":6389},[2147],[54,10519],{}," are controlled operations that prepare ",[54,10522,10524,10546],{"className":10523},[2079],[54,10525,10527],{"className":10526},[2083],[2085,10528,10529],{"xmlns":2087},[2089,10530,10531,10543],{},[2092,10532,10533,10535,10541],{},[2098,10534,2305],{"mathvariant":2304},[2724,10536,10537,10539],{},[2098,10538,3349],{},[2098,10540,2849],{},[2245,10542,2312],{"stretchy":2311},[2105,10544,10545],{"encoding":2107},"|\\psi_i\\rangle",[54,10547,10549],{"className":10548,"ariaHidden":2113},[2112],[54,10550,10552,10555,10558,10598],{"className":10551},[2117],[54,10553],{"className":10554,"style":2325},[2121],[54,10556,2305],{"className":10557},[2126],[54,10559,10561,10564],{"className":10560},[2126],[54,10562,3349],{"className":10563,"style":2332},[2126,2167],[54,10565,10567],{"className":10566},[2135],[54,10568,10570,10590],{"className":10569},[2139,2756],[54,10571,10573,10587],{"className":10572},[2143],[54,10574,10576],{"className":10575,"style":6358},[2147],[54,10577,10578,10581],{"style":4128},[54,10579],{"className":10580,"style":2156},[2155],[54,10582,10584],{"className":10583},[2160,2161,2162,2163],[54,10585,2849],{"className":10586},[2126,2167,2163],[54,10588,2781],{"className":10589},[2780],[54,10591,10593],{"className":10592},[2143],[54,10594,10596],{"className":10595,"style":2788},[2147],[54,10597],{},[54,10599,2312],{"className":10600},[2336],[54,10602,10604,10626],{"className":10603},[2079],[54,10605,10607],{"className":10606},[2083],[2085,10608,10609],{"xmlns":2087},[2089,10610,10611,10623],{},[2092,10612,10613,10615,10621],{},[2098,10614,2305],{"mathvariant":2304},[2724,10616,10617,10619],{},[2098,10618,3349],{},[2098,10620,1928],{},[2245,10622,2312],{"stretchy":2311},[2105,10624,10625],{"encoding":2107},"|\\psi_j\\rangle",[54,10627,10629],{"className":10628,"ariaHidden":2113},[2112],[54,10630,10632,10635,10638,10678],{"className":10631},[2117],[54,10633],{"className":10634,"style":6507},[2121],[54,10636,2305],{"className":10637},[2126],[54,10639,10641,10644],{"className":10640},[2126],[54,10642,3349],{"className":10643,"style":2332},[2126,2167],[54,10645,10647],{"className":10646},[2135],[54,10648,10650,10670],{"className":10649},[2139,2756],[54,10651,10653,10667],{"className":10652},[2143],[54,10654,10656],{"className":10655,"style":6358},[2147],[54,10657,10658,10661],{"style":4128},[54,10659],{"className":10660,"style":2156},[2155],[54,10662,10664],{"className":10663},[2160,2161,2162,2163],[54,10665,1928],{"className":10666,"style":6379},[2126,2167,2163],[54,10668,2781],{"className":10669},[2780],[54,10671,10673],{"className":10672},[2143],[54,10674,10676],{"className":10675,"style":6389},[2147],[54,10677],{},[54,10679,2312],{"className":10680},[2336]," vectors of the unitary Krylov space, with ",[54,10683,10685,10769],{"className":10684},[2079],[54,10686,10688],{"className":10687},[2083],[2085,10689,10690],{"xmlns":2087},[2089,10691,10692,10766],{},[2092,10693,10694,10696,10702,10704,10706,10724,10726,10728,10730,10732,10750,10756,10758,10760],{},[2098,10695,2305],{"mathvariant":2304},[2724,10697,10698,10700],{},[2098,10699,3349],{},[2098,10701,3182],{},[2245,10703,2312],{"stretchy":2311},[2245,10705,122],{},[2095,10707,10708,10710],{},[2098,10709,2842],{},[2092,10711,10712,10714,10716,10718,10720,10722],{},[2245,10713,2247],{},[2098,10715,2849],{},[2098,10717,2694],{},[2098,10719,3182],{},[2098,10721,4515],{},[2098,10723,2854],{},[2098,10725,2305],{"mathvariant":2304},[2098,10727,3349],{},[2245,10729,2312],{"stretchy":2311},[2245,10731,122],{},[2095,10733,10734,10736],{},[2098,10735,2842],{},[2092,10737,10738,10740,10742,10744,10746,10748],{},[2245,10739,2247],{},[2098,10741,2849],{},[2098,10743,2694],{},[2098,10745,3182],{},[2098,10747,4515],{},[2098,10749,2854],{},[2724,10751,10752,10754],{},[2098,10753,2835],{},[2098,10755,3349],{},[2098,10757,2305],{"mathvariant":2304},[2249,10759,150],{},[2095,10761,10762,10764],{},[2245,10763,2312],{"stretchy":2311},[2098,10765,7237],{},[2105,10767,10768],{"encoding":2107},"|\\psi_k\\rangle = e^{-i H k dt } \\vert \\psi \\rangle = e^{-i H k dt } U_{\\psi} \\vert 0 \\rangle^N",[54,10770,10772,10834,10906],{"className":10771,"ariaHidden":2113},[2112],[54,10773,10775,10778,10781,10822,10825,10828,10831],{"className":10774},[2117],[54,10776],{"className":10777,"style":2325},[2121],[54,10779,2305],{"className":10780},[2126],[54,10782,10784,10787],{"className":10783},[2126],[54,10785,3349],{"className":10786,"style":2332},[2126,2167],[54,10788,10790],{"className":10789},[2135],[54,10791,10793,10814],{"className":10792},[2139,2756],[54,10794,10796,10811],{"className":10795},[2143],[54,10797,10800],{"className":10798,"style":10799},[2147],"height:0.3361em;",[54,10801,10802,10805],{"style":4128},[54,10803],{"className":10804,"style":2156},[2155],[54,10806,10808],{"className":10807},[2160,2161,2162,2163],[54,10809,3182],{"className":10810,"style":3248},[2126,2167,2163],[54,10812,2781],{"className":10813},[2780],[54,10815,10817],{"className":10816},[2143],[54,10818,10820],{"className":10819,"style":2788},[2147],[54,10821],{},[54,10823,2312],{"className":10824},[2336],[54,10826],{"className":10827,"style":2475},[2270],[54,10829,122],{"className":10830},[2479],[54,10832],{"className":10833,"style":2475},[2270],[54,10835,10837,10841,10888,10891,10894,10897,10900,10903],{"className":10836},[2117],[54,10838],{"className":10839,"style":10840},[2121],"height:1.0991em;vertical-align:-0.25em;",[54,10842,10844,10847],{"className":10843},[2126],[54,10845,2842],{"className":10846},[2126,2167],[54,10848,10850],{"className":10849},[2135],[54,10851,10853],{"className":10852},[2139],[54,10854,10856],{"className":10855},[2143],[54,10857,10859],{"className":10858,"style":3218},[2147],[54,10860,10861,10864],{"style":2151},[54,10862],{"className":10863,"style":2156},[2155],[54,10865,10867],{"className":10866},[2160,2161,2162,2163],[54,10868,10870,10873,10876,10879,10882,10885],{"className":10869},[2126,2163],[54,10871,2247],{"className":10872},[2126,2163],[54,10874,2849],{"className":10875},[2126,2167,2163],[54,10877,2694],{"className":10878,"style":2709},[2126,2167,2163],[54,10880,3182],{"className":10881,"style":3248},[2126,2167,2163],[54,10883,4515],{"className":10884},[2126,2167,2163],[54,10886,2854],{"className":10887},[2126,2167,2163],[54,10889,2305],{"className":10890},[2126],[54,10892,3349],{"className":10893,"style":2332},[2126,2167],[54,10895,2312],{"className":10896},[2336],[54,10898],{"className":10899,"style":2475},[2270],[54,10901,122],{"className":10902},[2479],[54,10904],{"className":10905,"style":2475},[2270],[54,10907,10909,10913,10960,11004,11007],{"className":10908},[2117],[54,10910],{"className":10911,"style":10912},[2121],"height:1.1352em;vertical-align:-0.2861em;",[54,10914,10916,10919],{"className":10915},[2126],[54,10917,2842],{"className":10918},[2126,2167],[54,10920,10922],{"className":10921},[2135],[54,10923,10925],{"className":10924},[2139],[54,10926,10928],{"className":10927},[2143],[54,10929,10931],{"className":10930,"style":3218},[2147],[54,10932,10933,10936],{"style":2151},[54,10934],{"className":10935,"style":2156},[2155],[54,10937,10939],{"className":10938},[2160,2161,2162,2163],[54,10940,10942,10945,10948,10951,10954,10957],{"className":10941},[2126,2163],[54,10943,2247],{"className":10944},[2126,2163],[54,10946,2849],{"className":10947},[2126,2167,2163],[54,10949,2694],{"className":10950,"style":2709},[2126,2167,2163],[54,10952,3182],{"className":10953,"style":3248},[2126,2167,2163],[54,10955,4515],{"className":10956},[2126,2167,2163],[54,10958,2854],{"className":10959},[2126,2167,2163],[54,10961,10963,10966],{"className":10962},[2126],[54,10964,2835],{"className":10965,"style":2870},[2126,2167],[54,10967,10969],{"className":10968},[2135],[54,10970,10972,10996],{"className":10971},[2139,2756],[54,10973,10975,10993],{"className":10974},[2143],[54,10976,10978],{"className":10977,"style":10799},[2147],[54,10979,10981,10984],{"style":10980},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[54,10982],{"className":10983,"style":2156},[2155],[54,10985,10987],{"className":10986},[2160,2161,2162,2163],[54,10988,10990],{"className":10989},[2126,2163],[54,10991,3349],{"className":10992,"style":2332},[2126,2167,2163],[54,10994,2781],{"className":10995},[2780],[54,10997,10999],{"className":10998},[2143],[54,11000,11002],{"className":11001,"style":6389},[2147],[54,11003],{},[54,11005,9291],{"className":11006},[2126],[54,11008,11010,11013],{"className":11009},[2336],[54,11011,2312],{"className":11012},[2336],[54,11014,11016],{"className":11015},[2135],[54,11017,11019],{"className":11018},[2139],[54,11020,11022],{"className":11021},[2143],[54,11023,11025],{"className":11024,"style":2886},[2147],[54,11026,11027,11030],{"style":2151},[54,11028],{"className":11029,"style":2156},[2155],[54,11031,11033],{"className":11032},[2160,2161,2162,2163],[54,11034,7237],{"className":11035,"style":2870},[2126,2167,2163],". To measure ",[54,11038,11040,11053],{"className":11039},[2079],[54,11041,11043],{"className":11042},[2083],[2085,11044,11045],{"xmlns":2087},[2089,11046,11047,11051],{},[2092,11048,11049],{},[2098,11050,7287],{},[2105,11052,7287],{"encoding":2107},[54,11054,11056],{"className":11055,"ariaHidden":2113},[2112],[54,11057,11059,11062],{"className":11058},[2117],[54,11060],{"className":11061,"style":2122},[2121],[54,11063,7287],{"className":11064,"style":7432},[2126,2167],", first apply ",[54,11067,11069,11082],{"className":11068},[2079],[54,11070,11072],{"className":11071},[2083],[2085,11073,11074],{"xmlns":2087},[2089,11075,11076,11080],{},[2092,11077,11078],{},[2098,11079,2694],{},[2105,11081,2694],{"encoding":2107},[54,11083,11085],{"className":11084,"ariaHidden":2113},[2112],[54,11086,11088,11091],{"className":11087},[2117],[54,11089],{"className":11090,"style":2122},[2121],[54,11092,2694],{"className":11093,"style":2709},[2126,2167],[54,11095,11097],{"className":11096},[2340],[54,11098,11100,11206],{"className":11099},[2079],[54,11101,11103],{"className":11102},[2083],[2085,11104,11105],{"xmlns":2087,"display":2349},[2089,11106,11107,11203],{},[9008,11108,11109],{"rowspacing":9010,"columnspacing":9011},[9013,11110,11111],{},[9016,11112,11113],{},[9019,11114,11115],{"scriptlevel":150,"displaystyle":2113},[2092,11116,11117,11119,11121,11127,11129,11131,11133,11135,11137,11143,11145,11147,11149,11151,11157,11159,11161,11163,11169,11171,11173,11175,11177,11179,11185,11187,11189,11191,11193,11199,11201],{},[2245,11118,9043],{},[2270,11120],{"width":9011},[9047,11122,11123,11125],{},[2249,11124,174],{},[2249,11126,133],{},[2098,11128,2305],{"mathvariant":2304},[2249,11130,150],{},[2245,11132,2312],{"stretchy":2311},[2245,11134,129],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2098,11136,2305],{"mathvariant":2304},[2724,11138,11139,11141],{},[2098,11140,3349],{},[2098,11142,1928],{},[2245,11144,2312],{"stretchy":2311},[2245,11146,1663],{},[2098,11148,2730],{},[2098,11150,2305],{"mathvariant":2304},[2724,11152,11153,11155],{},[2098,11154,3349],{},[2098,11156,2849],{},[2245,11158,2312],{"stretchy":2311},[2245,11160,3205],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2245,11162,1663],{},[9047,11164,11165,11167],{},[2249,11166,174],{},[2249,11168,133],{},[2098,11170,2305],{"mathvariant":2304},[2249,11172,174],{},[2245,11174,2312],{"stretchy":2311},[2245,11176,129],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2098,11178,2305],{"mathvariant":2304},[2724,11180,11181,11183],{},[2098,11182,3349],{},[2098,11184,1928],{},[2245,11186,2312],{"stretchy":2311},[2245,11188,2247],{},[2098,11190,2730],{},[2098,11192,2305],{"mathvariant":2304},[2724,11194,11195,11197],{},[2098,11196,3349],{},[2098,11198,2849],{},[2245,11200,2312],{"stretchy":2311},[2245,11202,3205],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2105,11204,11205],{"encoding":2107},"\\begin{equation*}\n \\longrightarrow\\quad\\frac{1}{2}|0\\rangle\\Big( |\\psi_j\\rangle + P|\\psi_i\\rangle\\Big) + \\frac{1}{2}|1\\rangle\\Big(|\\psi_j\\rangle - P|\\psi_i\\rangle\\Big)\n\\end{equation*}",[54,11207,11209],{"className":11208,"ariaHidden":2113},[2112],[54,11210,11212,11216],{"className":11211},[2117],[54,11213],{"className":11214,"style":11215},[2121],"height:2.0074em;vertical-align:-0.7537em;",[54,11217,11219],{"className":11218},[2126],[54,11220,11222],{"className":11221},[9008],[54,11223,11225],{"className":11224},[9267],[54,11226,11228,11635],{"className":11227},[2139,2756],[54,11229,11231,11632],{"className":11230},[2143],[54,11232,11235],{"className":11233,"style":11234},[2147],"height:1.2537em;",[54,11236,11238,11241],{"style":11237},"top:-3.2537em;",[54,11239],{"className":11240,"style":9284},[2155],[54,11242,11244,11247,11250,11253,11317,11320,11323,11329,11332,11372,11375,11378,11381,11384,11387,11390,11430,11433,11439,11442,11445,11448,11510,11513,11516,11522,11525,11565,11568,11571,11574,11577,11580,11583,11623,11626],{"className":11243},[2126],[54,11245,9043],{"className":11246},[2479],[54,11248],{"className":11249,"style":9330},[2270],[54,11251],{"className":11252,"style":2475},[2270],[54,11254,11256,11259,11314],{"className":11255},[2126],[54,11257],{"className":11258},[2497,9349],[54,11260,11262],{"className":11261},[9047],[54,11263,11265,11305],{"className":11264},[2139,2756],[54,11266,11268,11302],{"className":11267},[2143],[54,11269,11271,11283,11291],{"className":11270,"style":9362},[2147],[54,11272,11274,11277],{"style":11273},"top:-2.314em;",[54,11275],{"className":11276,"style":3866},[2155],[54,11278,11280],{"className":11279},[2126],[54,11281,133],{"className":11282},[2126],[54,11284,11285,11288],{"style":9434},[54,11286],{"className":11287,"style":3866},[2155],[54,11289],{"className":11290,"style":9442},[9441],[54,11292,11293,11296],{"style":9445},[54,11294],{"className":11295,"style":3866},[2155],[54,11297,11299],{"className":11298},[2126],[54,11300,174],{"className":11301},[2126],[54,11303,2781],{"className":11304},[2780],[54,11306,11308],{"className":11307},[2143],[54,11309,11312],{"className":11310,"style":11311},[2147],"height:0.686em;",[54,11313],{},[54,11315],{"className":11316},[2336,9349],[54,11318,9291],{"className":11319},[2126],[54,11321,2312],{"className":11322},[2336],[54,11324,11326],{"className":11325},[2126],[54,11327,129],{"className":11328},[2503,9476],[54,11330,2305],{"className":11331},[2126],[54,11333,11335,11338],{"className":11334},[2126],[54,11336,3349],{"className":11337,"style":2332},[2126,2167],[54,11339,11341],{"className":11340},[2135],[54,11342,11344,11364],{"className":11343},[2139,2756],[54,11345,11347,11361],{"className":11346},[2143],[54,11348,11350],{"className":11349,"style":6358},[2147],[54,11351,11352,11355],{"style":4128},[54,11353],{"className":11354,"style":2156},[2155],[54,11356,11358],{"className":11357},[2160,2161,2162,2163],[54,11359,1928],{"className":11360,"style":6379},[2126,2167,2163],[54,11362,2781],{"className":11363},[2780],[54,11365,11367],{"className":11366},[2143],[54,11368,11370],{"className":11369,"style":6389},[2147],[54,11371],{},[54,11373,2312],{"className":11374},[2336],[54,11376],{"className":11377,"style":2271},[2270],[54,11379,1663],{"className":11380},[2275],[54,11382],{"className":11383,"style":2271},[2270],[54,11385,2730],{"className":11386,"style":2776},[2126,2167],[54,11388,2305],{"className":11389},[2126],[54,11391,11393,11396],{"className":11392},[2126],[54,11394,3349],{"className":11395,"style":2332},[2126,2167],[54,11397,11399],{"className":11398},[2135],[54,11400,11402,11422],{"className":11401},[2139,2756],[54,11403,11405,11419],{"className":11404},[2143],[54,11406,11408],{"className":11407,"style":6358},[2147],[54,11409,11410,11413],{"style":4128},[54,11411],{"className":11412,"style":2156},[2155],[54,11414,11416],{"className":11415},[2160,2161,2162,2163],[54,11417,2849],{"className":11418},[2126,2167,2163],[54,11420,2781],{"className":11421},[2780],[54,11423,11425],{"className":11424},[2143],[54,11426,11428],{"className":11427,"style":2788},[2147],[54,11429],{},[54,11431,2312],{"className":11432},[2336],[54,11434,11436],{"className":11435},[2126],[54,11437,3205],{"className":11438},[2503,9476],[54,11440],{"className":11441,"style":2271},[2270],[54,11443,1663],{"className":11444},[2275],[54,11446],{"className":11447,"style":2271},[2270],[54,11449,11451,11454,11507],{"className":11450},[2126],[54,11452],{"className":11453},[2497,9349],[54,11455,11457],{"className":11456},[9047],[54,11458,11460,11499],{"className":11459},[2139,2756],[54,11461,11463,11496],{"className":11462},[2143],[54,11464,11466,11477,11485],{"className":11465,"style":9362},[2147],[54,11467,11468,11471],{"style":11273},[54,11469],{"className":11470,"style":3866},[2155],[54,11472,11474],{"className":11473},[2126],[54,11475,133],{"className":11476},[2126],[54,11478,11479,11482],{"style":9434},[54,11480],{"className":11481,"style":3866},[2155],[54,11483],{"className":11484,"style":9442},[9441],[54,11486,11487,11490],{"style":9445},[54,11488],{"className":11489,"style":3866},[2155],[54,11491,11493],{"className":11492},[2126],[54,11494,174],{"className":11495},[2126],[54,11497,2781],{"className":11498},[2780],[54,11500,11502],{"className":11501},[2143],[54,11503,11505],{"className":11504,"style":11311},[2147],[54,11506],{},[54,11508],{"className":11509},[2336,9349],[54,11511,9495],{"className":11512},[2126],[54,11514,2312],{"className":11515},[2336],[54,11517,11519],{"className":11518},[2126],[54,11520,129],{"className":11521},[2503,9476],[54,11523,2305],{"className":11524},[2126],[54,11526,11528,11531],{"className":11527},[2126],[54,11529,3349],{"className":11530,"style":2332},[2126,2167],[54,11532,11534],{"className":11533},[2135],[54,11535,11537,11557],{"className":11536},[2139,2756],[54,11538,11540,11554],{"className":11539},[2143],[54,11541,11543],{"className":11542,"style":6358},[2147],[54,11544,11545,11548],{"style":4128},[54,11546],{"className":11547,"style":2156},[2155],[54,11549,11551],{"className":11550},[2160,2161,2162,2163],[54,11552,1928],{"className":11553,"style":6379},[2126,2167,2163],[54,11555,2781],{"className":11556},[2780],[54,11558,11560],{"className":11559},[2143],[54,11561,11563],{"className":11562,"style":6389},[2147],[54,11564],{},[54,11566,2312],{"className":11567},[2336],[54,11569],{"className":11570,"style":2271},[2270],[54,11572,2247],{"className":11573},[2275],[54,11575],{"className":11576,"style":2271},[2270],[54,11578,2730],{"className":11579,"style":2776},[2126,2167],[54,11581,2305],{"className":11582},[2126],[54,11584,11586,11589],{"className":11585},[2126],[54,11587,3349],{"className":11588,"style":2332},[2126,2167],[54,11590,11592],{"className":11591},[2135],[54,11593,11595,11615],{"className":11594},[2139,2756],[54,11596,11598,11612],{"className":11597},[2143],[54,11599,11601],{"className":11600,"style":6358},[2147],[54,11602,11603,11606],{"style":4128},[54,11604],{"className":11605,"style":2156},[2155],[54,11607,11609],{"className":11608},[2160,2161,2162,2163],[54,11610,2849],{"className":11611},[2126,2167,2163],[54,11613,2781],{"className":11614},[2780],[54,11616,11618],{"className":11617},[2143],[54,11619,11621],{"className":11620,"style":2788},[2147],[54,11622],{},[54,11624,2312],{"className":11625},[2336],[54,11627,11629],{"className":11628},[2126],[54,11630,3205],{"className":11631},[2503,9476],[54,11633,2781],{"className":11634},[2780],[54,11636,11638],{"className":11637},[2143],[54,11639,11642],{"className":11640,"style":11641},[2147],"height:0.7537em;",[54,11643],{},[13,11645,11646],{},"... then measure:",[54,11648,11650],{"className":11649},[2340],[54,11651,11653,11835],{"className":11652},[2079],[54,11654,11656],{"className":11655},[2083],[2085,11657,11658],{"xmlns":2087,"display":2349},[2089,11659,11660,11832],{},[9008,11661,11662],{"rowspacing":9010,"columnspacing":9011},[9013,11663,11664],{},[9016,11665,11666],{},[9019,11667,11668],{"scriptlevel":150,"displaystyle":2113},[9008,11669,11673,11781],{"rowspacing":11670,"columnalign":11671,"columnspacing":11672},"0.25em","right left","0em",[9013,11674,11675,11692],{},[9016,11676,11677],{},[9019,11678,11679],{"scriptlevel":150,"displaystyle":2113},[2092,11680,11681,11684,11686,11688,11690],{},[2245,11682,11683],{},"⇒",[2270,11685],{"width":9011},[2245,11687,3981],{"stretchy":2311},[2098,11689,7287],{},[2245,11691,2312],{"stretchy":2311},[9016,11693,11694],{},[9019,11695,11696],{"scriptlevel":150,"displaystyle":2113},[2092,11697,11698,11700,11702,11709,11712,11715,11717,11723,11725,11727,11729,11731,11737,11739,11745,11747,11749,11751,11757,11759,11761,11763,11765,11771,11773,11779],{},[2092,11699],{},[2245,11701,122],{},[9047,11703,11704,11706],{},[2249,11705,174],{},[2249,11707,11708],{},"4",[2245,11710,129],{"fence":2311,"stretchy":2113,"minsize":11711,"maxsize":11711},"3em",[2245,11713,11714],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},"∥",[2098,11716,2305],{"mathvariant":2304},[2724,11718,11719,11721],{},[2098,11720,3349],{},[2098,11722,1928],{},[2245,11724,2312],{"stretchy":2311},[2245,11726,1663],{},[2098,11728,2730],{},[2098,11730,2305],{"mathvariant":2304},[2724,11732,11733,11735],{},[2098,11734,3349],{},[2098,11736,2849],{},[2245,11738,2312],{"stretchy":2311},[2095,11740,11741,11743],{},[2245,11742,11714],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2249,11744,133],{},[2245,11746,2247],{},[2245,11748,11714],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2098,11750,2305],{"mathvariant":2304},[2724,11752,11753,11755],{},[2098,11754,3349],{},[2098,11756,1928],{},[2245,11758,2312],{"stretchy":2311},[2245,11760,2247],{},[2098,11762,2730],{},[2098,11764,2305],{"mathvariant":2304},[2724,11766,11767,11769],{},[2098,11768,3349],{},[2098,11770,2849],{},[2245,11772,2312],{"stretchy":2311},[2095,11774,11775,11777],{},[2245,11776,11714],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2249,11778,133],{},[2245,11780,3205],{"fence":2311,"stretchy":2113,"minsize":11711,"maxsize":11711},[9013,11782,11783,11789],{},[9016,11784,11785],{},[9019,11786,11787],{"scriptlevel":150,"displaystyle":2113},[2092,11788],{},[9016,11790,11791],{},[9019,11792,11793],{"scriptlevel":150,"displaystyle":2113},[2092,11794,11795,11797,11799,11802,11805,11807,11813,11815,11817,11819,11825,11827,11830],{},[2092,11796],{},[2245,11798,122],{},[5723,11800,11801],{},"Re",[2245,11803,11804],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},"[",[2245,11806,3981],{"stretchy":2311},[2724,11808,11809,11811],{},[2098,11810,3349],{},[2098,11812,1928],{},[2098,11814,2305],{"mathvariant":2304},[2098,11816,2730],{},[2098,11818,2305],{"mathvariant":2304},[2724,11820,11821,11823],{},[2098,11822,3349],{},[2098,11824,2849],{},[2245,11826,2312],{"stretchy":2311},[2245,11828,11829],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},"]",[2098,11831,29],{"mathvariant":2304},[2105,11833,11834],{"encoding":2107},"\\begin{equation*}\n\\begin{split}\n \\Rightarrow\\quad\\langle X\\rangle &= \\frac{1}{4}\\Bigg(\\Big\\|| \\psi_j\\rangle + P|\\psi_i\\rangle \\Big\\|^2-\\Big\\||\\psi_j\\rangle - P|\\psi_i\\rangle\\Big\\|^2\\Bigg) \\\\\n &= \\text{Re}\\Big[\\langle\\psi_j| P|\\psi_i\\rangle\\Big].\n\\end{split}\n\\end{equation*}",[54,11836,11838],{"className":11837,"ariaHidden":2113},[2112],[54,11839,11841,11845],{"className":11840},[2117],[54,11842],{"className":11843,"style":11844},[2121],"height:5.1em;vertical-align:-2.3em;",[54,11846,11848],{"className":11847},[2126],[54,11849,11851],{"className":11850},[9008],[54,11852,11854],{"className":11853},[9267],[54,11855,11857,12627],{"className":11856},[2139,2756],[54,11858,11860,12624],{"className":11859},[2143],[54,11861,11864],{"className":11862,"style":11863},[2147],"height:2.8em;",[54,11865,11867,11871],{"style":11866},"top:-4.8em;",[54,11868],{"className":11869,"style":11870},[2155],"height:4.8em;",[54,11872,11874],{"className":11873},[2126],[54,11875,11877],{"className":11876},[2126],[54,11878,11880,11941],{"className":11879},[9008],[54,11881,11884],{"className":11882},[11883],"col-align-r",[54,11885,11887,11932],{"className":11886},[2139,2756],[54,11888,11890,11929],{"className":11889},[2143],[54,11891,11893,11920],{"className":11892,"style":11863},[2147],[54,11894,11895,11899],{"style":11866},[54,11896],{"className":11897,"style":11898},[2155],"height:3.75em;",[54,11900,11902,11905,11908,11911,11914,11917],{"className":11901},[2126],[54,11903,11683],{"className":11904},[2479],[54,11906],{"className":11907,"style":9330},[2270],[54,11909],{"className":11910,"style":2475},[2270],[54,11912,3981],{"className":11913},[2497],[54,11915,7287],{"className":11916,"style":7432},[2126,2167],[54,11918,2312],{"className":11919},[2336],[54,11921,11923,11926],{"style":11922},"top:-2.1em;",[54,11924],{"className":11925,"style":11898},[2155],[54,11927],{"className":11928},[2126],[54,11930,2781],{"className":11931},[2780],[54,11933,11935],{"className":11934},[2143],[54,11936,11939],{"className":11937,"style":11938},[2147],"height:2.3em;",[54,11940],{},[54,11942,11945],{"className":11943},[11944],"col-align-l",[54,11946,11948,12616],{"className":11947},[2139,2756],[54,11949,11951,12613],{"className":11950},[2143],[54,11952,11954,12477],{"className":11953,"style":11863},[2147],[54,11955,11956,11959],{"style":11866},[54,11957],{"className":11958,"style":11898},[2155],[54,11960,11962,11965,11968,11971,11974,12036,12043,12089,12092,12132,12135,12138,12141,12144,12147,12150,12190,12193,12258,12261,12264,12267,12304,12307,12347,12350,12353,12356,12359,12362,12365,12405,12408,12471],{"className":11961},[2126],[54,11963],{"className":11964},[2126],[54,11966],{"className":11967,"style":2475},[2270],[54,11969,122],{"className":11970},[2479],[54,11972],{"className":11973,"style":2475},[2270],[54,11975,11977,11980,12033],{"className":11976},[2126],[54,11978],{"className":11979},[2497,9349],[54,11981,11983],{"className":11982},[9047],[54,11984,11986,12025],{"className":11985},[2139,2756],[54,11987,11989,12022],{"className":11988},[2143],[54,11990,11992,12003,12011],{"className":11991,"style":9362},[2147],[54,11993,11994,11997],{"style":11273},[54,11995],{"className":11996,"style":3866},[2155],[54,11998,12000],{"className":11999},[2126],[54,12001,11708],{"className":12002},[2126],[54,12004,12005,12008],{"style":9434},[54,12006],{"className":12007,"style":3866},[2155],[54,12009],{"className":12010,"style":9442},[9441],[54,12012,12013,12016],{"style":9445},[54,12014],{"className":12015,"style":3866},[2155],[54,12017,12019],{"className":12018},[2126],[54,12020,174],{"className":12021},[2126],[54,12023,2781],{"className":12024},[2780],[54,12026,12028],{"className":12027},[2143],[54,12029,12031],{"className":12030,"style":11311},[2147],[54,12032],{},[54,12034],{"className":12035},[2336,9349],[54,12037,12039],{"className":12038},[2126],[54,12040,129],{"className":12041},[2503,12042],"size4",[54,12044,12046],{"className":12045},[2126],[54,12047,12050],{"className":12048},[2503,12049],"mult",[54,12051,12053,12080],{"className":12052},[2139,2756],[54,12054,12056,12077],{"className":12055},[2143],[54,12057,12060],{"className":12058,"style":12059},[2147],"height:1.15em;",[54,12061,12063,12067],{"style":12062},"top:-3.15em;",[54,12064],{"className":12065,"style":12066},[2155],"height:3.8em;",[54,12068,12070],{"style":12069},"width:0.556em;height:1.8em;",[5837,12071,12074],{"xmlns":5839,"width":12072,"height":9058,"viewBox":12073},"0.556em","0 0 556 1800",[5846,12075],{"d":12076},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z\nM367 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M410 15 H367 v585 v600 v585 h43z",[54,12078,2781],{"className":12079},[2780],[54,12081,12083],{"className":12082},[2143],[54,12084,12087],{"className":12085,"style":12086},[2147],"height:0.65em;",[54,12088],{},[54,12090,2305],{"className":12091},[2126],[54,12093,12095,12098],{"className":12094},[2126],[54,12096,3349],{"className":12097,"style":2332},[2126,2167],[54,12099,12101],{"className":12100},[2135],[54,12102,12104,12124],{"className":12103},[2139,2756],[54,12105,12107,12121],{"className":12106},[2143],[54,12108,12110],{"className":12109,"style":6358},[2147],[54,12111,12112,12115],{"style":4128},[54,12113],{"className":12114,"style":2156},[2155],[54,12116,12118],{"className":12117},[2160,2161,2162,2163],[54,12119,1928],{"className":12120,"style":6379},[2126,2167,2163],[54,12122,2781],{"className":12123},[2780],[54,12125,12127],{"className":12126},[2143],[54,12128,12130],{"className":12129,"style":6389},[2147],[54,12131],{},[54,12133,2312],{"className":12134},[2336],[54,12136],{"className":12137,"style":2271},[2270],[54,12139,1663],{"className":12140},[2275],[54,12142],{"className":12143,"style":2271},[2270],[54,12145,2730],{"className":12146,"style":2776},[2126,2167],[54,12148,2305],{"className":12149},[2126],[54,12151,12153,12156],{"className":12152},[2126],[54,12154,3349],{"className":12155,"style":2332},[2126,2167],[54,12157,12159],{"className":12158},[2135],[54,12160,12162,12182],{"className":12161},[2139,2756],[54,12163,12165,12179],{"className":12164},[2143],[54,12166,12168],{"className":12167,"style":6358},[2147],[54,12169,12170,12173],{"style":4128},[54,12171],{"className":12172,"style":2156},[2155],[54,12174,12176],{"className":12175},[2160,2161,2162,2163],[54,12177,2849],{"className":12178},[2126,2167,2163],[54,12180,2781],{"className":12181},[2780],[54,12183,12185],{"className":12184},[2143],[54,12186,12188],{"className":12187,"style":2788},[2147],[54,12189],{},[54,12191,2312],{"className":12192},[2336],[54,12194,12196,12233],{"className":12195},[2126],[54,12197,12199],{"className":12198},[2126],[54,12200,12202],{"className":12201},[2503,12049],[54,12203,12205,12225],{"className":12204},[2139,2756],[54,12206,12208,12222],{"className":12207},[2143],[54,12209,12211],{"className":12210,"style":12059},[2147],[54,12212,12213,12216],{"style":12062},[54,12214],{"className":12215,"style":12066},[2155],[54,12217,12218],{"style":12069},[5837,12219,12220],{"xmlns":5839,"width":12072,"height":9058,"viewBox":12073},[5846,12221],{"d":12076},[54,12223,2781],{"className":12224},[2780],[54,12226,12228],{"className":12227},[2143],[54,12229,12231],{"className":12230,"style":12086},[2147],[54,12232],{},[54,12234,12236],{"className":12235},[2135],[54,12237,12239],{"className":12238},[2139],[54,12240,12242],{"className":12241},[2143],[54,12243,12246],{"className":12244,"style":12245},[2147],"height:1.354em;",[54,12247,12249,12252],{"style":12248},"top:-3.6029em;margin-right:0.05em;",[54,12250],{"className":12251,"style":2156},[2155],[54,12253,12255],{"className":12254},[2160,2161,2162,2163],[54,12256,133],{"className":12257},[2126,2163],[54,12259],{"className":12260,"style":2271},[2270],[54,12262,2247],{"className":12263},[2275],[54,12265],{"className":12266,"style":2271},[2270],[54,12268,12270],{"className":12269},[2126],[54,12271,12273],{"className":12272},[2503,12049],[54,12274,12276,12296],{"className":12275},[2139,2756],[54,12277,12279,12293],{"className":12278},[2143],[54,12280,12282],{"className":12281,"style":12059},[2147],[54,12283,12284,12287],{"style":12062},[54,12285],{"className":12286,"style":12066},[2155],[54,12288,12289],{"style":12069},[5837,12290,12291],{"xmlns":5839,"width":12072,"height":9058,"viewBox":12073},[5846,12292],{"d":12076},[54,12294,2781],{"className":12295},[2780],[54,12297,12299],{"className":12298},[2143],[54,12300,12302],{"className":12301,"style":12086},[2147],[54,12303],{},[54,12305,2305],{"className":12306},[2126],[54,12308,12310,12313],{"className":12309},[2126],[54,12311,3349],{"className":12312,"style":2332},[2126,2167],[54,12314,12316],{"className":12315},[2135],[54,12317,12319,12339],{"className":12318},[2139,2756],[54,12320,12322,12336],{"className":12321},[2143],[54,12323,12325],{"className":12324,"style":6358},[2147],[54,12326,12327,12330],{"style":4128},[54,12328],{"className":12329,"style":2156},[2155],[54,12331,12333],{"className":12332},[2160,2161,2162,2163],[54,12334,1928],{"className":12335,"style":6379},[2126,2167,2163],[54,12337,2781],{"className":12338},[2780],[54,12340,12342],{"className":12341},[2143],[54,12343,12345],{"className":12344,"style":6389},[2147],[54,12346],{},[54,12348,2312],{"className":12349},[2336],[54,12351],{"className":12352,"style":2271},[2270],[54,12354,2247],{"className":12355},[2275],[54,12357],{"className":12358,"style":2271},[2270],[54,12360,2730],{"className":12361,"style":2776},[2126,2167],[54,12363,2305],{"className":12364},[2126],[54,12366,12368,12371],{"className":12367},[2126],[54,12369,3349],{"className":12370,"style":2332},[2126,2167],[54,12372,12374],{"className":12373},[2135],[54,12375,12377,12397],{"className":12376},[2139,2756],[54,12378,12380,12394],{"className":12379},[2143],[54,12381,12383],{"className":12382,"style":6358},[2147],[54,12384,12385,12388],{"style":4128},[54,12386],{"className":12387,"style":2156},[2155],[54,12389,12391],{"className":12390},[2160,2161,2162,2163],[54,12392,2849],{"className":12393},[2126,2167,2163],[54,12395,2781],{"className":12396},[2780],[54,12398,12400],{"className":12399},[2143],[54,12401,12403],{"className":12402,"style":2788},[2147],[54,12404],{},[54,12406,2312],{"className":12407},[2336],[54,12409,12411,12448],{"className":12410},[2126],[54,12412,12414],{"className":12413},[2126],[54,12415,12417],{"className":12416},[2503,12049],[54,12418,12420,12440],{"className":12419},[2139,2756],[54,12421,12423,12437],{"className":12422},[2143],[54,12424,12426],{"className":12425,"style":12059},[2147],[54,12427,12428,12431],{"style":12062},[54,12429],{"className":12430,"style":12066},[2155],[54,12432,12433],{"style":12069},[5837,12434,12435],{"xmlns":5839,"width":12072,"height":9058,"viewBox":12073},[5846,12436],{"d":12076},[54,12438,2781],{"className":12439},[2780],[54,12441,12443],{"className":12442},[2143],[54,12444,12446],{"className":12445,"style":12086},[2147],[54,12447],{},[54,12449,12451],{"className":12450},[2135],[54,12452,12454],{"className":12453},[2139],[54,12455,12457],{"className":12456},[2143],[54,12458,12460],{"className":12459,"style":12245},[2147],[54,12461,12462,12465],{"style":12248},[54,12463],{"className":12464,"style":2156},[2155],[54,12466,12468],{"className":12467},[2160,2161,2162,2163],[54,12469,133],{"className":12470},[2126,2163],[54,12472,12474],{"className":12473},[2126],[54,12475,3205],{"className":12476},[2503,12042],[54,12478,12479,12482],{"style":11922},[54,12480],{"className":12481,"style":11898},[2155],[54,12483,12485,12488,12491,12494,12497,12503,12509,12512,12552,12555,12558,12561,12601,12604,12610],{"className":12484},[2126],[54,12486],{"className":12487},[2126],[54,12489],{"className":12490,"style":2475},[2270],[54,12492,122],{"className":12493},[2479],[54,12495],{"className":12496,"style":2475},[2270],[54,12498,12500],{"className":12499},[2126,6595],[54,12501,11801],{"className":12502},[2126],[54,12504,12506],{"className":12505},[2126],[54,12507,11804],{"className":12508},[2503,9476],[54,12510,3981],{"className":12511},[2497],[54,12513,12515,12518],{"className":12514},[2126],[54,12516,3349],{"className":12517,"style":2332},[2126,2167],[54,12519,12521],{"className":12520},[2135],[54,12522,12524,12544],{"className":12523},[2139,2756],[54,12525,12527,12541],{"className":12526},[2143],[54,12528,12530],{"className":12529,"style":6358},[2147],[54,12531,12532,12535],{"style":4128},[54,12533],{"className":12534,"style":2156},[2155],[54,12536,12538],{"className":12537},[2160,2161,2162,2163],[54,12539,1928],{"className":12540,"style":6379},[2126,2167,2163],[54,12542,2781],{"className":12543},[2780],[54,12545,12547],{"className":12546},[2143],[54,12548,12550],{"className":12549,"style":6389},[2147],[54,12551],{},[54,12553,2305],{"className":12554},[2126],[54,12556,2730],{"className":12557,"style":2776},[2126,2167],[54,12559,2305],{"className":12560},[2126],[54,12562,12564,12567],{"className":12563},[2126],[54,12565,3349],{"className":12566,"style":2332},[2126,2167],[54,12568,12570],{"className":12569},[2135],[54,12571,12573,12593],{"className":12572},[2139,2756],[54,12574,12576,12590],{"className":12575},[2143],[54,12577,12579],{"className":12578,"style":6358},[2147],[54,12580,12581,12584],{"style":4128},[54,12582],{"className":12583,"style":2156},[2155],[54,12585,12587],{"className":12586},[2160,2161,2162,2163],[54,12588,2849],{"className":12589},[2126,2167,2163],[54,12591,2781],{"className":12592},[2780],[54,12594,12596],{"className":12595},[2143],[54,12597,12599],{"className":12598,"style":2788},[2147],[54,12600],{},[54,12602,2312],{"className":12603},[2336],[54,12605,12607],{"className":12606},[2126],[54,12608,11829],{"className":12609},[2503,9476],[54,12611,29],{"className":12612},[2126],[54,12614,2781],{"className":12615},[2780],[54,12617,12619],{"className":12618},[2143],[54,12620,12622],{"className":12621,"style":11938},[2147],[54,12623],{},[54,12625,2781],{"className":12626},[2780],[54,12628,12630],{"className":12629},[2143],[54,12631,12633],{"className":12632,"style":11938},[2147],[54,12634],{},[13,12636,12637,12638,12997,12998,13026],{},"From the identity ",[54,12639,12641,12728],{"className":12640},[2079],[54,12642,12644],{"className":12643},[2083],[2085,12645,12646],{"xmlns":2087},[2089,12647,12648,12725],{},[2092,12649,12650,12652,12654,12656,12659,12665,12667,12669,12671,12673,12675,12677,12679,12681,12683,12685,12687,12689,12691,12697,12699,12701,12703,12709,12711,12713,12715,12717,12719,12721,12723],{},[2098,12651,2305],{"mathvariant":2304},[2098,12653,20],{},[2245,12655,1663],{},[2098,12657,12658],{},"b",[2095,12660,12661,12663],{},[2098,12662,11714],{"mathvariant":2304},[2249,12664,133],{},[2245,12666,122],{},[2245,12668,3981],{"stretchy":2311},[2098,12670,20],{},[2245,12672,1663],{},[2098,12674,12658],{},[2098,12676,2305],{"mathvariant":2304},[2098,12678,20],{},[2245,12680,1663],{},[2098,12682,12658],{},[2245,12684,2312],{"stretchy":2311},[2245,12686,122],{},[2098,12688,11714],{"mathvariant":2304},[2098,12690,20],{},[2095,12692,12693,12695],{},[2098,12694,11714],{"mathvariant":2304},[2249,12696,133],{},[2245,12698,1663],{},[2098,12700,11714],{"mathvariant":2304},[2098,12702,12658],{},[2095,12704,12705,12707],{},[2098,12706,11714],{"mathvariant":2304},[2249,12708,133],{},[2245,12710,1663],{},[2249,12712,133],{},[5723,12714,11801],{},[2245,12716,3981],{"stretchy":2311},[2098,12718,20],{},[2098,12720,2305],{"mathvariant":2304},[2098,12722,12658],{},[2245,12724,2312],{"stretchy":2311},[2105,12726,12727],{"encoding":2107},"|a + b\\|^2 = \\langle a + b | a + b \\rangle = \\|a\\|^2 + \\|b\\|^2 + 2\\text{Re}\\langle a | b \\rangle",[54,12729,12731,12752,12801,12822,12846,12867,12917,12967],{"className":12730,"ariaHidden":2113},[2112],[54,12732,12734,12737,12740,12743,12746,12749],{"className":12733},[2117],[54,12735],{"className":12736,"style":2325},[2121],[54,12738,2305],{"className":12739},[2126],[54,12741,20],{"className":12742},[2126,2167],[54,12744],{"className":12745,"style":2271},[2270],[54,12747,1663],{"className":12748},[2275],[54,12750],{"className":12751,"style":2271},[2270],[54,12753,12755,12759,12762,12792,12795,12798],{"className":12754},[2117],[54,12756],{"className":12757,"style":12758},[2121],"height:1.0641em;vertical-align:-0.25em;",[54,12760,12658],{"className":12761},[2126,2167],[54,12763,12765,12768],{"className":12764},[2126],[54,12766,11714],{"className":12767},[2126],[54,12769,12771],{"className":12770},[2135],[54,12772,12774],{"className":12773},[2139],[54,12775,12777],{"className":12776},[2143],[54,12778,12781],{"className":12779,"style":12780},[2147],"height:0.8141em;",[54,12782,12783,12786],{"style":2151},[54,12784],{"className":12785,"style":2156},[2155],[54,12787,12789],{"className":12788},[2160,2161,2162,2163],[54,12790,133],{"className":12791},[2126,2163],[54,12793],{"className":12794,"style":2475},[2270],[54,12796,122],{"className":12797},[2479],[54,12799],{"className":12800,"style":2475},[2270],[54,12802,12804,12807,12810,12813,12816,12819],{"className":12803},[2117],[54,12805],{"className":12806,"style":2325},[2121],[54,12808,3981],{"className":12809},[2497],[54,12811,20],{"className":12812},[2126,2167],[54,12814],{"className":12815,"style":2271},[2270],[54,12817,1663],{"className":12818},[2275],[54,12820],{"className":12821,"style":2271},[2270],[54,12823,12825,12828,12831,12834,12837,12840,12843],{"className":12824},[2117],[54,12826],{"className":12827,"style":2325},[2121],[54,12829,12658],{"className":12830},[2126,2167],[54,12832,2305],{"className":12833},[2126],[54,12835,20],{"className":12836},[2126,2167],[54,12838],{"className":12839,"style":2271},[2270],[54,12841,1663],{"className":12842},[2275],[54,12844],{"className":12845,"style":2271},[2270],[54,12847,12849,12852,12855,12858,12861,12864],{"className":12848},[2117],[54,12850],{"className":12851,"style":2325},[2121],[54,12853,12658],{"className":12854},[2126,2167],[54,12856,2312],{"className":12857},[2336],[54,12859],{"className":12860,"style":2475},[2270],[54,12862,122],{"className":12863},[2479],[54,12865],{"className":12866,"style":2475},[2270],[54,12868,12870,12873,12876,12879,12908,12911,12914],{"className":12869},[2117],[54,12871],{"className":12872,"style":12758},[2121],[54,12874,11714],{"className":12875},[2126],[54,12877,20],{"className":12878},[2126,2167],[54,12880,12882,12885],{"className":12881},[2126],[54,12883,11714],{"className":12884},[2126],[54,12886,12888],{"className":12887},[2135],[54,12889,12891],{"className":12890},[2139],[54,12892,12894],{"className":12893},[2143],[54,12895,12897],{"className":12896,"style":12780},[2147],[54,12898,12899,12902],{"style":2151},[54,12900],{"className":12901,"style":2156},[2155],[54,12903,12905],{"className":12904},[2160,2161,2162,2163],[54,12906,133],{"className":12907},[2126,2163],[54,12909],{"className":12910,"style":2271},[2270],[54,12912,1663],{"className":12913},[2275],[54,12915],{"className":12916,"style":2271},[2270],[54,12918,12920,12923,12926,12929,12958,12961,12964],{"className":12919},[2117],[54,12921],{"className":12922,"style":12758},[2121],[54,12924,11714],{"className":12925},[2126],[54,12927,12658],{"className":12928},[2126,2167],[54,12930,12932,12935],{"className":12931},[2126],[54,12933,11714],{"className":12934},[2126],[54,12936,12938],{"className":12937},[2135],[54,12939,12941],{"className":12940},[2139],[54,12942,12944],{"className":12943},[2143],[54,12945,12947],{"className":12946,"style":12780},[2147],[54,12948,12949,12952],{"style":2151},[54,12950],{"className":12951,"style":2156},[2155],[54,12953,12955],{"className":12954},[2160,2161,2162,2163],[54,12956,133],{"className":12957},[2126,2163],[54,12959],{"className":12960,"style":2271},[2270],[54,12962,1663],{"className":12963},[2275],[54,12965],{"className":12966,"style":2271},[2270],[54,12968,12970,12973,12976,12982,12985,12988,12991,12994],{"className":12969},[2117],[54,12971],{"className":12972,"style":2325},[2121],[54,12974,133],{"className":12975},[2126],[54,12977,12979],{"className":12978},[2126,6595],[54,12980,11801],{"className":12981},[2126],[54,12983,3981],{"className":12984},[2497],[54,12986,20],{"className":12987},[2126,2167],[54,12989,2305],{"className":12990},[2126],[54,12992,12658],{"className":12993},[2126,2167],[54,12995,2312],{"className":12996},[2336],". Similarly, measuring ",[54,12999,13001,13014],{"className":13000},[2079],[54,13002,13004],{"className":13003},[2083],[2085,13005,13006],{"xmlns":2087},[2089,13007,13008,13012],{},[2092,13009,13010],{},[2098,13011,7302],{},[2105,13013,7302],{"encoding":2107},[54,13015,13017],{"className":13016,"ariaHidden":2113},[2112],[54,13018,13020,13023],{"className":13019},[2117],[54,13021],{"className":13022,"style":2122},[2121],[54,13024,7302],{"className":13025,"style":2271},[2126,2167]," yields",[54,13028,13030],{"className":13029},[2340],[54,13031,13033,13092],{"className":13032},[2079],[54,13034,13036],{"className":13035},[2083],[2085,13037,13038],{"xmlns":2087,"display":2349},[2089,13039,13040,13089],{},[9008,13041,13042],{"rowspacing":9010,"columnspacing":9011},[9013,13043,13044],{},[9016,13045,13046],{},[9019,13047,13048],{"scriptlevel":150,"displaystyle":2113},[2092,13049,13050,13052,13054,13056,13058,13061,13063,13065,13071,13073,13075,13077,13083,13085,13087],{},[2245,13051,3981],{"stretchy":2311},[2098,13053,7302],{},[2245,13055,2312],{"stretchy":2311},[2245,13057,122],{},[5723,13059,13060],{},"Im",[2245,13062,11804],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2245,13064,3981],{"stretchy":2311},[2724,13066,13067,13069],{},[2098,13068,3349],{},[2098,13070,1928],{},[2098,13072,2305],{"mathvariant":2304},[2098,13074,2730],{},[2098,13076,2305],{"mathvariant":2304},[2724,13078,13079,13081],{},[2098,13080,3349],{},[2098,13082,2849],{},[2245,13084,2312],{"stretchy":2311},[2245,13086,11829],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2098,13088,29],{"mathvariant":2304},[2105,13090,13091],{"encoding":2107},"\\begin{equation*}\n \\langle Y\\rangle = \\text{Im}\\Big[\\langle\\psi_j| P|\\psi_i\\rangle\\Big].\n\\end{equation*}",[54,13093,13095],{"className":13094,"ariaHidden":2113},[2112],[54,13096,13098,13102],{"className":13097},[2117],[54,13099],{"className":13100,"style":13101},[2121],"height:1.8em;vertical-align:-0.65em;",[54,13103,13105],{"className":13104},[2126],[54,13106,13108],{"className":13107},[9008],[54,13109,13111],{"className":13110},[9267],[54,13112,13114,13266],{"className":13113},[2139,2756],[54,13115,13117,13263],{"className":13116},[2143],[54,13118,13120],{"className":13119,"style":12059},[2147],[54,13121,13122,13126],{"style":12062},[54,13123],{"className":13124,"style":13125},[2155],"height:3.15em;",[54,13127,13129,13132,13135,13138,13141,13144,13147,13153,13159,13162,13202,13205,13208,13211,13251,13254,13260],{"className":13128},[2126],[54,13130,3981],{"className":13131},[2497],[54,13133,7302],{"className":13134,"style":2271},[2126,2167],[54,13136,2312],{"className":13137},[2336],[54,13139],{"className":13140,"style":2475},[2270],[54,13142,122],{"className":13143},[2479],[54,13145],{"className":13146,"style":2475},[2270],[54,13148,13150],{"className":13149},[2126,6595],[54,13151,13060],{"className":13152},[2126],[54,13154,13156],{"className":13155},[2126],[54,13157,11804],{"className":13158},[2503,9476],[54,13160,3981],{"className":13161},[2497],[54,13163,13165,13168],{"className":13164},[2126],[54,13166,3349],{"className":13167,"style":2332},[2126,2167],[54,13169,13171],{"className":13170},[2135],[54,13172,13174,13194],{"className":13173},[2139,2756],[54,13175,13177,13191],{"className":13176},[2143],[54,13178,13180],{"className":13179,"style":6358},[2147],[54,13181,13182,13185],{"style":4128},[54,13183],{"className":13184,"style":2156},[2155],[54,13186,13188],{"className":13187},[2160,2161,2162,2163],[54,13189,1928],{"className":13190,"style":6379},[2126,2167,2163],[54,13192,2781],{"className":13193},[2780],[54,13195,13197],{"className":13196},[2143],[54,13198,13200],{"className":13199,"style":6389},[2147],[54,13201],{},[54,13203,2305],{"className":13204},[2126],[54,13206,2730],{"className":13207,"style":2776},[2126,2167],[54,13209,2305],{"className":13210},[2126],[54,13212,13214,13217],{"className":13213},[2126],[54,13215,3349],{"className":13216,"style":2332},[2126,2167],[54,13218,13220],{"className":13219},[2135],[54,13221,13223,13243],{"className":13222},[2139,2756],[54,13224,13226,13240],{"className":13225},[2143],[54,13227,13229],{"className":13228,"style":6358},[2147],[54,13230,13231,13234],{"style":4128},[54,13232],{"className":13233,"style":2156},[2155],[54,13235,13237],{"className":13236},[2160,2161,2162,2163],[54,13238,2849],{"className":13239},[2126,2167,2163],[54,13241,2781],{"className":13242},[2780],[54,13244,13246],{"className":13245},[2143],[54,13247,13249],{"className":13248,"style":2788},[2147],[54,13250],{},[54,13252,2312],{"className":13253},[2336],[54,13255,13257],{"className":13256},[2126],[54,13258,11829],{"className":13259},[2503,9476],[54,13261,29],{"className":13262},[2126],[54,13264,2781],{"className":13265},[2780],[54,13267,13269],{"className":13268},[2143],[54,13270,13272],{"className":13271,"style":12086},[2147],[54,13273],{},[44,13275,13277],{"className":46,"code":13276,"language":48,"meta":49,"style":49},"## Create the time-evo op circuit\nevol_gate = PauliEvolutionGate(\n H_op, time=dt, synthesis=LieTrotter(reps=num_trotter_steps)\n)\n\n## Create the time-evo op dagger circuit\nevol_gate_d = PauliEvolutionGate(\n H_op, time=dt, synthesis=LieTrotter(reps=num_trotter_steps)\n)\nevol_gate_d = evol_gate_d.inverse()\n\n# Put pieces together\nqc_reg = QuantumRegister(n_qubits)\nqc_temp = QuantumCircuit(qc_reg)\nqc_temp.compose(qc_state_prep, inplace=True)\nfor _ in range(num_trotter_steps):\n qc_temp.append(evol_gate, qargs=qc_reg)\nfor _ in range(num_trotter_steps):\n qc_temp.append(evol_gate_d, qargs=qc_reg)\nqc_temp.compose(qc_state_prep.inverse(), inplace=True)\n\n# Create controlled version of the circuit\ncontrolled_U = qc_temp.to_gate().control(1)\n\n# Create hadamard test circuit for real part\nqr = QuantumRegister(n_qubits + 1)\nqc_real = QuantumCircuit(qr)\nqc_real.h(0)\nqc_real.append(controlled_U, list(range(n_qubits + 1)))\nqc_real.h(0)\n\nprint(\n \"Circuit for calculating the real part of the overlap in S via Hadamard test\"\n)\nqc_real.draw(\"mpl\", fold=-1, scale=0.5)\n",[51,13278,13279,13283,13293,13318,13322,13326,13331,13342,13366,13370,13384,13388,13393,13404,13416,13437,13450,13466,13478,13493,13515,13519,13524,13549,13553,13558,13574,13585,13598,13623,13635,13639,13645,13650,13654],{"__ignoreMap":49},[54,13280,13281],{"class":56,"line":57},[54,13282,8884],{"class":111},[54,13284,13285,13287,13289,13291],{"class":56,"line":74},[54,13286,8889],{"class":64},[54,13288,122],{"class":121},[54,13290,8894],{"class":125},[54,13292,932],{"class":64},[54,13294,13295,13297,13299,13301,13304,13306,13308,13310,13312,13314,13316],{"class":56,"line":88},[54,13296,8901],{"class":64},[54,13298,8904],{"class":1711},[54,13300,122],{"class":121},[54,13302,13303],{"class":64},"dt, ",[54,13305,8912],{"class":1711},[54,13307,122],{"class":121},[54,13309,8917],{"class":125},[54,13311,129],{"class":64},[54,13313,8922],{"class":1711},[54,13315,122],{"class":121},[54,13317,8927],{"class":64},[54,13319,13320],{"class":56,"line":101},[54,13321,136],{"class":64},[54,13323,13324],{"class":56,"line":108},[54,13325,105],{"emptyLinePlaceholder":104},[54,13327,13328],{"class":56,"line":115},[54,13329,13330],{"class":111},"## Create the time-evo op dagger circuit\n",[54,13332,13333,13336,13338,13340],{"class":56,"line":139},[54,13334,13335],{"class":64},"evol_gate_d ",[54,13337,122],{"class":121},[54,13339,8894],{"class":125},[54,13341,932],{"class":64},[54,13343,13344,13346,13348,13350,13352,13354,13356,13358,13360,13362,13364],{"class":56,"line":159},[54,13345,8901],{"class":64},[54,13347,8904],{"class":1711},[54,13349,122],{"class":121},[54,13351,13303],{"class":64},[54,13353,8912],{"class":1711},[54,13355,122],{"class":121},[54,13357,8917],{"class":125},[54,13359,129],{"class":64},[54,13361,8922],{"class":1711},[54,13363,122],{"class":121},[54,13365,8927],{"class":64},[54,13367,13368],{"class":56,"line":183},[54,13369,136],{"class":64},[54,13371,13372,13374,13376,13379,13382],{"class":56,"line":210},[54,13373,13335],{"class":64},[54,13375,122],{"class":121},[54,13377,13378],{"class":64}," evol_gate_d.",[54,13380,13381],{"class":125},"inverse",[54,13383,330],{"class":64},[54,13385,13386],{"class":56,"line":227},[54,13387,105],{"emptyLinePlaceholder":104},[54,13389,13390],{"class":56,"line":232},[54,13391,13392],{"class":111},"# Put pieces together\n",[54,13394,13395,13398,13400,13402],{"class":56,"line":246},[54,13396,13397],{"class":64},"qc_reg ",[54,13399,122],{"class":121},[54,13401,8945],{"class":125},[54,13403,8339],{"class":64},[54,13405,13406,13409,13411,13413],{"class":56,"line":259},[54,13407,13408],{"class":64},"qc_temp ",[54,13410,122],{"class":121},[54,13412,126],{"class":125},[54,13414,13415],{"class":64},"(qc_reg)\n",[54,13417,13418,13421,13424,13427,13430,13432,13435],{"class":56,"line":264},[54,13419,13420],{"class":64},"qc_temp.",[54,13422,13423],{"class":125},"compose",[54,13425,13426],{"class":64},"(qc_state_prep, ",[54,13428,13429],{"class":1711},"inplace",[54,13431,122],{"class":121},[54,13433,13434],{"class":132},"True",[54,13436,136],{"class":64},[54,13438,13439,13441,13443,13445,13447],{"class":56,"line":270},[54,13440,1186],{"class":60},[54,13442,7772],{"class":64},[54,13444,1192],{"class":60},[54,13446,7777],{"class":121},[54,13448,13449],{"class":64},"(num_trotter_steps):\n",[54,13451,13452,13455,13457,13459,13461,13463],{"class":56,"line":285},[54,13453,13454],{"class":64}," qc_temp.",[54,13456,1495],{"class":125},[54,13458,8969],{"class":64},[54,13460,8972],{"class":1711},[54,13462,122],{"class":121},[54,13464,13465],{"class":64},"qc_reg)\n",[54,13467,13468,13470,13472,13474,13476],{"class":56,"line":297},[54,13469,1186],{"class":60},[54,13471,7772],{"class":64},[54,13473,1192],{"class":60},[54,13475,7777],{"class":121},[54,13477,13449],{"class":64},[54,13479,13480,13482,13484,13487,13489,13491],{"class":56,"line":310},[54,13481,13454],{"class":64},[54,13483,1495],{"class":125},[54,13485,13486],{"class":64},"(evol_gate_d, ",[54,13488,8972],{"class":1711},[54,13490,122],{"class":121},[54,13492,13465],{"class":64},[54,13494,13495,13497,13499,13502,13504,13507,13509,13511,13513],{"class":56,"line":324},[54,13496,13420],{"class":64},[54,13498,13423],{"class":125},[54,13500,13501],{"class":64},"(qc_state_prep.",[54,13503,13381],{"class":125},[54,13505,13506],{"class":64},"(), ",[54,13508,13429],{"class":1711},[54,13510,122],{"class":121},[54,13512,13434],{"class":132},[54,13514,136],{"class":64},[54,13516,13517],{"class":56,"line":333},[54,13518,105],{"emptyLinePlaceholder":104},[54,13520,13521],{"class":56,"line":345},[54,13522,13523],{"class":111},"# Create controlled version of the circuit\n",[54,13525,13526,13529,13531,13534,13537,13540,13543,13545,13547],{"class":56,"line":358},[54,13527,13528],{"class":64},"controlled_U ",[54,13530,122],{"class":121},[54,13532,13533],{"class":64}," qc_temp.",[54,13535,13536],{"class":125},"to_gate",[54,13538,13539],{"class":64},"().",[54,13541,13542],{"class":125},"control",[54,13544,129],{"class":64},[54,13546,174],{"class":132},[54,13548,136],{"class":64},[54,13550,13551],{"class":56,"line":908},[54,13552,105],{"emptyLinePlaceholder":104},[54,13554,13555],{"class":56,"line":913},[54,13556,13557],{"class":111},"# Create hadamard test circuit for real part\n",[54,13559,13560,13562,13564,13566,13568,13570,13572],{"class":56,"line":918},[54,13561,8940],{"class":64},[54,13563,122],{"class":121},[54,13565,8945],{"class":125},[54,13567,1660],{"class":64},[54,13569,1663],{"class":121},[54,13571,982],{"class":132},[54,13573,136],{"class":64},[54,13575,13576,13579,13581,13583],{"class":56,"line":923},[54,13577,13578],{"class":64},"qc_real ",[54,13580,122],{"class":121},[54,13582,126],{"class":125},[54,13584,8959],{"class":64},[54,13586,13587,13590,13592,13594,13596],{"class":56,"line":935},[54,13588,13589],{"class":64},"qc_real.",[54,13591,145],{"class":125},[54,13593,129],{"class":64},[54,13595,150],{"class":132},[54,13597,136],{"class":64},[54,13599,13600,13602,13604,13607,13610,13612,13614,13616,13618,13620],{"class":56,"line":945},[54,13601,13589],{"class":64},[54,13603,1495],{"class":125},[54,13605,13606],{"class":64},"(controlled_U, ",[54,13608,13609],{"class":121},"list",[54,13611,129],{"class":64},[54,13613,1760],{"class":121},[54,13615,1660],{"class":64},[54,13617,1663],{"class":121},[54,13619,982],{"class":132},[54,13621,13622],{"class":64},")))\n",[54,13624,13625,13627,13629,13631,13633],{"class":56,"line":953},[54,13626,13589],{"class":64},[54,13628,145],{"class":125},[54,13630,129],{"class":64},[54,13632,150],{"class":132},[54,13634,136],{"class":64},[54,13636,13637],{"class":56,"line":968},[54,13638,105],{"emptyLinePlaceholder":104},[54,13640,13641,13643],{"class":56,"line":987},[54,13642,8081],{"class":121},[54,13644,932],{"class":64},[54,13646,13647],{"class":56,"line":1005},[54,13648,13649],{"class":279}," \"Circuit for calculating the real part of the overlap in S via Hadamard test\"\n",[54,13651,13652],{"class":56,"line":1021},[54,13653,136],{"class":64},[54,13655,13656,13658,13660,13662,13664,13666,13669,13672,13674,13676,13678,13680,13682],{"class":56,"line":1027},[54,13657,13589],{"class":64},[54,13659,8720],{"class":125},[54,13661,129],{"class":64},[54,13663,8725],{"class":279},[54,13665,171],{"class":64},[54,13667,13668],{"class":1711},"fold",[54,13670,13671],{"class":121},"=-",[54,13673,174],{"class":132},[54,13675,171],{"class":64},[54,13677,8730],{"class":1711},[54,13679,122],{"class":121},[54,13681,8735],{"class":132},[54,13683,136],{"class":64},[44,13685,13688],{"className":13686,"code":13687,"language":6595,"meta":49},[8087],"Circuit for calculating the real part of the overlap in S via Hadamard test\n",[51,13689,13687],{"__ignoreMap":49},[380,13691],{"alt":8740,"src":13692},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-02.avif",[13,13694,13695],{},"The Hadamard test circuit can be a deep circuit once we decompose to native gates (which will increase even more if we account for the topology of the device)",[44,13697,13699],{"className":46,"code":13698,"language":48,"meta":49,"style":49},"print(\n \"Number of layers of 2Q operations\",\n qc_real.decompose(reps=2).depth(lambda x: x[0].num_qubits == 2),\n)\n",[51,13700,13701,13707,13714,13759],{"__ignoreMap":49},[54,13702,13703,13705],{"class":56,"line":57},[54,13704,8081],{"class":121},[54,13706,932],{"class":64},[54,13708,13709,13712],{"class":56,"line":74},[54,13710,13711],{"class":279}," \"Number of layers of 2Q operations\"",[54,13713,965],{"class":64},[54,13715,13716,13719,13722,13724,13726,13728,13730,13733,13736,13738,13741,13744,13747,13749,13752,13754,13756],{"class":56,"line":88},[54,13717,13718],{"class":64}," qc_real.",[54,13720,13721],{"class":125},"decompose",[54,13723,129],{"class":64},[54,13725,8922],{"class":1711},[54,13727,122],{"class":121},[54,13729,133],{"class":132},[54,13731,13732],{"class":64},").",[54,13734,13735],{"class":125},"depth",[54,13737,129],{"class":64},[54,13739,13740],{"class":60},"lambda",[54,13742,13743],{"class":938}," x",[54,13745,13746],{"class":64},": x[",[54,13748,150],{"class":132},[54,13750,13751],{"class":64},"].num_qubits ",[54,13753,1279],{"class":121},[54,13755,1872],{"class":132},[54,13757,13758],{"class":64},"),\n",[54,13760,13761],{"class":56,"line":101},[54,13762,136],{"class":64},[44,13764,13767],{"className":13765,"code":13766,"language":6595,"meta":49},[8087],"Number of layers of 2Q operations 112753\n",[51,13768,13766],{"__ignoreMap":49},[31,13770,13772],{"id":13771},"step-2-optimize-problem-for-quantum-hardware-execution","Step 2: Optimize problem for quantum hardware execution",[2069,13774,13776],{"id":13775},"efficient-hadamard-test","Efficient Hadamard test",[13,13778,13779],{},"We can optimize the deep circuits for the Hadamard test that we have obtained by introducing some approximations and relying on some assumption about the model Hamiltonian. For example, consider the following circuit for the Hadamard test:",[13,13781,13782],{},[6264,13783],{"alt":13784,"src":13785},"fig3.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-08.avif",[13,13787,13788,13789,13860,13861,13927,13928,13956,13957,14022,14023,14065,14066,14345,14346,14387,14388,14429],{},"Assume we can classically calculate ",[54,13790,13792,13810],{"className":13791},[2079],[54,13793,13795],{"className":13794},[2083],[2085,13796,13797],{"xmlns":2087},[2089,13798,13799,13807],{},[2092,13800,13801],{},[2724,13802,13803,13805],{},[2098,13804,5738],{},[2249,13806,150],{},[2105,13808,13809],{"encoding":2107},"E_0",[54,13811,13813],{"className":13812,"ariaHidden":2113},[2112],[54,13814,13816,13819],{"className":13815},[2117],[54,13817],{"className":13818,"style":2743},[2121],[54,13820,13822,13825],{"className":13821},[2126],[54,13823,5738],{"className":13824,"style":5448},[2126,2167],[54,13826,13828],{"className":13827},[2135],[54,13829,13831,13852],{"className":13830},[2139,2756],[54,13832,13834,13849],{"className":13833},[2143],[54,13835,13838],{"className":13836,"style":13837},[2147],"height:0.3011em;",[54,13839,13840,13843],{"style":5570},[54,13841],{"className":13842,"style":2156},[2155],[54,13844,13846],{"className":13845},[2160,2161,2162,2163],[54,13847,150],{"className":13848},[2126,2163],[54,13850,2781],{"className":13851},[2780],[54,13853,13855],{"className":13854},[2143],[54,13856,13858],{"className":13857,"style":2788},[2147],[54,13859],{},", the eigenvalue of ",[54,13862,13864,13886],{"className":13863},[2079],[54,13865,13867],{"className":13866},[2083],[2085,13868,13869],{"xmlns":2087},[2089,13870,13871,13883],{},[2092,13872,13873,13875,13877],{},[2098,13874,2305],{"mathvariant":2304},[2249,13876,150],{},[2095,13878,13879,13881],{},[2245,13880,2312],{"stretchy":2311},[2098,13882,7237],{},[2105,13884,13885],{"encoding":2107},"|0\\rangle^N",[54,13887,13889],{"className":13888,"ariaHidden":2113},[2112],[54,13890,13892,13895,13898],{"className":13891},[2117],[54,13893],{"className":13894,"style":4806},[2121],[54,13896,9291],{"className":13897},[2126],[54,13899,13901,13904],{"className":13900},[2336],[54,13902,2312],{"className":13903},[2336],[54,13905,13907],{"className":13906},[2135],[54,13908,13910],{"className":13909},[2139],[54,13911,13913],{"className":13912},[2143],[54,13914,13916],{"className":13915,"style":2886},[2147],[54,13917,13918,13921],{"style":2151},[54,13919],{"className":13920,"style":2156},[2155],[54,13922,13924],{"className":13923},[2160,2161,2162,2163],[54,13925,7237],{"className":13926,"style":2870},[2126,2167,2163]," under the Hamiltonian ",[54,13929,13931,13944],{"className":13930},[2079],[54,13932,13934],{"className":13933},[2083],[2085,13935,13936],{"xmlns":2087},[2089,13937,13938,13942],{},[2092,13939,13940],{},[2098,13941,2694],{},[2105,13943,2694],{"encoding":2107},[54,13945,13947],{"className":13946,"ariaHidden":2113},[2112],[54,13948,13950,13953],{"className":13949},[2117],[54,13951],{"className":13952,"style":2122},[2121],[54,13954,2694],{"className":13955,"style":2709},[2126,2167],".\nThis is satisfied when the Hamiltonian preserves the U(1) symmetry. Although this may seem like a strong assumption, there are many cases where it is safe to assume that there is a vacuum state (in this case it maps to the ",[54,13958,13960,13981],{"className":13959},[2079],[54,13961,13963],{"className":13962},[2083],[2085,13964,13965],{"xmlns":2087},[2089,13966,13967,13979],{},[2092,13968,13969,13971,13973],{},[2098,13970,2305],{"mathvariant":2304},[2249,13972,150],{},[2095,13974,13975,13977],{},[2245,13976,2312],{"stretchy":2311},[2098,13978,7237],{},[2105,13980,13885],{"encoding":2107},[54,13982,13984],{"className":13983,"ariaHidden":2113},[2112],[54,13985,13987,13990,13993],{"className":13986},[2117],[54,13988],{"className":13989,"style":4806},[2121],[54,13991,9291],{"className":13992},[2126],[54,13994,13996,13999],{"className":13995},[2336],[54,13997,2312],{"className":13998},[2336],[54,14000,14002],{"className":14001},[2135],[54,14003,14005],{"className":14004},[2139],[54,14006,14008],{"className":14007},[2143],[54,14009,14011],{"className":14010,"style":2886},[2147],[54,14012,14013,14016],{"style":2151},[54,14014],{"className":14015,"style":2156},[2155],[54,14017,14019],{"className":14018},[2160,2161,2162,2163],[54,14020,7237],{"className":14021,"style":2870},[2126,2167,2163]," state) which is unaffected by the action of the Hamiltonian. This is true for example for chemistry Hamiltonians that describe stable molecule (where the number of electrons is conserved).\nGiven that the gate ",[54,14024,14026,14044],{"className":14025},[2079],[54,14027,14029],{"className":14028},[2083],[2085,14030,14031],{"xmlns":2087},[2089,14032,14033,14041],{},[2092,14034,14035,14037,14039],{},[5723,14036,6569],{},[5723,14038,6572],{},[2098,14040,3349],{},[2105,14042,14043],{"encoding":2107},"\\text{Prep} \\; \\psi",[54,14045,14047],{"className":14046,"ariaHidden":2113},[2112],[54,14048,14050,14053,14059,14062],{"className":14049},[2117],[54,14051],{"className":14052,"style":6591},[2121],[54,14054,14056],{"className":14055},[2126,6595],[54,14057,6569],{"className":14058},[2126],[54,14060],{"className":14061,"style":2475},[2270],[54,14063,3349],{"className":14064,"style":2332},[2126,2167],", prepares the desired reference state ",[54,14067,14069,14147],{"className":14068},[2079],[54,14070,14072],{"className":14071},[2083],[2085,14073,14074],{"xmlns":2087},[2089,14075,14076,14144],{},[2092,14077,14078,14094,14096,14098,14100,14102,14110,14112,14130,14136],{},[14079,14080,14081,14083,14092],"mpadded",{},[2098,14082,2305],{"mathvariant":2304},[2092,14084,14085,14087,14090],{},[2098,14086,13],{},[2098,14088,14089],{},"s",[2098,14091,2849],{},[2245,14093,2312],{"stretchy":2311},[2245,14095,122],{},[5723,14097,6569],{},[5723,14099,6572],{},[2098,14101,3349],{},[14079,14103,14104,14106,14108],{},[2098,14105,2305],{"mathvariant":2304},[2249,14107,150],{},[2245,14109,2312],{"stretchy":2311},[2245,14111,122],{},[2095,14113,14114,14116],{},[2098,14115,2842],{},[2092,14117,14118,14120,14122,14124,14126,14128],{},[2245,14119,2247],{},[2098,14121,2849],{},[2098,14123,2694],{},[2249,14125,150],{},[2098,14127,4515],{},[2098,14129,2854],{},[2724,14131,14132,14134],{},[2098,14133,2835],{},[2098,14135,3349],{},[14079,14137,14138,14140,14142],{},[2098,14139,2305],{"mathvariant":2304},[2249,14141,150],{},[2245,14143,2312],{"stretchy":2311},[2105,14145,14146],{"encoding":2107},"\\ket{psi} = \\text{Prep} \\; \\psi \\ket{0} = e^{-i H 0 dt} U_{\\psi} \\ket{0}",[54,14148,14150,14186,14231],{"className":14149,"ariaHidden":2113},[2112],[54,14151,14153,14156,14177,14180,14183],{"className":14152},[2117],[54,14154],{"className":14155,"style":2325},[2121],[54,14157,14159,14162,14174],{"className":14158},[2493],[54,14160,2305],{"className":14161},[2126],[54,14163,14165,14168,14171],{"className":14164},[2126],[54,14166,13],{"className":14167},[2126,2167],[54,14169,14089],{"className":14170},[2126,2167],[54,14172,2849],{"className":14173},[2126,2167],[54,14175,2312],{"className":14176},[2336],[54,14178],{"className":14179,"style":2475},[2270],[54,14181,122],{"className":14182},[2479],[54,14184],{"className":14185,"style":2475},[2270],[54,14187,14189,14192,14198,14201,14204,14207,14222,14225,14228],{"className":14188},[2117],[54,14190],{"className":14191,"style":2325},[2121],[54,14193,14195],{"className":14194},[2126,6595],[54,14196,6569],{"className":14197},[2126],[54,14199],{"className":14200,"style":2475},[2270],[54,14202,3349],{"className":14203,"style":2332},[2126,2167],[54,14205],{"className":14206,"style":2521},[2270],[54,14208,14210,14213,14219],{"className":14209},[2493],[54,14211,2305],{"className":14212},[2126],[54,14214,14216],{"className":14215},[2126],[54,14217,150],{"className":14218},[2126],[54,14220,2312],{"className":14221},[2336],[54,14223],{"className":14224,"style":2475},[2270],[54,14226,122],{"className":14227},[2479],[54,14229],{"className":14230,"style":2475},[2270],[54,14232,14234,14237,14284,14327,14330],{"className":14233},[2117],[54,14235],{"className":14236,"style":10912},[2121],[54,14238,14240,14243],{"className":14239},[2126],[54,14241,2842],{"className":14242},[2126,2167],[54,14244,14246],{"className":14245},[2135],[54,14247,14249],{"className":14248},[2139],[54,14250,14252],{"className":14251},[2143],[54,14253,14255],{"className":14254,"style":3218},[2147],[54,14256,14257,14260],{"style":2151},[54,14258],{"className":14259,"style":2156},[2155],[54,14261,14263],{"className":14262},[2160,2161,2162,2163],[54,14264,14266,14269,14272,14275,14278,14281],{"className":14265},[2126,2163],[54,14267,2247],{"className":14268},[2126,2163],[54,14270,2849],{"className":14271},[2126,2167,2163],[54,14273,2694],{"className":14274,"style":2709},[2126,2167,2163],[54,14276,150],{"className":14277},[2126,2163],[54,14279,4515],{"className":14280},[2126,2167,2163],[54,14282,2854],{"className":14283},[2126,2167,2163],[54,14285,14287,14290],{"className":14286},[2126],[54,14288,2835],{"className":14289,"style":2870},[2126,2167],[54,14291,14293],{"className":14292},[2135],[54,14294,14296,14319],{"className":14295},[2139,2756],[54,14297,14299,14316],{"className":14298},[2143],[54,14300,14302],{"className":14301,"style":10799},[2147],[54,14303,14304,14307],{"style":10980},[54,14305],{"className":14306,"style":2156},[2155],[54,14308,14310],{"className":14309},[2160,2161,2162,2163],[54,14311,14313],{"className":14312},[2126,2163],[54,14314,3349],{"className":14315,"style":2332},[2126,2167,2163],[54,14317,2781],{"className":14318},[2780],[54,14320,14322],{"className":14321},[2143],[54,14323,14325],{"className":14324,"style":6389},[2147],[54,14326],{},[54,14328],{"className":14329,"style":2521},[2270],[54,14331,14333,14336,14342],{"className":14332},[2493],[54,14334,2305],{"className":14335},[2126],[54,14337,14339],{"className":14338},[2126],[54,14340,150],{"className":14341},[2126],[54,14343,2312],{"className":14344},[2336],", for example, to prepare the HF state for chemistry ",[54,14347,14349,14366],{"className":14348},[2079],[54,14350,14352],{"className":14351},[2083],[2085,14353,14354],{"xmlns":2087},[2089,14355,14356,14364],{},[2092,14357,14358,14360,14362],{},[5723,14359,6569],{},[5723,14361,6572],{},[2098,14363,3349],{},[2105,14365,14043],{"encoding":2107},[54,14367,14369],{"className":14368,"ariaHidden":2113},[2112],[54,14370,14372,14375,14381,14384],{"className":14371},[2117],[54,14373],{"className":14374,"style":6591},[2121],[54,14376,14378],{"className":14377},[2126,6595],[54,14379,6569],{"className":14380},[2126],[54,14382],{"className":14383,"style":2475},[2270],[54,14385,3349],{"className":14386,"style":2332},[2126,2167]," would be a product of single-qubit NOTs, so controlled-",[54,14389,14391,14408],{"className":14390},[2079],[54,14392,14394],{"className":14393},[2083],[2085,14395,14396],{"xmlns":2087},[2089,14397,14398,14406],{},[2092,14399,14400,14402,14404],{},[5723,14401,6569],{},[5723,14403,6572],{},[2098,14405,3349],{},[2105,14407,14043],{"encoding":2107},[54,14409,14411],{"className":14410,"ariaHidden":2113},[2112],[54,14412,14414,14417,14423,14426],{"className":14413},[2117],[54,14415],{"className":14416,"style":6591},[2121],[54,14418,14420],{"className":14419},[2126,6595],[54,14421,6569],{"className":14422},[2126],[54,14424],{"className":14425,"style":2475},[2270],[54,14427,3349],{"className":14428,"style":2332},[2126,2167]," is just a product of CNOTs.\nThen the circuit above implements the following state prior to measurement:",[54,14431,14433],{"className":14432},[2340],[54,14434,14436,15056],{"className":14435},[2079],[54,14437,14439],{"className":14438},[2083],[2085,14440,14441],{"xmlns":2087,"display":2349},[2089,14442,14443,15053],{},[9008,14444,14445],{"rowspacing":9010,"columnspacing":9011},[9013,14446,14447,14451,15046,15049],{},[9016,14448],{"className":14449},[14450],"mtr-glue",[9016,14452,14453],{},[9019,14454,14455],{"scriptlevel":150,"displaystyle":2113},[9008,14456,14457,14561,14628,14715,14798,14916],{"rowspacing":11670,"columnalign":11671,"columnspacing":11672},[9013,14458,14459,14497],{},[9016,14460,14461],{},[9019,14462,14463],{"scriptlevel":150,"displaystyle":2113},[2092,14464,14465,14473,14485],{},[14079,14466,14467,14469,14471],{},[2098,14468,2305],{"mathvariant":2304},[2249,14470,150],{},[2245,14472,2312],{"stretchy":2311},[2095,14474,14475,14483],{},[14079,14476,14477,14479,14481],{},[2098,14478,2305],{"mathvariant":2304},[2249,14480,150],{},[2245,14482,2312],{"stretchy":2311},[2098,14484,7237],{},[3827,14486,14487,14491],{},[2245,14488,14490],{"stretchy":2113,"minsize":14489},"3.0em","→",[14079,14492,14495],{"width":14493,"lspace":14494},"+0.6em","0.3em",[2098,14496,2694],{},[9016,14498,14499],{},[9019,14500,14501],{"scriptlevel":150,"displaystyle":2113},[2092,14502,14503,14505,14513],{},[2092,14504],{},[9047,14506,14507,14509],{},[2249,14508,174],{},[9052,14510,14511],{},[2249,14512,133],{},[2092,14514,14515,14517,14525,14537,14539,14547,14559],{},[2245,14516,129],{"fence":2113},[14079,14518,14519,14521,14523],{},[2098,14520,2305],{"mathvariant":2304},[2249,14522,150],{},[2245,14524,2312],{"stretchy":2311},[2095,14526,14527,14535],{},[14079,14528,14529,14531,14533],{},[2098,14530,2305],{"mathvariant":2304},[2249,14532,150],{},[2245,14534,2312],{"stretchy":2311},[2098,14536,7237],{},[2245,14538,1663],{},[14079,14540,14541,14543,14545],{},[2098,14542,2305],{"mathvariant":2304},[2249,14544,174],{},[2245,14546,2312],{"stretchy":2311},[2095,14548,14549,14557],{},[14079,14550,14551,14553,14555],{},[2098,14552,2305],{"mathvariant":2304},[2249,14554,150],{},[2245,14556,2312],{"stretchy":2311},[2098,14558,7237],{},[2245,14560,3205],{"fence":2113},[9013,14562,14563,14576],{},[9016,14564,14565],{},[9019,14566,14567],{"scriptlevel":150,"displaystyle":2113},[3827,14568,14569,14571],{},[2245,14570,14490],{"stretchy":2113,"minsize":14489},[14079,14572,14573],{"width":14493,"lspace":14494},[5723,14574,14575],{},"1-ctrl-init",[9016,14577,14578],{},[9019,14579,14580],{"scriptlevel":150,"displaystyle":2113},[2092,14581,14582,14584,14592],{},[2092,14583],{},[9047,14585,14586,14588],{},[2249,14587,174],{},[9052,14589,14590],{},[2249,14591,133],{},[2092,14593,14594,14596,14598,14600,14602,14604,14606,14612,14614,14616,14618,14620,14622,14624,14626],{},[2245,14595,129],{"fence":2113},[2098,14597,2305],{"mathvariant":2304},[2249,14599,150],{},[2245,14601,2312],{"stretchy":2311},[2098,14603,2305],{"mathvariant":2304},[2249,14605,150],{},[2095,14607,14608,14610],{},[2245,14609,2312],{"stretchy":2311},[2098,14611,7237],{},[2245,14613,1663],{},[2098,14615,2305],{"mathvariant":2304},[2249,14617,174],{},[2245,14619,2312],{"stretchy":2311},[2098,14621,2305],{"mathvariant":2304},[2098,14623,3349],{},[2245,14625,2312],{"stretchy":2311},[2245,14627,3205],{"fence":2113},[9013,14629,14630,14642],{},[9016,14631,14632],{},[9019,14633,14634],{"scriptlevel":150,"displaystyle":2113},[3827,14635,14636,14638],{},[2245,14637,14490],{"stretchy":2113,"minsize":14489},[14079,14639,14640],{"width":14493,"lspace":14494},[2098,14641,2835],{},[9016,14643,14644],{},[9019,14645,14646],{"scriptlevel":150,"displaystyle":2113},[2092,14647,14648,14650,14658],{},[2092,14649],{},[9047,14651,14652,14654],{},[2249,14653,174],{},[9052,14655,14656],{},[2249,14657,133],{},[2092,14659,14660,14662,14673,14681,14693,14695,14703,14705,14713],{},[2245,14661,129],{"fence":2113},[2095,14663,14664,14666],{},[2098,14665,2842],{},[2092,14667,14668,14670],{},[2098,14669,2849],{},[2098,14671,14672],{},"ϕ",[14079,14674,14675,14677,14679],{},[2098,14676,2305],{"mathvariant":2304},[2249,14678,150],{},[2245,14680,2312],{"stretchy":2311},[2095,14682,14683,14691],{},[14079,14684,14685,14687,14689],{},[2098,14686,2305],{"mathvariant":2304},[2249,14688,150],{},[2245,14690,2312],{"stretchy":2311},[2098,14692,7237],{},[2245,14694,1663],{},[14079,14696,14697,14699,14701],{},[2098,14698,2305],{"mathvariant":2304},[2249,14700,174],{},[2245,14702,2312],{"stretchy":2311},[2098,14704,2835],{},[14079,14706,14707,14709,14711],{},[2098,14708,2305],{"mathvariant":2304},[2098,14710,3349],{},[2245,14712,2312],{"stretchy":2311},[2245,14714,3205],{"fence":2113},[9013,14716,14717,14730],{},[9016,14718,14719],{},[9019,14720,14721],{"scriptlevel":150,"displaystyle":2113},[3827,14722,14723,14725],{},[2245,14724,14490],{"stretchy":2113,"minsize":14489},[14079,14726,14727],{"width":14493,"lspace":14494},[5723,14728,14729],{},"0-ctrl-init",[9016,14731,14732],{},[9019,14733,14734],{"scriptlevel":150,"displaystyle":2113},[2092,14735,14736,14738,14746],{},[2092,14737],{},[9047,14739,14740,14742],{},[2249,14741,174],{},[9052,14743,14744],{},[2249,14745,133],{},[2092,14747,14748,14750,14760,14768,14776,14778,14786,14788,14796],{},[2245,14749,129],{"fence":2113},[2095,14751,14752,14754],{},[2098,14753,2842],{},[2092,14755,14756,14758],{},[2098,14757,2849],{},[2098,14759,14672],{},[14079,14761,14762,14764,14766],{},[2098,14763,2305],{"mathvariant":2304},[2249,14765,150],{},[2245,14767,2312],{"stretchy":2311},[14079,14769,14770,14772,14774],{},[2098,14771,2305],{"mathvariant":2304},[2098,14773,3349],{},[2245,14775,2312],{"stretchy":2311},[2245,14777,1663],{},[14079,14779,14780,14782,14784],{},[2098,14781,2305],{"mathvariant":2304},[2249,14783,174],{},[2245,14785,2312],{"stretchy":2311},[2098,14787,2835],{},[14079,14789,14790,14792,14794],{},[2098,14791,2305],{"mathvariant":2304},[2098,14793,3349],{},[2245,14795,2312],{"stretchy":2311},[2245,14797,3205],{"fence":2113},[9013,14799,14800,14806],{},[9016,14801,14802],{},[9019,14803,14804],{"scriptlevel":150,"displaystyle":2113},[2245,14805,122],{"lspace":11672,"rspace":11672},[9016,14807,14808],{},[9019,14809,14810],{"scriptlevel":150,"displaystyle":2113},[2092,14811,14812,14814,14820],{},[2092,14813],{},[9047,14815,14816,14818],{},[2249,14817,174],{},[2249,14819,133],{},[2092,14821,14822,14824,14832,14868,14870,14878,14914],{},[2245,14823,129],{"fence":2113},[14079,14825,14826,14828,14830],{},[2098,14827,2305],{"mathvariant":2304},[2245,14829,1663],{"lspace":11672,"rspace":11672},[2245,14831,2312],{"stretchy":2311},[2092,14833,14834,14836,14846,14854,14856,14858,14866],{},[2245,14835,129],{"fence":2113},[2095,14837,14838,14840],{},[2098,14839,2842],{},[2092,14841,14842,14844],{},[2098,14843,2849],{},[2098,14845,14672],{},[14079,14847,14848,14850,14852],{},[2098,14849,2305],{"mathvariant":2304},[2098,14851,3349],{},[2245,14853,2312],{"stretchy":2311},[2245,14855,1663],{},[2098,14857,2835],{},[14079,14859,14860,14862,14864],{},[2098,14861,2305],{"mathvariant":2304},[2098,14863,3349],{},[2245,14865,2312],{"stretchy":2311},[2245,14867,3205],{"fence":2113},[2245,14869,1663],{},[14079,14871,14872,14874,14876],{},[2098,14873,2305],{"mathvariant":2304},[2245,14875,2247],{"lspace":11672,"rspace":11672},[2245,14877,2312],{"stretchy":2311},[2092,14879,14880,14882,14892,14900,14902,14904,14912],{},[2245,14881,129],{"fence":2113},[2095,14883,14884,14886],{},[2098,14885,2842],{},[2092,14887,14888,14890],{},[2098,14889,2849],{},[2098,14891,14672],{},[14079,14893,14894,14896,14898],{},[2098,14895,2305],{"mathvariant":2304},[2098,14897,3349],{},[2245,14899,2312],{"stretchy":2311},[2245,14901,2247],{},[2098,14903,2835],{},[14079,14905,14906,14908,14910],{},[2098,14907,2305],{"mathvariant":2304},[2098,14909,3349],{},[2245,14911,2312],{"stretchy":2311},[2245,14913,3205],{"fence":2113},[2245,14915,3205],{"fence":2113},[9013,14917,14918,14924],{},[9016,14919,14920],{},[9019,14921,14922],{"scriptlevel":150,"displaystyle":2113},[2245,14923,122],{"lspace":11672,"rspace":11672},[9016,14925,14926],{},[9019,14927,14928],{"scriptlevel":150,"displaystyle":2113},[2092,14929,14930,14932,14938],{},[2092,14931],{},[9047,14933,14934,14936],{},[2249,14935,174],{},[2249,14937,133],{},[2092,14939,14940,14942,14954,14992,14994,15006,15044],{},[2245,14941,129],{"fence":2113},[14079,14943,14944,14946,14952],{},[2098,14945,2305],{"mathvariant":2304},[2092,14947,14948,14950],{},[2245,14949,1663],{},[2098,14951,2849],{},[2245,14953,2312],{"stretchy":2311},[2092,14955,14956,14958,14968,14976,14978,14980,14982,14990],{},[2245,14957,129],{"fence":2113},[2095,14959,14960,14962],{},[2098,14961,2842],{},[2092,14963,14964,14966],{},[2098,14965,2849],{},[2098,14967,14672],{},[14079,14969,14970,14972,14974],{},[2098,14971,2305],{"mathvariant":2304},[2098,14973,3349],{},[2245,14975,2312],{"stretchy":2311},[2245,14977,2247],{},[2098,14979,2849],{},[2098,14981,2835],{},[14079,14983,14984,14986,14988],{},[2098,14985,2305],{"mathvariant":2304},[2098,14987,3349],{},[2245,14989,2312],{"stretchy":2311},[2245,14991,3205],{"fence":2113},[2245,14993,1663],{},[14079,14995,14996,14998,15004],{},[2098,14997,2305],{"mathvariant":2304},[2092,14999,15000,15002],{},[2245,15001,2247],{},[2098,15003,2849],{},[2245,15005,2312],{"stretchy":2311},[2092,15007,15008,15010,15020,15028,15030,15032,15034,15042],{},[2245,15009,129],{"fence":2113},[2095,15011,15012,15014],{},[2098,15013,2842],{},[2092,15015,15016,15018],{},[2098,15017,2849],{},[2098,15019,14672],{},[14079,15021,15022,15024,15026],{},[2098,15023,2305],{"mathvariant":2304},[2098,15025,3349],{},[2245,15027,2312],{"stretchy":2311},[2245,15029,1663],{},[2098,15031,2849],{},[2098,15033,2835],{},[14079,15035,15036,15038,15040],{},[2098,15037,2305],{"mathvariant":2304},[2098,15039,3349],{},[2245,15041,2312],{"stretchy":2311},[2245,15043,3205],{"fence":2113},[2245,15045,3205],{"fence":2113},[9016,15047],{"className":15048},[14450],[9016,15050],{"className":15051},[15052],"mml-eqn-num",[2105,15054,15055],{"encoding":2107},"\\begin{equation}\n\\begin{split}\n \\ket{0} \\ket{0}^N\\xrightarrow{H}&\\frac{1}{\\sqrt{2}}\n \\left(\n \\ket{0}\\ket{0}^N+ \\ket{1} \\ket{0}^N\n \\right)\\\\\n \\xrightarrow{\\text{1-ctrl-init}}&\\frac{1}{\\sqrt{2}}\\left(|0\\rangle|0\\rangle^N+|1\\rangle|\\psi\\rangle\\right)\\\\\n \\xrightarrow{U}&\\frac{1}{\\sqrt{2}}\\left(e^{i\\phi}\\ket{0}\\ket{0}^N+\\ket{1} U\\ket{\\psi}\\right)\\\\\n \\xrightarrow{\\text{0-ctrl-init}}&\\frac{1}{\\sqrt{2}}\n \\left(\n e^{i\\phi}\\ket{0} \\ket{\\psi}\n +\\ket{1} U\\ket{\\psi}\n \\right)\\\\\n =&\\frac{1}{2}\n \\left(\n \\ket{+}\\left(e^{i\\phi}\\ket{\\psi}+U\\ket{\\psi}\\right)\n +\\ket{-}\\left(e^{i\\phi}\\ket{\\psi}-U\\ket{\\psi}\\right)\n \\right)\\\\\n =&\\frac{1}{2}\n \\left(\n \\ket{+i}\\left(e^{i\\phi}\\ket{\\psi}-iU\\ket{\\psi}\\right)\n +\\ket{-i}\\left(e^{i\\phi}\\ket{\\psi}+iU\\ket{\\psi}\\right)\n \\right)\n\\end{split}\n\\end{equation}",[54,15057,15059,17164],{"className":15058,"ariaHidden":2113},[2112],[54,15060,15062,15066],{"className":15061},[2117],[54,15063],{"className":15064,"style":15065},[2121],"height:14.5206em;vertical-align:-7.0103em;",[54,15067,15069],{"className":15068},[9008],[54,15070,15072],{"className":15071},[9267],[54,15073,15075,17156],{"className":15074},[2139,2756],[54,15076,15078,17153],{"className":15077},[2143],[54,15079,15082],{"className":15080,"style":15081},[2147],"height:7.5103em;",[54,15083,15085,15089],{"style":15084},"top:-9.5103em;",[54,15086],{"className":15087,"style":15088},[2155],"height:9.5103em;",[54,15090,15092],{"className":15091},[2126],[54,15093,15095],{"className":15094},[2126],[54,15096,15098,15461],{"className":15097},[9008],[54,15099,15101],{"className":15100},[11883],[54,15102,15104,15452],{"className":15103},[2139,2756],[54,15105,15107,15449],{"className":15106},[2143],[54,15108,15110,15241,15304,15363,15425,15437],{"className":15109,"style":15081},[2147],[54,15111,15112,15115],{"style":15084},[54,15113],{"className":15114,"style":9284},[2155],[54,15116,15118,15133,15136,15177,15180],{"className":15117},[2126],[54,15119,15121,15124,15130],{"className":15120},[2493],[54,15122,2305],{"className":15123},[2126],[54,15125,15127],{"className":15126},[2126],[54,15128,150],{"className":15129},[2126],[54,15131,2312],{"className":15132},[2336],[54,15134],{"className":15135,"style":2521},[2270],[54,15137,15139,15154],{"className":15138},[2493],[54,15140,15142,15145,15151],{"className":15141},[2493],[54,15143,2305],{"className":15144},[2126],[54,15146,15148],{"className":15147},[2126],[54,15149,150],{"className":15150},[2126],[54,15152,2312],{"className":15153},[2336],[54,15155,15157],{"className":15156},[2135],[54,15158,15160],{"className":15159},[2139],[54,15161,15163],{"className":15162},[2143],[54,15164,15166],{"className":15165,"style":7378},[2147],[54,15167,15168,15171],{"style":7401},[54,15169],{"className":15170,"style":2156},[2155],[54,15172,15174],{"className":15173},[2160,2161,2162,2163],[54,15175,7237],{"className":15176,"style":2870},[2126,2167,2163],[54,15178],{"className":15179,"style":2475},[2270],[54,15181,15184],{"className":15182},[2479,15183],"x-arrow",[54,15185,15187,15232],{"className":15186},[2139,2756],[54,15188,15190,15229],{"className":15189},[2143],[54,15191,15194,15210],{"className":15192,"style":15193},[2147],"height:1.1003em;",[54,15195,15197,15200],{"style":15196},"top:-3.322em;",[54,15198],{"className":15199,"style":2156},[2155],[54,15201,15204],{"className":15202},[2160,2161,2162,2163,15203],"x-arrow-pad",[54,15205,15207],{"className":15206},[2126,2163],[54,15208,2694],{"className":15209,"style":2709},[2126,2167,2163],[54,15211,15214,15217],{"className":15212,"style":15213},[9389],"top:-2.689em;",[54,15215],{"className":15216,"style":2156},[2155],[54,15218,15221],{"className":15219,"style":15220},[9409],"height:0.522em;min-width:1.469em;",[5837,15222,15226],{"xmlns":5839,"width":9413,"height":15223,"viewBox":15224,"preserveAspectRatio":15225},"0.522em","0 0 400000 522","xMaxYMin slice",[5846,15227],{"d":15228},"M0 241v40h399891c-47.3 35.3-84 78-110 128\n-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20\n 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7\n 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85\n-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n 151.7 139 205zm0 0v40h399900v-40z",[54,15230,2781],{"className":15231},[2780],[54,15233,15235],{"className":15234},[2143],[54,15236,15239],{"className":15237,"style":15238},[2147],"height:0.011em;",[54,15240],{},[54,15242,15244,15247],{"style":15243},"top:-6.9589em;",[54,15245],{"className":15246,"style":9284},[2155],[54,15248,15250],{"className":15249},[2126],[54,15251,15253],{"className":15252},[2479,15183],[54,15254,15256,15296],{"className":15255},[2139,2756],[54,15257,15259,15293],{"className":15258},[2143],[54,15260,15263,15280],{"className":15261,"style":15262},[2147],"height:1.1081em;",[54,15264,15265,15268],{"style":15196},[54,15266],{"className":15267,"style":2156},[2155],[54,15269,15271],{"className":15270},[2160,2161,2162,2163,15203],[54,15272,15274],{"className":15273},[2126,2163],[54,15275,15277],{"className":15276},[2126,6595,2163],[54,15278,14575],{"className":15279},[2126,2163],[54,15281,15283,15286],{"className":15282,"style":15213},[9389],[54,15284],{"className":15285,"style":2156},[2155],[54,15287,15289],{"className":15288,"style":15220},[9409],[5837,15290,15291],{"xmlns":5839,"width":9413,"height":15223,"viewBox":15224,"preserveAspectRatio":15225},[5846,15292],{"d":15228},[54,15294,2781],{"className":15295},[2780],[54,15297,15299],{"className":15298},[2143],[54,15300,15302],{"className":15301,"style":15238},[2147],[54,15303],{},[54,15305,15307,15310],{"style":15306},"top:-4.4074em;",[54,15308],{"className":15309,"style":9284},[2155],[54,15311,15313],{"className":15312},[2126],[54,15314,15316],{"className":15315},[2479,15183],[54,15317,15319,15355],{"className":15318},[2139,2756],[54,15320,15322,15352],{"className":15321},[2143],[54,15323,15325,15339],{"className":15324,"style":15193},[2147],[54,15326,15327,15330],{"style":15196},[54,15328],{"className":15329,"style":2156},[2155],[54,15331,15333],{"className":15332},[2160,2161,2162,2163,15203],[54,15334,15336],{"className":15335},[2126,2163],[54,15337,2835],{"className":15338,"style":2870},[2126,2167,2163],[54,15340,15342,15345],{"className":15341,"style":15213},[9389],[54,15343],{"className":15344,"style":2156},[2155],[54,15346,15348],{"className":15347,"style":15220},[9409],[5837,15349,15350],{"xmlns":5839,"width":9413,"height":15223,"viewBox":15224,"preserveAspectRatio":15225},[5846,15351],{"d":15228},[54,15353,2781],{"className":15354},[2780],[54,15356,15358],{"className":15357},[2143],[54,15359,15361],{"className":15360,"style":15238},[2147],[54,15362],{},[54,15364,15366,15369],{"style":15365},"top:-1.856em;",[54,15367],{"className":15368,"style":9284},[2155],[54,15370,15372],{"className":15371},[2126],[54,15373,15375],{"className":15374},[2479,15183],[54,15376,15378,15417],{"className":15377},[2139,2756],[54,15379,15381,15414],{"className":15380},[2143],[54,15382,15384,15401],{"className":15383,"style":15262},[2147],[54,15385,15386,15389],{"style":15196},[54,15387],{"className":15388,"style":2156},[2155],[54,15390,15392],{"className":15391},[2160,2161,2162,2163,15203],[54,15393,15395],{"className":15394},[2126,2163],[54,15396,15398],{"className":15397},[2126,6595,2163],[54,15399,14729],{"className":15400},[2126,2163],[54,15402,15404,15407],{"className":15403,"style":15213},[9389],[54,15405],{"className":15406,"style":2156},[2155],[54,15408,15410],{"className":15409,"style":15220},[9409],[5837,15411,15412],{"xmlns":5839,"width":9413,"height":15223,"viewBox":15224,"preserveAspectRatio":15225},[5846,15413],{"d":15228},[54,15415,2781],{"className":15416},[2780],[54,15418,15420],{"className":15419},[2143],[54,15421,15423],{"className":15422,"style":15238},[2147],[54,15424],{},[54,15426,15428,15431],{"style":15427},"top:0.6954em;",[54,15429],{"className":15430,"style":9284},[2155],[54,15432,15434],{"className":15433},[2126],[54,15435,122],{"className":15436},[2479],[54,15438,15440,15443],{"style":15439},"top:3.0029em;",[54,15441],{"className":15442,"style":9284},[2155],[54,15444,15446],{"className":15445},[2126],[54,15447,122],{"className":15448},[2479],[54,15450,2781],{"className":15451},[2780],[54,15453,15455],{"className":15454},[2143],[54,15456,15459],{"className":15457,"style":15458},[2147],"height:7.0103em;",[54,15460],{},[54,15462,15464],{"className":15463},[11944],[54,15465,15467,17145],{"className":15466},[2139,2756],[54,15468,15470,17142],{"className":15469},[2143],[54,15471,15473,15735,15932,16212,16466,16798],{"className":15472,"style":15081},[2147],[54,15474,15475,15478],{"style":15084},[54,15476],{"className":15477,"style":9284},[2155],[54,15479,15481,15484,15590,15593],{"className":15480},[2126],[54,15482],{"className":15483},[2126],[54,15485,15487,15490,15587],{"className":15486},[2126],[54,15488],{"className":15489},[2497,9349],[54,15491,15493],{"className":15492},[9047],[54,15494,15496,15579],{"className":15495},[2139,2756],[54,15497,15499,15576],{"className":15498},[2143],[54,15500,15502,15557,15565],{"className":15501,"style":9362},[2147],[54,15503,15504,15507],{"style":9365},[54,15505],{"className":15506,"style":3866},[2155],[54,15508,15510],{"className":15509},[2126],[54,15511,15513],{"className":15512},[2126,9375],[54,15514,15516,15549],{"className":15515},[2139,2756],[54,15517,15519,15546],{"className":15518},[2143],[54,15520,15522,15534],{"className":15521,"style":9385},[2147],[54,15523,15525,15528],{"className":15524,"style":3862},[9389],[54,15526],{"className":15527,"style":3866},[2155],[54,15529,15531],{"className":15530,"style":9396},[2126],[54,15532,133],{"className":15533},[2126],[54,15535,15536,15539],{"style":9402},[54,15537],{"className":15538,"style":3866},[2155],[54,15540,15542],{"className":15541,"style":9410},[9409],[5837,15543,15544],{"xmlns":5839,"width":9413,"height":9414,"viewBox":9415,"preserveAspectRatio":9416},[5846,15545],{"d":9419},[54,15547,2781],{"className":15548},[2780],[54,15550,15552],{"className":15551},[2143],[54,15553,15555],{"className":15554,"style":9429},[2147],[54,15556],{},[54,15558,15559,15562],{"style":9434},[54,15560],{"className":15561,"style":3866},[2155],[54,15563],{"className":15564,"style":9442},[9441],[54,15566,15567,15570],{"style":9445},[54,15568],{"className":15569,"style":3866},[2155],[54,15571,15573],{"className":15572},[2126],[54,15574,174],{"className":15575},[2126],[54,15577,2781],{"className":15578},[2780],[54,15580,15582],{"className":15581},[2143],[54,15583,15585],{"className":15584,"style":9464},[2147],[54,15586],{},[54,15588],{"className":15589},[2336,9349],[54,15591],{"className":15592,"style":2521},[2270],[54,15594,15596,15602,15617,15620,15661,15664,15667,15670,15685,15688,15729],{"className":15595},[2493],[54,15597,15599],{"className":15598,"style":2499},[2497,2498],[54,15600,129],{"className":15601},[2503,9476],[54,15603,15605,15608,15614],{"className":15604},[2493],[54,15606,2305],{"className":15607},[2126],[54,15609,15611],{"className":15610},[2126],[54,15612,150],{"className":15613},[2126],[54,15615,2312],{"className":15616},[2336],[54,15618],{"className":15619,"style":2521},[2270],[54,15621,15623,15638],{"className":15622},[2493],[54,15624,15626,15629,15635],{"className":15625},[2493],[54,15627,2305],{"className":15628},[2126],[54,15630,15632],{"className":15631},[2126],[54,15633,150],{"className":15634},[2126],[54,15636,2312],{"className":15637},[2336],[54,15639,15641],{"className":15640},[2135],[54,15642,15644],{"className":15643},[2139],[54,15645,15647],{"className":15646},[2143],[54,15648,15650],{"className":15649,"style":7378},[2147],[54,15651,15652,15655],{"style":7401},[54,15653],{"className":15654,"style":2156},[2155],[54,15656,15658],{"className":15657},[2160,2161,2162,2163],[54,15659,7237],{"className":15660,"style":2870},[2126,2167,2163],[54,15662],{"className":15663,"style":2271},[2270],[54,15665,1663],{"className":15666},[2275],[54,15668],{"className":15669,"style":2271},[2270],[54,15671,15673,15676,15682],{"className":15672},[2493],[54,15674,2305],{"className":15675},[2126],[54,15677,15679],{"className":15678},[2126],[54,15680,174],{"className":15681},[2126],[54,15683,2312],{"className":15684},[2336],[54,15686],{"className":15687,"style":2521},[2270],[54,15689,15691,15706],{"className":15690},[2493],[54,15692,15694,15697,15703],{"className":15693},[2493],[54,15695,2305],{"className":15696},[2126],[54,15698,15700],{"className":15699},[2126],[54,15701,150],{"className":15702},[2126],[54,15704,2312],{"className":15705},[2336],[54,15707,15709],{"className":15708},[2135],[54,15710,15712],{"className":15711},[2139],[54,15713,15715],{"className":15714},[2143],[54,15716,15718],{"className":15717,"style":7378},[2147],[54,15719,15720,15723],{"style":7401},[54,15721],{"className":15722,"style":2156},[2155],[54,15724,15726],{"className":15725},[2160,2161,2162,2163],[54,15727,7237],{"className":15728,"style":2870},[2126,2167,2163],[54,15730,15732],{"className":15731,"style":2499},[2336,2498],[54,15733,3205],{"className":15734},[2503,9476],[54,15736,15737,15740],{"style":15243},[54,15738],{"className":15739,"style":9284},[2155],[54,15741,15743,15746,15852,15855],{"className":15742},[2126],[54,15744],{"className":15745},[2126],[54,15747,15749,15752,15849],{"className":15748},[2126],[54,15750],{"className":15751},[2497,9349],[54,15753,15755],{"className":15754},[9047],[54,15756,15758,15841],{"className":15757},[2139,2756],[54,15759,15761,15838],{"className":15760},[2143],[54,15762,15764,15819,15827],{"className":15763,"style":9362},[2147],[54,15765,15766,15769],{"style":9365},[54,15767],{"className":15768,"style":3866},[2155],[54,15770,15772],{"className":15771},[2126],[54,15773,15775],{"className":15774},[2126,9375],[54,15776,15778,15811],{"className":15777},[2139,2756],[54,15779,15781,15808],{"className":15780},[2143],[54,15782,15784,15796],{"className":15783,"style":9385},[2147],[54,15785,15787,15790],{"className":15786,"style":3862},[9389],[54,15788],{"className":15789,"style":3866},[2155],[54,15791,15793],{"className":15792,"style":9396},[2126],[54,15794,133],{"className":15795},[2126],[54,15797,15798,15801],{"style":9402},[54,15799],{"className":15800,"style":3866},[2155],[54,15802,15804],{"className":15803,"style":9410},[9409],[5837,15805,15806],{"xmlns":5839,"width":9413,"height":9414,"viewBox":9415,"preserveAspectRatio":9416},[5846,15807],{"d":9419},[54,15809,2781],{"className":15810},[2780],[54,15812,15814],{"className":15813},[2143],[54,15815,15817],{"className":15816,"style":9429},[2147],[54,15818],{},[54,15820,15821,15824],{"style":9434},[54,15822],{"className":15823,"style":3866},[2155],[54,15825],{"className":15826,"style":9442},[9441],[54,15828,15829,15832],{"style":9445},[54,15830],{"className":15831,"style":3866},[2155],[54,15833,15835],{"className":15834},[2126],[54,15836,174],{"className":15837},[2126],[54,15839,2781],{"className":15840},[2780],[54,15842,15844],{"className":15843},[2143],[54,15845,15847],{"className":15846,"style":9464},[2147],[54,15848],{},[54,15850],{"className":15851},[2336,9349],[54,15853],{"className":15854,"style":2521},[2270],[54,15856,15858,15864,15867,15870,15873,15902,15905,15908,15911,15914,15917,15920,15923,15926],{"className":15857},[2493],[54,15859,15861],{"className":15860,"style":2499},[2497,2498],[54,15862,129],{"className":15863},[2503,2504],[54,15865,9291],{"className":15866},[2126],[54,15868,2312],{"className":15869},[2336],[54,15871,9291],{"className":15872},[2126],[54,15874,15876,15879],{"className":15875},[2336],[54,15877,2312],{"className":15878},[2336],[54,15880,15882],{"className":15881},[2135],[54,15883,15885],{"className":15884},[2139],[54,15886,15888],{"className":15887},[2143],[54,15889,15891],{"className":15890,"style":4325},[2147],[54,15892,15893,15896],{"style":2462},[54,15894],{"className":15895,"style":2156},[2155],[54,15897,15899],{"className":15898},[2160,2161,2162,2163],[54,15900,7237],{"className":15901,"style":2870},[2126,2167,2163],[54,15903],{"className":15904,"style":2271},[2270],[54,15906,1663],{"className":15907},[2275],[54,15909],{"className":15910,"style":2271},[2270],[54,15912,9495],{"className":15913},[2126],[54,15915,2312],{"className":15916},[2336],[54,15918,2305],{"className":15919},[2126],[54,15921,3349],{"className":15922,"style":2332},[2126,2167],[54,15924,2312],{"className":15925},[2336],[54,15927,15929],{"className":15928,"style":2499},[2336,2498],[54,15930,3205],{"className":15931},[2503,2504],[54,15933,15934,15937],{"style":15306},[54,15935],{"className":15936,"style":9284},[2155],[54,15938,15940,15943,16049,16052],{"className":15939},[2126],[54,15941],{"className":15942},[2126],[54,15944,15946,15949,16046],{"className":15945},[2126],[54,15947],{"className":15948},[2497,9349],[54,15950,15952],{"className":15951},[9047],[54,15953,15955,16038],{"className":15954},[2139,2756],[54,15956,15958,16035],{"className":15957},[2143],[54,15959,15961,16016,16024],{"className":15960,"style":9362},[2147],[54,15962,15963,15966],{"style":9365},[54,15964],{"className":15965,"style":3866},[2155],[54,15967,15969],{"className":15968},[2126],[54,15970,15972],{"className":15971},[2126,9375],[54,15973,15975,16008],{"className":15974},[2139,2756],[54,15976,15978,16005],{"className":15977},[2143],[54,15979,15981,15993],{"className":15980,"style":9385},[2147],[54,15982,15984,15987],{"className":15983,"style":3862},[9389],[54,15985],{"className":15986,"style":3866},[2155],[54,15988,15990],{"className":15989,"style":9396},[2126],[54,15991,133],{"className":15992},[2126],[54,15994,15995,15998],{"style":9402},[54,15996],{"className":15997,"style":3866},[2155],[54,15999,16001],{"className":16000,"style":9410},[9409],[5837,16002,16003],{"xmlns":5839,"width":9413,"height":9414,"viewBox":9415,"preserveAspectRatio":9416},[5846,16004],{"d":9419},[54,16006,2781],{"className":16007},[2780],[54,16009,16011],{"className":16010},[2143],[54,16012,16014],{"className":16013,"style":9429},[2147],[54,16015],{},[54,16017,16018,16021],{"style":9434},[54,16019],{"className":16020,"style":3866},[2155],[54,16022],{"className":16023,"style":9442},[9441],[54,16025,16026,16029],{"style":9445},[54,16027],{"className":16028,"style":3866},[2155],[54,16030,16032],{"className":16031},[2126],[54,16033,174],{"className":16034},[2126],[54,16036,2781],{"className":16037},[2780],[54,16039,16041],{"className":16040},[2143],[54,16042,16044],{"className":16043,"style":9464},[2147],[54,16045],{},[54,16047],{"className":16048},[2336,9349],[54,16050],{"className":16051,"style":2521},[2270],[54,16053,16055,16061,16096,16099,16114,16117,16158,16161,16164,16167,16182,16185,16188,16191,16206],{"className":16054},[2493],[54,16056,16058],{"className":16057,"style":2499},[2497,2498],[54,16059,129],{"className":16060},[2503,9476],[54,16062,16064,16067],{"className":16063},[2126],[54,16065,2842],{"className":16066},[2126,2167],[54,16068,16070],{"className":16069},[2135],[54,16071,16073],{"className":16072},[2139],[54,16074,16076],{"className":16075},[2143],[54,16077,16079],{"className":16078,"style":4596},[2147],[54,16080,16081,16084],{"style":2462},[54,16082],{"className":16083,"style":2156},[2155],[54,16085,16087],{"className":16086},[2160,2161,2162,2163],[54,16088,16090,16093],{"className":16089},[2126,2163],[54,16091,2849],{"className":16092},[2126,2167,2163],[54,16094,14672],{"className":16095},[2126,2167,2163],[54,16097],{"className":16098,"style":2521},[2270],[54,16100,16102,16105,16111],{"className":16101},[2493],[54,16103,2305],{"className":16104},[2126],[54,16106,16108],{"className":16107},[2126],[54,16109,150],{"className":16110},[2126],[54,16112,2312],{"className":16113},[2336],[54,16115],{"className":16116,"style":2521},[2270],[54,16118,16120,16135],{"className":16119},[2493],[54,16121,16123,16126,16132],{"className":16122},[2493],[54,16124,2305],{"className":16125},[2126],[54,16127,16129],{"className":16128},[2126],[54,16130,150],{"className":16131},[2126],[54,16133,2312],{"className":16134},[2336],[54,16136,16138],{"className":16137},[2135],[54,16139,16141],{"className":16140},[2139],[54,16142,16144],{"className":16143},[2143],[54,16145,16147],{"className":16146,"style":7378},[2147],[54,16148,16149,16152],{"style":7401},[54,16150],{"className":16151,"style":2156},[2155],[54,16153,16155],{"className":16154},[2160,2161,2162,2163],[54,16156,7237],{"className":16157,"style":2870},[2126,2167,2163],[54,16159],{"className":16160,"style":2271},[2270],[54,16162,1663],{"className":16163},[2275],[54,16165],{"className":16166,"style":2271},[2270],[54,16168,16170,16173,16179],{"className":16169},[2493],[54,16171,2305],{"className":16172},[2126],[54,16174,16176],{"className":16175},[2126],[54,16177,174],{"className":16178},[2126],[54,16180,2312],{"className":16181},[2336],[54,16183],{"className":16184,"style":2521},[2270],[54,16186,2835],{"className":16187,"style":2870},[2126,2167],[54,16189],{"className":16190,"style":2521},[2270],[54,16192,16194,16197,16203],{"className":16193},[2493],[54,16195,2305],{"className":16196},[2126],[54,16198,16200],{"className":16199},[2126],[54,16201,3349],{"className":16202,"style":2332},[2126,2167],[54,16204,2312],{"className":16205},[2336],[54,16207,16209],{"className":16208,"style":2499},[2336,2498],[54,16210,3205],{"className":16211},[2503,9476],[54,16213,16214,16217],{"style":15365},[54,16215],{"className":16216,"style":9284},[2155],[54,16218,16220,16223,16329,16332],{"className":16219},[2126],[54,16221],{"className":16222},[2126],[54,16224,16226,16229,16326],{"className":16225},[2126],[54,16227],{"className":16228},[2497,9349],[54,16230,16232],{"className":16231},[9047],[54,16233,16235,16318],{"className":16234},[2139,2756],[54,16236,16238,16315],{"className":16237},[2143],[54,16239,16241,16296,16304],{"className":16240,"style":9362},[2147],[54,16242,16243,16246],{"style":9365},[54,16244],{"className":16245,"style":3866},[2155],[54,16247,16249],{"className":16248},[2126],[54,16250,16252],{"className":16251},[2126,9375],[54,16253,16255,16288],{"className":16254},[2139,2756],[54,16256,16258,16285],{"className":16257},[2143],[54,16259,16261,16273],{"className":16260,"style":9385},[2147],[54,16262,16264,16267],{"className":16263,"style":3862},[9389],[54,16265],{"className":16266,"style":3866},[2155],[54,16268,16270],{"className":16269,"style":9396},[2126],[54,16271,133],{"className":16272},[2126],[54,16274,16275,16278],{"style":9402},[54,16276],{"className":16277,"style":3866},[2155],[54,16279,16281],{"className":16280,"style":9410},[9409],[5837,16282,16283],{"xmlns":5839,"width":9413,"height":9414,"viewBox":9415,"preserveAspectRatio":9416},[5846,16284],{"d":9419},[54,16286,2781],{"className":16287},[2780],[54,16289,16291],{"className":16290},[2143],[54,16292,16294],{"className":16293,"style":9429},[2147],[54,16295],{},[54,16297,16298,16301],{"style":9434},[54,16299],{"className":16300,"style":3866},[2155],[54,16302],{"className":16303,"style":9442},[9441],[54,16305,16306,16309],{"style":9445},[54,16307],{"className":16308,"style":3866},[2155],[54,16310,16312],{"className":16311},[2126],[54,16313,174],{"className":16314},[2126],[54,16316,2781],{"className":16317},[2780],[54,16319,16321],{"className":16320},[2143],[54,16322,16324],{"className":16323,"style":9464},[2147],[54,16325],{},[54,16327],{"className":16328},[2336,9349],[54,16330],{"className":16331,"style":2521},[2270],[54,16333,16335,16341,16376,16379,16394,16397,16412,16415,16418,16421,16436,16439,16442,16445,16460],{"className":16334},[2493],[54,16336,16338],{"className":16337,"style":2499},[2497,2498],[54,16339,129],{"className":16340},[2503,2504],[54,16342,16344,16347],{"className":16343},[2126],[54,16345,2842],{"className":16346},[2126,2167],[54,16348,16350],{"className":16349},[2135],[54,16351,16353],{"className":16352},[2139],[54,16354,16356],{"className":16355},[2143],[54,16357,16359],{"className":16358,"style":4596},[2147],[54,16360,16361,16364],{"style":2462},[54,16362],{"className":16363,"style":2156},[2155],[54,16365,16367],{"className":16366},[2160,2161,2162,2163],[54,16368,16370,16373],{"className":16369},[2126,2163],[54,16371,2849],{"className":16372},[2126,2167,2163],[54,16374,14672],{"className":16375},[2126,2167,2163],[54,16377],{"className":16378,"style":2521},[2270],[54,16380,16382,16385,16391],{"className":16381},[2493],[54,16383,2305],{"className":16384},[2126],[54,16386,16388],{"className":16387},[2126],[54,16389,150],{"className":16390},[2126],[54,16392,2312],{"className":16393},[2336],[54,16395],{"className":16396,"style":2521},[2270],[54,16398,16400,16403,16409],{"className":16399},[2493],[54,16401,2305],{"className":16402},[2126],[54,16404,16406],{"className":16405},[2126],[54,16407,3349],{"className":16408,"style":2332},[2126,2167],[54,16410,2312],{"className":16411},[2336],[54,16413],{"className":16414,"style":2271},[2270],[54,16416,1663],{"className":16417},[2275],[54,16419],{"className":16420,"style":2271},[2270],[54,16422,16424,16427,16433],{"className":16423},[2493],[54,16425,2305],{"className":16426},[2126],[54,16428,16430],{"className":16429},[2126],[54,16431,174],{"className":16432},[2126],[54,16434,2312],{"className":16435},[2336],[54,16437],{"className":16438,"style":2521},[2270],[54,16440,2835],{"className":16441,"style":2870},[2126,2167],[54,16443],{"className":16444,"style":2521},[2270],[54,16446,16448,16451,16457],{"className":16447},[2493],[54,16449,2305],{"className":16450},[2126],[54,16452,16454],{"className":16453},[2126],[54,16455,3349],{"className":16456,"style":2332},[2126,2167],[54,16458,2312],{"className":16459},[2336],[54,16461,16463],{"className":16462,"style":2499},[2336,2498],[54,16464,3205],{"className":16465},[2503,2504],[54,16467,16468,16471],{"style":15427},[54,16469],{"className":16470,"style":9284},[2155],[54,16472,16474,16477,16539,16542],{"className":16473},[2126],[54,16475],{"className":16476},[2126],[54,16478,16480,16483,16536],{"className":16479},[2126],[54,16481],{"className":16482},[2497,9349],[54,16484,16486],{"className":16485},[9047],[54,16487,16489,16528],{"className":16488},[2139,2756],[54,16490,16492,16525],{"className":16491},[2143],[54,16493,16495,16506,16514],{"className":16494,"style":9362},[2147],[54,16496,16497,16500],{"style":11273},[54,16498],{"className":16499,"style":3866},[2155],[54,16501,16503],{"className":16502},[2126],[54,16504,133],{"className":16505},[2126],[54,16507,16508,16511],{"style":9434},[54,16509],{"className":16510,"style":3866},[2155],[54,16512],{"className":16513,"style":9442},[9441],[54,16515,16516,16519],{"style":9445},[54,16517],{"className":16518,"style":3866},[2155],[54,16520,16522],{"className":16521},[2126],[54,16523,174],{"className":16524},[2126],[54,16526,2781],{"className":16527},[2780],[54,16529,16531],{"className":16530},[2143],[54,16532,16534],{"className":16533,"style":11311},[2147],[54,16535],{},[54,16537],{"className":16538},[2336,9349],[54,16540],{"className":16541,"style":2521},[2270],[54,16543,16545,16551,16566,16569,16667,16670,16673,16676,16691,16694,16792],{"className":16544},[2493],[54,16546,16548],{"className":16547,"style":2499},[2497,2498],[54,16549,129],{"className":16550},[2503,2504],[54,16552,16554,16557,16563],{"className":16553},[2493],[54,16555,2305],{"className":16556},[2126],[54,16558,16560],{"className":16559},[2126],[54,16561,1663],{"className":16562},[2126],[54,16564,2312],{"className":16565},[2336],[54,16567],{"className":16568,"style":2521},[2270],[54,16570,16572,16578,16613,16616,16631,16634,16637,16640,16643,16646,16661],{"className":16571},[2493],[54,16573,16575],{"className":16574,"style":2499},[2497,2498],[54,16576,129],{"className":16577},[2503,2504],[54,16579,16581,16584],{"className":16580},[2126],[54,16582,2842],{"className":16583},[2126,2167],[54,16585,16587],{"className":16586},[2135],[54,16588,16590],{"className":16589},[2139],[54,16591,16593],{"className":16592},[2143],[54,16594,16596],{"className":16595,"style":4596},[2147],[54,16597,16598,16601],{"style":2462},[54,16599],{"className":16600,"style":2156},[2155],[54,16602,16604],{"className":16603},[2160,2161,2162,2163],[54,16605,16607,16610],{"className":16606},[2126,2163],[54,16608,2849],{"className":16609},[2126,2167,2163],[54,16611,14672],{"className":16612},[2126,2167,2163],[54,16614],{"className":16615,"style":2521},[2270],[54,16617,16619,16622,16628],{"className":16618},[2493],[54,16620,2305],{"className":16621},[2126],[54,16623,16625],{"className":16624},[2126],[54,16626,3349],{"className":16627,"style":2332},[2126,2167],[54,16629,2312],{"className":16630},[2336],[54,16632],{"className":16633,"style":2271},[2270],[54,16635,1663],{"className":16636},[2275],[54,16638],{"className":16639,"style":2271},[2270],[54,16641,2835],{"className":16642,"style":2870},[2126,2167],[54,16644],{"className":16645,"style":2521},[2270],[54,16647,16649,16652,16658],{"className":16648},[2493],[54,16650,2305],{"className":16651},[2126],[54,16653,16655],{"className":16654},[2126],[54,16656,3349],{"className":16657,"style":2332},[2126,2167],[54,16659,2312],{"className":16660},[2336],[54,16662,16664],{"className":16663,"style":2499},[2336,2498],[54,16665,3205],{"className":16666},[2503,2504],[54,16668],{"className":16669,"style":2271},[2270],[54,16671,1663],{"className":16672},[2275],[54,16674],{"className":16675,"style":2271},[2270],[54,16677,16679,16682,16688],{"className":16678},[2493],[54,16680,2305],{"className":16681},[2126],[54,16683,16685],{"className":16684},[2126],[54,16686,2247],{"className":16687},[2126],[54,16689,2312],{"className":16690},[2336],[54,16692],{"className":16693,"style":2521},[2270],[54,16695,16697,16703,16738,16741,16756,16759,16762,16765,16768,16771,16786],{"className":16696},[2493],[54,16698,16700],{"className":16699,"style":2499},[2497,2498],[54,16701,129],{"className":16702},[2503,2504],[54,16704,16706,16709],{"className":16705},[2126],[54,16707,2842],{"className":16708},[2126,2167],[54,16710,16712],{"className":16711},[2135],[54,16713,16715],{"className":16714},[2139],[54,16716,16718],{"className":16717},[2143],[54,16719,16721],{"className":16720,"style":4596},[2147],[54,16722,16723,16726],{"style":2462},[54,16724],{"className":16725,"style":2156},[2155],[54,16727,16729],{"className":16728},[2160,2161,2162,2163],[54,16730,16732,16735],{"className":16731},[2126,2163],[54,16733,2849],{"className":16734},[2126,2167,2163],[54,16736,14672],{"className":16737},[2126,2167,2163],[54,16739],{"className":16740,"style":2521},[2270],[54,16742,16744,16747,16753],{"className":16743},[2493],[54,16745,2305],{"className":16746},[2126],[54,16748,16750],{"className":16749},[2126],[54,16751,3349],{"className":16752,"style":2332},[2126,2167],[54,16754,2312],{"className":16755},[2336],[54,16757],{"className":16758,"style":2271},[2270],[54,16760,2247],{"className":16761},[2275],[54,16763],{"className":16764,"style":2271},[2270],[54,16766,2835],{"className":16767,"style":2870},[2126,2167],[54,16769],{"className":16770,"style":2521},[2270],[54,16772,16774,16777,16783],{"className":16773},[2493],[54,16775,2305],{"className":16776},[2126],[54,16778,16780],{"className":16779},[2126],[54,16781,3349],{"className":16782,"style":2332},[2126,2167],[54,16784,2312],{"className":16785},[2336],[54,16787,16789],{"className":16788,"style":2499},[2336,2498],[54,16790,3205],{"className":16791},[2503,2504],[54,16793,16795],{"className":16794,"style":2499},[2336,2498],[54,16796,3205],{"className":16797},[2503,2504],[54,16799,16800,16803],{"style":15439},[54,16801],{"className":16802,"style":9284},[2155],[54,16804,16806,16809,16871,16874],{"className":16805},[2126],[54,16807],{"className":16808},[2126],[54,16810,16812,16815,16868],{"className":16811},[2126],[54,16813],{"className":16814},[2497,9349],[54,16816,16818],{"className":16817},[9047],[54,16819,16821,16860],{"className":16820},[2139,2756],[54,16822,16824,16857],{"className":16823},[2143],[54,16825,16827,16838,16846],{"className":16826,"style":9362},[2147],[54,16828,16829,16832],{"style":11273},[54,16830],{"className":16831,"style":3866},[2155],[54,16833,16835],{"className":16834},[2126],[54,16836,133],{"className":16837},[2126],[54,16839,16840,16843],{"style":9434},[54,16841],{"className":16842,"style":3866},[2155],[54,16844],{"className":16845,"style":9442},[9441],[54,16847,16848,16851],{"style":9445},[54,16849],{"className":16850,"style":3866},[2155],[54,16852,16854],{"className":16853},[2126],[54,16855,174],{"className":16856},[2126],[54,16858,2781],{"className":16859},[2780],[54,16861,16863],{"className":16862},[2143],[54,16864,16866],{"className":16865,"style":11311},[2147],[54,16867],{},[54,16869],{"className":16870},[2336,9349],[54,16872],{"className":16873,"style":2521},[2270],[54,16875,16877,16883,16901,16904,17005,17008,17011,17014,17032,17035,17136],{"className":16876},[2493],[54,16878,16880],{"className":16879,"style":2499},[2497,2498],[54,16881,129],{"className":16882},[2503,2504],[54,16884,16886,16889,16898],{"className":16885},[2493],[54,16887,2305],{"className":16888},[2126],[54,16890,16892,16895],{"className":16891},[2126],[54,16893,1663],{"className":16894},[2126],[54,16896,2849],{"className":16897},[2126,2167],[54,16899,2312],{"className":16900},[2336],[54,16902],{"className":16903,"style":2521},[2270],[54,16905,16907,16913,16948,16951,16966,16969,16972,16975,16978,16981,16984,16999],{"className":16906},[2493],[54,16908,16910],{"className":16909,"style":2499},[2497,2498],[54,16911,129],{"className":16912},[2503,2504],[54,16914,16916,16919],{"className":16915},[2126],[54,16917,2842],{"className":16918},[2126,2167],[54,16920,16922],{"className":16921},[2135],[54,16923,16925],{"className":16924},[2139],[54,16926,16928],{"className":16927},[2143],[54,16929,16931],{"className":16930,"style":4596},[2147],[54,16932,16933,16936],{"style":2462},[54,16934],{"className":16935,"style":2156},[2155],[54,16937,16939],{"className":16938},[2160,2161,2162,2163],[54,16940,16942,16945],{"className":16941},[2126,2163],[54,16943,2849],{"className":16944},[2126,2167,2163],[54,16946,14672],{"className":16947},[2126,2167,2163],[54,16949],{"className":16950,"style":2521},[2270],[54,16952,16954,16957,16963],{"className":16953},[2493],[54,16955,2305],{"className":16956},[2126],[54,16958,16960],{"className":16959},[2126],[54,16961,3349],{"className":16962,"style":2332},[2126,2167],[54,16964,2312],{"className":16965},[2336],[54,16967],{"className":16968,"style":2271},[2270],[54,16970,2247],{"className":16971},[2275],[54,16973],{"className":16974,"style":2271},[2270],[54,16976,2849],{"className":16977},[2126,2167],[54,16979,2835],{"className":16980,"style":2870},[2126,2167],[54,16982],{"className":16983,"style":2521},[2270],[54,16985,16987,16990,16996],{"className":16986},[2493],[54,16988,2305],{"className":16989},[2126],[54,16991,16993],{"className":16992},[2126],[54,16994,3349],{"className":16995,"style":2332},[2126,2167],[54,16997,2312],{"className":16998},[2336],[54,17000,17002],{"className":17001,"style":2499},[2336,2498],[54,17003,3205],{"className":17004},[2503,2504],[54,17006],{"className":17007,"style":2271},[2270],[54,17009,1663],{"className":17010},[2275],[54,17012],{"className":17013,"style":2271},[2270],[54,17015,17017,17020,17029],{"className":17016},[2493],[54,17018,2305],{"className":17019},[2126],[54,17021,17023,17026],{"className":17022},[2126],[54,17024,2247],{"className":17025},[2126],[54,17027,2849],{"className":17028},[2126,2167],[54,17030,2312],{"className":17031},[2336],[54,17033],{"className":17034,"style":2521},[2270],[54,17036,17038,17044,17079,17082,17097,17100,17103,17106,17109,17112,17115,17130],{"className":17037},[2493],[54,17039,17041],{"className":17040,"style":2499},[2497,2498],[54,17042,129],{"className":17043},[2503,2504],[54,17045,17047,17050],{"className":17046},[2126],[54,17048,2842],{"className":17049},[2126,2167],[54,17051,17053],{"className":17052},[2135],[54,17054,17056],{"className":17055},[2139],[54,17057,17059],{"className":17058},[2143],[54,17060,17062],{"className":17061,"style":4596},[2147],[54,17063,17064,17067],{"style":2462},[54,17065],{"className":17066,"style":2156},[2155],[54,17068,17070],{"className":17069},[2160,2161,2162,2163],[54,17071,17073,17076],{"className":17072},[2126,2163],[54,17074,2849],{"className":17075},[2126,2167,2163],[54,17077,14672],{"className":17078},[2126,2167,2163],[54,17080],{"className":17081,"style":2521},[2270],[54,17083,17085,17088,17094],{"className":17084},[2493],[54,17086,2305],{"className":17087},[2126],[54,17089,17091],{"className":17090},[2126],[54,17092,3349],{"className":17093,"style":2332},[2126,2167],[54,17095,2312],{"className":17096},[2336],[54,17098],{"className":17099,"style":2271},[2270],[54,17101,1663],{"className":17102},[2275],[54,17104],{"className":17105,"style":2271},[2270],[54,17107,2849],{"className":17108},[2126,2167],[54,17110,2835],{"className":17111,"style":2870},[2126,2167],[54,17113],{"className":17114,"style":2521},[2270],[54,17116,17118,17121,17127],{"className":17117},[2493],[54,17119,2305],{"className":17120},[2126],[54,17122,17124],{"className":17123},[2126],[54,17125,3349],{"className":17126,"style":2332},[2126,2167],[54,17128,2312],{"className":17129},[2336],[54,17131,17133],{"className":17132,"style":2499},[2336,2498],[54,17134,3205],{"className":17135},[2503,2504],[54,17137,17139],{"className":17138,"style":2499},[2336,2498],[54,17140,3205],{"className":17141},[2503,2504],[54,17143,2781],{"className":17144},[2780],[54,17146,17148],{"className":17147},[2143],[54,17149,17151],{"className":17150,"style":15458},[2147],[54,17152],{},[54,17154,2781],{"className":17155},[2780],[54,17157,17159],{"className":17158},[2143],[54,17160,17162],{"className":17161,"style":15458},[2147],[54,17163],{},[54,17165,17168],{"className":17166},[17167],"tag",[54,17169,17171,17189],{"className":17170},[2139,2756],[54,17172,17174,17186],{"className":17173},[2143],[54,17175,17177],{"className":17176,"style":15081},[2147],[54,17178,17179,17182],{"style":15084},[54,17180],{"className":17181,"style":15088},[2155],[54,17183],{"className":17184},[17185],"eqn-num",[54,17187,2781],{"className":17188},[2780],[54,17190,17192],{"className":17191},[2143],[54,17193,17195],{"className":17194,"style":15458},[2147],[54,17196],{},[13,17198,17199,17200,17404],{},"where we have used the classical simulable phase shift ",[54,17201,17203,17253],{"className":17202},[2079],[54,17204,17206],{"className":17205},[2083],[2085,17207,17208],{"xmlns":2087},[2089,17209,17210,17250],{},[2092,17211,17212,17214,17226,17228,17238],{},[2098,17213,2835],{},[2095,17215,17216,17224],{},[14079,17217,17218,17220,17222],{},[2098,17219,2305],{"mathvariant":2304},[2249,17221,150],{},[2245,17223,2312],{"stretchy":2311},[2098,17225,7237],{},[2245,17227,122],{},[2095,17229,17230,17232],{},[2098,17231,2842],{},[2092,17233,17234,17236],{},[2098,17235,2849],{},[2098,17237,14672],{},[2095,17239,17240,17248],{},[14079,17241,17242,17244,17246],{},[2098,17243,2305],{"mathvariant":2304},[2249,17245,150],{},[2245,17247,2312],{"stretchy":2311},[2098,17249,7237],{},[2105,17251,17252],{"encoding":2107}," U\\ket{0}^N = e^{i\\phi}\\ket{0}^N",[54,17254,17256,17319],{"className":17255,"ariaHidden":2113},[2112],[54,17257,17259,17263,17266,17269,17310,17313,17316],{"className":17258},[2117],[54,17260],{"className":17261,"style":17262},[2121],"height:1.2312em;vertical-align:-0.25em;",[54,17264,2835],{"className":17265,"style":2870},[2126,2167],[54,17267],{"className":17268,"style":2521},[2270],[54,17270,17272,17287],{"className":17271},[2493],[54,17273,17275,17278,17284],{"className":17274},[2493],[54,17276,2305],{"className":17277},[2126],[54,17279,17281],{"className":17280},[2126],[54,17282,150],{"className":17283},[2126],[54,17285,2312],{"className":17286},[2336],[54,17288,17290],{"className":17289},[2135],[54,17291,17293],{"className":17292},[2139],[54,17294,17296],{"className":17295},[2143],[54,17297,17299],{"className":17298,"style":7378},[2147],[54,17300,17301,17304],{"style":7401},[54,17302],{"className":17303,"style":2156},[2155],[54,17305,17307],{"className":17306},[2160,2161,2162,2163],[54,17308,7237],{"className":17309,"style":2870},[2126,2167,2163],[54,17311],{"className":17312,"style":2475},[2270],[54,17314,122],{"className":17315},[2479],[54,17317],{"className":17318,"style":2475},[2270],[54,17320,17322,17325,17360,17363],{"className":17321},[2117],[54,17323],{"className":17324,"style":17262},[2121],[54,17326,17328,17331],{"className":17327},[2126],[54,17329,2842],{"className":17330},[2126,2167],[54,17332,17334],{"className":17333},[2135],[54,17335,17337],{"className":17336},[2139],[54,17338,17340],{"className":17339},[2143],[54,17341,17343],{"className":17342,"style":3218},[2147],[54,17344,17345,17348],{"style":2151},[54,17346],{"className":17347,"style":2156},[2155],[54,17349,17351],{"className":17350},[2160,2161,2162,2163],[54,17352,17354,17357],{"className":17353},[2126,2163],[54,17355,2849],{"className":17356},[2126,2167,2163],[54,17358,14672],{"className":17359},[2126,2167,2163],[54,17361],{"className":17362,"style":2521},[2270],[54,17364,17366,17381],{"className":17365},[2493],[54,17367,17369,17372,17378],{"className":17368},[2493],[54,17370,2305],{"className":17371},[2126],[54,17373,17375],{"className":17374},[2126],[54,17376,150],{"className":17377},[2126],[54,17379,2312],{"className":17380},[2336],[54,17382,17384],{"className":17383},[2135],[54,17385,17387],{"className":17386},[2139],[54,17388,17390],{"className":17389},[2143],[54,17391,17393],{"className":17392,"style":7378},[2147],[54,17394,17395,17398],{"style":7401},[54,17396],{"className":17397,"style":2156},[2155],[54,17399,17401],{"className":17400},[2160,2161,2162,2163],[54,17402,7237],{"className":17403,"style":2870},[2126,2167,2163]," in the third line. Therefore the expectation values are obtained as",[54,17406,17408],{"className":17407},[2340],[54,17409,17411,17721],{"className":17410},[2079],[54,17412,17414],{"className":17413},[2083],[2085,17415,17416],{"xmlns":2087,"display":2349},[2089,17417,17418,17718],{},[9008,17419,17420],{"rowspacing":9010,"columnspacing":9011},[9013,17421,17422,17425,17712,17715],{},[9016,17423],{"className":17424},[14450],[9016,17426,17427],{},[9019,17428,17429],{"scriptlevel":150,"displaystyle":2113},[9008,17430,17431,17549,17652],{"rowspacing":11670,"columnalign":11671,"columnspacing":11672},[9013,17432,17433,17450],{},[9016,17434,17435],{},[9019,17436,17437],{"scriptlevel":150,"displaystyle":2113},[2092,17438,17439,17441,17443,17446,17448],{},[2245,17440,3981],{"stretchy":2311},[2098,17442,7287],{},[2245,17444,17445],{},"⊗",[2098,17447,2730],{},[2245,17449,2312],{"stretchy":2311},[9016,17451,17452],{},[9019,17453,17454],{"scriptlevel":150,"displaystyle":2113},[2092,17455,17456,17458,17460,17466,17468,17511,17513],{},[2092,17457],{},[2245,17459,122],{},[9047,17461,17462,17464],{},[2249,17463,174],{},[2249,17465,11708],{},[2245,17467,129],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2092,17469,17470,17472,17484,17492,17494,17502,17509],{},[2245,17471,129],{"fence":2113},[2095,17473,17474,17476],{},[2098,17475,2842],{},[2092,17477,17478,17480,17482],{},[2245,17479,2247],{},[2098,17481,2849],{},[2098,17483,14672],{},[14079,17485,17486,17488,17490],{},[2245,17487,3981],{"stretchy":2311},[2098,17489,3349],{},[2098,17491,2305],{"mathvariant":2304},[2245,17493,1663],{},[14079,17495,17496,17498,17500],{},[2245,17497,3981],{"stretchy":2311},[2098,17499,3349],{},[2098,17501,2305],{"mathvariant":2304},[2095,17503,17504,17506],{},[2098,17505,2835],{},[2245,17507,17508],{},"†",[2245,17510,3205],{"fence":2113},[2098,17512,2730],{},[2092,17514,17515,17517,17527,17535,17537,17539,17547],{},[2245,17516,129],{"fence":2113},[2095,17518,17519,17521],{},[2098,17520,2842],{},[2092,17522,17523,17525],{},[2098,17524,2849],{},[2098,17526,14672],{},[14079,17528,17529,17531,17533],{},[2098,17530,2305],{"mathvariant":2304},[2098,17532,3349],{},[2245,17534,2312],{"stretchy":2311},[2245,17536,1663],{},[2098,17538,2835],{},[14079,17540,17541,17543,17545],{},[2098,17542,2305],{"mathvariant":2304},[2098,17544,3349],{},[2245,17546,2312],{"stretchy":2311},[2245,17548,3205],{"fence":2113},[9013,17550,17551,17557],{},[9016,17552,17553],{},[9019,17554,17555],{"scriptlevel":150,"displaystyle":2113},[2092,17556],{},[9016,17558,17559],{},[9019,17560,17561],{"scriptlevel":150,"displaystyle":2113},[2092,17562,17563,17565,17568,17570,17612,17614,17650],{},[2092,17564],{},[2270,17566],{"width":17567},"2em",[2245,17569,2247],{},[2092,17571,17572,17574,17586,17594,17596,17604,17610],{},[2245,17573,129],{"fence":2113},[2095,17575,17576,17578],{},[2098,17577,2842],{},[2092,17579,17580,17582,17584],{},[2245,17581,2247],{},[2098,17583,2849],{},[2098,17585,14672],{},[14079,17587,17588,17590,17592],{},[2245,17589,3981],{"stretchy":2311},[2098,17591,3349],{},[2098,17593,2305],{"mathvariant":2304},[2245,17595,2247],{},[14079,17597,17598,17600,17602],{},[2245,17599,3981],{"stretchy":2311},[2098,17601,3349],{},[2098,17603,2305],{"mathvariant":2304},[2095,17605,17606,17608],{},[2098,17607,2835],{},[2245,17609,17508],{},[2245,17611,3205],{"fence":2113},[2098,17613,2730],{},[2092,17615,17616,17618,17628,17636,17638,17640,17648],{},[2245,17617,129],{"fence":2113},[2095,17619,17620,17622],{},[2098,17621,2842],{},[2092,17623,17624,17626],{},[2098,17625,2849],{},[2098,17627,14672],{},[14079,17629,17630,17632,17634],{},[2098,17631,2305],{"mathvariant":2304},[2098,17633,3349],{},[2245,17635,2312],{"stretchy":2311},[2245,17637,2247],{},[2098,17639,2835],{},[14079,17641,17642,17644,17646],{},[2098,17643,2305],{"mathvariant":2304},[2098,17645,3349],{},[2245,17647,2312],{"stretchy":2311},[2245,17649,3205],{"fence":2113},[2245,17651,3205],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[9013,17653,17654,17660],{},[9016,17655,17656],{},[9019,17657,17658],{"scriptlevel":150,"displaystyle":2113},[2092,17659],{},[9016,17661,17662],{},[9019,17663,17664],{"scriptlevel":150,"displaystyle":2113},[2092,17665,17666,17668,17670,17672,17710],{},[2092,17667],{},[2245,17669,122],{},[5723,17671,11801],{},[2092,17673,17674,17676,17688,17696,17698,17700,17708],{},[2245,17675,11804],{"fence":2113},[2095,17677,17678,17680],{},[2098,17679,2842],{},[2092,17681,17682,17684,17686],{},[2245,17683,2247],{},[2098,17685,2849],{},[2098,17687,14672],{},[14079,17689,17690,17692,17694],{},[2245,17691,3981],{"stretchy":2311},[2098,17693,3349],{},[2098,17695,2305],{"mathvariant":2304},[2098,17697,2730],{},[2098,17699,2835],{},[14079,17701,17702,17704,17706],{},[2098,17703,2305],{"mathvariant":2304},[2098,17705,3349],{},[2245,17707,2312],{"stretchy":2311},[2245,17709,11829],{"fence":2113},[2245,17711,2375],{"separator":2113},[9016,17713],{"className":17714},[14450],[9016,17716],{"className":17717},[15052],[2105,17719,17720],{"encoding":2107},"\\begin{equation}\n\\begin{split}\n \\langle X\\otimes P\\rangle&=\\frac{1}{4}\n \\Big(\n \\left(e^{-i\\phi}\\bra{\\psi}+\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}+U\\ket{\\psi}\\right)\n \\\\\n &\\qquad-\\left(e^{-i\\phi}\\bra{\\psi}-\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}-U\\ket{\\psi}\\right)\n \\Big)\\\\\n &=\\text{Re}\\left[e^{-i\\phi}\\bra{\\psi}PU\\ket{\\psi}\\right],\n\\end{split}\n\\end{equation}",[54,17722,17724,18593],{"className":17723,"ariaHidden":2113},[2112],[54,17725,17727,17731],{"className":17726},[2117],[54,17728],{"className":17729,"style":17730},[2121],"height:5.6666em;vertical-align:-2.5833em;",[54,17732,17734],{"className":17733},[9008],[54,17735,17737],{"className":17736},[9267],[54,17738,17740,18585],{"className":17739},[2139,2756],[54,17741,17743,18582],{"className":17742},[2143],[54,17744,17747],{"className":17745,"style":17746},[2147],"height:3.0833em;",[54,17748,17750,17754],{"style":17749},"top:-5.0833em;",[54,17751],{"className":17752,"style":17753},[2155],"height:5.0833em;",[54,17755,17757],{"className":17756},[2126],[54,17758,17760],{"className":17759},[2126],[54,17761,17763,17834],{"className":17762},[9008],[54,17764,17766],{"className":17765},[11883],[54,17767,17769,17825],{"className":17768},[2139,2756],[54,17770,17772,17822],{"className":17771},[2143],[54,17773,17775,17804,17813],{"className":17774,"style":17746},[2147],[54,17776,17777,17780],{"style":17749},[54,17778],{"className":17779,"style":9284},[2155],[54,17781,17783,17786,17789,17792,17795,17798,17801],{"className":17782},[2126],[54,17784,3981],{"className":17785},[2497],[54,17787,7287],{"className":17788,"style":7432},[2126,2167],[54,17790],{"className":17791,"style":2271},[2270],[54,17793,17445],{"className":17794},[2275],[54,17796],{"className":17797,"style":2271},[2270],[54,17799,2730],{"className":17800,"style":2776},[2126,2167],[54,17802,2312],{"className":17803},[2336],[54,17805,17807,17810],{"style":17806},"top:-2.9473em;",[54,17808],{"className":17809,"style":9284},[2155],[54,17811],{"className":17812},[2126],[54,17814,17816,17819],{"style":17815},"top:-1.0982em;",[54,17817],{"className":17818,"style":9284},[2155],[54,17820],{"className":17821},[2126],[54,17823,2781],{"className":17824},[2780],[54,17826,17828],{"className":17827},[2143],[54,17829,17832],{"className":17830,"style":17831},[2147],"height:2.5833em;",[54,17833],{},[54,17835,17837],{"className":17836},[11944],[54,17838,17840,18574],{"className":17839},[2139,2756],[54,17841,17843,18571],{"className":17842},[2143],[54,17844,17846,18171,18438],{"className":17845,"style":17746},[2147],[54,17847,17848,17851],{"style":17749},[54,17849],{"className":17850,"style":9284},[2155],[54,17852,17854,17857,17860,17863,17866,17928,17934,17937,18064,18067,18070,18073],{"className":17853},[2126],[54,17855],{"className":17856},[2126],[54,17858],{"className":17859,"style":2475},[2270],[54,17861,122],{"className":17862},[2479],[54,17864],{"className":17865,"style":2475},[2270],[54,17867,17869,17872,17925],{"className":17868},[2126],[54,17870],{"className":17871},[2497,9349],[54,17873,17875],{"className":17874},[9047],[54,17876,17878,17917],{"className":17877},[2139,2756],[54,17879,17881,17914],{"className":17880},[2143],[54,17882,17884,17895,17903],{"className":17883,"style":9362},[2147],[54,17885,17886,17889],{"style":11273},[54,17887],{"className":17888,"style":3866},[2155],[54,17890,17892],{"className":17891},[2126],[54,17893,11708],{"className":17894},[2126],[54,17896,17897,17900],{"style":9434},[54,17898],{"className":17899,"style":3866},[2155],[54,17901],{"className":17902,"style":9442},[9441],[54,17904,17905,17908],{"style":9445},[54,17906],{"className":17907,"style":3866},[2155],[54,17909,17911],{"className":17910},[2126],[54,17912,174],{"className":17913},[2126],[54,17915,2781],{"className":17916},[2780],[54,17918,17920],{"className":17919},[2143],[54,17921,17923],{"className":17922,"style":11311},[2147],[54,17924],{},[54,17926],{"className":17927},[2336,9349],[54,17929,17931],{"className":17930},[2126],[54,17932,129],{"className":17933},[2503,9476],[54,17935],{"className":17936,"style":2521},[2270],[54,17938,17940,17946,17984,17987,18002,18005,18008,18011,18026,18029,18058],{"className":17939},[2493],[54,17941,17943],{"className":17942,"style":2499},[2497,2498],[54,17944,129],{"className":17945},[2503,2504],[54,17947,17949,17952],{"className":17948},[2126],[54,17950,2842],{"className":17951},[2126,2167],[54,17953,17955],{"className":17954},[2135],[54,17956,17958],{"className":17957},[2139],[54,17959,17961],{"className":17960},[2143],[54,17962,17964],{"className":17963,"style":4596},[2147],[54,17965,17966,17969],{"style":2462},[54,17967],{"className":17968,"style":2156},[2155],[54,17970,17972],{"className":17971},[2160,2161,2162,2163],[54,17973,17975,17978,17981],{"className":17974},[2126,2163],[54,17976,2247],{"className":17977},[2126,2163],[54,17979,2849],{"className":17980},[2126,2167,2163],[54,17982,14672],{"className":17983},[2126,2167,2163],[54,17985],{"className":17986,"style":2521},[2270],[54,17988,17990,17993,17999],{"className":17989},[2493],[54,17991,3981],{"className":17992},[2497],[54,17994,17996],{"className":17995},[2126],[54,17997,3349],{"className":17998,"style":2332},[2126,2167],[54,18000,2305],{"className":18001},[2126],[54,18003],{"className":18004,"style":2271},[2270],[54,18006,1663],{"className":18007},[2275],[54,18009],{"className":18010,"style":2271},[2270],[54,18012,18014,18017,18023],{"className":18013},[2493],[54,18015,3981],{"className":18016},[2497],[54,18018,18020],{"className":18019},[2126],[54,18021,3349],{"className":18022,"style":2332},[2126,2167],[54,18024,2305],{"className":18025},[2126],[54,18027],{"className":18028,"style":2521},[2270],[54,18030,18032,18035],{"className":18031},[2126],[54,18033,2835],{"className":18034,"style":2870},[2126,2167],[54,18036,18038],{"className":18037},[2135],[54,18039,18041],{"className":18040},[2139],[54,18042,18044],{"className":18043},[2143],[54,18045,18047],{"className":18046,"style":4596},[2147],[54,18048,18049,18052],{"style":2462},[54,18050],{"className":18051,"style":2156},[2155],[54,18053,18055],{"className":18054},[2160,2161,2162,2163],[54,18056,17508],{"className":18057},[2275,2163],[54,18059,18061],{"className":18060,"style":2499},[2336,2498],[54,18062,3205],{"className":18063},[2503,2504],[54,18065],{"className":18066,"style":2521},[2270],[54,18068,2730],{"className":18069,"style":2776},[2126,2167],[54,18071],{"className":18072,"style":2521},[2270],[54,18074,18076,18082,18117,18120,18135,18138,18141,18144,18147,18150,18165],{"className":18075},[2493],[54,18077,18079],{"className":18078,"style":2499},[2497,2498],[54,18080,129],{"className":18081},[2503,2504],[54,18083,18085,18088],{"className":18084},[2126],[54,18086,2842],{"className":18087},[2126,2167],[54,18089,18091],{"className":18090},[2135],[54,18092,18094],{"className":18093},[2139],[54,18095,18097],{"className":18096},[2143],[54,18098,18100],{"className":18099,"style":4596},[2147],[54,18101,18102,18105],{"style":2462},[54,18103],{"className":18104,"style":2156},[2155],[54,18106,18108],{"className":18107},[2160,2161,2162,2163],[54,18109,18111,18114],{"className":18110},[2126,2163],[54,18112,2849],{"className":18113},[2126,2167,2163],[54,18115,14672],{"className":18116},[2126,2167,2163],[54,18118],{"className":18119,"style":2521},[2270],[54,18121,18123,18126,18132],{"className":18122},[2493],[54,18124,2305],{"className":18125},[2126],[54,18127,18129],{"className":18128},[2126],[54,18130,3349],{"className":18131,"style":2332},[2126,2167],[54,18133,2312],{"className":18134},[2336],[54,18136],{"className":18137,"style":2271},[2270],[54,18139,1663],{"className":18140},[2275],[54,18142],{"className":18143,"style":2271},[2270],[54,18145,2835],{"className":18146,"style":2870},[2126,2167],[54,18148],{"className":18149,"style":2521},[2270],[54,18151,18153,18156,18162],{"className":18152},[2493],[54,18154,2305],{"className":18155},[2126],[54,18157,18159],{"className":18158},[2126],[54,18160,3349],{"className":18161,"style":2332},[2126,2167],[54,18163,2312],{"className":18164},[2336],[54,18166,18168],{"className":18167,"style":2499},[2336,2498],[54,18169,3205],{"className":18170},[2503,2504],[54,18172,18173,18176],{"style":17806},[54,18174],{"className":18175,"style":9284},[2155],[54,18177,18179,18182,18186,18189,18192,18195,18322,18325,18328,18331,18429,18432],{"className":18178},[2126],[54,18180],{"className":18181},[2126],[54,18183],{"className":18184,"style":18185},[2270],"margin-right:2em;",[54,18187],{"className":18188,"style":2271},[2270],[54,18190,2247],{"className":18191},[2275],[54,18193],{"className":18194,"style":2271},[2270],[54,18196,18198,18204,18242,18245,18260,18263,18266,18269,18284,18287,18316],{"className":18197},[2493],[54,18199,18201],{"className":18200,"style":2499},[2497,2498],[54,18202,129],{"className":18203},[2503,2504],[54,18205,18207,18210],{"className":18206},[2126],[54,18208,2842],{"className":18209},[2126,2167],[54,18211,18213],{"className":18212},[2135],[54,18214,18216],{"className":18215},[2139],[54,18217,18219],{"className":18218},[2143],[54,18220,18222],{"className":18221,"style":4596},[2147],[54,18223,18224,18227],{"style":2462},[54,18225],{"className":18226,"style":2156},[2155],[54,18228,18230],{"className":18229},[2160,2161,2162,2163],[54,18231,18233,18236,18239],{"className":18232},[2126,2163],[54,18234,2247],{"className":18235},[2126,2163],[54,18237,2849],{"className":18238},[2126,2167,2163],[54,18240,14672],{"className":18241},[2126,2167,2163],[54,18243],{"className":18244,"style":2521},[2270],[54,18246,18248,18251,18257],{"className":18247},[2493],[54,18249,3981],{"className":18250},[2497],[54,18252,18254],{"className":18253},[2126],[54,18255,3349],{"className":18256,"style":2332},[2126,2167],[54,18258,2305],{"className":18259},[2126],[54,18261],{"className":18262,"style":2271},[2270],[54,18264,2247],{"className":18265},[2275],[54,18267],{"className":18268,"style":2271},[2270],[54,18270,18272,18275,18281],{"className":18271},[2493],[54,18273,3981],{"className":18274},[2497],[54,18276,18278],{"className":18277},[2126],[54,18279,3349],{"className":18280,"style":2332},[2126,2167],[54,18282,2305],{"className":18283},[2126],[54,18285],{"className":18286,"style":2521},[2270],[54,18288,18290,18293],{"className":18289},[2126],[54,18291,2835],{"className":18292,"style":2870},[2126,2167],[54,18294,18296],{"className":18295},[2135],[54,18297,18299],{"className":18298},[2139],[54,18300,18302],{"className":18301},[2143],[54,18303,18305],{"className":18304,"style":4596},[2147],[54,18306,18307,18310],{"style":2462},[54,18308],{"className":18309,"style":2156},[2155],[54,18311,18313],{"className":18312},[2160,2161,2162,2163],[54,18314,17508],{"className":18315},[2275,2163],[54,18317,18319],{"className":18318,"style":2499},[2336,2498],[54,18320,3205],{"className":18321},[2503,2504],[54,18323],{"className":18324,"style":2521},[2270],[54,18326,2730],{"className":18327,"style":2776},[2126,2167],[54,18329],{"className":18330,"style":2521},[2270],[54,18332,18334,18340,18375,18378,18393,18396,18399,18402,18405,18408,18423],{"className":18333},[2493],[54,18335,18337],{"className":18336,"style":2499},[2497,2498],[54,18338,129],{"className":18339},[2503,2504],[54,18341,18343,18346],{"className":18342},[2126],[54,18344,2842],{"className":18345},[2126,2167],[54,18347,18349],{"className":18348},[2135],[54,18350,18352],{"className":18351},[2139],[54,18353,18355],{"className":18354},[2143],[54,18356,18358],{"className":18357,"style":4596},[2147],[54,18359,18360,18363],{"style":2462},[54,18361],{"className":18362,"style":2156},[2155],[54,18364,18366],{"className":18365},[2160,2161,2162,2163],[54,18367,18369,18372],{"className":18368},[2126,2163],[54,18370,2849],{"className":18371},[2126,2167,2163],[54,18373,14672],{"className":18374},[2126,2167,2163],[54,18376],{"className":18377,"style":2521},[2270],[54,18379,18381,18384,18390],{"className":18380},[2493],[54,18382,2305],{"className":18383},[2126],[54,18385,18387],{"className":18386},[2126],[54,18388,3349],{"className":18389,"style":2332},[2126,2167],[54,18391,2312],{"className":18392},[2336],[54,18394],{"className":18395,"style":2271},[2270],[54,18397,2247],{"className":18398},[2275],[54,18400],{"className":18401,"style":2271},[2270],[54,18403,2835],{"className":18404,"style":2870},[2126,2167],[54,18406],{"className":18407,"style":2521},[2270],[54,18409,18411,18414,18420],{"className":18410},[2493],[54,18412,2305],{"className":18413},[2126],[54,18415,18417],{"className":18416},[2126],[54,18418,3349],{"className":18419,"style":2332},[2126,2167],[54,18421,2312],{"className":18422},[2336],[54,18424,18426],{"className":18425,"style":2499},[2336,2498],[54,18427,3205],{"className":18428},[2503,2504],[54,18430],{"className":18431,"style":2521},[2270],[54,18433,18435],{"className":18434},[2126],[54,18436,3205],{"className":18437},[2503,9476],[54,18439,18440,18443],{"style":17815},[54,18441],{"className":18442,"style":9284},[2155],[54,18444,18446,18449,18452,18455,18458,18464,18467,18565,18568],{"className":18445},[2126],[54,18447],{"className":18448},[2126],[54,18450],{"className":18451,"style":2475},[2270],[54,18453,122],{"className":18454},[2479],[54,18456],{"className":18457,"style":2475},[2270],[54,18459,18461],{"className":18460},[2126,6595],[54,18462,11801],{"className":18463},[2126],[54,18465],{"className":18466,"style":2521},[2270],[54,18468,18470,18476,18514,18517,18532,18535,18538,18541,18544,18559],{"className":18469},[2493],[54,18471,18473],{"className":18472,"style":2499},[2497,2498],[54,18474,11804],{"className":18475},[2503,2504],[54,18477,18479,18482],{"className":18478},[2126],[54,18480,2842],{"className":18481},[2126,2167],[54,18483,18485],{"className":18484},[2135],[54,18486,18488],{"className":18487},[2139],[54,18489,18491],{"className":18490},[2143],[54,18492,18494],{"className":18493,"style":4596},[2147],[54,18495,18496,18499],{"style":2462},[54,18497],{"className":18498,"style":2156},[2155],[54,18500,18502],{"className":18501},[2160,2161,2162,2163],[54,18503,18505,18508,18511],{"className":18504},[2126,2163],[54,18506,2247],{"className":18507},[2126,2163],[54,18509,2849],{"className":18510},[2126,2167,2163],[54,18512,14672],{"className":18513},[2126,2167,2163],[54,18515],{"className":18516,"style":2521},[2270],[54,18518,18520,18523,18529],{"className":18519},[2493],[54,18521,3981],{"className":18522},[2497],[54,18524,18526],{"className":18525},[2126],[54,18527,3349],{"className":18528,"style":2332},[2126,2167],[54,18530,2305],{"className":18531},[2126],[54,18533],{"className":18534,"style":2521},[2270],[54,18536,2730],{"className":18537,"style":2776},[2126,2167],[54,18539,2835],{"className":18540,"style":2870},[2126,2167],[54,18542],{"className":18543,"style":2521},[2270],[54,18545,18547,18550,18556],{"className":18546},[2493],[54,18548,2305],{"className":18549},[2126],[54,18551,18553],{"className":18552},[2126],[54,18554,3349],{"className":18555,"style":2332},[2126,2167],[54,18557,2312],{"className":18558},[2336],[54,18560,18562],{"className":18561,"style":2499},[2336,2498],[54,18563,11829],{"className":18564},[2503,2504],[54,18566],{"className":18567,"style":2521},[2270],[54,18569,2375],{"className":18570},[2517],[54,18572,2781],{"className":18573},[2780],[54,18575,18577],{"className":18576},[2143],[54,18578,18580],{"className":18579,"style":17831},[2147],[54,18581],{},[54,18583,2781],{"className":18584},[2780],[54,18586,18588],{"className":18587},[2143],[54,18589,18591],{"className":18590,"style":17831},[2147],[54,18592],{},[54,18594,18596],{"className":18595},[17167],[54,18597,18599,18616],{"className":18598},[2139,2756],[54,18600,18602,18613],{"className":18601},[2143],[54,18603,18605],{"className":18604,"style":17746},[2147],[54,18606,18607,18610],{"style":17749},[54,18608],{"className":18609,"style":17753},[2155],[54,18611],{"className":18612},[17185],[54,18614,2781],{"className":18615},[2780],[54,18617,18619],{"className":18618},[2143],[54,18620,18622],{"className":18621,"style":17831},[2147],[54,18623],{},[54,18625,18627],{"className":18626},[2340],[54,18628,18630,18945],{"className":18629},[2079],[54,18631,18633],{"className":18632},[2083],[2085,18634,18635],{"xmlns":2087,"display":2349},[2089,18636,18637,18942],{},[9008,18638,18639],{"rowspacing":9010,"columnspacing":9011},[9013,18640,18641,18644,18936,18939],{},[9016,18642],{"className":18643},[14450],[9016,18645,18646],{},[9019,18647,18648],{"scriptlevel":150,"displaystyle":2113},[9008,18649,18650,18770,18876],{"rowspacing":11670,"columnalign":11671,"columnspacing":11672},[9013,18651,18652,18668],{},[9016,18653,18654],{},[9019,18655,18656],{"scriptlevel":150,"displaystyle":2113},[2092,18657,18658,18660,18662,18664,18666],{},[2245,18659,3981],{"stretchy":2311},[2098,18661,7302],{},[2245,18663,17445],{},[2098,18665,2730],{},[2245,18667,2312],{"stretchy":2311},[9016,18669,18670],{},[9019,18671,18672],{"scriptlevel":150,"displaystyle":2113},[2092,18673,18674,18676,18678,18684,18686,18730,18732],{},[2092,18675],{},[2245,18677,122],{},[9047,18679,18680,18682],{},[2249,18681,174],{},[2249,18683,11708],{},[2245,18685,129],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[2092,18687,18688,18690,18702,18710,18712,18714,18722,18728],{},[2245,18689,129],{"fence":2113},[2095,18691,18692,18694],{},[2098,18693,2842],{},[2092,18695,18696,18698,18700],{},[2245,18697,2247],{},[2098,18699,2849],{},[2098,18701,14672],{},[14079,18703,18704,18706,18708],{},[2245,18705,3981],{"stretchy":2311},[2098,18707,3349],{},[2098,18709,2305],{"mathvariant":2304},[2245,18711,1663],{},[2098,18713,2849],{},[14079,18715,18716,18718,18720],{},[2245,18717,3981],{"stretchy":2311},[2098,18719,3349],{},[2098,18721,2305],{"mathvariant":2304},[2095,18723,18724,18726],{},[2098,18725,2835],{},[2245,18727,17508],{},[2245,18729,3205],{"fence":2113},[2098,18731,2730],{},[2092,18733,18734,18736,18746,18754,18756,18758,18760,18768],{},[2245,18735,129],{"fence":2113},[2095,18737,18738,18740],{},[2098,18739,2842],{},[2092,18741,18742,18744],{},[2098,18743,2849],{},[2098,18745,14672],{},[14079,18747,18748,18750,18752],{},[2098,18749,2305],{"mathvariant":2304},[2098,18751,3349],{},[2245,18753,2312],{"stretchy":2311},[2245,18755,2247],{},[2098,18757,2849],{},[2098,18759,2835],{},[14079,18761,18762,18764,18766],{},[2098,18763,2305],{"mathvariant":2304},[2098,18765,3349],{},[2245,18767,2312],{"stretchy":2311},[2245,18769,3205],{"fence":2113},[9013,18771,18772,18778],{},[9016,18773,18774],{},[9019,18775,18776],{"scriptlevel":150,"displaystyle":2113},[2092,18777],{},[9016,18779,18780],{},[9019,18781,18782],{"scriptlevel":150,"displaystyle":2113},[2092,18783,18784,18786,18788,18790,18834,18836,18874],{},[2092,18785],{},[2270,18787],{"width":17567},[2245,18789,2247],{},[2092,18791,18792,18794,18806,18814,18816,18818,18826,18832],{},[2245,18793,129],{"fence":2113},[2095,18795,18796,18798],{},[2098,18797,2842],{},[2092,18799,18800,18802,18804],{},[2245,18801,2247],{},[2098,18803,2849],{},[2098,18805,14672],{},[14079,18807,18808,18810,18812],{},[2245,18809,3981],{"stretchy":2311},[2098,18811,3349],{},[2098,18813,2305],{"mathvariant":2304},[2245,18815,2247],{},[2098,18817,2849],{},[14079,18819,18820,18822,18824],{},[2245,18821,3981],{"stretchy":2311},[2098,18823,3349],{},[2098,18825,2305],{"mathvariant":2304},[2095,18827,18828,18830],{},[2098,18829,2835],{},[2245,18831,17508],{},[2245,18833,3205],{"fence":2113},[2098,18835,2730],{},[2092,18837,18838,18840,18850,18858,18860,18862,18864,18872],{},[2245,18839,129],{"fence":2113},[2095,18841,18842,18844],{},[2098,18843,2842],{},[2092,18845,18846,18848],{},[2098,18847,2849],{},[2098,18849,14672],{},[14079,18851,18852,18854,18856],{},[2098,18853,2305],{"mathvariant":2304},[2098,18855,3349],{},[2245,18857,2312],{"stretchy":2311},[2245,18859,1663],{},[2098,18861,2849],{},[2098,18863,2835],{},[14079,18865,18866,18868,18870],{},[2098,18867,2305],{"mathvariant":2304},[2098,18869,3349],{},[2245,18871,2312],{"stretchy":2311},[2245,18873,3205],{"fence":2113},[2245,18875,3205],{"fence":2311,"stretchy":2113,"minsize":9058,"maxsize":9058},[9013,18877,18878,18884],{},[9016,18879,18880],{},[9019,18881,18882],{"scriptlevel":150,"displaystyle":2113},[2092,18883],{},[9016,18885,18886],{},[9019,18887,18888],{"scriptlevel":150,"displaystyle":2113},[2092,18889,18890,18892,18894,18896,18934],{},[2092,18891],{},[2245,18893,122],{},[5723,18895,13060],{},[2092,18897,18898,18900,18912,18920,18922,18924,18932],{},[2245,18899,11804],{"fence":2113},[2095,18901,18902,18904],{},[2098,18903,2842],{},[2092,18905,18906,18908,18910],{},[2245,18907,2247],{},[2098,18909,2849],{},[2098,18911,14672],{},[14079,18913,18914,18916,18918],{},[2245,18915,3981],{"stretchy":2311},[2098,18917,3349],{},[2098,18919,2305],{"mathvariant":2304},[2098,18921,2730],{},[2098,18923,2835],{},[14079,18925,18926,18928,18930],{},[2098,18927,2305],{"mathvariant":2304},[2098,18929,3349],{},[2245,18931,2312],{"stretchy":2311},[2245,18933,11829],{"fence":2113},[2098,18935,29],{"mathvariant":2304},[9016,18937],{"className":18938},[14450],[9016,18940],{"className":18941},[15052],[2105,18943,18944],{"encoding":2107},"\\begin{equation}\n\\begin{split}\n \\langle Y\\otimes P\\rangle&=\\frac{1}{4}\n \\Big(\n \\left(e^{-i\\phi}\\bra{\\psi}+i\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}-iU\\ket{\\psi}\\right)\n \\\\\n &\\qquad-\\left(e^{-i\\phi}\\bra{\\psi}-i\\bra{\\psi}U^\\dagger\\right)P\\left(e^{i\\phi}\\ket{\\psi}+iU\\ket{\\psi}\\right)\n \\Big)\\\\\n &=\\text{Im}\\left[e^{-i\\phi}\\bra{\\psi}PU\\ket{\\psi}\\right].\n\\end{split}\n\\end{equation}",[54,18946,18948,19827],{"className":18947,"ariaHidden":2113},[2112],[54,18949,18951,18954],{"className":18950},[2117],[54,18952],{"className":18953,"style":17730},[2121],[54,18955,18957],{"className":18956},[9008],[54,18958,18960],{"className":18959},[9267],[54,18961,18963,19819],{"className":18962},[2139,2756],[54,18964,18966,19816],{"className":18965},[2143],[54,18967,18969],{"className":18968,"style":17746},[2147],[54,18970,18971,18974],{"style":17749},[54,18972],{"className":18973,"style":17753},[2155],[54,18975,18977],{"className":18976},[2126],[54,18978,18980],{"className":18979},[2126],[54,18981,18983,19051],{"className":18982},[9008],[54,18984,18986],{"className":18985},[11883],[54,18987,18989,19043],{"className":18988},[2139,2756],[54,18990,18992,19040],{"className":18991},[2143],[54,18993,18995,19024,19032],{"className":18994,"style":17746},[2147],[54,18996,18997,19000],{"style":17749},[54,18998],{"className":18999,"style":9284},[2155],[54,19001,19003,19006,19009,19012,19015,19018,19021],{"className":19002},[2126],[54,19004,3981],{"className":19005},[2497],[54,19007,7302],{"className":19008,"style":2271},[2126,2167],[54,19010],{"className":19011,"style":2271},[2270],[54,19013,17445],{"className":19014},[2275],[54,19016],{"className":19017,"style":2271},[2270],[54,19019,2730],{"className":19020,"style":2776},[2126,2167],[54,19022,2312],{"className":19023},[2336],[54,19025,19026,19029],{"style":17806},[54,19027],{"className":19028,"style":9284},[2155],[54,19030],{"className":19031},[2126],[54,19033,19034,19037],{"style":17815},[54,19035],{"className":19036,"style":9284},[2155],[54,19038],{"className":19039},[2126],[54,19041,2781],{"className":19042},[2780],[54,19044,19046],{"className":19045},[2143],[54,19047,19049],{"className":19048,"style":17831},[2147],[54,19050],{},[54,19052,19054],{"className":19053},[11944],[54,19055,19057,19808],{"className":19056},[2139,2756],[54,19058,19060,19805],{"className":19059},[2143],[54,19061,19063,19397,19672],{"className":19062,"style":17746},[2147],[54,19064,19065,19068],{"style":17749},[54,19066],{"className":19067,"style":9284},[2155],[54,19069,19071,19074,19077,19080,19083,19145,19151,19154,19287,19290,19293,19296],{"className":19070},[2126],[54,19072],{"className":19073},[2126],[54,19075],{"className":19076,"style":2475},[2270],[54,19078,122],{"className":19079},[2479],[54,19081],{"className":19082,"style":2475},[2270],[54,19084,19086,19089,19142],{"className":19085},[2126],[54,19087],{"className":19088},[2497,9349],[54,19090,19092],{"className":19091},[9047],[54,19093,19095,19134],{"className":19094},[2139,2756],[54,19096,19098,19131],{"className":19097},[2143],[54,19099,19101,19112,19120],{"className":19100,"style":9362},[2147],[54,19102,19103,19106],{"style":11273},[54,19104],{"className":19105,"style":3866},[2155],[54,19107,19109],{"className":19108},[2126],[54,19110,11708],{"className":19111},[2126],[54,19113,19114,19117],{"style":9434},[54,19115],{"className":19116,"style":3866},[2155],[54,19118],{"className":19119,"style":9442},[9441],[54,19121,19122,19125],{"style":9445},[54,19123],{"className":19124,"style":3866},[2155],[54,19126,19128],{"className":19127},[2126],[54,19129,174],{"className":19130},[2126],[54,19132,2781],{"className":19133},[2780],[54,19135,19137],{"className":19136},[2143],[54,19138,19140],{"className":19139,"style":11311},[2147],[54,19141],{},[54,19143],{"className":19144},[2336,9349],[54,19146,19148],{"className":19147},[2126],[54,19149,129],{"className":19150},[2503,9476],[54,19152],{"className":19153,"style":2521},[2270],[54,19155,19157,19163,19201,19204,19219,19222,19225,19228,19231,19234,19249,19252,19281],{"className":19156},[2493],[54,19158,19160],{"className":19159,"style":2499},[2497,2498],[54,19161,129],{"className":19162},[2503,2504],[54,19164,19166,19169],{"className":19165},[2126],[54,19167,2842],{"className":19168},[2126,2167],[54,19170,19172],{"className":19171},[2135],[54,19173,19175],{"className":19174},[2139],[54,19176,19178],{"className":19177},[2143],[54,19179,19181],{"className":19180,"style":4596},[2147],[54,19182,19183,19186],{"style":2462},[54,19184],{"className":19185,"style":2156},[2155],[54,19187,19189],{"className":19188},[2160,2161,2162,2163],[54,19190,19192,19195,19198],{"className":19191},[2126,2163],[54,19193,2247],{"className":19194},[2126,2163],[54,19196,2849],{"className":19197},[2126,2167,2163],[54,19199,14672],{"className":19200},[2126,2167,2163],[54,19202],{"className":19203,"style":2521},[2270],[54,19205,19207,19210,19216],{"className":19206},[2493],[54,19208,3981],{"className":19209},[2497],[54,19211,19213],{"className":19212},[2126],[54,19214,3349],{"className":19215,"style":2332},[2126,2167],[54,19217,2305],{"className":19218},[2126],[54,19220],{"className":19221,"style":2271},[2270],[54,19223,1663],{"className":19224},[2275],[54,19226],{"className":19227,"style":2271},[2270],[54,19229,2849],{"className":19230},[2126,2167],[54,19232],{"className":19233,"style":2521},[2270],[54,19235,19237,19240,19246],{"className":19236},[2493],[54,19238,3981],{"className":19239},[2497],[54,19241,19243],{"className":19242},[2126],[54,19244,3349],{"className":19245,"style":2332},[2126,2167],[54,19247,2305],{"className":19248},[2126],[54,19250],{"className":19251,"style":2521},[2270],[54,19253,19255,19258],{"className":19254},[2126],[54,19256,2835],{"className":19257,"style":2870},[2126,2167],[54,19259,19261],{"className":19260},[2135],[54,19262,19264],{"className":19263},[2139],[54,19265,19267],{"className":19266},[2143],[54,19268,19270],{"className":19269,"style":4596},[2147],[54,19271,19272,19275],{"style":2462},[54,19273],{"className":19274,"style":2156},[2155],[54,19276,19278],{"className":19277},[2160,2161,2162,2163],[54,19279,17508],{"className":19280},[2275,2163],[54,19282,19284],{"className":19283,"style":2499},[2336,2498],[54,19285,3205],{"className":19286},[2503,2504],[54,19288],{"className":19289,"style":2521},[2270],[54,19291,2730],{"className":19292,"style":2776},[2126,2167],[54,19294],{"className":19295,"style":2521},[2270],[54,19297,19299,19305,19340,19343,19358,19361,19364,19367,19370,19373,19376,19391],{"className":19298},[2493],[54,19300,19302],{"className":19301,"style":2499},[2497,2498],[54,19303,129],{"className":19304},[2503,2504],[54,19306,19308,19311],{"className":19307},[2126],[54,19309,2842],{"className":19310},[2126,2167],[54,19312,19314],{"className":19313},[2135],[54,19315,19317],{"className":19316},[2139],[54,19318,19320],{"className":19319},[2143],[54,19321,19323],{"className":19322,"style":4596},[2147],[54,19324,19325,19328],{"style":2462},[54,19326],{"className":19327,"style":2156},[2155],[54,19329,19331],{"className":19330},[2160,2161,2162,2163],[54,19332,19334,19337],{"className":19333},[2126,2163],[54,19335,2849],{"className":19336},[2126,2167,2163],[54,19338,14672],{"className":19339},[2126,2167,2163],[54,19341],{"className":19342,"style":2521},[2270],[54,19344,19346,19349,19355],{"className":19345},[2493],[54,19347,2305],{"className":19348},[2126],[54,19350,19352],{"className":19351},[2126],[54,19353,3349],{"className":19354,"style":2332},[2126,2167],[54,19356,2312],{"className":19357},[2336],[54,19359],{"className":19360,"style":2271},[2270],[54,19362,2247],{"className":19363},[2275],[54,19365],{"className":19366,"style":2271},[2270],[54,19368,2849],{"className":19369},[2126,2167],[54,19371,2835],{"className":19372,"style":2870},[2126,2167],[54,19374],{"className":19375,"style":2521},[2270],[54,19377,19379,19382,19388],{"className":19378},[2493],[54,19380,2305],{"className":19381},[2126],[54,19383,19385],{"className":19384},[2126],[54,19386,3349],{"className":19387,"style":2332},[2126,2167],[54,19389,2312],{"className":19390},[2336],[54,19392,19394],{"className":19393,"style":2499},[2336,2498],[54,19395,3205],{"className":19396},[2503,2504],[54,19398,19399,19402],{"style":17806},[54,19400],{"className":19401,"style":9284},[2155],[54,19403,19405,19408,19411,19414,19417,19420,19553,19556,19559,19562,19663,19666],{"className":19404},[2126],[54,19406],{"className":19407},[2126],[54,19409],{"className":19410,"style":18185},[2270],[54,19412],{"className":19413,"style":2271},[2270],[54,19415,2247],{"className":19416},[2275],[54,19418],{"className":19419,"style":2271},[2270],[54,19421,19423,19429,19467,19470,19485,19488,19491,19494,19497,19500,19515,19518,19547],{"className":19422},[2493],[54,19424,19426],{"className":19425,"style":2499},[2497,2498],[54,19427,129],{"className":19428},[2503,2504],[54,19430,19432,19435],{"className":19431},[2126],[54,19433,2842],{"className":19434},[2126,2167],[54,19436,19438],{"className":19437},[2135],[54,19439,19441],{"className":19440},[2139],[54,19442,19444],{"className":19443},[2143],[54,19445,19447],{"className":19446,"style":4596},[2147],[54,19448,19449,19452],{"style":2462},[54,19450],{"className":19451,"style":2156},[2155],[54,19453,19455],{"className":19454},[2160,2161,2162,2163],[54,19456,19458,19461,19464],{"className":19457},[2126,2163],[54,19459,2247],{"className":19460},[2126,2163],[54,19462,2849],{"className":19463},[2126,2167,2163],[54,19465,14672],{"className":19466},[2126,2167,2163],[54,19468],{"className":19469,"style":2521},[2270],[54,19471,19473,19476,19482],{"className":19472},[2493],[54,19474,3981],{"className":19475},[2497],[54,19477,19479],{"className":19478},[2126],[54,19480,3349],{"className":19481,"style":2332},[2126,2167],[54,19483,2305],{"className":19484},[2126],[54,19486],{"className":19487,"style":2271},[2270],[54,19489,2247],{"className":19490},[2275],[54,19492],{"className":19493,"style":2271},[2270],[54,19495,2849],{"className":19496},[2126,2167],[54,19498],{"className":19499,"style":2521},[2270],[54,19501,19503,19506,19512],{"className":19502},[2493],[54,19504,3981],{"className":19505},[2497],[54,19507,19509],{"className":19508},[2126],[54,19510,3349],{"className":19511,"style":2332},[2126,2167],[54,19513,2305],{"className":19514},[2126],[54,19516],{"className":19517,"style":2521},[2270],[54,19519,19521,19524],{"className":19520},[2126],[54,19522,2835],{"className":19523,"style":2870},[2126,2167],[54,19525,19527],{"className":19526},[2135],[54,19528,19530],{"className":19529},[2139],[54,19531,19533],{"className":19532},[2143],[54,19534,19536],{"className":19535,"style":4596},[2147],[54,19537,19538,19541],{"style":2462},[54,19539],{"className":19540,"style":2156},[2155],[54,19542,19544],{"className":19543},[2160,2161,2162,2163],[54,19545,17508],{"className":19546},[2275,2163],[54,19548,19550],{"className":19549,"style":2499},[2336,2498],[54,19551,3205],{"className":19552},[2503,2504],[54,19554],{"className":19555,"style":2521},[2270],[54,19557,2730],{"className":19558,"style":2776},[2126,2167],[54,19560],{"className":19561,"style":2521},[2270],[54,19563,19565,19571,19606,19609,19624,19627,19630,19633,19636,19639,19642,19657],{"className":19564},[2493],[54,19566,19568],{"className":19567,"style":2499},[2497,2498],[54,19569,129],{"className":19570},[2503,2504],[54,19572,19574,19577],{"className":19573},[2126],[54,19575,2842],{"className":19576},[2126,2167],[54,19578,19580],{"className":19579},[2135],[54,19581,19583],{"className":19582},[2139],[54,19584,19586],{"className":19585},[2143],[54,19587,19589],{"className":19588,"style":4596},[2147],[54,19590,19591,19594],{"style":2462},[54,19592],{"className":19593,"style":2156},[2155],[54,19595,19597],{"className":19596},[2160,2161,2162,2163],[54,19598,19600,19603],{"className":19599},[2126,2163],[54,19601,2849],{"className":19602},[2126,2167,2163],[54,19604,14672],{"className":19605},[2126,2167,2163],[54,19607],{"className":19608,"style":2521},[2270],[54,19610,19612,19615,19621],{"className":19611},[2493],[54,19613,2305],{"className":19614},[2126],[54,19616,19618],{"className":19617},[2126],[54,19619,3349],{"className":19620,"style":2332},[2126,2167],[54,19622,2312],{"className":19623},[2336],[54,19625],{"className":19626,"style":2271},[2270],[54,19628,1663],{"className":19629},[2275],[54,19631],{"className":19632,"style":2271},[2270],[54,19634,2849],{"className":19635},[2126,2167],[54,19637,2835],{"className":19638,"style":2870},[2126,2167],[54,19640],{"className":19641,"style":2521},[2270],[54,19643,19645,19648,19654],{"className":19644},[2493],[54,19646,2305],{"className":19647},[2126],[54,19649,19651],{"className":19650},[2126],[54,19652,3349],{"className":19653,"style":2332},[2126,2167],[54,19655,2312],{"className":19656},[2336],[54,19658,19660],{"className":19659,"style":2499},[2336,2498],[54,19661,3205],{"className":19662},[2503,2504],[54,19664],{"className":19665,"style":2521},[2270],[54,19667,19669],{"className":19668},[2126],[54,19670,3205],{"className":19671},[2503,9476],[54,19673,19674,19677],{"style":17815},[54,19675],{"className":19676,"style":9284},[2155],[54,19678,19680,19683,19686,19689,19692,19698,19701,19799,19802],{"className":19679},[2126],[54,19681],{"className":19682},[2126],[54,19684],{"className":19685,"style":2475},[2270],[54,19687,122],{"className":19688},[2479],[54,19690],{"className":19691,"style":2475},[2270],[54,19693,19695],{"className":19694},[2126,6595],[54,19696,13060],{"className":19697},[2126],[54,19699],{"className":19700,"style":2521},[2270],[54,19702,19704,19710,19748,19751,19766,19769,19772,19775,19778,19793],{"className":19703},[2493],[54,19705,19707],{"className":19706,"style":2499},[2497,2498],[54,19708,11804],{"className":19709},[2503,2504],[54,19711,19713,19716],{"className":19712},[2126],[54,19714,2842],{"className":19715},[2126,2167],[54,19717,19719],{"className":19718},[2135],[54,19720,19722],{"className":19721},[2139],[54,19723,19725],{"className":19724},[2143],[54,19726,19728],{"className":19727,"style":4596},[2147],[54,19729,19730,19733],{"style":2462},[54,19731],{"className":19732,"style":2156},[2155],[54,19734,19736],{"className":19735},[2160,2161,2162,2163],[54,19737,19739,19742,19745],{"className":19738},[2126,2163],[54,19740,2247],{"className":19741},[2126,2163],[54,19743,2849],{"className":19744},[2126,2167,2163],[54,19746,14672],{"className":19747},[2126,2167,2163],[54,19749],{"className":19750,"style":2521},[2270],[54,19752,19754,19757,19763],{"className":19753},[2493],[54,19755,3981],{"className":19756},[2497],[54,19758,19760],{"className":19759},[2126],[54,19761,3349],{"className":19762,"style":2332},[2126,2167],[54,19764,2305],{"className":19765},[2126],[54,19767],{"className":19768,"style":2521},[2270],[54,19770,2730],{"className":19771,"style":2776},[2126,2167],[54,19773,2835],{"className":19774,"style":2870},[2126,2167],[54,19776],{"className":19777,"style":2521},[2270],[54,19779,19781,19784,19790],{"className":19780},[2493],[54,19782,2305],{"className":19783},[2126],[54,19785,19787],{"className":19786},[2126],[54,19788,3349],{"className":19789,"style":2332},[2126,2167],[54,19791,2312],{"className":19792},[2336],[54,19794,19796],{"className":19795,"style":2499},[2336,2498],[54,19797,11829],{"className":19798},[2503,2504],[54,19800],{"className":19801,"style":2521},[2270],[54,19803,29],{"className":19804},[2126],[54,19806,2781],{"className":19807},[2780],[54,19809,19811],{"className":19810},[2143],[54,19812,19814],{"className":19813,"style":17831},[2147],[54,19815],{},[54,19817,2781],{"className":19818},[2780],[54,19820,19822],{"className":19821},[2143],[54,19823,19825],{"className":19824,"style":17831},[2147],[54,19826],{},[54,19828,19830],{"className":19829},[17167],[54,19831,19833,19850],{"className":19832},[2139,2756],[54,19834,19836,19847],{"className":19835},[2143],[54,19837,19839],{"className":19838,"style":17746},[2147],[54,19840,19841,19844],{"style":17749},[54,19842],{"className":19843,"style":17753},[2155],[54,19845],{"className":19846},[17185],[54,19848,2781],{"className":19849},[2780],[54,19851,19853],{"className":19852},[2143],[54,19854,19856],{"className":19855,"style":17831},[2147],[54,19857],{},[13,19859,19860,19861,19902],{},"Using these assumptions we were able to write the expectation values of operators of interest with fewer controlled operations. In fact, we only need to implement the controlled state preparation ",[54,19862,19864,19881],{"className":19863},[2079],[54,19865,19867],{"className":19866},[2083],[2085,19868,19869],{"xmlns":2087},[2089,19870,19871,19879],{},[2092,19872,19873,19875,19877],{},[5723,19874,6569],{},[5723,19876,6572],{},[2098,19878,3349],{},[2105,19880,14043],{"encoding":2107},[54,19882,19884],{"className":19883,"ariaHidden":2113},[2112],[54,19885,19887,19890,19896,19899],{"className":19886},[2117],[54,19888],{"className":19889,"style":6591},[2121],[54,19891,19893],{"className":19892},[2126,6595],[54,19894,6569],{"className":19895},[2126],[54,19897],{"className":19898,"style":2475},[2270],[54,19900,3349],{"className":19901,"style":2332},[2126,2167]," and not controlled time evolutions. Reframing our calculation as above will allow us to greatly reduce the depth of the resulting circuits.",[2069,19904,19906],{"id":19905},"decompose-time-evolution-operator-with-trotter-decomposition","Decompose time-evolution operator with Trotter decomposition",[13,19908,19909,19910,171,19991,171,20072,20153,20154,20183,20184,20308,20309,20366,20367,20400,20401,20450,20451,20454],{},"Instead of implementing the time-evolution operator exactly we can use the Trotter decomposition to implement an approximation of it. Repeating several times a certain order Trotter decomposition gives us further reduction of the error introduced from the approximation. In the following, we directly build the Trotter implementation in the most efficient way for the interaction graph of the Hamiltonian we are considering (nearest neighbor interactions only). In practice we insert Pauli rotations ",[54,19911,19913,19936],{"className":19912},[2079],[54,19914,19916],{"className":19915},[2083],[2085,19917,19918],{"xmlns":2087},[2089,19919,19920,19933],{},[2092,19921,19922],{},[2724,19923,19924,19927],{},[2098,19925,19926],{},"R",[2092,19928,19929,19931],{},[2098,19930,8695],{},[2098,19932,8695],{},[2105,19934,19935],{"encoding":2107},"R_{xx}",[54,19937,19939],{"className":19938,"ariaHidden":2113},[2112],[54,19940,19942,19945],{"className":19941},[2117],[54,19943],{"className":19944,"style":2743},[2121],[54,19946,19948,19952],{"className":19947},[2126],[54,19949,19926],{"className":19950,"style":19951},[2126,2167],"margin-right:0.0077em;",[54,19953,19955],{"className":19954},[2135],[54,19956,19958,19983],{"className":19957},[2139,2756],[54,19959,19961,19980],{"className":19960},[2143],[54,19962,19964],{"className":19963,"style":4063},[2147],[54,19965,19967,19970],{"style":19966},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[54,19968],{"className":19969,"style":2156},[2155],[54,19971,19973],{"className":19972},[2160,2161,2162,2163],[54,19974,19976],{"className":19975},[2126,2163],[54,19977,19979],{"className":19978},[2126,2167,2163],"xx",[54,19981,2781],{"className":19982},[2780],[54,19984,19986],{"className":19985},[2143],[54,19987,19989],{"className":19988,"style":2788},[2147],[54,19990],{},[54,19992,19994,20017],{"className":19993},[2079],[54,19995,19997],{"className":19996},[2083],[2085,19998,19999],{"xmlns":2087},[2089,20000,20001,20014],{},[2092,20002,20003],{},[2724,20004,20005,20007],{},[2098,20006,19926],{},[2092,20008,20009,20012],{},[2098,20010,20011],{},"y",[2098,20013,20011],{},[2105,20015,20016],{"encoding":2107},"R_{yy}",[54,20018,20020],{"className":20019,"ariaHidden":2113},[2112],[54,20021,20023,20026],{"className":20022},[2117],[54,20024],{"className":20025,"style":7523},[2121],[54,20027,20029,20032],{"className":20028},[2126],[54,20030,19926],{"className":20031,"style":19951},[2126,2167],[54,20033,20035],{"className":20034},[2135],[54,20036,20038,20064],{"className":20037},[2139,2756],[54,20039,20041,20061],{"className":20040},[2143],[54,20042,20044],{"className":20043,"style":4063},[2147],[54,20045,20046,20049],{"style":19966},[54,20047],{"className":20048,"style":2156},[2155],[54,20050,20052],{"className":20051},[2160,2161,2162,2163],[54,20053,20055,20058],{"className":20054},[2126,2163],[54,20056,20011],{"className":20057,"style":2332},[2126,2167,2163],[54,20059,20011],{"className":20060,"style":2332},[2126,2167,2163],[54,20062,2781],{"className":20063},[2780],[54,20065,20067],{"className":20066},[2143],[54,20068,20070],{"className":20069,"style":6389},[2147],[54,20071],{},[54,20073,20075,20097],{"className":20074},[2079],[54,20076,20078],{"className":20077},[2083],[2085,20079,20080],{"xmlns":2087},[2089,20081,20082,20094],{},[2092,20083,20084],{},[2724,20085,20086,20088],{},[2098,20087,19926],{},[2092,20089,20090,20092],{},[2098,20091,215],{},[2098,20093,215],{},[2105,20095,20096],{"encoding":2107},"R_{zz}",[54,20098,20100],{"className":20099,"ariaHidden":2113},[2112],[54,20101,20103,20106],{"className":20102},[2117],[54,20104],{"className":20105,"style":2743},[2121],[54,20107,20109,20112],{"className":20108},[2126],[54,20110,19926],{"className":20111,"style":19951},[2126,2167],[54,20113,20115],{"className":20114},[2135],[54,20116,20118,20145],{"className":20117},[2139,2756],[54,20119,20121,20142],{"className":20120},[2143],[54,20122,20124],{"className":20123,"style":4063},[2147],[54,20125,20126,20129],{"style":19966},[54,20127],{"className":20128,"style":2156},[2155],[54,20130,20132],{"className":20131},[2160,2161,2162,2163],[54,20133,20135,20139],{"className":20134},[2126,2163],[54,20136,215],{"className":20137,"style":20138},[2126,2167,2163],"margin-right:0.044em;",[54,20140,215],{"className":20141,"style":20138},[2126,2167,2163],[54,20143,2781],{"className":20144},[2780],[54,20146,20148],{"className":20147},[2143],[54,20149,20151],{"className":20150,"style":2788},[2147],[54,20152],{}," with a parametrized angle ",[54,20155,20157,20170],{"className":20156},[2079],[54,20158,20160],{"className":20159},[2083],[2085,20161,20162],{"xmlns":2087},[2089,20163,20164,20168],{},[2092,20165,20166],{},[2098,20167,2854],{},[2105,20169,2854],{"encoding":2107},[54,20171,20173],{"className":20172,"ariaHidden":2113},[2112],[54,20174,20176,20180],{"className":20175},[2117],[54,20177],{"className":20178,"style":20179},[2121],"height:0.6151em;",[54,20181,2854],{"className":20182},[2126,2167]," which correspond to the approximate implementation of ",[54,20185,20187,20231],{"className":20186},[2079],[54,20188,20190],{"className":20189},[2083],[2085,20191,20192],{"xmlns":2087},[2089,20193,20194,20228],{},[2092,20195,20196],{},[2095,20197,20198,20200],{},[2098,20199,2842],{},[2092,20201,20202,20204,20206,20208,20210,20212,20214,20216,20218,20220,20222,20224,20226],{},[2245,20203,2247],{},[2098,20205,2849],{},[2245,20207,129],{"stretchy":2311},[2098,20209,7287],{},[2098,20211,7287],{},[2245,20213,1663],{},[2098,20215,7302],{},[2098,20217,7302],{},[2245,20219,1663],{},[2098,20221,7320],{},[2098,20223,7320],{},[2245,20225,3205],{"stretchy":2311},[2098,20227,2854],{},[2105,20229,20230],{"encoding":2107},"e^{-i (XX + YY + ZZ) t}",[54,20232,20234],{"className":20233,"ariaHidden":2113},[2112],[54,20235,20237,20240],{"className":20236},[2117],[54,20238],{"className":20239,"style":3264},[2121],[54,20241,20243,20246],{"className":20242},[2126],[54,20244,2842],{"className":20245},[2126,2167],[54,20247,20249],{"className":20248},[2135],[54,20250,20252],{"className":20251},[2139],[54,20253,20255],{"className":20254},[2143],[54,20256,20258],{"className":20257,"style":3264},[2147],[54,20259,20260,20263],{"style":2151},[54,20261],{"className":20262,"style":2156},[2155],[54,20264,20266],{"className":20265},[2160,2161,2162,2163],[54,20267,20269,20272,20275,20278,20281,20284,20287,20290,20293,20296,20299,20302,20305],{"className":20268},[2126,2163],[54,20270,2247],{"className":20271},[2126,2163],[54,20273,2849],{"className":20274},[2126,2167,2163],[54,20276,129],{"className":20277},[2497,2163],[54,20279,7287],{"className":20280,"style":7432},[2126,2167,2163],[54,20282,7287],{"className":20283,"style":7432},[2126,2167,2163],[54,20285,1663],{"className":20286},[2275,2163],[54,20288,7302],{"className":20289,"style":2271},[2126,2167,2163],[54,20291,7302],{"className":20292,"style":2271},[2126,2167,2163],[54,20294,1663],{"className":20295},[2275,2163],[54,20297,7320],{"className":20298,"style":7630},[2126,2167,2163],[54,20300,7320],{"className":20301,"style":7630},[2126,2167,2163],[54,20303,3205],{"className":20304},[2336,2163],[54,20306,2854],{"className":20307},[2126,2167,2163],". Given the difference in definition of the Pauli rotations and the time-evolution that we are trying to implement, we'll have to use the parameter ",[54,20310,20312,20333],{"className":20311},[2079],[54,20313,20315],{"className":20314},[2083],[2085,20316,20317],{"xmlns":2087},[2089,20318,20319,20330],{},[2092,20320,20321,20323,20326,20328],{},[2249,20322,133],{},[2245,20324,20325],{},"∗",[2098,20327,4515],{},[2098,20329,2854],{},[2105,20331,20332],{"encoding":2107},"2*dt",[54,20334,20336,20354],{"className":20335,"ariaHidden":2113},[2112],[54,20337,20339,20342,20345,20348,20351],{"className":20338},[2117],[54,20340],{"className":20341,"style":2285},[2121],[54,20343,133],{"className":20344},[2126],[54,20346],{"className":20347,"style":2271},[2270],[54,20349,20325],{"className":20350},[2275],[54,20352],{"className":20353,"style":2271},[2270],[54,20355,20357,20360,20363],{"className":20356},[2117],[54,20358],{"className":20359,"style":4990},[2121],[54,20361,4515],{"className":20362},[2126,2167],[54,20364,2854],{"className":20365},[2126,2167]," to achieve a time-evolution of ",[54,20368,20370,20385],{"className":20369},[2079],[54,20371,20373],{"className":20372},[2083],[2085,20374,20375],{"xmlns":2087},[2089,20376,20377,20383],{},[2092,20378,20379,20381],{},[2098,20380,4515],{},[2098,20382,2854],{},[2105,20384,5019],{"encoding":2107},[54,20386,20388],{"className":20387,"ariaHidden":2113},[2112],[54,20389,20391,20394,20397],{"className":20390},[2117],[54,20392],{"className":20393,"style":4990},[2121],[54,20395,4515],{"className":20396},[2126,2167],[54,20398,2854],{"className":20399},[2126,2167],". Furthermore, we reverse the order of the operations for odd number of repetitions of the Trotter steps, which is functionally equivalent but allows for synthesizing adjacent operations in a single ",[54,20402,20404,20426],{"className":20403},[2079],[54,20405,20407],{"className":20406},[2083],[2085,20408,20409],{"xmlns":2087},[2089,20410,20411,20423],{},[2092,20412,20413,20415,20417,20419,20421],{},[2098,20414,5413],{},[2098,20416,2835],{},[2245,20418,129],{"stretchy":2311},[2249,20420,133],{},[2245,20422,3205],{"stretchy":2311},[2105,20424,20425],{"encoding":2107},"SU(2)",[54,20427,20429],{"className":20428,"ariaHidden":2113},[2112],[54,20430,20432,20435,20438,20441,20444,20447],{"className":20431},[2117],[54,20433],{"className":20434,"style":2325},[2121],[54,20436,5413],{"className":20437,"style":5448},[2126,2167],[54,20439,2835],{"className":20440,"style":2870},[2126,2167],[54,20442,129],{"className":20443},[2497],[54,20445,133],{"className":20446},[2126],[54,20448,3205],{"className":20449},[2336]," unitary. This gives a much shallower circuit than what is obtained using the generic ",[51,20452,20453],{},"PauliEvolutionGate()"," functionality.",[44,20456,20458],{"className":46,"code":20457,"language":48,"meta":49,"style":49},"t = Parameter(\"t\")\n\n# Create instruction for rotation about XX+YY-ZZ:\nRxyz_circ = QuantumCircuit(2)\nRxyz_circ.rxx(t, 0, 1)\nRxyz_circ.ryy(t, 0, 1)\nRxyz_circ.rzz(t, 0, 1)\nRxyz_instr = Rxyz_circ.to_instruction(label=\"RXX+YY+ZZ\")\n\ninteraction_list = [\n [[i, i + 1] for i in range(0, n_qubits - 1, 2)],\n [[i, i + 1] for i in range(1, n_qubits - 1, 2)],\n] # linear chain\n\nqr = QuantumRegister(n_qubits)\ntrotter_step_circ = QuantumCircuit(qr)\nfor i, color in enumerate(interaction_list):\n for interaction in color:\n trotter_step_circ.append(Rxyz_instr, interaction)\n if i \u003C len(interaction_list) - 1:\n trotter_step_circ.barrier()\nreverse_trotter_step_circ = trotter_step_circ.reverse_ops()\n\nqc_evol = QuantumCircuit(qr)\nfor step in range(num_trotter_steps):\n if step % 2 == 0:\n qc_evol = qc_evol.compose(trotter_step_circ)\n else:\n qc_evol = qc_evol.compose(reverse_trotter_step_circ)\n\nqc_evol.decompose().draw(\"mpl\", fold=-1, scale=0.5)\n",[51,20459,20460,20474,20478,20483,20498,20517,20534,20551,20576,20580,20589,20626,20660,20668,20672,20682,20693,20707,20719,20729,20748,20757,20772,20776,20786,20799,20817,20832,20838,20851,20855],{"__ignoreMap":49},[54,20461,20462,20464,20466,20468,20470,20472],{"class":56,"line":57},[54,20463,8863],{"class":64},[54,20465,122],{"class":121},[54,20467,8868],{"class":125},[54,20469,129],{"class":64},[54,20471,8873],{"class":279},[54,20473,136],{"class":64},[54,20475,20476],{"class":56,"line":74},[54,20477,105],{"emptyLinePlaceholder":104},[54,20479,20480],{"class":56,"line":88},[54,20481,20482],{"class":111},"# Create instruction for rotation about XX+YY-ZZ:\n",[54,20484,20485,20488,20490,20492,20494,20496],{"class":56,"line":101},[54,20486,20487],{"class":64},"Rxyz_circ ",[54,20489,122],{"class":121},[54,20491,126],{"class":125},[54,20493,129],{"class":64},[54,20495,133],{"class":132},[54,20497,136],{"class":64},[54,20499,20500,20503,20506,20509,20511,20513,20515],{"class":56,"line":108},[54,20501,20502],{"class":64},"Rxyz_circ.",[54,20504,20505],{"class":125},"rxx",[54,20507,20508],{"class":64},"(t, ",[54,20510,150],{"class":132},[54,20512,171],{"class":64},[54,20514,174],{"class":132},[54,20516,136],{"class":64},[54,20518,20519,20521,20524,20526,20528,20530,20532],{"class":56,"line":115},[54,20520,20502],{"class":64},[54,20522,20523],{"class":125},"ryy",[54,20525,20508],{"class":64},[54,20527,150],{"class":132},[54,20529,171],{"class":64},[54,20531,174],{"class":132},[54,20533,136],{"class":64},[54,20535,20536,20538,20541,20543,20545,20547,20549],{"class":56,"line":139},[54,20537,20502],{"class":64},[54,20539,20540],{"class":125},"rzz",[54,20542,20508],{"class":64},[54,20544,150],{"class":132},[54,20546,171],{"class":64},[54,20548,174],{"class":132},[54,20550,136],{"class":64},[54,20552,20553,20556,20558,20561,20564,20566,20569,20571,20574],{"class":56,"line":159},[54,20554,20555],{"class":64},"Rxyz_instr ",[54,20557,122],{"class":121},[54,20559,20560],{"class":64}," Rxyz_circ.",[54,20562,20563],{"class":125},"to_instruction",[54,20565,129],{"class":64},[54,20567,20568],{"class":1711},"label",[54,20570,122],{"class":121},[54,20572,20573],{"class":279},"\"RXX+YY+ZZ\"",[54,20575,136],{"class":64},[54,20577,20578],{"class":56,"line":183},[54,20579,105],{"emptyLinePlaceholder":104},[54,20581,20582,20585,20587],{"class":56,"line":210},[54,20583,20584],{"class":64},"interaction_list ",[54,20586,122],{"class":121},[54,20588,1733],{"class":64},[54,20590,20591,20594,20596,20598,20600,20602,20604,20606,20608,20610,20612,20615,20617,20619,20621,20623],{"class":56,"line":227},[54,20592,20593],{"class":64}," [[i, i ",[54,20595,1663],{"class":121},[54,20597,982],{"class":132},[54,20599,7762],{"class":64},[54,20601,1186],{"class":60},[54,20603,7802],{"class":64},[54,20605,1192],{"class":60},[54,20607,7777],{"class":121},[54,20609,129],{"class":64},[54,20611,150],{"class":132},[54,20613,20614],{"class":64},", n_qubits ",[54,20616,1923],{"class":121},[54,20618,982],{"class":132},[54,20620,171],{"class":64},[54,20622,133],{"class":132},[54,20624,20625],{"class":64},")],\n",[54,20627,20628,20630,20632,20634,20636,20638,20640,20642,20644,20646,20648,20650,20652,20654,20656,20658],{"class":56,"line":232},[54,20629,20593],{"class":64},[54,20631,1663],{"class":121},[54,20633,982],{"class":132},[54,20635,7762],{"class":64},[54,20637,1186],{"class":60},[54,20639,7802],{"class":64},[54,20641,1192],{"class":60},[54,20643,7777],{"class":121},[54,20645,129],{"class":64},[54,20647,174],{"class":132},[54,20649,20614],{"class":64},[54,20651,1923],{"class":121},[54,20653,982],{"class":132},[54,20655,171],{"class":64},[54,20657,133],{"class":132},[54,20659,20625],{"class":64},[54,20661,20662,20665],{"class":56,"line":246},[54,20663,20664],{"class":64},"] ",[54,20666,20667],{"class":111},"# linear chain\n",[54,20669,20670],{"class":56,"line":259},[54,20671,105],{"emptyLinePlaceholder":104},[54,20673,20674,20676,20678,20680],{"class":56,"line":264},[54,20675,8940],{"class":64},[54,20677,122],{"class":121},[54,20679,8945],{"class":125},[54,20681,8339],{"class":64},[54,20683,20684,20687,20689,20691],{"class":56,"line":270},[54,20685,20686],{"class":64},"trotter_step_circ ",[54,20688,122],{"class":121},[54,20690,126],{"class":125},[54,20692,8959],{"class":64},[54,20694,20695,20697,20700,20702,20704],{"class":56,"line":285},[54,20696,1186],{"class":60},[54,20698,20699],{"class":64}," i, color ",[54,20701,1192],{"class":60},[54,20703,1513],{"class":121},[54,20705,20706],{"class":64},"(interaction_list):\n",[54,20708,20709,20711,20714,20716],{"class":56,"line":297},[54,20710,1472],{"class":60},[54,20712,20713],{"class":64}," interaction ",[54,20715,1192],{"class":60},[54,20717,20718],{"class":64}," color:\n",[54,20720,20721,20724,20726],{"class":56,"line":310},[54,20722,20723],{"class":64}," trotter_step_circ.",[54,20725,1495],{"class":125},[54,20727,20728],{"class":64},"(Rxyz_instr, interaction)\n",[54,20730,20731,20733,20735,20737,20739,20742,20744,20746],{"class":56,"line":324},[54,20732,1273],{"class":60},[54,20734,7802],{"class":64},[54,20736,5233],{"class":121},[54,20738,1857],{"class":121},[54,20740,20741],{"class":64},"(interaction_list) ",[54,20743,1923],{"class":121},[54,20745,982],{"class":132},[54,20747,1284],{"class":64},[54,20749,20750,20752,20755],{"class":56,"line":333},[54,20751,20723],{"class":64},[54,20753,20754],{"class":125},"barrier",[54,20756,330],{"class":64},[54,20758,20759,20762,20764,20767,20770],{"class":56,"line":345},[54,20760,20761],{"class":64},"reverse_trotter_step_circ ",[54,20763,122],{"class":121},[54,20765,20766],{"class":64}," trotter_step_circ.",[54,20768,20769],{"class":125},"reverse_ops",[54,20771,330],{"class":64},[54,20773,20774],{"class":56,"line":358},[54,20775,105],{"emptyLinePlaceholder":104},[54,20777,20778,20780,20782,20784],{"class":56,"line":908},[54,20779,8952],{"class":64},[54,20781,122],{"class":121},[54,20783,126],{"class":125},[54,20785,8959],{"class":64},[54,20787,20788,20790,20793,20795,20797],{"class":56,"line":913},[54,20789,1186],{"class":60},[54,20791,20792],{"class":64}," step ",[54,20794,1192],{"class":60},[54,20796,7777],{"class":121},[54,20798,13449],{"class":64},[54,20800,20801,20803,20805,20807,20809,20812,20815],{"class":56,"line":918},[54,20802,1273],{"class":60},[54,20804,20792],{"class":64},[54,20806,8325],{"class":121},[54,20808,1872],{"class":132},[54,20810,20811],{"class":121}," ==",[54,20813,20814],{"class":132}," 0",[54,20816,1284],{"class":64},[54,20818,20819,20822,20824,20827,20829],{"class":56,"line":923},[54,20820,20821],{"class":64}," qc_evol ",[54,20823,122],{"class":121},[54,20825,20826],{"class":64}," qc_evol.",[54,20828,13423],{"class":125},[54,20830,20831],{"class":64},"(trotter_step_circ)\n",[54,20833,20834,20836],{"class":56,"line":935},[54,20835,1356],{"class":60},[54,20837,1284],{"class":64},[54,20839,20840,20842,20844,20846,20848],{"class":56,"line":945},[54,20841,20821],{"class":64},[54,20843,122],{"class":121},[54,20845,20826],{"class":64},[54,20847,13423],{"class":125},[54,20849,20850],{"class":64},"(reverse_trotter_step_circ)\n",[54,20852,20853],{"class":56,"line":953},[54,20854,105],{"emptyLinePlaceholder":104},[54,20856,20857,20859,20861,20863,20865,20867,20869,20871,20873,20875,20877,20879,20881,20883,20885],{"class":56,"line":968},[54,20858,8964],{"class":64},[54,20860,13721],{"class":125},[54,20862,13539],{"class":64},[54,20864,8720],{"class":125},[54,20866,129],{"class":64},[54,20868,8725],{"class":279},[54,20870,171],{"class":64},[54,20872,13668],{"class":1711},[54,20874,13671],{"class":121},[54,20876,174],{"class":132},[54,20878,171],{"class":64},[54,20880,8730],{"class":1711},[54,20882,122],{"class":121},[54,20884,8735],{"class":132},[54,20886,136],{"class":64},[380,20888],{"alt":8740,"src":20889},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-03.avif",[2069,20891,20893],{"id":20892},"use-an-optimized-circuit-for-state-preparation","Use an optimized circuit for state preparation",[44,20895,20897],{"className":46,"code":20896,"language":48,"meta":49,"style":49},"control = 0\nexcitation = int(n_qubits \u002F 2) + 1\ncontrolled_state_prep = QuantumCircuit(n_qubits + 1)\ncontrolled_state_prep.cx(control, excitation)\ncontrolled_state_prep.draw(\"mpl\", fold=-1, scale=0.5)\n",[51,20898,20899,20908,20929,20946,20956],{"__ignoreMap":49},[54,20900,20901,20904,20906],{"class":56,"line":57},[54,20902,20903],{"class":64},"control ",[54,20905,122],{"class":121},[54,20907,1802],{"class":132},[54,20909,20910,20913,20915,20917,20919,20921,20923,20925,20927],{"class":56,"line":74},[54,20911,20912],{"class":64},"excitation ",[54,20914,122],{"class":121},[54,20916,1651],{"class":121},[54,20918,1660],{"class":64},[54,20920,194],{"class":121},[54,20922,1872],{"class":132},[54,20924,204],{"class":64},[54,20926,1663],{"class":121},[54,20928,1640],{"class":132},[54,20930,20931,20934,20936,20938,20940,20942,20944],{"class":56,"line":88},[54,20932,20933],{"class":64},"controlled_state_prep ",[54,20935,122],{"class":121},[54,20937,126],{"class":125},[54,20939,1660],{"class":64},[54,20941,1663],{"class":121},[54,20943,982],{"class":132},[54,20945,136],{"class":64},[54,20947,20948,20951,20953],{"class":56,"line":101},[54,20949,20950],{"class":64},"controlled_state_prep.",[54,20952,164],{"class":125},[54,20954,20955],{"class":64},"(control, excitation)\n",[54,20957,20958,20960,20962,20964,20966,20968,20970,20972,20974,20976,20978,20980,20982],{"class":56,"line":108},[54,20959,20950],{"class":64},[54,20961,8720],{"class":125},[54,20963,129],{"class":64},[54,20965,8725],{"class":279},[54,20967,171],{"class":64},[54,20969,13668],{"class":1711},[54,20971,13671],{"class":121},[54,20973,174],{"class":132},[54,20975,171],{"class":64},[54,20977,8730],{"class":1711},[54,20979,122],{"class":121},[54,20981,8735],{"class":132},[54,20983,136],{"class":64},[380,20985],{"alt":8740,"src":20986},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-04.avif",[2069,20988,20990,20991,10355,21051,21111],{"id":20989},"template-circuits-for-calculating-matrix-elements-of-stildess-and-htildehh-via-hadamard-test","Template circuits for calculating matrix elements of ",[54,20992,20994,21011],{"className":20993},[2079],[54,20995,20997],{"className":20996},[2083],[2085,20998,20999],{"xmlns":2087},[2089,21000,21001,21009],{},[2092,21002,21003],{},[3827,21004,21005,21007],{"accent":2113},[2098,21006,5413],{},[2245,21008,3833],{},[2105,21010,5418],{"encoding":2107},[54,21012,21014],{"className":21013,"ariaHidden":2113},[2112],[54,21015,21017,21020],{"className":21016},[2117],[54,21018],{"className":21019,"style":3846},[2121],[54,21021,21023],{"className":21022},[2126,3850],[54,21024,21026],{"className":21025},[2139],[54,21027,21029],{"className":21028},[2143],[54,21030,21032,21040],{"className":21031,"style":3846},[2147],[54,21033,21034,21037],{"style":3862},[54,21035],{"className":21036,"style":3866},[2155],[54,21038,5413],{"className":21039,"style":5448},[2126,2167],[54,21041,21042,21045],{"style":3872},[54,21043],{"className":21044,"style":3866},[2155],[54,21046,21048],{"className":21047,"style":5457},[3879],[54,21049,3833],{"className":21050},[2126],[54,21052,21054,21071],{"className":21053},[2079],[54,21055,21057],{"className":21056},[2083],[2085,21058,21059],{"xmlns":2087},[2089,21060,21061,21069],{},[2092,21062,21063],{},[3827,21064,21065,21067],{"accent":2113},[2098,21066,2694],{},[2245,21068,3833],{},[2105,21070,3836],{"encoding":2107},[54,21072,21074],{"className":21073,"ariaHidden":2113},[2112],[54,21075,21077,21080],{"className":21076},[2117],[54,21078],{"className":21079,"style":3846},[2121],[54,21081,21083],{"className":21082},[2126,3850],[54,21084,21086],{"className":21085},[2139],[54,21087,21089],{"className":21088},[2143],[54,21090,21092,21100],{"className":21091,"style":3846},[2147],[54,21093,21094,21097],{"style":3862},[54,21095],{"className":21096,"style":3866},[2155],[54,21098,2694],{"className":21099,"style":2709},[2126,2167],[54,21101,21102,21105],{"style":3872},[54,21103],{"className":21104,"style":3866},[2155],[54,21106,21108],{"className":21107,"style":3880},[3879],[54,21109,3833],{"className":21110},[2126]," via Hadamard test",[13,21113,21114],{},"The only difference between the circuits used in the Hadamard test will be the phase in the time-evolution operator and the observables measured. Therefore we can prepare a template circuit which represent the generic circuit for the Hadamard test, with placeholders for the gates that depend on the time-evolution operator.",[44,21116,21118],{"className":46,"code":21117,"language":48,"meta":49,"style":49},"# Parameters for the template circuits\nparameters = []\nfor idx in range(1, krylov_dim):\n parameters.append(2 * dt_circ * (idx))\n",[51,21119,21120,21125,21134,21152],{"__ignoreMap":49},[54,21121,21122],{"class":56,"line":57},[54,21123,21124],{"class":111},"# Parameters for the template circuits\n",[54,21126,21127,21130,21132],{"class":56,"line":74},[54,21128,21129],{"class":64},"parameters ",[54,21131,122],{"class":121},[54,21133,1466],{"class":64},[54,21135,21136,21138,21141,21143,21145,21147,21149],{"class":56,"line":88},[54,21137,1186],{"class":60},[54,21139,21140],{"class":64}," idx ",[54,21142,1192],{"class":60},[54,21144,7777],{"class":121},[54,21146,129],{"class":64},[54,21148,174],{"class":132},[54,21150,21151],{"class":64},", krylov_dim):\n",[54,21153,21154,21157,21159,21161,21163,21165,21168,21170],{"class":56,"line":101},[54,21155,21156],{"class":64}," parameters.",[54,21158,1495],{"class":125},[54,21160,129],{"class":64},[54,21162,133],{"class":132},[54,21164,7785],{"class":121},[54,21166,21167],{"class":64}," dt_circ ",[54,21169,1947],{"class":121},[54,21171,21172],{"class":64}," (idx))\n",[44,21174,21176],{"className":46,"code":21175,"language":48,"meta":49,"style":49},"# Create modified hadamard test circuit\nqr = QuantumRegister(n_qubits + 1)\nqc = QuantumCircuit(qr)\nqc.h(0)\nqc.compose(controlled_state_prep, list(range(n_qubits + 1)), inplace=True)\nqc.barrier()\nqc.compose(qc_evol, list(range(1, n_qubits + 1)), inplace=True)\nqc.barrier()\nqc.x(0)\nqc.compose(\n controlled_state_prep.inverse(), list(range(n_qubits + 1)), inplace=True\n)\nqc.x(0)\n\nqc.decompose().draw(\"mpl\", fold=-1)\n",[51,21177,21178,21183,21199,21209,21221,21253,21261,21296,21304,21316,21324,21354,21358,21370,21374],{"__ignoreMap":49},[54,21179,21180],{"class":56,"line":57},[54,21181,21182],{"class":111},"# Create modified hadamard test circuit\n",[54,21184,21185,21187,21189,21191,21193,21195,21197],{"class":56,"line":74},[54,21186,8940],{"class":64},[54,21188,122],{"class":121},[54,21190,8945],{"class":125},[54,21192,1660],{"class":64},[54,21194,1663],{"class":121},[54,21196,982],{"class":132},[54,21198,136],{"class":64},[54,21200,21201,21203,21205,21207],{"class":56,"line":88},[54,21202,118],{"class":64},[54,21204,122],{"class":121},[54,21206,126],{"class":125},[54,21208,8959],{"class":64},[54,21210,21211,21213,21215,21217,21219],{"class":56,"line":101},[54,21212,142],{"class":64},[54,21214,145],{"class":125},[54,21216,129],{"class":64},[54,21218,150],{"class":132},[54,21220,136],{"class":64},[54,21222,21223,21225,21227,21230,21232,21234,21236,21238,21240,21242,21245,21247,21249,21251],{"class":56,"line":108},[54,21224,142],{"class":64},[54,21226,13423],{"class":125},[54,21228,21229],{"class":64},"(controlled_state_prep, ",[54,21231,13609],{"class":121},[54,21233,129],{"class":64},[54,21235,1760],{"class":121},[54,21237,1660],{"class":64},[54,21239,1663],{"class":121},[54,21241,982],{"class":132},[54,21243,21244],{"class":64},")), ",[54,21246,13429],{"class":1711},[54,21248,122],{"class":121},[54,21250,13434],{"class":132},[54,21252,136],{"class":64},[54,21254,21255,21257,21259],{"class":56,"line":115},[54,21256,142],{"class":64},[54,21258,20754],{"class":125},[54,21260,330],{"class":64},[54,21262,21263,21265,21267,21270,21272,21274,21276,21278,21280,21282,21284,21286,21288,21290,21292,21294],{"class":56,"line":139},[54,21264,142],{"class":64},[54,21266,13423],{"class":125},[54,21268,21269],{"class":64},"(qc_evol, ",[54,21271,13609],{"class":121},[54,21273,129],{"class":64},[54,21275,1760],{"class":121},[54,21277,129],{"class":64},[54,21279,174],{"class":132},[54,21281,20614],{"class":64},[54,21283,1663],{"class":121},[54,21285,982],{"class":132},[54,21287,21244],{"class":64},[54,21289,13429],{"class":1711},[54,21291,122],{"class":121},[54,21293,13434],{"class":132},[54,21295,136],{"class":64},[54,21297,21298,21300,21302],{"class":56,"line":159},[54,21299,142],{"class":64},[54,21301,20754],{"class":125},[54,21303,330],{"class":64},[54,21305,21306,21308,21310,21312,21314],{"class":56,"line":183},[54,21307,142],{"class":64},[54,21309,8695],{"class":125},[54,21311,129],{"class":64},[54,21313,150],{"class":132},[54,21315,136],{"class":64},[54,21317,21318,21320,21322],{"class":56,"line":210},[54,21319,142],{"class":64},[54,21321,13423],{"class":125},[54,21323,932],{"class":64},[54,21325,21326,21329,21331,21333,21335,21337,21339,21341,21343,21345,21347,21349,21351],{"class":56,"line":227},[54,21327,21328],{"class":64}," controlled_state_prep.",[54,21330,13381],{"class":125},[54,21332,13506],{"class":64},[54,21334,13609],{"class":121},[54,21336,129],{"class":64},[54,21338,1760],{"class":121},[54,21340,1660],{"class":64},[54,21342,1663],{"class":121},[54,21344,982],{"class":132},[54,21346,21244],{"class":64},[54,21348,13429],{"class":1711},[54,21350,122],{"class":121},[54,21352,21353],{"class":132},"True\n",[54,21355,21356],{"class":56,"line":232},[54,21357,136],{"class":64},[54,21359,21360,21362,21364,21366,21368],{"class":56,"line":246},[54,21361,142],{"class":64},[54,21363,8695],{"class":125},[54,21365,129],{"class":64},[54,21367,150],{"class":132},[54,21369,136],{"class":64},[54,21371,21372],{"class":56,"line":259},[54,21373,105],{"emptyLinePlaceholder":104},[54,21375,21376,21378,21380,21382,21384,21386,21388,21390,21392,21394,21396],{"class":56,"line":264},[54,21377,142],{"class":64},[54,21379,13721],{"class":125},[54,21381,13539],{"class":64},[54,21383,8720],{"class":125},[54,21385,129],{"class":64},[54,21387,8725],{"class":279},[54,21389,171],{"class":64},[54,21391,13668],{"class":1711},[54,21393,13671],{"class":121},[54,21395,174],{"class":132},[54,21397,136],{"class":64},[380,21399],{"alt":8740,"src":21400},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-05.avif",[44,21402,21404],{"className":46,"code":21403,"language":48,"meta":49,"style":49},"print(\n \"The optimized circuit has 2Q gates depth: \",\n qc.decompose().decompose().depth(lambda x: x[0].num_qubits == 2),\n)\n",[51,21405,21406,21412,21419,21452],{"__ignoreMap":49},[54,21407,21408,21410],{"class":56,"line":57},[54,21409,8081],{"class":121},[54,21411,932],{"class":64},[54,21413,21414,21417],{"class":56,"line":74},[54,21415,21416],{"class":279}," \"The optimized circuit has 2Q gates depth: \"",[54,21418,965],{"class":64},[54,21420,21421,21424,21426,21428,21430,21432,21434,21436,21438,21440,21442,21444,21446,21448,21450],{"class":56,"line":88},[54,21422,21423],{"class":64}," qc.",[54,21425,13721],{"class":125},[54,21427,13539],{"class":64},[54,21429,13721],{"class":125},[54,21431,13539],{"class":64},[54,21433,13735],{"class":125},[54,21435,129],{"class":64},[54,21437,13740],{"class":60},[54,21439,13743],{"class":938},[54,21441,13746],{"class":64},[54,21443,150],{"class":132},[54,21445,13751],{"class":64},[54,21447,1279],{"class":121},[54,21449,1872],{"class":132},[54,21451,13758],{"class":64},[54,21453,21454],{"class":56,"line":101},[54,21455,136],{"class":64},[44,21457,21460],{"className":21458,"code":21459,"language":6595,"meta":49},[8087],"The optimized circuit has 2Q gates depth: 74\n",[51,21461,21459],{"__ignoreMap":49},[13,21463,21464],{},"We have considerably reduced the depth of the Hadamard test with a combination of Trotter approximation and uncontrolled unitaries",[31,21466,21468],{"id":21467},"step-3-execute-using-qiskit-primitives","Step 3: Execute using Qiskit primitives",[13,21470,21471],{},"Instantiate the backend and set runtime parameters",[44,21473,21475],{"className":46,"code":21474,"language":48,"meta":49,"style":49},"service = QiskitRuntimeService()\nbackend = service.least_busy(operational=True, simulator=False)\nif (\n \"if_else\" not in backend.target.operation_names\n): # Needed as \"op_name\" could be \"if_else\"\n backend.target.add_instruction(IfElseOp, name=\"if_else\")\nprint(backend.name)\n",[51,21476,21477,21489,21523,21529,21543,21551,21572],{"__ignoreMap":49},[54,21478,21479,21482,21484,21487],{"class":56,"line":57},[54,21480,21481],{"class":64},"service ",[54,21483,122],{"class":121},[54,21485,21486],{"class":125}," QiskitRuntimeService",[54,21488,330],{"class":64},[54,21490,21491,21494,21496,21499,21502,21504,21507,21509,21511,21513,21516,21518,21521],{"class":56,"line":74},[54,21492,21493],{"class":64},"backend ",[54,21495,122],{"class":121},[54,21497,21498],{"class":64}," service.",[54,21500,21501],{"class":125},"least_busy",[54,21503,129],{"class":64},[54,21505,21506],{"class":1711},"operational",[54,21508,122],{"class":121},[54,21510,13434],{"class":132},[54,21512,171],{"class":64},[54,21514,21515],{"class":1711},"simulator",[54,21517,122],{"class":121},[54,21519,21520],{"class":132},"False",[54,21522,136],{"class":64},[54,21524,21525,21527],{"class":56,"line":88},[54,21526,1201],{"class":60},[54,21528,890],{"class":64},[54,21530,21531,21534,21537,21540],{"class":56,"line":101},[54,21532,21533],{"class":279}," \"if_else\"",[54,21535,21536],{"class":60}," not",[54,21538,21539],{"class":60}," in",[54,21541,21542],{"class":64}," backend.target.operation_names\n",[54,21544,21545,21548],{"class":56,"line":108},[54,21546,21547],{"class":64},"): ",[54,21549,21550],{"class":111},"# Needed as \"op_name\" could be \"if_else\"\n",[54,21552,21553,21556,21559,21562,21565,21567,21570],{"class":56,"line":115},[54,21554,21555],{"class":64}," backend.target.",[54,21557,21558],{"class":125},"add_instruction",[54,21560,21561],{"class":64},"(IfElseOp, ",[54,21563,21564],{"class":1711},"name",[54,21566,122],{"class":121},[54,21568,21569],{"class":279},"\"if_else\"",[54,21571,136],{"class":64},[54,21573,21574,21576],{"class":56,"line":139},[54,21575,8081],{"class":121},[54,21577,21578],{"class":64},"(backend.name)\n",[2069,21580,21582],{"id":21581},"transpiling-to-a-qpu","Transpiling to a QPU",[13,21584,21585],{},"First, let's pick subsets of the coupling map with \"good\" performing qubits (where \"good\" is pretty arbitrary here, we mostly want to avoid really poor performing qubits) and create a new target for transpilation",[44,21587,21589],{"className":46,"code":21588,"language":48,"meta":49,"style":49},"target = backend.target\ncmap = target.build_coupling_map(filter_idle_qubits=True)\ncmap_list = list(cmap.get_edges())\n\ncust_cmap_list = copy.deepcopy(cmap_list)\nfor q in range(target.num_qubits):\n meas_err = target[\"measure\"][(q,)].error\n t2 = target.qubit_properties[q].t2 * 1e6\n if meas_err > 0.02 or t2 \u003C 100:\n for q_pair in cmap_list:\n if q in q_pair:\n try:\n cust_cmap_list.remove(q_pair)\n except:\n continue\n\nfor q in cmap_list:\n op_name = list(target.operation_names_for_qargs(q))[0]\n twoq_gate_err = target[f\"{op_name}\"][q].error\n if twoq_gate_err > 0.005:\n for q_pair in cmap_list:\n if q == q_pair:\n try:\n cust_cmap_list.remove(q)\n except:\n continue\n\n\ncust_cmap = CouplingMap(cust_cmap_list)\ncust_target = Target.from_configuration(\n basis_gates=backend.configuration().basis_gates,\n coupling_map=cust_cmap,\n)\n",[51,21590,21591,21601,21625,21644,21648,21664,21678,21694,21709,21734,21746,21757,21764,21775,21782,21787,21791,21801,21823,21850,21864,21874,21884,21890,21899,21905,21909,21913,21917,21930,21945,21961,21971],{"__ignoreMap":49},[54,21592,21593,21596,21598],{"class":56,"line":57},[54,21594,21595],{"class":64},"target ",[54,21597,122],{"class":121},[54,21599,21600],{"class":64}," backend.target\n",[54,21602,21603,21606,21608,21611,21614,21616,21619,21621,21623],{"class":56,"line":74},[54,21604,21605],{"class":64},"cmap ",[54,21607,122],{"class":121},[54,21609,21610],{"class":64}," target.",[54,21612,21613],{"class":125},"build_coupling_map",[54,21615,129],{"class":64},[54,21617,21618],{"class":1711},"filter_idle_qubits",[54,21620,122],{"class":121},[54,21622,13434],{"class":132},[54,21624,136],{"class":64},[54,21626,21627,21630,21632,21635,21638,21641],{"class":56,"line":88},[54,21628,21629],{"class":64},"cmap_list ",[54,21631,122],{"class":121},[54,21633,21634],{"class":121}," list",[54,21636,21637],{"class":64},"(cmap.",[54,21639,21640],{"class":125},"get_edges",[54,21642,21643],{"class":64},"())\n",[54,21645,21646],{"class":56,"line":101},[54,21647,105],{"emptyLinePlaceholder":104},[54,21649,21650,21653,21655,21658,21661],{"class":56,"line":108},[54,21651,21652],{"class":64},"cust_cmap_list ",[54,21654,122],{"class":121},[54,21656,21657],{"class":64}," copy.",[54,21659,21660],{"class":125},"deepcopy",[54,21662,21663],{"class":64},"(cmap_list)\n",[54,21665,21666,21668,21671,21673,21675],{"class":56,"line":115},[54,21667,1186],{"class":60},[54,21669,21670],{"class":64}," q ",[54,21672,1192],{"class":60},[54,21674,7777],{"class":121},[54,21676,21677],{"class":64},"(target.num_qubits):\n",[54,21679,21680,21683,21685,21688,21691],{"class":56,"line":139},[54,21681,21682],{"class":64}," meas_err ",[54,21684,122],{"class":121},[54,21686,21687],{"class":64}," target[",[54,21689,21690],{"class":279},"\"measure\"",[54,21692,21693],{"class":64},"][(q,)].error\n",[54,21695,21696,21699,21701,21704,21706],{"class":56,"line":159},[54,21697,21698],{"class":64}," t2 ",[54,21700,122],{"class":121},[54,21702,21703],{"class":64}," target.qubit_properties[q].t2 ",[54,21705,1947],{"class":121},[54,21707,21708],{"class":132}," 1e6\n",[54,21710,21711,21713,21716,21718,21721,21724,21727,21729,21732],{"class":56,"line":183},[54,21712,1273],{"class":60},[54,21714,21715],{"class":64}," meas_err ",[54,21717,1207],{"class":121},[54,21719,21720],{"class":132}," 0.02",[54,21722,21723],{"class":60}," or",[54,21725,21726],{"class":64}," t2 ",[54,21728,5233],{"class":121},[54,21730,21731],{"class":132}," 100",[54,21733,1284],{"class":64},[54,21735,21736,21738,21741,21743],{"class":56,"line":210},[54,21737,1823],{"class":60},[54,21739,21740],{"class":64}," q_pair ",[54,21742,1192],{"class":60},[54,21744,21745],{"class":64}," cmap_list:\n",[54,21747,21748,21750,21752,21754],{"class":56,"line":227},[54,21749,1854],{"class":60},[54,21751,21670],{"class":64},[54,21753,1192],{"class":60},[54,21755,21756],{"class":64}," q_pair:\n",[54,21758,21759,21762],{"class":56,"line":232},[54,21760,21761],{"class":60}," try",[54,21763,1284],{"class":64},[54,21765,21766,21769,21772],{"class":56,"line":246},[54,21767,21768],{"class":64}," cust_cmap_list.",[54,21770,21771],{"class":125},"remove",[54,21773,21774],{"class":64},"(q_pair)\n",[54,21776,21777,21780],{"class":56,"line":259},[54,21778,21779],{"class":60}," except",[54,21781,1284],{"class":64},[54,21783,21784],{"class":56,"line":264},[54,21785,21786],{"class":60}," continue\n",[54,21788,21789],{"class":56,"line":270},[54,21790,105],{"emptyLinePlaceholder":104},[54,21792,21793,21795,21797,21799],{"class":56,"line":285},[54,21794,1186],{"class":60},[54,21796,21670],{"class":64},[54,21798,1192],{"class":60},[54,21800,21745],{"class":64},[54,21802,21803,21806,21808,21810,21813,21816,21819,21821],{"class":56,"line":297},[54,21804,21805],{"class":64}," op_name ",[54,21807,122],{"class":121},[54,21809,21634],{"class":121},[54,21811,21812],{"class":64},"(target.",[54,21814,21815],{"class":125},"operation_names_for_qargs",[54,21817,21818],{"class":64},"(q))[",[54,21820,150],{"class":132},[54,21822,1350],{"class":64},[54,21824,21825,21828,21830,21832,21835,21838,21840,21843,21845,21847],{"class":56,"line":310},[54,21826,21827],{"class":64}," twoq_gate_err ",[54,21829,122],{"class":121},[54,21831,21687],{"class":64},[54,21833,21834],{"class":60},"f",[54,21836,21837],{"class":279},"\"",[54,21839,2366],{"class":132},[54,21841,21842],{"class":64},"op_name",[54,21844,2428],{"class":132},[54,21846,21837],{"class":279},[54,21848,21849],{"class":64},"][q].error\n",[54,21851,21852,21854,21857,21859,21862],{"class":56,"line":324},[54,21853,1273],{"class":60},[54,21855,21856],{"class":64}," twoq_gate_err ",[54,21858,1207],{"class":121},[54,21860,21861],{"class":132}," 0.005",[54,21863,1284],{"class":64},[54,21865,21866,21868,21870,21872],{"class":56,"line":333},[54,21867,1823],{"class":60},[54,21869,21740],{"class":64},[54,21871,1192],{"class":60},[54,21873,21745],{"class":64},[54,21875,21876,21878,21880,21882],{"class":56,"line":345},[54,21877,1854],{"class":60},[54,21879,21670],{"class":64},[54,21881,1279],{"class":121},[54,21883,21756],{"class":64},[54,21885,21886,21888],{"class":56,"line":358},[54,21887,21761],{"class":60},[54,21889,1284],{"class":64},[54,21891,21892,21894,21896],{"class":56,"line":908},[54,21893,21768],{"class":64},[54,21895,21771],{"class":125},[54,21897,21898],{"class":64},"(q)\n",[54,21900,21901,21903],{"class":56,"line":913},[54,21902,21779],{"class":60},[54,21904,1284],{"class":64},[54,21906,21907],{"class":56,"line":918},[54,21908,21786],{"class":60},[54,21910,21911],{"class":56,"line":923},[54,21912,105],{"emptyLinePlaceholder":104},[54,21914,21915],{"class":56,"line":935},[54,21916,105],{"emptyLinePlaceholder":104},[54,21918,21919,21922,21924,21927],{"class":56,"line":945},[54,21920,21921],{"class":64},"cust_cmap ",[54,21923,122],{"class":121},[54,21925,21926],{"class":125}," CouplingMap",[54,21928,21929],{"class":64},"(cust_cmap_list)\n",[54,21931,21932,21935,21937,21940,21943],{"class":56,"line":953},[54,21933,21934],{"class":64},"cust_target ",[54,21936,122],{"class":121},[54,21938,21939],{"class":64}," Target.",[54,21941,21942],{"class":125},"from_configuration",[54,21944,932],{"class":64},[54,21946,21947,21950,21952,21955,21958],{"class":56,"line":968},[54,21948,21949],{"class":1711}," basis_gates",[54,21951,122],{"class":121},[54,21953,21954],{"class":64},"backend.",[54,21956,21957],{"class":125},"configuration",[54,21959,21960],{"class":64},"().basis_gates,\n",[54,21962,21963,21966,21968],{"class":56,"line":987},[54,21964,21965],{"class":1711}," coupling_map",[54,21967,122],{"class":121},[54,21969,21970],{"class":64},"cust_cmap,\n",[54,21972,21973],{"class":56,"line":1005},[54,21974,136],{"class":64},[13,21976,21977],{},"Then transpile the virtual circuit to the best physical layout in this new target",[44,21979,21981],{"className":46,"code":21980,"language":48,"meta":49,"style":49},"basis_gates = list(target.operation_names)\npm = generate_preset_pass_manager(\n optimization_level=3,\n target=cust_target,\n basis_gates=basis_gates,\n)\n\nqc_trans = pm.run(qc)\n\nprint(\"depth\", qc_trans.depth(lambda x: x[0].num_qubits == 2))\nprint(\"num 2q ops\", qc_trans.count_ops())\nprint(\n \"physical qubits\",\n sorted(\n [\n idx\n for idx, qb in qc_trans.layout.initial_layout.get_physical_bits().items()\n if qb._register.name != \"ancilla\"\n ]\n ),\n)\n",[51,21982,21983,21995,22007,22018,22028,22037,22041,22045,22060,22064,22097,22113,22119,22126,22133,22138,22143,22166,22179,22184,22189],{"__ignoreMap":49},[54,21984,21985,21988,21990,21992],{"class":56,"line":57},[54,21986,21987],{"class":64},"basis_gates ",[54,21989,122],{"class":121},[54,21991,21634],{"class":121},[54,21993,21994],{"class":64},"(target.operation_names)\n",[54,21996,21997,22000,22002,22005],{"class":56,"line":74},[54,21998,21999],{"class":64},"pm ",[54,22001,122],{"class":121},[54,22003,22004],{"class":125}," generate_preset_pass_manager",[54,22006,932],{"class":64},[54,22008,22009,22012,22014,22016],{"class":56,"line":88},[54,22010,22011],{"class":1711}," optimization_level",[54,22013,122],{"class":121},[54,22015,7782],{"class":132},[54,22017,965],{"class":64},[54,22019,22020,22023,22025],{"class":56,"line":101},[54,22021,22022],{"class":1711}," target",[54,22024,122],{"class":121},[54,22026,22027],{"class":64},"cust_target,\n",[54,22029,22030,22032,22034],{"class":56,"line":108},[54,22031,21949],{"class":1711},[54,22033,122],{"class":121},[54,22035,22036],{"class":64},"basis_gates,\n",[54,22038,22039],{"class":56,"line":115},[54,22040,136],{"class":64},[54,22042,22043],{"class":56,"line":139},[54,22044,105],{"emptyLinePlaceholder":104},[54,22046,22047,22050,22052,22055,22058],{"class":56,"line":159},[54,22048,22049],{"class":64},"qc_trans ",[54,22051,122],{"class":121},[54,22053,22054],{"class":64}," pm.",[54,22056,22057],{"class":125},"run",[54,22059,243],{"class":64},[54,22061,22062],{"class":56,"line":183},[54,22063,105],{"emptyLinePlaceholder":104},[54,22065,22066,22068,22070,22073,22076,22078,22080,22082,22084,22086,22088,22090,22092,22094],{"class":56,"line":210},[54,22067,8081],{"class":121},[54,22069,129],{"class":64},[54,22071,22072],{"class":279},"\"depth\"",[54,22074,22075],{"class":64},", qc_trans.",[54,22077,13735],{"class":125},[54,22079,129],{"class":64},[54,22081,13740],{"class":60},[54,22083,13743],{"class":938},[54,22085,13746],{"class":64},[54,22087,150],{"class":132},[54,22089,13751],{"class":64},[54,22091,1279],{"class":121},[54,22093,1872],{"class":132},[54,22095,22096],{"class":64},"))\n",[54,22098,22099,22101,22103,22106,22108,22111],{"class":56,"line":227},[54,22100,8081],{"class":121},[54,22102,129],{"class":64},[54,22104,22105],{"class":279},"\"num 2q ops\"",[54,22107,22075],{"class":64},[54,22109,22110],{"class":125},"count_ops",[54,22112,21643],{"class":64},[54,22114,22115,22117],{"class":56,"line":232},[54,22116,8081],{"class":121},[54,22118,932],{"class":64},[54,22120,22121,22124],{"class":56,"line":246},[54,22122,22123],{"class":279}," \"physical qubits\"",[54,22125,965],{"class":64},[54,22127,22128,22131],{"class":56,"line":259},[54,22129,22130],{"class":121}," sorted",[54,22132,932],{"class":64},[54,22134,22135],{"class":56,"line":264},[54,22136,22137],{"class":64}," [\n",[54,22139,22140],{"class":56,"line":270},[54,22141,22142],{"class":64}," idx\n",[54,22144,22145,22148,22151,22153,22156,22159,22161,22164],{"class":56,"line":285},[54,22146,22147],{"class":60}," for",[54,22149,22150],{"class":64}," idx, qb ",[54,22152,1192],{"class":60},[54,22154,22155],{"class":64}," qc_trans.layout.initial_layout.",[54,22157,22158],{"class":125},"get_physical_bits",[54,22160,13539],{"class":64},[54,22162,22163],{"class":125},"items",[54,22165,330],{"class":64},[54,22167,22168,22170,22173,22176],{"class":56,"line":297},[54,22169,1854],{"class":60},[54,22171,22172],{"class":64}," qb._register.name ",[54,22174,22175],{"class":121},"!=",[54,22177,22178],{"class":279}," \"ancilla\"\n",[54,22180,22181],{"class":56,"line":310},[54,22182,22183],{"class":64}," ]\n",[54,22185,22186],{"class":56,"line":324},[54,22187,22188],{"class":64}," ),\n",[54,22190,22191],{"class":56,"line":333},[54,22192,136],{"class":64},[44,22194,22197],{"className":22195,"code":22196,"language":6595,"meta":49},[8087],"depth 52\nnum 2q ops OrderedDict([('rz', 2058), ('sx', 1703), ('cz', 728), ('x', 84), ('barrier', 8)])\nphysical qubits [91, 92, 93, 94, 95, 98, 99, 108, 109, 110, 111, 113, 114, 115, 119, 127, 132, 133, 134, 135, 137, 139, 147, 148, 149, 150, 151, 152, 153, 154, 155]\n",[51,22198,22196],{"__ignoreMap":49},[2069,22200,22202],{"id":22201},"create-pubs-for-execution-with-estimator","Create PUBs for execution with Estimator",[44,22204,22206],{"className":46,"code":22205,"language":48,"meta":49,"style":49},"# Define observables to measure for S\nobservable_S_real = \"I\" * (n_qubits) + \"X\"\nobservable_S_imag = \"I\" * (n_qubits) + \"Y\"\n\nobservable_op_real = SparsePauliOp(\n observable_S_real\n) # define a sparse pauli operator for the observable\nobservable_op_imag = SparsePauliOp(observable_S_imag)\n\nlayout = qc_trans.layout # get layout of transpiled circuit\nobservable_op_real = observable_op_real.apply_layout(\n layout\n) # apply physical layout to the observable\nobservable_op_imag = observable_op_imag.apply_layout(layout)\nobservable_S_real = (\n observable_op_real.paulis.to_labels()\n) # get the label of the physical observable\nobservable_S_imag = observable_op_imag.paulis.to_labels()\n\nobservables_S = [[observable_S_real], [observable_S_imag]]\n\n\n# Define observables to measure for H\n# Hamiltonian terms to measure\nobservable_list = []\nfor pauli, coeff in zip(H_op.paulis, H_op.coeffs):\n # print(pauli)\n observable_H_real = pauli[::-1].to_label() + \"X\"\n observable_H_imag = pauli[::-1].to_label() + \"Y\"\n observable_list.append([observable_H_real])\n observable_list.append([observable_H_imag])\n\nlayout = qc_trans.layout\n\nobservable_trans_list = []\nfor observable in observable_list:\n observable_op = SparsePauliOp(observable)\n observable_op = observable_op.apply_layout(layout)\n observable_trans_list.append([observable_op.paulis.to_labels()])\n\nobservables_H = observable_trans_list\n\n\n# Define a sweep over parameter values\nparams = np.vstack(parameters).T\n\n\n# Estimate the expectation value for all combinations of\n# observables and parameter values, where the pub result will have\n# shape (# observables, # parameter values).\npub = (qc_trans, observables_S + observables_H, params)\n",[51,22207,22208,22213,22232,22249,22253,22265,22270,22278,22290,22294,22307,22321,22326,22333,22347,22355,22365,22372,22385,22389,22399,22403,22407,22412,22417,22426,22440,22445,22471,22494,22504,22513,22517,22526,22530,22539,22551,22563,22576,22591,22595,22605,22609,22613,22618,22633,22637,22641,22646,22651,22656],{"__ignoreMap":49},[54,22209,22210],{"class":56,"line":57},[54,22211,22212],{"class":111},"# Define observables to measure for S\n",[54,22214,22215,22218,22220,22223,22225,22228,22230],{"class":56,"line":74},[54,22216,22217],{"class":64},"observable_S_real ",[54,22219,122],{"class":121},[54,22221,22222],{"class":279}," \"I\"",[54,22224,7785],{"class":121},[54,22226,22227],{"class":64}," (n_qubits) ",[54,22229,1663],{"class":121},[54,22231,7877],{"class":279},[54,22233,22234,22237,22239,22241,22243,22245,22247],{"class":56,"line":88},[54,22235,22236],{"class":64},"observable_S_imag ",[54,22238,122],{"class":121},[54,22240,22222],{"class":279},[54,22242,7785],{"class":121},[54,22244,22227],{"class":64},[54,22246,1663],{"class":121},[54,22248,7944],{"class":279},[54,22250,22251],{"class":56,"line":101},[54,22252,105],{"emptyLinePlaceholder":104},[54,22254,22255,22258,22260,22263],{"class":56,"line":108},[54,22256,22257],{"class":64},"observable_op_real ",[54,22259,122],{"class":121},[54,22261,22262],{"class":125}," SparsePauliOp",[54,22264,932],{"class":64},[54,22266,22267],{"class":56,"line":115},[54,22268,22269],{"class":64}," observable_S_real\n",[54,22271,22272,22275],{"class":56,"line":139},[54,22273,22274],{"class":64},") ",[54,22276,22277],{"class":111},"# define a sparse pauli operator for the observable\n",[54,22279,22280,22283,22285,22287],{"class":56,"line":159},[54,22281,22282],{"class":64},"observable_op_imag ",[54,22284,122],{"class":121},[54,22286,22262],{"class":125},[54,22288,22289],{"class":64},"(observable_S_imag)\n",[54,22291,22292],{"class":56,"line":183},[54,22293,105],{"emptyLinePlaceholder":104},[54,22295,22296,22299,22301,22304],{"class":56,"line":210},[54,22297,22298],{"class":64},"layout ",[54,22300,122],{"class":121},[54,22302,22303],{"class":64}," qc_trans.layout ",[54,22305,22306],{"class":111},"# get layout of transpiled circuit\n",[54,22308,22309,22311,22313,22316,22319],{"class":56,"line":227},[54,22310,22257],{"class":64},[54,22312,122],{"class":121},[54,22314,22315],{"class":64}," observable_op_real.",[54,22317,22318],{"class":125},"apply_layout",[54,22320,932],{"class":64},[54,22322,22323],{"class":56,"line":232},[54,22324,22325],{"class":64}," layout\n",[54,22327,22328,22330],{"class":56,"line":246},[54,22329,22274],{"class":64},[54,22331,22332],{"class":111},"# apply physical layout to the observable\n",[54,22334,22335,22337,22339,22342,22344],{"class":56,"line":259},[54,22336,22282],{"class":64},[54,22338,122],{"class":121},[54,22340,22341],{"class":64}," observable_op_imag.",[54,22343,22318],{"class":125},[54,22345,22346],{"class":64},"(layout)\n",[54,22348,22349,22351,22353],{"class":56,"line":264},[54,22350,22217],{"class":64},[54,22352,122],{"class":121},[54,22354,890],{"class":64},[54,22356,22357,22360,22363],{"class":56,"line":270},[54,22358,22359],{"class":64}," observable_op_real.paulis.",[54,22361,22362],{"class":125},"to_labels",[54,22364,330],{"class":64},[54,22366,22367,22369],{"class":56,"line":285},[54,22368,22274],{"class":64},[54,22370,22371],{"class":111},"# get the label of the physical observable\n",[54,22373,22374,22376,22378,22381,22383],{"class":56,"line":297},[54,22375,22236],{"class":64},[54,22377,122],{"class":121},[54,22379,22380],{"class":64}," observable_op_imag.paulis.",[54,22382,22362],{"class":125},[54,22384,330],{"class":64},[54,22386,22387],{"class":56,"line":310},[54,22388,105],{"emptyLinePlaceholder":104},[54,22390,22391,22394,22396],{"class":56,"line":324},[54,22392,22393],{"class":64},"observables_S ",[54,22395,122],{"class":121},[54,22397,22398],{"class":64}," [[observable_S_real], [observable_S_imag]]\n",[54,22400,22401],{"class":56,"line":333},[54,22402,105],{"emptyLinePlaceholder":104},[54,22404,22405],{"class":56,"line":345},[54,22406,105],{"emptyLinePlaceholder":104},[54,22408,22409],{"class":56,"line":358},[54,22410,22411],{"class":111},"# Define observables to measure for H\n",[54,22413,22414],{"class":56,"line":908},[54,22415,22416],{"class":111},"# Hamiltonian terms to measure\n",[54,22418,22419,22422,22424],{"class":56,"line":913},[54,22420,22421],{"class":64},"observable_list ",[54,22423,122],{"class":121},[54,22425,1466],{"class":64},[54,22427,22428,22430,22433,22435,22437],{"class":56,"line":918},[54,22429,1186],{"class":60},[54,22431,22432],{"class":64}," pauli, coeff ",[54,22434,1192],{"class":60},[54,22436,1195],{"class":121},[54,22438,22439],{"class":64},"(H_op.paulis, H_op.coeffs):\n",[54,22441,22442],{"class":56,"line":923},[54,22443,22444],{"class":111}," # print(pauli)\n",[54,22446,22447,22450,22452,22455,22457,22459,22462,22465,22467,22469],{"class":56,"line":935},[54,22448,22449],{"class":64}," observable_H_real ",[54,22451,122],{"class":121},[54,22453,22454],{"class":64}," pauli[::",[54,22456,1923],{"class":121},[54,22458,174],{"class":132},[54,22460,22461],{"class":64},"].",[54,22463,22464],{"class":125},"to_label",[54,22466,1233],{"class":64},[54,22468,1663],{"class":121},[54,22470,7877],{"class":279},[54,22472,22473,22476,22478,22480,22482,22484,22486,22488,22490,22492],{"class":56,"line":945},[54,22474,22475],{"class":64}," observable_H_imag ",[54,22477,122],{"class":121},[54,22479,22454],{"class":64},[54,22481,1923],{"class":121},[54,22483,174],{"class":132},[54,22485,22461],{"class":64},[54,22487,22464],{"class":125},[54,22489,1233],{"class":64},[54,22491,1663],{"class":121},[54,22493,7944],{"class":279},[54,22495,22496,22499,22501],{"class":56,"line":953},[54,22497,22498],{"class":64}," observable_list.",[54,22500,1495],{"class":125},[54,22502,22503],{"class":64},"([observable_H_real])\n",[54,22505,22506,22508,22510],{"class":56,"line":968},[54,22507,22498],{"class":64},[54,22509,1495],{"class":125},[54,22511,22512],{"class":64},"([observable_H_imag])\n",[54,22514,22515],{"class":56,"line":987},[54,22516,105],{"emptyLinePlaceholder":104},[54,22518,22519,22521,22523],{"class":56,"line":1005},[54,22520,22298],{"class":64},[54,22522,122],{"class":121},[54,22524,22525],{"class":64}," qc_trans.layout\n",[54,22527,22528],{"class":56,"line":1021},[54,22529,105],{"emptyLinePlaceholder":104},[54,22531,22532,22535,22537],{"class":56,"line":1027},[54,22533,22534],{"class":64},"observable_trans_list ",[54,22536,122],{"class":121},[54,22538,1466],{"class":64},[54,22540,22541,22543,22546,22548],{"class":56,"line":1033},[54,22542,1186],{"class":60},[54,22544,22545],{"class":64}," observable ",[54,22547,1192],{"class":60},[54,22549,22550],{"class":64}," observable_list:\n",[54,22552,22553,22556,22558,22560],{"class":56,"line":1038},[54,22554,22555],{"class":64}," observable_op ",[54,22557,122],{"class":121},[54,22559,22262],{"class":125},[54,22561,22562],{"class":64},"(observable)\n",[54,22564,22565,22567,22569,22572,22574],{"class":56,"line":1044},[54,22566,22555],{"class":64},[54,22568,122],{"class":121},[54,22570,22571],{"class":64}," observable_op.",[54,22573,22318],{"class":125},[54,22575,22346],{"class":64},[54,22577,22578,22581,22583,22586,22588],{"class":56,"line":1050},[54,22579,22580],{"class":64}," observable_trans_list.",[54,22582,1495],{"class":125},[54,22584,22585],{"class":64},"([observable_op.paulis.",[54,22587,22362],{"class":125},[54,22589,22590],{"class":64},"()])\n",[54,22592,22593],{"class":56,"line":1056},[54,22594,105],{"emptyLinePlaceholder":104},[54,22596,22597,22600,22602],{"class":56,"line":1062},[54,22598,22599],{"class":64},"observables_H ",[54,22601,122],{"class":121},[54,22603,22604],{"class":64}," observable_trans_list\n",[54,22606,22607],{"class":56,"line":1068},[54,22608,105],{"emptyLinePlaceholder":104},[54,22610,22611],{"class":56,"line":1074},[54,22612,105],{"emptyLinePlaceholder":104},[54,22614,22615],{"class":56,"line":1080},[54,22616,22617],{"class":111},"# Define a sweep over parameter values\n",[54,22619,22620,22623,22625,22627,22630],{"class":56,"line":1086},[54,22621,22622],{"class":64},"params ",[54,22624,122],{"class":121},[54,22626,1172],{"class":64},[54,22628,22629],{"class":125},"vstack",[54,22631,22632],{"class":64},"(parameters).T\n",[54,22634,22635],{"class":56,"line":1092},[54,22636,105],{"emptyLinePlaceholder":104},[54,22638,22639],{"class":56,"line":1098},[54,22640,105],{"emptyLinePlaceholder":104},[54,22642,22643],{"class":56,"line":1104},[54,22644,22645],{"class":111},"# Estimate the expectation value for all combinations of\n",[54,22647,22648],{"class":56,"line":1109},[54,22649,22650],{"class":111},"# observables and parameter values, where the pub result will have\n",[54,22652,22653],{"class":56,"line":1115},[54,22654,22655],{"class":111},"# shape (# observables, # parameter values).\n",[54,22657,22658,22661,22663,22666,22668],{"class":56,"line":1121},[54,22659,22660],{"class":64},"pub ",[54,22662,122],{"class":121},[54,22664,22665],{"class":64}," (qc_trans, observables_S ",[54,22667,1663],{"class":121},[54,22669,22670],{"class":64}," observables_H, params)\n",[2069,22672,22674],{"id":22673},"run-circuits","Run circuits",[13,22676,22677,22678,22729],{},"Circuits for ",[54,22679,22681,22699],{"className":22680},[2079],[54,22682,22684],{"className":22683},[2083],[2085,22685,22686],{"xmlns":2087},[2089,22687,22688,22696],{},[2092,22689,22690,22692,22694],{},[2098,22691,2854],{},[2245,22693,122],{},[2249,22695,150],{},[2105,22697,22698],{"encoding":2107},"t=0",[54,22700,22702,22720],{"className":22701,"ariaHidden":2113},[2112],[54,22703,22705,22708,22711,22714,22717],{"className":22704},[2117],[54,22706],{"className":22707,"style":20179},[2121],[54,22709,2854],{"className":22710},[2126,2167],[54,22712],{"className":22713,"style":2475},[2270],[54,22715,122],{"className":22716},[2479],[54,22718],{"className":22719,"style":2475},[2270],[54,22721,22723,22726],{"className":22722},[2117],[54,22724],{"className":22725,"style":2285},[2121],[54,22727,150],{"className":22728},[2126]," are classically calculable",[44,22731,22733],{"className":46,"code":22732,"language":48,"meta":49,"style":49},"qc_cliff = qc.assign_parameters({t: 0})\n\n\n# Get expectation values from experiment\nS_expval_real = StabilizerState(qc_cliff).expectation_value(\n Pauli(\"I\" * (n_qubits) + \"X\")\n)\nS_expval_imag = StabilizerState(qc_cliff).expectation_value(\n Pauli(\"I\" * (n_qubits) + \"Y\")\n)\n\n# Get expectation values\nS_expval = S_expval_real + 1j * S_expval_imag\n\nH_expval = 0\nfor obs_idx, (pauli, coeff) in enumerate(zip(H_op.paulis, H_op.coeffs)):\n # Get expectation values from experiment\n expval_real = StabilizerState(qc_cliff).expectation_value(\n Pauli(pauli[::-1].to_label() + \"X\")\n )\n expval_imag = StabilizerState(qc_cliff).expectation_value(\n Pauli(pauli[::-1].to_label() + \"Y\")\n )\n expval = expval_real + 1j * expval_imag\n\n # Fill-in matrix elements\n H_expval += coeff * expval\n\n\nprint(H_expval)\n",[51,22734,22735,22756,22760,22764,22769,22787,22807,22811,22826,22845,22849,22853,22858,22879,22883,22892,22911,22916,22931,22955,22959,22974,22996,23000,23021,23025,23030,23045,23049,23053],{"__ignoreMap":49},[54,22736,22737,22740,22742,22745,22748,22751,22753],{"class":56,"line":57},[54,22738,22739],{"class":64},"qc_cliff ",[54,22741,122],{"class":121},[54,22743,22744],{"class":64}," qc.",[54,22746,22747],{"class":125},"assign_parameters",[54,22749,22750],{"class":64},"({t: ",[54,22752,150],{"class":132},[54,22754,22755],{"class":64},"})\n",[54,22757,22758],{"class":56,"line":74},[54,22759,105],{"emptyLinePlaceholder":104},[54,22761,22762],{"class":56,"line":88},[54,22763,105],{"emptyLinePlaceholder":104},[54,22765,22766],{"class":56,"line":101},[54,22767,22768],{"class":111},"# Get expectation values from experiment\n",[54,22770,22771,22774,22776,22779,22782,22785],{"class":56,"line":108},[54,22772,22773],{"class":64},"S_expval_real ",[54,22775,122],{"class":121},[54,22777,22778],{"class":125}," StabilizerState",[54,22780,22781],{"class":64},"(qc_cliff).",[54,22783,22784],{"class":125},"expectation_value",[54,22786,932],{"class":64},[54,22788,22789,22792,22794,22796,22798,22800,22802,22805],{"class":56,"line":115},[54,22790,22791],{"class":125}," Pauli",[54,22793,129],{"class":64},[54,22795,7759],{"class":279},[54,22797,7785],{"class":121},[54,22799,22227],{"class":64},[54,22801,1663],{"class":121},[54,22803,22804],{"class":279}," \"X\"",[54,22806,136],{"class":64},[54,22808,22809],{"class":56,"line":139},[54,22810,136],{"class":64},[54,22812,22813,22816,22818,22820,22822,22824],{"class":56,"line":159},[54,22814,22815],{"class":64},"S_expval_imag ",[54,22817,122],{"class":121},[54,22819,22778],{"class":125},[54,22821,22781],{"class":64},[54,22823,22784],{"class":125},[54,22825,932],{"class":64},[54,22827,22828,22830,22832,22834,22836,22838,22840,22843],{"class":56,"line":183},[54,22829,22791],{"class":125},[54,22831,129],{"class":64},[54,22833,7759],{"class":279},[54,22835,7785],{"class":121},[54,22837,22227],{"class":64},[54,22839,1663],{"class":121},[54,22841,22842],{"class":279}," \"Y\"",[54,22844,136],{"class":64},[54,22846,22847],{"class":56,"line":210},[54,22848,136],{"class":64},[54,22850,22851],{"class":56,"line":227},[54,22852,105],{"emptyLinePlaceholder":104},[54,22854,22855],{"class":56,"line":232},[54,22856,22857],{"class":111},"# Get expectation values\n",[54,22859,22860,22863,22865,22868,22870,22872,22874,22876],{"class":56,"line":246},[54,22861,22862],{"class":64},"S_expval ",[54,22864,122],{"class":121},[54,22866,22867],{"class":64}," S_expval_real ",[54,22869,1663],{"class":121},[54,22871,982],{"class":132},[54,22873,1928],{"class":60},[54,22875,7785],{"class":121},[54,22877,22878],{"class":64}," S_expval_imag\n",[54,22880,22881],{"class":56,"line":259},[54,22882,105],{"emptyLinePlaceholder":104},[54,22884,22885,22888,22890],{"class":56,"line":264},[54,22886,22887],{"class":64},"H_expval ",[54,22889,122],{"class":121},[54,22891,1802],{"class":132},[54,22893,22894,22896,22899,22901,22903,22905,22908],{"class":56,"line":270},[54,22895,1186],{"class":60},[54,22897,22898],{"class":64}," obs_idx, (pauli, coeff) ",[54,22900,1192],{"class":60},[54,22902,1513],{"class":121},[54,22904,129],{"class":64},[54,22906,22907],{"class":121},"zip",[54,22909,22910],{"class":64},"(H_op.paulis, H_op.coeffs)):\n",[54,22912,22913],{"class":56,"line":285},[54,22914,22915],{"class":111}," # Get expectation values from experiment\n",[54,22917,22918,22921,22923,22925,22927,22929],{"class":56,"line":297},[54,22919,22920],{"class":64}," expval_real ",[54,22922,122],{"class":121},[54,22924,22778],{"class":125},[54,22926,22781],{"class":64},[54,22928,22784],{"class":125},[54,22930,932],{"class":64},[54,22932,22933,22936,22939,22941,22943,22945,22947,22949,22951,22953],{"class":56,"line":310},[54,22934,22935],{"class":125}," Pauli",[54,22937,22938],{"class":64},"(pauli[::",[54,22940,1923],{"class":121},[54,22942,174],{"class":132},[54,22944,22461],{"class":64},[54,22946,22464],{"class":125},[54,22948,1233],{"class":64},[54,22950,1663],{"class":121},[54,22952,22804],{"class":279},[54,22954,136],{"class":64},[54,22956,22957],{"class":56,"line":324},[54,22958,1216],{"class":64},[54,22960,22961,22964,22966,22968,22970,22972],{"class":56,"line":333},[54,22962,22963],{"class":64}," expval_imag ",[54,22965,122],{"class":121},[54,22967,22778],{"class":125},[54,22969,22781],{"class":64},[54,22971,22784],{"class":125},[54,22973,932],{"class":64},[54,22975,22976,22978,22980,22982,22984,22986,22988,22990,22992,22994],{"class":56,"line":345},[54,22977,22935],{"class":125},[54,22979,22938],{"class":64},[54,22981,1923],{"class":121},[54,22983,174],{"class":132},[54,22985,22461],{"class":64},[54,22987,22464],{"class":125},[54,22989,1233],{"class":64},[54,22991,1663],{"class":121},[54,22993,22842],{"class":279},[54,22995,136],{"class":64},[54,22997,22998],{"class":56,"line":358},[54,22999,1216],{"class":64},[54,23001,23002,23005,23007,23010,23012,23014,23016,23018],{"class":56,"line":908},[54,23003,23004],{"class":64}," expval ",[54,23006,122],{"class":121},[54,23008,23009],{"class":64}," expval_real ",[54,23011,1663],{"class":121},[54,23013,982],{"class":132},[54,23015,1928],{"class":60},[54,23017,7785],{"class":121},[54,23019,23020],{"class":64}," expval_imag\n",[54,23022,23023],{"class":56,"line":913},[54,23024,105],{"emptyLinePlaceholder":104},[54,23026,23027],{"class":56,"line":918},[54,23028,23029],{"class":111}," # Fill-in matrix elements\n",[54,23031,23032,23035,23037,23040,23042],{"class":56,"line":923},[54,23033,23034],{"class":64}," H_expval ",[54,23036,1841],{"class":121},[54,23038,23039],{"class":64}," coeff ",[54,23041,1947],{"class":121},[54,23043,23044],{"class":64}," expval\n",[54,23046,23047],{"class":56,"line":935},[54,23048,105],{"emptyLinePlaceholder":104},[54,23050,23051],{"class":56,"line":945},[54,23052,105],{"emptyLinePlaceholder":104},[54,23054,23055,23057],{"class":56,"line":953},[54,23056,8081],{"class":121},[54,23058,23059],{"class":64},"(H_expval)\n",[44,23061,23064],{"className":23062,"code":23063,"language":6595,"meta":49},[8087],"(25+0j)\n",[51,23065,23063],{"__ignoreMap":49},[13,23067,23068,23069,10355,23097,23157],{},"Execute circuits for ",[54,23070,23072,23085],{"className":23071},[2079],[54,23073,23075],{"className":23074},[2083],[2085,23076,23077],{"xmlns":2087},[2089,23078,23079,23083],{},[2092,23080,23081],{},[2098,23082,5413],{},[2105,23084,5413],{"encoding":2107},[54,23086,23088],{"className":23087,"ariaHidden":2113},[2112],[54,23089,23091,23094],{"className":23090},[2117],[54,23092],{"className":23093,"style":2122},[2121],[54,23095,5413],{"className":23096,"style":5448},[2126,2167],[54,23098,23100,23117],{"className":23099},[2079],[54,23101,23103],{"className":23102},[2083],[2085,23104,23105],{"xmlns":2087},[2089,23106,23107,23115],{},[2092,23108,23109],{},[3827,23110,23111,23113],{"accent":2113},[2098,23112,2694],{},[2245,23114,3833],{},[2105,23116,3836],{"encoding":2107},[54,23118,23120],{"className":23119,"ariaHidden":2113},[2112],[54,23121,23123,23126],{"className":23122},[2117],[54,23124],{"className":23125,"style":3846},[2121],[54,23127,23129],{"className":23128},[2126,3850],[54,23130,23132],{"className":23131},[2139],[54,23133,23135],{"className":23134},[2143],[54,23136,23138,23146],{"className":23137,"style":3846},[2147],[54,23139,23140,23143],{"style":3862},[54,23141],{"className":23142,"style":3866},[2155],[54,23144,2694],{"className":23145,"style":2709},[2126,2167],[54,23147,23148,23151],{"style":3872},[54,23149],{"className":23150,"style":3866},[2155],[54,23152,23154],{"className":23153,"style":3880},[3879],[54,23155,3833],{"className":23156},[2126]," with Estimator",[44,23159,23161],{"className":46,"code":23160,"language":48,"meta":49,"style":49},"# Experiment options\nnum_randomizations = 300\nnum_randomizations_learning = 30\nshots_per_randomization = 100\nnoise_factors = [1, 1.2, 1.4]\nlearning_pair_depths = [0, 4, 24, 48]\n\n\nexperimental_opts = {}\nexperimental_opts[\"resilience\"] = {\n \"measure_mitigation\": True,\n \"measure_noise_learning\": {\n \"num_randomizations\": num_randomizations_learning,\n \"shots_per_randomization\": shots_per_randomization,\n },\n \"zne_mitigation\": True,\n \"zne\": {\"noise_factors\": noise_factors},\n \"layer_noise_learning\": {\n \"max_layers_to_learn\": 10,\n \"layer_pair_depths\": learning_pair_depths,\n \"shots_per_randomization\": shots_per_randomization,\n \"num_randomizations\": num_randomizations_learning,\n },\n \"zne\": {\n \"amplifier\": \"pea\",\n \"extrapolated_noise_factors\": [0] + noise_factors,\n },\n}\nexperimental_opts[\"twirling\"] = {\n \"num_randomizations\": num_randomizations,\n \"shots_per_randomization\": shots_per_randomization,\n \"strategy\": \"all\",\n}\n\nestimator = Estimator(mode=backend, options=experimental_opts)\n\n\njob = estimator.run([pub])\n",[51,23162,23163,23168,23178,23187,23197,23220,23247,23251,23255,23265,23280,23291,23299,23307,23315,23320,23331,23345,23352,23364,23372,23378,23384,23388,23394,23406,23423,23427,23432,23445,23453,23460,23472,23476,23480,23508,23512,23516],{"__ignoreMap":49},[54,23164,23165],{"class":56,"line":57},[54,23166,23167],{"class":111},"# Experiment options\n",[54,23169,23170,23173,23175],{"class":56,"line":74},[54,23171,23172],{"class":64},"num_randomizations ",[54,23174,122],{"class":121},[54,23176,23177],{"class":132}," 300\n",[54,23179,23180,23183,23185],{"class":56,"line":88},[54,23181,23182],{"class":64},"num_randomizations_learning ",[54,23184,122],{"class":121},[54,23186,7725],{"class":132},[54,23188,23189,23192,23194],{"class":56,"line":101},[54,23190,23191],{"class":64},"shots_per_randomization ",[54,23193,122],{"class":121},[54,23195,23196],{"class":132}," 100\n",[54,23198,23199,23202,23204,23206,23208,23210,23213,23215,23218],{"class":56,"line":108},[54,23200,23201],{"class":64},"noise_factors ",[54,23203,122],{"class":121},[54,23205,7983],{"class":64},[54,23207,174],{"class":132},[54,23209,171],{"class":64},[54,23211,23212],{"class":132},"1.2",[54,23214,171],{"class":64},[54,23216,23217],{"class":132},"1.4",[54,23219,1350],{"class":64},[54,23221,23222,23225,23227,23229,23231,23233,23235,23237,23240,23242,23245],{"class":56,"line":115},[54,23223,23224],{"class":64},"learning_pair_depths ",[54,23226,122],{"class":121},[54,23228,7983],{"class":64},[54,23230,150],{"class":132},[54,23232,171],{"class":64},[54,23234,11708],{"class":132},[54,23236,171],{"class":64},[54,23238,23239],{"class":132},"24",[54,23241,171],{"class":64},[54,23243,23244],{"class":132},"48",[54,23246,1350],{"class":64},[54,23248,23249],{"class":56,"line":139},[54,23250,105],{"emptyLinePlaceholder":104},[54,23252,23253],{"class":56,"line":159},[54,23254,105],{"emptyLinePlaceholder":104},[54,23256,23257,23260,23262],{"class":56,"line":183},[54,23258,23259],{"class":64},"experimental_opts ",[54,23261,122],{"class":121},[54,23263,23264],{"class":64}," {}\n",[54,23266,23267,23270,23273,23275,23277],{"class":56,"line":210},[54,23268,23269],{"class":64},"experimental_opts[",[54,23271,23272],{"class":279},"\"resilience\"",[54,23274,7762],{"class":64},[54,23276,122],{"class":121},[54,23278,23279],{"class":64}," {\n",[54,23281,23282,23285,23287,23289],{"class":56,"line":227},[54,23283,23284],{"class":279}," \"measure_mitigation\"",[54,23286,959],{"class":64},[54,23288,13434],{"class":132},[54,23290,965],{"class":64},[54,23292,23293,23296],{"class":56,"line":232},[54,23294,23295],{"class":279}," \"measure_noise_learning\"",[54,23297,23298],{"class":64},": {\n",[54,23300,23301,23304],{"class":56,"line":246},[54,23302,23303],{"class":279}," \"num_randomizations\"",[54,23305,23306],{"class":64},": num_randomizations_learning,\n",[54,23308,23309,23312],{"class":56,"line":259},[54,23310,23311],{"class":279}," \"shots_per_randomization\"",[54,23313,23314],{"class":64},": shots_per_randomization,\n",[54,23316,23317],{"class":56,"line":264},[54,23318,23319],{"class":64}," },\n",[54,23321,23322,23325,23327,23329],{"class":56,"line":270},[54,23323,23324],{"class":279}," \"zne_mitigation\"",[54,23326,959],{"class":64},[54,23328,13434],{"class":132},[54,23330,965],{"class":64},[54,23332,23333,23336,23339,23342],{"class":56,"line":285},[54,23334,23335],{"class":279}," \"zne\"",[54,23337,23338],{"class":64},": {",[54,23340,23341],{"class":279},"\"noise_factors\"",[54,23343,23344],{"class":64},": noise_factors},\n",[54,23346,23347,23350],{"class":56,"line":297},[54,23348,23349],{"class":279}," \"layer_noise_learning\"",[54,23351,23298],{"class":64},[54,23353,23354,23357,23359,23362],{"class":56,"line":310},[54,23355,23356],{"class":279}," \"max_layers_to_learn\"",[54,23358,959],{"class":64},[54,23360,23361],{"class":132},"10",[54,23363,965],{"class":64},[54,23365,23366,23369],{"class":56,"line":324},[54,23367,23368],{"class":279}," \"layer_pair_depths\"",[54,23370,23371],{"class":64},": learning_pair_depths,\n",[54,23373,23374,23376],{"class":56,"line":333},[54,23375,23311],{"class":279},[54,23377,23314],{"class":64},[54,23379,23380,23382],{"class":56,"line":345},[54,23381,23303],{"class":279},[54,23383,23306],{"class":64},[54,23385,23386],{"class":56,"line":358},[54,23387,23319],{"class":64},[54,23389,23390,23392],{"class":56,"line":908},[54,23391,23335],{"class":279},[54,23393,23298],{"class":64},[54,23395,23396,23399,23401,23404],{"class":56,"line":913},[54,23397,23398],{"class":279}," \"amplifier\"",[54,23400,959],{"class":64},[54,23402,23403],{"class":279},"\"pea\"",[54,23405,965],{"class":64},[54,23407,23408,23411,23414,23416,23418,23420],{"class":56,"line":918},[54,23409,23410],{"class":279}," \"extrapolated_noise_factors\"",[54,23412,23413],{"class":64},": [",[54,23415,150],{"class":132},[54,23417,7762],{"class":64},[54,23419,1663],{"class":121},[54,23421,23422],{"class":64}," noise_factors,\n",[54,23424,23425],{"class":56,"line":923},[54,23426,23319],{"class":64},[54,23428,23429],{"class":56,"line":935},[54,23430,23431],{"class":64},"}\n",[54,23433,23434,23436,23439,23441,23443],{"class":56,"line":945},[54,23435,23269],{"class":64},[54,23437,23438],{"class":279},"\"twirling\"",[54,23440,7762],{"class":64},[54,23442,122],{"class":121},[54,23444,23279],{"class":64},[54,23446,23447,23450],{"class":56,"line":953},[54,23448,23449],{"class":279}," \"num_randomizations\"",[54,23451,23452],{"class":64},": num_randomizations,\n",[54,23454,23455,23458],{"class":56,"line":968},[54,23456,23457],{"class":279}," \"shots_per_randomization\"",[54,23459,23314],{"class":64},[54,23461,23462,23465,23467,23470],{"class":56,"line":987},[54,23463,23464],{"class":279}," \"strategy\"",[54,23466,959],{"class":64},[54,23468,23469],{"class":279},"\"all\"",[54,23471,965],{"class":64},[54,23473,23474],{"class":56,"line":1005},[54,23475,23431],{"class":64},[54,23477,23478],{"class":56,"line":1021},[54,23479,105],{"emptyLinePlaceholder":104},[54,23481,23482,23485,23487,23490,23492,23495,23497,23500,23503,23505],{"class":56,"line":1027},[54,23483,23484],{"class":64},"estimator ",[54,23486,122],{"class":121},[54,23488,23489],{"class":125}," Estimator",[54,23491,129],{"class":64},[54,23493,23494],{"class":1711},"mode",[54,23496,122],{"class":121},[54,23498,23499],{"class":64},"backend, ",[54,23501,23502],{"class":1711},"options",[54,23504,122],{"class":121},[54,23506,23507],{"class":64},"experimental_opts)\n",[54,23509,23510],{"class":56,"line":1033},[54,23511,105],{"emptyLinePlaceholder":104},[54,23513,23514],{"class":56,"line":1038},[54,23515,105],{"emptyLinePlaceholder":104},[54,23517,23518,23521,23523,23526,23528],{"class":56,"line":1044},[54,23519,23520],{"class":64},"job ",[54,23522,122],{"class":121},[54,23524,23525],{"class":64}," estimator.",[54,23527,22057],{"class":125},[54,23529,23530],{"class":64},"([pub])\n",[31,23532,23534],{"id":23533},"step-4-post-process-and-return-result-in-desired-classical-format","Step 4: Post-process and return result in desired classical format",[44,23536,23538],{"className":46,"code":23537,"language":48,"meta":49,"style":49},"results = job.result()[0]\n",[51,23539,23540],{"__ignoreMap":49},[54,23541,23542,23545,23547,23550,23553,23556,23558],{"class":56,"line":57},[54,23543,23544],{"class":64},"results ",[54,23546,122],{"class":121},[54,23548,23549],{"class":64}," job.",[54,23551,23552],{"class":125},"result",[54,23554,23555],{"class":64},"()[",[54,23557,150],{"class":132},[54,23559,1350],{"class":64},[2069,23561,23563],{"id":23562},"calculate-effective-hamiltonian-and-overlap-matrices","Calculate Effective Hamiltonian and Overlap matrices",[13,23565,23566,23567,23603],{},"First calculate the phase accumulated by the ",[54,23568,23570,23588],{"className":23569},[2079],[54,23571,23573],{"className":23572},[2083],[2085,23574,23575],{"xmlns":2087},[2089,23576,23577,23585],{},[2092,23578,23579,23581,23583],{},[2098,23580,2305],{"mathvariant":2304},[2249,23582,150],{},[2245,23584,2312],{"stretchy":2311},[2105,23586,23587],{"encoding":2107},"\\vert 0 \\rangle",[54,23589,23591],{"className":23590,"ariaHidden":2113},[2112],[54,23592,23594,23597,23600],{"className":23593},[2117],[54,23595],{"className":23596,"style":2325},[2121],[54,23598,9291],{"className":23599},[2126],[54,23601,2312],{"className":23602},[2336]," state during the uncontrolled time evolution",[44,23605,23607],{"className":46,"code":23606,"language":48,"meta":49,"style":49},"prefactors = [\n np.exp(-1j * sum([c for p, c in H_op.to_list() if \"Z\" in p]) * i * dt)\n for i in range(1, krylov_dim)\n]\n",[51,23608,23609,23618,23673,23690],{"__ignoreMap":49},[54,23610,23611,23614,23616],{"class":56,"line":57},[54,23612,23613],{"class":64},"prefactors ",[54,23615,122],{"class":121},[54,23617,1733],{"class":64},[54,23619,23620,23623,23626,23628,23630,23632,23634,23636,23638,23641,23643,23646,23648,23650,23652,23654,23656,23659,23661,23664,23666,23668,23670],{"class":56,"line":74},[54,23621,23622],{"class":64}," np.",[54,23624,23625],{"class":125},"exp",[54,23627,129],{"class":64},[54,23629,1923],{"class":121},[54,23631,174],{"class":132},[54,23633,1928],{"class":60},[54,23635,7785],{"class":121},[54,23637,8371],{"class":121},[54,23639,23640],{"class":64},"([c ",[54,23642,1186],{"class":60},[54,23644,23645],{"class":64}," p, c ",[54,23647,1192],{"class":60},[54,23649,1480],{"class":64},[54,23651,1483],{"class":125},[54,23653,1233],{"class":64},[54,23655,1201],{"class":60},[54,23657,23658],{"class":279}," \"Z\"",[54,23660,21539],{"class":60},[54,23662,23663],{"class":64}," p]) ",[54,23665,1947],{"class":121},[54,23667,7802],{"class":64},[54,23669,1947],{"class":121},[54,23671,23672],{"class":64}," dt)\n",[54,23674,23675,23677,23679,23681,23683,23685,23687],{"class":56,"line":88},[54,23676,1472],{"class":60},[54,23678,7802],{"class":64},[54,23680,1192],{"class":60},[54,23682,7777],{"class":121},[54,23684,129],{"class":64},[54,23686,174],{"class":132},[54,23688,23689],{"class":64},", krylov_dim)\n",[54,23691,23692],{"class":56,"line":101},[54,23693,1350],{"class":64},[13,23695,23696,23697],{},"Once we have the results of the circuit executions we can post-process the data to calculate the matrix elements of ",[54,23698,23700,23713],{"className":23699},[2079],[54,23701,23703],{"className":23702},[2083],[2085,23704,23705],{"xmlns":2087},[2089,23706,23707,23711],{},[2092,23708,23709],{},[2098,23710,5413],{},[2105,23712,5413],{"encoding":2107},[54,23714,23716],{"className":23715,"ariaHidden":2113},[2112],[54,23717,23719,23722],{"className":23718},[2117],[54,23720],{"className":23721,"style":2122},[2121],[54,23723,5413],{"className":23724,"style":5448},[2126,2167],[44,23726,23728],{"className":46,"code":23727,"language":48,"meta":49,"style":49},"# Assemble S, the overlap matrix of dimension D:\nS_first_row = np.zeros(krylov_dim, dtype=complex)\nS_first_row[0] = 1 + 0j\n\n# Add in ancilla-only measurements:\nfor i in range(krylov_dim - 1):\n # Get expectation values from experiment\n expval_real = results.data.evs[0][0][\n i\n ] # automatic extrapolated evs if ZNE is used\n expval_imag = results.data.evs[1][0][\n i\n ] # automatic extrapolated evs if ZNE is used\n\n # Get expectation values\n expval = expval_real + 1j * expval_imag\n S_first_row[i + 1] += prefactors[i] * expval\n\nS_first_row_list = S_first_row.tolist() # for saving purposes\n\n\nS_circ = np.zeros((krylov_dim, krylov_dim), dtype=complex)\n\n# Distribute entries from first row across matrix:\nfor i, j in it.product(range(krylov_dim), repeat=2):\n if i >= j:\n S_circ[j, i] = S_first_row[i - j]\n else:\n S_circ[j, i] = np.conj(S_first_row[j - i])\n",[51,23729,23730,23735,23755,23775,23779,23784,23803,23807,23825,23830,23837,23853,23857,23863,23867,23872,23890,23910,23914,23933,23937,23941,23961,23965,23970,24000,24012,24027,24033],{"__ignoreMap":49},[54,23731,23732],{"class":56,"line":57},[54,23733,23734],{"class":111},"# Assemble S, the overlap matrix of dimension D:\n",[54,23736,23737,23740,23742,23744,23746,23749,23751,23753],{"class":56,"line":74},[54,23738,23739],{"class":64},"S_first_row ",[54,23741,122],{"class":121},[54,23743,1172],{"class":64},[54,23745,1705],{"class":125},[54,23747,23748],{"class":64},"(krylov_dim, ",[54,23750,1712],{"class":1711},[54,23752,1715],{"class":121},[54,23754,136],{"class":64},[54,23756,23757,23760,23762,23764,23766,23768,23770,23772],{"class":56,"line":88},[54,23758,23759],{"class":64},"S_first_row[",[54,23761,150],{"class":132},[54,23763,7762],{"class":64},[54,23765,122],{"class":121},[54,23767,982],{"class":132},[54,23769,7869],{"class":121},[54,23771,20814],{"class":132},[54,23773,23774],{"class":60},"j\n",[54,23776,23777],{"class":56,"line":101},[54,23778,105],{"emptyLinePlaceholder":104},[54,23780,23781],{"class":56,"line":108},[54,23782,23783],{"class":111},"# Add in ancilla-only measurements:\n",[54,23785,23786,23788,23790,23792,23794,23797,23799,23801],{"class":56,"line":115},[54,23787,1186],{"class":60},[54,23789,7802],{"class":64},[54,23791,1192],{"class":60},[54,23793,7777],{"class":121},[54,23795,23796],{"class":64},"(krylov_dim ",[54,23798,1923],{"class":121},[54,23800,982],{"class":132},[54,23802,1439],{"class":64},[54,23804,23805],{"class":56,"line":139},[54,23806,22915],{"class":111},[54,23808,23809,23811,23813,23816,23818,23820,23822],{"class":56,"line":159},[54,23810,22920],{"class":64},[54,23812,122],{"class":121},[54,23814,23815],{"class":64}," results.data.evs[",[54,23817,150],{"class":132},[54,23819,1311],{"class":64},[54,23821,150],{"class":132},[54,23823,23824],{"class":64},"][\n",[54,23826,23827],{"class":56,"line":183},[54,23828,23829],{"class":64}," i\n",[54,23831,23832,23834],{"class":56,"line":210},[54,23833,1783],{"class":64},[54,23835,23836],{"class":111},"# automatic extrapolated evs if ZNE is used\n",[54,23838,23839,23841,23843,23845,23847,23849,23851],{"class":56,"line":227},[54,23840,22963],{"class":64},[54,23842,122],{"class":121},[54,23844,23815],{"class":64},[54,23846,174],{"class":132},[54,23848,1311],{"class":64},[54,23850,150],{"class":132},[54,23852,23824],{"class":64},[54,23854,23855],{"class":56,"line":232},[54,23856,23829],{"class":64},[54,23858,23859,23861],{"class":56,"line":246},[54,23860,1783],{"class":64},[54,23862,23836],{"class":111},[54,23864,23865],{"class":56,"line":259},[54,23866,105],{"emptyLinePlaceholder":104},[54,23868,23869],{"class":56,"line":264},[54,23870,23871],{"class":111}," # Get expectation values\n",[54,23873,23874,23876,23878,23880,23882,23884,23886,23888],{"class":56,"line":270},[54,23875,23004],{"class":64},[54,23877,122],{"class":121},[54,23879,23009],{"class":64},[54,23881,1663],{"class":121},[54,23883,982],{"class":132},[54,23885,1928],{"class":60},[54,23887,7785],{"class":121},[54,23889,23020],{"class":64},[54,23891,23892,23895,23897,23899,23901,23903,23906,23908],{"class":56,"line":285},[54,23893,23894],{"class":64}," S_first_row[i ",[54,23896,1663],{"class":121},[54,23898,982],{"class":132},[54,23900,7762],{"class":64},[54,23902,1841],{"class":121},[54,23904,23905],{"class":64}," prefactors[i] ",[54,23907,1947],{"class":121},[54,23909,23044],{"class":64},[54,23911,23912],{"class":56,"line":297},[54,23913,105],{"emptyLinePlaceholder":104},[54,23915,23916,23919,23921,23924,23927,23930],{"class":56,"line":310},[54,23917,23918],{"class":64},"S_first_row_list ",[54,23920,122],{"class":121},[54,23922,23923],{"class":64}," S_first_row.",[54,23925,23926],{"class":125},"tolist",[54,23928,23929],{"class":64},"() ",[54,23931,23932],{"class":111},"# for saving purposes\n",[54,23934,23935],{"class":56,"line":324},[54,23936,105],{"emptyLinePlaceholder":104},[54,23938,23939],{"class":56,"line":333},[54,23940,105],{"emptyLinePlaceholder":104},[54,23942,23943,23946,23948,23950,23952,23955,23957,23959],{"class":56,"line":345},[54,23944,23945],{"class":64},"S_circ ",[54,23947,122],{"class":121},[54,23949,1172],{"class":64},[54,23951,1705],{"class":125},[54,23953,23954],{"class":64},"((krylov_dim, krylov_dim), ",[54,23956,1712],{"class":1711},[54,23958,1715],{"class":121},[54,23960,136],{"class":64},[54,23962,23963],{"class":56,"line":358},[54,23964,105],{"emptyLinePlaceholder":104},[54,23966,23967],{"class":56,"line":908},[54,23968,23969],{"class":111},"# Distribute entries from first row across matrix:\n",[54,23971,23972,23974,23977,23979,23981,23984,23986,23988,23991,23994,23996,23998],{"class":56,"line":913},[54,23973,1186],{"class":60},[54,23975,23976],{"class":64}," i, j ",[54,23978,1192],{"class":60},[54,23980,1752],{"class":64},[54,23982,23983],{"class":125},"product",[54,23985,129],{"class":64},[54,23987,1760],{"class":121},[54,23989,23990],{"class":64},"(krylov_dim), ",[54,23992,23993],{"class":1711},"repeat",[54,23995,122],{"class":121},[54,23997,133],{"class":132},[54,23999,1439],{"class":64},[54,24001,24002,24004,24006,24009],{"class":56,"line":918},[54,24003,1273],{"class":60},[54,24005,7802],{"class":64},[54,24007,24008],{"class":121},">=",[54,24010,24011],{"class":64}," j:\n",[54,24013,24014,24017,24019,24022,24024],{"class":56,"line":923},[54,24015,24016],{"class":64}," S_circ[j, i] ",[54,24018,122],{"class":121},[54,24020,24021],{"class":64}," S_first_row[i ",[54,24023,1923],{"class":121},[54,24025,24026],{"class":64}," j]\n",[54,24028,24029,24031],{"class":56,"line":935},[54,24030,1356],{"class":60},[54,24032,1284],{"class":64},[54,24034,24035,24037,24039,24041,24043,24046,24048],{"class":56,"line":945},[54,24036,24016],{"class":64},[54,24038,122],{"class":121},[54,24040,1172],{"class":64},[54,24042,1230],{"class":125},[54,24044,24045],{"class":64},"(S_first_row[j ",[54,24047,1923],{"class":121},[54,24049,24050],{"class":64}," i])\n",[44,24052,24054],{"className":46,"code":24053,"language":48,"meta":49,"style":49},"Matrix(S_circ)\n",[51,24055,24056],{"__ignoreMap":49},[54,24057,24058,24061],{"class":56,"line":57},[54,24059,24060],{"class":125},"Matrix",[54,24062,24063],{"class":64},"(S_circ)\n",[13,24065,24066,24067],{},"And the matrix elements of ",[54,24068,24070,24087],{"className":24069},[2079],[54,24071,24073],{"className":24072},[2083],[2085,24074,24075],{"xmlns":2087},[2089,24076,24077,24085],{},[2092,24078,24079],{},[3827,24080,24081,24083],{"accent":2113},[2098,24082,2694],{},[2245,24084,3833],{},[2105,24086,3836],{"encoding":2107},[54,24088,24090],{"className":24089,"ariaHidden":2113},[2112],[54,24091,24093,24096],{"className":24092},[2117],[54,24094],{"className":24095,"style":3846},[2121],[54,24097,24099],{"className":24098},[2126,3850],[54,24100,24102],{"className":24101},[2139],[54,24103,24105],{"className":24104},[2143],[54,24106,24108,24116],{"className":24107,"style":3846},[2147],[54,24109,24110,24113],{"style":3862},[54,24111],{"className":24112,"style":3866},[2155],[54,24114,2694],{"className":24115,"style":2709},[2126,2167],[54,24117,24118,24121],{"style":3872},[54,24119],{"className":24120,"style":3866},[2155],[54,24122,24124],{"className":24123,"style":3880},[3879],[54,24125,3833],{"className":24126},[2126],[44,24128,24130],{"className":46,"code":24129,"language":48,"meta":49,"style":49},"# Assemble S, the overlap matrix of dimension D:\nH_first_row = np.zeros(krylov_dim, dtype=complex)\nH_first_row[0] = H_expval\n\nfor obs_idx, (pauli, coeff) in enumerate(zip(H_op.paulis, H_op.coeffs)):\n # Add in ancilla-only measurements:\n for i in range(krylov_dim - 1):\n # Get expectation values from experiment\n expval_real = results.data.evs[2 + 2 * obs_idx][0][\n i\n ] # automatic extrapolated evs if ZNE is used\n expval_imag = results.data.evs[2 + 2 * obs_idx + 1][0][\n i\n ] # automatic extrapolated evs if ZNE is used\n\n # Get expectation values\n expval = expval_real + 1j * expval_imag\n H_first_row[i + 1] += prefactors[i] * coeff * expval\n\nH_first_row_list = H_first_row.tolist()\n\nH_eff_circ = np.zeros((krylov_dim, krylov_dim), dtype=complex)\n\n# Distribute entries from first row across matrix:\nfor i, j in it.product(range(krylov_dim), repeat=2):\n if i >= j:\n H_eff_circ[j, i] = H_first_row[i - j]\n else:\n H_eff_circ[j, i] = np.conj(H_first_row[j - i])\n",[51,24131,24132,24136,24155,24169,24173,24189,24194,24212,24217,24241,24246,24253,24283,24287,24293,24297,24302,24321,24344,24348,24362,24366,24385,24389,24393,24419,24429,24443,24449],{"__ignoreMap":49},[54,24133,24134],{"class":56,"line":57},[54,24135,23734],{"class":111},[54,24137,24138,24141,24143,24145,24147,24149,24151,24153],{"class":56,"line":74},[54,24139,24140],{"class":64},"H_first_row ",[54,24142,122],{"class":121},[54,24144,1172],{"class":64},[54,24146,1705],{"class":125},[54,24148,23748],{"class":64},[54,24150,1712],{"class":1711},[54,24152,1715],{"class":121},[54,24154,136],{"class":64},[54,24156,24157,24160,24162,24164,24166],{"class":56,"line":88},[54,24158,24159],{"class":64},"H_first_row[",[54,24161,150],{"class":132},[54,24163,7762],{"class":64},[54,24165,122],{"class":121},[54,24167,24168],{"class":64}," H_expval\n",[54,24170,24171],{"class":56,"line":101},[54,24172,105],{"emptyLinePlaceholder":104},[54,24174,24175,24177,24179,24181,24183,24185,24187],{"class":56,"line":108},[54,24176,1186],{"class":60},[54,24178,22898],{"class":64},[54,24180,1192],{"class":60},[54,24182,1513],{"class":121},[54,24184,129],{"class":64},[54,24186,22907],{"class":121},[54,24188,22910],{"class":64},[54,24190,24191],{"class":56,"line":115},[54,24192,24193],{"class":111}," # Add in ancilla-only measurements:\n",[54,24195,24196,24198,24200,24202,24204,24206,24208,24210],{"class":56,"line":139},[54,24197,1472],{"class":60},[54,24199,7802],{"class":64},[54,24201,1192],{"class":60},[54,24203,7777],{"class":121},[54,24205,23796],{"class":64},[54,24207,1923],{"class":121},[54,24209,982],{"class":132},[54,24211,1439],{"class":64},[54,24213,24214],{"class":56,"line":159},[54,24215,24216],{"class":111}," # Get expectation values from experiment\n",[54,24218,24219,24222,24224,24226,24228,24230,24232,24234,24237,24239],{"class":56,"line":183},[54,24220,24221],{"class":64}," expval_real ",[54,24223,122],{"class":121},[54,24225,23815],{"class":64},[54,24227,133],{"class":132},[54,24229,7869],{"class":121},[54,24231,1872],{"class":132},[54,24233,7785],{"class":121},[54,24235,24236],{"class":64}," obs_idx][",[54,24238,150],{"class":132},[54,24240,23824],{"class":64},[54,24242,24243],{"class":56,"line":210},[54,24244,24245],{"class":64}," i\n",[54,24247,24248,24251],{"class":56,"line":227},[54,24249,24250],{"class":64}," ] ",[54,24252,23836],{"class":111},[54,24254,24255,24258,24260,24262,24264,24266,24268,24270,24273,24275,24277,24279,24281],{"class":56,"line":232},[54,24256,24257],{"class":64}," expval_imag ",[54,24259,122],{"class":121},[54,24261,23815],{"class":64},[54,24263,133],{"class":132},[54,24265,7869],{"class":121},[54,24267,1872],{"class":132},[54,24269,7785],{"class":121},[54,24271,24272],{"class":64}," obs_idx ",[54,24274,1663],{"class":121},[54,24276,982],{"class":132},[54,24278,1311],{"class":64},[54,24280,150],{"class":132},[54,24282,23824],{"class":64},[54,24284,24285],{"class":56,"line":246},[54,24286,24245],{"class":64},[54,24288,24289,24291],{"class":56,"line":259},[54,24290,24250],{"class":64},[54,24292,23836],{"class":111},[54,24294,24295],{"class":56,"line":264},[54,24296,105],{"emptyLinePlaceholder":104},[54,24298,24299],{"class":56,"line":270},[54,24300,24301],{"class":111}," # Get expectation values\n",[54,24303,24304,24307,24309,24311,24313,24315,24317,24319],{"class":56,"line":285},[54,24305,24306],{"class":64}," expval ",[54,24308,122],{"class":121},[54,24310,23009],{"class":64},[54,24312,1663],{"class":121},[54,24314,982],{"class":132},[54,24316,1928],{"class":60},[54,24318,7785],{"class":121},[54,24320,23020],{"class":64},[54,24322,24323,24326,24328,24330,24332,24334,24336,24338,24340,24342],{"class":56,"line":297},[54,24324,24325],{"class":64}," H_first_row[i ",[54,24327,1663],{"class":121},[54,24329,982],{"class":132},[54,24331,7762],{"class":64},[54,24333,1841],{"class":121},[54,24335,23905],{"class":64},[54,24337,1947],{"class":121},[54,24339,23039],{"class":64},[54,24341,1947],{"class":121},[54,24343,23044],{"class":64},[54,24345,24346],{"class":56,"line":310},[54,24347,105],{"emptyLinePlaceholder":104},[54,24349,24350,24353,24355,24358,24360],{"class":56,"line":324},[54,24351,24352],{"class":64},"H_first_row_list ",[54,24354,122],{"class":121},[54,24356,24357],{"class":64}," H_first_row.",[54,24359,23926],{"class":125},[54,24361,330],{"class":64},[54,24363,24364],{"class":56,"line":333},[54,24365,105],{"emptyLinePlaceholder":104},[54,24367,24368,24371,24373,24375,24377,24379,24381,24383],{"class":56,"line":345},[54,24369,24370],{"class":64},"H_eff_circ ",[54,24372,122],{"class":121},[54,24374,1172],{"class":64},[54,24376,1705],{"class":125},[54,24378,23954],{"class":64},[54,24380,1712],{"class":1711},[54,24382,1715],{"class":121},[54,24384,136],{"class":64},[54,24386,24387],{"class":56,"line":358},[54,24388,105],{"emptyLinePlaceholder":104},[54,24390,24391],{"class":56,"line":908},[54,24392,23969],{"class":111},[54,24394,24395,24397,24399,24401,24403,24405,24407,24409,24411,24413,24415,24417],{"class":56,"line":913},[54,24396,1186],{"class":60},[54,24398,23976],{"class":64},[54,24400,1192],{"class":60},[54,24402,1752],{"class":64},[54,24404,23983],{"class":125},[54,24406,129],{"class":64},[54,24408,1760],{"class":121},[54,24410,23990],{"class":64},[54,24412,23993],{"class":1711},[54,24414,122],{"class":121},[54,24416,133],{"class":132},[54,24418,1439],{"class":64},[54,24420,24421,24423,24425,24427],{"class":56,"line":918},[54,24422,1273],{"class":60},[54,24424,7802],{"class":64},[54,24426,24008],{"class":121},[54,24428,24011],{"class":64},[54,24430,24431,24434,24436,24439,24441],{"class":56,"line":923},[54,24432,24433],{"class":64}," H_eff_circ[j, i] ",[54,24435,122],{"class":121},[54,24437,24438],{"class":64}," H_first_row[i ",[54,24440,1923],{"class":121},[54,24442,24026],{"class":64},[54,24444,24445,24447],{"class":56,"line":935},[54,24446,1356],{"class":60},[54,24448,1284],{"class":64},[54,24450,24451,24453,24455,24457,24459,24462,24464],{"class":56,"line":945},[54,24452,24433],{"class":64},[54,24454,122],{"class":121},[54,24456,1172],{"class":64},[54,24458,1230],{"class":125},[54,24460,24461],{"class":64},"(H_first_row[j ",[54,24463,1923],{"class":121},[54,24465,24050],{"class":64},[44,24467,24469],{"className":46,"code":24468,"language":48,"meta":49,"style":49},"Matrix(H_eff_circ)\n",[51,24470,24471],{"__ignoreMap":49},[54,24472,24473,24475],{"class":56,"line":57},[54,24474,24060],{"class":125},[54,24476,24477],{"class":64},"(H_eff_circ)\n",[13,24479,24480,24481,24541],{},"Finally, we can solve the generalized eigenvalue problem for ",[54,24482,24484,24501],{"className":24483},[2079],[54,24485,24487],{"className":24486},[2083],[2085,24488,24489],{"xmlns":2087},[2089,24490,24491,24499],{},[2092,24492,24493],{},[3827,24494,24495,24497],{"accent":2113},[2098,24496,2694],{},[2245,24498,3833],{},[2105,24500,3836],{"encoding":2107},[54,24502,24504],{"className":24503,"ariaHidden":2113},[2112],[54,24505,24507,24510],{"className":24506},[2117],[54,24508],{"className":24509,"style":3846},[2121],[54,24511,24513],{"className":24512},[2126,3850],[54,24514,24516],{"className":24515},[2139],[54,24517,24519],{"className":24518},[2143],[54,24520,24522,24530],{"className":24521,"style":3846},[2147],[54,24523,24524,24527],{"style":3862},[54,24525],{"className":24526,"style":3866},[2155],[54,24528,2694],{"className":24529,"style":2709},[2126,2167],[54,24531,24532,24535],{"style":3872},[54,24533],{"className":24534,"style":3866},[2155],[54,24536,24538],{"className":24537,"style":3880},[3879],[54,24539,3833],{"className":24540},[2126],":",[13,24543,24544],{},[54,24545,24547,24583],{"className":24546},[2079],[54,24548,24550],{"className":24549},[2083],[2085,24551,24552],{"xmlns":2087},[2089,24553,24554,24580],{},[2092,24555,24556,24562,24568,24570,24572,24574],{},[3827,24557,24558,24560],{"accent":2113},[2098,24559,2694],{},[2245,24561,3833],{},[3827,24563,24564,24566],{"accent":2113},[2098,24565,5730],{},[2245,24567,5733],{},[2245,24569,122],{},[2098,24571,5730],{},[2098,24573,5413],{},[3827,24575,24576,24578],{"accent":2113},[2098,24577,5730],{},[2245,24579,5733],{},[2105,24581,24582],{"encoding":2107},"\\tilde{H} \\vec{c} = c S \\vec{c}",[54,24584,24586,24667],{"className":24585,"ariaHidden":2113},[2112],[54,24587,24589,24592,24623,24658,24661,24664],{"className":24588},[2117],[54,24590],{"className":24591,"style":3846},[2121],[54,24593,24595],{"className":24594},[2126,3850],[54,24596,24598],{"className":24597},[2139],[54,24599,24601],{"className":24600},[2143],[54,24602,24604,24612],{"className":24603,"style":3846},[2147],[54,24605,24606,24609],{"style":3862},[54,24607],{"className":24608,"style":3866},[2155],[54,24610,2694],{"className":24611,"style":2709},[2126,2167],[54,24613,24614,24617],{"style":3872},[54,24615],{"className":24616,"style":3866},[2155],[54,24618,24620],{"className":24619,"style":3880},[3879],[54,24621,3833],{"className":24622},[2126],[54,24624,24626],{"className":24625},[2126,3850],[54,24627,24629],{"className":24628},[2139],[54,24630,24632],{"className":24631},[2143],[54,24633,24635,24643],{"className":24634,"style":5813},[2147],[54,24636,24637,24640],{"style":3862},[54,24638],{"className":24639,"style":3866},[2155],[54,24641,5730],{"className":24642},[2126,2167],[54,24644,24645,24648],{"style":3862},[54,24646],{"className":24647,"style":3866},[2155],[54,24649,24651],{"className":24650,"style":5830},[3879],[54,24652,24654],{"className":24653,"style":5835},[5834],[5837,24655,24656],{"xmlns":5839,"width":5840,"height":5841,"style":5842,"viewBox":5843,"preserveAspectRatio":5844},[5846,24657],{"d":5848},[54,24659],{"className":24660,"style":2475},[2270],[54,24662,122],{"className":24663},[2479],[54,24665],{"className":24666,"style":2475},[2270],[54,24668,24670,24673,24676,24679],{"className":24669},[2117],[54,24671],{"className":24672,"style":5813},[2121],[54,24674,5730],{"className":24675},[2126,2167],[54,24677,5413],{"className":24678,"style":5448},[2126,2167],[54,24680,24682],{"className":24681},[2126,3850],[54,24683,24685],{"className":24684},[2139],[54,24686,24688],{"className":24687},[2143],[54,24689,24691,24699],{"className":24690,"style":5813},[2147],[54,24692,24693,24696],{"style":3862},[54,24694],{"className":24695,"style":3866},[2155],[54,24697,5730],{"className":24698},[2126,2167],[54,24700,24701,24704],{"style":3862},[54,24702],{"className":24703,"style":3866},[2155],[54,24705,24707],{"className":24706,"style":5830},[3879],[54,24708,24710],{"className":24709,"style":5835},[5834],[5837,24711,24712],{"xmlns":5839,"width":5840,"height":5841,"style":5842,"viewBox":5843,"preserveAspectRatio":5844},[5846,24713],{"d":5848},[13,24715,24716,24717],{},"and get an estimate of the ground state energy ",[54,24718,24720,24744],{"className":24719},[2079],[54,24721,24723],{"className":24722},[2083],[2085,24724,24725],{"xmlns":2087},[2089,24726,24727,24741],{},[2092,24728,24729],{},[2724,24730,24731,24733],{},[2098,24732,5730],{},[2092,24734,24735,24737,24739],{},[2098,24736,3973],{},[2098,24738,2849],{},[2098,24740,3976],{},[2105,24742,24743],{"encoding":2107},"c_{min}",[54,24745,24747],{"className":24746,"ariaHidden":2113},[2112],[54,24748,24750,24754],{"className":24749},[2117],[54,24751],{"className":24752,"style":24753},[2121],"height:0.5806em;vertical-align:-0.15em;",[54,24755,24757,24760],{"className":24756},[2126],[54,24758,5730],{"className":24759},[2126,2167],[54,24761,24763],{"className":24762},[2135],[54,24764,24766,24790],{"className":24765},[2139,2756],[54,24767,24769,24787],{"className":24768},[2143],[54,24770,24772],{"className":24771,"style":6358},[2147],[54,24773,24774,24777],{"style":4954},[54,24775],{"className":24776,"style":2156},[2155],[54,24778,24780],{"className":24779},[2160,2161,2162,2163],[54,24781,24783],{"className":24782},[2126,2163],[54,24784,24786],{"className":24785},[2126,2167,2163],"min",[54,24788,2781],{"className":24789},[2780],[54,24791,24793],{"className":24792},[2143],[54,24794,24796],{"className":24795,"style":2788},[2147],[54,24797],{},[44,24799,24801],{"className":46,"code":24800,"language":48,"meta":49,"style":49},"gnd_en_circ_est_list = []\nfor d in range(1, krylov_dim + 1):\n # Solve generalized eigenvalue problem for different size of the Krylov space\n gnd_en_circ_est = solve_regularized_gen_eig(\n H_eff_circ[:d, :d], S_circ[:d, :d], threshold=9e-1\n )\n gnd_en_circ_est_list.append(gnd_en_circ_est)\n print(\"The estimated ground state energy is: \", gnd_en_circ_est)\n",[51,24802,24803,24812,24836,24841,24852,24865,24869,24879],{"__ignoreMap":49},[54,24804,24805,24808,24810],{"class":56,"line":57},[54,24806,24807],{"class":64},"gnd_en_circ_est_list ",[54,24809,122],{"class":121},[54,24811,1466],{"class":64},[54,24813,24814,24816,24819,24821,24823,24825,24827,24830,24832,24834],{"class":56,"line":74},[54,24815,1186],{"class":60},[54,24817,24818],{"class":64}," d ",[54,24820,1192],{"class":60},[54,24822,7777],{"class":121},[54,24824,129],{"class":64},[54,24826,174],{"class":132},[54,24828,24829],{"class":64},", krylov_dim ",[54,24831,1663],{"class":121},[54,24833,982],{"class":132},[54,24835,1439],{"class":64},[54,24837,24838],{"class":56,"line":88},[54,24839,24840],{"class":111}," # Solve generalized eigenvalue problem for different size of the Krylov space\n",[54,24842,24843,24846,24848,24850],{"class":56,"line":101},[54,24844,24845],{"class":64}," gnd_en_circ_est ",[54,24847,122],{"class":121},[54,24849,929],{"class":125},[54,24851,932],{"class":64},[54,24853,24854,24857,24860,24862],{"class":56,"line":108},[54,24855,24856],{"class":64}," H_eff_circ[:d, :d], S_circ[:d, :d], ",[54,24858,24859],{"class":1711},"threshold",[54,24861,122],{"class":121},[54,24863,24864],{"class":132},"9e-1\n",[54,24866,24867],{"class":56,"line":115},[54,24868,1216],{"class":64},[54,24870,24871,24874,24876],{"class":56,"line":139},[54,24872,24873],{"class":64}," gnd_en_circ_est_list.",[54,24875,1495],{"class":125},[54,24877,24878],{"class":64},"(gnd_en_circ_est)\n",[54,24880,24881,24883,24885,24888],{"class":56,"line":159},[54,24882,1616],{"class":121},[54,24884,129],{"class":64},[54,24886,24887],{"class":279},"\"The estimated ground state energy is: \"",[54,24889,24890],{"class":64},", gnd_en_circ_est)\n",[44,24892,24895],{"className":24893,"code":24894,"language":6595,"meta":49},[8087],"The estimated ground state energy is: 25.0\nThe estimated ground state energy is: 22.572154819954875\nThe estimated ground state energy is: 21.691509219286587\nThe estimated ground state energy is: 21.23882298756386\nThe estimated ground state energy is: 20.965499325470294\n",[51,24896,24894],{"__ignoreMap":49},[13,24898,24899],{},"For a single-particle sector, we can efficiently calculate the ground state of this sector of the Hamiltonian classically",[44,24901,24903],{"className":46,"code":24902,"language":48,"meta":49,"style":49},"gs_en = single_particle_gs(H_op, n_qubits)\n",[51,24904,24905],{"__ignoreMap":49},[54,24906,24907,24910,24912,24914],{"class":56,"line":57},[54,24908,24909],{"class":64},"gs_en ",[54,24911,122],{"class":121},[54,24913,1426],{"class":125},[54,24915,24916],{"class":64},"(H_op, n_qubits)\n",[44,24918,24921],{"className":24919,"code":24920,"language":6595,"meta":49},[8087],"n_sys_qubits 30\nn_exc 1 , subspace dimension 31\nsingle particle ground state energy: 21.021912418526906\n",[51,24922,24920],{"__ignoreMap":49},[44,24924,24926],{"className":46,"code":24925,"language":48,"meta":49,"style":49},"plt.plot(\n range(1, krylov_dim + 1),\n gnd_en_circ_est_list,\n color=\"blue\",\n linestyle=\"-.\",\n label=\"KQD estimate\",\n)\nplt.plot(\n range(1, krylov_dim + 1),\n [gs_en] * krylov_dim,\n color=\"red\",\n linestyle=\"-\",\n label=\"exact\",\n)\nplt.xticks(range(1, krylov_dim + 1), range(1, krylov_dim + 1))\nplt.legend()\nplt.xlabel(\"Krylov space dimension\")\nplt.ylabel(\"Energy\")\nplt.title(\n \"Estimating Ground state energy with Krylov Quantum Diagonalization\"\n)\nplt.show()\n",[51,24927,24928,24938,24955,24960,24972,24984,24996,25000,25008,25024,25034,25045,25056,25067,25071,25108,25117,25131,25145,25154,25159,25163],{"__ignoreMap":49},[54,24929,24930,24933,24936],{"class":56,"line":57},[54,24931,24932],{"class":64},"plt.",[54,24934,24935],{"class":125},"plot",[54,24937,932],{"class":64},[54,24939,24940,24943,24945,24947,24949,24951,24953],{"class":56,"line":74},[54,24941,24942],{"class":121}," range",[54,24944,129],{"class":64},[54,24946,174],{"class":132},[54,24948,24829],{"class":64},[54,24950,1663],{"class":121},[54,24952,982],{"class":132},[54,24954,13758],{"class":64},[54,24956,24957],{"class":56,"line":88},[54,24958,24959],{"class":64}," gnd_en_circ_est_list,\n",[54,24961,24962,24965,24967,24970],{"class":56,"line":101},[54,24963,24964],{"class":1711}," color",[54,24966,122],{"class":121},[54,24968,24969],{"class":279},"\"blue\"",[54,24971,965],{"class":64},[54,24973,24974,24977,24979,24982],{"class":56,"line":108},[54,24975,24976],{"class":1711}," linestyle",[54,24978,122],{"class":121},[54,24980,24981],{"class":279},"\"-.\"",[54,24983,965],{"class":64},[54,24985,24986,24989,24991,24994],{"class":56,"line":115},[54,24987,24988],{"class":1711}," label",[54,24990,122],{"class":121},[54,24992,24993],{"class":279},"\"KQD estimate\"",[54,24995,965],{"class":64},[54,24997,24998],{"class":56,"line":139},[54,24999,136],{"class":64},[54,25001,25002,25004,25006],{"class":56,"line":159},[54,25003,24932],{"class":64},[54,25005,24935],{"class":125},[54,25007,932],{"class":64},[54,25009,25010,25012,25014,25016,25018,25020,25022],{"class":56,"line":183},[54,25011,24942],{"class":121},[54,25013,129],{"class":64},[54,25015,174],{"class":132},[54,25017,24829],{"class":64},[54,25019,1663],{"class":121},[54,25021,982],{"class":132},[54,25023,13758],{"class":64},[54,25025,25026,25029,25031],{"class":56,"line":210},[54,25027,25028],{"class":64}," [gs_en] ",[54,25030,1947],{"class":121},[54,25032,25033],{"class":64}," krylov_dim,\n",[54,25035,25036,25038,25040,25043],{"class":56,"line":227},[54,25037,24964],{"class":1711},[54,25039,122],{"class":121},[54,25041,25042],{"class":279},"\"red\"",[54,25044,965],{"class":64},[54,25046,25047,25049,25051,25054],{"class":56,"line":232},[54,25048,24976],{"class":1711},[54,25050,122],{"class":121},[54,25052,25053],{"class":279},"\"-\"",[54,25055,965],{"class":64},[54,25057,25058,25060,25062,25065],{"class":56,"line":246},[54,25059,24988],{"class":1711},[54,25061,122],{"class":121},[54,25063,25064],{"class":279},"\"exact\"",[54,25066,965],{"class":64},[54,25068,25069],{"class":56,"line":259},[54,25070,136],{"class":64},[54,25072,25073,25075,25078,25080,25082,25084,25086,25088,25090,25092,25094,25096,25098,25100,25102,25104,25106],{"class":56,"line":264},[54,25074,24932],{"class":64},[54,25076,25077],{"class":125},"xticks",[54,25079,129],{"class":64},[54,25081,1760],{"class":121},[54,25083,129],{"class":64},[54,25085,174],{"class":132},[54,25087,24829],{"class":64},[54,25089,1663],{"class":121},[54,25091,982],{"class":132},[54,25093,1769],{"class":64},[54,25095,1760],{"class":121},[54,25097,129],{"class":64},[54,25099,174],{"class":132},[54,25101,24829],{"class":64},[54,25103,1663],{"class":121},[54,25105,982],{"class":132},[54,25107,22096],{"class":64},[54,25109,25110,25112,25115],{"class":56,"line":270},[54,25111,24932],{"class":64},[54,25113,25114],{"class":125},"legend",[54,25116,330],{"class":64},[54,25118,25119,25121,25124,25126,25129],{"class":56,"line":285},[54,25120,24932],{"class":64},[54,25122,25123],{"class":125},"xlabel",[54,25125,129],{"class":64},[54,25127,25128],{"class":279},"\"Krylov space dimension\"",[54,25130,136],{"class":64},[54,25132,25133,25135,25138,25140,25143],{"class":56,"line":297},[54,25134,24932],{"class":64},[54,25136,25137],{"class":125},"ylabel",[54,25139,129],{"class":64},[54,25141,25142],{"class":279},"\"Energy\"",[54,25144,136],{"class":64},[54,25146,25147,25149,25152],{"class":56,"line":310},[54,25148,24932],{"class":64},[54,25150,25151],{"class":125},"title",[54,25153,932],{"class":64},[54,25155,25156],{"class":56,"line":324},[54,25157,25158],{"class":279}," \"Estimating Ground state energy with Krylov Quantum Diagonalization\"\n",[54,25160,25161],{"class":56,"line":333},[54,25162,136],{"class":64},[54,25164,25165,25167,25170],{"class":56,"line":345},[54,25166,24932],{"class":64},[54,25168,25169],{"class":125},"show",[54,25171,330],{"class":64},[31,25173,25175],{"id":25174},"appendix-krylov-subspace-from-real-time-evolutions","Appendix: Krylov subspace from real time-evolutions",[13,25177,25178],{},"The unitary Krylov space is defined as",[54,25180,25182],{"className":25181},[2340],[54,25183,25185,25293],{"className":25184},[2079],[54,25186,25188],{"className":25187},[2083],[2085,25189,25190],{"xmlns":2087,"display":2349},[2089,25191,25192,25290],{},[2092,25193,25194,25200,25202,25204,25206,25208,25210,25212,25214,25216,25218],{},[2724,25195,25196,25198],{},[2098,25197,2101],{"mathvariant":2100},[2098,25199,2835],{},[2245,25201,129],{"stretchy":2311},[2098,25203,2694],{},[2245,25205,2375],{"separator":2113},[2098,25207,2305],{"mathvariant":2304},[2098,25209,3349],{},[2245,25211,2312],{"stretchy":2311},[2245,25213,3205],{"stretchy":2311},[2245,25215,122],{},[5723,25217,54],{},[2092,25219,25220,25222,25224,25226,25228,25230,25249,25251,25253,25255,25257,25260,25262,25282,25284,25286,25288],{},[2245,25221,2366],{"fence":2113},[2098,25223,2305],{"mathvariant":2304},[2098,25225,3349],{},[2245,25227,2312],{"stretchy":2311},[2245,25229,2375],{"separator":2113},[2095,25231,25232,25234],{},[2098,25233,2842],{},[2092,25235,25236,25238,25240,25242,25245,25247],{},[2245,25237,2247],{},[2098,25239,2849],{},[2098,25241,2694],{},[5723,25243,25244],{}," ",[2098,25246,4515],{},[2098,25248,2854],{},[2098,25250,2305],{"mathvariant":2304},[2098,25252,3349],{},[2245,25254,2312],{"stretchy":2311},[2245,25256,2375],{"separator":2113},[2245,25258,25259],{},"…",[2245,25261,2375],{"separator":2113},[2095,25263,25264,25266],{},[2098,25265,2842],{},[2092,25267,25268,25270,25272,25274,25276,25278,25280],{},[2245,25269,2247],{},[2098,25271,2849],{},[2098,25273,1772],{},[2098,25275,2694],{},[5723,25277,25244],{},[2098,25279,4515],{},[2098,25281,2854],{},[2098,25283,2305],{"mathvariant":2304},[2098,25285,3349],{},[2245,25287,2312],{"stretchy":2311},[2245,25289,2428],{"fence":2113},[2105,25291,25292],{"encoding":2107},"\\mathcal{K}_U(H, |\\psi\\rangle) = \\text{span}\\left\\{ |\\psi\\rangle, e^{-iH\\,dt} |\\psi\\rangle, \\dots, e^{-irH\\,dt} |\\psi\\rangle \\right\\}",[54,25294,25296,25373],{"className":25295,"ariaHidden":2113},[2112],[54,25297,25299,25302,25342,25345,25348,25351,25354,25357,25360,25364,25367,25370],{"className":25298},[2117],[54,25300],{"className":25301,"style":2325},[2121],[54,25303,25305,25308],{"className":25304},[2126],[54,25306,2101],{"className":25307,"style":2131},[2126,2130],[54,25309,25311],{"className":25310},[2135],[54,25312,25314,25334],{"className":25313},[2139,2756],[54,25315,25317,25331],{"className":25316},[2143],[54,25318,25320],{"className":25319,"style":2763},[2147],[54,25321,25322,25325],{"style":2766},[54,25323],{"className":25324,"style":2156},[2155],[54,25326,25328],{"className":25327},[2160,2161,2162,2163],[54,25329,2835],{"className":25330,"style":2870},[2126,2167,2163],[54,25332,2781],{"className":25333},[2780],[54,25335,25337],{"className":25336},[2143],[54,25338,25340],{"className":25339,"style":2788},[2147],[54,25341],{},[54,25343,129],{"className":25344},[2497],[54,25346,2694],{"className":25347,"style":2709},[2126,2167],[54,25349,2375],{"className":25350},[2517],[54,25352],{"className":25353,"style":2521},[2270],[54,25355,2305],{"className":25356},[2126],[54,25358,3349],{"className":25359,"style":2332},[2126,2167],[54,25361,25363],{"className":25362},[2336],"⟩)",[54,25365],{"className":25366,"style":2475},[2270],[54,25368,122],{"className":25369},[2479],[54,25371],{"className":25372,"style":2475},[2270],[54,25374,25376,25380,25386,25389],{"className":25375},[2117],[54,25377],{"className":25378,"style":25379},[2121],"height:1.2491em;vertical-align:-0.35em;",[54,25381,25383],{"className":25382},[2126,6595],[54,25384,54],{"className":25385},[2126],[54,25387],{"className":25388,"style":2521},[2270],[54,25390,25392,25398,25401,25404,25407,25410,25413,25461,25464,25467,25470,25473,25476,25479,25482,25485,25488,25538,25541,25544,25547],{"className":25391},[2493],[54,25393,25395],{"className":25394,"style":2499},[2497,2498],[54,25396,2366],{"className":25397},[2503,2504],[54,25399,2305],{"className":25400},[2126],[54,25402,3349],{"className":25403,"style":2332},[2126,2167],[54,25405,2312],{"className":25406},[2336],[54,25408,2375],{"className":25409},[2517],[54,25411],{"className":25412,"style":2521},[2270],[54,25414,25416,25419],{"className":25415},[2126],[54,25417,2842],{"className":25418},[2126,2167],[54,25420,25422],{"className":25421},[2135],[54,25423,25425],{"className":25424},[2139],[54,25426,25428],{"className":25427},[2143],[54,25429,25431],{"className":25430,"style":4596},[2147],[54,25432,25433,25436],{"style":2462},[54,25434],{"className":25435,"style":2156},[2155],[54,25437,25439],{"className":25438},[2160,2161,2162,2163],[54,25440,25442,25445,25448,25451,25455,25458],{"className":25441},[2126,2163],[54,25443,2247],{"className":25444},[2126,2163],[54,25446,2849],{"className":25447},[2126,2167,2163],[54,25449,2694],{"className":25450,"style":2709},[2126,2167,2163],[54,25452],{"className":25453,"style":25454},[2270,2163],"margin-right:0.1952em;",[54,25456,4515],{"className":25457},[2126,2167,2163],[54,25459,2854],{"className":25460},[2126,2167,2163],[54,25462,2305],{"className":25463},[2126],[54,25465,3349],{"className":25466,"style":2332},[2126,2167],[54,25468,2312],{"className":25469},[2336],[54,25471,2375],{"className":25472},[2517],[54,25474],{"className":25475,"style":2521},[2270],[54,25477,25259],{"className":25478},[2493],[54,25480],{"className":25481,"style":2521},[2270],[54,25483,2375],{"className":25484},[2517],[54,25486],{"className":25487,"style":2521},[2270],[54,25489,25491,25494],{"className":25490},[2126],[54,25492,2842],{"className":25493},[2126,2167],[54,25495,25497],{"className":25496},[2135],[54,25498,25500],{"className":25499},[2139],[54,25501,25503],{"className":25502},[2143],[54,25504,25506],{"className":25505,"style":4596},[2147],[54,25507,25508,25511],{"style":2462},[54,25509],{"className":25510,"style":2156},[2155],[54,25512,25514],{"className":25513},[2160,2161,2162,2163],[54,25515,25517,25520,25523,25526,25529,25532,25535],{"className":25516},[2126,2163],[54,25518,2247],{"className":25519},[2126,2163],[54,25521,2849],{"className":25522},[2126,2167,2163],[54,25524,1772],{"className":25525,"style":2168},[2126,2167,2163],[54,25527,2694],{"className":25528,"style":2709},[2126,2167,2163],[54,25530],{"className":25531,"style":25454},[2270,2163],[54,25533,4515],{"className":25534},[2126,2167,2163],[54,25536,2854],{"className":25537},[2126,2167,2163],[54,25539,2305],{"className":25540},[2126],[54,25542,3349],{"className":25543,"style":2332},[2126,2167],[54,25545,2312],{"className":25546},[2336],[54,25548,25550],{"className":25549,"style":2499},[2336,2498],[54,25551,2428],{"className":25552},[2503,2504],[13,25554,25555,25556,25589,25590,25618,25619,25678],{},"for some timestep ",[54,25557,25559,25574],{"className":25558},[2079],[54,25560,25562],{"className":25561},[2083],[2085,25563,25564],{"xmlns":2087},[2089,25565,25566,25572],{},[2092,25567,25568,25570],{},[2098,25569,4515],{},[2098,25571,2854],{},[2105,25573,5019],{"encoding":2107},[54,25575,25577],{"className":25576,"ariaHidden":2113},[2112],[54,25578,25580,25583,25586],{"className":25579},[2117],[54,25581],{"className":25582,"style":4990},[2121],[54,25584,4515],{"className":25585},[2126,2167],[54,25587,2854],{"className":25588},[2126,2167]," that we will determine later. Temporarily assume ",[54,25591,25593,25606],{"className":25592},[2079],[54,25594,25596],{"className":25595},[2083],[2085,25597,25598],{"xmlns":2087},[2089,25599,25600,25604],{},[2092,25601,25602],{},[2098,25603,1772],{},[2105,25605,1772],{"encoding":2107},[54,25607,25609],{"className":25608,"ariaHidden":2113},[2112],[54,25610,25612,25615],{"className":25611},[2117],[54,25613],{"className":25614,"style":2195},[2121],[54,25616,1772],{"className":25617,"style":2168},[2126,2167]," is even: then define ",[54,25620,25622,25644],{"className":25621},[2079],[54,25623,25625],{"className":25624},[2083],[2085,25626,25627],{"xmlns":2087},[2089,25628,25629,25641],{},[2092,25630,25631,25633,25635,25637,25639],{},[2098,25632,4515],{},[2245,25634,122],{},[2098,25636,1772],{},[2098,25638,194],{"mathvariant":2304},[2249,25640,133],{},[2105,25642,25643],{"encoding":2107},"d=r\u002F2",[54,25645,25647,25665],{"className":25646,"ariaHidden":2113},[2112],[54,25648,25650,25653,25656,25659,25662],{"className":25649},[2117],[54,25651],{"className":25652,"style":4990},[2121],[54,25654,4515],{"className":25655},[2126,2167],[54,25657],{"className":25658,"style":2475},[2270],[54,25660,122],{"className":25661},[2479],[54,25663],{"className":25664,"style":2475},[2270],[54,25666,25668,25671,25674],{"className":25667},[2117],[54,25669],{"className":25670,"style":2325},[2121],[54,25672,1772],{"className":25673,"style":2168},[2126,2167],[54,25675,25677],{"className":25676},[2126],"\u002F2",". Notice that when we project the Hamiltonian into the Krylov space above, it is indistinguishable from the Krylov space",[54,25680,25682],{"className":25681},[2340],[54,25683,25685,25863],{"className":25684},[2079],[54,25686,25688],{"className":25687},[2083],[2085,25689,25690],{"xmlns":2087,"display":2349},[2089,25691,25692,25860],{},[2092,25693,25694,25700,25702,25704,25706,25708,25710,25712,25714,25716,25718,25858],{},[2724,25695,25696,25698],{},[2098,25697,2101],{"mathvariant":2100},[2098,25699,2835],{},[2245,25701,129],{"stretchy":2311},[2098,25703,2694],{},[2245,25705,2375],{"separator":2113},[2098,25707,2305],{"mathvariant":2304},[2098,25709,3349],{},[2245,25711,2312],{"stretchy":2311},[2245,25713,3205],{"stretchy":2311},[2245,25715,122],{},[5723,25717,54],{},[2092,25719,25720,25722,25744,25746,25748,25750,25752,25778,25780,25782,25784,25786,25788,25790,25818,25820,25822,25824,25826,25850,25852,25854,25856],{},[2245,25721,2366],{"fence":2113},[2095,25723,25724,25726],{},[2098,25725,2842],{},[2092,25727,25728,25730,25732,25734,25736,25738,25740,25742],{},[2098,25729,2849],{},[5723,25731,25244],{},[2098,25733,4515],{},[5723,25735,25244],{},[2098,25737,2694],{},[5723,25739,25244],{},[2098,25741,4515],{},[2098,25743,2854],{},[2098,25745,2305],{"mathvariant":2304},[2098,25747,3349],{},[2245,25749,2312],{"stretchy":2311},[2245,25751,2375],{"separator":2113},[2095,25753,25754,25756],{},[2098,25755,2842],{},[2092,25757,25758,25760,25762,25764,25766,25768,25770,25772,25774,25776],{},[2098,25759,2849],{},[2245,25761,129],{"stretchy":2311},[2098,25763,4515],{},[2245,25765,2247],{},[2249,25767,174],{},[2245,25769,3205],{"stretchy":2311},[2098,25771,2694],{},[5723,25773,25244],{},[2098,25775,4515],{},[2098,25777,2854],{},[2098,25779,2305],{"mathvariant":2304},[2098,25781,3349],{},[2245,25783,2312],{"stretchy":2311},[2245,25785,2375],{"separator":2113},[2245,25787,25259],{},[2245,25789,2375],{"separator":2113},[2095,25791,25792,25794],{},[2098,25793,2842],{},[2092,25795,25796,25798,25800,25802,25804,25806,25808,25810,25812,25814,25816],{},[2245,25797,2247],{},[2098,25799,2849],{},[2245,25801,129],{"stretchy":2311},[2098,25803,4515],{},[2245,25805,2247],{},[2249,25807,174],{},[2245,25809,3205],{"stretchy":2311},[2098,25811,2694],{},[5723,25813,25244],{},[2098,25815,4515],{},[2098,25817,2854],{},[2098,25819,2305],{"mathvariant":2304},[2098,25821,3349],{},[2245,25823,2312],{"stretchy":2311},[2245,25825,2375],{"separator":2113},[2095,25827,25828,25830],{},[2098,25829,2842],{},[2092,25831,25832,25834,25836,25838,25840,25842,25844,25846,25848],{},[2245,25833,2247],{},[2098,25835,2849],{},[5723,25837,25244],{},[2098,25839,4515],{},[5723,25841,25244],{},[2098,25843,2694],{},[5723,25845,25244],{},[2098,25847,4515],{},[2098,25849,2854],{},[2098,25851,2305],{"mathvariant":2304},[2098,25853,3349],{},[2245,25855,2312],{"stretchy":2311},[2245,25857,2428],{"fence":2113},[2245,25859,2375],{"separator":2113},[2105,25861,25862],{"encoding":2107},"\\mathcal{K}_U(H, |\\psi\\rangle) = \\text{span}\\left\\{ e^{i\\,d\\,H\\,dt}|\\psi\\rangle, e^{i(d-1)H\\,dt} |\\psi\\rangle, \\dots, e^{-i(d-1)H\\,dt} |\\psi\\rangle, e^{-i\\,d\\,H\\,dt} |\\psi\\rangle \\right\\},",[54,25864,25866,25942],{"className":25865,"ariaHidden":2113},[2112],[54,25867,25869,25872,25912,25915,25918,25921,25924,25927,25930,25933,25936,25939],{"className":25868},[2117],[54,25870],{"className":25871,"style":2325},[2121],[54,25873,25875,25878],{"className":25874},[2126],[54,25876,2101],{"className":25877,"style":2131},[2126,2130],[54,25879,25881],{"className":25880},[2135],[54,25882,25884,25904],{"className":25883},[2139,2756],[54,25885,25887,25901],{"className":25886},[2143],[54,25888,25890],{"className":25889,"style":2763},[2147],[54,25891,25892,25895],{"style":2766},[54,25893],{"className":25894,"style":2156},[2155],[54,25896,25898],{"className":25897},[2160,2161,2162,2163],[54,25899,2835],{"className":25900,"style":2870},[2126,2167,2163],[54,25902,2781],{"className":25903},[2780],[54,25905,25907],{"className":25906},[2143],[54,25908,25910],{"className":25909,"style":2788},[2147],[54,25911],{},[54,25913,129],{"className":25914},[2497],[54,25916,2694],{"className":25917,"style":2709},[2126,2167],[54,25919,2375],{"className":25920},[2517],[54,25922],{"className":25923,"style":2521},[2270],[54,25925,2305],{"className":25926},[2126],[54,25928,3349],{"className":25929,"style":2332},[2126,2167],[54,25931,25363],{"className":25932},[2336],[54,25934],{"className":25935,"style":2475},[2270],[54,25937,122],{"className":25938},[2479],[54,25940],{"className":25941,"style":2475},[2270],[54,25943,25945,25948,25954,25957,26269,26272],{"className":25944},[2117],[54,25946],{"className":25947,"style":13101},[2121],[54,25949,25951],{"className":25950},[2126,6595],[54,25952,54],{"className":25953},[2126],[54,25955],{"className":25956,"style":2521},[2270],[54,25958,25960,25966,26019,26022,26025,26028,26031,26034,26094,26097,26100,26103,26106,26109,26112,26115,26118,26121,26183,26186,26189,26192,26195,26198,26254,26257,26260,26263],{"className":25959},[2493],[54,25961,25963],{"className":25962,"style":2499},[2497,2498],[54,25964,2366],{"className":25965},[2503,9476],[54,25967,25969,25972],{"className":25968},[2126],[54,25970,2842],{"className":25971},[2126,2167],[54,25973,25975],{"className":25974},[2135],[54,25976,25978],{"className":25977},[2139],[54,25979,25981],{"className":25980},[2143],[54,25982,25984],{"className":25983,"style":4596},[2147],[54,25985,25986,25989],{"style":2462},[54,25987],{"className":25988,"style":2156},[2155],[54,25990,25992],{"className":25991},[2160,2161,2162,2163],[54,25993,25995,25998,26001,26004,26007,26010,26013,26016],{"className":25994},[2126,2163],[54,25996,2849],{"className":25997},[2126,2167,2163],[54,25999],{"className":26000,"style":25454},[2270,2163],[54,26002,4515],{"className":26003},[2126,2167,2163],[54,26005],{"className":26006,"style":25454},[2270,2163],[54,26008,2694],{"className":26009,"style":2709},[2126,2167,2163],[54,26011],{"className":26012,"style":25454},[2270,2163],[54,26014,4515],{"className":26015},[2126,2167,2163],[54,26017,2854],{"className":26018},[2126,2167,2163],[54,26020,2305],{"className":26021},[2126],[54,26023,3349],{"className":26024,"style":2332},[2126,2167],[54,26026,2312],{"className":26027},[2336],[54,26029,2375],{"className":26030},[2517],[54,26032],{"className":26033,"style":2521},[2270],[54,26035,26037,26040],{"className":26036},[2126],[54,26038,2842],{"className":26039},[2126,2167],[54,26041,26043],{"className":26042},[2135],[54,26044,26046],{"className":26045},[2139],[54,26047,26049],{"className":26048},[2143],[54,26050,26053],{"className":26051,"style":26052},[2147],"height:0.938em;",[54,26054,26055,26058],{"style":2462},[54,26056],{"className":26057,"style":2156},[2155],[54,26059,26061],{"className":26060},[2160,2161,2162,2163],[54,26062,26064,26067,26070,26073,26076,26079,26082,26085,26088,26091],{"className":26063},[2126,2163],[54,26065,2849],{"className":26066},[2126,2167,2163],[54,26068,129],{"className":26069},[2497,2163],[54,26071,4515],{"className":26072},[2126,2167,2163],[54,26074,2247],{"className":26075},[2275,2163],[54,26077,174],{"className":26078},[2126,2163],[54,26080,3205],{"className":26081},[2336,2163],[54,26083,2694],{"className":26084,"style":2709},[2126,2167,2163],[54,26086],{"className":26087,"style":25454},[2270,2163],[54,26089,4515],{"className":26090},[2126,2167,2163],[54,26092,2854],{"className":26093},[2126,2167,2163],[54,26095,2305],{"className":26096},[2126],[54,26098,3349],{"className":26099,"style":2332},[2126,2167],[54,26101,2312],{"className":26102},[2336],[54,26104,2375],{"className":26105},[2517],[54,26107],{"className":26108,"style":2521},[2270],[54,26110,25259],{"className":26111},[2493],[54,26113],{"className":26114,"style":2521},[2270],[54,26116,2375],{"className":26117},[2517],[54,26119],{"className":26120,"style":2521},[2270],[54,26122,26124,26127],{"className":26123},[2126],[54,26125,2842],{"className":26126},[2126,2167],[54,26128,26130],{"className":26129},[2135],[54,26131,26133],{"className":26132},[2139],[54,26134,26136],{"className":26135},[2143],[54,26137,26139],{"className":26138,"style":26052},[2147],[54,26140,26141,26144],{"style":2462},[54,26142],{"className":26143,"style":2156},[2155],[54,26145,26147],{"className":26146},[2160,2161,2162,2163],[54,26148,26150,26153,26156,26159,26162,26165,26168,26171,26174,26177,26180],{"className":26149},[2126,2163],[54,26151,2247],{"className":26152},[2126,2163],[54,26154,2849],{"className":26155},[2126,2167,2163],[54,26157,129],{"className":26158},[2497,2163],[54,26160,4515],{"className":26161},[2126,2167,2163],[54,26163,2247],{"className":26164},[2275,2163],[54,26166,174],{"className":26167},[2126,2163],[54,26169,3205],{"className":26170},[2336,2163],[54,26172,2694],{"className":26173,"style":2709},[2126,2167,2163],[54,26175],{"className":26176,"style":25454},[2270,2163],[54,26178,4515],{"className":26179},[2126,2167,2163],[54,26181,2854],{"className":26182},[2126,2167,2163],[54,26184,2305],{"className":26185},[2126],[54,26187,3349],{"className":26188,"style":2332},[2126,2167],[54,26190,2312],{"className":26191},[2336],[54,26193,2375],{"className":26194},[2517],[54,26196],{"className":26197,"style":2521},[2270],[54,26199,26201,26204],{"className":26200},[2126],[54,26202,2842],{"className":26203},[2126,2167],[54,26205,26207],{"className":26206},[2135],[54,26208,26210],{"className":26209},[2139],[54,26211,26213],{"className":26212},[2143],[54,26214,26216],{"className":26215,"style":4596},[2147],[54,26217,26218,26221],{"style":2462},[54,26219],{"className":26220,"style":2156},[2155],[54,26222,26224],{"className":26223},[2160,2161,2162,2163],[54,26225,26227,26230,26233,26236,26239,26242,26245,26248,26251],{"className":26226},[2126,2163],[54,26228,2247],{"className":26229},[2126,2163],[54,26231,2849],{"className":26232},[2126,2167,2163],[54,26234],{"className":26235,"style":25454},[2270,2163],[54,26237,4515],{"className":26238},[2126,2167,2163],[54,26240],{"className":26241,"style":25454},[2270,2163],[54,26243,2694],{"className":26244,"style":2709},[2126,2167,2163],[54,26246],{"className":26247,"style":25454},[2270,2163],[54,26249,4515],{"className":26250},[2126,2167,2163],[54,26252,2854],{"className":26253},[2126,2167,2163],[54,26255,2305],{"className":26256},[2126],[54,26258,3349],{"className":26259,"style":2332},[2126,2167],[54,26261,2312],{"className":26262},[2336],[54,26264,26266],{"className":26265,"style":2499},[2336,2498],[54,26267,2428],{"className":26268},[2503,9476],[54,26270],{"className":26271,"style":2521},[2270],[54,26273,2375],{"className":26274},[2517],[13,26276,26277,26278,26306],{},"that is, where all the time-evolutions are shifted backward by ",[54,26279,26281,26294],{"className":26280},[2079],[54,26282,26284],{"className":26283},[2083],[2085,26285,26286],{"xmlns":2087},[2089,26287,26288,26292],{},[2092,26289,26290],{},[2098,26291,4515],{},[2105,26293,4515],{"encoding":2107},[54,26295,26297],{"className":26296,"ariaHidden":2113},[2112],[54,26298,26300,26303],{"className":26299},[2117],[54,26301],{"className":26302,"style":4990},[2121],[54,26304,4515],{"className":26305},[2126,2167]," timesteps.\nThe reason it is indistinguishable is because the matrix elements",[54,26308,26310],{"className":26309},[2340],[54,26311,26313,26445],{"className":26312},[2079],[54,26314,26316],{"className":26315},[2083],[2085,26317,26318],{"xmlns":2087,"display":2349},[2089,26319,26320,26442],{},[2092,26321,26322,26338,26340,26342,26344,26346,26368,26370,26394,26396,26398,26400,26402,26404,26406,26408,26410,26436,26438,26440],{},[2724,26323,26324,26330],{},[3827,26325,26326,26328],{"accent":2113},[2098,26327,2694],{},[2245,26329,3833],{},[2092,26331,26332,26334,26336],{},[2098,26333,1928],{},[2245,26335,2375],{"separator":2113},[2098,26337,3182],{},[2245,26339,122],{},[2245,26341,3981],{"stretchy":2311},[2098,26343,3349],{},[2098,26345,2305],{"mathvariant":2304},[2095,26347,26348,26350],{},[2098,26349,2842],{},[2092,26351,26352,26354,26356,26358,26360,26362,26364,26366],{},[2098,26353,2849],{},[5723,26355,25244],{},[2098,26357,1928],{},[5723,26359,25244],{},[2098,26361,2694],{},[5723,26363,25244],{},[2098,26365,4515],{},[2098,26367,2854],{},[2098,26369,2694],{},[2095,26371,26372,26374],{},[2098,26373,2842],{},[2092,26375,26376,26378,26380,26382,26384,26386,26388,26390,26392],{},[2245,26377,2247],{},[2098,26379,2849],{},[5723,26381,25244],{},[2098,26383,3182],{},[5723,26385,25244],{},[2098,26387,2694],{},[5723,26389,25244],{},[2098,26391,4515],{},[2098,26393,2854],{},[2098,26395,2305],{"mathvariant":2304},[2098,26397,3349],{},[2245,26399,2312],{"stretchy":2311},[2245,26401,122],{},[2245,26403,3981],{"stretchy":2311},[2098,26405,3349],{},[2098,26407,2305],{"mathvariant":2304},[2098,26409,2694],{},[2095,26411,26412,26414],{},[2098,26413,2842],{},[2092,26415,26416,26418,26420,26422,26424,26426,26428,26430,26432,26434],{},[2098,26417,2849],{},[2245,26419,129],{"stretchy":2311},[2098,26421,1928],{},[2245,26423,2247],{},[2098,26425,3182],{},[2245,26427,3205],{"stretchy":2311},[2098,26429,2694],{},[5723,26431,25244],{},[2098,26433,4515],{},[2098,26435,2854],{},[2098,26437,2305],{"mathvariant":2304},[2098,26439,3349],{},[2245,26441,2312],{"stretchy":2311},[2105,26443,26444],{"encoding":2107},"\\tilde{H}_{j,k} = \\langle\\psi|e^{i\\,j\\,H\\,dt}He^{-i\\,k\\,H\\,dt}|\\psi\\rangle=\\langle\\psi|He^{i(j-k)H\\,dt}|\\psi\\rangle",[54,26446,26448,26540,26685],{"className":26447,"ariaHidden":2113},[2112],[54,26449,26451,26454,26531,26534,26537],{"className":26450},[2117],[54,26452],{"className":26453,"style":6311},[2121],[54,26455,26457,26488],{"className":26456},[2126],[54,26458,26460],{"className":26459},[2126,3850],[54,26461,26463],{"className":26462},[2139],[54,26464,26466],{"className":26465},[2143],[54,26467,26469,26477],{"className":26468,"style":3846},[2147],[54,26470,26471,26474],{"style":3862},[54,26472],{"className":26473,"style":3866},[2155],[54,26475,2694],{"className":26476,"style":2709},[2126,2167],[54,26478,26479,26482],{"style":3872},[54,26480],{"className":26481,"style":3866},[2155],[54,26483,26485],{"className":26484,"style":3880},[3879],[54,26486,3833],{"className":26487},[2126],[54,26489,26491],{"className":26490},[2135],[54,26492,26494,26523],{"className":26493},[2139,2756],[54,26495,26497,26520],{"className":26496},[2143],[54,26498,26500],{"className":26499,"style":10799},[2147],[54,26501,26502,26505],{"style":4066},[54,26503],{"className":26504,"style":2156},[2155],[54,26506,26508],{"className":26507},[2160,2161,2162,2163],[54,26509,26511,26514,26517],{"className":26510},[2126,2163],[54,26512,1928],{"className":26513,"style":6379},[2126,2167,2163],[54,26515,2375],{"className":26516},[2517,2163],[54,26518,3182],{"className":26519,"style":3248},[2126,2167,2163],[54,26521,2781],{"className":26522},[2780],[54,26524,26526],{"className":26525},[2143],[54,26527,26529],{"className":26528,"style":6389},[2147],[54,26530],{},[54,26532],{"className":26533,"style":2475},[2270],[54,26535,122],{"className":26536},[2479],[54,26538],{"className":26539,"style":2475},[2270],[54,26541,26543,26546,26549,26552,26555,26608,26611,26667,26670,26673,26676,26679,26682],{"className":26542},[2117],[54,26544],{"className":26545,"style":4568},[2121],[54,26547,3981],{"className":26548},[2497],[54,26550,3349],{"className":26551,"style":2332},[2126,2167],[54,26553,2305],{"className":26554},[2126],[54,26556,26558,26561],{"className":26557},[2126],[54,26559,2842],{"className":26560},[2126,2167],[54,26562,26564],{"className":26563},[2135],[54,26565,26567],{"className":26566},[2139],[54,26568,26570],{"className":26569},[2143],[54,26571,26573],{"className":26572,"style":4596},[2147],[54,26574,26575,26578],{"style":2462},[54,26576],{"className":26577,"style":2156},[2155],[54,26579,26581],{"className":26580},[2160,2161,2162,2163],[54,26582,26584,26587,26590,26593,26596,26599,26602,26605],{"className":26583},[2126,2163],[54,26585,2849],{"className":26586},[2126,2167,2163],[54,26588],{"className":26589,"style":25454},[2270,2163],[54,26591,1928],{"className":26592,"style":6379},[2126,2167,2163],[54,26594],{"className":26595,"style":25454},[2270,2163],[54,26597,2694],{"className":26598,"style":2709},[2126,2167,2163],[54,26600],{"className":26601,"style":25454},[2270,2163],[54,26603,4515],{"className":26604},[2126,2167,2163],[54,26606,2854],{"className":26607},[2126,2167,2163],[54,26609,2694],{"className":26610,"style":2709},[2126,2167],[54,26612,26614,26617],{"className":26613},[2126],[54,26615,2842],{"className":26616},[2126,2167],[54,26618,26620],{"className":26619},[2135],[54,26621,26623],{"className":26622},[2139],[54,26624,26626],{"className":26625},[2143],[54,26627,26629],{"className":26628,"style":4596},[2147],[54,26630,26631,26634],{"style":2462},[54,26632],{"className":26633,"style":2156},[2155],[54,26635,26637],{"className":26636},[2160,2161,2162,2163],[54,26638,26640,26643,26646,26649,26652,26655,26658,26661,26664],{"className":26639},[2126,2163],[54,26641,2247],{"className":26642},[2126,2163],[54,26644,2849],{"className":26645},[2126,2167,2163],[54,26647],{"className":26648,"style":25454},[2270,2163],[54,26650,3182],{"className":26651,"style":3248},[2126,2167,2163],[54,26653],{"className":26654,"style":25454},[2270,2163],[54,26656,2694],{"className":26657,"style":2709},[2126,2167,2163],[54,26659],{"className":26660,"style":25454},[2270,2163],[54,26662,4515],{"className":26663},[2126,2167,2163],[54,26665,2854],{"className":26666},[2126,2167,2163],[54,26668,2305],{"className":26669},[2126],[54,26671,3349],{"className":26672,"style":2332},[2126,2167],[54,26674,2312],{"className":26675},[2336],[54,26677],{"className":26678,"style":2475},[2270],[54,26680,122],{"className":26681},[2479],[54,26683],{"className":26684,"style":2475},[2270],[54,26686,26688,26692,26695,26698,26701,26704,26763,26766,26769],{"className":26687},[2117],[54,26689],{"className":26690,"style":26691},[2121],"height:1.188em;vertical-align:-0.25em;",[54,26693,3981],{"className":26694},[2497],[54,26696,3349],{"className":26697,"style":2332},[2126,2167],[54,26699,2305],{"className":26700},[2126],[54,26702,2694],{"className":26703,"style":2709},[2126,2167],[54,26705,26707,26710],{"className":26706},[2126],[54,26708,2842],{"className":26709},[2126,2167],[54,26711,26713],{"className":26712},[2135],[54,26714,26716],{"className":26715},[2139],[54,26717,26719],{"className":26718},[2143],[54,26720,26722],{"className":26721,"style":26052},[2147],[54,26723,26724,26727],{"style":2462},[54,26725],{"className":26726,"style":2156},[2155],[54,26728,26730],{"className":26729},[2160,2161,2162,2163],[54,26731,26733,26736,26739,26742,26745,26748,26751,26754,26757,26760],{"className":26732},[2126,2163],[54,26734,2849],{"className":26735},[2126,2167,2163],[54,26737,129],{"className":26738},[2497,2163],[54,26740,1928],{"className":26741,"style":6379},[2126,2167,2163],[54,26743,2247],{"className":26744},[2275,2163],[54,26746,3182],{"className":26747,"style":3248},[2126,2167,2163],[54,26749,3205],{"className":26750},[2336,2163],[54,26752,2694],{"className":26753,"style":2709},[2126,2167,2163],[54,26755],{"className":26756,"style":25454},[2270,2163],[54,26758,4515],{"className":26759},[2126,2167,2163],[54,26761,2854],{"className":26762},[2126,2167,2163],[54,26764,2305],{"className":26765},[2126],[54,26767,3349],{"className":26768,"style":2332},[2126,2167],[54,26770,2312],{"className":26771},[2336],[13,26773,26774,26775,26803,26804,29],{},"are invariant under overall shifts of the evolution time, since the time-evolutions commute with the Hamiltonian. For odd ",[54,26776,26778,26791],{"className":26777},[2079],[54,26779,26781],{"className":26780},[2083],[2085,26782,26783],{"xmlns":2087},[2089,26784,26785,26789],{},[2092,26786,26787],{},[2098,26788,1772],{},[2105,26790,1772],{"encoding":2107},[54,26792,26794],{"className":26793,"ariaHidden":2113},[2112],[54,26795,26797,26800],{"className":26796},[2117],[54,26798],{"className":26799,"style":2195},[2121],[54,26801,1772],{"className":26802,"style":2168},[2126,2167],", we can use the analysis for ",[54,26805,26807,26824],{"className":26806},[2079],[54,26808,26810],{"className":26809},[2083],[2085,26811,26812],{"xmlns":2087},[2089,26813,26814,26822],{},[2092,26815,26816,26818,26820],{},[2098,26817,1772],{},[2245,26819,2247],{},[2249,26821,174],{},[2105,26823,2253],{"encoding":2107},[54,26825,26827,26845],{"className":26826,"ariaHidden":2113},[2112],[54,26828,26830,26833,26836,26839,26842],{"className":26829},[2117],[54,26831],{"className":26832,"style":2263},[2121],[54,26834,1772],{"className":26835,"style":2168},[2126,2167],[54,26837],{"className":26838,"style":2271},[2270],[54,26840,2247],{"className":26841},[2275],[54,26843],{"className":26844,"style":2271},[2270],[54,26846,26848,26851],{"className":26847},[2117],[54,26849],{"className":26850,"style":2285},[2121],[54,26852,174],{"className":26853},[2126],[13,26855,26856,26857,24541],{},"We want to show that somewhere in this Krylov space, there is guaranteed to be a low-energy state. We do so by way of the following result, which is derived from Theorem 3.1 in ",[20,26858,26859],{"href":640},"[3]",[13,26861,26862,26866,26867,26896,26897,26925],{},[26863,26864,26865],"strong",{},"Claim 1:"," there exists a function ",[54,26868,26870,26883],{"className":26869},[2079],[54,26871,26873],{"className":26872},[2083],[2085,26874,26875],{"xmlns":2087},[2089,26876,26877,26881],{},[2092,26878,26879],{},[2098,26880,21834],{},[2105,26882,21834],{"encoding":2107},[54,26884,26886],{"className":26885,"ariaHidden":2113},[2112],[54,26887,26889,26892],{"className":26888},[2117],[54,26890],{"className":26891,"style":6591},[2121],[54,26893,21834],{"className":26894,"style":26895},[2126,2167],"margin-right:0.1076em;"," such that for energies ",[54,26898,26900,26913],{"className":26899},[2079],[54,26901,26903],{"className":26902},[2083],[2085,26904,26905],{"xmlns":2087},[2089,26906,26907,26911],{},[2092,26908,26909],{},[2098,26910,5738],{},[2105,26912,5738],{"encoding":2107},[54,26914,26916],{"className":26915,"ariaHidden":2113},[2112],[54,26917,26919,26922],{"className":26918},[2117],[54,26920],{"className":26921,"style":2122},[2121],[54,26923,5738],{"className":26924,"style":5448},[2126,2167]," in the spectral range of the Hamiltonian (that is, between the ground state energy and the maximum energy)...",[26927,26928,26929,27038,27346],"ol",{},[659,26930,26931],{},[54,26932,26934,26962],{"className":26933},[2079],[54,26935,26937],{"className":26936},[2083],[2085,26938,26939],{"xmlns":2087},[2089,26940,26941,26959],{},[2092,26942,26943,26945,26947,26953,26955,26957],{},[2098,26944,21834],{},[2245,26946,129],{"stretchy":2311},[2724,26948,26949,26951],{},[2098,26950,5738],{},[2249,26952,150],{},[2245,26954,3205],{"stretchy":2311},[2245,26956,122],{},[2249,26958,174],{},[2105,26960,26961],{"encoding":2107},"f(E_0)=1",[54,26963,26965,27029],{"className":26964,"ariaHidden":2113},[2112],[54,26966,26968,26971,26974,26977,27017,27020,27023,27026],{"className":26967},[2117],[54,26969],{"className":26970,"style":2325},[2121],[54,26972,21834],{"className":26973,"style":26895},[2126,2167],[54,26975,129],{"className":26976},[2497],[54,26978,26980,26983],{"className":26979},[2126],[54,26981,5738],{"className":26982,"style":5448},[2126,2167],[54,26984,26986],{"className":26985},[2135],[54,26987,26989,27009],{"className":26988},[2139,2756],[54,26990,26992,27006],{"className":26991},[2143],[54,26993,26995],{"className":26994,"style":13837},[2147],[54,26996,26997,27000],{"style":5570},[54,26998],{"className":26999,"style":2156},[2155],[54,27001,27003],{"className":27002},[2160,2161,2162,2163],[54,27004,150],{"className":27005},[2126,2163],[54,27007,2781],{"className":27008},[2780],[54,27010,27012],{"className":27011},[2143],[54,27013,27015],{"className":27014,"style":2788},[2147],[54,27016],{},[54,27018,3205],{"className":27019},[2336],[54,27021],{"className":27022,"style":2475},[2270],[54,27024,122],{"className":27025},[2479],[54,27027],{"className":27028,"style":2475},[2270],[54,27030,27032,27035],{"className":27031},[2117],[54,27033],{"className":27034,"style":2285},[2121],[54,27036,174],{"className":27037},[2126],[659,27039,27040,27200,27201,27229,27230,27275,27276,27345],{},[54,27041,27043,27093],{"className":27042},[2079],[54,27044,27046],{"className":27045},[2083],[2085,27047,27048],{"xmlns":2087},[2089,27049,27050,27090],{},[2092,27051,27052,27054,27056,27058,27060,27062,27064,27067,27069],{},[2098,27053,2305],{"mathvariant":2304},[2098,27055,21834],{},[2245,27057,129],{"stretchy":2311},[2098,27059,5738],{},[2245,27061,3205],{"stretchy":2311},[2098,27063,2305],{"mathvariant":2304},[2245,27065,27066],{},"≤",[2249,27068,133],{},[2095,27070,27071,27084],{},[2092,27072,27073,27075,27077,27079,27082],{},[2245,27074,129],{"fence":2113},[2249,27076,174],{},[2245,27078,1663],{},[2098,27080,27081],{},"δ",[2245,27083,3205],{"fence":2113},[2092,27085,27086,27088],{},[2245,27087,2247],{},[2098,27089,4515],{},[2105,27091,27092],{"encoding":2107},"|f(E)|\\le2\\left(1 + \\delta\\right)^{-d}",[54,27094,27096,27129],{"className":27095,"ariaHidden":2113},[2112],[54,27097,27099,27102,27105,27108,27111,27114,27117,27120,27123,27126],{"className":27098},[2117],[54,27100],{"className":27101,"style":2325},[2121],[54,27103,2305],{"className":27104},[2126],[54,27106,21834],{"className":27107,"style":26895},[2126,2167],[54,27109,129],{"className":27110},[2497],[54,27112,5738],{"className":27113,"style":5448},[2126,2167],[54,27115,3205],{"className":27116},[2336],[54,27118,2305],{"className":27119},[2126],[54,27121],{"className":27122,"style":2475},[2270],[54,27124,27066],{"className":27125},[2479],[54,27127],{"className":27128,"style":2475},[2270],[54,27130,27132,27136,27139,27142],{"className":27131},[2117],[54,27133],{"className":27134,"style":27135},[2121],"height:1.239em;vertical-align:-0.25em;",[54,27137,133],{"className":27138},[2126],[54,27140],{"className":27141,"style":2521},[2270],[54,27143,27145,27170],{"className":27144},[2493],[54,27146,27148,27151,27154,27157,27160,27163,27167],{"className":27147},[2493],[54,27149,129],{"className":27150,"style":2499},[2497,2498],[54,27152,174],{"className":27153},[2126],[54,27155],{"className":27156,"style":2271},[2270],[54,27158,1663],{"className":27159},[2275],[54,27161],{"className":27162,"style":2271},[2270],[54,27164,27081],{"className":27165,"style":27166},[2126,2167],"margin-right:0.0379em;",[54,27168,3205],{"className":27169,"style":2499},[2336,2498],[54,27171,27173],{"className":27172},[2135],[54,27174,27176],{"className":27175},[2139],[54,27177,27179],{"className":27178},[2143],[54,27180,27183],{"className":27181,"style":27182},[2147],"height:0.989em;",[54,27184,27185,27188],{"style":7401},[54,27186],{"className":27187,"style":2156},[2155],[54,27189,27191],{"className":27190},[2160,2161,2162,2163],[54,27192,27194,27197],{"className":27193},[2126,2163],[54,27195,2247],{"className":27196},[2126,2163],[54,27198,4515],{"className":27199},[2126,2167,2163]," for all values of ",[54,27202,27204,27217],{"className":27203},[2079],[54,27205,27207],{"className":27206},[2083],[2085,27208,27209],{"xmlns":2087},[2089,27210,27211,27215],{},[2092,27212,27213],{},[2098,27214,5738],{},[2105,27216,5738],{"encoding":2107},[54,27218,27220],{"className":27219,"ariaHidden":2113},[2112],[54,27221,27223,27226],{"className":27222},[2117],[54,27224],{"className":27225,"style":2122},[2121],[54,27227,5738],{"className":27228,"style":5448},[2126,2167]," that lie ",[54,27231,27233,27250],{"className":27232},[2079],[54,27234,27236],{"className":27235},[2083],[2085,27237,27238],{"xmlns":2087},[2089,27239,27240,27247],{},[2092,27241,27242,27245],{},[2245,27243,27244],{},"≥",[2098,27246,27081],{},[2105,27248,27249],{"encoding":2107},"\\ge\\delta",[54,27251,27253,27266],{"className":27252,"ariaHidden":2113},[2112],[54,27254,27256,27260,27263],{"className":27255},[2117],[54,27257],{"className":27258,"style":27259},[2121],"height:0.7719em;vertical-align:-0.136em;",[54,27261,27244],{"className":27262},[2479],[54,27264],{"className":27265,"style":2475},[2270],[54,27267,27269,27272],{"className":27268},[2117],[54,27270],{"className":27271,"style":4990},[2121],[54,27273,27081],{"className":27274,"style":27166},[2126,2167]," away from ",[54,27277,27279,27296],{"className":27278},[2079],[54,27280,27282],{"className":27281},[2083],[2085,27283,27284],{"xmlns":2087},[2089,27285,27286,27294],{},[2092,27287,27288],{},[2724,27289,27290,27292],{},[2098,27291,5738],{},[2249,27293,150],{},[2105,27295,13809],{"encoding":2107},[54,27297,27299],{"className":27298,"ariaHidden":2113},[2112],[54,27300,27302,27305],{"className":27301},[2117],[54,27303],{"className":27304,"style":2743},[2121],[54,27306,27308,27311],{"className":27307},[2126],[54,27309,5738],{"className":27310,"style":5448},[2126,2167],[54,27312,27314],{"className":27313},[2135],[54,27315,27317,27337],{"className":27316},[2139,2756],[54,27318,27320,27334],{"className":27319},[2143],[54,27321,27323],{"className":27322,"style":13837},[2147],[54,27324,27325,27328],{"style":5570},[54,27326],{"className":27327,"style":2156},[2155],[54,27329,27331],{"className":27330},[2160,2161,2162,2163],[54,27332,150],{"className":27333},[2126,2163],[54,27335,2781],{"className":27336},[2780],[54,27338,27340],{"className":27339},[2143],[54,27341,27343],{"className":27342,"style":2788},[2147],[54,27344],{},", that is, it is exponentially suppressed",[659,27347,27348,27392,27393,27480,27481],{},[54,27349,27351,27371],{"className":27350},[2079],[54,27352,27354],{"className":27353},[2083],[2085,27355,27356],{"xmlns":2087},[2089,27357,27358,27368],{},[2092,27359,27360,27362,27364,27366],{},[2098,27361,21834],{},[2245,27363,129],{"stretchy":2311},[2098,27365,5738],{},[2245,27367,3205],{"stretchy":2311},[2105,27369,27370],{"encoding":2107},"f(E)",[54,27372,27374],{"className":27373,"ariaHidden":2113},[2112],[54,27375,27377,27380,27383,27386,27389],{"className":27376},[2117],[54,27378],{"className":27379,"style":2325},[2121],[54,27381,21834],{"className":27382,"style":26895},[2126,2167],[54,27384,129],{"className":27385},[2497],[54,27387,5738],{"className":27388,"style":5448},[2126,2167],[54,27390,3205],{"className":27391},[2336]," is a linear combination of ",[54,27394,27396,27426],{"className":27395},[2079],[54,27397,27399],{"className":27398},[2083],[2085,27400,27401],{"xmlns":2087},[2089,27402,27403,27423],{},[2092,27404,27405],{},[2095,27406,27407,27409],{},[2098,27408,2842],{},[2092,27410,27411,27413,27415,27417,27419,27421],{},[2098,27412,2849],{},[2098,27414,1928],{},[2098,27416,5738],{},[5723,27418,25244],{},[2098,27420,4515],{},[2098,27422,2854],{},[2105,27424,27425],{"encoding":2107},"e^{ijE\\,dt}",[54,27427,27429],{"className":27428,"ariaHidden":2113},[2112],[54,27430,27432,27435],{"className":27431},[2117],[54,27433],{"className":27434,"style":3218},[2121],[54,27436,27438,27441],{"className":27437},[2126],[54,27439,2842],{"className":27440},[2126,2167],[54,27442,27444],{"className":27443},[2135],[54,27445,27447],{"className":27446},[2139],[54,27448,27450],{"className":27449},[2143],[54,27451,27453],{"className":27452,"style":3218},[2147],[54,27454,27455,27458],{"style":2151},[54,27456],{"className":27457,"style":2156},[2155],[54,27459,27461],{"className":27460},[2160,2161,2162,2163],[54,27462,27464,27468,27471,27474,27477],{"className":27463},[2126,2163],[54,27465,27467],{"className":27466,"style":6379},[2126,2167,2163],"ij",[54,27469,5738],{"className":27470,"style":5448},[2126,2167,2163],[54,27472],{"className":27473,"style":25454},[2270,2163],[54,27475,4515],{"className":27476},[2126,2167,2163],[54,27478,2854],{"className":27479},[2126,2167,2163]," for ",[54,27482,27484,27534],{"className":27483},[2079],[54,27485,27487],{"className":27486},[2083],[2085,27488,27489],{"xmlns":2087},[2089,27490,27491,27531],{},[2092,27492,27493,27495,27497,27499,27501,27503,27505,27507,27509,27511,27513,27515,27517,27519,27521,27523,27525,27527,27529],{},[2098,27494,1928],{},[2245,27496,122],{},[2245,27498,2247],{},[2098,27500,4515],{},[2245,27502,2375],{"separator":2113},[2245,27504,2247],{},[2098,27506,4515],{},[2245,27508,1663],{},[2249,27510,174],{},[2245,27512,2375],{"separator":2113},[2098,27514,29],{"mathvariant":2304},[2098,27516,29],{"mathvariant":2304},[2098,27518,29],{"mathvariant":2304},[2245,27520,2375],{"separator":2113},[2098,27522,4515],{},[2245,27524,2247],{},[2249,27526,174],{},[2245,27528,2375],{"separator":2113},[2098,27530,4515],{},[2105,27532,27533],{"encoding":2107},"j=-d,-d+1,...,d-1,d",[54,27535,27537,27556,27589,27625],{"className":27536,"ariaHidden":2113},[2112],[54,27538,27540,27544,27547,27550,27553],{"className":27539},[2117],[54,27541],{"className":27542,"style":27543},[2121],"height:0.854em;vertical-align:-0.1944em;",[54,27545,1928],{"className":27546,"style":6379},[2126,2167],[54,27548],{"className":27549,"style":2475},[2270],[54,27551,122],{"className":27552},[2479],[54,27554],{"className":27555,"style":2475},[2270],[54,27557,27559,27562,27565,27568,27571,27574,27577,27580,27583,27586],{"className":27558},[2117],[54,27560],{"className":27561,"style":6591},[2121],[54,27563,2247],{"className":27564},[2126],[54,27566,4515],{"className":27567},[2126,2167],[54,27569,2375],{"className":27570},[2517],[54,27572],{"className":27573,"style":2521},[2270],[54,27575,2247],{"className":27576},[2126],[54,27578,4515],{"className":27579},[2126,2167],[54,27581],{"className":27582,"style":2271},[2270],[54,27584,1663],{"className":27585},[2275],[54,27587],{"className":27588,"style":2271},[2270],[54,27590,27592,27595,27598,27601,27604,27607,27610,27613,27616,27619,27622],{"className":27591},[2117],[54,27593],{"className":27594,"style":6591},[2121],[54,27596,174],{"className":27597},[2126],[54,27599,2375],{"className":27600},[2517],[54,27602],{"className":27603,"style":2521},[2270],[54,27605,2588],{"className":27606},[2126],[54,27608,2375],{"className":27609},[2517],[54,27611],{"className":27612,"style":2521},[2270],[54,27614,4515],{"className":27615},[2126,2167],[54,27617],{"className":27618,"style":2271},[2270],[54,27620,2247],{"className":27621},[2275],[54,27623],{"className":27624,"style":2271},[2270],[54,27626,27628,27631,27634,27637,27640],{"className":27627},[2117],[54,27629],{"className":27630,"style":6591},[2121],[54,27632,174],{"className":27633},[2126],[54,27635,2375],{"className":27636},[2517],[54,27638],{"className":27639,"style":2521},[2270],[54,27641,4515],{"className":27642},[2126,2167],[13,27644,27645,27646,27705,27706,27745],{},"We give a proof below, but that can be safely skipped unless one wants to understand the full, rigorous argument. For now we focus on the implications of the above claim. By property 3 above, we can see that the shifted Krylov space above contains the state ",[54,27647,27649,27675],{"className":27648},[2079],[54,27650,27652],{"className":27651},[2083],[2085,27653,27654],{"xmlns":2087},[2089,27655,27656,27672],{},[2092,27657,27658,27660,27662,27664,27666,27668,27670],{},[2098,27659,21834],{},[2245,27661,129],{"stretchy":2311},[2098,27663,2694],{},[2245,27665,3205],{"stretchy":2311},[2098,27667,2305],{"mathvariant":2304},[2098,27669,3349],{},[2245,27671,2312],{"stretchy":2311},[2105,27673,27674],{"encoding":2107},"f(H)|\\psi\\rangle",[54,27676,27678],{"className":27677,"ariaHidden":2113},[2112],[54,27679,27681,27684,27687,27690,27693,27696,27699,27702],{"className":27680},[2117],[54,27682],{"className":27683,"style":2325},[2121],[54,27685,21834],{"className":27686,"style":26895},[2126,2167],[54,27688,129],{"className":27689},[2497],[54,27691,2694],{"className":27692,"style":2709},[2126,2167],[54,27694,3205],{"className":27695},[2336],[54,27697,2305],{"className":27698},[2126],[54,27700,3349],{"className":27701,"style":2332},[2126,2167],[54,27703,2312],{"className":27704},[2336],". This is our low-energy state. To see why, write ",[54,27707,27709,27727],{"className":27708},[2079],[54,27710,27712],{"className":27711},[2083],[2085,27713,27714],{"xmlns":2087},[2089,27715,27716,27724],{},[2092,27717,27718,27720,27722],{},[2098,27719,2305],{"mathvariant":2304},[2098,27721,3349],{},[2245,27723,2312],{"stretchy":2311},[2105,27725,27726],{"encoding":2107},"|\\psi\\rangle",[54,27728,27730],{"className":27729,"ariaHidden":2113},[2112],[54,27731,27733,27736,27739,27742],{"className":27732},[2117],[54,27734],{"className":27735,"style":2325},[2121],[54,27737,2305],{"className":27738},[2126],[54,27740,3349],{"className":27741,"style":2332},[2126,2167],[54,27743,2312],{"className":27744},[2336]," in the energy eigenbasis:",[54,27747,27749],{"className":27748},[2340],[54,27750,27752,27806],{"className":27751},[2079],[54,27753,27755],{"className":27754},[2083],[2085,27756,27757],{"xmlns":2087,"display":2349},[2089,27758,27759,27803],{},[2092,27760,27761,27763,27765,27767,27769,27784,27791,27793,27799,27801],{},[2098,27762,2305],{"mathvariant":2304},[2098,27764,3349],{},[2245,27766,2312],{"stretchy":2311},[2245,27768,122],{},[27770,27771,27772,27774,27782],"munderover",{},[2245,27773,7083],{},[2092,27775,27776,27778,27780],{},[2098,27777,3182],{},[2245,27779,122],{},[2249,27781,150],{},[2098,27783,7237],{},[2724,27785,27786,27789],{},[2098,27787,27788],{},"γ",[2098,27790,3182],{},[2098,27792,2305],{"mathvariant":2304},[2724,27794,27795,27797],{},[2098,27796,5738],{},[2098,27798,3182],{},[2245,27800,2312],{"stretchy":2311},[2245,27802,2375],{"separator":2113},[2105,27804,27805],{"encoding":2107},"|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_k|E_k\\rangle,",[54,27807,27809,27833],{"className":27808,"ariaHidden":2113},[2112],[54,27810,27812,27815,27818,27821,27824,27827,27830],{"className":27811},[2117],[54,27813],{"className":27814,"style":2325},[2121],[54,27816,2305],{"className":27817},[2126],[54,27819,3349],{"className":27820,"style":2332},[2126,2167],[54,27822,2312],{"className":27823},[2336],[54,27825],{"className":27826,"style":2475},[2270],[54,27828,122],{"className":27829},[2479],[54,27831],{"className":27832,"style":2475},[2270],[54,27834,27836,27840,27915,27918,27960,27963,28003,28006],{"className":27835},[2117],[54,27837],{"className":27838,"style":27839},[2121],"height:3.1304em;vertical-align:-1.3021em;",[54,27841,27844],{"className":27842},[7126,27843],"op-limits",[54,27845,27847,27906],{"className":27846},[2139,2756],[54,27848,27850,27903],{"className":27849},[2143],[54,27851,27854,27876,27888],{"className":27852,"style":27853},[2147],"height:1.8283em;",[54,27855,27857,27861],{"style":27856},"top:-1.8479em;margin-left:0em;",[54,27858],{"className":27859,"style":27860},[2155],"height:3.05em;",[54,27862,27864],{"className":27863},[2160,2161,2162,2163],[54,27865,27867,27870,27873],{"className":27866},[2126,2163],[54,27868,3182],{"className":27869,"style":3248},[2126,2167,2163],[54,27871,122],{"className":27872},[2479,2163],[54,27874,150],{"className":27875},[2126,2163],[54,27877,27879,27882],{"style":27878},"top:-3.05em;",[54,27880],{"className":27881,"style":27860},[2155],[54,27883,27884],{},[54,27885,7083],{"className":27886},[7126,7130,27887],"large-op",[54,27889,27891,27894],{"style":27890},"top:-4.3em;margin-left:0em;",[54,27892],{"className":27893,"style":27860},[2155],[54,27895,27897],{"className":27896},[2160,2161,2162,2163],[54,27898,27900],{"className":27899},[2126,2163],[54,27901,7237],{"className":27902,"style":2870},[2126,2167,2163],[54,27904,2781],{"className":27905},[2780],[54,27907,27909],{"className":27908},[2143],[54,27910,27913],{"className":27911,"style":27912},[2147],"height:1.3021em;",[54,27914],{},[54,27916],{"className":27917,"style":2521},[2270],[54,27919,27921,27925],{"className":27920},[2126],[54,27922,27788],{"className":27923,"style":27924},[2126,2167],"margin-right:0.0556em;",[54,27926,27928],{"className":27927},[2135],[54,27929,27931,27952],{"className":27930},[2139,2756],[54,27932,27934,27949],{"className":27933},[2143],[54,27935,27937],{"className":27936,"style":10799},[2147],[54,27938,27940,27943],{"style":27939},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[54,27941],{"className":27942,"style":2156},[2155],[54,27944,27946],{"className":27945},[2160,2161,2162,2163],[54,27947,3182],{"className":27948,"style":3248},[2126,2167,2163],[54,27950,2781],{"className":27951},[2780],[54,27953,27955],{"className":27954},[2143],[54,27956,27958],{"className":27957,"style":2788},[2147],[54,27959],{},[54,27961,2305],{"className":27962},[2126],[54,27964,27966,27969],{"className":27965},[2126],[54,27967,5738],{"className":27968,"style":5448},[2126,2167],[54,27970,27972],{"className":27971},[2135],[54,27973,27975,27995],{"className":27974},[2139,2756],[54,27976,27978,27992],{"className":27977},[2143],[54,27979,27981],{"className":27980,"style":10799},[2147],[54,27982,27983,27986],{"style":5570},[54,27984],{"className":27985,"style":2156},[2155],[54,27987,27989],{"className":27988},[2160,2161,2162,2163],[54,27990,3182],{"className":27991,"style":3248},[2126,2167,2163],[54,27993,2781],{"className":27994},[2780],[54,27996,27998],{"className":27997},[2143],[54,27999,28001],{"className":28000,"style":2788},[2147],[54,28002],{},[54,28004,2312],{"className":28005},[2336],[54,28007,2375],{"className":28008},[2517],[13,28010,28011,28012,28092,28093,28164,28165,28203,28204,28262],{},"where ",[54,28013,28015,28037],{"className":28014},[2079],[54,28016,28018],{"className":28017},[2083],[2085,28019,28020],{"xmlns":2087},[2089,28021,28022,28034],{},[2092,28023,28024,28026,28032],{},[2098,28025,2305],{"mathvariant":2304},[2724,28027,28028,28030],{},[2098,28029,5738],{},[2098,28031,3182],{},[2245,28033,2312],{"stretchy":2311},[2105,28035,28036],{"encoding":2107},"|E_k\\rangle",[54,28038,28040],{"className":28039,"ariaHidden":2113},[2112],[54,28041,28043,28046,28049,28089],{"className":28042},[2117],[54,28044],{"className":28045,"style":2325},[2121],[54,28047,2305],{"className":28048},[2126],[54,28050,28052,28055],{"className":28051},[2126],[54,28053,5738],{"className":28054,"style":5448},[2126,2167],[54,28056,28058],{"className":28057},[2135],[54,28059,28061,28081],{"className":28060},[2139,2756],[54,28062,28064,28078],{"className":28063},[2143],[54,28065,28067],{"className":28066,"style":10799},[2147],[54,28068,28069,28072],{"style":5570},[54,28070],{"className":28071,"style":2156},[2155],[54,28073,28075],{"className":28074},[2160,2161,2162,2163],[54,28076,3182],{"className":28077,"style":3248},[2126,2167,2163],[54,28079,2781],{"className":28080},[2780],[54,28082,28084],{"className":28083},[2143],[54,28085,28087],{"className":28086,"style":2788},[2147],[54,28088],{},[54,28090,2312],{"className":28091},[2336]," is the kth energy eigenstate and ",[54,28094,28096,28114],{"className":28095},[2079],[54,28097,28099],{"className":28098},[2083],[2085,28100,28101],{"xmlns":2087},[2089,28102,28103,28111],{},[2092,28104,28105],{},[2724,28106,28107,28109],{},[2098,28108,27788],{},[2098,28110,3182],{},[2105,28112,28113],{"encoding":2107},"\\gamma_k",[54,28115,28117],{"className":28116,"ariaHidden":2113},[2112],[54,28118,28120,28124],{"className":28119},[2117],[54,28121],{"className":28122,"style":28123},[2121],"height:0.625em;vertical-align:-0.1944em;",[54,28125,28127,28130],{"className":28126},[2126],[54,28128,27788],{"className":28129,"style":27924},[2126,2167],[54,28131,28133],{"className":28132},[2135],[54,28134,28136,28156],{"className":28135},[2139,2756],[54,28137,28139,28153],{"className":28138},[2143],[54,28140,28142],{"className":28141,"style":10799},[2147],[54,28143,28144,28147],{"style":27939},[54,28145],{"className":28146,"style":2156},[2155],[54,28148,28150],{"className":28149},[2160,2161,2162,2163],[54,28151,3182],{"className":28152,"style":3248},[2126,2167,2163],[54,28154,2781],{"className":28155},[2780],[54,28157,28159],{"className":28158},[2143],[54,28160,28162],{"className":28161,"style":2788},[2147],[54,28163],{}," is its amplitude in the initial state ",[54,28166,28168,28185],{"className":28167},[2079],[54,28169,28171],{"className":28170},[2083],[2085,28172,28173],{"xmlns":2087},[2089,28174,28175,28183],{},[2092,28176,28177,28179,28181],{},[2098,28178,2305],{"mathvariant":2304},[2098,28180,3349],{},[2245,28182,2312],{"stretchy":2311},[2105,28184,27726],{"encoding":2107},[54,28186,28188],{"className":28187,"ariaHidden":2113},[2112],[54,28189,28191,28194,28197,28200],{"className":28190},[2117],[54,28192],{"className":28193,"style":2325},[2121],[54,28195,2305],{"className":28196},[2126],[54,28198,3349],{"className":28199,"style":2332},[2126,2167],[54,28201,2312],{"className":28202},[2336],". Expressed in terms of this, ",[54,28205,28207,28232],{"className":28206},[2079],[54,28208,28210],{"className":28209},[2083],[2085,28211,28212],{"xmlns":2087},[2089,28213,28214,28230],{},[2092,28215,28216,28218,28220,28222,28224,28226,28228],{},[2098,28217,21834],{},[2245,28219,129],{"stretchy":2311},[2098,28221,2694],{},[2245,28223,3205],{"stretchy":2311},[2098,28225,2305],{"mathvariant":2304},[2098,28227,3349],{},[2245,28229,2312],{"stretchy":2311},[2105,28231,27674],{"encoding":2107},[54,28233,28235],{"className":28234,"ariaHidden":2113},[2112],[54,28236,28238,28241,28244,28247,28250,28253,28256,28259],{"className":28237},[2117],[54,28239],{"className":28240,"style":2325},[2121],[54,28242,21834],{"className":28243,"style":26895},[2126,2167],[54,28245,129],{"className":28246},[2497],[54,28248,2694],{"className":28249,"style":2709},[2126,2167],[54,28251,3205],{"className":28252},[2336],[54,28254,2305],{"className":28255},[2126],[54,28257,3349],{"className":28258,"style":2332},[2126,2167],[54,28260,2312],{"className":28261},[2336]," is given by",[54,28264,28266],{"className":28265},[2340],[54,28267,28269,28341],{"className":28268},[2079],[54,28270,28272],{"className":28271},[2083],[2085,28273,28274],{"xmlns":2087,"display":2349},[2089,28275,28276,28338],{},[2092,28277,28278,28280,28282,28284,28286,28288,28290,28292,28294,28308,28314,28316,28318,28324,28326,28328,28334,28336],{},[2098,28279,21834],{},[2245,28281,129],{"stretchy":2311},[2098,28283,2694],{},[2245,28285,3205],{"stretchy":2311},[2098,28287,2305],{"mathvariant":2304},[2098,28289,3349],{},[2245,28291,2312],{"stretchy":2311},[2245,28293,122],{},[27770,28295,28296,28298,28306],{},[2245,28297,7083],{},[2092,28299,28300,28302,28304],{},[2098,28301,3182],{},[2245,28303,122],{},[2249,28305,150],{},[2098,28307,7237],{},[2724,28309,28310,28312],{},[2098,28311,27788],{},[2098,28313,3182],{},[2098,28315,21834],{},[2245,28317,129],{"stretchy":2311},[2724,28319,28320,28322],{},[2098,28321,5738],{},[2098,28323,3182],{},[2245,28325,3205],{"stretchy":2311},[2098,28327,2305],{"mathvariant":2304},[2724,28329,28330,28332],{},[2098,28331,5738],{},[2098,28333,3182],{},[2245,28335,2312],{"stretchy":2311},[2245,28337,2375],{"separator":2113},[2105,28339,28340],{"encoding":2107},"f(H)|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_kf(E_k)|E_k\\rangle,",[54,28342,28344,28380],{"className":28343,"ariaHidden":2113},[2112],[54,28345,28347,28350,28353,28356,28359,28362,28365,28368,28371,28374,28377],{"className":28346},[2117],[54,28348],{"className":28349,"style":2325},[2121],[54,28351,21834],{"className":28352,"style":26895},[2126,2167],[54,28354,129],{"className":28355},[2497],[54,28357,2694],{"className":28358,"style":2709},[2126,2167],[54,28360,3205],{"className":28361},[2336],[54,28363,2305],{"className":28364},[2126],[54,28366,3349],{"className":28367,"style":2332},[2126,2167],[54,28369,2312],{"className":28370},[2336],[54,28372],{"className":28373,"style":2475},[2270],[54,28375,122],{"className":28376},[2479],[54,28378],{"className":28379,"style":2475},[2270],[54,28381,28383,28386,28453,28456,28496,28499,28502,28542,28545,28548,28588,28591],{"className":28382},[2117],[54,28384],{"className":28385,"style":27839},[2121],[54,28387,28389],{"className":28388},[7126,27843],[54,28390,28392,28445],{"className":28391},[2139,2756],[54,28393,28395,28442],{"className":28394},[2143],[54,28396,28398,28418,28428],{"className":28397,"style":27853},[2147],[54,28399,28400,28403],{"style":27856},[54,28401],{"className":28402,"style":27860},[2155],[54,28404,28406],{"className":28405},[2160,2161,2162,2163],[54,28407,28409,28412,28415],{"className":28408},[2126,2163],[54,28410,3182],{"className":28411,"style":3248},[2126,2167,2163],[54,28413,122],{"className":28414},[2479,2163],[54,28416,150],{"className":28417},[2126,2163],[54,28419,28420,28423],{"style":27878},[54,28421],{"className":28422,"style":27860},[2155],[54,28424,28425],{},[54,28426,7083],{"className":28427},[7126,7130,27887],[54,28429,28430,28433],{"style":27890},[54,28431],{"className":28432,"style":27860},[2155],[54,28434,28436],{"className":28435},[2160,2161,2162,2163],[54,28437,28439],{"className":28438},[2126,2163],[54,28440,7237],{"className":28441,"style":2870},[2126,2167,2163],[54,28443,2781],{"className":28444},[2780],[54,28446,28448],{"className":28447},[2143],[54,28449,28451],{"className":28450,"style":27912},[2147],[54,28452],{},[54,28454],{"className":28455,"style":2521},[2270],[54,28457,28459,28462],{"className":28458},[2126],[54,28460,27788],{"className":28461,"style":27924},[2126,2167],[54,28463,28465],{"className":28464},[2135],[54,28466,28468,28488],{"className":28467},[2139,2756],[54,28469,28471,28485],{"className":28470},[2143],[54,28472,28474],{"className":28473,"style":10799},[2147],[54,28475,28476,28479],{"style":27939},[54,28477],{"className":28478,"style":2156},[2155],[54,28480,28482],{"className":28481},[2160,2161,2162,2163],[54,28483,3182],{"className":28484,"style":3248},[2126,2167,2163],[54,28486,2781],{"className":28487},[2780],[54,28489,28491],{"className":28490},[2143],[54,28492,28494],{"className":28493,"style":2788},[2147],[54,28495],{},[54,28497,21834],{"className":28498,"style":26895},[2126,2167],[54,28500,129],{"className":28501},[2497],[54,28503,28505,28508],{"className":28504},[2126],[54,28506,5738],{"className":28507,"style":5448},[2126,2167],[54,28509,28511],{"className":28510},[2135],[54,28512,28514,28534],{"className":28513},[2139,2756],[54,28515,28517,28531],{"className":28516},[2143],[54,28518,28520],{"className":28519,"style":10799},[2147],[54,28521,28522,28525],{"style":5570},[54,28523],{"className":28524,"style":2156},[2155],[54,28526,28528],{"className":28527},[2160,2161,2162,2163],[54,28529,3182],{"className":28530,"style":3248},[2126,2167,2163],[54,28532,2781],{"className":28533},[2780],[54,28535,28537],{"className":28536},[2143],[54,28538,28540],{"className":28539,"style":2788},[2147],[54,28541],{},[54,28543,3205],{"className":28544},[2336],[54,28546,2305],{"className":28547},[2126],[54,28549,28551,28554],{"className":28550},[2126],[54,28552,5738],{"className":28553,"style":5448},[2126,2167],[54,28555,28557],{"className":28556},[2135],[54,28558,28560,28580],{"className":28559},[2139,2756],[54,28561,28563,28577],{"className":28562},[2143],[54,28564,28566],{"className":28565,"style":10799},[2147],[54,28567,28568,28571],{"style":5570},[54,28569],{"className":28570,"style":2156},[2155],[54,28572,28574],{"className":28573},[2160,2161,2162,2163],[54,28575,3182],{"className":28576,"style":3248},[2126,2167,2163],[54,28578,2781],{"className":28579},[2780],[54,28581,28583],{"className":28582},[2143],[54,28584,28586],{"className":28585,"style":2788},[2147],[54,28587],{},[54,28589,2312],{"className":28590},[2336],[54,28592,2375],{"className":28593},[2517],[13,28595,28596,28597,28625,28626,28696,28697,28776],{},"using the fact that we can replace ",[54,28598,28600,28613],{"className":28599},[2079],[54,28601,28603],{"className":28602},[2083],[2085,28604,28605],{"xmlns":2087},[2089,28606,28607,28611],{},[2092,28608,28609],{},[2098,28610,2694],{},[2105,28612,2694],{"encoding":2107},[54,28614,28616],{"className":28615,"ariaHidden":2113},[2112],[54,28617,28619,28622],{"className":28618},[2117],[54,28620],{"className":28621,"style":2122},[2121],[54,28623,2694],{"className":28624,"style":2709},[2126,2167]," by ",[54,28627,28629,28647],{"className":28628},[2079],[54,28630,28632],{"className":28631},[2083],[2085,28633,28634],{"xmlns":2087},[2089,28635,28636,28644],{},[2092,28637,28638],{},[2724,28639,28640,28642],{},[2098,28641,5738],{},[2098,28643,3182],{},[2105,28645,28646],{"encoding":2107},"E_k",[54,28648,28650],{"className":28649,"ariaHidden":2113},[2112],[54,28651,28653,28656],{"className":28652},[2117],[54,28654],{"className":28655,"style":2743},[2121],[54,28657,28659,28662],{"className":28658},[2126],[54,28660,5738],{"className":28661,"style":5448},[2126,2167],[54,28663,28665],{"className":28664},[2135],[54,28666,28668,28688],{"className":28667},[2139,2756],[54,28669,28671,28685],{"className":28670},[2143],[54,28672,28674],{"className":28673,"style":10799},[2147],[54,28675,28676,28679],{"style":5570},[54,28677],{"className":28678,"style":2156},[2155],[54,28680,28682],{"className":28681},[2160,2161,2162,2163],[54,28683,3182],{"className":28684,"style":3248},[2126,2167,2163],[54,28686,2781],{"className":28687},[2780],[54,28689,28691],{"className":28690},[2143],[54,28692,28694],{"className":28693,"style":2788},[2147],[54,28695],{}," when it acts on the eigenstate ",[54,28698,28700,28721],{"className":28699},[2079],[54,28701,28703],{"className":28702},[2083],[2085,28704,28705],{"xmlns":2087},[2089,28706,28707,28719],{},[2092,28708,28709,28711,28717],{},[2098,28710,2305],{"mathvariant":2304},[2724,28712,28713,28715],{},[2098,28714,5738],{},[2098,28716,3182],{},[2245,28718,2312],{"stretchy":2311},[2105,28720,28036],{"encoding":2107},[54,28722,28724],{"className":28723,"ariaHidden":2113},[2112],[54,28725,28727,28730,28733,28773],{"className":28726},[2117],[54,28728],{"className":28729,"style":2325},[2121],[54,28731,2305],{"className":28732},[2126],[54,28734,28736,28739],{"className":28735},[2126],[54,28737,5738],{"className":28738,"style":5448},[2126,2167],[54,28740,28742],{"className":28741},[2135],[54,28743,28745,28765],{"className":28744},[2139,2756],[54,28746,28748,28762],{"className":28747},[2143],[54,28749,28751],{"className":28750,"style":10799},[2147],[54,28752,28753,28756],{"style":5570},[54,28754],{"className":28755,"style":2156},[2155],[54,28757,28759],{"className":28758},[2160,2161,2162,2163],[54,28760,3182],{"className":28761,"style":3248},[2126,2167,2163],[54,28763,2781],{"className":28764},[2780],[54,28766,28768],{"className":28767},[2143],[54,28769,28771],{"className":28770,"style":2788},[2147],[54,28772],{},[54,28774,2312],{"className":28775},[2336],". The energy error of this state is therefore",[54,28778,28780],{"className":28779},[2340],[54,28781,28783,28872],{"className":28782},[2079],[54,28784,28786],{"className":28785},[2083],[2085,28787,28788],{"xmlns":2087,"display":2349},[2089,28789,28790,28869],{},[2092,28791,28792,28795,28797],{},[5723,28793,28794],{},"energy error",[2245,28796,122],{},[9047,28798,28799,28843],{},[2092,28800,28801,28803,28805,28807,28809,28811,28813,28815,28817,28819,28821,28827,28829,28831,28833,28835,28837,28839,28841],{},[2245,28802,3981],{"stretchy":2311},[2098,28804,3349],{},[2098,28806,2305],{"mathvariant":2304},[2098,28808,21834],{},[2245,28810,129],{"stretchy":2311},[2098,28812,2694],{},[2245,28814,3205],{"stretchy":2311},[2245,28816,129],{"stretchy":2311},[2098,28818,2694],{},[2245,28820,2247],{},[2724,28822,28823,28825],{},[2098,28824,5738],{},[2249,28826,150],{},[2245,28828,3205],{"stretchy":2311},[2098,28830,21834],{},[2245,28832,129],{"stretchy":2311},[2098,28834,2694],{},[2245,28836,3205],{"stretchy":2311},[2098,28838,2305],{"mathvariant":2304},[2098,28840,3349],{},[2245,28842,2312],{"stretchy":2311},[2092,28844,28845,28847,28849,28851,28853,28855,28857,28863,28865,28867],{},[2245,28846,3981],{"stretchy":2311},[2098,28848,3349],{},[2098,28850,2305],{"mathvariant":2304},[2098,28852,21834],{},[2245,28854,129],{"stretchy":2311},[2098,28856,2694],{},[2095,28858,28859,28861],{},[2245,28860,3205],{"stretchy":2311},[2249,28862,133],{},[2098,28864,2305],{"mathvariant":2304},[2098,28866,3349],{},[2245,28868,2312],{"stretchy":2311},[2105,28870,28871],{"encoding":2107},"\\text{energy error} = \\frac{\\langle\\psi|f(H)(H-E_0)f(H)|\\psi\\rangle}{\\langle\\psi|f(H)^2|\\psi\\rangle}",[54,28873,28875,28896],{"className":28874,"ariaHidden":2113},[2112],[54,28876,28878,28881,28887,28890,28893],{"className":28877},[2117],[54,28879],{"className":28880,"style":28123},[2121],[54,28882,28884],{"className":28883},[2126,6595],[54,28885,28794],{"className":28886},[2126],[54,28888],{"className":28889,"style":2475},[2270],[54,28891,122],{"className":28892},[2479],[54,28894],{"className":28895,"style":2475},[2270],[54,28897,28899,28903],{"className":28898},[2117],[54,28900],{"className":28901,"style":28902},[2121],"height:2.363em;vertical-align:-0.936em;",[54,28904,28906,28909,29116],{"className":28905},[2126],[54,28907],{"className":28908},[2497,9349],[54,28910,28912],{"className":28911},[9047],[54,28913,28915,29107],{"className":28914},[2139,2756],[54,28916,28918,29104],{"className":28917},[2143],[54,28919,28922,28988,28996],{"className":28920,"style":28921},[2147],"height:1.427em;",[54,28923,28924,28927],{"style":11273},[54,28925],{"className":28926,"style":3866},[2155],[54,28928,28930,28933,28936,28939,28942,28945,28948,28979,28982,28985],{"className":28929},[2126],[54,28931,3981],{"className":28932},[2497],[54,28934,3349],{"className":28935,"style":2332},[2126,2167],[54,28937,2305],{"className":28938},[2126],[54,28940,21834],{"className":28941,"style":26895},[2126,2167],[54,28943,129],{"className":28944},[2497],[54,28946,2694],{"className":28947,"style":2709},[2126,2167],[54,28949,28951,28954],{"className":28950},[2336],[54,28952,3205],{"className":28953},[2336],[54,28955,28957],{"className":28956},[2135],[54,28958,28960],{"className":28959},[2139],[54,28961,28963],{"className":28962},[2143],[54,28964,28967],{"className":28965,"style":28966},[2147],"height:0.7401em;",[54,28968,28970,28973],{"style":28969},"top:-2.989em;margin-right:0.05em;",[54,28971],{"className":28972,"style":2156},[2155],[54,28974,28976],{"className":28975},[2160,2161,2162,2163],[54,28977,133],{"className":28978},[2126,2163],[54,28980,2305],{"className":28981},[2126],[54,28983,3349],{"className":28984,"style":2332},[2126,2167],[54,28986,2312],{"className":28987},[2336],[54,28989,28990,28993],{"style":9434},[54,28991],{"className":28992,"style":3866},[2155],[54,28994],{"className":28995,"style":9442},[9441],[54,28997,28998,29001],{"style":9445},[54,28999],{"className":29000,"style":3866},[2155],[54,29002,29004,29007,29010,29013,29016,29019,29022,29025,29028,29031,29034,29037,29040,29080,29083,29086,29089,29092,29095,29098,29101],{"className":29003},[2126],[54,29005,3981],{"className":29006},[2497],[54,29008,3349],{"className":29009,"style":2332},[2126,2167],[54,29011,2305],{"className":29012},[2126],[54,29014,21834],{"className":29015,"style":26895},[2126,2167],[54,29017,129],{"className":29018},[2497],[54,29020,2694],{"className":29021,"style":2709},[2126,2167],[54,29023,3205],{"className":29024},[2336],[54,29026,129],{"className":29027},[2497],[54,29029,2694],{"className":29030,"style":2709},[2126,2167],[54,29032],{"className":29033,"style":2271},[2270],[54,29035,2247],{"className":29036},[2275],[54,29038],{"className":29039,"style":2271},[2270],[54,29041,29043,29046],{"className":29042},[2126],[54,29044,5738],{"className":29045,"style":5448},[2126,2167],[54,29047,29049],{"className":29048},[2135],[54,29050,29052,29072],{"className":29051},[2139,2756],[54,29053,29055,29069],{"className":29054},[2143],[54,29056,29058],{"className":29057,"style":13837},[2147],[54,29059,29060,29063],{"style":5570},[54,29061],{"className":29062,"style":2156},[2155],[54,29064,29066],{"className":29065},[2160,2161,2162,2163],[54,29067,150],{"className":29068},[2126,2163],[54,29070,2781],{"className":29071},[2780],[54,29073,29075],{"className":29074},[2143],[54,29076,29078],{"className":29077,"style":2788},[2147],[54,29079],{},[54,29081,3205],{"className":29082},[2336],[54,29084,21834],{"className":29085,"style":26895},[2126,2167],[54,29087,129],{"className":29088},[2497],[54,29090,2694],{"className":29091,"style":2709},[2126,2167],[54,29093,3205],{"className":29094},[2336],[54,29096,2305],{"className":29097},[2126],[54,29099,3349],{"className":29100,"style":2332},[2126,2167],[54,29102,2312],{"className":29103},[2336],[54,29105,2781],{"className":29106},[2780],[54,29108,29110],{"className":29109},[2143],[54,29111,29114],{"className":29112,"style":29113},[2147],"height:0.936em;",[54,29115],{},[54,29117],{"className":29118},[2336,9349],[54,29120,29122],{"className":29121},[2340],[54,29123,29125,29253],{"className":29124},[2079],[54,29126,29128],{"className":29127},[2083],[2085,29129,29130],{"xmlns":2087,"display":2349},[2089,29131,29132,29250],{},[2092,29133,29134,29136,29248],{},[2245,29135,122],{},[9047,29137,29138,29202],{},[2092,29139,29140,29154,29156,29162,29168,29170,29172,29178,29184,29186,29192,29194,29200],{},[27770,29141,29142,29144,29152],{},[2245,29143,7083],{},[2092,29145,29146,29148,29150],{},[2098,29147,3182],{},[2245,29149,122],{},[2249,29151,150],{},[2098,29153,7237],{},[2098,29155,2305],{"mathvariant":2304},[2724,29157,29158,29160],{},[2098,29159,27788],{},[2098,29161,3182],{},[2095,29163,29164,29166],{},[2098,29165,2305],{"mathvariant":2304},[2249,29167,133],{},[2098,29169,21834],{},[2245,29171,129],{"stretchy":2311},[2724,29173,29174,29176],{},[2098,29175,5738],{},[2098,29177,3182],{},[2095,29179,29180,29182],{},[2245,29181,3205],{"stretchy":2311},[2249,29183,133],{},[2245,29185,129],{"stretchy":2311},[2724,29187,29188,29190],{},[2098,29189,5738],{},[2098,29191,3182],{},[2245,29193,2247],{},[2724,29195,29196,29198],{},[2098,29197,5738],{},[2249,29199,150],{},[2245,29201,3205],{"stretchy":2311},[2092,29203,29204,29218,29220,29226,29232,29234,29236,29242],{},[27770,29205,29206,29208,29216],{},[2245,29207,7083],{},[2092,29209,29210,29212,29214],{},[2098,29211,3182],{},[2245,29213,122],{},[2249,29215,150],{},[2098,29217,7237],{},[2098,29219,2305],{"mathvariant":2304},[2724,29221,29222,29224],{},[2098,29223,27788],{},[2098,29225,3182],{},[2095,29227,29228,29230],{},[2098,29229,2305],{"mathvariant":2304},[2249,29231,133],{},[2098,29233,21834],{},[2245,29235,129],{"stretchy":2311},[2724,29237,29238,29240],{},[2098,29239,5738],{},[2098,29241,3182],{},[2095,29243,29244,29246],{},[2245,29245,3205],{"stretchy":2311},[2249,29247,133],{},[2098,29249,29],{"mathvariant":2304},[2105,29251,29252],{"encoding":2107},"= \\frac{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2(E_k-E_0)}{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2}.",[54,29254,29256,29268],{"className":29255,"ariaHidden":2113},[2112],[54,29257,29259,29262,29265],{"className":29258},[2117],[54,29260],{"className":29261,"style":4284},[2121],[54,29263,122],{"className":29264},[2479],[54,29266],{"className":29267,"style":2475},[2270],[54,29269,29271,29275,29856],{"className":29270},[2117],[54,29272],{"className":29273,"style":29274},[2121],"height:2.8419em;vertical-align:-1.1709em;",[54,29276,29278,29281,29853],{"className":29277},[2126],[54,29279],{"className":29280},[2497,9349],[54,29282,29284],{"className":29283},[9047],[54,29285,29287,29844],{"className":29286},[2139,2756],[54,29288,29290,29841],{"className":29289},[2143],[54,29291,29294,29516,29524],{"className":29292,"style":29293},[2147],"height:1.6709em;",[54,29295,29297,29300],{"style":29296},"top:-2.1288em;",[54,29298],{"className":29299,"style":3866},[2155],[54,29301,29303,29366,29369,29372,29412,29441,29444,29447,29487],{"className":29302},[2126],[54,29304,29306,29309],{"className":29305},[7126],[54,29307,7083],{"className":29308,"style":7132},[7126,7130,7131],[54,29310,29312],{"className":29311},[2135],[54,29313,29315,29358],{"className":29314},[2139,2756],[54,29316,29318,29355],{"className":29317},[2143],[54,29319,29321,29341],{"className":29320,"style":7378},[2147],[54,29322,29323,29326],{"style":7148},[54,29324],{"className":29325,"style":2156},[2155],[54,29327,29329],{"className":29328},[2160,2161,2162,2163],[54,29330,29332,29335,29338],{"className":29331},[2126,2163],[54,29333,3182],{"className":29334,"style":3248},[2126,2167,2163],[54,29336,122],{"className":29337},[2479,2163],[54,29339,150],{"className":29340},[2126,2163],[54,29342,29343,29346],{"style":7401},[54,29344],{"className":29345,"style":2156},[2155],[54,29347,29349],{"className":29348},[2160,2161,2162,2163],[54,29350,29352],{"className":29351},[2126,2163],[54,29353,7237],{"className":29354,"style":2870},[2126,2167,2163],[54,29356,2781],{"className":29357},[2780],[54,29359,29361],{"className":29360},[2143],[54,29362,29364],{"className":29363,"style":7167},[2147],[54,29365],{},[54,29367],{"className":29368,"style":2521},[2270],[54,29370,2305],{"className":29371},[2126],[54,29373,29375,29378],{"className":29374},[2126],[54,29376,27788],{"className":29377,"style":27924},[2126,2167],[54,29379,29381],{"className":29380},[2135],[54,29382,29384,29404],{"className":29383},[2139,2756],[54,29385,29387,29401],{"className":29386},[2143],[54,29388,29390],{"className":29389,"style":10799},[2147],[54,29391,29392,29395],{"style":27939},[54,29393],{"className":29394,"style":2156},[2155],[54,29396,29398],{"className":29397},[2160,2161,2162,2163],[54,29399,3182],{"className":29400,"style":3248},[2126,2167,2163],[54,29402,2781],{"className":29403},[2780],[54,29405,29407],{"className":29406},[2143],[54,29408,29410],{"className":29409,"style":2788},[2147],[54,29411],{},[54,29413,29415,29418],{"className":29414},[2126],[54,29416,2305],{"className":29417},[2126],[54,29419,29421],{"className":29420},[2135],[54,29422,29424],{"className":29423},[2139],[54,29425,29427],{"className":29426},[2143],[54,29428,29430],{"className":29429,"style":28966},[2147],[54,29431,29432,29435],{"style":28969},[54,29433],{"className":29434,"style":2156},[2155],[54,29436,29438],{"className":29437},[2160,2161,2162,2163],[54,29439,133],{"className":29440},[2126,2163],[54,29442,21834],{"className":29443,"style":26895},[2126,2167],[54,29445,129],{"className":29446},[2497],[54,29448,29450,29453],{"className":29449},[2126],[54,29451,5738],{"className":29452,"style":5448},[2126,2167],[54,29454,29456],{"className":29455},[2135],[54,29457,29459,29479],{"className":29458},[2139,2756],[54,29460,29462,29476],{"className":29461},[2143],[54,29463,29465],{"className":29464,"style":10799},[2147],[54,29466,29467,29470],{"style":5570},[54,29468],{"className":29469,"style":2156},[2155],[54,29471,29473],{"className":29472},[2160,2161,2162,2163],[54,29474,3182],{"className":29475,"style":3248},[2126,2167,2163],[54,29477,2781],{"className":29478},[2780],[54,29480,29482],{"className":29481},[2143],[54,29483,29485],{"className":29484,"style":2788},[2147],[54,29486],{},[54,29488,29490,29493],{"className":29489},[2336],[54,29491,3205],{"className":29492},[2336],[54,29494,29496],{"className":29495},[2135],[54,29497,29499],{"className":29498},[2139],[54,29500,29502],{"className":29501},[2143],[54,29503,29505],{"className":29504,"style":28966},[2147],[54,29506,29507,29510],{"style":28969},[54,29508],{"className":29509,"style":2156},[2155],[54,29511,29513],{"className":29512},[2160,2161,2162,2163],[54,29514,133],{"className":29515},[2126,2163],[54,29517,29518,29521],{"style":9434},[54,29519],{"className":29520,"style":3866},[2155],[54,29522],{"className":29523,"style":9442},[9441],[54,29525,29527,29530],{"style":29526},"top:-3.6897em;",[54,29528],{"className":29529,"style":3866},[2155],[54,29531,29533,29596,29599,29602,29642,29671,29674,29677,29717,29746,29749,29789,29792,29795,29798,29838],{"className":29532},[2126],[54,29534,29536,29539],{"className":29535},[7126],[54,29537,7083],{"className":29538,"style":7132},[7126,7130,7131],[54,29540,29542],{"className":29541},[2135],[54,29543,29545,29588],{"className":29544},[2139,2756],[54,29546,29548,29585],{"className":29547},[2143],[54,29549,29551,29571],{"className":29550,"style":7378},[2147],[54,29552,29553,29556],{"style":7148},[54,29554],{"className":29555,"style":2156},[2155],[54,29557,29559],{"className":29558},[2160,2161,2162,2163],[54,29560,29562,29565,29568],{"className":29561},[2126,2163],[54,29563,3182],{"className":29564,"style":3248},[2126,2167,2163],[54,29566,122],{"className":29567},[2479,2163],[54,29569,150],{"className":29570},[2126,2163],[54,29572,29573,29576],{"style":7401},[54,29574],{"className":29575,"style":2156},[2155],[54,29577,29579],{"className":29578},[2160,2161,2162,2163],[54,29580,29582],{"className":29581},[2126,2163],[54,29583,7237],{"className":29584,"style":2870},[2126,2167,2163],[54,29586,2781],{"className":29587},[2780],[54,29589,29591],{"className":29590},[2143],[54,29592,29594],{"className":29593,"style":7167},[2147],[54,29595],{},[54,29597],{"className":29598,"style":2521},[2270],[54,29600,2305],{"className":29601},[2126],[54,29603,29605,29608],{"className":29604},[2126],[54,29606,27788],{"className":29607,"style":27924},[2126,2167],[54,29609,29611],{"className":29610},[2135],[54,29612,29614,29634],{"className":29613},[2139,2756],[54,29615,29617,29631],{"className":29616},[2143],[54,29618,29620],{"className":29619,"style":10799},[2147],[54,29621,29622,29625],{"style":27939},[54,29623],{"className":29624,"style":2156},[2155],[54,29626,29628],{"className":29627},[2160,2161,2162,2163],[54,29629,3182],{"className":29630,"style":3248},[2126,2167,2163],[54,29632,2781],{"className":29633},[2780],[54,29635,29637],{"className":29636},[2143],[54,29638,29640],{"className":29639,"style":2788},[2147],[54,29641],{},[54,29643,29645,29648],{"className":29644},[2126],[54,29646,2305],{"className":29647},[2126],[54,29649,29651],{"className":29650},[2135],[54,29652,29654],{"className":29653},[2139],[54,29655,29657],{"className":29656},[2143],[54,29658,29660],{"className":29659,"style":12780},[2147],[54,29661,29662,29665],{"style":2151},[54,29663],{"className":29664,"style":2156},[2155],[54,29666,29668],{"className":29667},[2160,2161,2162,2163],[54,29669,133],{"className":29670},[2126,2163],[54,29672,21834],{"className":29673,"style":26895},[2126,2167],[54,29675,129],{"className":29676},[2497],[54,29678,29680,29683],{"className":29679},[2126],[54,29681,5738],{"className":29682,"style":5448},[2126,2167],[54,29684,29686],{"className":29685},[2135],[54,29687,29689,29709],{"className":29688},[2139,2756],[54,29690,29692,29706],{"className":29691},[2143],[54,29693,29695],{"className":29694,"style":10799},[2147],[54,29696,29697,29700],{"style":5570},[54,29698],{"className":29699,"style":2156},[2155],[54,29701,29703],{"className":29702},[2160,2161,2162,2163],[54,29704,3182],{"className":29705,"style":3248},[2126,2167,2163],[54,29707,2781],{"className":29708},[2780],[54,29710,29712],{"className":29711},[2143],[54,29713,29715],{"className":29714,"style":2788},[2147],[54,29716],{},[54,29718,29720,29723],{"className":29719},[2336],[54,29721,3205],{"className":29722},[2336],[54,29724,29726],{"className":29725},[2135],[54,29727,29729],{"className":29728},[2139],[54,29730,29732],{"className":29731},[2143],[54,29733,29735],{"className":29734,"style":12780},[2147],[54,29736,29737,29740],{"style":2151},[54,29738],{"className":29739,"style":2156},[2155],[54,29741,29743],{"className":29742},[2160,2161,2162,2163],[54,29744,133],{"className":29745},[2126,2163],[54,29747,129],{"className":29748},[2497],[54,29750,29752,29755],{"className":29751},[2126],[54,29753,5738],{"className":29754,"style":5448},[2126,2167],[54,29756,29758],{"className":29757},[2135],[54,29759,29761,29781],{"className":29760},[2139,2756],[54,29762,29764,29778],{"className":29763},[2143],[54,29765,29767],{"className":29766,"style":10799},[2147],[54,29768,29769,29772],{"style":5570},[54,29770],{"className":29771,"style":2156},[2155],[54,29773,29775],{"className":29774},[2160,2161,2162,2163],[54,29776,3182],{"className":29777,"style":3248},[2126,2167,2163],[54,29779,2781],{"className":29780},[2780],[54,29782,29784],{"className":29783},[2143],[54,29785,29787],{"className":29786,"style":2788},[2147],[54,29788],{},[54,29790],{"className":29791,"style":2271},[2270],[54,29793,2247],{"className":29794},[2275],[54,29796],{"className":29797,"style":2271},[2270],[54,29799,29801,29804],{"className":29800},[2126],[54,29802,5738],{"className":29803,"style":5448},[2126,2167],[54,29805,29807],{"className":29806},[2135],[54,29808,29810,29830],{"className":29809},[2139,2756],[54,29811,29813,29827],{"className":29812},[2143],[54,29814,29816],{"className":29815,"style":13837},[2147],[54,29817,29818,29821],{"style":5570},[54,29819],{"className":29820,"style":2156},[2155],[54,29822,29824],{"className":29823},[2160,2161,2162,2163],[54,29825,150],{"className":29826},[2126,2163],[54,29828,2781],{"className":29829},[2780],[54,29831,29833],{"className":29832},[2143],[54,29834,29836],{"className":29835,"style":2788},[2147],[54,29837],{},[54,29839,3205],{"className":29840},[2336],[54,29842,2781],{"className":29843},[2780],[54,29845,29847],{"className":29846},[2143],[54,29848,29851],{"className":29849,"style":29850},[2147],"height:1.1709em;",[54,29852],{},[54,29854],{"className":29855},[2336,9349],[54,29857,29],{"className":29858},[2126],[13,29860,29861,29862,30017,30018,24541],{},"To turn this into an upper bound that is easier to understand, we first separate the sum in the numerator into terms with ",[54,29863,29865,29895],{"className":29864},[2079],[54,29866,29868],{"className":29867},[2083],[2085,29869,29870],{"xmlns":2087},[2089,29871,29872,29892],{},[2092,29873,29874,29880,29882,29888,29890],{},[2724,29875,29876,29878],{},[2098,29877,5738],{},[2098,29879,3182],{},[2245,29881,2247],{},[2724,29883,29884,29886],{},[2098,29885,5738],{},[2249,29887,150],{},[2245,29889,27066],{},[2098,29891,27081],{},[2105,29893,29894],{"encoding":2107},"E_k-E_0\\le\\delta",[54,29896,29898,29953,30008],{"className":29897,"ariaHidden":2113},[2112],[54,29899,29901,29904,29944,29947,29950],{"className":29900},[2117],[54,29902],{"className":29903,"style":2743},[2121],[54,29905,29907,29910],{"className":29906},[2126],[54,29908,5738],{"className":29909,"style":5448},[2126,2167],[54,29911,29913],{"className":29912},[2135],[54,29914,29916,29936],{"className":29915},[2139,2756],[54,29917,29919,29933],{"className":29918},[2143],[54,29920,29922],{"className":29921,"style":10799},[2147],[54,29923,29924,29927],{"style":5570},[54,29925],{"className":29926,"style":2156},[2155],[54,29928,29930],{"className":29929},[2160,2161,2162,2163],[54,29931,3182],{"className":29932,"style":3248},[2126,2167,2163],[54,29934,2781],{"className":29935},[2780],[54,29937,29939],{"className":29938},[2143],[54,29940,29942],{"className":29941,"style":2788},[2147],[54,29943],{},[54,29945],{"className":29946,"style":2271},[2270],[54,29948,2247],{"className":29949},[2275],[54,29951],{"className":29952,"style":2271},[2270],[54,29954,29956,29959,29999,30002,30005],{"className":29955},[2117],[54,29957],{"className":29958,"style":2743},[2121],[54,29960,29962,29965],{"className":29961},[2126],[54,29963,5738],{"className":29964,"style":5448},[2126,2167],[54,29966,29968],{"className":29967},[2135],[54,29969,29971,29991],{"className":29970},[2139,2756],[54,29972,29974,29988],{"className":29973},[2143],[54,29975,29977],{"className":29976,"style":13837},[2147],[54,29978,29979,29982],{"style":5570},[54,29980],{"className":29981,"style":2156},[2155],[54,29983,29985],{"className":29984},[2160,2161,2162,2163],[54,29986,150],{"className":29987},[2126,2163],[54,29989,2781],{"className":29990},[2780],[54,29992,29994],{"className":29993},[2143],[54,29995,29997],{"className":29996,"style":2788},[2147],[54,29998],{},[54,30000],{"className":30001,"style":2475},[2270],[54,30003,27066],{"className":30004},[2479],[54,30006],{"className":30007,"style":2475},[2270],[54,30009,30011,30014],{"className":30010},[2117],[54,30012],{"className":30013,"style":4990},[2121],[54,30015,27081],{"className":30016,"style":27166},[2126,2167]," and terms with ",[54,30019,30021,30051],{"className":30020},[2079],[54,30022,30024],{"className":30023},[2083],[2085,30025,30026],{"xmlns":2087},[2089,30027,30028,30048],{},[2092,30029,30030,30036,30038,30044,30046],{},[2724,30031,30032,30034],{},[2098,30033,5738],{},[2098,30035,3182],{},[2245,30037,2247],{},[2724,30039,30040,30042],{},[2098,30041,5738],{},[2249,30043,150],{},[2245,30045,1207],{},[2098,30047,27081],{},[2105,30049,30050],{"encoding":2107},"E_k-E_0>\\delta",[54,30052,30054,30109,30164],{"className":30053,"ariaHidden":2113},[2112],[54,30055,30057,30060,30100,30103,30106],{"className":30056},[2117],[54,30058],{"className":30059,"style":2743},[2121],[54,30061,30063,30066],{"className":30062},[2126],[54,30064,5738],{"className":30065,"style":5448},[2126,2167],[54,30067,30069],{"className":30068},[2135],[54,30070,30072,30092],{"className":30071},[2139,2756],[54,30073,30075,30089],{"className":30074},[2143],[54,30076,30078],{"className":30077,"style":10799},[2147],[54,30079,30080,30083],{"style":5570},[54,30081],{"className":30082,"style":2156},[2155],[54,30084,30086],{"className":30085},[2160,2161,2162,2163],[54,30087,3182],{"className":30088,"style":3248},[2126,2167,2163],[54,30090,2781],{"className":30091},[2780],[54,30093,30095],{"className":30094},[2143],[54,30096,30098],{"className":30097,"style":2788},[2147],[54,30099],{},[54,30101],{"className":30102,"style":2271},[2270],[54,30104,2247],{"className":30105},[2275],[54,30107],{"className":30108,"style":2271},[2270],[54,30110,30112,30115,30155,30158,30161],{"className":30111},[2117],[54,30113],{"className":30114,"style":2743},[2121],[54,30116,30118,30121],{"className":30117},[2126],[54,30119,5738],{"className":30120,"style":5448},[2126,2167],[54,30122,30124],{"className":30123},[2135],[54,30125,30127,30147],{"className":30126},[2139,2756],[54,30128,30130,30144],{"className":30129},[2143],[54,30131,30133],{"className":30132,"style":13837},[2147],[54,30134,30135,30138],{"style":5570},[54,30136],{"className":30137,"style":2156},[2155],[54,30139,30141],{"className":30140},[2160,2161,2162,2163],[54,30142,150],{"className":30143},[2126,2163],[54,30145,2781],{"className":30146},[2780],[54,30148,30150],{"className":30149},[2143],[54,30151,30153],{"className":30152,"style":2788},[2147],[54,30154],{},[54,30156],{"className":30157,"style":2475},[2270],[54,30159,1207],{"className":30160},[2479],[54,30162],{"className":30163,"style":2475},[2270],[54,30165,30167,30170],{"className":30166},[2117],[54,30168],{"className":30169,"style":4990},[2121],[54,30171,27081],{"className":30172,"style":27166},[2126,2167],[54,30174,30176],{"className":30175},[2340],[54,30177,30179,30444],{"className":30178},[2079],[54,30180,30182],{"className":30181},[2083],[2085,30183,30184],{"xmlns":2087,"display":2349},[2089,30185,30186,30441],{},[2092,30187,30188,30190,30192,30315,30317,30439],{},[5723,30189,28794],{},[2245,30191,122],{},[9047,30193,30194,30269],{},[2092,30195,30196,30221,30223,30229,30235,30237,30239,30245,30251,30253,30259,30261,30267],{},[30197,30198,30199,30201],"munder",{},[2245,30200,7083],{},[2092,30202,30203,30209,30211,30217,30219],{},[2724,30204,30205,30207],{},[2098,30206,5738],{},[2098,30208,3182],{},[2245,30210,27066],{},[2724,30212,30213,30215],{},[2098,30214,5738],{},[2249,30216,150],{},[2245,30218,1663],{},[2098,30220,27081],{},[2098,30222,2305],{"mathvariant":2304},[2724,30224,30225,30227],{},[2098,30226,27788],{},[2098,30228,3182],{},[2095,30230,30231,30233],{},[2098,30232,2305],{"mathvariant":2304},[2249,30234,133],{},[2098,30236,21834],{},[2245,30238,129],{"stretchy":2311},[2724,30240,30241,30243],{},[2098,30242,5738],{},[2098,30244,3182],{},[2095,30246,30247,30249],{},[2245,30248,3205],{"stretchy":2311},[2249,30250,133],{},[2245,30252,129],{"stretchy":2311},[2724,30254,30255,30257],{},[2098,30256,5738],{},[2098,30258,3182],{},[2245,30260,2247],{},[2724,30262,30263,30265],{},[2098,30264,5738],{},[2249,30266,150],{},[2245,30268,3205],{"stretchy":2311},[2092,30270,30271,30285,30287,30293,30299,30301,30303,30309],{},[27770,30272,30273,30275,30283],{},[2245,30274,7083],{},[2092,30276,30277,30279,30281],{},[2098,30278,3182],{},[2245,30280,122],{},[2249,30282,150],{},[2098,30284,7237],{},[2098,30286,2305],{"mathvariant":2304},[2724,30288,30289,30291],{},[2098,30290,27788],{},[2098,30292,3182],{},[2095,30294,30295,30297],{},[2098,30296,2305],{"mathvariant":2304},[2249,30298,133],{},[2098,30300,21834],{},[2245,30302,129],{"stretchy":2311},[2724,30304,30305,30307],{},[2098,30306,5738],{},[2098,30308,3182],{},[2095,30310,30311,30313],{},[2245,30312,3205],{"stretchy":2311},[2249,30314,133],{},[2245,30316,1663],{},[9047,30318,30319,30393],{},[2092,30320,30321,30345,30347,30353,30359,30361,30363,30369,30375,30377,30383,30385,30391],{},[30197,30322,30323,30325],{},[2245,30324,7083],{},[2092,30326,30327,30333,30335,30341,30343],{},[2724,30328,30329,30331],{},[2098,30330,5738],{},[2098,30332,3182],{},[2245,30334,1207],{},[2724,30336,30337,30339],{},[2098,30338,5738],{},[2249,30340,150],{},[2245,30342,1663],{},[2098,30344,27081],{},[2098,30346,2305],{"mathvariant":2304},[2724,30348,30349,30351],{},[2098,30350,27788],{},[2098,30352,3182],{},[2095,30354,30355,30357],{},[2098,30356,2305],{"mathvariant":2304},[2249,30358,133],{},[2098,30360,21834],{},[2245,30362,129],{"stretchy":2311},[2724,30364,30365,30367],{},[2098,30366,5738],{},[2098,30368,3182],{},[2095,30370,30371,30373],{},[2245,30372,3205],{"stretchy":2311},[2249,30374,133],{},[2245,30376,129],{"stretchy":2311},[2724,30378,30379,30381],{},[2098,30380,5738],{},[2098,30382,3182],{},[2245,30384,2247],{},[2724,30386,30387,30389],{},[2098,30388,5738],{},[2249,30390,150],{},[2245,30392,3205],{"stretchy":2311},[2092,30394,30395,30409,30411,30417,30423,30425,30427,30433],{},[27770,30396,30397,30399,30407],{},[2245,30398,7083],{},[2092,30400,30401,30403,30405],{},[2098,30402,3182],{},[2245,30404,122],{},[2249,30406,150],{},[2098,30408,7237],{},[2098,30410,2305],{"mathvariant":2304},[2724,30412,30413,30415],{},[2098,30414,27788],{},[2098,30416,3182],{},[2095,30418,30419,30421],{},[2098,30420,2305],{"mathvariant":2304},[2249,30422,133],{},[2098,30424,21834],{},[2245,30426,129],{"stretchy":2311},[2724,30428,30429,30431],{},[2098,30430,5738],{},[2098,30432,3182],{},[2095,30434,30435,30437],{},[2245,30436,3205],{"stretchy":2311},[2249,30438,133],{},[2098,30440,29],{"mathvariant":2304},[2105,30442,30443],{"encoding":2107},"\\text{energy error} = \\frac{\\sum_{E_k\\le E_0+\\delta}|\\gamma_k|^2f(E_k)^2(E_k-E_0)}{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2} + \\frac{\\sum_{E_k> E_0+\\delta}|\\gamma_k|^2f(E_k)^2(E_k-E_0)}{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2}.",[54,30445,30447,30468,31136],{"className":30446,"ariaHidden":2113},[2112],[54,30448,30450,30453,30459,30462,30465],{"className":30449},[2117],[54,30451],{"className":30452,"style":28123},[2121],[54,30454,30456],{"className":30455},[2126,6595],[54,30457,28794],{"className":30458},[2126],[54,30460],{"className":30461,"style":2475},[2270],[54,30463,122],{"className":30464},[2479],[54,30466],{"className":30467,"style":2475},[2270],[54,30469,30471,30475,31127,31130,31133],{"className":30470},[2117],[54,30472],{"className":30473,"style":30474},[2121],"height:2.7806em;vertical-align:-1.1709em;",[54,30476,30478,30481,31124],{"className":30477},[2126],[54,30479],{"className":30480},[2497,9349],[54,30482,30484],{"className":30483},[9047],[54,30485,30487,31116],{"className":30486},[2139,2756],[54,30488,30490,31113],{"className":30489},[2143],[54,30491,30494,30715,30723],{"className":30492,"style":30493},[2147],"height:1.6097em;",[54,30495,30496,30499],{"style":29296},[54,30497],{"className":30498,"style":3866},[2155],[54,30500,30502,30565,30568,30571,30611,30640,30643,30646,30686],{"className":30501},[2126],[54,30503,30505,30508],{"className":30504},[7126],[54,30506,7083],{"className":30507,"style":7132},[7126,7130,7131],[54,30509,30511],{"className":30510},[2135],[54,30512,30514,30557],{"className":30513},[2139,2756],[54,30515,30517,30554],{"className":30516},[2143],[54,30518,30520,30540],{"className":30519,"style":7378},[2147],[54,30521,30522,30525],{"style":7148},[54,30523],{"className":30524,"style":2156},[2155],[54,30526,30528],{"className":30527},[2160,2161,2162,2163],[54,30529,30531,30534,30537],{"className":30530},[2126,2163],[54,30532,3182],{"className":30533,"style":3248},[2126,2167,2163],[54,30535,122],{"className":30536},[2479,2163],[54,30538,150],{"className":30539},[2126,2163],[54,30541,30542,30545],{"style":7401},[54,30543],{"className":30544,"style":2156},[2155],[54,30546,30548],{"className":30547},[2160,2161,2162,2163],[54,30549,30551],{"className":30550},[2126,2163],[54,30552,7237],{"className":30553,"style":2870},[2126,2167,2163],[54,30555,2781],{"className":30556},[2780],[54,30558,30560],{"className":30559},[2143],[54,30561,30563],{"className":30562,"style":7167},[2147],[54,30564],{},[54,30566],{"className":30567,"style":2521},[2270],[54,30569,2305],{"className":30570},[2126],[54,30572,30574,30577],{"className":30573},[2126],[54,30575,27788],{"className":30576,"style":27924},[2126,2167],[54,30578,30580],{"className":30579},[2135],[54,30581,30583,30603],{"className":30582},[2139,2756],[54,30584,30586,30600],{"className":30585},[2143],[54,30587,30589],{"className":30588,"style":10799},[2147],[54,30590,30591,30594],{"style":27939},[54,30592],{"className":30593,"style":2156},[2155],[54,30595,30597],{"className":30596},[2160,2161,2162,2163],[54,30598,3182],{"className":30599,"style":3248},[2126,2167,2163],[54,30601,2781],{"className":30602},[2780],[54,30604,30606],{"className":30605},[2143],[54,30607,30609],{"className":30608,"style":2788},[2147],[54,30610],{},[54,30612,30614,30617],{"className":30613},[2126],[54,30615,2305],{"className":30616},[2126],[54,30618,30620],{"className":30619},[2135],[54,30621,30623],{"className":30622},[2139],[54,30624,30626],{"className":30625},[2143],[54,30627,30629],{"className":30628,"style":28966},[2147],[54,30630,30631,30634],{"style":28969},[54,30632],{"className":30633,"style":2156},[2155],[54,30635,30637],{"className":30636},[2160,2161,2162,2163],[54,30638,133],{"className":30639},[2126,2163],[54,30641,21834],{"className":30642,"style":26895},[2126,2167],[54,30644,129],{"className":30645},[2497],[54,30647,30649,30652],{"className":30648},[2126],[54,30650,5738],{"className":30651,"style":5448},[2126,2167],[54,30653,30655],{"className":30654},[2135],[54,30656,30658,30678],{"className":30657},[2139,2756],[54,30659,30661,30675],{"className":30660},[2143],[54,30662,30664],{"className":30663,"style":10799},[2147],[54,30665,30666,30669],{"style":5570},[54,30667],{"className":30668,"style":2156},[2155],[54,30670,30672],{"className":30671},[2160,2161,2162,2163],[54,30673,3182],{"className":30674,"style":3248},[2126,2167,2163],[54,30676,2781],{"className":30677},[2780],[54,30679,30681],{"className":30680},[2143],[54,30682,30684],{"className":30683,"style":2788},[2147],[54,30685],{},[54,30687,30689,30692],{"className":30688},[2336],[54,30690,3205],{"className":30691},[2336],[54,30693,30695],{"className":30694},[2135],[54,30696,30698],{"className":30697},[2139],[54,30699,30701],{"className":30700},[2143],[54,30702,30704],{"className":30703,"style":28966},[2147],[54,30705,30706,30709],{"style":28969},[54,30707],{"className":30708,"style":2156},[2155],[54,30710,30712],{"className":30711},[2160,2161,2162,2163],[54,30713,133],{"className":30714},[2126,2163],[54,30716,30717,30720],{"style":9434},[54,30718],{"className":30719,"style":3866},[2155],[54,30721],{"className":30722,"style":9442},[9441],[54,30724,30726,30729],{"style":30725},"top:-3.7956em;",[54,30727],{"className":30728,"style":3866},[2155],[54,30730,30732,30868,30871,30874,30914,30943,30946,30949,30989,31018,31021,31061,31064,31067,31070,31110],{"className":30731},[2126],[54,30733,30735,30738],{"className":30734},[7126],[54,30736,7083],{"className":30737,"style":7132},[7126,7130,7131],[54,30739,30741],{"className":30740},[2135],[54,30742,30744,30859],{"className":30743},[2139,2756],[54,30745,30747,30856],{"className":30746},[2143],[54,30748,30751],{"className":30749,"style":30750},[2147],"height:0.1864em;",[54,30752,30753,30756],{"style":7148},[54,30754],{"className":30755,"style":2156},[2155],[54,30757,30759],{"className":30758},[2160,2161,2162,2163],[54,30760,30762,30805,30808,30850,30853],{"className":30761},[2126,2163],[54,30763,30765,30768],{"className":30764},[2126,2163],[54,30766,5738],{"className":30767,"style":5448},[2126,2167,2163],[54,30769,30771],{"className":30770},[2135],[54,30772,30774,30796],{"className":30773},[2139,2756],[54,30775,30777,30793],{"className":30776},[2143],[54,30778,30781],{"className":30779,"style":30780},[2147],"height:0.3448em;",[54,30782,30784,30787],{"style":30783},"top:-2.3488em;margin-left:-0.0576em;margin-right:0.0714em;",[54,30785],{"className":30786,"style":4368},[2155],[54,30788,30790],{"className":30789},[2160,4372,2504,2163],[54,30791,3182],{"className":30792,"style":3248},[2126,2167,2163],[54,30794,2781],{"className":30795},[2780],[54,30797,30799],{"className":30798},[2143],[54,30800,30803],{"className":30801,"style":30802},[2147],"height:0.1512em;",[54,30804],{},[54,30806,27066],{"className":30807},[2479,2163],[54,30809,30811,30814],{"className":30810},[2126,2163],[54,30812,5738],{"className":30813,"style":5448},[2126,2167,2163],[54,30815,30817],{"className":30816},[2135],[54,30818,30820,30842],{"className":30819},[2139,2756],[54,30821,30823,30839],{"className":30822},[2143],[54,30824,30827],{"className":30825,"style":30826},[2147],"height:0.3173em;",[54,30828,30830,30833],{"style":30829},"top:-2.357em;margin-left:-0.0576em;margin-right:0.0714em;",[54,30831],{"className":30832,"style":4368},[2155],[54,30834,30836],{"className":30835},[2160,4372,2504,2163],[54,30837,150],{"className":30838},[2126,2163],[54,30840,2781],{"className":30841},[2780],[54,30843,30845],{"className":30844},[2143],[54,30846,30848],{"className":30847,"style":4385},[2147],[54,30849],{},[54,30851,1663],{"className":30852},[2275,2163],[54,30854,27081],{"className":30855,"style":27166},[2126,2167,2163],[54,30857,2781],{"className":30858},[2780],[54,30860,30862],{"className":30861},[2143],[54,30863,30866],{"className":30864,"style":30865},[2147],"height:0.4056em;",[54,30867],{},[54,30869],{"className":30870,"style":2521},[2270],[54,30872,2305],{"className":30873},[2126],[54,30875,30877,30880],{"className":30876},[2126],[54,30878,27788],{"className":30879,"style":27924},[2126,2167],[54,30881,30883],{"className":30882},[2135],[54,30884,30886,30906],{"className":30885},[2139,2756],[54,30887,30889,30903],{"className":30888},[2143],[54,30890,30892],{"className":30891,"style":10799},[2147],[54,30893,30894,30897],{"style":27939},[54,30895],{"className":30896,"style":2156},[2155],[54,30898,30900],{"className":30899},[2160,2161,2162,2163],[54,30901,3182],{"className":30902,"style":3248},[2126,2167,2163],[54,30904,2781],{"className":30905},[2780],[54,30907,30909],{"className":30908},[2143],[54,30910,30912],{"className":30911,"style":2788},[2147],[54,30913],{},[54,30915,30917,30920],{"className":30916},[2126],[54,30918,2305],{"className":30919},[2126],[54,30921,30923],{"className":30922},[2135],[54,30924,30926],{"className":30925},[2139],[54,30927,30929],{"className":30928},[2143],[54,30930,30932],{"className":30931,"style":12780},[2147],[54,30933,30934,30937],{"style":2151},[54,30935],{"className":30936,"style":2156},[2155],[54,30938,30940],{"className":30939},[2160,2161,2162,2163],[54,30941,133],{"className":30942},[2126,2163],[54,30944,21834],{"className":30945,"style":26895},[2126,2167],[54,30947,129],{"className":30948},[2497],[54,30950,30952,30955],{"className":30951},[2126],[54,30953,5738],{"className":30954,"style":5448},[2126,2167],[54,30956,30958],{"className":30957},[2135],[54,30959,30961,30981],{"className":30960},[2139,2756],[54,30962,30964,30978],{"className":30963},[2143],[54,30965,30967],{"className":30966,"style":10799},[2147],[54,30968,30969,30972],{"style":5570},[54,30970],{"className":30971,"style":2156},[2155],[54,30973,30975],{"className":30974},[2160,2161,2162,2163],[54,30976,3182],{"className":30977,"style":3248},[2126,2167,2163],[54,30979,2781],{"className":30980},[2780],[54,30982,30984],{"className":30983},[2143],[54,30985,30987],{"className":30986,"style":2788},[2147],[54,30988],{},[54,30990,30992,30995],{"className":30991},[2336],[54,30993,3205],{"className":30994},[2336],[54,30996,30998],{"className":30997},[2135],[54,30999,31001],{"className":31000},[2139],[54,31002,31004],{"className":31003},[2143],[54,31005,31007],{"className":31006,"style":12780},[2147],[54,31008,31009,31012],{"style":2151},[54,31010],{"className":31011,"style":2156},[2155],[54,31013,31015],{"className":31014},[2160,2161,2162,2163],[54,31016,133],{"className":31017},[2126,2163],[54,31019,129],{"className":31020},[2497],[54,31022,31024,31027],{"className":31023},[2126],[54,31025,5738],{"className":31026,"style":5448},[2126,2167],[54,31028,31030],{"className":31029},[2135],[54,31031,31033,31053],{"className":31032},[2139,2756],[54,31034,31036,31050],{"className":31035},[2143],[54,31037,31039],{"className":31038,"style":10799},[2147],[54,31040,31041,31044],{"style":5570},[54,31042],{"className":31043,"style":2156},[2155],[54,31045,31047],{"className":31046},[2160,2161,2162,2163],[54,31048,3182],{"className":31049,"style":3248},[2126,2167,2163],[54,31051,2781],{"className":31052},[2780],[54,31054,31056],{"className":31055},[2143],[54,31057,31059],{"className":31058,"style":2788},[2147],[54,31060],{},[54,31062],{"className":31063,"style":2271},[2270],[54,31065,2247],{"className":31066},[2275],[54,31068],{"className":31069,"style":2271},[2270],[54,31071,31073,31076],{"className":31072},[2126],[54,31074,5738],{"className":31075,"style":5448},[2126,2167],[54,31077,31079],{"className":31078},[2135],[54,31080,31082,31102],{"className":31081},[2139,2756],[54,31083,31085,31099],{"className":31084},[2143],[54,31086,31088],{"className":31087,"style":13837},[2147],[54,31089,31090,31093],{"style":5570},[54,31091],{"className":31092,"style":2156},[2155],[54,31094,31096],{"className":31095},[2160,2161,2162,2163],[54,31097,150],{"className":31098},[2126,2163],[54,31100,2781],{"className":31101},[2780],[54,31103,31105],{"className":31104},[2143],[54,31106,31108],{"className":31107,"style":2788},[2147],[54,31109],{},[54,31111,3205],{"className":31112},[2336],[54,31114,2781],{"className":31115},[2780],[54,31117,31119],{"className":31118},[2143],[54,31120,31122],{"className":31121,"style":29850},[2147],[54,31123],{},[54,31125],{"className":31126},[2336,9349],[54,31128],{"className":31129,"style":2271},[2270],[54,31131,1663],{"className":31132},[2275],[54,31134],{"className":31135,"style":2271},[2270],[54,31137,31139,31142,31785],{"className":31138},[2117],[54,31140],{"className":31141,"style":30474},[2121],[54,31143,31145,31148,31782],{"className":31144},[2126],[54,31146],{"className":31147},[2497,9349],[54,31149,31151],{"className":31150},[9047],[54,31152,31154,31774],{"className":31153},[2139,2756],[54,31155,31157,31771],{"className":31156},[2143],[54,31158,31160,31381,31389],{"className":31159,"style":30493},[2147],[54,31161,31162,31165],{"style":29296},[54,31163],{"className":31164,"style":3866},[2155],[54,31166,31168,31231,31234,31237,31277,31306,31309,31312,31352],{"className":31167},[2126],[54,31169,31171,31174],{"className":31170},[7126],[54,31172,7083],{"className":31173,"style":7132},[7126,7130,7131],[54,31175,31177],{"className":31176},[2135],[54,31178,31180,31223],{"className":31179},[2139,2756],[54,31181,31183,31220],{"className":31182},[2143],[54,31184,31186,31206],{"className":31185,"style":7378},[2147],[54,31187,31188,31191],{"style":7148},[54,31189],{"className":31190,"style":2156},[2155],[54,31192,31194],{"className":31193},[2160,2161,2162,2163],[54,31195,31197,31200,31203],{"className":31196},[2126,2163],[54,31198,3182],{"className":31199,"style":3248},[2126,2167,2163],[54,31201,122],{"className":31202},[2479,2163],[54,31204,150],{"className":31205},[2126,2163],[54,31207,31208,31211],{"style":7401},[54,31209],{"className":31210,"style":2156},[2155],[54,31212,31214],{"className":31213},[2160,2161,2162,2163],[54,31215,31217],{"className":31216},[2126,2163],[54,31218,7237],{"className":31219,"style":2870},[2126,2167,2163],[54,31221,2781],{"className":31222},[2780],[54,31224,31226],{"className":31225},[2143],[54,31227,31229],{"className":31228,"style":7167},[2147],[54,31230],{},[54,31232],{"className":31233,"style":2521},[2270],[54,31235,2305],{"className":31236},[2126],[54,31238,31240,31243],{"className":31239},[2126],[54,31241,27788],{"className":31242,"style":27924},[2126,2167],[54,31244,31246],{"className":31245},[2135],[54,31247,31249,31269],{"className":31248},[2139,2756],[54,31250,31252,31266],{"className":31251},[2143],[54,31253,31255],{"className":31254,"style":10799},[2147],[54,31256,31257,31260],{"style":27939},[54,31258],{"className":31259,"style":2156},[2155],[54,31261,31263],{"className":31262},[2160,2161,2162,2163],[54,31264,3182],{"className":31265,"style":3248},[2126,2167,2163],[54,31267,2781],{"className":31268},[2780],[54,31270,31272],{"className":31271},[2143],[54,31273,31275],{"className":31274,"style":2788},[2147],[54,31276],{},[54,31278,31280,31283],{"className":31279},[2126],[54,31281,2305],{"className":31282},[2126],[54,31284,31286],{"className":31285},[2135],[54,31287,31289],{"className":31288},[2139],[54,31290,31292],{"className":31291},[2143],[54,31293,31295],{"className":31294,"style":28966},[2147],[54,31296,31297,31300],{"style":28969},[54,31298],{"className":31299,"style":2156},[2155],[54,31301,31303],{"className":31302},[2160,2161,2162,2163],[54,31304,133],{"className":31305},[2126,2163],[54,31307,21834],{"className":31308,"style":26895},[2126,2167],[54,31310,129],{"className":31311},[2497],[54,31313,31315,31318],{"className":31314},[2126],[54,31316,5738],{"className":31317,"style":5448},[2126,2167],[54,31319,31321],{"className":31320},[2135],[54,31322,31324,31344],{"className":31323},[2139,2756],[54,31325,31327,31341],{"className":31326},[2143],[54,31328,31330],{"className":31329,"style":10799},[2147],[54,31331,31332,31335],{"style":5570},[54,31333],{"className":31334,"style":2156},[2155],[54,31336,31338],{"className":31337},[2160,2161,2162,2163],[54,31339,3182],{"className":31340,"style":3248},[2126,2167,2163],[54,31342,2781],{"className":31343},[2780],[54,31345,31347],{"className":31346},[2143],[54,31348,31350],{"className":31349,"style":2788},[2147],[54,31351],{},[54,31353,31355,31358],{"className":31354},[2336],[54,31356,3205],{"className":31357},[2336],[54,31359,31361],{"className":31360},[2135],[54,31362,31364],{"className":31363},[2139],[54,31365,31367],{"className":31366},[2143],[54,31368,31370],{"className":31369,"style":28966},[2147],[54,31371,31372,31375],{"style":28969},[54,31373],{"className":31374,"style":2156},[2155],[54,31376,31378],{"className":31377},[2160,2161,2162,2163],[54,31379,133],{"className":31380},[2126,2163],[54,31382,31383,31386],{"style":9434},[54,31384],{"className":31385,"style":3866},[2155],[54,31387],{"className":31388,"style":9442},[9441],[54,31390,31391,31394],{"style":30725},[54,31392],{"className":31393,"style":3866},[2155],[54,31395,31397,31526,31529,31532,31572,31601,31604,31607,31647,31676,31679,31719,31722,31725,31728,31768],{"className":31396},[2126],[54,31398,31400,31403],{"className":31399},[7126],[54,31401,7083],{"className":31402,"style":7132},[7126,7130,7131],[54,31404,31406],{"className":31405},[2135],[54,31407,31409,31518],{"className":31408},[2139,2756],[54,31410,31412,31515],{"className":31411},[2143],[54,31413,31415],{"className":31414,"style":30750},[2147],[54,31416,31417,31420],{"style":7148},[54,31418],{"className":31419,"style":2156},[2155],[54,31421,31423],{"className":31422},[2160,2161,2162,2163],[54,31424,31426,31466,31469,31509,31512],{"className":31425},[2126,2163],[54,31427,31429,31432],{"className":31428},[2126,2163],[54,31430,5738],{"className":31431,"style":5448},[2126,2167,2163],[54,31433,31435],{"className":31434},[2135],[54,31436,31438,31458],{"className":31437},[2139,2756],[54,31439,31441,31455],{"className":31440},[2143],[54,31442,31444],{"className":31443,"style":30780},[2147],[54,31445,31446,31449],{"style":30783},[54,31447],{"className":31448,"style":4368},[2155],[54,31450,31452],{"className":31451},[2160,4372,2504,2163],[54,31453,3182],{"className":31454,"style":3248},[2126,2167,2163],[54,31456,2781],{"className":31457},[2780],[54,31459,31461],{"className":31460},[2143],[54,31462,31464],{"className":31463,"style":30802},[2147],[54,31465],{},[54,31467,1207],{"className":31468},[2479,2163],[54,31470,31472,31475],{"className":31471},[2126,2163],[54,31473,5738],{"className":31474,"style":5448},[2126,2167,2163],[54,31476,31478],{"className":31477},[2135],[54,31479,31481,31501],{"className":31480},[2139,2756],[54,31482,31484,31498],{"className":31483},[2143],[54,31485,31487],{"className":31486,"style":30826},[2147],[54,31488,31489,31492],{"style":30829},[54,31490],{"className":31491,"style":4368},[2155],[54,31493,31495],{"className":31494},[2160,4372,2504,2163],[54,31496,150],{"className":31497},[2126,2163],[54,31499,2781],{"className":31500},[2780],[54,31502,31504],{"className":31503},[2143],[54,31505,31507],{"className":31506,"style":4385},[2147],[54,31508],{},[54,31510,1663],{"className":31511},[2275,2163],[54,31513,27081],{"className":31514,"style":27166},[2126,2167,2163],[54,31516,2781],{"className":31517},[2780],[54,31519,31521],{"className":31520},[2143],[54,31522,31524],{"className":31523,"style":30865},[2147],[54,31525],{},[54,31527],{"className":31528,"style":2521},[2270],[54,31530,2305],{"className":31531},[2126],[54,31533,31535,31538],{"className":31534},[2126],[54,31536,27788],{"className":31537,"style":27924},[2126,2167],[54,31539,31541],{"className":31540},[2135],[54,31542,31544,31564],{"className":31543},[2139,2756],[54,31545,31547,31561],{"className":31546},[2143],[54,31548,31550],{"className":31549,"style":10799},[2147],[54,31551,31552,31555],{"style":27939},[54,31553],{"className":31554,"style":2156},[2155],[54,31556,31558],{"className":31557},[2160,2161,2162,2163],[54,31559,3182],{"className":31560,"style":3248},[2126,2167,2163],[54,31562,2781],{"className":31563},[2780],[54,31565,31567],{"className":31566},[2143],[54,31568,31570],{"className":31569,"style":2788},[2147],[54,31571],{},[54,31573,31575,31578],{"className":31574},[2126],[54,31576,2305],{"className":31577},[2126],[54,31579,31581],{"className":31580},[2135],[54,31582,31584],{"className":31583},[2139],[54,31585,31587],{"className":31586},[2143],[54,31588,31590],{"className":31589,"style":12780},[2147],[54,31591,31592,31595],{"style":2151},[54,31593],{"className":31594,"style":2156},[2155],[54,31596,31598],{"className":31597},[2160,2161,2162,2163],[54,31599,133],{"className":31600},[2126,2163],[54,31602,21834],{"className":31603,"style":26895},[2126,2167],[54,31605,129],{"className":31606},[2497],[54,31608,31610,31613],{"className":31609},[2126],[54,31611,5738],{"className":31612,"style":5448},[2126,2167],[54,31614,31616],{"className":31615},[2135],[54,31617,31619,31639],{"className":31618},[2139,2756],[54,31620,31622,31636],{"className":31621},[2143],[54,31623,31625],{"className":31624,"style":10799},[2147],[54,31626,31627,31630],{"style":5570},[54,31628],{"className":31629,"style":2156},[2155],[54,31631,31633],{"className":31632},[2160,2161,2162,2163],[54,31634,3182],{"className":31635,"style":3248},[2126,2167,2163],[54,31637,2781],{"className":31638},[2780],[54,31640,31642],{"className":31641},[2143],[54,31643,31645],{"className":31644,"style":2788},[2147],[54,31646],{},[54,31648,31650,31653],{"className":31649},[2336],[54,31651,3205],{"className":31652},[2336],[54,31654,31656],{"className":31655},[2135],[54,31657,31659],{"className":31658},[2139],[54,31660,31662],{"className":31661},[2143],[54,31663,31665],{"className":31664,"style":12780},[2147],[54,31666,31667,31670],{"style":2151},[54,31668],{"className":31669,"style":2156},[2155],[54,31671,31673],{"className":31672},[2160,2161,2162,2163],[54,31674,133],{"className":31675},[2126,2163],[54,31677,129],{"className":31678},[2497],[54,31680,31682,31685],{"className":31681},[2126],[54,31683,5738],{"className":31684,"style":5448},[2126,2167],[54,31686,31688],{"className":31687},[2135],[54,31689,31691,31711],{"className":31690},[2139,2756],[54,31692,31694,31708],{"className":31693},[2143],[54,31695,31697],{"className":31696,"style":10799},[2147],[54,31698,31699,31702],{"style":5570},[54,31700],{"className":31701,"style":2156},[2155],[54,31703,31705],{"className":31704},[2160,2161,2162,2163],[54,31706,3182],{"className":31707,"style":3248},[2126,2167,2163],[54,31709,2781],{"className":31710},[2780],[54,31712,31714],{"className":31713},[2143],[54,31715,31717],{"className":31716,"style":2788},[2147],[54,31718],{},[54,31720],{"className":31721,"style":2271},[2270],[54,31723,2247],{"className":31724},[2275],[54,31726],{"className":31727,"style":2271},[2270],[54,31729,31731,31734],{"className":31730},[2126],[54,31732,5738],{"className":31733,"style":5448},[2126,2167],[54,31735,31737],{"className":31736},[2135],[54,31738,31740,31760],{"className":31739},[2139,2756],[54,31741,31743,31757],{"className":31742},[2143],[54,31744,31746],{"className":31745,"style":13837},[2147],[54,31747,31748,31751],{"style":5570},[54,31749],{"className":31750,"style":2156},[2155],[54,31752,31754],{"className":31753},[2160,2161,2162,2163],[54,31755,150],{"className":31756},[2126,2163],[54,31758,2781],{"className":31759},[2780],[54,31761,31763],{"className":31762},[2143],[54,31764,31766],{"className":31765,"style":2788},[2147],[54,31767],{},[54,31769,3205],{"className":31770},[2336],[54,31772,2781],{"className":31773},[2780],[54,31775,31777],{"className":31776},[2143],[54,31778,31780],{"className":31779,"style":29850},[2147],[54,31781],{},[54,31783],{"className":31784},[2336,9349],[54,31786,29],{"className":31787},[2126],[13,31789,31790,31791,2375],{},"We can upper bound the first term by ",[54,31792,31794,31808],{"className":31793},[2079],[54,31795,31797],{"className":31796},[2083],[2085,31798,31799],{"xmlns":2087},[2089,31800,31801,31805],{},[2092,31802,31803],{},[2098,31804,27081],{},[2105,31806,31807],{"encoding":2107},"\\delta",[54,31809,31811],{"className":31810,"ariaHidden":2113},[2112],[54,31812,31814,31817],{"className":31813},[2117],[54,31815],{"className":31816,"style":4990},[2121],[54,31818,27081],{"className":31819,"style":27166},[2126,2167],[54,31821,31823],{"className":31822},[2340],[54,31824,31826,32074],{"className":31825},[2079],[54,31827,31829],{"className":31828},[2083],[2085,31830,31831],{"xmlns":2087,"display":2349},[2089,31832,31833,32071],{},[2092,31834,31835,31957,31959,32065,32067,32069],{},[9047,31836,31837,31911],{},[2092,31838,31839,31863,31865,31871,31877,31879,31881,31887,31893,31895,31901,31903,31909],{},[30197,31840,31841,31843],{},[2245,31842,7083],{},[2092,31844,31845,31851,31853,31859,31861],{},[2724,31846,31847,31849],{},[2098,31848,5738],{},[2098,31850,3182],{},[2245,31852,27066],{},[2724,31854,31855,31857],{},[2098,31856,5738],{},[2249,31858,150],{},[2245,31860,1663],{},[2098,31862,27081],{},[2098,31864,2305],{"mathvariant":2304},[2724,31866,31867,31869],{},[2098,31868,27788],{},[2098,31870,3182],{},[2095,31872,31873,31875],{},[2098,31874,2305],{"mathvariant":2304},[2249,31876,133],{},[2098,31878,21834],{},[2245,31880,129],{"stretchy":2311},[2724,31882,31883,31885],{},[2098,31884,5738],{},[2098,31886,3182],{},[2095,31888,31889,31891],{},[2245,31890,3205],{"stretchy":2311},[2249,31892,133],{},[2245,31894,129],{"stretchy":2311},[2724,31896,31897,31899],{},[2098,31898,5738],{},[2098,31900,3182],{},[2245,31902,2247],{},[2724,31904,31905,31907],{},[2098,31906,5738],{},[2249,31908,150],{},[2245,31910,3205],{"stretchy":2311},[2092,31912,31913,31927,31929,31935,31941,31943,31945,31951],{},[27770,31914,31915,31917,31925],{},[2245,31916,7083],{},[2092,31918,31919,31921,31923],{},[2098,31920,3182],{},[2245,31922,122],{},[2249,31924,150],{},[2098,31926,7237],{},[2098,31928,2305],{"mathvariant":2304},[2724,31930,31931,31933],{},[2098,31932,27788],{},[2098,31934,3182],{},[2095,31936,31937,31939],{},[2098,31938,2305],{"mathvariant":2304},[2249,31940,133],{},[2098,31942,21834],{},[2245,31944,129],{"stretchy":2311},[2724,31946,31947,31949],{},[2098,31948,5738],{},[2098,31950,3182],{},[2095,31952,31953,31955],{},[2245,31954,3205],{"stretchy":2311},[2249,31956,133],{},[2245,31958,5233],{},[9047,31960,31961,32019],{},[2092,31962,31963,31965,31989,31991,31997,32003,32005,32007,32013],{},[2098,31964,27081],{},[30197,31966,31967,31969],{},[2245,31968,7083],{},[2092,31970,31971,31977,31979,31985,31987],{},[2724,31972,31973,31975],{},[2098,31974,5738],{},[2098,31976,3182],{},[2245,31978,27066],{},[2724,31980,31981,31983],{},[2098,31982,5738],{},[2249,31984,150],{},[2245,31986,1663],{},[2098,31988,27081],{},[2098,31990,2305],{"mathvariant":2304},[2724,31992,31993,31995],{},[2098,31994,27788],{},[2098,31996,3182],{},[2095,31998,31999,32001],{},[2098,32000,2305],{"mathvariant":2304},[2249,32002,133],{},[2098,32004,21834],{},[2245,32006,129],{"stretchy":2311},[2724,32008,32009,32011],{},[2098,32010,5738],{},[2098,32012,3182],{},[2095,32014,32015,32017],{},[2245,32016,3205],{"stretchy":2311},[2249,32018,133],{},[2092,32020,32021,32035,32037,32043,32049,32051,32053,32059],{},[27770,32022,32023,32025,32033],{},[2245,32024,7083],{},[2092,32026,32027,32029,32031],{},[2098,32028,3182],{},[2245,32030,122],{},[2249,32032,150],{},[2098,32034,7237],{},[2098,32036,2305],{"mathvariant":2304},[2724,32038,32039,32041],{},[2098,32040,27788],{},[2098,32042,3182],{},[2095,32044,32045,32047],{},[2098,32046,2305],{"mathvariant":2304},[2249,32048,133],{},[2098,32050,21834],{},[2245,32052,129],{"stretchy":2311},[2724,32054,32055,32057],{},[2098,32056,5738],{},[2098,32058,3182],{},[2095,32060,32061,32063],{},[2245,32062,3205],{"stretchy":2311},[2249,32064,133],{},[2245,32066,27066],{},[2098,32068,27081],{},[2245,32070,2375],{"separator":2113},[2105,32072,32073],{"encoding":2107},"\\frac{\\sum_{E_k\\le E_0+\\delta}|\\gamma_k|^2f(E_k)^2(E_k-E_0)}{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2} \u003C \\frac{\\delta\\sum_{E_k\\le E_0+\\delta}|\\gamma_k|^2f(E_k)^2}{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2} \\le \\delta,",[54,32075,32077,32735,33304],{"className":32076,"ariaHidden":2113},[2112],[54,32078,32080,32083,32726,32729,32732],{"className":32079},[2117],[54,32081],{"className":32082,"style":30474},[2121],[54,32084,32086,32089,32723],{"className":32085},[2126],[54,32087],{"className":32088},[2497,9349],[54,32090,32092],{"className":32091},[9047],[54,32093,32095,32715],{"className":32094},[2139,2756],[54,32096,32098,32712],{"className":32097},[2143],[54,32099,32101,32322,32330],{"className":32100,"style":30493},[2147],[54,32102,32103,32106],{"style":29296},[54,32104],{"className":32105,"style":3866},[2155],[54,32107,32109,32172,32175,32178,32218,32247,32250,32253,32293],{"className":32108},[2126],[54,32110,32112,32115],{"className":32111},[7126],[54,32113,7083],{"className":32114,"style":7132},[7126,7130,7131],[54,32116,32118],{"className":32117},[2135],[54,32119,32121,32164],{"className":32120},[2139,2756],[54,32122,32124,32161],{"className":32123},[2143],[54,32125,32127,32147],{"className":32126,"style":7378},[2147],[54,32128,32129,32132],{"style":7148},[54,32130],{"className":32131,"style":2156},[2155],[54,32133,32135],{"className":32134},[2160,2161,2162,2163],[54,32136,32138,32141,32144],{"className":32137},[2126,2163],[54,32139,3182],{"className":32140,"style":3248},[2126,2167,2163],[54,32142,122],{"className":32143},[2479,2163],[54,32145,150],{"className":32146},[2126,2163],[54,32148,32149,32152],{"style":7401},[54,32150],{"className":32151,"style":2156},[2155],[54,32153,32155],{"className":32154},[2160,2161,2162,2163],[54,32156,32158],{"className":32157},[2126,2163],[54,32159,7237],{"className":32160,"style":2870},[2126,2167,2163],[54,32162,2781],{"className":32163},[2780],[54,32165,32167],{"className":32166},[2143],[54,32168,32170],{"className":32169,"style":7167},[2147],[54,32171],{},[54,32173],{"className":32174,"style":2521},[2270],[54,32176,2305],{"className":32177},[2126],[54,32179,32181,32184],{"className":32180},[2126],[54,32182,27788],{"className":32183,"style":27924},[2126,2167],[54,32185,32187],{"className":32186},[2135],[54,32188,32190,32210],{"className":32189},[2139,2756],[54,32191,32193,32207],{"className":32192},[2143],[54,32194,32196],{"className":32195,"style":10799},[2147],[54,32197,32198,32201],{"style":27939},[54,32199],{"className":32200,"style":2156},[2155],[54,32202,32204],{"className":32203},[2160,2161,2162,2163],[54,32205,3182],{"className":32206,"style":3248},[2126,2167,2163],[54,32208,2781],{"className":32209},[2780],[54,32211,32213],{"className":32212},[2143],[54,32214,32216],{"className":32215,"style":2788},[2147],[54,32217],{},[54,32219,32221,32224],{"className":32220},[2126],[54,32222,2305],{"className":32223},[2126],[54,32225,32227],{"className":32226},[2135],[54,32228,32230],{"className":32229},[2139],[54,32231,32233],{"className":32232},[2143],[54,32234,32236],{"className":32235,"style":28966},[2147],[54,32237,32238,32241],{"style":28969},[54,32239],{"className":32240,"style":2156},[2155],[54,32242,32244],{"className":32243},[2160,2161,2162,2163],[54,32245,133],{"className":32246},[2126,2163],[54,32248,21834],{"className":32249,"style":26895},[2126,2167],[54,32251,129],{"className":32252},[2497],[54,32254,32256,32259],{"className":32255},[2126],[54,32257,5738],{"className":32258,"style":5448},[2126,2167],[54,32260,32262],{"className":32261},[2135],[54,32263,32265,32285],{"className":32264},[2139,2756],[54,32266,32268,32282],{"className":32267},[2143],[54,32269,32271],{"className":32270,"style":10799},[2147],[54,32272,32273,32276],{"style":5570},[54,32274],{"className":32275,"style":2156},[2155],[54,32277,32279],{"className":32278},[2160,2161,2162,2163],[54,32280,3182],{"className":32281,"style":3248},[2126,2167,2163],[54,32283,2781],{"className":32284},[2780],[54,32286,32288],{"className":32287},[2143],[54,32289,32291],{"className":32290,"style":2788},[2147],[54,32292],{},[54,32294,32296,32299],{"className":32295},[2336],[54,32297,3205],{"className":32298},[2336],[54,32300,32302],{"className":32301},[2135],[54,32303,32305],{"className":32304},[2139],[54,32306,32308],{"className":32307},[2143],[54,32309,32311],{"className":32310,"style":28966},[2147],[54,32312,32313,32316],{"style":28969},[54,32314],{"className":32315,"style":2156},[2155],[54,32317,32319],{"className":32318},[2160,2161,2162,2163],[54,32320,133],{"className":32321},[2126,2163],[54,32323,32324,32327],{"style":9434},[54,32325],{"className":32326,"style":3866},[2155],[54,32328],{"className":32329,"style":9442},[9441],[54,32331,32332,32335],{"style":30725},[54,32333],{"className":32334,"style":3866},[2155],[54,32336,32338,32467,32470,32473,32513,32542,32545,32548,32588,32617,32620,32660,32663,32666,32669,32709],{"className":32337},[2126],[54,32339,32341,32344],{"className":32340},[7126],[54,32342,7083],{"className":32343,"style":7132},[7126,7130,7131],[54,32345,32347],{"className":32346},[2135],[54,32348,32350,32459],{"className":32349},[2139,2756],[54,32351,32353,32456],{"className":32352},[2143],[54,32354,32356],{"className":32355,"style":30750},[2147],[54,32357,32358,32361],{"style":7148},[54,32359],{"className":32360,"style":2156},[2155],[54,32362,32364],{"className":32363},[2160,2161,2162,2163],[54,32365,32367,32407,32410,32450,32453],{"className":32366},[2126,2163],[54,32368,32370,32373],{"className":32369},[2126,2163],[54,32371,5738],{"className":32372,"style":5448},[2126,2167,2163],[54,32374,32376],{"className":32375},[2135],[54,32377,32379,32399],{"className":32378},[2139,2756],[54,32380,32382,32396],{"className":32381},[2143],[54,32383,32385],{"className":32384,"style":30780},[2147],[54,32386,32387,32390],{"style":30783},[54,32388],{"className":32389,"style":4368},[2155],[54,32391,32393],{"className":32392},[2160,4372,2504,2163],[54,32394,3182],{"className":32395,"style":3248},[2126,2167,2163],[54,32397,2781],{"className":32398},[2780],[54,32400,32402],{"className":32401},[2143],[54,32403,32405],{"className":32404,"style":30802},[2147],[54,32406],{},[54,32408,27066],{"className":32409},[2479,2163],[54,32411,32413,32416],{"className":32412},[2126,2163],[54,32414,5738],{"className":32415,"style":5448},[2126,2167,2163],[54,32417,32419],{"className":32418},[2135],[54,32420,32422,32442],{"className":32421},[2139,2756],[54,32423,32425,32439],{"className":32424},[2143],[54,32426,32428],{"className":32427,"style":30826},[2147],[54,32429,32430,32433],{"style":30829},[54,32431],{"className":32432,"style":4368},[2155],[54,32434,32436],{"className":32435},[2160,4372,2504,2163],[54,32437,150],{"className":32438},[2126,2163],[54,32440,2781],{"className":32441},[2780],[54,32443,32445],{"className":32444},[2143],[54,32446,32448],{"className":32447,"style":4385},[2147],[54,32449],{},[54,32451,1663],{"className":32452},[2275,2163],[54,32454,27081],{"className":32455,"style":27166},[2126,2167,2163],[54,32457,2781],{"className":32458},[2780],[54,32460,32462],{"className":32461},[2143],[54,32463,32465],{"className":32464,"style":30865},[2147],[54,32466],{},[54,32468],{"className":32469,"style":2521},[2270],[54,32471,2305],{"className":32472},[2126],[54,32474,32476,32479],{"className":32475},[2126],[54,32477,27788],{"className":32478,"style":27924},[2126,2167],[54,32480,32482],{"className":32481},[2135],[54,32483,32485,32505],{"className":32484},[2139,2756],[54,32486,32488,32502],{"className":32487},[2143],[54,32489,32491],{"className":32490,"style":10799},[2147],[54,32492,32493,32496],{"style":27939},[54,32494],{"className":32495,"style":2156},[2155],[54,32497,32499],{"className":32498},[2160,2161,2162,2163],[54,32500,3182],{"className":32501,"style":3248},[2126,2167,2163],[54,32503,2781],{"className":32504},[2780],[54,32506,32508],{"className":32507},[2143],[54,32509,32511],{"className":32510,"style":2788},[2147],[54,32512],{},[54,32514,32516,32519],{"className":32515},[2126],[54,32517,2305],{"className":32518},[2126],[54,32520,32522],{"className":32521},[2135],[54,32523,32525],{"className":32524},[2139],[54,32526,32528],{"className":32527},[2143],[54,32529,32531],{"className":32530,"style":12780},[2147],[54,32532,32533,32536],{"style":2151},[54,32534],{"className":32535,"style":2156},[2155],[54,32537,32539],{"className":32538},[2160,2161,2162,2163],[54,32540,133],{"className":32541},[2126,2163],[54,32543,21834],{"className":32544,"style":26895},[2126,2167],[54,32546,129],{"className":32547},[2497],[54,32549,32551,32554],{"className":32550},[2126],[54,32552,5738],{"className":32553,"style":5448},[2126,2167],[54,32555,32557],{"className":32556},[2135],[54,32558,32560,32580],{"className":32559},[2139,2756],[54,32561,32563,32577],{"className":32562},[2143],[54,32564,32566],{"className":32565,"style":10799},[2147],[54,32567,32568,32571],{"style":5570},[54,32569],{"className":32570,"style":2156},[2155],[54,32572,32574],{"className":32573},[2160,2161,2162,2163],[54,32575,3182],{"className":32576,"style":3248},[2126,2167,2163],[54,32578,2781],{"className":32579},[2780],[54,32581,32583],{"className":32582},[2143],[54,32584,32586],{"className":32585,"style":2788},[2147],[54,32587],{},[54,32589,32591,32594],{"className":32590},[2336],[54,32592,3205],{"className":32593},[2336],[54,32595,32597],{"className":32596},[2135],[54,32598,32600],{"className":32599},[2139],[54,32601,32603],{"className":32602},[2143],[54,32604,32606],{"className":32605,"style":12780},[2147],[54,32607,32608,32611],{"style":2151},[54,32609],{"className":32610,"style":2156},[2155],[54,32612,32614],{"className":32613},[2160,2161,2162,2163],[54,32615,133],{"className":32616},[2126,2163],[54,32618,129],{"className":32619},[2497],[54,32621,32623,32626],{"className":32622},[2126],[54,32624,5738],{"className":32625,"style":5448},[2126,2167],[54,32627,32629],{"className":32628},[2135],[54,32630,32632,32652],{"className":32631},[2139,2756],[54,32633,32635,32649],{"className":32634},[2143],[54,32636,32638],{"className":32637,"style":10799},[2147],[54,32639,32640,32643],{"style":5570},[54,32641],{"className":32642,"style":2156},[2155],[54,32644,32646],{"className":32645},[2160,2161,2162,2163],[54,32647,3182],{"className":32648,"style":3248},[2126,2167,2163],[54,32650,2781],{"className":32651},[2780],[54,32653,32655],{"className":32654},[2143],[54,32656,32658],{"className":32657,"style":2788},[2147],[54,32659],{},[54,32661],{"className":32662,"style":2271},[2270],[54,32664,2247],{"className":32665},[2275],[54,32667],{"className":32668,"style":2271},[2270],[54,32670,32672,32675],{"className":32671},[2126],[54,32673,5738],{"className":32674,"style":5448},[2126,2167],[54,32676,32678],{"className":32677},[2135],[54,32679,32681,32701],{"className":32680},[2139,2756],[54,32682,32684,32698],{"className":32683},[2143],[54,32685,32687],{"className":32686,"style":13837},[2147],[54,32688,32689,32692],{"style":5570},[54,32690],{"className":32691,"style":2156},[2155],[54,32693,32695],{"className":32694},[2160,2161,2162,2163],[54,32696,150],{"className":32697},[2126,2163],[54,32699,2781],{"className":32700},[2780],[54,32702,32704],{"className":32703},[2143],[54,32705,32707],{"className":32706,"style":2788},[2147],[54,32708],{},[54,32710,3205],{"className":32711},[2336],[54,32713,2781],{"className":32714},[2780],[54,32716,32718],{"className":32717},[2143],[54,32719,32721],{"className":32720,"style":29850},[2147],[54,32722],{},[54,32724],{"className":32725},[2336,9349],[54,32727],{"className":32728,"style":2475},[2270],[54,32730,5233],{"className":32731},[2479],[54,32733],{"className":32734,"style":2475},[2270],[54,32736,32738,32741,33295,33298,33301],{"className":32737},[2117],[54,32739],{"className":32740,"style":30474},[2121],[54,32742,32744,32747,33292],{"className":32743},[2126],[54,32745],{"className":32746},[2497,9349],[54,32748,32750],{"className":32749},[9047],[54,32751,32753,33284],{"className":32752},[2139,2756],[54,32754,32756,33281],{"className":32755},[2143],[54,32757,32759,32980,32988],{"className":32758,"style":30493},[2147],[54,32760,32761,32764],{"style":29296},[54,32762],{"className":32763,"style":3866},[2155],[54,32765,32767,32830,32833,32836,32876,32905,32908,32911,32951],{"className":32766},[2126],[54,32768,32770,32773],{"className":32769},[7126],[54,32771,7083],{"className":32772,"style":7132},[7126,7130,7131],[54,32774,32776],{"className":32775},[2135],[54,32777,32779,32822],{"className":32778},[2139,2756],[54,32780,32782,32819],{"className":32781},[2143],[54,32783,32785,32805],{"className":32784,"style":7378},[2147],[54,32786,32787,32790],{"style":7148},[54,32788],{"className":32789,"style":2156},[2155],[54,32791,32793],{"className":32792},[2160,2161,2162,2163],[54,32794,32796,32799,32802],{"className":32795},[2126,2163],[54,32797,3182],{"className":32798,"style":3248},[2126,2167,2163],[54,32800,122],{"className":32801},[2479,2163],[54,32803,150],{"className":32804},[2126,2163],[54,32806,32807,32810],{"style":7401},[54,32808],{"className":32809,"style":2156},[2155],[54,32811,32813],{"className":32812},[2160,2161,2162,2163],[54,32814,32816],{"className":32815},[2126,2163],[54,32817,7237],{"className":32818,"style":2870},[2126,2167,2163],[54,32820,2781],{"className":32821},[2780],[54,32823,32825],{"className":32824},[2143],[54,32826,32828],{"className":32827,"style":7167},[2147],[54,32829],{},[54,32831],{"className":32832,"style":2521},[2270],[54,32834,2305],{"className":32835},[2126],[54,32837,32839,32842],{"className":32838},[2126],[54,32840,27788],{"className":32841,"style":27924},[2126,2167],[54,32843,32845],{"className":32844},[2135],[54,32846,32848,32868],{"className":32847},[2139,2756],[54,32849,32851,32865],{"className":32850},[2143],[54,32852,32854],{"className":32853,"style":10799},[2147],[54,32855,32856,32859],{"style":27939},[54,32857],{"className":32858,"style":2156},[2155],[54,32860,32862],{"className":32861},[2160,2161,2162,2163],[54,32863,3182],{"className":32864,"style":3248},[2126,2167,2163],[54,32866,2781],{"className":32867},[2780],[54,32869,32871],{"className":32870},[2143],[54,32872,32874],{"className":32873,"style":2788},[2147],[54,32875],{},[54,32877,32879,32882],{"className":32878},[2126],[54,32880,2305],{"className":32881},[2126],[54,32883,32885],{"className":32884},[2135],[54,32886,32888],{"className":32887},[2139],[54,32889,32891],{"className":32890},[2143],[54,32892,32894],{"className":32893,"style":28966},[2147],[54,32895,32896,32899],{"style":28969},[54,32897],{"className":32898,"style":2156},[2155],[54,32900,32902],{"className":32901},[2160,2161,2162,2163],[54,32903,133],{"className":32904},[2126,2163],[54,32906,21834],{"className":32907,"style":26895},[2126,2167],[54,32909,129],{"className":32910},[2497],[54,32912,32914,32917],{"className":32913},[2126],[54,32915,5738],{"className":32916,"style":5448},[2126,2167],[54,32918,32920],{"className":32919},[2135],[54,32921,32923,32943],{"className":32922},[2139,2756],[54,32924,32926,32940],{"className":32925},[2143],[54,32927,32929],{"className":32928,"style":10799},[2147],[54,32930,32931,32934],{"style":5570},[54,32932],{"className":32933,"style":2156},[2155],[54,32935,32937],{"className":32936},[2160,2161,2162,2163],[54,32938,3182],{"className":32939,"style":3248},[2126,2167,2163],[54,32941,2781],{"className":32942},[2780],[54,32944,32946],{"className":32945},[2143],[54,32947,32949],{"className":32948,"style":2788},[2147],[54,32950],{},[54,32952,32954,32957],{"className":32953},[2336],[54,32955,3205],{"className":32956},[2336],[54,32958,32960],{"className":32959},[2135],[54,32961,32963],{"className":32962},[2139],[54,32964,32966],{"className":32965},[2143],[54,32967,32969],{"className":32968,"style":28966},[2147],[54,32970,32971,32974],{"style":28969},[54,32972],{"className":32973,"style":2156},[2155],[54,32975,32977],{"className":32976},[2160,2161,2162,2163],[54,32978,133],{"className":32979},[2126,2163],[54,32981,32982,32985],{"style":9434},[54,32983],{"className":32984,"style":3866},[2155],[54,32986],{"className":32987,"style":9442},[9441],[54,32989,32990,32993],{"style":30725},[54,32991],{"className":32992,"style":3866},[2155],[54,32994,32996,32999,33002,33131,33134,33137,33177,33206,33209,33212,33252],{"className":32995},[2126],[54,32997,27081],{"className":32998,"style":27166},[2126,2167],[54,33000],{"className":33001,"style":2521},[2270],[54,33003,33005,33008],{"className":33004},[7126],[54,33006,7083],{"className":33007,"style":7132},[7126,7130,7131],[54,33009,33011],{"className":33010},[2135],[54,33012,33014,33123],{"className":33013},[2139,2756],[54,33015,33017,33120],{"className":33016},[2143],[54,33018,33020],{"className":33019,"style":30750},[2147],[54,33021,33022,33025],{"style":7148},[54,33023],{"className":33024,"style":2156},[2155],[54,33026,33028],{"className":33027},[2160,2161,2162,2163],[54,33029,33031,33071,33074,33114,33117],{"className":33030},[2126,2163],[54,33032,33034,33037],{"className":33033},[2126,2163],[54,33035,5738],{"className":33036,"style":5448},[2126,2167,2163],[54,33038,33040],{"className":33039},[2135],[54,33041,33043,33063],{"className":33042},[2139,2756],[54,33044,33046,33060],{"className":33045},[2143],[54,33047,33049],{"className":33048,"style":30780},[2147],[54,33050,33051,33054],{"style":30783},[54,33052],{"className":33053,"style":4368},[2155],[54,33055,33057],{"className":33056},[2160,4372,2504,2163],[54,33058,3182],{"className":33059,"style":3248},[2126,2167,2163],[54,33061,2781],{"className":33062},[2780],[54,33064,33066],{"className":33065},[2143],[54,33067,33069],{"className":33068,"style":30802},[2147],[54,33070],{},[54,33072,27066],{"className":33073},[2479,2163],[54,33075,33077,33080],{"className":33076},[2126,2163],[54,33078,5738],{"className":33079,"style":5448},[2126,2167,2163],[54,33081,33083],{"className":33082},[2135],[54,33084,33086,33106],{"className":33085},[2139,2756],[54,33087,33089,33103],{"className":33088},[2143],[54,33090,33092],{"className":33091,"style":30826},[2147],[54,33093,33094,33097],{"style":30829},[54,33095],{"className":33096,"style":4368},[2155],[54,33098,33100],{"className":33099},[2160,4372,2504,2163],[54,33101,150],{"className":33102},[2126,2163],[54,33104,2781],{"className":33105},[2780],[54,33107,33109],{"className":33108},[2143],[54,33110,33112],{"className":33111,"style":4385},[2147],[54,33113],{},[54,33115,1663],{"className":33116},[2275,2163],[54,33118,27081],{"className":33119,"style":27166},[2126,2167,2163],[54,33121,2781],{"className":33122},[2780],[54,33124,33126],{"className":33125},[2143],[54,33127,33129],{"className":33128,"style":30865},[2147],[54,33130],{},[54,33132],{"className":33133,"style":2521},[2270],[54,33135,2305],{"className":33136},[2126],[54,33138,33140,33143],{"className":33139},[2126],[54,33141,27788],{"className":33142,"style":27924},[2126,2167],[54,33144,33146],{"className":33145},[2135],[54,33147,33149,33169],{"className":33148},[2139,2756],[54,33150,33152,33166],{"className":33151},[2143],[54,33153,33155],{"className":33154,"style":10799},[2147],[54,33156,33157,33160],{"style":27939},[54,33158],{"className":33159,"style":2156},[2155],[54,33161,33163],{"className":33162},[2160,2161,2162,2163],[54,33164,3182],{"className":33165,"style":3248},[2126,2167,2163],[54,33167,2781],{"className":33168},[2780],[54,33170,33172],{"className":33171},[2143],[54,33173,33175],{"className":33174,"style":2788},[2147],[54,33176],{},[54,33178,33180,33183],{"className":33179},[2126],[54,33181,2305],{"className":33182},[2126],[54,33184,33186],{"className":33185},[2135],[54,33187,33189],{"className":33188},[2139],[54,33190,33192],{"className":33191},[2143],[54,33193,33195],{"className":33194,"style":12780},[2147],[54,33196,33197,33200],{"style":2151},[54,33198],{"className":33199,"style":2156},[2155],[54,33201,33203],{"className":33202},[2160,2161,2162,2163],[54,33204,133],{"className":33205},[2126,2163],[54,33207,21834],{"className":33208,"style":26895},[2126,2167],[54,33210,129],{"className":33211},[2497],[54,33213,33215,33218],{"className":33214},[2126],[54,33216,5738],{"className":33217,"style":5448},[2126,2167],[54,33219,33221],{"className":33220},[2135],[54,33222,33224,33244],{"className":33223},[2139,2756],[54,33225,33227,33241],{"className":33226},[2143],[54,33228,33230],{"className":33229,"style":10799},[2147],[54,33231,33232,33235],{"style":5570},[54,33233],{"className":33234,"style":2156},[2155],[54,33236,33238],{"className":33237},[2160,2161,2162,2163],[54,33239,3182],{"className":33240,"style":3248},[2126,2167,2163],[54,33242,2781],{"className":33243},[2780],[54,33245,33247],{"className":33246},[2143],[54,33248,33250],{"className":33249,"style":2788},[2147],[54,33251],{},[54,33253,33255,33258],{"className":33254},[2336],[54,33256,3205],{"className":33257},[2336],[54,33259,33261],{"className":33260},[2135],[54,33262,33264],{"className":33263},[2139],[54,33265,33267],{"className":33266},[2143],[54,33268,33270],{"className":33269,"style":12780},[2147],[54,33271,33272,33275],{"style":2151},[54,33273],{"className":33274,"style":2156},[2155],[54,33276,33278],{"className":33277},[2160,2161,2162,2163],[54,33279,133],{"className":33280},[2126,2163],[54,33282,2781],{"className":33283},[2780],[54,33285,33287],{"className":33286},[2143],[54,33288,33290],{"className":33289,"style":29850},[2147],[54,33291],{},[54,33293],{"className":33294},[2336,9349],[54,33296],{"className":33297,"style":2475},[2270],[54,33299,27066],{"className":33300},[2479],[54,33302],{"className":33303,"style":2475},[2270],[54,33305,33307,33310,33313],{"className":33306},[2117],[54,33308],{"className":33309,"style":6591},[2121],[54,33311,27081],{"className":33312,"style":27166},[2126,2167],[54,33314,2375],{"className":33315},[2517],[13,33317,33318,33319,33473,33474,33543,33544,33654,33655,33792],{},"where the first step follows because ",[54,33320,33322,33351],{"className":33321},[2079],[54,33323,33325],{"className":33324},[2083],[2085,33326,33327],{"xmlns":2087},[2089,33328,33329,33349],{},[2092,33330,33331,33337,33339,33345,33347],{},[2724,33332,33333,33335],{},[2098,33334,5738],{},[2098,33336,3182],{},[2245,33338,2247],{},[2724,33340,33341,33343],{},[2098,33342,5738],{},[2249,33344,150],{},[2245,33346,27066],{},[2098,33348,27081],{},[2105,33350,29894],{"encoding":2107},[54,33352,33354,33409,33464],{"className":33353,"ariaHidden":2113},[2112],[54,33355,33357,33360,33400,33403,33406],{"className":33356},[2117],[54,33358],{"className":33359,"style":2743},[2121],[54,33361,33363,33366],{"className":33362},[2126],[54,33364,5738],{"className":33365,"style":5448},[2126,2167],[54,33367,33369],{"className":33368},[2135],[54,33370,33372,33392],{"className":33371},[2139,2756],[54,33373,33375,33389],{"className":33374},[2143],[54,33376,33378],{"className":33377,"style":10799},[2147],[54,33379,33380,33383],{"style":5570},[54,33381],{"className":33382,"style":2156},[2155],[54,33384,33386],{"className":33385},[2160,2161,2162,2163],[54,33387,3182],{"className":33388,"style":3248},[2126,2167,2163],[54,33390,2781],{"className":33391},[2780],[54,33393,33395],{"className":33394},[2143],[54,33396,33398],{"className":33397,"style":2788},[2147],[54,33399],{},[54,33401],{"className":33402,"style":2271},[2270],[54,33404,2247],{"className":33405},[2275],[54,33407],{"className":33408,"style":2271},[2270],[54,33410,33412,33415,33455,33458,33461],{"className":33411},[2117],[54,33413],{"className":33414,"style":2743},[2121],[54,33416,33418,33421],{"className":33417},[2126],[54,33419,5738],{"className":33420,"style":5448},[2126,2167],[54,33422,33424],{"className":33423},[2135],[54,33425,33427,33447],{"className":33426},[2139,2756],[54,33428,33430,33444],{"className":33429},[2143],[54,33431,33433],{"className":33432,"style":13837},[2147],[54,33434,33435,33438],{"style":5570},[54,33436],{"className":33437,"style":2156},[2155],[54,33439,33441],{"className":33440},[2160,2161,2162,2163],[54,33442,150],{"className":33443},[2126,2163],[54,33445,2781],{"className":33446},[2780],[54,33448,33450],{"className":33449},[2143],[54,33451,33453],{"className":33452,"style":2788},[2147],[54,33454],{},[54,33456],{"className":33457,"style":2475},[2270],[54,33459,27066],{"className":33460},[2479],[54,33462],{"className":33463,"style":2475},[2270],[54,33465,33467,33470],{"className":33466},[2117],[54,33468],{"className":33469,"style":4990},[2121],[54,33471,27081],{"className":33472,"style":27166},[2126,2167]," for every ",[54,33475,33477,33494],{"className":33476},[2079],[54,33478,33480],{"className":33479},[2083],[2085,33481,33482],{"xmlns":2087},[2089,33483,33484,33492],{},[2092,33485,33486],{},[2724,33487,33488,33490],{},[2098,33489,5738],{},[2098,33491,3182],{},[2105,33493,28646],{"encoding":2107},[54,33495,33497],{"className":33496,"ariaHidden":2113},[2112],[54,33498,33500,33503],{"className":33499},[2117],[54,33501],{"className":33502,"style":2743},[2121],[54,33504,33506,33509],{"className":33505},[2126],[54,33507,5738],{"className":33508,"style":5448},[2126,2167],[54,33510,33512],{"className":33511},[2135],[54,33513,33515,33535],{"className":33514},[2139,2756],[54,33516,33518,33532],{"className":33517},[2143],[54,33519,33521],{"className":33520,"style":10799},[2147],[54,33522,33523,33526],{"style":5570},[54,33524],{"className":33525,"style":2156},[2155],[54,33527,33529],{"className":33528},[2160,2161,2162,2163],[54,33530,3182],{"className":33531,"style":3248},[2126,2167,2163],[54,33533,2781],{"className":33534},[2780],[54,33536,33538],{"className":33537},[2143],[54,33539,33541],{"className":33540,"style":2788},[2147],[54,33542],{}," in the sum, and the second step follows because the sum in the numerator is a subset of the sum in the denominator. For the second term, first we lower bound the denominator by ",[54,33545,33547,33573],{"className":33546},[2079],[54,33548,33550],{"className":33549},[2083],[2085,33551,33552],{"xmlns":2087},[2089,33553,33554,33570],{},[2092,33555,33556,33558,33564],{},[2098,33557,2305],{"mathvariant":2304},[2724,33559,33560,33562],{},[2098,33561,27788],{},[2249,33563,150],{},[2095,33565,33566,33568],{},[2098,33567,2305],{"mathvariant":2304},[2249,33569,133],{},[2105,33571,33572],{"encoding":2107},"|\\gamma_0|^2",[54,33574,33576],{"className":33575,"ariaHidden":2113},[2112],[54,33577,33579,33582,33585,33625],{"className":33578},[2117],[54,33580],{"className":33581,"style":12758},[2121],[54,33583,2305],{"className":33584},[2126],[54,33586,33588,33591],{"className":33587},[2126],[54,33589,27788],{"className":33590,"style":27924},[2126,2167],[54,33592,33594],{"className":33593},[2135],[54,33595,33597,33617],{"className":33596},[2139,2756],[54,33598,33600,33614],{"className":33599},[2143],[54,33601,33603],{"className":33602,"style":13837},[2147],[54,33604,33605,33608],{"style":27939},[54,33606],{"className":33607,"style":2156},[2155],[54,33609,33611],{"className":33610},[2160,2161,2162,2163],[54,33612,150],{"className":33613},[2126,2163],[54,33615,2781],{"className":33616},[2780],[54,33618,33620],{"className":33619},[2143],[54,33621,33623],{"className":33622,"style":2788},[2147],[54,33624],{},[54,33626,33628,33631],{"className":33627},[2126],[54,33629,2305],{"className":33630},[2126],[54,33632,33634],{"className":33633},[2135],[54,33635,33637],{"className":33636},[2139],[54,33638,33640],{"className":33639},[2143],[54,33641,33643],{"className":33642,"style":12780},[2147],[54,33644,33645,33648],{"style":2151},[54,33646],{"className":33647,"style":2156},[2155],[54,33649,33651],{"className":33650},[2160,2161,2162,2163],[54,33652,133],{"className":33653},[2126,2163],", since ",[54,33656,33658,33690],{"className":33657},[2079],[54,33659,33661],{"className":33660},[2083],[2085,33662,33663],{"xmlns":2087},[2089,33664,33665,33687],{},[2092,33666,33667,33669,33671,33677,33683,33685],{},[2098,33668,21834],{},[2245,33670,129],{"stretchy":2311},[2724,33672,33673,33675],{},[2098,33674,5738],{},[2249,33676,150],{},[2095,33678,33679,33681],{},[2245,33680,3205],{"stretchy":2311},[2249,33682,133],{},[2245,33684,122],{},[2249,33686,174],{},[2105,33688,33689],{"encoding":2107},"f(E_0)^2=1",[54,33691,33693,33783],{"className":33692,"ariaHidden":2113},[2112],[54,33694,33696,33699,33702,33705,33745,33774,33777,33780],{"className":33695},[2117],[54,33697],{"className":33698,"style":12758},[2121],[54,33700,21834],{"className":33701,"style":26895},[2126,2167],[54,33703,129],{"className":33704},[2497],[54,33706,33708,33711],{"className":33707},[2126],[54,33709,5738],{"className":33710,"style":5448},[2126,2167],[54,33712,33714],{"className":33713},[2135],[54,33715,33717,33737],{"className":33716},[2139,2756],[54,33718,33720,33734],{"className":33719},[2143],[54,33721,33723],{"className":33722,"style":13837},[2147],[54,33724,33725,33728],{"style":5570},[54,33726],{"className":33727,"style":2156},[2155],[54,33729,33731],{"className":33730},[2160,2161,2162,2163],[54,33732,150],{"className":33733},[2126,2163],[54,33735,2781],{"className":33736},[2780],[54,33738,33740],{"className":33739},[2143],[54,33741,33743],{"className":33742,"style":2788},[2147],[54,33744],{},[54,33746,33748,33751],{"className":33747},[2336],[54,33749,3205],{"className":33750},[2336],[54,33752,33754],{"className":33753},[2135],[54,33755,33757],{"className":33756},[2139],[54,33758,33760],{"className":33759},[2143],[54,33761,33763],{"className":33762,"style":12780},[2147],[54,33764,33765,33768],{"style":2151},[54,33766],{"className":33767,"style":2156},[2155],[54,33769,33771],{"className":33770},[2160,2161,2162,2163],[54,33772,133],{"className":33773},[2126,2163],[54,33775],{"className":33776,"style":2475},[2270],[54,33778,122],{"className":33779},[2479],[54,33781],{"className":33782,"style":2475},[2270],[54,33784,33786,33789],{"className":33785},[2117],[54,33787],{"className":33788,"style":2285},[2121],[54,33790,174],{"className":33791},[2126],": adding everything back together, this gives",[54,33794,33796],{"className":33795},[2340],[54,33797,33799,33913],{"className":33798},[2079],[54,33800,33802],{"className":33801},[2083],[2085,33803,33804],{"xmlns":2087,"display":2349},[2089,33805,33806,33910],{},[2092,33807,33808,33810,33812,33814,33816,33836,33860,33862,33868,33874,33876,33878,33884,33890,33892,33898,33900,33906,33908],{},[5723,33809,28794],{},[2245,33811,27066],{},[2098,33813,27081],{},[2245,33815,1663],{},[9047,33817,33818,33820],{},[2249,33819,174],{},[2092,33821,33822,33824,33830],{},[2098,33823,2305],{"mathvariant":2304},[2724,33825,33826,33828],{},[2098,33827,27788],{},[2249,33829,150],{},[2095,33831,33832,33834],{},[2098,33833,2305],{"mathvariant":2304},[2249,33835,133],{},[30197,33837,33838,33840],{},[2245,33839,7083],{},[2092,33841,33842,33848,33850,33856,33858],{},[2724,33843,33844,33846],{},[2098,33845,5738],{},[2098,33847,3182],{},[2245,33849,1207],{},[2724,33851,33852,33854],{},[2098,33853,5738],{},[2249,33855,150],{},[2245,33857,1663],{},[2098,33859,27081],{},[2098,33861,2305],{"mathvariant":2304},[2724,33863,33864,33866],{},[2098,33865,27788],{},[2098,33867,3182],{},[2095,33869,33870,33872],{},[2098,33871,2305],{"mathvariant":2304},[2249,33873,133],{},[2098,33875,21834],{},[2245,33877,129],{"stretchy":2311},[2724,33879,33880,33882],{},[2098,33881,5738],{},[2098,33883,3182],{},[2095,33885,33886,33888],{},[2245,33887,3205],{"stretchy":2311},[2249,33889,133],{},[2245,33891,129],{"stretchy":2311},[2724,33893,33894,33896],{},[2098,33895,5738],{},[2098,33897,3182],{},[2245,33899,2247],{},[2724,33901,33902,33904],{},[2098,33903,5738],{},[2249,33905,150],{},[2245,33907,3205],{"stretchy":2311},[2098,33909,29],{"mathvariant":2304},[2105,33911,33912],{"encoding":2107},"\\text{energy error} \\le \\delta + \\frac{1}{|\\gamma_0|^2}\\sum_{E_k>E_0+\\delta}|\\gamma_k|^2f(E_k)^2(E_k-E_0).",[54,33914,33916,33938,33957,34435],{"className":33915,"ariaHidden":2113},[2112],[54,33917,33919,33923,33929,33932,33935],{"className":33918},[2117],[54,33920],{"className":33921,"style":33922},[2121],"height:0.8304em;vertical-align:-0.1944em;",[54,33924,33926],{"className":33925},[2126,6595],[54,33927,28794],{"className":33928},[2126],[54,33930],{"className":33931,"style":2475},[2270],[54,33933,27066],{"className":33934},[2479],[54,33936],{"className":33937,"style":2475},[2270],[54,33939,33941,33945,33948,33951,33954],{"className":33940},[2117],[54,33942],{"className":33943,"style":33944},[2121],"height:0.7778em;vertical-align:-0.0833em;",[54,33946,27081],{"className":33947,"style":27166},[2126,2167],[54,33949],{"className":33950,"style":2271},[2270],[54,33952,1663],{"className":33953},[2275],[54,33955],{"className":33956,"style":2271},[2270],[54,33958,33960,33964,34095,34098,34233,34236,34239,34279,34308,34311,34314,34354,34383,34386,34426,34429,34432],{"className":33959},[2117],[54,33961],{"className":33962,"style":33963},[2121],"height:2.7294em;vertical-align:-1.408em;",[54,33965,33967,33970,34092],{"className":33966},[2126],[54,33968],{"className":33969},[2497,9349],[54,33971,33973],{"className":33972},[9047],[54,33974,33976,34084],{"className":33975},[2139,2756],[54,33977,33979,34081],{"className":33978},[2143],[54,33980,33982,34062,34070],{"className":33981,"style":9362},[2147],[54,33983,33984,33987],{"style":11273},[54,33985],{"className":33986,"style":3866},[2155],[54,33988,33990,33993,34033],{"className":33989},[2126],[54,33991,2305],{"className":33992},[2126],[54,33994,33996,33999],{"className":33995},[2126],[54,33997,27788],{"className":33998,"style":27924},[2126,2167],[54,34000,34002],{"className":34001},[2135],[54,34003,34005,34025],{"className":34004},[2139,2756],[54,34006,34008,34022],{"className":34007},[2143],[54,34009,34011],{"className":34010,"style":13837},[2147],[54,34012,34013,34016],{"style":27939},[54,34014],{"className":34015,"style":2156},[2155],[54,34017,34019],{"className":34018},[2160,2161,2162,2163],[54,34020,150],{"className":34021},[2126,2163],[54,34023,2781],{"className":34024},[2780],[54,34026,34028],{"className":34027},[2143],[54,34029,34031],{"className":34030,"style":2788},[2147],[54,34032],{},[54,34034,34036,34039],{"className":34035},[2126],[54,34037,2305],{"className":34038},[2126],[54,34040,34042],{"className":34041},[2135],[54,34043,34045],{"className":34044},[2139],[54,34046,34048],{"className":34047},[2143],[54,34049,34051],{"className":34050,"style":28966},[2147],[54,34052,34053,34056],{"style":28969},[54,34054],{"className":34055,"style":2156},[2155],[54,34057,34059],{"className":34058},[2160,2161,2162,2163],[54,34060,133],{"className":34061},[2126,2163],[54,34063,34064,34067],{"style":9434},[54,34065],{"className":34066,"style":3866},[2155],[54,34068],{"className":34069,"style":9442},[9441],[54,34071,34072,34075],{"style":9445},[54,34073],{"className":34074,"style":3866},[2155],[54,34076,34078],{"className":34077},[2126],[54,34079,174],{"className":34080},[2126],[54,34082,2781],{"className":34083},[2780],[54,34085,34087],{"className":34086},[2143],[54,34088,34090],{"className":34089,"style":29113},[2147],[54,34091],{},[54,34093],{"className":34094},[2336,9349],[54,34096],{"className":34097,"style":2521},[2270],[54,34099,34101],{"className":34100},[7126,27843],[54,34102,34104,34224],{"className":34103},[2139,2756],[54,34105,34107,34221],{"className":34106},[2143],[54,34108,34111,34211],{"className":34109,"style":34110},[2147],"height:1.05em;",[54,34112,34113,34116],{"style":27856},[54,34114],{"className":34115,"style":27860},[2155],[54,34117,34119],{"className":34118},[2160,2161,2162,2163],[54,34120,34122,34162,34165,34205,34208],{"className":34121},[2126,2163],[54,34123,34125,34128],{"className":34124},[2126,2163],[54,34126,5738],{"className":34127,"style":5448},[2126,2167,2163],[54,34129,34131],{"className":34130},[2135],[54,34132,34134,34154],{"className":34133},[2139,2756],[54,34135,34137,34151],{"className":34136},[2143],[54,34138,34140],{"className":34139,"style":30780},[2147],[54,34141,34142,34145],{"style":30783},[54,34143],{"className":34144,"style":4368},[2155],[54,34146,34148],{"className":34147},[2160,4372,2504,2163],[54,34149,3182],{"className":34150,"style":3248},[2126,2167,2163],[54,34152,2781],{"className":34153},[2780],[54,34155,34157],{"className":34156},[2143],[54,34158,34160],{"className":34159,"style":30802},[2147],[54,34161],{},[54,34163,1207],{"className":34164},[2479,2163],[54,34166,34168,34171],{"className":34167},[2126,2163],[54,34169,5738],{"className":34170,"style":5448},[2126,2167,2163],[54,34172,34174],{"className":34173},[2135],[54,34175,34177,34197],{"className":34176},[2139,2756],[54,34178,34180,34194],{"className":34179},[2143],[54,34181,34183],{"className":34182,"style":30826},[2147],[54,34184,34185,34188],{"style":30829},[54,34186],{"className":34187,"style":4368},[2155],[54,34189,34191],{"className":34190},[2160,4372,2504,2163],[54,34192,150],{"className":34193},[2126,2163],[54,34195,2781],{"className":34196},[2780],[54,34198,34200],{"className":34199},[2143],[54,34201,34203],{"className":34202,"style":4385},[2147],[54,34204],{},[54,34206,1663],{"className":34207},[2275,2163],[54,34209,27081],{"className":34210,"style":27166},[2126,2167,2163],[54,34212,34213,34216],{"style":27878},[54,34214],{"className":34215,"style":27860},[2155],[54,34217,34218],{},[54,34219,7083],{"className":34220},[7126,7130,27887],[54,34222,2781],{"className":34223},[2780],[54,34225,34227],{"className":34226},[2143],[54,34228,34231],{"className":34229,"style":34230},[2147],"height:1.408em;",[54,34232],{},[54,34234],{"className":34235,"style":2521},[2270],[54,34237,2305],{"className":34238},[2126],[54,34240,34242,34245],{"className":34241},[2126],[54,34243,27788],{"className":34244,"style":27924},[2126,2167],[54,34246,34248],{"className":34247},[2135],[54,34249,34251,34271],{"className":34250},[2139,2756],[54,34252,34254,34268],{"className":34253},[2143],[54,34255,34257],{"className":34256,"style":10799},[2147],[54,34258,34259,34262],{"style":27939},[54,34260],{"className":34261,"style":2156},[2155],[54,34263,34265],{"className":34264},[2160,2161,2162,2163],[54,34266,3182],{"className":34267,"style":3248},[2126,2167,2163],[54,34269,2781],{"className":34270},[2780],[54,34272,34274],{"className":34273},[2143],[54,34275,34277],{"className":34276,"style":2788},[2147],[54,34278],{},[54,34280,34282,34285],{"className":34281},[2126],[54,34283,2305],{"className":34284},[2126],[54,34286,34288],{"className":34287},[2135],[54,34289,34291],{"className":34290},[2139],[54,34292,34294],{"className":34293},[2143],[54,34295,34297],{"className":34296,"style":2558},[2147],[54,34298,34299,34302],{"style":2462},[54,34300],{"className":34301,"style":2156},[2155],[54,34303,34305],{"className":34304},[2160,2161,2162,2163],[54,34306,133],{"className":34307},[2126,2163],[54,34309,21834],{"className":34310,"style":26895},[2126,2167],[54,34312,129],{"className":34313},[2497],[54,34315,34317,34320],{"className":34316},[2126],[54,34318,5738],{"className":34319,"style":5448},[2126,2167],[54,34321,34323],{"className":34322},[2135],[54,34324,34326,34346],{"className":34325},[2139,2756],[54,34327,34329,34343],{"className":34328},[2143],[54,34330,34332],{"className":34331,"style":10799},[2147],[54,34333,34334,34337],{"style":5570},[54,34335],{"className":34336,"style":2156},[2155],[54,34338,34340],{"className":34339},[2160,2161,2162,2163],[54,34341,3182],{"className":34342,"style":3248},[2126,2167,2163],[54,34344,2781],{"className":34345},[2780],[54,34347,34349],{"className":34348},[2143],[54,34350,34352],{"className":34351,"style":2788},[2147],[54,34353],{},[54,34355,34357,34360],{"className":34356},[2336],[54,34358,3205],{"className":34359},[2336],[54,34361,34363],{"className":34362},[2135],[54,34364,34366],{"className":34365},[2139],[54,34367,34369],{"className":34368},[2143],[54,34370,34372],{"className":34371,"style":2558},[2147],[54,34373,34374,34377],{"style":2462},[54,34375],{"className":34376,"style":2156},[2155],[54,34378,34380],{"className":34379},[2160,2161,2162,2163],[54,34381,133],{"className":34382},[2126,2163],[54,34384,129],{"className":34385},[2497],[54,34387,34389,34392],{"className":34388},[2126],[54,34390,5738],{"className":34391,"style":5448},[2126,2167],[54,34393,34395],{"className":34394},[2135],[54,34396,34398,34418],{"className":34397},[2139,2756],[54,34399,34401,34415],{"className":34400},[2143],[54,34402,34404],{"className":34403,"style":10799},[2147],[54,34405,34406,34409],{"style":5570},[54,34407],{"className":34408,"style":2156},[2155],[54,34410,34412],{"className":34411},[2160,2161,2162,2163],[54,34413,3182],{"className":34414,"style":3248},[2126,2167,2163],[54,34416,2781],{"className":34417},[2780],[54,34419,34421],{"className":34420},[2143],[54,34422,34424],{"className":34423,"style":2788},[2147],[54,34425],{},[54,34427],{"className":34428,"style":2271},[2270],[54,34430,2247],{"className":34431},[2275],[54,34433],{"className":34434,"style":2271},[2270],[54,34436,34438,34441,34481,34484],{"className":34437},[2117],[54,34439],{"className":34440,"style":2325},[2121],[54,34442,34444,34447],{"className":34443},[2126],[54,34445,5738],{"className":34446,"style":5448},[2126,2167],[54,34448,34450],{"className":34449},[2135],[54,34451,34453,34473],{"className":34452},[2139,2756],[54,34454,34456,34470],{"className":34455},[2143],[54,34457,34459],{"className":34458,"style":13837},[2147],[54,34460,34461,34464],{"style":5570},[54,34462],{"className":34463,"style":2156},[2155],[54,34465,34467],{"className":34466},[2160,2161,2162,2163],[54,34468,150],{"className":34469},[2126,2163],[54,34471,2781],{"className":34472},[2780],[54,34474,34476],{"className":34475},[2143],[54,34477,34479],{"className":34478,"style":2788},[2147],[54,34480],{},[54,34482,3205],{"className":34483},[2336],[54,34485,29],{"className":34486},[2126],[13,34488,34489,34490,34559,34560,34588,34589,34810,34811,34979,34980,35269],{},"To simplify what is left, notice that for all these ",[54,34491,34493,34510],{"className":34492},[2079],[54,34494,34496],{"className":34495},[2083],[2085,34497,34498],{"xmlns":2087},[2089,34499,34500,34508],{},[2092,34501,34502],{},[2724,34503,34504,34506],{},[2098,34505,5738],{},[2098,34507,3182],{},[2105,34509,28646],{"encoding":2107},[54,34511,34513],{"className":34512,"ariaHidden":2113},[2112],[54,34514,34516,34519],{"className":34515},[2117],[54,34517],{"className":34518,"style":2743},[2121],[54,34520,34522,34525],{"className":34521},[2126],[54,34523,5738],{"className":34524,"style":5448},[2126,2167],[54,34526,34528],{"className":34527},[2135],[54,34529,34531,34551],{"className":34530},[2139,2756],[54,34532,34534,34548],{"className":34533},[2143],[54,34535,34537],{"className":34536,"style":10799},[2147],[54,34538,34539,34542],{"style":5570},[54,34540],{"className":34541,"style":2156},[2155],[54,34543,34545],{"className":34544},[2160,2161,2162,2163],[54,34546,3182],{"className":34547,"style":3248},[2126,2167,2163],[54,34549,2781],{"className":34550},[2780],[54,34552,34554],{"className":34553},[2143],[54,34555,34557],{"className":34556,"style":2788},[2147],[54,34558],{},", by the definition of ",[54,34561,34563,34576],{"className":34562},[2079],[54,34564,34566],{"className":34565},[2083],[2085,34567,34568],{"xmlns":2087},[2089,34569,34570,34574],{},[2092,34571,34572],{},[2098,34573,21834],{},[2105,34575,21834],{"encoding":2107},[54,34577,34579],{"className":34578,"ariaHidden":2113},[2112],[54,34580,34582,34585],{"className":34581},[2117],[54,34583],{"className":34584,"style":6591},[2121],[54,34586,21834],{"className":34587,"style":26895},[2126,2167]," we know that ",[54,34590,34592,34646],{"className":34591},[2079],[54,34593,34595],{"className":34594},[2083],[2085,34596,34597],{"xmlns":2087},[2089,34598,34599,34643],{},[2092,34600,34601,34603,34605,34611,34617,34619,34621],{},[2098,34602,21834],{},[2245,34604,129],{"stretchy":2311},[2724,34606,34607,34609],{},[2098,34608,5738],{},[2098,34610,3182],{},[2095,34612,34613,34615],{},[2245,34614,3205],{"stretchy":2311},[2249,34616,133],{},[2245,34618,27066],{},[2249,34620,11708],{},[2095,34622,34623,34635],{},[2092,34624,34625,34627,34629,34631,34633],{},[2245,34626,129],{"fence":2113},[2249,34628,174],{},[2245,34630,1663],{},[2098,34632,27081],{},[2245,34634,3205],{"fence":2113},[2092,34636,34637,34639,34641],{},[2245,34638,2247],{},[2249,34640,133],{},[2098,34642,4515],{},[2105,34644,34645],{"encoding":2107},"f(E_k)^2 \\le 4\\left(1 + \\delta\\right)^{-2d}",[54,34647,34649,34739],{"className":34648,"ariaHidden":2113},[2112],[54,34650,34652,34655,34658,34661,34701,34730,34733,34736],{"className":34651},[2117],[54,34653],{"className":34654,"style":12758},[2121],[54,34656,21834],{"className":34657,"style":26895},[2126,2167],[54,34659,129],{"className":34660},[2497],[54,34662,34664,34667],{"className":34663},[2126],[54,34665,5738],{"className":34666,"style":5448},[2126,2167],[54,34668,34670],{"className":34669},[2135],[54,34671,34673,34693],{"className":34672},[2139,2756],[54,34674,34676,34690],{"className":34675},[2143],[54,34677,34679],{"className":34678,"style":10799},[2147],[54,34680,34681,34684],{"style":5570},[54,34682],{"className":34683,"style":2156},[2155],[54,34685,34687],{"className":34686},[2160,2161,2162,2163],[54,34688,3182],{"className":34689,"style":3248},[2126,2167,2163],[54,34691,2781],{"className":34692},[2780],[54,34694,34696],{"className":34695},[2143],[54,34697,34699],{"className":34698,"style":2788},[2147],[54,34700],{},[54,34702,34704,34707],{"className":34703},[2336],[54,34705,3205],{"className":34706},[2336],[54,34708,34710],{"className":34709},[2135],[54,34711,34713],{"className":34712},[2139],[54,34714,34716],{"className":34715},[2143],[54,34717,34719],{"className":34718,"style":12780},[2147],[54,34720,34721,34724],{"style":2151},[54,34722],{"className":34723,"style":2156},[2155],[54,34725,34727],{"className":34726},[2160,2161,2162,2163],[54,34728,133],{"className":34729},[2126,2163],[54,34731],{"className":34732,"style":2475},[2270],[54,34734,27066],{"className":34735},[2479],[54,34737],{"className":34738,"style":2475},[2270],[54,34740,34742,34745,34748,34751],{"className":34741},[2117],[54,34743],{"className":34744,"style":27135},[2121],[54,34746,11708],{"className":34747},[2126],[54,34749],{"className":34750,"style":2521},[2270],[54,34752,34754,34778],{"className":34753},[2493],[54,34755,34757,34760,34763,34766,34769,34772,34775],{"className":34756},[2493],[54,34758,129],{"className":34759,"style":2499},[2497,2498],[54,34761,174],{"className":34762},[2126],[54,34764],{"className":34765,"style":2271},[2270],[54,34767,1663],{"className":34768},[2275],[54,34770],{"className":34771,"style":2271},[2270],[54,34773,27081],{"className":34774,"style":27166},[2126,2167],[54,34776,3205],{"className":34777,"style":2499},[2336,2498],[54,34779,34781],{"className":34780},[2135],[54,34782,34784],{"className":34783},[2139],[54,34785,34787],{"className":34786},[2143],[54,34788,34790],{"className":34789,"style":27182},[2147],[54,34791,34792,34795],{"style":7401},[54,34793],{"className":34794,"style":2156},[2155],[54,34796,34798],{"className":34797},[2160,2161,2162,2163],[54,34799,34801,34804,34807],{"className":34800},[2126,2163],[54,34802,2247],{"className":34803},[2126,2163],[54,34805,133],{"className":34806},[2126,2163],[54,34808,4515],{"className":34809},[2126,2167,2163],". Additionally upper bounding ",[54,34812,34814,34850],{"className":34813},[2079],[54,34815,34817],{"className":34816},[2083],[2085,34818,34819],{"xmlns":2087},[2089,34820,34821,34847],{},[2092,34822,34823,34829,34831,34837,34839,34841,34843,34845],{},[2724,34824,34825,34827],{},[2098,34826,5738],{},[2098,34828,3182],{},[2245,34830,2247],{},[2724,34832,34833,34835],{},[2098,34834,5738],{},[2249,34836,150],{},[2245,34838,5233],{},[2249,34840,133],{},[2098,34842,11714],{"mathvariant":2304},[2098,34844,2694],{},[2098,34846,11714],{"mathvariant":2304},[2105,34848,34849],{"encoding":2107},"E_k-E_0\u003C2\\|H\\|",[54,34851,34853,34908,34963],{"className":34852,"ariaHidden":2113},[2112],[54,34854,34856,34859,34899,34902,34905],{"className":34855},[2117],[54,34857],{"className":34858,"style":2743},[2121],[54,34860,34862,34865],{"className":34861},[2126],[54,34863,5738],{"className":34864,"style":5448},[2126,2167],[54,34866,34868],{"className":34867},[2135],[54,34869,34871,34891],{"className":34870},[2139,2756],[54,34872,34874,34888],{"className":34873},[2143],[54,34875,34877],{"className":34876,"style":10799},[2147],[54,34878,34879,34882],{"style":5570},[54,34880],{"className":34881,"style":2156},[2155],[54,34883,34885],{"className":34884},[2160,2161,2162,2163],[54,34886,3182],{"className":34887,"style":3248},[2126,2167,2163],[54,34889,2781],{"className":34890},[2780],[54,34892,34894],{"className":34893},[2143],[54,34895,34897],{"className":34896,"style":2788},[2147],[54,34898],{},[54,34900],{"className":34901,"style":2271},[2270],[54,34903,2247],{"className":34904},[2275],[54,34906],{"className":34907,"style":2271},[2270],[54,34909,34911,34914,34954,34957,34960],{"className":34910},[2117],[54,34912],{"className":34913,"style":2743},[2121],[54,34915,34917,34920],{"className":34916},[2126],[54,34918,5738],{"className":34919,"style":5448},[2126,2167],[54,34921,34923],{"className":34922},[2135],[54,34924,34926,34946],{"className":34925},[2139,2756],[54,34927,34929,34943],{"className":34928},[2143],[54,34930,34932],{"className":34931,"style":13837},[2147],[54,34933,34934,34937],{"style":5570},[54,34935],{"className":34936,"style":2156},[2155],[54,34938,34940],{"className":34939},[2160,2161,2162,2163],[54,34941,150],{"className":34942},[2126,2163],[54,34944,2781],{"className":34945},[2780],[54,34947,34949],{"className":34948},[2143],[54,34950,34952],{"className":34951,"style":2788},[2147],[54,34953],{},[54,34955],{"className":34956,"style":2475},[2270],[54,34958,5233],{"className":34959},[2479],[54,34961],{"className":34962,"style":2475},[2270],[54,34964,34966,34969,34973,34976],{"className":34965},[2117],[54,34967],{"className":34968,"style":2325},[2121],[54,34970,34972],{"className":34971},[2126],"2∥",[54,34974,2694],{"className":34975,"style":2709},[2126,2167],[54,34977,11714],{"className":34978},[2126]," and upper bounding ",[54,34981,34983,35037],{"className":34982},[2079],[54,34984,34986],{"className":34985},[2083],[2085,34987,34988],{"xmlns":2087},[2089,34989,34990,35034],{},[2092,34991,34992,35016,35018,35024,35030,35032],{},[2724,34993,34994,34996],{},[2245,34995,7083],{},[2092,34997,34998,35004,35006,35012,35014],{},[2724,34999,35000,35002],{},[2098,35001,5738],{},[2098,35003,3182],{},[2245,35005,1207],{},[2724,35007,35008,35010],{},[2098,35009,5738],{},[2249,35011,150],{},[2245,35013,1663],{},[2098,35015,27081],{},[2098,35017,2305],{"mathvariant":2304},[2724,35019,35020,35022],{},[2098,35021,27788],{},[2098,35023,3182],{},[2095,35025,35026,35028],{},[2098,35027,2305],{"mathvariant":2304},[2249,35029,133],{},[2245,35031,5233],{},[2249,35033,174],{},[2105,35035,35036],{"encoding":2107},"\\sum_{E_k>E_0+\\delta}|\\gamma_k|^2\u003C1",[54,35038,35040,35260],{"className":35039,"ariaHidden":2113},[2112],[54,35041,35043,35047,35176,35179,35182,35222,35251,35254,35257],{"className":35042},[2117],[54,35044],{"className":35045,"style":35046},[2121],"height:1.2197em;vertical-align:-0.4056em;",[54,35048,35050,35053],{"className":35049},[7126],[54,35051,7083],{"className":35052,"style":7132},[7126,7130,7131],[54,35054,35056],{"className":35055},[2135],[54,35057,35059,35168],{"className":35058},[2139,2756],[54,35060,35062,35165],{"className":35061},[2143],[54,35063,35065],{"className":35064,"style":30750},[2147],[54,35066,35067,35070],{"style":7148},[54,35068],{"className":35069,"style":2156},[2155],[54,35071,35073],{"className":35072},[2160,2161,2162,2163],[54,35074,35076,35116,35119,35159,35162],{"className":35075},[2126,2163],[54,35077,35079,35082],{"className":35078},[2126,2163],[54,35080,5738],{"className":35081,"style":5448},[2126,2167,2163],[54,35083,35085],{"className":35084},[2135],[54,35086,35088,35108],{"className":35087},[2139,2756],[54,35089,35091,35105],{"className":35090},[2143],[54,35092,35094],{"className":35093,"style":30780},[2147],[54,35095,35096,35099],{"style":30783},[54,35097],{"className":35098,"style":4368},[2155],[54,35100,35102],{"className":35101},[2160,4372,2504,2163],[54,35103,3182],{"className":35104,"style":3248},[2126,2167,2163],[54,35106,2781],{"className":35107},[2780],[54,35109,35111],{"className":35110},[2143],[54,35112,35114],{"className":35113,"style":30802},[2147],[54,35115],{},[54,35117,1207],{"className":35118},[2479,2163],[54,35120,35122,35125],{"className":35121},[2126,2163],[54,35123,5738],{"className":35124,"style":5448},[2126,2167,2163],[54,35126,35128],{"className":35127},[2135],[54,35129,35131,35151],{"className":35130},[2139,2756],[54,35132,35134,35148],{"className":35133},[2143],[54,35135,35137],{"className":35136,"style":30826},[2147],[54,35138,35139,35142],{"style":30829},[54,35140],{"className":35141,"style":4368},[2155],[54,35143,35145],{"className":35144},[2160,4372,2504,2163],[54,35146,150],{"className":35147},[2126,2163],[54,35149,2781],{"className":35150},[2780],[54,35152,35154],{"className":35153},[2143],[54,35155,35157],{"className":35156,"style":4385},[2147],[54,35158],{},[54,35160,1663],{"className":35161},[2275,2163],[54,35163,27081],{"className":35164,"style":27166},[2126,2167,2163],[54,35166,2781],{"className":35167},[2780],[54,35169,35171],{"className":35170},[2143],[54,35172,35174],{"className":35173,"style":30865},[2147],[54,35175],{},[54,35177],{"className":35178,"style":2521},[2270],[54,35180,2305],{"className":35181},[2126],[54,35183,35185,35188],{"className":35184},[2126],[54,35186,27788],{"className":35187,"style":27924},[2126,2167],[54,35189,35191],{"className":35190},[2135],[54,35192,35194,35214],{"className":35193},[2139,2756],[54,35195,35197,35211],{"className":35196},[2143],[54,35198,35200],{"className":35199,"style":10799},[2147],[54,35201,35202,35205],{"style":27939},[54,35203],{"className":35204,"style":2156},[2155],[54,35206,35208],{"className":35207},[2160,2161,2162,2163],[54,35209,3182],{"className":35210,"style":3248},[2126,2167,2163],[54,35212,2781],{"className":35213},[2780],[54,35215,35217],{"className":35216},[2143],[54,35218,35220],{"className":35219,"style":2788},[2147],[54,35221],{},[54,35223,35225,35228],{"className":35224},[2126],[54,35226,2305],{"className":35227},[2126],[54,35229,35231],{"className":35230},[2135],[54,35232,35234],{"className":35233},[2139],[54,35235,35237],{"className":35236},[2143],[54,35238,35240],{"className":35239,"style":12780},[2147],[54,35241,35242,35245],{"style":2151},[54,35243],{"className":35244,"style":2156},[2155],[54,35246,35248],{"className":35247},[2160,2161,2162,2163],[54,35249,133],{"className":35250},[2126,2163],[54,35252],{"className":35253,"style":2475},[2270],[54,35255,5233],{"className":35256},[2479],[54,35258],{"className":35259,"style":2475},[2270],[54,35261,35263,35266],{"className":35262},[2117],[54,35264],{"className":35265,"style":2285},[2121],[54,35267,174],{"className":35268},[2126]," gives",[54,35271,35273],{"className":35272},[2340],[54,35274,35276,35347],{"className":35275},[2079],[54,35277,35279],{"className":35278},[2083],[2085,35280,35281],{"xmlns":2087,"display":2349},[2089,35282,35283,35344],{},[2092,35284,35285,35287,35289,35291,35293,35314,35316,35318,35320,35342],{},[5723,35286,28794],{},[2245,35288,27066],{},[2098,35290,27081],{},[2245,35292,1663],{},[9047,35294,35295,35298],{},[2249,35296,35297],{},"8",[2092,35299,35300,35302,35308],{},[2098,35301,2305],{"mathvariant":2304},[2724,35303,35304,35306],{},[2098,35305,27788],{},[2249,35307,150],{},[2095,35309,35310,35312],{},[2098,35311,2305],{"mathvariant":2304},[2249,35313,133],{},[2098,35315,11714],{"mathvariant":2304},[2098,35317,2694],{},[2098,35319,11714],{"mathvariant":2304},[2095,35321,35322,35334],{},[2092,35323,35324,35326,35328,35330,35332],{},[2245,35325,129],{"fence":2113},[2249,35327,174],{},[2245,35329,1663],{},[2098,35331,27081],{},[2245,35333,3205],{"fence":2113},[2092,35335,35336,35338,35340],{},[2245,35337,2247],{},[2249,35339,133],{},[2098,35341,4515],{},[2098,35343,29],{"mathvariant":2304},[2105,35345,35346],{"encoding":2107},"\\text{energy error} \\le \\delta + \\frac{8}{|\\gamma_0|^2}\\|H\\|\\left(1 + \\delta\\right)^{-2d}.",[54,35348,35350,35371,35389],{"className":35349,"ariaHidden":2113},[2112],[54,35351,35353,35356,35362,35365,35368],{"className":35352},[2117],[54,35354],{"className":35355,"style":33922},[2121],[54,35357,35359],{"className":35358},[2126,6595],[54,35360,28794],{"className":35361},[2126],[54,35363],{"className":35364,"style":2475},[2270],[54,35366,27066],{"className":35367},[2479],[54,35369],{"className":35370,"style":2475},[2270],[54,35372,35374,35377,35380,35383,35386],{"className":35373},[2117],[54,35375],{"className":35376,"style":33944},[2121],[54,35378,27081],{"className":35379,"style":27166},[2126,2167],[54,35381],{"className":35382,"style":2271},[2270],[54,35384,1663],{"className":35385},[2275],[54,35387],{"className":35388,"style":2271},[2270],[54,35390,35392,35396,35527,35530,35533,35536,35539,35598,35601],{"className":35391},[2117],[54,35393],{"className":35394,"style":35395},[2121],"height:2.2574em;vertical-align:-0.936em;",[54,35397,35399,35402,35524],{"className":35398},[2126],[54,35400],{"className":35401},[2497,9349],[54,35403,35405],{"className":35404},[9047],[54,35406,35408,35516],{"className":35407},[2139,2756],[54,35409,35411,35513],{"className":35410},[2143],[54,35412,35414,35494,35502],{"className":35413,"style":9362},[2147],[54,35415,35416,35419],{"style":11273},[54,35417],{"className":35418,"style":3866},[2155],[54,35420,35422,35425,35465],{"className":35421},[2126],[54,35423,2305],{"className":35424},[2126],[54,35426,35428,35431],{"className":35427},[2126],[54,35429,27788],{"className":35430,"style":27924},[2126,2167],[54,35432,35434],{"className":35433},[2135],[54,35435,35437,35457],{"className":35436},[2139,2756],[54,35438,35440,35454],{"className":35439},[2143],[54,35441,35443],{"className":35442,"style":13837},[2147],[54,35444,35445,35448],{"style":27939},[54,35446],{"className":35447,"style":2156},[2155],[54,35449,35451],{"className":35450},[2160,2161,2162,2163],[54,35452,150],{"className":35453},[2126,2163],[54,35455,2781],{"className":35456},[2780],[54,35458,35460],{"className":35459},[2143],[54,35461,35463],{"className":35462,"style":2788},[2147],[54,35464],{},[54,35466,35468,35471],{"className":35467},[2126],[54,35469,2305],{"className":35470},[2126],[54,35472,35474],{"className":35473},[2135],[54,35475,35477],{"className":35476},[2139],[54,35478,35480],{"className":35479},[2143],[54,35481,35483],{"className":35482,"style":28966},[2147],[54,35484,35485,35488],{"style":28969},[54,35486],{"className":35487,"style":2156},[2155],[54,35489,35491],{"className":35490},[2160,2161,2162,2163],[54,35492,133],{"className":35493},[2126,2163],[54,35495,35496,35499],{"style":9434},[54,35497],{"className":35498,"style":3866},[2155],[54,35500],{"className":35501,"style":9442},[9441],[54,35503,35504,35507],{"style":9445},[54,35505],{"className":35506,"style":3866},[2155],[54,35508,35510],{"className":35509},[2126],[54,35511,35297],{"className":35512},[2126],[54,35514,2781],{"className":35515},[2780],[54,35517,35519],{"className":35518},[2143],[54,35520,35522],{"className":35521,"style":29113},[2147],[54,35523],{},[54,35525],{"className":35526},[2336,9349],[54,35528,11714],{"className":35529},[2126],[54,35531,2694],{"className":35532,"style":2709},[2126,2167],[54,35534,11714],{"className":35535},[2126],[54,35537],{"className":35538,"style":2521},[2270],[54,35540,35542,35566],{"className":35541},[2493],[54,35543,35545,35548,35551,35554,35557,35560,35563],{"className":35544},[2493],[54,35546,129],{"className":35547,"style":2499},[2497,2498],[54,35549,174],{"className":35550},[2126],[54,35552],{"className":35553,"style":2271},[2270],[54,35555,1663],{"className":35556},[2275],[54,35558],{"className":35559,"style":2271},[2270],[54,35561,27081],{"className":35562,"style":27166},[2126,2167],[54,35564,3205],{"className":35565,"style":2499},[2336,2498],[54,35567,35569],{"className":35568},[2135],[54,35570,35572],{"className":35571},[2139],[54,35573,35575],{"className":35574},[2143],[54,35576,35578],{"className":35577,"style":27182},[2147],[54,35579,35580,35583],{"style":7401},[54,35581],{"className":35582,"style":2156},[2155],[54,35584,35586],{"className":35585},[2160,2161,2162,2163],[54,35587,35589,35592,35595],{"className":35588},[2126,2163],[54,35590,2247],{"className":35591},[2126,2163],[54,35593,133],{"className":35594},[2126,2163],[54,35596,4515],{"className":35597},[2126,2167,2163],[54,35599],{"className":35600,"style":2521},[2270],[54,35602,29],{"className":35603},[2126],[13,35605,35606,35607,35659,35660,35688,35689,35745,35746,35901,35902,35930],{},"This holds for any ",[54,35608,35610,35628],{"className":35609},[2079],[54,35611,35613],{"className":35612},[2083],[2085,35614,35615],{"xmlns":2087},[2089,35616,35617,35625],{},[2092,35618,35619,35621,35623],{},[2098,35620,27081],{},[2245,35622,1207],{},[2249,35624,150],{},[2105,35626,35627],{"encoding":2107},"\\delta>0",[54,35629,35631,35650],{"className":35630,"ariaHidden":2113},[2112],[54,35632,35634,35638,35641,35644,35647],{"className":35633},[2117],[54,35635],{"className":35636,"style":35637},[2121],"height:0.7335em;vertical-align:-0.0391em;",[54,35639,27081],{"className":35640,"style":27166},[2126,2167],[54,35642],{"className":35643,"style":2475},[2270],[54,35645,1207],{"className":35646},[2479],[54,35648],{"className":35649,"style":2475},[2270],[54,35651,35653,35656],{"className":35652},[2117],[54,35654],{"className":35655,"style":2285},[2121],[54,35657,150],{"className":35658},[2126],", so if we set ",[54,35661,35663,35676],{"className":35662},[2079],[54,35664,35666],{"className":35665},[2083],[2085,35667,35668],{"xmlns":2087},[2089,35669,35670,35674],{},[2092,35671,35672],{},[2098,35673,27081],{},[2105,35675,31807],{"encoding":2107},[54,35677,35679],{"className":35678,"ariaHidden":2113},[2112],[54,35680,35682,35685],{"className":35681},[2117],[54,35683],{"className":35684,"style":4990},[2121],[54,35686,27081],{"className":35687,"style":27166},[2126,2167]," equal to our goal error, then the error bound above converges towards that exponentially with the Krylov dimension ",[54,35690,35692,35712],{"className":35691},[2079],[54,35693,35695],{"className":35694},[2083],[2085,35696,35697],{"xmlns":2087},[2089,35698,35699,35709],{},[2092,35700,35701,35703,35705,35707],{},[2249,35702,133],{},[2098,35704,4515],{},[2245,35706,122],{},[2098,35708,1772],{},[2105,35710,35711],{"encoding":2107},"2d=r",[54,35713,35715,35736],{"className":35714,"ariaHidden":2113},[2112],[54,35716,35718,35721,35724,35727,35730,35733],{"className":35717},[2117],[54,35719],{"className":35720,"style":4990},[2121],[54,35722,133],{"className":35723},[2126],[54,35725,4515],{"className":35726},[2126,2167],[54,35728],{"className":35729,"style":2475},[2270],[54,35731,122],{"className":35732},[2479],[54,35734],{"className":35735,"style":2475},[2270],[54,35737,35739,35742],{"className":35738},[2117],[54,35740],{"className":35741,"style":2195},[2121],[54,35743,1772],{"className":35744,"style":2168},[2126,2167],". Also note that if ",[54,35747,35749,35779],{"className":35748},[2079],[54,35750,35752],{"className":35751},[2083],[2085,35753,35754],{"xmlns":2087},[2089,35755,35756,35776],{},[2092,35757,35758,35760,35762,35768,35770],{},[2098,35759,27081],{},[2245,35761,5233],{},[2724,35763,35764,35766],{},[2098,35765,5738],{},[2249,35767,174],{},[2245,35769,2247],{},[2724,35771,35772,35774],{},[2098,35773,5738],{},[2249,35775,150],{},[2105,35777,35778],{"encoding":2107},"\\delta\u003CE_1-E_0",[54,35780,35782,35800,35855],{"className":35781,"ariaHidden":2113},[2112],[54,35783,35785,35788,35791,35794,35797],{"className":35784},[2117],[54,35786],{"className":35787,"style":35637},[2121],[54,35789,27081],{"className":35790,"style":27166},[2126,2167],[54,35792],{"className":35793,"style":2475},[2270],[54,35795,5233],{"className":35796},[2479],[54,35798],{"className":35799,"style":2475},[2270],[54,35801,35803,35806,35846,35849,35852],{"className":35802},[2117],[54,35804],{"className":35805,"style":2743},[2121],[54,35807,35809,35812],{"className":35808},[2126],[54,35810,5738],{"className":35811,"style":5448},[2126,2167],[54,35813,35815],{"className":35814},[2135],[54,35816,35818,35838],{"className":35817},[2139,2756],[54,35819,35821,35835],{"className":35820},[2143],[54,35822,35824],{"className":35823,"style":13837},[2147],[54,35825,35826,35829],{"style":5570},[54,35827],{"className":35828,"style":2156},[2155],[54,35830,35832],{"className":35831},[2160,2161,2162,2163],[54,35833,174],{"className":35834},[2126,2163],[54,35836,2781],{"className":35837},[2780],[54,35839,35841],{"className":35840},[2143],[54,35842,35844],{"className":35843,"style":2788},[2147],[54,35845],{},[54,35847],{"className":35848,"style":2271},[2270],[54,35850,2247],{"className":35851},[2275],[54,35853],{"className":35854,"style":2271},[2270],[54,35856,35858,35861],{"className":35857},[2117],[54,35859],{"className":35860,"style":2743},[2121],[54,35862,35864,35867],{"className":35863},[2126],[54,35865,5738],{"className":35866,"style":5448},[2126,2167],[54,35868,35870],{"className":35869},[2135],[54,35871,35873,35893],{"className":35872},[2139,2756],[54,35874,35876,35890],{"className":35875},[2143],[54,35877,35879],{"className":35878,"style":13837},[2147],[54,35880,35881,35884],{"style":5570},[54,35882],{"className":35883,"style":2156},[2155],[54,35885,35887],{"className":35886},[2160,2161,2162,2163],[54,35888,150],{"className":35889},[2126,2163],[54,35891,2781],{"className":35892},[2780],[54,35894,35896],{"className":35895},[2143],[54,35897,35899],{"className":35898,"style":2788},[2147],[54,35900],{}," then the ",[54,35903,35905,35918],{"className":35904},[2079],[54,35906,35908],{"className":35907},[2083],[2085,35909,35910],{"xmlns":2087},[2089,35911,35912,35916],{},[2092,35913,35914],{},[2098,35915,27081],{},[2105,35917,31807],{"encoding":2107},[54,35919,35921],{"className":35920,"ariaHidden":2113},[2112],[54,35922,35924,35927],{"className":35923},[2117],[54,35925],{"className":35926,"style":4990},[2121],[54,35928,27081],{"className":35929,"style":27166},[2126,2167]," term actually goes away entirely in the above bound.",[13,35932,35933,35934,35992],{},"To complete the argument, we first note that the above is just the energy error of the particular state ",[54,35935,35937,35962],{"className":35936},[2079],[54,35938,35940],{"className":35939},[2083],[2085,35941,35942],{"xmlns":2087},[2089,35943,35944,35960],{},[2092,35945,35946,35948,35950,35952,35954,35956,35958],{},[2098,35947,21834],{},[2245,35949,129],{"stretchy":2311},[2098,35951,2694],{},[2245,35953,3205],{"stretchy":2311},[2098,35955,2305],{"mathvariant":2304},[2098,35957,3349],{},[2245,35959,2312],{"stretchy":2311},[2105,35961,27674],{"encoding":2107},[54,35963,35965],{"className":35964,"ariaHidden":2113},[2112],[54,35966,35968,35971,35974,35977,35980,35983,35986,35989],{"className":35967},[2117],[54,35969],{"className":35970,"style":2325},[2121],[54,35972,21834],{"className":35973,"style":26895},[2126,2167],[54,35975,129],{"className":35976},[2497],[54,35978,2694],{"className":35979,"style":2709},[2126,2167],[54,35981,3205],{"className":35982},[2336],[54,35984,2305],{"className":35985},[2126],[54,35987,3349],{"className":35988,"style":2332},[2126,2167],[54,35990,2312],{"className":35991},[2336],", rather than the energy error of the lowest energy state in the Krylov space. However, by the (Rayleigh-Ritz) variational principle, the energy error of the lowest energy state in the Krylov space is upper bounded by the energy error of any state in the Krylov space, so the above is also an upper bound on the energy error of the lowest energy state, that is, the output of the Krylov quantum diagonalization algorithm.",[13,35994,35995,35996,10355,35998,36001],{},"A similar analysis as the above can be carried out that additionally accounts for noise and the thresholding procedure discussed in the notebook. See ",[20,35997,8103],{"href":640},[20,35999,36000],{"href":640},"[4]"," for this analysis.",[31,36003,36005],{"id":36004},"appendix-proof-of-claim-1","Appendix: proof of Claim 1",[13,36007,36008,36009,36011,36012,36086,36087,36175,36176,36204],{},"The following is mostly derived from ",[20,36010,26859],{"href":640},", Theorem 3.1: let ",[54,36013,36015,36037],{"className":36014},[2079],[54,36016,36018],{"className":36017},[2083],[2085,36019,36020],{"xmlns":2087},[2089,36021,36022,36034],{},[2092,36023,36024,36026,36028,36030,36032],{},[2249,36025,150],{},[2245,36027,5233],{},[2098,36029,20],{},[2245,36031,5233],{},[2098,36033,12658],{},[2105,36035,36036],{"encoding":2107},"0 \u003C a \u003C b",[54,36038,36040,36059,36077],{"className":36039,"ariaHidden":2113},[2112],[54,36041,36043,36047,36050,36053,36056],{"className":36042},[2117],[54,36044],{"className":36045,"style":36046},[2121],"height:0.6835em;vertical-align:-0.0391em;",[54,36048,150],{"className":36049},[2126],[54,36051],{"className":36052,"style":2475},[2270],[54,36054,5233],{"className":36055},[2479],[54,36057],{"className":36058,"style":2475},[2270],[54,36060,36062,36065,36068,36071,36074],{"className":36061},[2117],[54,36063],{"className":36064,"style":5251},[2121],[54,36066,20],{"className":36067},[2126,2167],[54,36069],{"className":36070,"style":2475},[2270],[54,36072,5233],{"className":36073},[2479],[54,36075],{"className":36076,"style":2475},[2270],[54,36078,36080,36083],{"className":36079},[2117],[54,36081],{"className":36082,"style":4990},[2121],[54,36084,12658],{"className":36085},[2126,2167]," and let ",[54,36088,36090,36111],{"className":36089},[2079],[54,36091,36093],{"className":36092},[2083],[2085,36094,36095],{"xmlns":2087},[2089,36096,36097,36108],{},[2092,36098,36099],{},[3331,36100,36101,36104,36106],{},[2098,36102,36103],{"mathvariant":2304},"Π",[2098,36105,4515],{},[2245,36107,20325],{},[2105,36109,36110],{"encoding":2107},"\\Pi^*_d",[54,36112,36114],{"className":36113,"ariaHidden":2113},[2112],[54,36115,36117,36121],{"className":36116},[2117],[54,36118],{"className":36119,"style":36120},[2121],"height:0.9718em;vertical-align:-0.2831em;",[54,36122,36124,36127],{"className":36123},[2126],[54,36125,36103],{"className":36126},[2126],[54,36128,36130],{"className":36129},[2135],[54,36131,36133,36166],{"className":36132},[2139,2756],[54,36134,36136,36163],{"className":36135},[2143],[54,36137,36140,36152],{"className":36138,"style":36139},[2147],"height:0.6887em;",[54,36141,36143,36146],{"style":36142},"top:-2.4169em;margin-left:0em;margin-right:0.05em;",[54,36144],{"className":36145,"style":2156},[2155],[54,36147,36149],{"className":36148},[2160,2161,2162,2163],[54,36150,4515],{"className":36151},[2126,2167,2163],[54,36153,36154,36157],{"style":2151},[54,36155],{"className":36156,"style":2156},[2155],[54,36158,36160],{"className":36159},[2160,2161,2162,2163],[54,36161,20325],{"className":36162},[2275,2163],[54,36164,2781],{"className":36165},[2780],[54,36167,36169],{"className":36168},[2143],[54,36170,36173],{"className":36171,"style":36172},[2147],"height:0.2831em;",[54,36174],{}," be the space of residual polynomials (polynomials whose value at 0 is 1) of degree at most ",[54,36177,36179,36192],{"className":36178},[2079],[54,36180,36182],{"className":36181},[2083],[2085,36183,36184],{"xmlns":2087},[2089,36185,36186,36190],{},[2092,36187,36188],{},[2098,36189,4515],{},[2105,36191,4515],{"encoding":2107},[54,36193,36195],{"className":36194,"ariaHidden":2113},[2112],[54,36196,36198,36201],{"className":36197},[2117],[54,36199],{"className":36200,"style":4990},[2121],[54,36202,4515],{"className":36203},[2126,2167],". The solution to",[54,36206,36208],{"className":36207},[2340],[54,36209,36211,36305],{"className":36210},[2079],[54,36212,36214],{"className":36213},[2083],[2085,36215,36216],{"xmlns":2087,"display":2349},[2089,36217,36218,36302],{},[2092,36219,36220,36223,36225,36227,36229,36231,36233,36235,36237,36239,36263,36288,36290,36292,36294,36296,36298,36300],{},[2098,36221,36222],{},"β",[2245,36224,129],{"stretchy":2311},[2098,36226,20],{},[2245,36228,2375],{"separator":2113},[2098,36230,12658],{},[2245,36232,2375],{"separator":2113},[2098,36234,4515],{},[2245,36236,3205],{"stretchy":2311},[2245,36238,122],{},[30197,36240,36241,36248],{},[2092,36242,36243,36245],{},[2098,36244,24786],{},[2245,36246,36247],{},"⁡",[2092,36249,36250,36252,36255],{},[2098,36251,13],{},[2245,36253,36254],{},"∈",[3331,36256,36257,36259,36261],{},[2098,36258,36103],{"mathvariant":2304},[2098,36260,4515],{},[2245,36262,20325],{},[30197,36264,36265,36272],{},[2092,36266,36267,36270],{},[2098,36268,36269],{},"max",[2245,36271,36247],{},[2092,36273,36274,36276,36278,36280,36282,36284,36286],{},[2098,36275,8695],{},[2245,36277,36254],{},[2245,36279,11804],{"stretchy":2311},[2098,36281,20],{},[2245,36283,2375],{"separator":2113},[2098,36285,12658],{},[2245,36287,11829],{"stretchy":2311},[2098,36289,2305],{"mathvariant":2304},[2098,36291,13],{},[2245,36293,129],{"stretchy":2311},[2098,36295,8695],{},[2245,36297,3205],{"stretchy":2311},[2098,36299,2305],{"mathvariant":2304},[2270,36301],{"width":9011},[2105,36303,36304],{"encoding":2107},"\\beta(a, b, d) = \\min_{p \\in \\Pi^*_d} \\max_{x \\in [a, b]} |p(x)| \\quad",[54,36306,36308,36354],{"className":36307,"ariaHidden":2113},[2112],[54,36309,36311,36314,36318,36321,36324,36327,36330,36333,36336,36339,36342,36345,36348,36351],{"className":36310},[2117],[54,36312],{"className":36313,"style":2325},[2121],[54,36315,36222],{"className":36316,"style":36317},[2126,2167],"margin-right:0.0528em;",[54,36319,129],{"className":36320},[2497],[54,36322,20],{"className":36323},[2126,2167],[54,36325,2375],{"className":36326},[2517],[54,36328],{"className":36329,"style":2521},[2270],[54,36331,12658],{"className":36332},[2126,2167],[54,36334,2375],{"className":36335},[2517],[54,36337],{"className":36338,"style":2521},[2270],[54,36340,4515],{"className":36341},[2126,2167],[54,36343,3205],{"className":36344},[2336],[54,36346],{"className":36347,"style":2475},[2270],[54,36349,122],{"className":36350},[2479],[54,36352],{"className":36353,"style":2475},[2270],[54,36355,36357,36361,36469,36472,36539,36542,36545,36548,36551,36554,36557,36560],{"className":36356},[2117],[54,36358],{"className":36359,"style":36360},[2121],"height:1.7374em;vertical-align:-0.9874em;",[54,36362,36364],{"className":36363},[7126,27843],[54,36365,36367,36460],{"className":36366},[2139,2756],[54,36368,36370,36457],{"className":36369},[2143],[54,36371,36374,36447],{"className":36372,"style":36373},[2147],"height:0.6679em;",[54,36375,36377,36380],{"style":36376},"top:-2.3557em;margin-left:0em;",[54,36378],{"className":36379,"style":3866},[2155],[54,36381,36383],{"className":36382},[2160,2161,2162,2163],[54,36384,36386,36389,36392],{"className":36385},[2126,2163],[54,36387,13],{"className":36388},[2126,2167,2163],[54,36390,36254],{"className":36391},[2479,2163],[54,36393,36395,36398],{"className":36394},[2126,2163],[54,36396,36103],{"className":36397},[2126,2163],[54,36399,36401],{"className":36400},[2135],[54,36402,36404,36438],{"className":36403},[2139,2756],[54,36405,36407,36435],{"className":36406},[2143],[54,36408,36411,36423],{"className":36409,"style":36410},[2147],"height:0.6771em;",[54,36412,36414,36417],{"style":36413},"top:-2.1528em;margin-left:0em;margin-right:0.0714em;",[54,36415],{"className":36416,"style":4368},[2155],[54,36418,36420],{"className":36419},[2160,4372,2504,2163],[54,36421,4515],{"className":36422},[2126,2167,2163],[54,36424,36426,36429],{"style":36425},"top:-2.8448em;margin-right:0.0714em;",[54,36427],{"className":36428,"style":4368},[2155],[54,36430,36432],{"className":36431},[2160,4372,2504,2163],[54,36433,20325],{"className":36434},[2275,2163],[54,36436,2781],{"className":36437},[2780],[54,36439,36441],{"className":36440},[2143],[54,36442,36445],{"className":36443,"style":36444},[2147],"height:0.3472em;",[54,36446],{},[54,36448,36449,36452],{"style":3862},[54,36450],{"className":36451,"style":3866},[2155],[54,36453,36454],{},[54,36455,24786],{"className":36456},[7126],[54,36458,2781],{"className":36459},[2780],[54,36461,36463],{"className":36462},[2143],[54,36464,36467],{"className":36465,"style":36466},[2147],"height:0.9874em;",[54,36468],{},[54,36470],{"className":36471,"style":2521},[2270],[54,36473,36475],{"className":36474},[7126,27843],[54,36476,36478,36530],{"className":36477},[2139,2756],[54,36479,36481,36527],{"className":36480},[2143],[54,36482,36484,36517],{"className":36483,"style":2195},[2147],[54,36485,36487,36490],{"style":36486},"top:-2.309em;margin-left:0em;",[54,36488],{"className":36489,"style":3866},[2155],[54,36491,36493],{"className":36492},[2160,2161,2162,2163],[54,36494,36496,36499,36502,36505,36508,36511,36514],{"className":36495},[2126,2163],[54,36497,8695],{"className":36498},[2126,2167,2163],[54,36500,36254],{"className":36501},[2479,2163],[54,36503,11804],{"className":36504},[2497,2163],[54,36506,20],{"className":36507},[2126,2167,2163],[54,36509,2375],{"className":36510},[2517,2163],[54,36512,12658],{"className":36513},[2126,2167,2163],[54,36515,11829],{"className":36516},[2336,2163],[54,36518,36519,36522],{"style":3862},[54,36520],{"className":36521,"style":3866},[2155],[54,36523,36524],{},[54,36525,36269],{"className":36526},[7126],[54,36528,2781],{"className":36529},[2780],[54,36531,36533],{"className":36532},[2143],[54,36534,36537],{"className":36535,"style":36536},[2147],"height:0.966em;",[54,36538],{},[54,36540],{"className":36541,"style":2521},[2270],[54,36543,2305],{"className":36544},[2126],[54,36546,13],{"className":36547},[2126,2167],[54,36549,129],{"className":36550},[2497],[54,36552,8695],{"className":36553},[2126,2167],[54,36555,3205],{"className":36556},[2336],[54,36558,2305],{"className":36559},[2126],[54,36561],{"className":36562,"style":9330},[2270],[13,36564,36565],{},"is",[54,36567,36569],{"className":36568},[2340],[54,36570,36572,36675],{"className":36571},[2079],[54,36573,36575],{"className":36574},[2083],[2085,36576,36577],{"xmlns":2087,"display":2349},[2089,36578,36579,36672],{},[2092,36580,36581,36587,36589,36591,36593,36595,36668,36670],{},[2095,36582,36583,36585],{},[2098,36584,13],{},[2245,36586,20325],{},[2245,36588,129],{"stretchy":2311},[2098,36590,8695],{},[2245,36592,3205],{"stretchy":2311},[2245,36594,122],{},[9047,36596,36597,36636],{},[2092,36598,36599,36606],{},[2724,36600,36601,36604],{},[2098,36602,36603],{},"T",[2098,36605,4515],{},[2092,36607,36608,36610,36634],{},[2245,36609,129],{"fence":2113},[9047,36611,36612,36626],{},[2092,36613,36614,36616,36618,36620,36622,36624],{},[2098,36615,12658],{},[2245,36617,1663],{},[2098,36619,20],{},[2245,36621,2247],{},[2249,36623,133],{},[2098,36625,8695],{},[2092,36627,36628,36630,36632],{},[2098,36629,12658],{},[2245,36631,2247],{},[2098,36633,20],{},[2245,36635,3205],{"fence":2113},[2092,36637,36638,36644],{},[2724,36639,36640,36642],{},[2098,36641,36603],{},[2098,36643,4515],{},[2092,36645,36646,36648,36666],{},[2245,36647,129],{"fence":2113},[9047,36649,36650,36658],{},[2092,36651,36652,36654,36656],{},[2098,36653,12658],{},[2245,36655,1663],{},[2098,36657,20],{},[2092,36659,36660,36662,36664],{},[2098,36661,12658],{},[2245,36663,2247],{},[2098,36665,20],{},[2245,36667,3205],{"fence":2113},[2245,36669,2375],{"separator":2113},[2270,36671],{"width":9011},[2105,36673,36674],{"encoding":2107},"p^*(x) = \\frac{T_d\\left(\\frac{b + a - 2x}{b - a}\\right)}{T_d\\left(\\frac{b + a}{b - a}\\right)}, \\quad",[54,36676,36678,36732],{"className":36677,"ariaHidden":2113},[2112],[54,36679,36681,36684,36714,36717,36720,36723,36726,36729],{"className":36680},[2117],[54,36682],{"className":36683,"style":2325},[2121],[54,36685,36687,36690],{"className":36686},[2126],[54,36688,13],{"className":36689},[2126,2167],[54,36691,36693],{"className":36692},[2135],[54,36694,36696],{"className":36695},[2139],[54,36697,36699],{"className":36698},[2143],[54,36700,36703],{"className":36701,"style":36702},[2147],"height:0.7387em;",[54,36704,36705,36708],{"style":2462},[54,36706],{"className":36707,"style":2156},[2155],[54,36709,36711],{"className":36710},[2160,2161,2162,2163],[54,36712,20325],{"className":36713},[2275,2163],[54,36715,129],{"className":36716},[2497],[54,36718,8695],{"className":36719},[2126,2167],[54,36721,3205],{"className":36722},[2336],[54,36724],{"className":36725,"style":2475},[2270],[54,36727,122],{"className":36728},[2479],[54,36730],{"className":36731,"style":2475},[2270],[54,36733,36735,36739,37088,37091],{"className":36734},[2117],[54,36736],{"className":36737,"style":36738},[2121],"height:2.8469em;vertical-align:-1.1734em;",[54,36740,36742,36745,37085],{"className":36741},[2126],[54,36743],{"className":36744},[2497,9349],[54,36746,36748],{"className":36747},[9047],[54,36749,36751,37076],{"className":36750},[2139,2756],[54,36752,36754,37073],{"className":36753},[2143],[54,36755,36758,36909,36917],{"className":36756,"style":36757},[2147],"height:1.6734em;",[54,36759,36761,36764],{"style":36760},"top:-2.2299em;",[54,36762],{"className":36763,"style":3866},[2155],[54,36765,36767,36807,36810],{"className":36766},[2126],[54,36768,36770,36773],{"className":36769},[2126],[54,36771,36603],{"className":36772,"style":2776},[2126,2167],[54,36774,36776],{"className":36775},[2135],[54,36777,36779,36799],{"className":36778},[2139,2756],[54,36780,36782,36796],{"className":36781},[2143],[54,36783,36785],{"className":36784,"style":10799},[2147],[54,36786,36787,36790],{"style":7193},[54,36788],{"className":36789,"style":2156},[2155],[54,36791,36793],{"className":36792},[2160,2161,2162,2163],[54,36794,4515],{"className":36795},[2126,2167,2163],[54,36797,2781],{"className":36798},[2780],[54,36800,36802],{"className":36801},[2143],[54,36803,36805],{"className":36804,"style":2788},[2147],[54,36806],{},[54,36808],{"className":36809,"style":2521},[2270],[54,36811,36813,36819,36903],{"className":36812},[2493],[54,36814,36816],{"className":36815,"style":2499},[2497,2498],[54,36817,129],{"className":36818},[2503,2504],[54,36820,36822,36825,36900],{"className":36821},[2126],[54,36823],{"className":36824},[2497,9349],[54,36826,36828],{"className":36827},[9047],[54,36829,36831,36891],{"className":36830},[2139,2756],[54,36832,36834,36888],{"className":36833},[2143],[54,36835,36838,36859,36867],{"className":36836,"style":36837},[2147],"height:0.8801em;",[54,36839,36841,36844],{"style":36840},"top:-2.655em;",[54,36842],{"className":36843,"style":3866},[2155],[54,36845,36847],{"className":36846},[2160,2161,2162,2163],[54,36848,36850,36853,36856],{"className":36849},[2126,2163],[54,36851,12658],{"className":36852},[2126,2167,2163],[54,36854,2247],{"className":36855},[2275,2163],[54,36857,20],{"className":36858},[2126,2167,2163],[54,36860,36861,36864],{"style":9434},[54,36862],{"className":36863,"style":3866},[2155],[54,36865],{"className":36866,"style":9442},[9441],[54,36868,36870,36873],{"style":36869},"top:-3.394em;",[54,36871],{"className":36872,"style":3866},[2155],[54,36874,36876],{"className":36875},[2160,2161,2162,2163],[54,36877,36879,36882,36885],{"className":36878},[2126,2163],[54,36880,12658],{"className":36881},[2126,2167,2163],[54,36883,1663],{"className":36884},[2275,2163],[54,36886,20],{"className":36887},[2126,2167,2163],[54,36889,2781],{"className":36890},[2780],[54,36892,36894],{"className":36893},[2143],[54,36895,36898],{"className":36896,"style":36897},[2147],"height:0.4033em;",[54,36899],{},[54,36901],{"className":36902},[2336,9349],[54,36904,36906],{"className":36905,"style":2499},[2336,2498],[54,36907,3205],{"className":36908},[2503,2504],[54,36910,36911,36914],{"style":9434},[54,36912],{"className":36913,"style":3866},[2155],[54,36915],{"className":36916,"style":9442},[9441],[54,36918,36920,36923],{"style":36919},"top:-3.7933em;",[54,36921],{"className":36922,"style":3866},[2155],[54,36924,36926,36966,36969],{"className":36925},[2126],[54,36927,36929,36932],{"className":36928},[2126],[54,36930,36603],{"className":36931,"style":2776},[2126,2167],[54,36933,36935],{"className":36934},[2135],[54,36936,36938,36958],{"className":36937},[2139,2756],[54,36939,36941,36955],{"className":36940},[2143],[54,36942,36944],{"className":36943,"style":10799},[2147],[54,36945,36946,36949],{"style":7193},[54,36947],{"className":36948,"style":2156},[2155],[54,36950,36952],{"className":36951},[2160,2161,2162,2163],[54,36953,4515],{"className":36954},[2126,2167,2163],[54,36956,2781],{"className":36957},[2780],[54,36959,36961],{"className":36960},[2143],[54,36962,36964],{"className":36963,"style":2788},[2147],[54,36965],{},[54,36967],{"className":36968,"style":2521},[2270],[54,36970,36972,36978,37067],{"className":36971},[2493],[54,36973,36975],{"className":36974,"style":2499},[2497,2498],[54,36976,129],{"className":36977},[2503,2504],[54,36979,36981,36984,37064],{"className":36980},[2126],[54,36982],{"className":36983},[2497,9349],[54,36985,36987],{"className":36986},[9047],[54,36988,36990,37056],{"className":36989},[2139,2756],[54,36991,36993,37053],{"className":36992},[2143],[54,36994,36996,37016,37024],{"className":36995,"style":36837},[2147],[54,36997,36998,37001],{"style":36840},[54,36999],{"className":37000,"style":3866},[2155],[54,37002,37004],{"className":37003},[2160,2161,2162,2163],[54,37005,37007,37010,37013],{"className":37006},[2126,2163],[54,37008,12658],{"className":37009},[2126,2167,2163],[54,37011,2247],{"className":37012},[2275,2163],[54,37014,20],{"className":37015},[2126,2167,2163],[54,37017,37018,37021],{"style":9434},[54,37019],{"className":37020,"style":3866},[2155],[54,37022],{"className":37023,"style":9442},[9441],[54,37025,37026,37029],{"style":36869},[54,37027],{"className":37028,"style":3866},[2155],[54,37030,37032],{"className":37031},[2160,2161,2162,2163],[54,37033,37035,37038,37041,37044,37047,37050],{"className":37034},[2126,2163],[54,37036,12658],{"className":37037},[2126,2167,2163],[54,37039,1663],{"className":37040},[2275,2163],[54,37042,20],{"className":37043},[2126,2167,2163],[54,37045,2247],{"className":37046},[2275,2163],[54,37048,133],{"className":37049},[2126,2163],[54,37051,8695],{"className":37052},[2126,2167,2163],[54,37054,2781],{"className":37055},[2780],[54,37057,37059],{"className":37058},[2143],[54,37060,37062],{"className":37061,"style":36897},[2147],[54,37063],{},[54,37065],{"className":37066},[2336,9349],[54,37068,37070],{"className":37069,"style":2499},[2336,2498],[54,37071,3205],{"className":37072},[2503,2504],[54,37074,2781],{"className":37075},[2780],[54,37077,37079],{"className":37078},[2143],[54,37080,37083],{"className":37081,"style":37082},[2147],"height:1.1734em;",[54,37084],{},[54,37086],{"className":37087},[2336,9349],[54,37089,2375],{"className":37090},[2517],[54,37092],{"className":37093,"style":9330},[2270],[13,37095,37096],{},"and the corresponding minimal value is",[54,37098,37100],{"className":37099},[2340],[54,37101,37103,37171],{"className":37102},[2079],[54,37104,37106],{"className":37105},[2083],[2085,37107,37108],{"xmlns":2087,"display":2349},[2089,37109,37110,37168],{},[2092,37111,37112,37114,37116,37118,37120,37122,37124,37126,37128,37130,37142,37166],{},[2098,37113,36222],{},[2245,37115,129],{"stretchy":2311},[2098,37117,20],{},[2245,37119,2375],{"separator":2113},[2098,37121,12658],{},[2245,37123,2375],{"separator":2113},[2098,37125,4515],{},[2245,37127,3205],{"stretchy":2311},[2245,37129,122],{},[3331,37131,37132,37134,37136],{},[2098,37133,36603],{},[2098,37135,4515],{},[2092,37137,37138,37140],{},[2245,37139,2247],{},[2249,37141,174],{},[2092,37143,37144,37146,37164],{},[2245,37145,129],{"fence":2113},[9047,37147,37148,37156],{},[2092,37149,37150,37152,37154],{},[2098,37151,12658],{},[2245,37153,1663],{},[2098,37155,20],{},[2092,37157,37158,37160,37162],{},[2098,37159,12658],{},[2245,37161,2247],{},[2098,37163,20],{},[2245,37165,3205],{"fence":2113},[2098,37167,29],{"mathvariant":2304},[2105,37169,37170],{"encoding":2107},"\\beta(a, b, d) = T_d^{-1}\\left(\\frac{b + a}{b - a}\\right).",[54,37172,37174,37219],{"className":37173,"ariaHidden":2113},[2112],[54,37175,37177,37180,37183,37186,37189,37192,37195,37198,37201,37204,37207,37210,37213,37216],{"className":37176},[2117],[54,37178],{"className":37179,"style":2325},[2121],[54,37181,36222],{"className":37182,"style":36317},[2126,2167],[54,37184,129],{"className":37185},[2497],[54,37187,20],{"className":37188},[2126,2167],[54,37190,2375],{"className":37191},[2517],[54,37193],{"className":37194,"style":2521},[2270],[54,37196,12658],{"className":37197},[2126,2167],[54,37199,2375],{"className":37200},[2517],[54,37202],{"className":37203,"style":2521},[2270],[54,37205,4515],{"className":37206},[2126,2167],[54,37208,3205],{"className":37209},[2336],[54,37211],{"className":37212,"style":2475},[2270],[54,37214,122],{"className":37215},[2479],[54,37217],{"className":37218,"style":2475},[2270],[54,37220,37222,37226,37285,37288,37391,37394],{"className":37221},[2117],[54,37223],{"className":37224,"style":37225},[2121],"height:2.4em;vertical-align:-0.95em;",[54,37227,37229,37232],{"className":37228},[2126],[54,37230,36603],{"className":37231,"style":2776},[2126,2167],[54,37233,37235],{"className":37234},[2135],[54,37236,37238,37276],{"className":37237},[2139,2756],[54,37239,37241,37273],{"className":37240},[2143],[54,37242,37244,37256],{"className":37243,"style":2558},[2147],[54,37245,37247,37250],{"style":37246},"top:-2.4086em;margin-left:-0.1389em;margin-right:0.05em;",[54,37248],{"className":37249,"style":2156},[2155],[54,37251,37253],{"className":37252},[2160,2161,2162,2163],[54,37254,4515],{"className":37255},[2126,2167,2163],[54,37257,37258,37261],{"style":2462},[54,37259],{"className":37260,"style":2156},[2155],[54,37262,37264],{"className":37263},[2160,2161,2162,2163],[54,37265,37267,37270],{"className":37266},[2126,2163],[54,37268,2247],{"className":37269},[2126,2163],[54,37271,174],{"className":37272},[2126,2163],[54,37274,2781],{"className":37275},[2780],[54,37277,37279],{"className":37278},[2143],[54,37280,37283],{"className":37281,"style":37282},[2147],"height:0.2914em;",[54,37284],{},[54,37286],{"className":37287,"style":2521},[2270],[54,37289,37291,37297,37385],{"className":37290},[2493],[54,37292,37294],{"className":37293,"style":2499},[2497,2498],[54,37295,129],{"className":37296},[2503,2162],[54,37298,37300,37303,37382],{"className":37299},[2126],[54,37301],{"className":37302},[2497,9349],[54,37304,37306],{"className":37305},[9047],[54,37307,37309,37373],{"className":37308},[2139,2756],[54,37310,37312,37370],{"className":37311},[2143],[54,37313,37316,37339,37347],{"className":37314,"style":37315},[2147],"height:1.3714em;",[54,37317,37318,37321],{"style":11273},[54,37319],{"className":37320,"style":3866},[2155],[54,37322,37324,37327,37330,37333,37336],{"className":37323},[2126],[54,37325,12658],{"className":37326},[2126,2167],[54,37328],{"className":37329,"style":2271},[2270],[54,37331,2247],{"className":37332},[2275],[54,37334],{"className":37335,"style":2271},[2270],[54,37337,20],{"className":37338},[2126,2167],[54,37340,37341,37344],{"style":9434},[54,37342],{"className":37343,"style":3866},[2155],[54,37345],{"className":37346,"style":9442},[9441],[54,37348,37349,37352],{"style":9445},[54,37350],{"className":37351,"style":3866},[2155],[54,37353,37355,37358,37361,37364,37367],{"className":37354},[2126],[54,37356,12658],{"className":37357},[2126,2167],[54,37359],{"className":37360,"style":2271},[2270],[54,37362,1663],{"className":37363},[2275],[54,37365],{"className":37366,"style":2271},[2270],[54,37368,20],{"className":37369},[2126,2167],[54,37371,2781],{"className":37372},[2780],[54,37374,37376],{"className":37375},[2143],[54,37377,37380],{"className":37378,"style":37379},[2147],"height:0.7693em;",[54,37381],{},[54,37383],{"className":37384},[2336,9349],[54,37386,37388],{"className":37387,"style":2499},[2336,2498],[54,37389,3205],{"className":37390},[2503,2162],[54,37392],{"className":37393,"style":2521},[2270],[54,37395,29],{"className":37396},[2126],[13,37398,37399,37400,37452],{},"We want to convert this into a function that can be expressed naturally in terms of complex exponentials, because those are the real time-evolutions that generate the quantum Krylov space.\nTo do so, it is convenient to introduce the following transformation of energies within the spectral range of the Hamiltonian to numbers in the range ",[54,37401,37403,37425],{"className":37402},[2079],[54,37404,37406],{"className":37405},[2083],[2085,37407,37408],{"xmlns":2087},[2089,37409,37410,37422],{},[2092,37411,37412,37414,37416,37418,37420],{},[2245,37413,11804],{"stretchy":2311},[2249,37415,150],{},[2245,37417,2375],{"separator":2113},[2249,37419,174],{},[2245,37421,11829],{"stretchy":2311},[2105,37423,37424],{"encoding":2107},"[0,1]",[54,37426,37428],{"className":37427,"ariaHidden":2113},[2112],[54,37429,37431,37434,37437,37440,37443,37446,37449],{"className":37430},[2117],[54,37432],{"className":37433,"style":2325},[2121],[54,37435,11804],{"className":37436},[2497],[54,37438,150],{"className":37439},[2126],[54,37441,2375],{"className":37442},[2517],[54,37444],{"className":37445,"style":2521},[2270],[54,37447,174],{"className":37448},[2126],[54,37450,11829],{"className":37451},[2336],": define",[54,37454,37456],{"className":37455},[2340],[54,37457,37459,37522],{"className":37458},[2079],[54,37460,37462],{"className":37461},[2083],[2085,37463,37464],{"xmlns":2087,"display":2349},[2089,37465,37466,37519],{},[2092,37467,37468,37471,37473,37475,37477,37479,37517],{},[2098,37469,37470],{},"g",[2245,37472,129],{"stretchy":2311},[2098,37474,5738],{},[2245,37476,3205],{"stretchy":2311},[2245,37478,122],{},[9047,37480,37481,37515],{},[2092,37482,37483,37485,37487,37490,37492,37495,37497,37499,37501,37507,37509,37511,37513],{},[2249,37484,174],{},[2245,37486,2247],{},[2098,37488,37489],{},"cos",[2245,37491,36247],{},[2245,37493,129],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},"1.2em",[2245,37496,129],{"stretchy":2311},[2098,37498,5738],{},[2245,37500,2247],{},[2724,37502,37503,37505],{},[2098,37504,5738],{},[2249,37506,150],{},[2245,37508,3205],{"stretchy":2311},[2098,37510,4515],{},[2098,37512,2854],{},[2245,37514,3205],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2249,37516,133],{},[2245,37518,2375],{"separator":2113},[2105,37520,37521],{"encoding":2107},"g(E) = \\frac{1-\\cos\\big((E-E_0)dt\\big)}{2},",[54,37523,37525,37552],{"className":37524,"ariaHidden":2113},[2112],[54,37526,37528,37531,37534,37537,37540,37543,37546,37549],{"className":37527},[2117],[54,37529],{"className":37530,"style":2325},[2121],[54,37532,37470],{"className":37533,"style":2332},[2126,2167],[54,37535,129],{"className":37536},[2497],[54,37538,5738],{"className":37539,"style":5448},[2126,2167],[54,37541,3205],{"className":37542},[2336],[54,37544],{"className":37545,"style":2475},[2270],[54,37547,122],{"className":37548},[2479],[54,37550],{"className":37551,"style":2475},[2270],[54,37553,37555,37559,37714],{"className":37554},[2117],[54,37556],{"className":37557,"style":37558},[2121],"height:2.276em;vertical-align:-0.686em;",[54,37560,37562,37565,37711],{"className":37561},[2126],[54,37563],{"className":37564},[2497,9349],[54,37566,37568],{"className":37567},[9047],[54,37569,37571,37703],{"className":37570},[2139,2756],[54,37572,37574,37700],{"className":37573},[2143],[54,37575,37578,37589,37597],{"className":37576,"style":37577},[2147],"height:1.59em;",[54,37579,37580,37583],{"style":11273},[54,37581],{"className":37582,"style":3866},[2155],[54,37584,37586],{"className":37585},[2126],[54,37587,133],{"className":37588},[2126],[54,37590,37591,37594],{"style":9434},[54,37592],{"className":37593,"style":3866},[2155],[54,37595],{"className":37596,"style":9442},[9441],[54,37598,37600,37603],{"style":37599},"top:-3.74em;",[54,37601],{"className":37602,"style":3866},[2155],[54,37604,37606,37609,37612,37615,37618,37621,37624,37630,37633,37636,37639,37642,37645,37685,37688,37691,37694],{"className":37605},[2126],[54,37607,174],{"className":37608},[2126],[54,37610],{"className":37611,"style":2271},[2270],[54,37613,2247],{"className":37614},[2275],[54,37616],{"className":37617,"style":2271},[2270],[54,37619,37489],{"className":37620},[7126],[54,37622],{"className":37623,"style":2521},[2270],[54,37625,37627],{"className":37626},[2126],[54,37628,129],{"className":37629},[2503,2504],[54,37631,129],{"className":37632},[2497],[54,37634,5738],{"className":37635,"style":5448},[2126,2167],[54,37637],{"className":37638,"style":2271},[2270],[54,37640,2247],{"className":37641},[2275],[54,37643],{"className":37644,"style":2271},[2270],[54,37646,37648,37651],{"className":37647},[2126],[54,37649,5738],{"className":37650,"style":5448},[2126,2167],[54,37652,37654],{"className":37653},[2135],[54,37655,37657,37677],{"className":37656},[2139,2756],[54,37658,37660,37674],{"className":37659},[2143],[54,37661,37663],{"className":37662,"style":13837},[2147],[54,37664,37665,37668],{"style":5570},[54,37666],{"className":37667,"style":2156},[2155],[54,37669,37671],{"className":37670},[2160,2161,2162,2163],[54,37672,150],{"className":37673},[2126,2163],[54,37675,2781],{"className":37676},[2780],[54,37678,37680],{"className":37679},[2143],[54,37681,37683],{"className":37682,"style":2788},[2147],[54,37684],{},[54,37686,3205],{"className":37687},[2336],[54,37689,4515],{"className":37690},[2126,2167],[54,37692,2854],{"className":37693},[2126,2167],[54,37695,37697],{"className":37696},[2126],[54,37698,3205],{"className":37699},[2503,2504],[54,37701,2781],{"className":37702},[2780],[54,37704,37706],{"className":37705},[2143],[54,37707,37709],{"className":37708,"style":11311},[2147],[54,37710],{},[54,37712],{"className":37713},[2336,9349],[54,37715,2375],{"className":37716},[2517],[13,37718,28011,37719,37752,37753,37959,37960,10355,38067,38111,38112,38140,38141,29],{},[54,37720,37722,37737],{"className":37721},[2079],[54,37723,37725],{"className":37724},[2083],[2085,37726,37727],{"xmlns":2087},[2089,37728,37729,37735],{},[2092,37730,37731,37733],{},[2098,37732,4515],{},[2098,37734,2854],{},[2105,37736,5019],{"encoding":2107},[54,37738,37740],{"className":37739,"ariaHidden":2113},[2112],[54,37741,37743,37746,37749],{"className":37742},[2117],[54,37744],{"className":37745,"style":4990},[2121],[54,37747,4515],{"className":37748},[2126,2167],[54,37750,2854],{"className":37751},[2126,2167]," is a timestep such that ",[54,37754,37756,37800],{"className":37755},[2079],[54,37757,37759],{"className":37758},[2083],[2085,37760,37761],{"xmlns":2087},[2089,37762,37763,37797],{},[2092,37764,37765,37767,37769,37771,37777,37779,37781,37783,37789,37791,37793,37795],{},[2245,37766,2247],{},[2098,37768,8119],{},[2245,37770,5233],{},[2724,37772,37773,37775],{},[2098,37774,5738],{},[2249,37776,150],{},[2098,37778,4515],{},[2098,37780,2854],{},[2245,37782,5233],{},[2724,37784,37785,37787],{},[2098,37786,5738],{},[5723,37788,36269],{},[2098,37790,4515],{},[2098,37792,2854],{},[2245,37794,5233],{},[2098,37796,8119],{},[2105,37798,37799],{"encoding":2107},"-\\pi \u003C E_0dt \u003C E_\\text{max}dt \u003C \\pi",[54,37801,37803,37824,37886,37950],{"className":37802,"ariaHidden":2113},[2112],[54,37804,37806,37809,37812,37815,37818,37821],{"className":37805},[2117],[54,37807],{"className":37808,"style":2263},[2121],[54,37810,2247],{"className":37811},[2126],[54,37813,8119],{"className":37814,"style":2332},[2126,2167],[54,37816],{"className":37817,"style":2475},[2270],[54,37819,5233],{"className":37820},[2479],[54,37822],{"className":37823,"style":2475},[2270],[54,37825,37827,37831,37871,37874,37877,37880,37883],{"className":37826},[2117],[54,37828],{"className":37829,"style":37830},[2121],"height:0.8444em;vertical-align:-0.15em;",[54,37832,37834,37837],{"className":37833},[2126],[54,37835,5738],{"className":37836,"style":5448},[2126,2167],[54,37838,37840],{"className":37839},[2135],[54,37841,37843,37863],{"className":37842},[2139,2756],[54,37844,37846,37860],{"className":37845},[2143],[54,37847,37849],{"className":37848,"style":13837},[2147],[54,37850,37851,37854],{"style":5570},[54,37852],{"className":37853,"style":2156},[2155],[54,37855,37857],{"className":37856},[2160,2161,2162,2163],[54,37858,150],{"className":37859},[2126,2163],[54,37861,2781],{"className":37862},[2780],[54,37864,37866],{"className":37865},[2143],[54,37867,37869],{"className":37868,"style":2788},[2147],[54,37870],{},[54,37872,4515],{"className":37873},[2126,2167],[54,37875,2854],{"className":37876},[2126,2167],[54,37878],{"className":37879,"style":2475},[2270],[54,37881,5233],{"className":37882},[2479],[54,37884],{"className":37885,"style":2475},[2270],[54,37887,37889,37892,37935,37938,37941,37944,37947],{"className":37888},[2117],[54,37890],{"className":37891,"style":37830},[2121],[54,37893,37895,37898],{"className":37894},[2126],[54,37896,5738],{"className":37897,"style":5448},[2126,2167],[54,37899,37901],{"className":37900},[2135],[54,37902,37904,37927],{"className":37903},[2139,2756],[54,37905,37907,37924],{"className":37906},[2143],[54,37908,37910],{"className":37909,"style":4063},[2147],[54,37911,37912,37915],{"style":5570},[54,37913],{"className":37914,"style":2156},[2155],[54,37916,37918],{"className":37917},[2160,2161,2162,2163],[54,37919,37921],{"className":37920},[2126,6595,2163],[54,37922,36269],{"className":37923},[2126,2163],[54,37925,2781],{"className":37926},[2780],[54,37928,37930],{"className":37929},[2143],[54,37931,37933],{"className":37932,"style":2788},[2147],[54,37934],{},[54,37936,4515],{"className":37937},[2126,2167],[54,37939,2854],{"className":37940},[2126,2167],[54,37942],{"className":37943,"style":2475},[2270],[54,37945,5233],{"className":37946},[2479],[54,37948],{"className":37949,"style":2475},[2270],[54,37951,37953,37956],{"className":37952},[2117],[54,37954],{"className":37955,"style":2195},[2121],[54,37957,8119],{"className":37958,"style":2332},[2126,2167],".\nNotice that ",[54,37961,37963,37991],{"className":37962},[2079],[54,37964,37966],{"className":37965},[2083],[2085,37967,37968],{"xmlns":2087},[2089,37969,37970,37988],{},[2092,37971,37972,37974,37976,37982,37984,37986],{},[2098,37973,37470],{},[2245,37975,129],{"stretchy":2311},[2724,37977,37978,37980],{},[2098,37979,5738],{},[2249,37981,150],{},[2245,37983,3205],{"stretchy":2311},[2245,37985,122],{},[2249,37987,150],{},[2105,37989,37990],{"encoding":2107},"g(E_0)=0",[54,37992,37994,38058],{"className":37993,"ariaHidden":2113},[2112],[54,37995,37997,38000,38003,38006,38046,38049,38052,38055],{"className":37996},[2117],[54,37998],{"className":37999,"style":2325},[2121],[54,38001,37470],{"className":38002,"style":2332},[2126,2167],[54,38004,129],{"className":38005},[2497],[54,38007,38009,38012],{"className":38008},[2126],[54,38010,5738],{"className":38011,"style":5448},[2126,2167],[54,38013,38015],{"className":38014},[2135],[54,38016,38018,38038],{"className":38017},[2139,2756],[54,38019,38021,38035],{"className":38020},[2143],[54,38022,38024],{"className":38023,"style":13837},[2147],[54,38025,38026,38029],{"style":5570},[54,38027],{"className":38028,"style":2156},[2155],[54,38030,38032],{"className":38031},[2160,2161,2162,2163],[54,38033,150],{"className":38034},[2126,2163],[54,38036,2781],{"className":38037},[2780],[54,38039,38041],{"className":38040},[2143],[54,38042,38044],{"className":38043,"style":2788},[2147],[54,38045],{},[54,38047,3205],{"className":38048},[2336],[54,38050],{"className":38051,"style":2475},[2270],[54,38053,122],{"className":38054},[2479],[54,38056],{"className":38057,"style":2475},[2270],[54,38059,38061,38064],{"className":38060},[2117],[54,38062],{"className":38063,"style":2285},[2121],[54,38065,150],{"className":38066},[2126],[54,38068,38070,38090],{"className":38069},[2079],[54,38071,38073],{"className":38072},[2083],[2085,38074,38075],{"xmlns":2087},[2089,38076,38077,38087],{},[2092,38078,38079,38081,38083,38085],{},[2098,38080,37470],{},[2245,38082,129],{"stretchy":2311},[2098,38084,5738],{},[2245,38086,3205],{"stretchy":2311},[2105,38088,38089],{"encoding":2107},"g(E)",[54,38091,38093],{"className":38092,"ariaHidden":2113},[2112],[54,38094,38096,38099,38102,38105,38108],{"className":38095},[2117],[54,38097],{"className":38098,"style":2325},[2121],[54,38100,37470],{"className":38101,"style":2332},[2126,2167],[54,38103,129],{"className":38104},[2497],[54,38106,5738],{"className":38107,"style":5448},[2126,2167],[54,38109,3205],{"className":38110},[2336]," grows as ",[54,38113,38115,38128],{"className":38114},[2079],[54,38116,38118],{"className":38117},[2083],[2085,38119,38120],{"xmlns":2087},[2089,38121,38122,38126],{},[2092,38123,38124],{},[2098,38125,5738],{},[2105,38127,5738],{"encoding":2107},[54,38129,38131],{"className":38130,"ariaHidden":2113},[2112],[54,38132,38134,38137],{"className":38133},[2117],[54,38135],{"className":38136,"style":2122},[2121],[54,38138,5738],{"className":38139,"style":5448},[2126,2167]," moves away from ",[54,38142,38144,38161],{"className":38143},[2079],[54,38145,38147],{"className":38146},[2083],[2085,38148,38149],{"xmlns":2087},[2089,38150,38151,38159],{},[2092,38152,38153],{},[2724,38154,38155,38157],{},[2098,38156,5738],{},[2249,38158,150],{},[2105,38160,13809],{"encoding":2107},[54,38162,38164],{"className":38163,"ariaHidden":2113},[2112],[54,38165,38167,38170],{"className":38166},[2117],[54,38168],{"className":38169,"style":2743},[2121],[54,38171,38173,38176],{"className":38172},[2126],[54,38174,5738],{"className":38175,"style":5448},[2126,2167],[54,38177,38179],{"className":38178},[2135],[54,38180,38182,38202],{"className":38181},[2139,2756],[54,38183,38185,38199],{"className":38184},[2143],[54,38186,38188],{"className":38187,"style":13837},[2147],[54,38189,38190,38193],{"style":5570},[54,38191],{"className":38192,"style":2156},[2155],[54,38194,38196],{"className":38195},[2160,2161,2162,2163],[54,38197,150],{"className":38198},[2126,2163],[54,38200,2781],{"className":38201},[2780],[54,38203,38205],{"className":38204},[2143],[54,38206,38208],{"className":38207,"style":2788},[2147],[54,38209],{},[13,38211,38212,38213,38287,38288,171,38417,38468],{},"Now using the polynomial ",[54,38214,38216,38240],{"className":38215},[2079],[54,38217,38219],{"className":38218},[2083],[2085,38220,38221],{"xmlns":2087},[2089,38222,38223,38237],{},[2092,38224,38225,38231,38233,38235],{},[2095,38226,38227,38229],{},[2098,38228,13],{},[2245,38230,20325],{},[2245,38232,129],{"stretchy":2311},[2098,38234,8695],{},[2245,38236,3205],{"stretchy":2311},[2105,38238,38239],{"encoding":2107},"p^*(x)",[54,38241,38243],{"className":38242,"ariaHidden":2113},[2112],[54,38244,38246,38249,38278,38281,38284],{"className":38245},[2117],[54,38247],{"className":38248,"style":2325},[2121],[54,38250,38252,38255],{"className":38251},[2126],[54,38253,13],{"className":38254},[2126,2167],[54,38256,38258],{"className":38257},[2135],[54,38259,38261],{"className":38260},[2139],[54,38262,38264],{"className":38263},[2143],[54,38265,38267],{"className":38266,"style":36139},[2147],[54,38268,38269,38272],{"style":2151},[54,38270],{"className":38271,"style":2156},[2155],[54,38273,38275],{"className":38274},[2160,2161,2162,2163],[54,38276,20325],{"className":38277},[2275,2163],[54,38279,129],{"className":38280},[2497],[54,38282,8695],{"className":38283},[2126,2167],[54,38285,3205],{"className":38286},[2336]," with the parameters a, b, d set to ",[54,38289,38291,38323],{"className":38290},[2079],[54,38292,38294],{"className":38293},[2083],[2085,38295,38296],{"xmlns":2087},[2089,38297,38298,38320],{},[2092,38299,38300,38302,38304,38306,38308,38314,38316,38318],{},[2098,38301,20],{},[2245,38303,122],{},[2098,38305,37470],{},[2245,38307,129],{"stretchy":2311},[2724,38309,38310,38312],{},[2098,38311,5738],{},[2249,38313,150],{},[2245,38315,1663],{},[2098,38317,27081],{},[2245,38319,3205],{"stretchy":2311},[2105,38321,38322],{"encoding":2107},"a = g(E_0 + \\delta)",[54,38324,38326,38344,38405],{"className":38325,"ariaHidden":2113},[2112],[54,38327,38329,38332,38335,38338,38341],{"className":38328},[2117],[54,38330],{"className":38331,"style":2195},[2121],[54,38333,20],{"className":38334},[2126,2167],[54,38336],{"className":38337,"style":2475},[2270],[54,38339,122],{"className":38340},[2479],[54,38342],{"className":38343,"style":2475},[2270],[54,38345,38347,38350,38353,38356,38396,38399,38402],{"className":38346},[2117],[54,38348],{"className":38349,"style":2325},[2121],[54,38351,37470],{"className":38352,"style":2332},[2126,2167],[54,38354,129],{"className":38355},[2497],[54,38357,38359,38362],{"className":38358},[2126],[54,38360,5738],{"className":38361,"style":5448},[2126,2167],[54,38363,38365],{"className":38364},[2135],[54,38366,38368,38388],{"className":38367},[2139,2756],[54,38369,38371,38385],{"className":38370},[2143],[54,38372,38374],{"className":38373,"style":13837},[2147],[54,38375,38376,38379],{"style":5570},[54,38377],{"className":38378,"style":2156},[2155],[54,38380,38382],{"className":38381},[2160,2161,2162,2163],[54,38383,150],{"className":38384},[2126,2163],[54,38386,2781],{"className":38387},[2780],[54,38389,38391],{"className":38390},[2143],[54,38392,38394],{"className":38393,"style":2788},[2147],[54,38395],{},[54,38397],{"className":38398,"style":2271},[2270],[54,38400,1663],{"className":38401},[2275],[54,38403],{"className":38404,"style":2271},[2270],[54,38406,38408,38411,38414],{"className":38407},[2117],[54,38409],{"className":38410,"style":2325},[2121],[54,38412,27081],{"className":38413,"style":27166},[2126,2167],[54,38415,3205],{"className":38416},[2336],[54,38418,38420,38438],{"className":38419},[2079],[54,38421,38423],{"className":38422},[2083],[2085,38424,38425],{"xmlns":2087},[2089,38426,38427,38435],{},[2092,38428,38429,38431,38433],{},[2098,38430,12658],{},[2245,38432,122],{},[2249,38434,174],{},[2105,38436,38437],{"encoding":2107},"b = 1",[54,38439,38441,38459],{"className":38440,"ariaHidden":2113},[2112],[54,38442,38444,38447,38450,38453,38456],{"className":38443},[2117],[54,38445],{"className":38446,"style":4990},[2121],[54,38448,12658],{"className":38449},[2126,2167],[54,38451],{"className":38452,"style":2475},[2270],[54,38454,122],{"className":38455},[2479],[54,38457],{"className":38458,"style":2475},[2270],[54,38460,38462,38465],{"className":38461},[2117],[54,38463],{"className":38464,"style":2285},[2121],[54,38466,174],{"className":38467},[2126],", and d = int(r\u002F2), we define the function:",[54,38470,38472],{"className":38471},[2340],[54,38473,38475,38677],{"className":38474},[2079],[54,38476,38478],{"className":38477},[2083],[2085,38479,38480],{"xmlns":2087,"display":2349},[2089,38481,38482,38674],{},[2092,38483,38484,38486,38488,38490,38492,38494,38500,38514,38516],{},[2098,38485,21834],{},[2245,38487,129],{"stretchy":2311},[2098,38489,5738],{},[2245,38491,3205],{"stretchy":2311},[2245,38493,122],{},[2095,38495,38496,38498],{},[2098,38497,13],{},[2245,38499,20325],{},[2092,38501,38502,38504,38506,38508,38510,38512],{},[2245,38503,129],{"fence":2113},[2098,38505,37470],{},[2245,38507,129],{"stretchy":2311},[2098,38509,5738],{},[2245,38511,3205],{"stretchy":2311},[2245,38513,3205],{"fence":2113},[2245,38515,122],{},[9047,38517,38518,38608],{},[2092,38519,38520,38526],{},[2724,38521,38522,38524],{},[2098,38523,36603],{},[2098,38525,4515],{},[2092,38527,38528,38530,38532,38534,38536,38606],{},[2245,38529,129],{"fence":2113},[2249,38531,174],{},[2245,38533,1663],{},[2249,38535,133],{},[9047,38537,38538,38584],{},[2092,38539,38540,38542,38544,38546,38548,38550,38552,38558,38560,38562,38564,38566,38568,38570,38572,38574,38576,38578,38580,38582],{},[2098,38541,37489],{},[2245,38543,36247],{},[2245,38545,129],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2245,38547,129],{"stretchy":2311},[2098,38549,5738],{},[2245,38551,2247],{},[2724,38553,38554,38556],{},[2098,38555,5738],{},[2249,38557,150],{},[2245,38559,3205],{"stretchy":2311},[2098,38561,4515],{},[2098,38563,2854],{},[2245,38565,3205],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2245,38567,2247],{},[2098,38569,37489],{},[2245,38571,36247],{},[2245,38573,129],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2098,38575,27081],{},[5723,38577,25244],{},[2098,38579,4515],{},[2098,38581,2854],{},[2245,38583,3205],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2092,38585,38586,38588,38590,38592,38594,38596,38598,38600,38602,38604],{},[2249,38587,174],{},[2245,38589,1663],{},[2098,38591,37489],{},[2245,38593,36247],{},[2245,38595,129],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2098,38597,27081],{},[5723,38599,25244],{},[2098,38601,4515],{},[2098,38603,2854],{},[2245,38605,3205],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2245,38607,3205],{"fence":2113},[2092,38609,38610,38616],{},[2724,38611,38612,38614],{},[2098,38613,36603],{},[2098,38615,4515],{},[2092,38617,38618,38620,38622,38624,38626,38672],{},[2245,38619,129],{"fence":2113},[2249,38621,174],{},[2245,38623,1663],{},[2249,38625,133],{},[9047,38627,38628,38650],{},[2092,38629,38630,38632,38634,38636,38638,38640,38642,38644,38646,38648],{},[2249,38631,174],{},[2245,38633,2247],{},[2098,38635,37489],{},[2245,38637,36247],{},[2245,38639,129],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2098,38641,27081],{},[5723,38643,25244],{},[2098,38645,4515],{},[2098,38647,2854],{},[2245,38649,3205],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2092,38651,38652,38654,38656,38658,38660,38662,38664,38666,38668,38670],{},[2249,38653,174],{},[2245,38655,1663],{},[2098,38657,37489],{},[2245,38659,36247],{},[2245,38661,129],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2098,38663,27081],{},[5723,38665,25244],{},[2098,38667,4515],{},[2098,38669,2854],{},[2245,38671,3205],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2245,38673,3205],{"fence":2113},[2105,38675,38676],{"encoding":2107},"f(E) = p^* \\left( g(E) \\right) = \\frac{T_d\\left(1 + 2\\frac{\\cos\\big((E-E_0)dt\\big) - \\cos\\big(\\delta\\,dt\\big)}{1 +\\cos\\big(\\delta\\,dt\\big)}\\right)}{T_d\\left(1 + 2\\frac{1-\\cos\\big(\\delta\\,dt\\big)}{1 + \\cos\\big(\\delta\\,dt\\big)}\\right)}",[54,38678,38680,38707,38775],{"className":38679,"ariaHidden":2113},[2112],[54,38681,38683,38686,38689,38692,38695,38698,38701,38704],{"className":38682},[2117],[54,38684],{"className":38685,"style":2325},[2121],[54,38687,21834],{"className":38688,"style":26895},[2126,2167],[54,38690,129],{"className":38691},[2497],[54,38693,5738],{"className":38694,"style":5448},[2126,2167],[54,38696,3205],{"className":38697},[2336],[54,38699],{"className":38700,"style":2475},[2270],[54,38702,122],{"className":38703},[2479],[54,38705],{"className":38706,"style":2475},[2270],[54,38708,38710,38713,38742,38745,38766,38769,38772],{"className":38709},[2117],[54,38711],{"className":38712,"style":2325},[2121],[54,38714,38716,38719],{"className":38715},[2126],[54,38717,13],{"className":38718},[2126,2167],[54,38720,38722],{"className":38721},[2135],[54,38723,38725],{"className":38724},[2139],[54,38726,38728],{"className":38727},[2143],[54,38729,38731],{"className":38730,"style":36702},[2147],[54,38732,38733,38736],{"style":2462},[54,38734],{"className":38735,"style":2156},[2155],[54,38737,38739],{"className":38738},[2160,2161,2162,2163],[54,38740,20325],{"className":38741},[2275,2163],[54,38743],{"className":38744,"style":2521},[2270],[54,38746,38748,38751,38754,38757,38760,38763],{"className":38747},[2493],[54,38749,129],{"className":38750,"style":2499},[2497,2498],[54,38752,37470],{"className":38753,"style":2332},[2126,2167],[54,38755,129],{"className":38756},[2497],[54,38758,5738],{"className":38759,"style":5448},[2126,2167],[54,38761,3205],{"className":38762},[2336],[54,38764,3205],{"className":38765,"style":2499},[2336,2498],[54,38767],{"className":38768,"style":2475},[2270],[54,38770,122],{"className":38771},[2479],[54,38773],{"className":38774,"style":2475},[2270],[54,38776,38778,38782],{"className":38777},[2117],[54,38779],{"className":38780,"style":38781},[2121],"height:5.32em;vertical-align:-2.41em;",[54,38783,38785,38788,39408],{"className":38784},[2126],[54,38786],{"className":38787},[2497,9349],[54,38789,38791],{"className":38790},[9047],[54,38792,38794,39399],{"className":38793},[2139,2756],[54,38795,38797,39396],{"className":38796},[2143],[54,38798,38801,39054,39062],{"className":38799,"style":38800},[2147],"height:2.91em;",[54,38802,38804,38808],{"style":38803},"top:-2.11em;",[54,38805],{"className":38806,"style":38807},[2155],"height:3.51em;",[54,38809,38811,38851,38854],{"className":38810},[2126],[54,38812,38814,38817],{"className":38813},[2126],[54,38815,36603],{"className":38816,"style":2776},[2126,2167],[54,38818,38820],{"className":38819},[2135],[54,38821,38823,38843],{"className":38822},[2139,2756],[54,38824,38826,38840],{"className":38825},[2143],[54,38827,38829],{"className":38828,"style":10799},[2147],[54,38830,38831,38834],{"style":7193},[54,38832],{"className":38833,"style":2156},[2155],[54,38835,38837],{"className":38836},[2160,2161,2162,2163],[54,38838,4515],{"className":38839},[2126,2167,2163],[54,38841,2781],{"className":38842},[2780],[54,38844,38846],{"className":38845},[2143],[54,38847,38849],{"className":38848,"style":2788},[2147],[54,38850],{},[54,38852],{"className":38853,"style":2521},[2270],[54,38855,38857,38863,38866,38869,38872,38875,38878,39048],{"className":38856},[2493],[54,38858,38860],{"className":38859,"style":2499},[2497,2498],[54,38861,129],{"className":38862},[2503,2162],[54,38864,174],{"className":38865},[2126],[54,38867],{"className":38868,"style":2271},[2270],[54,38870,1663],{"className":38871},[2275],[54,38873],{"className":38874,"style":2271},[2270],[54,38876,133],{"className":38877},[2126],[54,38879,38881,38884,39045],{"className":38880},[2126],[54,38882],{"className":38883},[2497,9349],[54,38885,38887],{"className":38886},[9047],[54,38888,38890,39036],{"className":38889},[2139,2756],[54,38891,38893,39033],{"className":38892},[2143],[54,38894,38897,38962,38970],{"className":38895,"style":38896},[2147],"height:1.51em;",[54,38898,38900,38903],{"style":38899},"top:-2.34em;",[54,38901],{"className":38902,"style":3866},[2155],[54,38904,38906],{"className":38905},[2160,2161,2162,2163],[54,38907,38909,38912,38915,38928,38931,38941,38944,38947,38950,38953],{"className":38908},[2126,2163],[54,38910,174],{"className":38911},[2126,2163],[54,38913,1663],{"className":38914},[2275,2163],[54,38916,38918,38921,38925],{"className":38917},[7126,2163],[54,38919,5730],{"className":38920},[2163],[54,38922,38924],{"className":38923},[2163],"o",[54,38926,14089],{"className":38927},[2163],[54,38929],{"className":38930,"style":25454},[2270,2163],[54,38932,38935],{"className":38933},[2126,2160,4372,38934,2163],"size6",[54,38936,38938],{"className":38937},[2503,2504,2163],[54,38939,129],{"className":38940},[2163],[54,38942,27081],{"className":38943,"style":27166},[2126,2167,2163],[54,38945],{"className":38946,"style":25454},[2270,2163],[54,38948,4515],{"className":38949},[2126,2167,2163],[54,38951,2854],{"className":38952},[2126,2167,2163],[54,38954,38956],{"className":38955},[2126,2160,4372,38934,2163],[54,38957,38959],{"className":38958},[2503,2504,2163],[54,38960,3205],{"className":38961},[2163],[54,38963,38964,38967],{"style":9434},[54,38965],{"className":38966,"style":3866},[2155],[54,38968],{"className":38969,"style":9442},[9441],[54,38971,38973,38976],{"style":38972},"top:-3.66em;",[54,38974],{"className":38975,"style":3866},[2155],[54,38977,38979],{"className":38978},[2160,2161,2162,2163],[54,38980,38982,38985,38988,39000,39003,39012,39015,39018,39021,39024],{"className":38981},[2126,2163],[54,38983,174],{"className":38984},[2126,2163],[54,38986,2247],{"className":38987},[2275,2163],[54,38989,38991,38994,38997],{"className":38990},[7126,2163],[54,38992,5730],{"className":38993},[2163],[54,38995,38924],{"className":38996},[2163],[54,38998,14089],{"className":38999},[2163],[54,39001],{"className":39002,"style":25454},[2270,2163],[54,39004,39006],{"className":39005},[2126,2160,4372,38934,2163],[54,39007,39009],{"className":39008},[2503,2504,2163],[54,39010,129],{"className":39011},[2163],[54,39013,27081],{"className":39014,"style":27166},[2126,2167,2163],[54,39016],{"className":39017,"style":25454},[2270,2163],[54,39019,4515],{"className":39020},[2126,2167,2163],[54,39022,2854],{"className":39023},[2126,2167,2163],[54,39025,39027],{"className":39026},[2126,2160,4372,38934,2163],[54,39028,39030],{"className":39029},[2503,2504,2163],[54,39031,3205],{"className":39032},[2163],[54,39034,2781],{"className":39035},[2780],[54,39037,39039],{"className":39038},[2143],[54,39040,39043],{"className":39041,"style":39042},[2147],"height:1.01em;",[54,39044],{},[54,39046],{"className":39047},[2336,9349],[54,39049,39051],{"className":39050,"style":2499},[2336,2498],[54,39052,3205],{"className":39053},[2503,2162],[54,39055,39056,39059],{"style":37599},[54,39057],{"className":39058,"style":38807},[2155],[54,39060],{"className":39061,"style":9442},[9441],[54,39063,39065,39068],{"style":39064},"top:-4.91em;",[54,39066],{"className":39067,"style":38807},[2155],[54,39069,39071,39111,39114],{"className":39070},[2126],[54,39072,39074,39077],{"className":39073},[2126],[54,39075,36603],{"className":39076,"style":2776},[2126,2167],[54,39078,39080],{"className":39079},[2135],[54,39081,39083,39103],{"className":39082},[2139,2756],[54,39084,39086,39100],{"className":39085},[2143],[54,39087,39089],{"className":39088,"style":10799},[2147],[54,39090,39091,39094],{"style":7193},[54,39092],{"className":39093,"style":2156},[2155],[54,39095,39097],{"className":39096},[2160,2161,2162,2163],[54,39098,4515],{"className":39099},[2126,2167,2163],[54,39101,2781],{"className":39102},[2780],[54,39104,39106],{"className":39105},[2143],[54,39107,39109],{"className":39108,"style":2788},[2147],[54,39110],{},[54,39112],{"className":39113,"style":2521},[2270],[54,39115,39117,39123,39126,39129,39132,39135,39138,39390],{"className":39116},[2493],[54,39118,39120],{"className":39119,"style":2499},[2497,2498],[54,39121,129],{"className":39122},[2503,2162],[54,39124,174],{"className":39125},[2126],[54,39127],{"className":39128,"style":2271},[2270],[54,39130,1663],{"className":39131},[2275],[54,39133],{"className":39134,"style":2271},[2270],[54,39136,133],{"className":39137},[2126],[54,39139,39141,39144,39387],{"className":39140},[2126],[54,39142],{"className":39143},[2497,9349],[54,39145,39147],{"className":39146},[9047],[54,39148,39150,39379],{"className":39149},[2139,2756],[54,39151,39153,39376],{"className":39152},[2143],[54,39154,39156,39218,39226],{"className":39155,"style":38896},[2147],[54,39157,39158,39161],{"style":38899},[54,39159],{"className":39160,"style":3866},[2155],[54,39162,39164],{"className":39163},[2160,2161,2162,2163],[54,39165,39167,39170,39173,39185,39188,39197,39200,39203,39206,39209],{"className":39166},[2126,2163],[54,39168,174],{"className":39169},[2126,2163],[54,39171,1663],{"className":39172},[2275,2163],[54,39174,39176,39179,39182],{"className":39175},[7126,2163],[54,39177,5730],{"className":39178},[2163],[54,39180,38924],{"className":39181},[2163],[54,39183,14089],{"className":39184},[2163],[54,39186],{"className":39187,"style":25454},[2270,2163],[54,39189,39191],{"className":39190},[2126,2160,4372,38934,2163],[54,39192,39194],{"className":39193},[2503,2504,2163],[54,39195,129],{"className":39196},[2163],[54,39198,27081],{"className":39199,"style":27166},[2126,2167,2163],[54,39201],{"className":39202,"style":25454},[2270,2163],[54,39204,4515],{"className":39205},[2126,2167,2163],[54,39207,2854],{"className":39208},[2126,2167,2163],[54,39210,39212],{"className":39211},[2126,2160,4372,38934,2163],[54,39213,39215],{"className":39214},[2503,2504,2163],[54,39216,3205],{"className":39217},[2163],[54,39219,39220,39223],{"style":9434},[54,39221],{"className":39222,"style":3866},[2155],[54,39224],{"className":39225,"style":9442},[9441],[54,39227,39228,39231],{"style":38972},[54,39229],{"className":39230,"style":3866},[2155],[54,39232,39234],{"className":39233},[2160,2161,2162,2163],[54,39235,39237,39249,39252,39261,39264,39267,39270,39310,39313,39316,39319,39328,39331,39343,39346,39355,39358,39361,39364,39367],{"className":39236},[2126,2163],[54,39238,39240,39243,39246],{"className":39239},[7126,2163],[54,39241,5730],{"className":39242},[2163],[54,39244,38924],{"className":39245},[2163],[54,39247,14089],{"className":39248},[2163],[54,39250],{"className":39251,"style":25454},[2270,2163],[54,39253,39255],{"className":39254},[2126,2160,4372,38934,2163],[54,39256,39258],{"className":39257},[2503,2504,2163],[54,39259,129],{"className":39260},[2163],[54,39262,129],{"className":39263},[2497,2163],[54,39265,5738],{"className":39266,"style":5448},[2126,2167,2163],[54,39268,2247],{"className":39269},[2275,2163],[54,39271,39273,39276],{"className":39272},[2126,2163],[54,39274,5738],{"className":39275,"style":5448},[2126,2167,2163],[54,39277,39279],{"className":39278},[2135],[54,39280,39282,39302],{"className":39281},[2139,2756],[54,39283,39285,39299],{"className":39284},[2143],[54,39286,39288],{"className":39287,"style":30826},[2147],[54,39289,39290,39293],{"style":30829},[54,39291],{"className":39292,"style":4368},[2155],[54,39294,39296],{"className":39295},[2160,4372,2504,2163],[54,39297,150],{"className":39298},[2126,2163],[54,39300,2781],{"className":39301},[2780],[54,39303,39305],{"className":39304},[2143],[54,39306,39308],{"className":39307,"style":4385},[2147],[54,39309],{},[54,39311,3205],{"className":39312},[2336,2163],[54,39314,4515],{"className":39315},[2126,2167,2163],[54,39317,2854],{"className":39318},[2126,2167,2163],[54,39320,39322],{"className":39321},[2126,2160,4372,38934,2163],[54,39323,39325],{"className":39324},[2503,2504,2163],[54,39326,3205],{"className":39327},[2163],[54,39329,2247],{"className":39330},[2275,2163],[54,39332,39334,39337,39340],{"className":39333},[7126,2163],[54,39335,5730],{"className":39336},[2163],[54,39338,38924],{"className":39339},[2163],[54,39341,14089],{"className":39342},[2163],[54,39344],{"className":39345,"style":25454},[2270,2163],[54,39347,39349],{"className":39348},[2126,2160,4372,38934,2163],[54,39350,39352],{"className":39351},[2503,2504,2163],[54,39353,129],{"className":39354},[2163],[54,39356,27081],{"className":39357,"style":27166},[2126,2167,2163],[54,39359],{"className":39360,"style":25454},[2270,2163],[54,39362,4515],{"className":39363},[2126,2167,2163],[54,39365,2854],{"className":39366},[2126,2167,2163],[54,39368,39370],{"className":39369},[2126,2160,4372,38934,2163],[54,39371,39373],{"className":39372},[2503,2504,2163],[54,39374,3205],{"className":39375},[2163],[54,39377,2781],{"className":39378},[2780],[54,39380,39382],{"className":39381},[2143],[54,39383,39385],{"className":39384,"style":39042},[2147],[54,39386],{},[54,39388],{"className":39389},[2336,9349],[54,39391,39393],{"className":39392,"style":2499},[2336,2498],[54,39394,3205],{"className":39395},[2503,2162],[54,39397,2781],{"className":39398},[2780],[54,39400,39402],{"className":39401},[2143],[54,39403,39406],{"className":39404,"style":39405},[2147],"height:2.41em;",[54,39407],{},[54,39409],{"className":39410},[2336,9349],[13,39412,28011,39413,39482,39483,39722,39723,39766,39767,39795,39796,27480,39881,40041,40042,40115,40116,40260,40261,40289,40290,40414],{},[54,39414,39416,39433],{"className":39415},[2079],[54,39417,39419],{"className":39418},[2083],[2085,39420,39421],{"xmlns":2087},[2089,39422,39423,39431],{},[2092,39424,39425],{},[2724,39426,39427,39429],{},[2098,39428,5738],{},[2249,39430,150],{},[2105,39432,13809],{"encoding":2107},[54,39434,39436],{"className":39435,"ariaHidden":2113},[2112],[54,39437,39439,39442],{"className":39438},[2117],[54,39440],{"className":39441,"style":2743},[2121],[54,39443,39445,39448],{"className":39444},[2126],[54,39446,5738],{"className":39447,"style":5448},[2126,2167],[54,39449,39451],{"className":39450},[2135],[54,39452,39454,39474],{"className":39453},[2139,2756],[54,39455,39457,39471],{"className":39456},[2143],[54,39458,39460],{"className":39459,"style":13837},[2147],[54,39461,39462,39465],{"style":5570},[54,39463],{"className":39464,"style":2156},[2155],[54,39466,39468],{"className":39467},[2160,2161,2162,2163],[54,39469,150],{"className":39470},[2126,2163],[54,39472,2781],{"className":39473},[2780],[54,39475,39477],{"className":39476},[2143],[54,39478,39480],{"className":39479,"style":2788},[2147],[54,39481],{}," is the ground state energy. We can see by inserting ",[54,39484,39486,39540],{"className":39485},[2079],[54,39487,39489],{"className":39488},[2083],[2085,39490,39491],{"xmlns":2087},[2089,39492,39493,39537],{},[2092,39494,39495,39497,39499,39501,39503,39505,39507],{},[2098,39496,37489],{},[2245,39498,36247],{},[2245,39500,129],{"stretchy":2311},[2098,39502,8695],{},[2245,39504,3205],{"stretchy":2311},[2245,39506,122],{},[9047,39508,39509,39535],{},[2092,39510,39511,39521,39523],{},[2095,39512,39513,39515],{},[2098,39514,2842],{},[2092,39516,39517,39519],{},[2098,39518,2849],{},[2098,39520,8695],{},[2245,39522,1663],{},[2095,39524,39525,39527],{},[2098,39526,2842],{},[2092,39528,39529,39531,39533],{},[2245,39530,2247],{},[2098,39532,2849],{},[2098,39534,8695],{},[2249,39536,133],{},[2105,39538,39539],{"encoding":2107},"\\cos(x)=\\frac{e^{ix}+e^{-ix}}{2}",[54,39541,39543,39570],{"className":39542,"ariaHidden":2113},[2112],[54,39544,39546,39549,39552,39555,39558,39561,39564,39567],{"className":39545},[2117],[54,39547],{"className":39548,"style":2325},[2121],[54,39550,37489],{"className":39551},[7126],[54,39553,129],{"className":39554},[2497],[54,39556,8695],{"className":39557},[2126,2167],[54,39559,3205],{"className":39560},[2336],[54,39562],{"className":39563,"style":2475},[2270],[54,39565,122],{"className":39566},[2479],[54,39568],{"className":39569,"style":2475},[2270],[54,39571,39573,39577],{"className":39572},[2117],[54,39574],{"className":39575,"style":39576},[2121],"height:1.3705em;vertical-align:-0.345em;",[54,39578,39580,39583,39719],{"className":39579},[2126],[54,39581],{"className":39582},[2497,9349],[54,39584,39586],{"className":39585},[9047],[54,39587,39589,39710],{"className":39588},[2139,2756],[54,39590,39592,39707],{"className":39591},[2143],[54,39593,39596,39610,39618],{"className":39594,"style":39595},[2147],"height:1.0255em;",[54,39597,39598,39601],{"style":36840},[54,39599],{"className":39600,"style":3866},[2155],[54,39602,39604],{"className":39603},[2160,2161,2162,2163],[54,39605,39607],{"className":39606},[2126,2163],[54,39608,133],{"className":39609},[2126,2163],[54,39611,39612,39615],{"style":9434},[54,39613],{"className":39614,"style":3866},[2155],[54,39616],{"className":39617,"style":9442},[9441],[54,39619,39620,39623],{"style":36869},[54,39621],{"className":39622,"style":3866},[2155],[54,39624,39626],{"className":39625},[2160,2161,2162,2163],[54,39627,39629,39666,39669],{"className":39628},[2126,2163],[54,39630,39632,39635],{"className":39631},[2126,2163],[54,39633,2842],{"className":39634},[2126,2167,2163],[54,39636,39638],{"className":39637},[2135],[54,39639,39641],{"className":39640},[2139],[54,39642,39644],{"className":39643},[2143],[54,39645,39648],{"className":39646,"style":39647},[2147],"height:0.9021em;",[54,39649,39651,39654],{"style":39650},"top:-2.931em;margin-right:0.0714em;",[54,39652],{"className":39653,"style":4368},[2155],[54,39655,39657],{"className":39656},[2160,4372,2504,2163],[54,39658,39660,39663],{"className":39659},[2126,2163],[54,39661,2849],{"className":39662},[2126,2167,2163],[54,39664,8695],{"className":39665},[2126,2167,2163],[54,39667,1663],{"className":39668},[2275,2163],[54,39670,39672,39675],{"className":39671},[2126,2163],[54,39673,2842],{"className":39674},[2126,2167,2163],[54,39676,39678],{"className":39677},[2135],[54,39679,39681],{"className":39680},[2139],[54,39682,39684],{"className":39683},[2143],[54,39685,39687],{"className":39686,"style":39647},[2147],[54,39688,39689,39692],{"style":39650},[54,39690],{"className":39691,"style":4368},[2155],[54,39693,39695],{"className":39694},[2160,4372,2504,2163],[54,39696,39698,39701,39704],{"className":39697},[2126,2163],[54,39699,2247],{"className":39700},[2126,2163],[54,39702,2849],{"className":39703},[2126,2167,2163],[54,39705,8695],{"className":39706},[2126,2167,2163],[54,39708,2781],{"className":39709},[2780],[54,39711,39713],{"className":39712},[2143],[54,39714,39717],{"className":39715,"style":39716},[2147],"height:0.345em;",[54,39718],{},[54,39720],{"className":39721},[2336,9349]," that ",[54,39724,39726,39745],{"className":39725},[2079],[54,39727,39729],{"className":39728},[2083],[2085,39730,39731],{"xmlns":2087},[2089,39732,39733,39743],{},[2092,39734,39735,39737,39739,39741],{},[2098,39736,21834],{},[2245,39738,129],{"stretchy":2311},[2098,39740,5738],{},[2245,39742,3205],{"stretchy":2311},[2105,39744,27370],{"encoding":2107},[54,39746,39748],{"className":39747,"ariaHidden":2113},[2112],[54,39749,39751,39754,39757,39760,39763],{"className":39750},[2117],[54,39752],{"className":39753,"style":2325},[2121],[54,39755,21834],{"className":39756,"style":26895},[2126,2167],[54,39758,129],{"className":39759},[2497],[54,39761,5738],{"className":39762,"style":5448},[2126,2167],[54,39764,3205],{"className":39765},[2336]," is a trigonometric polynomial of degree ",[54,39768,39770,39783],{"className":39769},[2079],[54,39771,39773],{"className":39772},[2083],[2085,39774,39775],{"xmlns":2087},[2089,39776,39777,39781],{},[2092,39778,39779],{},[2098,39780,4515],{},[2105,39782,4515],{"encoding":2107},[54,39784,39786],{"className":39785,"ariaHidden":2113},[2112],[54,39787,39789,39792],{"className":39788},[2117],[54,39790],{"className":39791,"style":4990},[2121],[54,39793,4515],{"className":39794},[2126,2167],", that is, a linear combination of ",[54,39797,39799,39828],{"className":39798},[2079],[54,39800,39802],{"className":39801},[2083],[2085,39803,39804],{"xmlns":2087},[2089,39805,39806,39826],{},[2092,39807,39808],{},[2095,39809,39810,39812],{},[2098,39811,2842],{},[2092,39813,39814,39816,39818,39820,39822,39824],{},[2098,39815,2849],{},[2098,39817,1928],{},[2098,39819,5738],{},[5723,39821,25244],{},[2098,39823,4515],{},[2098,39825,2854],{},[2105,39827,27425],{"encoding":2107},[54,39829,39831],{"className":39830,"ariaHidden":2113},[2112],[54,39832,39834,39837],{"className":39833},[2117],[54,39835],{"className":39836,"style":3218},[2121],[54,39838,39840,39843],{"className":39839},[2126],[54,39841,2842],{"className":39842},[2126,2167],[54,39844,39846],{"className":39845},[2135],[54,39847,39849],{"className":39848},[2139],[54,39850,39852],{"className":39851},[2143],[54,39853,39855],{"className":39854,"style":3218},[2147],[54,39856,39857,39860],{"style":2151},[54,39858],{"className":39859,"style":2156},[2155],[54,39861,39863],{"className":39862},[2160,2161,2162,2163],[54,39864,39866,39869,39872,39875,39878],{"className":39865},[2126,2163],[54,39867,27467],{"className":39868,"style":6379},[2126,2167,2163],[54,39870,5738],{"className":39871,"style":5448},[2126,2167,2163],[54,39873],{"className":39874,"style":25454},[2270,2163],[54,39876,4515],{"className":39877},[2126,2167,2163],[54,39879,2854],{"className":39880},[2126,2167,2163],[54,39882,39884,39933],{"className":39883},[2079],[54,39885,39887],{"className":39886},[2083],[2085,39888,39889],{"xmlns":2087},[2089,39890,39891,39931],{},[2092,39892,39893,39895,39897,39899,39901,39903,39905,39907,39909,39911,39913,39915,39917,39919,39921,39923,39925,39927,39929],{},[2098,39894,1928],{},[2245,39896,122],{},[2245,39898,2247],{},[2098,39900,4515],{},[2245,39902,2375],{"separator":2113},[2245,39904,2247],{},[2098,39906,4515],{},[2245,39908,1663],{},[2249,39910,174],{},[2245,39912,2375],{"separator":2113},[2098,39914,29],{"mathvariant":2304},[2098,39916,29],{"mathvariant":2304},[2098,39918,29],{"mathvariant":2304},[2245,39920,2375],{"separator":2113},[2098,39922,4515],{},[2245,39924,2247],{},[2249,39926,174],{},[2245,39928,2375],{"separator":2113},[2098,39930,4515],{},[2105,39932,27533],{"encoding":2107},[54,39934,39936,39954,39987,40023],{"className":39935,"ariaHidden":2113},[2112],[54,39937,39939,39942,39945,39948,39951],{"className":39938},[2117],[54,39940],{"className":39941,"style":27543},[2121],[54,39943,1928],{"className":39944,"style":6379},[2126,2167],[54,39946],{"className":39947,"style":2475},[2270],[54,39949,122],{"className":39950},[2479],[54,39952],{"className":39953,"style":2475},[2270],[54,39955,39957,39960,39963,39966,39969,39972,39975,39978,39981,39984],{"className":39956},[2117],[54,39958],{"className":39959,"style":6591},[2121],[54,39961,2247],{"className":39962},[2126],[54,39964,4515],{"className":39965},[2126,2167],[54,39967,2375],{"className":39968},[2517],[54,39970],{"className":39971,"style":2521},[2270],[54,39973,2247],{"className":39974},[2126],[54,39976,4515],{"className":39977},[2126,2167],[54,39979],{"className":39980,"style":2271},[2270],[54,39982,1663],{"className":39983},[2275],[54,39985],{"className":39986,"style":2271},[2270],[54,39988,39990,39993,39996,39999,40002,40005,40008,40011,40014,40017,40020],{"className":39989},[2117],[54,39991],{"className":39992,"style":6591},[2121],[54,39994,174],{"className":39995},[2126],[54,39997,2375],{"className":39998},[2517],[54,40000],{"className":40001,"style":2521},[2270],[54,40003,2588],{"className":40004},[2126],[54,40006,2375],{"className":40007},[2517],[54,40009],{"className":40010,"style":2521},[2270],[54,40012,4515],{"className":40013},[2126,2167],[54,40015],{"className":40016,"style":2271},[2270],[54,40018,2247],{"className":40019},[2275],[54,40021],{"className":40022,"style":2271},[2270],[54,40024,40026,40029,40032,40035,40038],{"className":40025},[2117],[54,40027],{"className":40028,"style":6591},[2121],[54,40030,174],{"className":40031},[2126],[54,40033,2375],{"className":40034},[2517],[54,40036],{"className":40037,"style":2521},[2270],[54,40039,4515],{"className":40040},[2126,2167],". Furthermore, from the definition of ",[54,40043,40045,40068],{"className":40044},[2079],[54,40046,40048],{"className":40047},[2083],[2085,40049,40050],{"xmlns":2087},[2089,40051,40052,40066],{},[2092,40053,40054,40060,40062,40064],{},[2095,40055,40056,40058],{},[2098,40057,13],{},[2245,40059,20325],{},[2245,40061,129],{"stretchy":2311},[2098,40063,8695],{},[2245,40065,3205],{"stretchy":2311},[2105,40067,38239],{"encoding":2107},[54,40069,40071],{"className":40070,"ariaHidden":2113},[2112],[54,40072,40074,40077,40106,40109,40112],{"className":40073},[2117],[54,40075],{"className":40076,"style":2325},[2121],[54,40078,40080,40083],{"className":40079},[2126],[54,40081,13],{"className":40082},[2126,2167],[54,40084,40086],{"className":40085},[2135],[54,40087,40089],{"className":40088},[2139],[54,40090,40092],{"className":40091},[2143],[54,40093,40095],{"className":40094,"style":36139},[2147],[54,40096,40097,40100],{"style":2151},[54,40098],{"className":40099,"style":2156},[2155],[54,40101,40103],{"className":40102},[2160,2161,2162,2163],[54,40104,20325],{"className":40105},[2275,2163],[54,40107,129],{"className":40108},[2497],[54,40110,8695],{"className":40111},[2126,2167],[54,40113,3205],{"className":40114},[2336]," above we have that ",[54,40117,40119,40157],{"className":40118},[2079],[54,40120,40122],{"className":40121},[2083],[2085,40123,40124],{"xmlns":2087},[2089,40125,40126,40154],{},[2092,40127,40128,40130,40132,40138,40140,40142,40144,40146,40148,40150,40152],{},[2098,40129,21834],{},[2245,40131,129],{"stretchy":2311},[2724,40133,40134,40136],{},[2098,40135,5738],{},[2249,40137,150],{},[2245,40139,3205],{"stretchy":2311},[2245,40141,122],{},[2098,40143,13],{},[2245,40145,129],{"stretchy":2311},[2249,40147,150],{},[2245,40149,3205],{"stretchy":2311},[2245,40151,122],{},[2249,40153,174],{},[2105,40155,40156],{"encoding":2107},"f(E_0)=p(0)=1",[54,40158,40160,40224,40251],{"className":40159,"ariaHidden":2113},[2112],[54,40161,40163,40166,40169,40172,40212,40215,40218,40221],{"className":40162},[2117],[54,40164],{"className":40165,"style":2325},[2121],[54,40167,21834],{"className":40168,"style":26895},[2126,2167],[54,40170,129],{"className":40171},[2497],[54,40173,40175,40178],{"className":40174},[2126],[54,40176,5738],{"className":40177,"style":5448},[2126,2167],[54,40179,40181],{"className":40180},[2135],[54,40182,40184,40204],{"className":40183},[2139,2756],[54,40185,40187,40201],{"className":40186},[2143],[54,40188,40190],{"className":40189,"style":13837},[2147],[54,40191,40192,40195],{"style":5570},[54,40193],{"className":40194,"style":2156},[2155],[54,40196,40198],{"className":40197},[2160,2161,2162,2163],[54,40199,150],{"className":40200},[2126,2163],[54,40202,2781],{"className":40203},[2780],[54,40205,40207],{"className":40206},[2143],[54,40208,40210],{"className":40209,"style":2788},[2147],[54,40211],{},[54,40213,3205],{"className":40214},[2336],[54,40216],{"className":40217,"style":2475},[2270],[54,40219,122],{"className":40220},[2479],[54,40222],{"className":40223,"style":2475},[2270],[54,40225,40227,40230,40233,40236,40239,40242,40245,40248],{"className":40226},[2117],[54,40228],{"className":40229,"style":2325},[2121],[54,40231,13],{"className":40232},[2126,2167],[54,40234,129],{"className":40235},[2497],[54,40237,150],{"className":40238},[2126],[54,40240,3205],{"className":40241},[2336],[54,40243],{"className":40244,"style":2475},[2270],[54,40246,122],{"className":40247},[2479],[54,40249],{"className":40250,"style":2475},[2270],[54,40252,40254,40257],{"className":40253},[2117],[54,40255],{"className":40256,"style":2285},[2121],[54,40258,174],{"className":40259},[2126]," and for any ",[54,40262,40264,40277],{"className":40263},[2079],[54,40265,40267],{"className":40266},[2083],[2085,40268,40269],{"xmlns":2087},[2089,40270,40271,40275],{},[2092,40272,40273],{},[2098,40274,5738],{},[2105,40276,5738],{"encoding":2107},[54,40278,40280],{"className":40279,"ariaHidden":2113},[2112],[54,40281,40283,40286],{"className":40282},[2117],[54,40284],{"className":40285,"style":2122},[2121],[54,40287,5738],{"className":40288,"style":5448},[2126,2167]," in the spectral range such that ",[54,40291,40293,40323],{"className":40292},[2079],[54,40294,40296],{"className":40295},[2083],[2085,40297,40298],{"xmlns":2087},[2089,40299,40300,40320],{},[2092,40301,40302,40304,40306,40308,40314,40316,40318],{},[2098,40303,2305],{"mathvariant":2304},[2098,40305,5738],{},[2245,40307,2247],{},[2724,40309,40310,40312],{},[2098,40311,5738],{},[2249,40313,150],{},[2098,40315,2305],{"mathvariant":2304},[2245,40317,1207],{},[2098,40319,27081],{},[2105,40321,40322],{"encoding":2107},"\\vert E-E_0 \\vert > \\delta",[54,40324,40326,40347,40405],{"className":40325,"ariaHidden":2113},[2112],[54,40327,40329,40332,40335,40338,40341,40344],{"className":40328},[2117],[54,40330],{"className":40331,"style":2325},[2121],[54,40333,2305],{"className":40334},[2126],[54,40336,5738],{"className":40337,"style":5448},[2126,2167],[54,40339],{"className":40340,"style":2271},[2270],[54,40342,2247],{"className":40343},[2275],[54,40345],{"className":40346,"style":2271},[2270],[54,40348,40350,40353,40393,40396,40399,40402],{"className":40349},[2117],[54,40351],{"className":40352,"style":2325},[2121],[54,40354,40356,40359],{"className":40355},[2126],[54,40357,5738],{"className":40358,"style":5448},[2126,2167],[54,40360,40362],{"className":40361},[2135],[54,40363,40365,40385],{"className":40364},[2139,2756],[54,40366,40368,40382],{"className":40367},[2143],[54,40369,40371],{"className":40370,"style":13837},[2147],[54,40372,40373,40376],{"style":5570},[54,40374],{"className":40375,"style":2156},[2155],[54,40377,40379],{"className":40378},[2160,2161,2162,2163],[54,40380,150],{"className":40381},[2126,2163],[54,40383,2781],{"className":40384},[2780],[54,40386,40388],{"className":40387},[2143],[54,40389,40391],{"className":40390,"style":2788},[2147],[54,40392],{},[54,40394,2305],{"className":40395},[2126],[54,40397],{"className":40398,"style":2475},[2270],[54,40400,1207],{"className":40401},[2479],[54,40403],{"className":40404,"style":2475},[2270],[54,40406,40408,40411],{"className":40407},[2117],[54,40409],{"className":40410,"style":4990},[2121],[54,40412,27081],{"className":40413,"style":27166},[2126,2167]," we have",[54,40416,40418],{"className":40417},[2340],[54,40419,40421,40535],{"className":40420},[2079],[54,40422,40424],{"className":40423},[2083],[2085,40425,40426],{"xmlns":2087,"display":2349},[2089,40427,40428,40532],{},[2092,40429,40430,40432,40434,40436,40438,40440,40442,40444,40446,40448,40450,40452,40454,40456,40458,40460,40462,40474],{},[2098,40431,2305],{"mathvariant":2304},[2098,40433,21834],{},[2245,40435,129],{"stretchy":2311},[2098,40437,5738],{},[2245,40439,3205],{"stretchy":2311},[2098,40441,2305],{"mathvariant":2304},[2245,40443,27066],{},[2098,40445,36222],{},[2245,40447,129],{"stretchy":2311},[2098,40449,20],{},[2245,40451,2375],{"separator":2113},[2098,40453,12658],{},[2245,40455,2375],{"separator":2113},[2098,40457,4515],{},[2245,40459,3205],{"stretchy":2311},[2245,40461,122],{},[3331,40463,40464,40466,40468],{},[2098,40465,36603],{},[2098,40467,4515],{},[2092,40469,40470,40472],{},[2245,40471,2247],{},[2249,40473,174],{},[2092,40475,40476,40478,40480,40482,40484,40530],{},[2245,40477,129],{"fence":2113},[2249,40479,174],{},[2245,40481,1663],{},[2249,40483,133],{},[9047,40485,40486,40508],{},[2092,40487,40488,40490,40492,40494,40496,40498,40500,40502,40504,40506],{},[2249,40489,174],{},[2245,40491,2247],{},[2098,40493,37489],{},[2245,40495,36247],{},[2245,40497,129],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2098,40499,27081],{},[5723,40501,25244],{},[2098,40503,4515],{},[2098,40505,2854],{},[2245,40507,3205],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2092,40509,40510,40512,40514,40516,40518,40520,40522,40524,40526,40528],{},[2249,40511,174],{},[2245,40513,1663],{},[2098,40515,37489],{},[2245,40517,36247],{},[2245,40519,129],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2098,40521,27081],{},[5723,40523,25244],{},[2098,40525,4515],{},[2098,40527,2854],{},[2245,40529,3205],{"fence":2311,"stretchy":2113,"minsize":37494,"maxsize":37494},[2245,40531,3205],{"fence":2113},[2105,40533,40534],{"encoding":2107},"|f(E)| \\le \\beta(a, b, d) = T_d^{-1}\\left(1 + 2\\frac{1-\\cos\\big(\\delta\\,dt\\big)}{1 + \\cos\\big(\\delta\\,dt\\big)}\\right)",[54,40536,40538,40571,40616],{"className":40537,"ariaHidden":2113},[2112],[54,40539,40541,40544,40547,40550,40553,40556,40559,40562,40565,40568],{"className":40540},[2117],[54,40542],{"className":40543,"style":2325},[2121],[54,40545,2305],{"className":40546},[2126],[54,40548,21834],{"className":40549,"style":26895},[2126,2167],[54,40551,129],{"className":40552},[2497],[54,40554,5738],{"className":40555,"style":5448},[2126,2167],[54,40557,3205],{"className":40558},[2336],[54,40560,2305],{"className":40561},[2126],[54,40563],{"className":40564,"style":2475},[2270],[54,40566,27066],{"className":40567},[2479],[54,40569],{"className":40570,"style":2475},[2270],[54,40572,40574,40577,40580,40583,40586,40589,40592,40595,40598,40601,40604,40607,40610,40613],{"className":40573},[2117],[54,40575],{"className":40576,"style":2325},[2121],[54,40578,36222],{"className":40579,"style":36317},[2126,2167],[54,40581,129],{"className":40582},[2497],[54,40584,20],{"className":40585},[2126,2167],[54,40587,2375],{"className":40588},[2517],[54,40590],{"className":40591,"style":2521},[2270],[54,40593,12658],{"className":40594},[2126,2167],[54,40596,2375],{"className":40597},[2517],[54,40599],{"className":40600,"style":2521},[2270],[54,40602,4515],{"className":40603},[2126,2167],[54,40605,3205],{"className":40606},[2336],[54,40608],{"className":40609,"style":2475},[2270],[54,40611,122],{"className":40612},[2479],[54,40614],{"className":40615,"style":2475},[2270],[54,40617,40619,40623,40680,40683],{"className":40618},[2117],[54,40620],{"className":40621,"style":40622},[2121],"height:3em;vertical-align:-1.25em;",[54,40624,40626,40629],{"className":40625},[2126],[54,40627,36603],{"className":40628,"style":2776},[2126,2167],[54,40630,40632],{"className":40631},[2135],[54,40633,40635,40672],{"className":40634},[2139,2756],[54,40636,40638,40669],{"className":40637},[2143],[54,40639,40641,40652],{"className":40640,"style":2558},[2147],[54,40642,40643,40646],{"style":37246},[54,40644],{"className":40645,"style":2156},[2155],[54,40647,40649],{"className":40648},[2160,2161,2162,2163],[54,40650,4515],{"className":40651},[2126,2167,2163],[54,40653,40654,40657],{"style":2462},[54,40655],{"className":40656,"style":2156},[2155],[54,40658,40660],{"className":40659},[2160,2161,2162,2163],[54,40661,40663,40666],{"className":40662},[2126,2163],[54,40664,2247],{"className":40665},[2126,2163],[54,40667,174],{"className":40668},[2126,2163],[54,40670,2781],{"className":40671},[2780],[54,40673,40675],{"className":40674},[2143],[54,40676,40678],{"className":40677,"style":37282},[2147],[54,40679],{},[54,40681],{"className":40682,"style":2521},[2270],[54,40684,40686,40692,40695,40698,40701,40704,40707,40849],{"className":40685},[2493],[54,40687,40689],{"className":40688,"style":2499},[2497,2498],[54,40690,129],{"className":40691},[2503,12042],[54,40693,174],{"className":40694},[2126],[54,40696],{"className":40697,"style":2271},[2270],[54,40699,1663],{"className":40700},[2275],[54,40702],{"className":40703,"style":2271},[2270],[54,40705,133],{"className":40706},[2126],[54,40708,40710,40713,40846],{"className":40709},[2126],[54,40711],{"className":40712},[2497,9349],[54,40714,40716],{"className":40715},[9047],[54,40717,40719,40837],{"className":40718},[2139,2756],[54,40720,40722,40834],{"className":40721},[2143],[54,40723,40725,40776,40784],{"className":40724,"style":37577},[2147],[54,40726,40728,40731],{"style":40727},"top:-2.26em;",[54,40729],{"className":40730,"style":3866},[2155],[54,40732,40734,40737,40740,40743,40746,40749,40752,40758,40761,40764,40767,40770],{"className":40733},[2126],[54,40735,174],{"className":40736},[2126],[54,40738],{"className":40739,"style":2271},[2270],[54,40741,1663],{"className":40742},[2275],[54,40744],{"className":40745,"style":2271},[2270],[54,40747,37489],{"className":40748},[7126],[54,40750],{"className":40751,"style":2521},[2270],[54,40753,40755],{"className":40754},[2126],[54,40756,129],{"className":40757},[2503,2504],[54,40759,27081],{"className":40760,"style":27166},[2126,2167],[54,40762],{"className":40763,"style":2521},[2270],[54,40765,4515],{"className":40766},[2126,2167],[54,40768,2854],{"className":40769},[2126,2167],[54,40771,40773],{"className":40772},[2126],[54,40774,3205],{"className":40775},[2503,2504],[54,40777,40778,40781],{"style":9434},[54,40779],{"className":40780,"style":3866},[2155],[54,40782],{"className":40783,"style":9442},[9441],[54,40785,40786,40789],{"style":37599},[54,40787],{"className":40788,"style":3866},[2155],[54,40790,40792,40795,40798,40801,40804,40807,40810,40816,40819,40822,40825,40828],{"className":40791},[2126],[54,40793,174],{"className":40794},[2126],[54,40796],{"className":40797,"style":2271},[2270],[54,40799,2247],{"className":40800},[2275],[54,40802],{"className":40803,"style":2271},[2270],[54,40805,37489],{"className":40806},[7126],[54,40808],{"className":40809,"style":2521},[2270],[54,40811,40813],{"className":40812},[2126],[54,40814,129],{"className":40815},[2503,2504],[54,40817,27081],{"className":40818,"style":27166},[2126,2167],[54,40820],{"className":40821,"style":2521},[2270],[54,40823,4515],{"className":40824},[2126,2167],[54,40826,2854],{"className":40827},[2126,2167],[54,40829,40831],{"className":40830},[2126],[54,40832,3205],{"className":40833},[2503,2504],[54,40835,2781],{"className":40836},[2780],[54,40838,40840],{"className":40839},[2143],[54,40841,40844],{"className":40842,"style":40843},[2147],"height:1.09em;",[54,40845],{},[54,40847],{"className":40848},[2336,9349],[54,40850,40852],{"className":40851,"style":2499},[2336,2498],[54,40853,3205],{"className":40854},[2503,12042],[54,40856,40858],{"className":40857},[2340],[54,40859,40861,40933],{"className":40860},[2079],[54,40862,40864],{"className":40863},[2083],[2085,40865,40866],{"xmlns":2087,"display":2349},[2089,40867,40868,40930],{},[2092,40869,40870,40872,40874,40894,40896,40898,40928],{},[2245,40871,27066],{},[2249,40873,133],{},[2095,40875,40876,40888],{},[2092,40877,40878,40880,40882,40884,40886],{},[2245,40879,129],{"fence":2113},[2249,40881,174],{},[2245,40883,1663],{},[2098,40885,27081],{},[2245,40887,3205],{"fence":2113},[2092,40889,40890,40892],{},[2245,40891,2247],{},[2098,40893,4515],{},[2245,40895,122],{},[2249,40897,133],{},[2095,40899,40900,40912],{},[2092,40901,40902,40904,40906,40908,40910],{},[2245,40903,129],{"fence":2113},[2249,40905,174],{},[2245,40907,1663],{},[2098,40909,27081],{},[2245,40911,3205],{"fence":2113},[2092,40913,40914,40916,40919,40921,40923,40925],{},[2245,40915,2247],{},[2245,40917,40918],{"stretchy":2311},"⌊",[2098,40920,3182],{},[2098,40922,194],{"mathvariant":2304},[2249,40924,133],{},[2245,40926,40927],{"stretchy":2311},"⌋",[2098,40929,29],{"mathvariant":2304},[2105,40931,40932],{"encoding":2107},"\\leq 2\\left(1 + \\delta\\right)^{-d} = 2\\left(1 + \\delta\\right)^{-\\lfloor k\u002F2\\rfloor}.",[54,40934,40936,40948,41025],{"className":40935,"ariaHidden":2113},[2112],[54,40937,40939,40942,40945],{"className":40938},[2117],[54,40940],{"className":40941,"style":27259},[2121],[54,40943,27066],{"className":40944},[2479],[54,40946],{"className":40947,"style":2475},[2270],[54,40949,40951,40954,40957,40960,41016,41019,41022],{"className":40950},[2117],[54,40952],{"className":40953,"style":27135},[2121],[54,40955,133],{"className":40956},[2126],[54,40958],{"className":40959,"style":2521},[2270],[54,40961,40963,40987],{"className":40962},[2493],[54,40964,40966,40969,40972,40975,40978,40981,40984],{"className":40965},[2493],[54,40967,129],{"className":40968,"style":2499},[2497,2498],[54,40970,174],{"className":40971},[2126],[54,40973],{"className":40974,"style":2271},[2270],[54,40976,1663],{"className":40977},[2275],[54,40979],{"className":40980,"style":2271},[2270],[54,40982,27081],{"className":40983,"style":27166},[2126,2167],[54,40985,3205],{"className":40986,"style":2499},[2336,2498],[54,40988,40990],{"className":40989},[2135],[54,40991,40993],{"className":40992},[2139],[54,40994,40996],{"className":40995},[2143],[54,40997,40999],{"className":40998,"style":27182},[2147],[54,41000,41001,41004],{"style":7401},[54,41002],{"className":41003,"style":2156},[2155],[54,41005,41007],{"className":41006},[2160,2161,2162,2163],[54,41008,41010,41013],{"className":41009},[2126,2163],[54,41011,2247],{"className":41012},[2126,2163],[54,41014,4515],{"className":41015},[2126,2167,2163],[54,41017],{"className":41018,"style":2475},[2270],[54,41020,122],{"className":41021},[2479],[54,41023],{"className":41024,"style":2475},[2270],[54,41026,41028,41032,41035,41038,41104,41107],{"className":41027},[2117],[54,41029],{"className":41030,"style":41031},[2121],"height:1.2779em;vertical-align:-0.25em;",[54,41033,133],{"className":41034},[2126],[54,41036],{"className":41037,"style":2521},[2270],[54,41039,41041,41065],{"className":41040},[2493],[54,41042,41044,41047,41050,41053,41056,41059,41062],{"className":41043},[2493],[54,41045,129],{"className":41046,"style":2499},[2497,2498],[54,41048,174],{"className":41049},[2126],[54,41051],{"className":41052,"style":2271},[2270],[54,41054,1663],{"className":41055},[2275],[54,41057],{"className":41058,"style":2271},[2270],[54,41060,27081],{"className":41061,"style":27166},[2126,2167],[54,41063,3205],{"className":41064,"style":2499},[2336,2498],[54,41066,41068],{"className":41067},[2135],[54,41069,41071],{"className":41070},[2139],[54,41072,41074],{"className":41073},[2143],[54,41075,41078],{"className":41076,"style":41077},[2147],"height:1.0279em;",[54,41079,41080,41083],{"style":7401},[54,41081],{"className":41082,"style":2156},[2155],[54,41084,41086],{"className":41085},[2160,2161,2162,2163],[54,41087,41089,41092,41095,41098,41101],{"className":41088},[2126,2163],[54,41090,2247],{"className":41091},[2126,2163],[54,41093,40918],{"className":41094},[2497,2163],[54,41096,3182],{"className":41097,"style":3248},[2126,2167,2163],[54,41099,25677],{"className":41100},[2126,2163],[54,41102,40927],{"className":41103},[2336,2163],[54,41105],{"className":41106,"style":2521},[2270],[54,41108,29],{"className":41109},[2126],[31,41111,41113],{"id":41112},"references","References",[13,41115,41116,41118,41119],{},[54,41117,174],{}," N. Yoshioka, M. Amico, W. Kirby et al. \"Diagonalization of large many-body Hamiltonians on a quantum processor\". ",[20,41120,41122],{"href":41121},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.14431","arXiv:2407.14431",[13,41124,41125,41127],{},[54,41126,133],{}," Ethan N. Epperly, Lin Lin, and Yuji Nakatsukasa. \"A theory of quantum subspace diagonalization\". SIAM Journal on Matrix Analysis and Applications 43, 1263–1290 (2022).",[13,41129,41130,41132],{},[54,41131,7782],{}," Å. Björck. \"Numerical methods in matrix computations\". Texts in Applied Mathematics. Springer International Publishing. (2014).",[13,41134,41135,41137],{},[54,41136,11708],{}," William Kirby. \"Analysis of quantum Krylov algorithms with errors\". Quantum 8, 1457 (2024).",[31,41139,41141],{"id":41140},"tutorial-survey","Tutorial survey",[13,41143,41144],{},"Please take this short survey to provide feedback on this tutorial. Your insights will help us improve our content offerings and user experience.",[13,41146,41147],{},[20,41148,41150],{"href":41149},"https:\u002F\u002Fyour.feedback.ibm.com\u002Fjfe\u002Fform\u002FSV_82nennpKIjjD8rQ","Link to survey",[31,41152,489],{"id":488},[13,41154,41155,41156,29],{},"Get ",[20,41157,591],{"href":590},[502,41159,41160],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}",{"title":49,"searchDepth":74,"depth":74,"links":41162},[41163,41164,41165,41166,41170,41177,41182,41185,41186,41187,41188,41189],{"id":615,"depth":74,"text":616},{"id":650,"depth":74,"text":651},{"id":675,"depth":74,"text":676},{"id":2066,"depth":74,"text":2067,"children":41167},[41168,41169],{"id":2071,"depth":88,"text":2072},{"id":3640,"depth":88,"text":3641},{"id":13771,"depth":74,"text":13772,"children":41171},[41172,41173,41174,41175],{"id":13775,"depth":88,"text":13776},{"id":19905,"depth":88,"text":19906},{"id":20892,"depth":88,"text":20893},{"id":20989,"depth":88,"text":41176},"Template circuits for calculating matrix elements of S~\\tilde{S}S~ and H~\\tilde{H}H~ via Hadamard test",{"id":21467,"depth":74,"text":21468,"children":41178},[41179,41180,41181],{"id":21581,"depth":88,"text":21582},{"id":22201,"depth":88,"text":22202},{"id":22673,"depth":88,"text":22674},{"id":23533,"depth":74,"text":23534,"children":41183},[41184],{"id":23562,"depth":88,"text":23563},{"id":25174,"depth":74,"text":25175},{"id":36004,"depth":74,"text":36005},{"id":41112,"depth":74,"text":41113},{"id":41140,"depth":74,"text":41141},{"id":488,"depth":74,"text":489},[518,519,520,41191],"Krylov diagonalization",[41193],{"username":579,"name":41194,"role":41195,"bio":41196,"links":41197},"Mirko Amico","Researcher, Extropic · formerly IBM Research","A researcher at Extropic, previously at IBM Research, and a Qiskit Pulse expert.",[41198],{"label":41199,"href":590},"Original tutorial ↗",{"username":579,"name":41194,"role":41201},"Researcher, Extropic","Implement the Krylov Quantum Diagonalization Algorithm (KQD) within the context of Qiskit patterns.","Deep dive · Algorithms",{},"\u002Fblog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization","15 min read",[],{"title":577,"description":41202},"blog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization",[41211,41212],"algorithms","tutorial","h_4WJ9crI4tBC8SOea5gUgWXCAFpFws4fEwgpxBRWsU",{"id":41215,"title":41216,"authors":41217,"body":41218,"breadcrumb":67212,"builders":67214,"byline":67219,"challenge":535,"courseAuthor":535,"courseLead":535,"dek":535,"description":67220,"draft":538,"extension":539,"eyebrow":67221,"finish":535,"fork":535,"hero":535,"heroAlt":535,"heroCta":535,"heroImage":535,"kind":541,"lessonCount":535,"meta":67222,"navigation":104,"newsItems":535,"next":535,"ogImage":67223,"order":535,"outcomes":535,"path":67224,"publishDate":545,"readingTime":546,"related":67225,"relatedProjects":535,"seo":67226,"stem":67227,"tags":67228,"track":535,"trackName":535,"__hash__":67230},"blog\u002Fblog\u002Fexpert-notes\u002Frabi-oscillation-visualization.md","Rabi Oscillations: Visualizing Excited-State Probability",[8],{"type":10,"value":41219,"toc":67195},[41220,41229,41232,41340,41401,41405,41408,42881,42885,42966,43177,43180,43331,43334,43458,43567,43667,43670,43674,43677,43798,43859,44017,44020,44242,44245,44982,45019,45023,45026,45360,45490,45800,45803,46030,46148,46152,46223,46694,46697,46927,46930,47109,47164,47365,47368,47371,47622,47655,47932,47935,48088,48120,48239,48242,48509,48512,48630,48634,48637,49003,49006,49139,49146,49150,49153,49295,49298,49840,49843,50069,50245,50249,50323,50594,50596,50660,50692,50858,50927,51149,51152,51155,51508,51510,51834,51837,51841,51844,51951,51983,52167,52210,52214,52345,52589,52592,52985,52988,53199,53202,53206,53679,54139,54258,54376,54379,54383,54485,54488,54587,54670,54855,54858,55058,55061,55162,55559,55563,55566,56051,56054,56058,56061,56323,56326,56539,56542,56644,56647,56676,56993,56999,57487,57579,57585,57762,57768,57938,57942,62844,62848,62854,63572,63576,64230,64234,64238,65401,65407,65608,65898,65902,66713,66717,67174,67178,67182,67184,67192],[13,41221,41222],{},[585,41223,587,41224,41228],{},[20,41225,41227],{"href":41226},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FRabi_Oscillation_Visualization_for_Excited_State_Probability.ipynb","the original notebook"," and shared under CC-BY-4.0. The text, code, and figures below are Onri's, as published.",[13,41230,41231],{},"This notebook provides a primer on the quantum dynamics of a driven two-level system by visualizing the excited-state probability under the rotating-wave approximation (RWA). The model treats a qubit as an ideal two-state system driven by a classical microwave tone. This formulation maps the probability of measuring the excited state as a function of drive frequency and pulse duration. The central computational object is the probability surface",[54,41233,41235],{"className":41234},[2340],[54,41236,41238,41269],{"className":41237},[2079],[54,41239,41241],{"className":41240},[2083],[2085,41242,41243],{"xmlns":2087,"display":2349},[2089,41244,41245,41266],{},[2092,41246,41247,41253,41255,41257,41259,41262,41264],{},[2724,41248,41249,41251],{},[2098,41250,2730],{},[2098,41252,2842],{},[2245,41254,129],{"stretchy":2311},[2098,41256,21834],{},[2245,41258,2375],{"separator":2113},[2098,41260,41261],{},"τ",[2245,41263,3205],{"stretchy":2311},[2245,41265,2375],{"separator":2113},[2105,41267,41268],{"encoding":2107},"P_e(f,\\tau),",[54,41270,41272],{"className":41271,"ariaHidden":2113},[2112],[54,41273,41275,41278,41318,41321,41324,41327,41330,41334,41337],{"className":41274},[2117],[54,41276],{"className":41277,"style":2325},[2121],[54,41279,41281,41284],{"className":41280},[2126],[54,41282,2730],{"className":41283,"style":2776},[2126,2167],[54,41285,41287],{"className":41286},[2135],[54,41288,41290,41310],{"className":41289},[2139,2756],[54,41291,41293,41307],{"className":41292},[2143],[54,41294,41296],{"className":41295,"style":4063},[2147],[54,41297,41298,41301],{"style":7193},[54,41299],{"className":41300,"style":2156},[2155],[54,41302,41304],{"className":41303},[2160,2161,2162,2163],[54,41305,2842],{"className":41306},[2126,2167,2163],[54,41308,2781],{"className":41309},[2780],[54,41311,41313],{"className":41312},[2143],[54,41314,41316],{"className":41315,"style":2788},[2147],[54,41317],{},[54,41319,129],{"className":41320},[2497],[54,41322,21834],{"className":41323,"style":26895},[2126,2167],[54,41325,2375],{"className":41326},[2517],[54,41328],{"className":41329,"style":2521},[2270],[54,41331,41261],{"className":41332,"style":41333},[2126,2167],"margin-right:0.1132em;",[54,41335,3205],{"className":41336},[2336],[54,41338,2375],{"className":41339},[2517],[13,41341,28011,41342,41370,41371,41400],{},[54,41343,41345,41358],{"className":41344},[2079],[54,41346,41348],{"className":41347},[2083],[2085,41349,41350],{"xmlns":2087},[2089,41351,41352,41356],{},[2092,41353,41354],{},[2098,41355,21834],{},[2105,41357,21834],{"encoding":2107},[54,41359,41361],{"className":41360,"ariaHidden":2113},[2112],[54,41362,41364,41367],{"className":41363},[2117],[54,41365],{"className":41366,"style":6591},[2121],[54,41368,21834],{"className":41369,"style":26895},[2126,2167]," denotes the applied drive frequency and ",[54,41372,41374,41388],{"className":41373},[2079],[54,41375,41377],{"className":41376},[2083],[2085,41378,41379],{"xmlns":2087},[2089,41380,41381,41385],{},[2092,41382,41383],{},[2098,41384,41261],{},[2105,41386,41387],{"encoding":2107},"\\tau",[54,41389,41391],{"className":41390,"ariaHidden":2113},[2112],[54,41392,41394,41397],{"className":41393},[2117],[54,41395],{"className":41396,"style":2195},[2121],[54,41398,41261],{"className":41399,"style":41333},[2126,2167]," denotes the microwave pulse duration. The notebook renders this probability surface with two-dimensional heatmaps, three-dimensional surfaces, one-dimensional cross sections, and Fast Fourier transform (FFT) spectrograms. Fourier analysis exposes the generalized Rabi frequency as a theoretical ridge in the frequency domain, which connects the time-domain quantum oscillation to its spectral structure.",[31,41402,41404],{"id":41403},"nomenclature-and-definitions","Nomenclature and Definitions",[13,41406,41407],{},"The following table defines the physical and mathematical quantities that recur throughout the visualization workflow.",[41409,41410,41411,41424],"table",{},[41412,41413,41414],"thead",{},[41415,41416,41417,41421],"tr",{},[41418,41419,41420],"th",{},"Symbol or Term",[41418,41422,41423],{},"Meaning",[41425,41426,41427,41474,41520,41566,41693,41770,41848,41883,42104,42201,42238,42284,42435,42512,42737,42772,42865,42873],"tbody",{},[41415,41428,41429,41471],{},[41430,41431,41432],"td",{},[54,41433,41435,41453],{"className":41434},[2079],[54,41436,41438],{"className":41437},[2083],[2085,41439,41440],{"xmlns":2087},[2089,41441,41442,41450],{},[2092,41443,41444,41446,41448],{},[2245,41445,2305],{"stretchy":2311},[2098,41447,37470],{},[2245,41449,2312],{"stretchy":2311},[2105,41451,41452],{"encoding":2107},"\\lvert g\\rangle",[54,41454,41456],{"className":41455,"ariaHidden":2113},[2112],[54,41457,41459,41462,41465,41468],{"className":41458},[2117],[54,41460],{"className":41461,"style":2325},[2121],[54,41463,2305],{"className":41464},[2497],[54,41466,37470],{"className":41467,"style":2332},[2126,2167],[54,41469,2312],{"className":41470},[2336],[41430,41472,41473],{},"Ground state of the qubit",[41415,41475,41476,41517],{},[41430,41477,41478],{},[54,41479,41481,41499],{"className":41480},[2079],[54,41482,41484],{"className":41483},[2083],[2085,41485,41486],{"xmlns":2087},[2089,41487,41488,41496],{},[2092,41489,41490,41492,41494],{},[2245,41491,2305],{"stretchy":2311},[2098,41493,2842],{},[2245,41495,2312],{"stretchy":2311},[2105,41497,41498],{"encoding":2107},"\\lvert e\\rangle",[54,41500,41502],{"className":41501,"ariaHidden":2113},[2112],[54,41503,41505,41508,41511,41514],{"className":41504},[2117],[54,41506],{"className":41507,"style":2325},[2121],[54,41509,2305],{"className":41510},[2497],[54,41512,2842],{"className":41513},[2126,2167],[54,41515,2312],{"className":41516},[2336],[41430,41518,41519],{},"Excited state of the qubit",[41415,41521,41522,41563],{},[41430,41523,41524],{},[54,41525,41527,41545],{"className":41526},[2079],[54,41528,41530],{"className":41529},[2083],[2085,41531,41532],{"xmlns":2087},[2089,41533,41534,41542],{},[2092,41535,41536,41538,41540],{},[2245,41537,2305],{"stretchy":2311},[2098,41539,3349],{},[2245,41541,2312],{"stretchy":2311},[2105,41543,41544],{"encoding":2107},"\\lvert \\psi\\rangle",[54,41546,41548],{"className":41547,"ariaHidden":2113},[2112],[54,41549,41551,41554,41557,41560],{"className":41550},[2117],[54,41552],{"className":41553,"style":2325},[2121],[54,41555,2305],{"className":41556},[2497],[54,41558,3349],{"className":41559,"style":2332},[2126,2167],[54,41561,2312],{"className":41562},[2336],[41430,41564,41565],{},"Quantum state vector in a two-dimensional Hilbert space",[41415,41567,41568,41614],{},[41430,41569,41570],{},[54,41571,41573,41592],{"className":41572},[2079],[54,41574,41576],{"className":41575},[2083],[2085,41577,41578],{"xmlns":2087},[2089,41579,41580,41589],{},[2092,41581,41582,41585,41587],{},[2098,41583,41584],{},"α",[2245,41586,2375],{"separator":2113},[2098,41588,36222],{},[2105,41590,41591],{"encoding":2107},"\\alpha,\\beta",[54,41593,41595],{"className":41594,"ariaHidden":2113},[2112],[54,41596,41598,41601,41605,41608,41611],{"className":41597},[2117],[54,41599],{"className":41600,"style":6591},[2121],[54,41602,41584],{"className":41603,"style":41604},[2126,2167],"margin-right:0.0037em;",[54,41606,2375],{"className":41607},[2517],[54,41609],{"className":41610,"style":2521},[2270],[54,41612,36222],{"className":41613,"style":36317},[2126,2167],[41430,41615,41616,41617,10355,41655],{},"Complex probability amplitudes for ",[54,41618,41620,41637],{"className":41619},[2079],[54,41621,41623],{"className":41622},[2083],[2085,41624,41625],{"xmlns":2087},[2089,41626,41627,41635],{},[2092,41628,41629,41631,41633],{},[2245,41630,2305],{"stretchy":2311},[2098,41632,37470],{},[2245,41634,2312],{"stretchy":2311},[2105,41636,41452],{"encoding":2107},[54,41638,41640],{"className":41639,"ariaHidden":2113},[2112],[54,41641,41643,41646,41649,41652],{"className":41642},[2117],[54,41644],{"className":41645,"style":2325},[2121],[54,41647,2305],{"className":41648},[2497],[54,41650,37470],{"className":41651,"style":2332},[2126,2167],[54,41653,2312],{"className":41654},[2336],[54,41656,41658,41675],{"className":41657},[2079],[54,41659,41661],{"className":41660},[2083],[2085,41662,41663],{"xmlns":2087},[2089,41664,41665,41673],{},[2092,41666,41667,41669,41671],{},[2245,41668,2305],{"stretchy":2311},[2098,41670,2842],{},[2245,41672,2312],{"stretchy":2311},[2105,41674,41498],{"encoding":2107},[54,41676,41678],{"className":41677,"ariaHidden":2113},[2112],[54,41679,41681,41684,41687,41690],{"className":41680},[2117],[54,41682],{"className":41683,"style":2325},[2121],[54,41685,2305],{"className":41686},[2497],[54,41688,2842],{"className":41689},[2126,2167],[54,41691,2312],{"className":41692},[2336],[41415,41694,41695,41767],{},[41430,41696,41697],{},[54,41698,41700,41718],{"className":41699},[2079],[54,41701,41703],{"className":41702},[2083],[2085,41704,41705],{"xmlns":2087},[2089,41706,41707,41715],{},[2092,41708,41709],{},[2724,41710,41711,41713],{},[2098,41712,2730],{},[2098,41714,2842],{},[2105,41716,41717],{"encoding":2107},"P_e",[54,41719,41721],{"className":41720,"ariaHidden":2113},[2112],[54,41722,41724,41727],{"className":41723},[2117],[54,41725],{"className":41726,"style":2743},[2121],[54,41728,41730,41733],{"className":41729},[2126],[54,41731,2730],{"className":41732,"style":2776},[2126,2167],[54,41734,41736],{"className":41735},[2135],[54,41737,41739,41759],{"className":41738},[2139,2756],[54,41740,41742,41756],{"className":41741},[2143],[54,41743,41745],{"className":41744,"style":4063},[2147],[54,41746,41747,41750],{"style":7193},[54,41748],{"className":41749,"style":2156},[2155],[54,41751,41753],{"className":41752},[2160,2161,2162,2163],[54,41754,2842],{"className":41755},[2126,2167,2163],[54,41757,2781],{"className":41758},[2780],[54,41760,41762],{"className":41761},[2143],[54,41763,41765],{"className":41764,"style":2788},[2147],[54,41766],{},[41430,41768,41769],{},"Probability of measuring the qubit in the excited state",[41415,41771,41772,41845],{},[41430,41773,41774],{},[54,41775,41777,41795],{"className":41776},[2079],[54,41778,41780],{"className":41779},[2083],[2085,41781,41782],{"xmlns":2087},[2089,41783,41784,41792],{},[2092,41785,41786],{},[2724,41787,41788,41790],{},[2098,41789,21834],{},[2249,41791,150],{},[2105,41793,41794],{"encoding":2107},"f_0",[54,41796,41798],{"className":41797,"ariaHidden":2113},[2112],[54,41799,41801,41804],{"className":41800},[2117],[54,41802],{"className":41803,"style":6591},[2121],[54,41805,41807,41810],{"className":41806},[2126],[54,41808,21834],{"className":41809,"style":26895},[2126,2167],[54,41811,41813],{"className":41812},[2135],[54,41814,41816,41837],{"className":41815},[2139,2756],[54,41817,41819,41834],{"className":41818},[2143],[54,41820,41822],{"className":41821,"style":13837},[2147],[54,41823,41825,41828],{"style":41824},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[54,41826],{"className":41827,"style":2156},[2155],[54,41829,41831],{"className":41830},[2160,2161,2162,2163],[54,41832,150],{"className":41833},[2126,2163],[54,41835,2781],{"className":41836},[2780],[54,41838,41840],{"className":41839},[2143],[54,41841,41843],{"className":41842,"style":2788},[2147],[54,41844],{},[41430,41846,41847],{},"Natural transition frequency of the qubit",[41415,41849,41850,41880],{},[41430,41851,41852],{},[54,41853,41855,41868],{"className":41854},[2079],[54,41856,41858],{"className":41857},[2083],[2085,41859,41860],{"xmlns":2087},[2089,41861,41862,41866],{},[2092,41863,41864],{},[2098,41865,21834],{},[2105,41867,21834],{"encoding":2107},[54,41869,41871],{"className":41870,"ariaHidden":2113},[2112],[54,41872,41874,41877],{"className":41873},[2117],[54,41875],{"className":41876,"style":6591},[2121],[54,41878,21834],{"className":41879,"style":26895},[2126,2167],[41430,41881,41882],{},"Applied drive frequency",[41415,41884,41885,41958],{},[41430,41886,41887],{},[54,41888,41890,41909],{"className":41889},[2079],[54,41891,41893],{"className":41892},[2083],[2085,41894,41895],{"xmlns":2087},[2089,41896,41897,41906],{},[2092,41898,41899],{},[2724,41900,41901,41904],{},[2098,41902,41903],{},"ω",[2249,41905,150],{},[2105,41907,41908],{"encoding":2107},"\\omega_0",[54,41910,41912],{"className":41911,"ariaHidden":2113},[2112],[54,41913,41915,41918],{"className":41914},[2117],[54,41916],{"className":41917,"style":24753},[2121],[54,41919,41921,41924],{"className":41920},[2126],[54,41922,41903],{"className":41923,"style":2332},[2126,2167],[54,41925,41927],{"className":41926},[2135],[54,41928,41930,41950],{"className":41929},[2139,2756],[54,41931,41933,41947],{"className":41932},[2143],[54,41934,41936],{"className":41935,"style":13837},[2147],[54,41937,41938,41941],{"style":4128},[54,41939],{"className":41940,"style":2156},[2155],[54,41942,41944],{"className":41943},[2160,2161,2162,2163],[54,41945,150],{"className":41946},[2126,2163],[54,41948,2781],{"className":41949},[2780],[54,41951,41953],{"className":41952},[2143],[54,41954,41956],{"className":41955,"style":2788},[2147],[54,41957],{},[41430,41959,41960,41961],{},"Angular transition frequency, ",[54,41962,41964,41994],{"className":41963},[2079],[54,41965,41967],{"className":41966},[2083],[2085,41968,41969],{"xmlns":2087},[2089,41970,41971,41991],{},[2092,41972,41973,41979,41981,41983,41985],{},[2724,41974,41975,41977],{},[2098,41976,41903],{},[2249,41978,150],{},[2245,41980,122],{},[2249,41982,133],{},[2098,41984,8119],{},[2724,41986,41987,41989],{},[2098,41988,21834],{},[2249,41990,150],{},[2105,41992,41993],{"encoding":2107},"\\omega_0=2\\pi f_0",[54,41995,41997,42052],{"className":41996,"ariaHidden":2113},[2112],[54,41998,42000,42003,42043,42046,42049],{"className":41999},[2117],[54,42001],{"className":42002,"style":24753},[2121],[54,42004,42006,42009],{"className":42005},[2126],[54,42007,41903],{"className":42008,"style":2332},[2126,2167],[54,42010,42012],{"className":42011},[2135],[54,42013,42015,42035],{"className":42014},[2139,2756],[54,42016,42018,42032],{"className":42017},[2143],[54,42019,42021],{"className":42020,"style":13837},[2147],[54,42022,42023,42026],{"style":4128},[54,42024],{"className":42025,"style":2156},[2155],[54,42027,42029],{"className":42028},[2160,2161,2162,2163],[54,42030,150],{"className":42031},[2126,2163],[54,42033,2781],{"className":42034},[2780],[54,42036,42038],{"className":42037},[2143],[54,42039,42041],{"className":42040,"style":2788},[2147],[54,42042],{},[54,42044],{"className":42045,"style":2475},[2270],[54,42047,122],{"className":42048},[2479],[54,42050],{"className":42051,"style":2475},[2270],[54,42053,42055,42058,42061,42064],{"className":42054},[2117],[54,42056],{"className":42057,"style":6591},[2121],[54,42059,133],{"className":42060},[2126],[54,42062,8119],{"className":42063,"style":2332},[2126,2167],[54,42065,42067,42070],{"className":42066},[2126],[54,42068,21834],{"className":42069,"style":26895},[2126,2167],[54,42071,42073],{"className":42072},[2135],[54,42074,42076,42096],{"className":42075},[2139,2756],[54,42077,42079,42093],{"className":42078},[2143],[54,42080,42082],{"className":42081,"style":13837},[2147],[54,42083,42084,42087],{"style":41824},[54,42085],{"className":42086,"style":2156},[2155],[54,42088,42090],{"className":42089},[2160,2161,2162,2163],[54,42091,150],{"className":42092},[2126,2163],[54,42094,2781],{"className":42095},[2780],[54,42097,42099],{"className":42098},[2143],[54,42100,42102],{"className":42101,"style":2788},[2147],[54,42103],{},[41415,42105,42106,42137],{},[41430,42107,42108],{},[54,42109,42111,42125],{"className":42110},[2079],[54,42112,42114],{"className":42113},[2083],[2085,42115,42116],{"xmlns":2087},[2089,42117,42118,42122],{},[2092,42119,42120],{},[2098,42121,41903],{},[2105,42123,42124],{"encoding":2107},"\\omega",[54,42126,42128],{"className":42127,"ariaHidden":2113},[2112],[54,42129,42131,42134],{"className":42130},[2117],[54,42132],{"className":42133,"style":2195},[2121],[54,42135,41903],{"className":42136,"style":2332},[2126,2167],[41430,42138,42139,42140],{},"Angular drive frequency, ",[54,42141,42143,42165],{"className":42142},[2079],[54,42144,42146],{"className":42145},[2083],[2085,42147,42148],{"xmlns":2087},[2089,42149,42150,42162],{},[2092,42151,42152,42154,42156,42158,42160],{},[2098,42153,41903],{},[2245,42155,122],{},[2249,42157,133],{},[2098,42159,8119],{},[2098,42161,21834],{},[2105,42163,42164],{"encoding":2107},"\\omega=2\\pi f",[54,42166,42168,42186],{"className":42167,"ariaHidden":2113},[2112],[54,42169,42171,42174,42177,42180,42183],{"className":42170},[2117],[54,42172],{"className":42173,"style":2195},[2121],[54,42175,41903],{"className":42176,"style":2332},[2126,2167],[54,42178],{"className":42179,"style":2475},[2270],[54,42181,122],{"className":42182},[2479],[54,42184],{"className":42185,"style":2475},[2270],[54,42187,42189,42192,42195,42198],{"className":42188},[2117],[54,42190],{"className":42191,"style":6591},[2121],[54,42193,133],{"className":42194},[2126],[54,42196,8119],{"className":42197,"style":2332},[2126,2167],[54,42199,21834],{"className":42200,"style":26895},[2126,2167],[41415,42202,42203,42235],{},[41430,42204,42205],{},[54,42206,42208,42223],{"className":42207},[2079],[54,42209,42211],{"className":42210},[2083],[2085,42212,42213],{"xmlns":2087},[2089,42214,42215,42220],{},[2092,42216,42217],{},[2098,42218,42219],{"mathvariant":2304},"Ω",[2105,42221,42222],{"encoding":2107},"\\Omega",[54,42224,42226],{"className":42225,"ariaHidden":2113},[2112],[54,42227,42229,42232],{"className":42228},[2117],[54,42230],{"className":42231,"style":2122},[2121],[54,42233,42219],{"className":42234},[2126],[41430,42236,42237],{},"On-resonance Rabi angular frequency",[41415,42239,42240,42281],{},[41430,42241,42242],{},[54,42243,42245,42265],{"className":42244},[2079],[54,42246,42248],{"className":42247},[2083],[2085,42249,42250],{"xmlns":2087},[2089,42251,42252,42262],{},[2092,42253,42254,42256,42258,42260],{},[2098,42255,42219],{"mathvariant":2304},[2098,42257,194],{"mathvariant":2304},[2249,42259,133],{},[2098,42261,8119],{},[2105,42263,42264],{"encoding":2107},"\\Omega\u002F2\\pi",[54,42266,42268],{"className":42267,"ariaHidden":2113},[2112],[54,42269,42271,42274,42278],{"className":42270},[2117],[54,42272],{"className":42273,"style":2325},[2121],[54,42275,42277],{"className":42276},[2126],"Ω\u002F2",[54,42279,8119],{"className":42280,"style":2332},[2126,2167],[41430,42282,42283],{},"On-resonance Rabi frequency in ordinary frequency units",[41415,42285,42286,42318],{},[41430,42287,42288],{},[54,42289,42291,42306],{"className":42290},[2079],[54,42292,42294],{"className":42293},[2083],[2085,42295,42296],{"xmlns":2087},[2089,42297,42298,42303],{},[2092,42299,42300],{},[2098,42301,42302],{"mathvariant":2304},"Δ",[2105,42304,42305],{"encoding":2107},"\\Delta",[54,42307,42309],{"className":42308,"ariaHidden":2113},[2112],[54,42310,42312,42315],{"className":42311},[2117],[54,42313],{"className":42314,"style":2122},[2121],[54,42316,42302],{"className":42317},[2126],[41430,42319,42320,42321],{},"Drive detuning, ",[54,42322,42324,42350],{"className":42323},[2079],[54,42325,42327],{"className":42326},[2083],[2085,42328,42329],{"xmlns":2087},[2089,42330,42331,42347],{},[2092,42332,42333,42335,42337,42339,42341],{},[2098,42334,42302],{"mathvariant":2304},[2245,42336,122],{},[2098,42338,41903],{},[2245,42340,2247],{},[2724,42342,42343,42345],{},[2098,42344,41903],{},[2249,42346,150],{},[2105,42348,42349],{"encoding":2107},"\\Delta=\\omega-\\omega_0",[54,42351,42353,42371,42389],{"className":42352,"ariaHidden":2113},[2112],[54,42354,42356,42359,42362,42365,42368],{"className":42355},[2117],[54,42357],{"className":42358,"style":2122},[2121],[54,42360,42302],{"className":42361},[2126],[54,42363],{"className":42364,"style":2475},[2270],[54,42366,122],{"className":42367},[2479],[54,42369],{"className":42370,"style":2475},[2270],[54,42372,42374,42377,42380,42383,42386],{"className":42373},[2117],[54,42375],{"className":42376,"style":2263},[2121],[54,42378,41903],{"className":42379,"style":2332},[2126,2167],[54,42381],{"className":42382,"style":2271},[2270],[54,42384,2247],{"className":42385},[2275],[54,42387],{"className":42388,"style":2271},[2270],[54,42390,42392,42395],{"className":42391},[2117],[54,42393],{"className":42394,"style":24753},[2121],[54,42396,42398,42401],{"className":42397},[2126],[54,42399,41903],{"className":42400,"style":2332},[2126,2167],[54,42402,42404],{"className":42403},[2135],[54,42405,42407,42427],{"className":42406},[2139,2756],[54,42408,42410,42424],{"className":42409},[2143],[54,42411,42413],{"className":42412,"style":13837},[2147],[54,42414,42415,42418],{"style":4128},[54,42416],{"className":42417,"style":2156},[2155],[54,42419,42421],{"className":42420},[2160,2161,2162,2163],[54,42422,150],{"className":42423},[2126,2163],[54,42425,2781],{"className":42426},[2780],[54,42428,42430],{"className":42429},[2143],[54,42431,42433],{"className":42432,"style":2788},[2147],[54,42434],{},[41415,42436,42437,42509],{},[41430,42438,42439],{},[54,42440,42442,42460],{"className":42441},[2079],[54,42443,42445],{"className":42444},[2083],[2085,42446,42447],{"xmlns":2087},[2089,42448,42449,42457],{},[2092,42450,42451],{},[2724,42452,42453,42455],{},[2098,42454,42219],{"mathvariant":2304},[2098,42456,19926],{},[2105,42458,42459],{"encoding":2107},"\\Omega_R",[54,42461,42463],{"className":42462,"ariaHidden":2113},[2112],[54,42464,42466,42469],{"className":42465},[2117],[54,42467],{"className":42468,"style":2743},[2121],[54,42470,42472,42475],{"className":42471},[2126],[54,42473,42219],{"className":42474},[2126],[54,42476,42478],{"className":42477},[2135],[54,42479,42481,42501],{"className":42480},[2139,2756],[54,42482,42484,42498],{"className":42483},[2143],[54,42485,42487],{"className":42486,"style":2763},[2147],[54,42488,42489,42492],{"style":4954},[54,42490],{"className":42491,"style":2156},[2155],[54,42493,42495],{"className":42494},[2160,2161,2162,2163],[54,42496,19926],{"className":42497,"style":19951},[2126,2167,2163],[54,42499,2781],{"className":42500},[2780],[54,42502,42504],{"className":42503},[2143],[54,42505,42507],{"className":42506,"style":2788},[2147],[54,42508],{},[41430,42510,42511],{},"Generalized Rabi angular frequency",[41415,42513,42514,42589],{},[41430,42515,42516],{},[54,42517,42519,42538],{"className":42518},[2079],[54,42520,42522],{"className":42521},[2083],[2085,42523,42524],{"xmlns":2087},[2089,42525,42526,42535],{},[2092,42527,42528],{},[2724,42529,42530,42533],{},[2098,42531,42532],{},"ν",[2098,42534,19926],{},[2105,42536,42537],{"encoding":2107},"\\nu_R",[54,42539,42541],{"className":42540,"ariaHidden":2113},[2112],[54,42542,42544,42547],{"className":42543},[2117],[54,42545],{"className":42546,"style":24753},[2121],[54,42548,42550,42554],{"className":42549},[2126],[54,42551,42532],{"className":42552,"style":42553},[2126,2167],"margin-right:0.0637em;",[54,42555,42557],{"className":42556},[2135],[54,42558,42560,42581],{"className":42559},[2139,2756],[54,42561,42563,42578],{"className":42562},[2143],[54,42564,42566],{"className":42565,"style":2763},[2147],[54,42567,42569,42572],{"style":42568},"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;",[54,42570],{"className":42571,"style":2156},[2155],[54,42573,42575],{"className":42574},[2160,2161,2162,2163],[54,42576,19926],{"className":42577,"style":19951},[2126,2167,2163],[54,42579,2781],{"className":42580},[2780],[54,42582,42584],{"className":42583},[2143],[54,42585,42587],{"className":42586,"style":2788},[2147],[54,42588],{},[41430,42590,42591,42592],{},"Generalized Rabi frequency, ",[54,42593,42595,42627],{"className":42594},[2079],[54,42596,42598],{"className":42597},[2083],[2085,42599,42600],{"xmlns":2087},[2089,42601,42602,42624],{},[2092,42603,42604,42610,42612,42618,42620,42622],{},[2724,42605,42606,42608],{},[2098,42607,42532],{},[2098,42609,19926],{},[2245,42611,122],{},[2724,42613,42614,42616],{},[2098,42615,42219],{"mathvariant":2304},[2098,42617,19926],{},[2098,42619,194],{"mathvariant":2304},[2249,42621,133],{},[2098,42623,8119],{},[2105,42625,42626],{"encoding":2107},"\\nu_R=\\Omega_R\u002F2\\pi",[54,42628,42630,42685],{"className":42629,"ariaHidden":2113},[2112],[54,42631,42633,42636,42676,42679,42682],{"className":42632},[2117],[54,42634],{"className":42635,"style":24753},[2121],[54,42637,42639,42642],{"className":42638},[2126],[54,42640,42532],{"className":42641,"style":42553},[2126,2167],[54,42643,42645],{"className":42644},[2135],[54,42646,42648,42668],{"className":42647},[2139,2756],[54,42649,42651,42665],{"className":42650},[2143],[54,42652,42654],{"className":42653,"style":2763},[2147],[54,42655,42656,42659],{"style":42568},[54,42657],{"className":42658,"style":2156},[2155],[54,42660,42662],{"className":42661},[2160,2161,2162,2163],[54,42663,19926],{"className":42664,"style":19951},[2126,2167,2163],[54,42666,2781],{"className":42667},[2780],[54,42669,42671],{"className":42670},[2143],[54,42672,42674],{"className":42673,"style":2788},[2147],[54,42675],{},[54,42677],{"className":42678,"style":2475},[2270],[54,42680,122],{"className":42681},[2479],[54,42683],{"className":42684,"style":2475},[2270],[54,42686,42688,42691,42731,42734],{"className":42687},[2117],[54,42689],{"className":42690,"style":2325},[2121],[54,42692,42694,42697],{"className":42693},[2126],[54,42695,42219],{"className":42696},[2126],[54,42698,42700],{"className":42699},[2135],[54,42701,42703,42723],{"className":42702},[2139,2756],[54,42704,42706,42720],{"className":42705},[2143],[54,42707,42709],{"className":42708,"style":2763},[2147],[54,42710,42711,42714],{"style":4954},[54,42712],{"className":42713,"style":2156},[2155],[54,42715,42717],{"className":42716},[2160,2161,2162,2163],[54,42718,19926],{"className":42719,"style":19951},[2126,2167,2163],[54,42721,2781],{"className":42722},[2780],[54,42724,42726],{"className":42725},[2143],[54,42727,42729],{"className":42728,"style":2788},[2147],[54,42730],{},[54,42732,25677],{"className":42733},[2126],[54,42735,8119],{"className":42736,"style":2332},[2126,2167],[41415,42738,42739,42769],{},[41430,42740,42741],{},[54,42742,42744,42757],{"className":42743},[2079],[54,42745,42747],{"className":42746},[2083],[2085,42748,42749],{"xmlns":2087},[2089,42750,42751,42755],{},[2092,42752,42753],{},[2098,42754,41261],{},[2105,42756,41387],{"encoding":2107},[54,42758,42760],{"className":42759,"ariaHidden":2113},[2112],[54,42761,42763,42766],{"className":42762},[2117],[54,42764],{"className":42765,"style":2195},[2121],[54,42767,41261],{"className":42768,"style":41333},[2126,2167],[41430,42770,42771],{},"Pulse duration",[41415,42773,42774,42862],{},[41430,42775,42776],{},[54,42777,42779,42799],{"className":42778},[2079],[54,42780,42782],{"className":42781},[2083],[2085,42783,42784],{"xmlns":2087},[2089,42785,42786,42796],{},[2092,42787,42788],{},[3331,42789,42790,42792,42794],{},[2098,42791,36603],{},[2249,42793,133],{},[2245,42795,20325],{},[2105,42797,42798],{"encoding":2107},"T_2^\\ast",[54,42800,42802],{"className":42801,"ariaHidden":2113},[2112],[54,42803,42805,42809],{"className":42804},[2117],[54,42806],{"className":42807,"style":42808},[2121],"height:0.9368em;vertical-align:-0.2481em;",[54,42810,42812,42815],{"className":42811},[2126],[54,42813,36603],{"className":42814,"style":2776},[2126,2167],[54,42816,42818],{"className":42817},[2135],[54,42819,42821,42853],{"className":42820},[2139,2756],[54,42822,42824,42850],{"className":42823},[2143],[54,42825,42827,42839],{"className":42826,"style":36139},[2147],[54,42828,42830,42833],{"style":42829},"top:-2.4519em;margin-left:-0.1389em;margin-right:0.05em;",[54,42831],{"className":42832,"style":2156},[2155],[54,42834,42836],{"className":42835},[2160,2161,2162,2163],[54,42837,133],{"className":42838},[2126,2163],[54,42840,42841,42844],{"style":2151},[54,42842],{"className":42843,"style":2156},[2155],[54,42845,42847],{"className":42846},[2160,2161,2162,2163],[54,42848,20325],{"className":42849},[2275,2163],[54,42851,2781],{"className":42852},[2780],[54,42854,42856],{"className":42855},[2143],[54,42857,42860],{"className":42858,"style":42859},[2147],"height:0.2481em;",[54,42861],{},[41430,42863,42864],{},"Phenomenological dephasing time used as an optional damping parameter",[41415,42866,42867,42870],{},[41430,42868,42869],{},"RWA",[41430,42871,42872],{},"Rotating-wave approximation",[41415,42874,42875,42878],{},[41430,42876,42877],{},"FFT",[41430,42879,42880],{},"Fast Fourier transform",[31,42882,42884],{"id":42883},"physical-model","Physical Model",[13,42886,42887,42888,42926,42927,42965],{},"A qubit can be modeled as a two-level quantum system with a ground state ",[54,42889,42891,42908],{"className":42890},[2079],[54,42892,42894],{"className":42893},[2083],[2085,42895,42896],{"xmlns":2087},[2089,42897,42898,42906],{},[2092,42899,42900,42902,42904],{},[2245,42901,2305],{"stretchy":2311},[2098,42903,37470],{},[2245,42905,2312],{"stretchy":2311},[2105,42907,41452],{"encoding":2107},[54,42909,42911],{"className":42910,"ariaHidden":2113},[2112],[54,42912,42914,42917,42920,42923],{"className":42913},[2117],[54,42915],{"className":42916,"style":2325},[2121],[54,42918,2305],{"className":42919},[2497],[54,42921,37470],{"className":42922,"style":2332},[2126,2167],[54,42924,2312],{"className":42925},[2336]," and an excited state ",[54,42928,42930,42947],{"className":42929},[2079],[54,42931,42933],{"className":42932},[2083],[2085,42934,42935],{"xmlns":2087},[2089,42936,42937,42945],{},[2092,42938,42939,42941,42943],{},[2245,42940,2305],{"stretchy":2311},[2098,42942,2842],{},[2245,42944,2312],{"stretchy":2311},[2105,42946,41498],{"encoding":2107},[54,42948,42950],{"className":42949,"ariaHidden":2113},[2112],[54,42951,42953,42956,42959,42962],{"className":42952},[2117],[54,42954],{"className":42955,"style":2325},[2121],[54,42957,2305],{"className":42958},[2497],[54,42960,2842],{"className":42961},[2126,2167],[54,42963,2312],{"className":42964},[2336],". In the absence of a microwave drive, the energy splitting is",[54,42967,42969],{"className":42968},[2340],[54,42970,42972,43011],{"className":42971},[2079],[54,42973,42975],{"className":42974},[2083],[2085,42976,42977],{"xmlns":2087,"display":2349},[2089,42978,42979,43008],{},[2092,42980,42981,42987,42989,42995,42997,43000,43006],{},[2724,42982,42983,42985],{},[2098,42984,5738],{},[2098,42986,2842],{},[2245,42988,2247],{},[2724,42990,42991,42993],{},[2098,42992,5738],{},[2098,42994,37470],{},[2245,42996,122],{},[2098,42998,42999],{"mathvariant":2304},"ℏ",[2724,43001,43002,43004],{},[2098,43003,41903],{},[2249,43005,150],{},[2245,43007,2375],{"separator":2113},[2105,43009,43010],{"encoding":2107},"E_e-E_g=\\hbar\\omega_0,",[54,43012,43014,43069,43124],{"className":43013,"ariaHidden":2113},[2112],[54,43015,43017,43020,43060,43063,43066],{"className":43016},[2117],[54,43018],{"className":43019,"style":2743},[2121],[54,43021,43023,43026],{"className":43022},[2126],[54,43024,5738],{"className":43025,"style":5448},[2126,2167],[54,43027,43029],{"className":43028},[2135],[54,43030,43032,43052],{"className":43031},[2139,2756],[54,43033,43035,43049],{"className":43034},[2143],[54,43036,43038],{"className":43037,"style":4063},[2147],[54,43039,43040,43043],{"style":5570},[54,43041],{"className":43042,"style":2156},[2155],[54,43044,43046],{"className":43045},[2160,2161,2162,2163],[54,43047,2842],{"className":43048},[2126,2167,2163],[54,43050,2781],{"className":43051},[2780],[54,43053,43055],{"className":43054},[2143],[54,43056,43058],{"className":43057,"style":2788},[2147],[54,43059],{},[54,43061],{"className":43062,"style":2271},[2270],[54,43064,2247],{"className":43065},[2275],[54,43067],{"className":43068,"style":2271},[2270],[54,43070,43072,43075,43115,43118,43121],{"className":43071},[2117],[54,43073],{"className":43074,"style":7523},[2121],[54,43076,43078,43081],{"className":43077},[2126],[54,43079,5738],{"className":43080,"style":5448},[2126,2167],[54,43082,43084],{"className":43083},[2135],[54,43085,43087,43107],{"className":43086},[2139,2756],[54,43088,43090,43104],{"className":43089},[2143],[54,43091,43093],{"className":43092,"style":4063},[2147],[54,43094,43095,43098],{"style":5570},[54,43096],{"className":43097,"style":2156},[2155],[54,43099,43101],{"className":43100},[2160,2161,2162,2163],[54,43102,37470],{"className":43103,"style":2332},[2126,2167,2163],[54,43105,2781],{"className":43106},[2780],[54,43108,43110],{"className":43109},[2143],[54,43111,43113],{"className":43112,"style":6389},[2147],[54,43114],{},[54,43116],{"className":43117,"style":2475},[2270],[54,43119,122],{"className":43120},[2479],[54,43122],{"className":43123,"style":2475},[2270],[54,43125,43127,43131,43134,43174],{"className":43126},[2117],[54,43128],{"className":43129,"style":43130},[2121],"height:0.8833em;vertical-align:-0.1944em;",[54,43132,42999],{"className":43133},[2126],[54,43135,43137,43140],{"className":43136},[2126],[54,43138,41903],{"className":43139,"style":2332},[2126,2167],[54,43141,43143],{"className":43142},[2135],[54,43144,43146,43166],{"className":43145},[2139,2756],[54,43147,43149,43163],{"className":43148},[2143],[54,43150,43152],{"className":43151,"style":13837},[2147],[54,43153,43154,43157],{"style":4128},[54,43155],{"className":43156,"style":2156},[2155],[54,43158,43160],{"className":43159},[2160,2161,2162,2163],[54,43161,150],{"className":43162},[2126,2163],[54,43164,2781],{"className":43165},[2780],[54,43167,43169],{"className":43168},[2143],[54,43170,43172],{"className":43171,"style":2788},[2147],[54,43173],{},[54,43175,2375],{"className":43176},[2517],[13,43178,43179],{},"where the angular transition frequency is",[54,43181,43183],{"className":43182},[2340],[54,43184,43186,43218],{"className":43185},[2079],[54,43187,43189],{"className":43188},[2083],[2085,43190,43191],{"xmlns":2087,"display":2349},[2089,43192,43193,43215],{},[2092,43194,43195,43201,43203,43205,43207,43213],{},[2724,43196,43197,43199],{},[2098,43198,41903],{},[2249,43200,150],{},[2245,43202,122],{},[2249,43204,133],{},[2098,43206,8119],{},[2724,43208,43209,43211],{},[2098,43210,21834],{},[2249,43212,150],{},[2098,43214,29],{"mathvariant":2304},[2105,43216,43217],{"encoding":2107},"\\omega_0=2\\pi f_0.",[54,43219,43221,43276],{"className":43220,"ariaHidden":2113},[2112],[54,43222,43224,43227,43267,43270,43273],{"className":43223},[2117],[54,43225],{"className":43226,"style":24753},[2121],[54,43228,43230,43233],{"className":43229},[2126],[54,43231,41903],{"className":43232,"style":2332},[2126,2167],[54,43234,43236],{"className":43235},[2135],[54,43237,43239,43259],{"className":43238},[2139,2756],[54,43240,43242,43256],{"className":43241},[2143],[54,43243,43245],{"className":43244,"style":13837},[2147],[54,43246,43247,43250],{"style":4128},[54,43248],{"className":43249,"style":2156},[2155],[54,43251,43253],{"className":43252},[2160,2161,2162,2163],[54,43254,150],{"className":43255},[2126,2163],[54,43257,2781],{"className":43258},[2780],[54,43260,43262],{"className":43261},[2143],[54,43263,43265],{"className":43264,"style":2788},[2147],[54,43266],{},[54,43268],{"className":43269,"style":2475},[2270],[54,43271,122],{"className":43272},[2479],[54,43274],{"className":43275,"style":2475},[2270],[54,43277,43279,43282,43285,43288,43328],{"className":43278},[2117],[54,43280],{"className":43281,"style":6591},[2121],[54,43283,133],{"className":43284},[2126],[54,43286,8119],{"className":43287,"style":2332},[2126,2167],[54,43289,43291,43294],{"className":43290},[2126],[54,43292,21834],{"className":43293,"style":26895},[2126,2167],[54,43295,43297],{"className":43296},[2135],[54,43298,43300,43320],{"className":43299},[2139,2756],[54,43301,43303,43317],{"className":43302},[2143],[54,43304,43306],{"className":43305,"style":13837},[2147],[54,43307,43308,43311],{"style":41824},[54,43309],{"className":43310,"style":2156},[2155],[54,43312,43314],{"className":43313},[2160,2161,2162,2163],[54,43315,150],{"className":43316},[2126,2163],[54,43318,2781],{"className":43319},[2780],[54,43321,43323],{"className":43322},[2143],[54,43324,43326],{"className":43325,"style":2788},[2147],[54,43327],{},[54,43329,29],{"className":43330},[2126],[13,43332,43333],{},"The default configuration uses a transition frequency of",[54,43335,43337],{"className":43336},[2340],[54,43338,43340,43376],{"className":43339},[2079],[54,43341,43343],{"className":43342},[2083],[2085,43344,43345],{"xmlns":2087,"display":2349},[2089,43346,43347,43373],{},[2092,43348,43349,43355,43357,43360,43362,43371],{},[2724,43350,43351,43353],{},[2098,43352,21834],{},[2249,43354,150],{},[2245,43356,122],{},[2249,43358,43359],{},"5.000",[5723,43361,5725],{},[2092,43363,43364,43367,43369],{},[2098,43365,43366],{"mathvariant":2304},"G",[2098,43368,2694],{"mathvariant":2304},[2098,43370,215],{"mathvariant":2304},[2098,43372,29],{"mathvariant":2304},[2105,43374,43375],{"encoding":2107},"f_0=5.000~\\mathrm{GHz}.",[54,43377,43379,43434],{"className":43378,"ariaHidden":2113},[2112],[54,43380,43382,43385,43425,43428,43431],{"className":43381},[2117],[54,43383],{"className":43384,"style":6591},[2121],[54,43386,43388,43391],{"className":43387},[2126],[54,43389,21834],{"className":43390,"style":26895},[2126,2167],[54,43392,43394],{"className":43393},[2135],[54,43395,43397,43417],{"className":43396},[2139,2756],[54,43398,43400,43414],{"className":43399},[2143],[54,43401,43403],{"className":43402,"style":13837},[2147],[54,43404,43405,43408],{"style":41824},[54,43406],{"className":43407,"style":2156},[2155],[54,43409,43411],{"className":43410},[2160,2161,2162,2163],[54,43412,150],{"className":43413},[2126,2163],[54,43415,2781],{"className":43416},[2780],[54,43418,43420],{"className":43419},[2143],[54,43421,43423],{"className":43422,"style":2788},[2147],[54,43424],{},[54,43426],{"className":43427,"style":2475},[2270],[54,43429,122],{"className":43430},[2479],[54,43432],{"className":43433,"style":2475},[2270],[54,43435,43437,43440,43443,43447,43455],{"className":43436},[2117],[54,43438],{"className":43439,"style":2122},[2121],[54,43441,43359],{"className":43442},[2126],[54,43444,5725],{"className":43445},[2270,43446],"nobreak",[54,43448,43450],{"className":43449},[2126],[54,43451,43454],{"className":43452},[2126,43453],"mathrm","GHz",[54,43456,29],{"className":43457},[2126],[13,43459,43460,43461,43489,43490,10355,43528,43566],{},"Applying a microwave drive with frequency ",[54,43462,43464,43477],{"className":43463},[2079],[54,43465,43467],{"className":43466},[2083],[2085,43468,43469],{"xmlns":2087},[2089,43470,43471,43475],{},[2092,43472,43473],{},[2098,43474,21834],{},[2105,43476,21834],{"encoding":2107},[54,43478,43480],{"className":43479,"ariaHidden":2113},[2112],[54,43481,43483,43486],{"className":43482},[2117],[54,43484],{"className":43485,"style":6591},[2121],[54,43487,21834],{"className":43488,"style":26895},[2126,2167]," induces coherent rotations between ",[54,43491,43493,43510],{"className":43492},[2079],[54,43494,43496],{"className":43495},[2083],[2085,43497,43498],{"xmlns":2087},[2089,43499,43500,43508],{},[2092,43501,43502,43504,43506],{},[2245,43503,2305],{"stretchy":2311},[2098,43505,37470],{},[2245,43507,2312],{"stretchy":2311},[2105,43509,41452],{"encoding":2107},[54,43511,43513],{"className":43512,"ariaHidden":2113},[2112],[54,43514,43516,43519,43522,43525],{"className":43515},[2117],[54,43517],{"className":43518,"style":2325},[2121],[54,43520,2305],{"className":43521},[2497],[54,43523,37470],{"className":43524,"style":2332},[2126,2167],[54,43526,2312],{"className":43527},[2336],[54,43529,43531,43548],{"className":43530},[2079],[54,43532,43534],{"className":43533},[2083],[2085,43535,43536],{"xmlns":2087},[2089,43537,43538,43546],{},[2092,43539,43540,43542,43544],{},[2245,43541,2305],{"stretchy":2311},[2098,43543,2842],{},[2245,43545,2312],{"stretchy":2311},[2105,43547,41498],{"encoding":2107},[54,43549,43551],{"className":43550,"ariaHidden":2113},[2112],[54,43552,43554,43557,43560,43563],{"className":43553},[2117],[54,43555],{"className":43556,"style":2325},[2121],[54,43558,2305],{"className":43559},[2497],[54,43561,2842],{"className":43562},[2126,2167],[54,43564,2312],{"className":43565},[2336],". Exact resonance occurs when",[54,43568,43570],{"className":43569},[2340],[54,43571,43573,43597],{"className":43572},[2079],[54,43574,43576],{"className":43575},[2083],[2085,43577,43578],{"xmlns":2087,"display":2349},[2089,43579,43580,43594],{},[2092,43581,43582,43584,43586,43592],{},[2098,43583,21834],{},[2245,43585,122],{},[2724,43587,43588,43590],{},[2098,43589,21834],{},[2249,43591,150],{},[2098,43593,29],{"mathvariant":2304},[2105,43595,43596],{"encoding":2107},"f=f_0.",[54,43598,43600,43618],{"className":43599,"ariaHidden":2113},[2112],[54,43601,43603,43606,43609,43612,43615],{"className":43602},[2117],[54,43604],{"className":43605,"style":6591},[2121],[54,43607,21834],{"className":43608,"style":26895},[2126,2167],[54,43610],{"className":43611,"style":2475},[2270],[54,43613,122],{"className":43614},[2479],[54,43616],{"className":43617,"style":2475},[2270],[54,43619,43621,43624,43664],{"className":43620},[2117],[54,43622],{"className":43623,"style":6591},[2121],[54,43625,43627,43630],{"className":43626},[2126],[54,43628,21834],{"className":43629,"style":26895},[2126,2167],[54,43631,43633],{"className":43632},[2135],[54,43634,43636,43656],{"className":43635},[2139,2756],[54,43637,43639,43653],{"className":43638},[2143],[54,43640,43642],{"className":43641,"style":13837},[2147],[54,43643,43644,43647],{"style":41824},[54,43645],{"className":43646,"style":2156},[2155],[54,43648,43650],{"className":43649},[2160,2161,2162,2163],[54,43651,150],{"className":43652},[2126,2163],[54,43654,2781],{"className":43655},[2780],[54,43657,43659],{"className":43658},[2143],[54,43660,43662],{"className":43661,"style":2788},[2147],[54,43663],{},[54,43665,29],{"className":43666},[2126],[13,43668,43669],{},"At resonance, the drive produces ideal Rabi oscillations. Away from resonance, detuning changes the oscillation frequency and reduces the maximum achievable excited-state probability. This same two-level control structure appears across superconducting qubits, spin qubits, trapped ions, neutral atoms, and nitrogen-vacancy centers whenever a selected transition can be treated as an effective qubit.",[31,43671,43673],{"id":43672},"mathematical-foundation","Mathematical Foundation",[13,43675,43676],{},"The quantum state is a normalized vector in a two-dimensional Hilbert space,",[54,43678,43680],{"className":43679},[2340],[54,43681,43683,43723],{"className":43682},[2079],[54,43684,43686],{"className":43685},[2083],[2085,43687,43688],{"xmlns":2087,"display":2349},[2089,43689,43690,43720],{},[2092,43691,43692,43694,43696,43698,43700,43702,43704,43706,43708,43710,43712,43714,43716,43718],{},[2245,43693,2305],{"stretchy":2311},[2098,43695,3349],{},[2245,43697,2312],{"stretchy":2311},[2245,43699,122],{},[2098,43701,41584],{},[2245,43703,2305],{"stretchy":2311},[2098,43705,37470],{},[2245,43707,2312],{"stretchy":2311},[2245,43709,1663],{},[2098,43711,36222],{},[2245,43713,2305],{"stretchy":2311},[2098,43715,2842],{},[2245,43717,2312],{"stretchy":2311},[2245,43719,2375],{"separator":2113},[2105,43721,43722],{"encoding":2107},"\\lvert \\psi\\rangle=\\alpha\\lvert g\\rangle+\\beta\\lvert e\\rangle,",[54,43724,43726,43750,43777],{"className":43725,"ariaHidden":2113},[2112],[54,43727,43729,43732,43735,43738,43741,43744,43747],{"className":43728},[2117],[54,43730],{"className":43731,"style":2325},[2121],[54,43733,2305],{"className":43734},[2497],[54,43736,3349],{"className":43737,"style":2332},[2126,2167],[54,43739,2312],{"className":43740},[2336],[54,43742],{"className":43743,"style":2475},[2270],[54,43745,122],{"className":43746},[2479],[54,43748],{"className":43749,"style":2475},[2270],[54,43751,43753,43756,43759,43762,43765,43768,43771,43774],{"className":43752},[2117],[54,43754],{"className":43755,"style":2325},[2121],[54,43757,41584],{"className":43758,"style":41604},[2126,2167],[54,43760,2305],{"className":43761},[2497],[54,43763,37470],{"className":43764,"style":2332},[2126,2167],[54,43766,2312],{"className":43767},[2336],[54,43769],{"className":43770,"style":2271},[2270],[54,43772,1663],{"className":43773},[2275],[54,43775],{"className":43776,"style":2271},[2270],[54,43778,43780,43783,43786,43789,43792,43795],{"className":43779},[2117],[54,43781],{"className":43782,"style":2325},[2121],[54,43784,36222],{"className":43785,"style":36317},[2126,2167],[54,43787,2305],{"className":43788},[2497],[54,43790,2842],{"className":43791},[2126,2167],[54,43793,2312],{"className":43794},[2336],[54,43796,2375],{"className":43797},[2517],[13,43799,28011,43800,10355,43829,43858],{},[54,43801,43803,43817],{"className":43802},[2079],[54,43804,43806],{"className":43805},[2083],[2085,43807,43808],{"xmlns":2087},[2089,43809,43810,43814],{},[2092,43811,43812],{},[2098,43813,41584],{},[2105,43815,43816],{"encoding":2107},"\\alpha",[54,43818,43820],{"className":43819,"ariaHidden":2113},[2112],[54,43821,43823,43826],{"className":43822},[2117],[54,43824],{"className":43825,"style":2195},[2121],[54,43827,41584],{"className":43828,"style":41604},[2126,2167],[54,43830,43832,43846],{"className":43831},[2079],[54,43833,43835],{"className":43834},[2083],[2085,43836,43837],{"xmlns":2087},[2089,43838,43839,43843],{},[2092,43840,43841],{},[2098,43842,36222],{},[2105,43844,43845],{"encoding":2107},"\\beta",[54,43847,43849],{"className":43848,"ariaHidden":2113},[2112],[54,43850,43852,43855],{"className":43851},[2117],[54,43853],{"className":43854,"style":6591},[2121],[54,43856,36222],{"className":43857,"style":36317},[2126,2167]," are complex probability amplitudes. Normalization requires",[54,43860,43862],{"className":43861},[2340],[54,43863,43865,43904],{"className":43864},[2079],[54,43866,43868],{"className":43867},[2083],[2085,43869,43870],{"xmlns":2087,"display":2349},[2089,43871,43872,43901],{},[2092,43873,43874,43876,43878,43884,43886,43888,43890,43896,43898],{},[2098,43875,2305],{"mathvariant":2304},[2098,43877,41584],{},[2095,43879,43880,43882],{},[2098,43881,2305],{"mathvariant":2304},[2249,43883,133],{},[2245,43885,1663],{},[2098,43887,2305],{"mathvariant":2304},[2098,43889,36222],{},[2095,43891,43892,43894],{},[2098,43893,2305],{"mathvariant":2304},[2249,43895,133],{},[2245,43897,122],{},[2249,43899,43900],{},"1.",[2105,43902,43903],{"encoding":2107},"|\\alpha|^2+|\\beta|^2=1.",[54,43905,43907,43958,44008],{"className":43906,"ariaHidden":2113},[2112],[54,43908,43910,43914,43917,43920,43949,43952,43955],{"className":43909},[2117],[54,43911],{"className":43912,"style":43913},[2121],"height:1.1141em;vertical-align:-0.25em;",[54,43915,2305],{"className":43916},[2126],[54,43918,41584],{"className":43919,"style":41604},[2126,2167],[54,43921,43923,43926],{"className":43922},[2126],[54,43924,2305],{"className":43925},[2126],[54,43927,43929],{"className":43928},[2135],[54,43930,43932],{"className":43931},[2139],[54,43933,43935],{"className":43934},[2143],[54,43936,43938],{"className":43937,"style":2558},[2147],[54,43939,43940,43943],{"style":2462},[54,43941],{"className":43942,"style":2156},[2155],[54,43944,43946],{"className":43945},[2160,2161,2162,2163],[54,43947,133],{"className":43948},[2126,2163],[54,43950],{"className":43951,"style":2271},[2270],[54,43953,1663],{"className":43954},[2275],[54,43956],{"className":43957,"style":2271},[2270],[54,43959,43961,43964,43967,43970,43999,44002,44005],{"className":43960},[2117],[54,43962],{"className":43963,"style":43913},[2121],[54,43965,2305],{"className":43966},[2126],[54,43968,36222],{"className":43969,"style":36317},[2126,2167],[54,43971,43973,43976],{"className":43972},[2126],[54,43974,2305],{"className":43975},[2126],[54,43977,43979],{"className":43978},[2135],[54,43980,43982],{"className":43981},[2139],[54,43983,43985],{"className":43984},[2143],[54,43986,43988],{"className":43987,"style":2558},[2147],[54,43989,43990,43993],{"style":2462},[54,43991],{"className":43992,"style":2156},[2155],[54,43994,43996],{"className":43995},[2160,2161,2162,2163],[54,43997,133],{"className":43998},[2126,2163],[54,44000],{"className":44001,"style":2475},[2270],[54,44003,122],{"className":44004},[2479],[54,44006],{"className":44007,"style":2475},[2270],[54,44009,44011,44014],{"className":44010},[2117],[54,44012],{"className":44013,"style":2285},[2121],[54,44015,43900],{"className":44016},[2126],[13,44018,44019],{},"The Born rule gives the probability of measuring the excited state,",[54,44021,44023],{"className":44022},[2340],[54,44024,44026,44078],{"className":44025},[2079],[54,44027,44029],{"className":44028},[2083],[2085,44030,44031],{"xmlns":2087,"display":2349},[2089,44032,44033,44075],{},[2092,44034,44035,44041,44043,44045,44047,44049,44051,44053,44055,44061,44063,44065,44067,44073],{},[2724,44036,44037,44039],{},[2098,44038,2730],{},[2098,44040,2842],{},[2245,44042,122],{},[2098,44044,2305],{"mathvariant":2304},[2245,44046,3981],{"stretchy":2311},[2098,44048,2842],{},[2245,44050,2305],{"stretchy":2311},[2098,44052,3349],{},[2245,44054,2312],{"stretchy":2311},[2095,44056,44057,44059],{},[2098,44058,2305],{"mathvariant":2304},[2249,44060,133],{},[2245,44062,122],{},[2098,44064,2305],{"mathvariant":2304},[2098,44066,36222],{},[2095,44068,44069,44071],{},[2098,44070,2305],{"mathvariant":2304},[2249,44072,133],{},[2098,44074,29],{"mathvariant":2304},[2105,44076,44077],{"encoding":2107},"P_e=|\\langle e\\lvert\\psi\\rangle|^2=|\\beta|^2.",[54,44079,44081,44136,44198],{"className":44080,"ariaHidden":2113},[2112],[54,44082,44084,44087,44127,44130,44133],{"className":44083},[2117],[54,44085],{"className":44086,"style":2743},[2121],[54,44088,44090,44093],{"className":44089},[2126],[54,44091,2730],{"className":44092,"style":2776},[2126,2167],[54,44094,44096],{"className":44095},[2135],[54,44097,44099,44119],{"className":44098},[2139,2756],[54,44100,44102,44116],{"className":44101},[2143],[54,44103,44105],{"className":44104,"style":4063},[2147],[54,44106,44107,44110],{"style":7193},[54,44108],{"className":44109,"style":2156},[2155],[54,44111,44113],{"className":44112},[2160,2161,2162,2163],[54,44114,2842],{"className":44115},[2126,2167,2163],[54,44117,2781],{"className":44118},[2780],[54,44120,44122],{"className":44121},[2143],[54,44123,44125],{"className":44124,"style":2788},[2147],[54,44126],{},[54,44128],{"className":44129,"style":2475},[2270],[54,44131,122],{"className":44132},[2479],[54,44134],{"className":44135,"style":2475},[2270],[54,44137,44139,44142,44145,44148,44151,44154,44157,44160,44189,44192,44195],{"className":44138},[2117],[54,44140],{"className":44141,"style":43913},[2121],[54,44143,2305],{"className":44144},[2126],[54,44146,3981],{"className":44147},[2497],[54,44149,2842],{"className":44150},[2126,2167],[54,44152,2305],{"className":44153},[2497],[54,44155,3349],{"className":44156,"style":2332},[2126,2167],[54,44158,2312],{"className":44159},[2336],[54,44161,44163,44166],{"className":44162},[2126],[54,44164,2305],{"className":44165},[2126],[54,44167,44169],{"className":44168},[2135],[54,44170,44172],{"className":44171},[2139],[54,44173,44175],{"className":44174},[2143],[54,44176,44178],{"className":44177,"style":2558},[2147],[54,44179,44180,44183],{"style":2462},[54,44181],{"className":44182,"style":2156},[2155],[54,44184,44186],{"className":44185},[2160,2161,2162,2163],[54,44187,133],{"className":44188},[2126,2163],[54,44190],{"className":44191,"style":2475},[2270],[54,44193,122],{"className":44194},[2479],[54,44196],{"className":44197,"style":2475},[2270],[54,44199,44201,44204,44207,44210,44239],{"className":44200},[2117],[54,44202],{"className":44203,"style":43913},[2121],[54,44205,2305],{"className":44206},[2126],[54,44208,36222],{"className":44209,"style":36317},[2126,2167],[54,44211,44213,44216],{"className":44212},[2126],[54,44214,2305],{"className":44215},[2126],[54,44217,44219],{"className":44218},[2135],[54,44220,44222],{"className":44221},[2139],[54,44223,44225],{"className":44224},[2143],[54,44226,44228],{"className":44227,"style":2558},[2147],[54,44229,44230,44233],{"style":2462},[54,44231],{"className":44232,"style":2156},[2155],[54,44234,44236],{"className":44235},[2160,2161,2162,2163],[54,44237,133],{"className":44238},[2126,2163],[54,44240,29],{"className":44241},[2126],[13,44243,44244],{},"Pauli matrices then provide a natural operator basis for the two-level system,",[54,44246,44248],{"className":44247},[2340],[54,44249,44251,44415],{"className":44250},[2079],[54,44252,44254],{"className":44253},[2083],[2085,44255,44256],{"xmlns":2087,"display":2349},[2089,44257,44258,44412],{},[2092,44259,44260,44267,44269,44306,44308,44310,44316,44318,44358,44360,44362,44368,44370,44410],{},[2724,44261,44262,44265],{},[2098,44263,44264],{},"σ",[2098,44266,8695],{},[2245,44268,122],{},[2092,44270,44271,44273,44304],{},[2245,44272,11804],{"fence":2113},[9008,44274,44276,44290],{"rowspacing":9010,"columnalign":44275,"columnspacing":9011},"center center",[9013,44277,44278,44284],{},[9016,44279,44280],{},[9019,44281,44282],{"scriptlevel":150,"displaystyle":2311},[2249,44283,150],{},[9016,44285,44286],{},[9019,44287,44288],{"scriptlevel":150,"displaystyle":2311},[2249,44289,174],{},[9013,44291,44292,44298],{},[9016,44293,44294],{},[9019,44295,44296],{"scriptlevel":150,"displaystyle":2311},[2249,44297,174],{},[9016,44299,44300],{},[9019,44301,44302],{"scriptlevel":150,"displaystyle":2311},[2249,44303,150],{},[2245,44305,11829],{"fence":2113},[2245,44307,2375],{"separator":2113},[2270,44309],{"width":17567},[2724,44311,44312,44314],{},[2098,44313,44264],{},[2098,44315,20011],{},[2245,44317,122],{},[2092,44319,44320,44322,44356],{},[2245,44321,11804],{"fence":2113},[9008,44323,44324,44342],{"rowspacing":9010,"columnalign":44275,"columnspacing":9011},[9013,44325,44326,44332],{},[9016,44327,44328],{},[9019,44329,44330],{"scriptlevel":150,"displaystyle":2311},[2249,44331,150],{},[9016,44333,44334],{},[9019,44335,44336],{"scriptlevel":150,"displaystyle":2311},[2092,44337,44338,44340],{},[2245,44339,2247],{},[2098,44341,2849],{},[9013,44343,44344,44350],{},[9016,44345,44346],{},[9019,44347,44348],{"scriptlevel":150,"displaystyle":2311},[2098,44349,2849],{},[9016,44351,44352],{},[9019,44353,44354],{"scriptlevel":150,"displaystyle":2311},[2249,44355,150],{},[2245,44357,11829],{"fence":2113},[2245,44359,2375],{"separator":2113},[2270,44361],{"width":17567},[2724,44363,44364,44366],{},[2098,44365,44264],{},[2098,44367,215],{},[2245,44369,122],{},[2092,44371,44372,44374,44408],{},[2245,44373,11804],{"fence":2113},[9008,44375,44376,44390],{"rowspacing":9010,"columnalign":44275,"columnspacing":9011},[9013,44377,44378,44384],{},[9016,44379,44380],{},[9019,44381,44382],{"scriptlevel":150,"displaystyle":2311},[2249,44383,174],{},[9016,44385,44386],{},[9019,44387,44388],{"scriptlevel":150,"displaystyle":2311},[2249,44389,150],{},[9013,44391,44392,44398],{},[9016,44393,44394],{},[9019,44395,44396],{"scriptlevel":150,"displaystyle":2311},[2249,44397,150],{},[9016,44399,44400],{},[9019,44401,44402],{"scriptlevel":150,"displaystyle":2311},[2092,44403,44404,44406],{},[2245,44405,2247],{},[2249,44407,174],{},[2245,44409,11829],{"fence":2113},[2098,44411,29],{"mathvariant":2304},[2105,44413,44414],{"encoding":2107},"\\sigma_x=\\begin{bmatrix}0 & 1 \\\\ 1 & 0\\end{bmatrix}, \\qquad\n\\sigma_y=\\begin{bmatrix}0 & -i \\\\ i & 0\\end{bmatrix}, \\qquad\n\\sigma_z=\\begin{bmatrix}1 & 0 \\\\ 0 & -1\\end{bmatrix}.",[54,44416,44418,44473,44663,44850],{"className":44417,"ariaHidden":2113},[2112],[54,44419,44421,44424,44464,44467,44470],{"className":44420},[2117],[54,44422],{"className":44423,"style":24753},[2121],[54,44425,44427,44430],{"className":44426},[2126],[54,44428,44264],{"className":44429,"style":2332},[2126,2167],[54,44431,44433],{"className":44432},[2135],[54,44434,44436,44456],{"className":44435},[2139,2756],[54,44437,44439,44453],{"className":44438},[2143],[54,44440,44442],{"className":44441,"style":4063},[2147],[54,44443,44444,44447],{"style":4128},[54,44445],{"className":44446,"style":2156},[2155],[54,44448,44450],{"className":44449},[2160,2161,2162,2163],[54,44451,8695],{"className":44452},[2126,2167,2163],[54,44454,2781],{"className":44455},[2780],[54,44457,44459],{"className":44458},[2143],[54,44460,44462],{"className":44461,"style":2788},[2147],[54,44463],{},[54,44465],{"className":44466,"style":2475},[2270],[54,44468,122],{"className":44469},[2479],[54,44471],{"className":44472,"style":2475},[2270],[54,44474,44476,44479,44602,44605,44608,44611,44614,44654,44657,44660],{"className":44475},[2117],[54,44477],{"className":44478,"style":37225},[2121],[54,44480,44482,44488,44596],{"className":44481},[2493],[54,44483,44485],{"className":44484,"style":2499},[2497,2498],[54,44486,11804],{"className":44487},[2503,2162],[54,44489,44491],{"className":44490},[2126],[54,44492,44494,44543,44548,44551],{"className":44493},[9008],[54,44495,44497],{"className":44496},[9267],[54,44498,44500,44534],{"className":44499},[2139,2756],[54,44501,44503,44531],{"className":44502},[2143],[54,44504,44507,44519],{"className":44505,"style":44506},[2147],"height:1.45em;",[54,44508,44510,44513],{"style":44509},"top:-3.61em;",[54,44511],{"className":44512,"style":3866},[2155],[54,44514,44516],{"className":44515},[2126],[54,44517,150],{"className":44518},[2126],[54,44520,44522,44525],{"style":44521},"top:-2.41em;",[54,44523],{"className":44524,"style":3866},[2155],[54,44526,44528],{"className":44527},[2126],[54,44529,174],{"className":44530},[2126],[54,44532,2781],{"className":44533},[2780],[54,44535,44537],{"className":44536},[2143],[54,44538,44541],{"className":44539,"style":44540},[2147],"height:0.95em;",[54,44542],{},[54,44544],{"className":44545,"style":44547},[44546],"arraycolsep","width:0.5em;",[54,44549],{"className":44550,"style":44547},[44546],[54,44552,44554],{"className":44553},[9267],[54,44555,44557,44588],{"className":44556},[2139,2756],[54,44558,44560,44585],{"className":44559},[2143],[54,44561,44563,44574],{"className":44562,"style":44506},[2147],[54,44564,44565,44568],{"style":44509},[54,44566],{"className":44567,"style":3866},[2155],[54,44569,44571],{"className":44570},[2126],[54,44572,174],{"className":44573},[2126],[54,44575,44576,44579],{"style":44521},[54,44577],{"className":44578,"style":3866},[2155],[54,44580,44582],{"className":44581},[2126],[54,44583,150],{"className":44584},[2126],[54,44586,2781],{"className":44587},[2780],[54,44589,44591],{"className":44590},[2143],[54,44592,44594],{"className":44593,"style":44540},[2147],[54,44595],{},[54,44597,44599],{"className":44598,"style":2499},[2336,2498],[54,44600,11829],{"className":44601},[2503,2162],[54,44603],{"className":44604,"style":2521},[2270],[54,44606,2375],{"className":44607},[2517],[54,44609],{"className":44610,"style":18185},[2270],[54,44612],{"className":44613,"style":2521},[2270],[54,44615,44617,44620],{"className":44616},[2126],[54,44618,44264],{"className":44619,"style":2332},[2126,2167],[54,44621,44623],{"className":44622},[2135],[54,44624,44626,44646],{"className":44625},[2139,2756],[54,44627,44629,44643],{"className":44628},[2143],[54,44630,44632],{"className":44631,"style":4063},[2147],[54,44633,44634,44637],{"style":4128},[54,44635],{"className":44636,"style":2156},[2155],[54,44638,44640],{"className":44639},[2160,2161,2162,2163],[54,44641,20011],{"className":44642,"style":2332},[2126,2167,2163],[54,44644,2781],{"className":44645},[2780],[54,44647,44649],{"className":44648},[2143],[54,44650,44652],{"className":44651,"style":6389},[2147],[54,44653],{},[54,44655],{"className":44656,"style":2475},[2270],[54,44658,122],{"className":44659},[2479],[54,44661],{"className":44662,"style":2475},[2270],[54,44664,44666,44669,44789,44792,44795,44798,44801,44841,44844,44847],{"className":44665},[2117],[54,44667],{"className":44668,"style":37225},[2121],[54,44670,44672,44678,44783],{"className":44671},[2493],[54,44673,44675],{"className":44674,"style":2499},[2497,2498],[54,44676,11804],{"className":44677},[2503,2162],[54,44679,44681],{"className":44680},[2126],[54,44682,44684,44729,44732,44735],{"className":44683},[9008],[54,44685,44687],{"className":44686},[9267],[54,44688,44690,44721],{"className":44689},[2139,2756],[54,44691,44693,44718],{"className":44692},[2143],[54,44694,44696,44707],{"className":44695,"style":44506},[2147],[54,44697,44698,44701],{"style":44509},[54,44699],{"className":44700,"style":3866},[2155],[54,44702,44704],{"className":44703},[2126],[54,44705,150],{"className":44706},[2126],[54,44708,44709,44712],{"style":44521},[54,44710],{"className":44711,"style":3866},[2155],[54,44713,44715],{"className":44714},[2126],[54,44716,2849],{"className":44717},[2126,2167],[54,44719,2781],{"className":44720},[2780],[54,44722,44724],{"className":44723},[2143],[54,44725,44727],{"className":44726,"style":44540},[2147],[54,44728],{},[54,44730],{"className":44731,"style":44547},[44546],[54,44733],{"className":44734,"style":44547},[44546],[54,44736,44738],{"className":44737},[9267],[54,44739,44741,44775],{"className":44740},[2139,2756],[54,44742,44744,44772],{"className":44743},[2143],[54,44745,44747,44761],{"className":44746,"style":44506},[2147],[54,44748,44749,44752],{"style":44509},[54,44750],{"className":44751,"style":3866},[2155],[54,44753,44755,44758],{"className":44754},[2126],[54,44756,2247],{"className":44757},[2126],[54,44759,2849],{"className":44760},[2126,2167],[54,44762,44763,44766],{"style":44521},[54,44764],{"className":44765,"style":3866},[2155],[54,44767,44769],{"className":44768},[2126],[54,44770,150],{"className":44771},[2126],[54,44773,2781],{"className":44774},[2780],[54,44776,44778],{"className":44777},[2143],[54,44779,44781],{"className":44780,"style":44540},[2147],[54,44782],{},[54,44784,44786],{"className":44785,"style":2499},[2336,2498],[54,44787,11829],{"className":44788},[2503,2162],[54,44790],{"className":44791,"style":2521},[2270],[54,44793,2375],{"className":44794},[2517],[54,44796],{"className":44797,"style":18185},[2270],[54,44799],{"className":44800,"style":2521},[2270],[54,44802,44804,44807],{"className":44803},[2126],[54,44805,44264],{"className":44806,"style":2332},[2126,2167],[54,44808,44810],{"className":44809},[2135],[54,44811,44813,44833],{"className":44812},[2139,2756],[54,44814,44816,44830],{"className":44815},[2143],[54,44817,44819],{"className":44818,"style":4063},[2147],[54,44820,44821,44824],{"style":4128},[54,44822],{"className":44823,"style":2156},[2155],[54,44825,44827],{"className":44826},[2160,2161,2162,2163],[54,44828,215],{"className":44829,"style":20138},[2126,2167,2163],[54,44831,2781],{"className":44832},[2780],[54,44834,44836],{"className":44835},[2143],[54,44837,44839],{"className":44838,"style":2788},[2147],[54,44840],{},[54,44842],{"className":44843,"style":2475},[2270],[54,44845,122],{"className":44846},[2479],[54,44848],{"className":44849,"style":2475},[2270],[54,44851,44853,44856,44976,44979],{"className":44852},[2117],[54,44854],{"className":44855,"style":37225},[2121],[54,44857,44859,44865,44970],{"className":44858},[2493],[54,44860,44862],{"className":44861,"style":2499},[2497,2498],[54,44863,11804],{"className":44864},[2503,2162],[54,44866,44868],{"className":44867},[2126],[54,44869,44871,44916,44919,44922],{"className":44870},[9008],[54,44872,44874],{"className":44873},[9267],[54,44875,44877,44908],{"className":44876},[2139,2756],[54,44878,44880,44905],{"className":44879},[2143],[54,44881,44883,44894],{"className":44882,"style":44506},[2147],[54,44884,44885,44888],{"style":44509},[54,44886],{"className":44887,"style":3866},[2155],[54,44889,44891],{"className":44890},[2126],[54,44892,174],{"className":44893},[2126],[54,44895,44896,44899],{"style":44521},[54,44897],{"className":44898,"style":3866},[2155],[54,44900,44902],{"className":44901},[2126],[54,44903,150],{"className":44904},[2126],[54,44906,2781],{"className":44907},[2780],[54,44909,44911],{"className":44910},[2143],[54,44912,44914],{"className":44913,"style":44540},[2147],[54,44915],{},[54,44917],{"className":44918,"style":44547},[44546],[54,44920],{"className":44921,"style":44547},[44546],[54,44923,44925],{"className":44924},[9267],[54,44926,44928,44962],{"className":44927},[2139,2756],[54,44929,44931,44959],{"className":44930},[2143],[54,44932,44934,44945],{"className":44933,"style":44506},[2147],[54,44935,44936,44939],{"style":44509},[54,44937],{"className":44938,"style":3866},[2155],[54,44940,44942],{"className":44941},[2126],[54,44943,150],{"className":44944},[2126],[54,44946,44947,44950],{"style":44521},[54,44948],{"className":44949,"style":3866},[2155],[54,44951,44953,44956],{"className":44952},[2126],[54,44954,2247],{"className":44955},[2126],[54,44957,174],{"className":44958},[2126],[54,44960,2781],{"className":44961},[2780],[54,44963,44965],{"className":44964},[2143],[54,44966,44968],{"className":44967,"style":44540},[2147],[54,44969],{},[54,44971,44973],{"className":44972,"style":2499},[2336,2498],[54,44974,11829],{"className":44975},[2503,2162],[54,44977],{"className":44978,"style":2521},[2270],[54,44980,29],{"className":44981},[2126],[13,44983,44984,44985,45018],{},"Within this representation, a driven two-level system behaves like a spin-",[54,44986,44988,45006],{"className":44987},[2079],[54,44989,44991],{"className":44990},[2083],[2085,44992,44993],{"xmlns":2087},[2089,44994,44995,45003],{},[2092,44996,44997,44999,45001],{},[2249,44998,174],{},[2098,45000,194],{"mathvariant":2304},[2249,45002,133],{},[2105,45004,45005],{"encoding":2107},"1\u002F2",[54,45007,45009],{"className":45008,"ariaHidden":2113},[2112],[54,45010,45012,45015],{"className":45011},[2117],[54,45013],{"className":45014,"style":2325},[2121],[54,45016,45005],{"className":45017},[2126]," particle precessing around an effective magnetic field. The drive strength supplies the transverse control component. Applied detuning supplies the longitudinal control component.",[31,45020,45022],{"id":45021},"rotating-frame-hamiltonian","Rotating-Frame Hamiltonian",[13,45024,45025],{},"In the laboratory frame, a minimal semiclassical Hamiltonian for the driven qubit is",[54,45027,45029],{"className":45028},[2340],[54,45030,45032,45100],{"className":45031},[2079],[54,45033,45035],{"className":45034},[2083],[2085,45036,45037],{"xmlns":2087,"display":2349},[2089,45038,45039,45097],{},[2092,45040,45041,45043,45045,45047,45049,45051,45065,45071,45073,45075,45077,45079,45081,45083,45085,45087,45089,45095],{},[2098,45042,2694],{},[2245,45044,129],{"stretchy":2311},[2098,45046,2854],{},[2245,45048,3205],{"stretchy":2311},[2245,45050,122],{},[9047,45052,45053,45063],{},[2092,45054,45055,45057],{},[2098,45056,42999],{"mathvariant":2304},[2724,45058,45059,45061],{},[2098,45060,41903],{},[2249,45062,150],{},[2249,45064,133],{},[2724,45066,45067,45069],{},[2098,45068,44264],{},[2098,45070,215],{},[2245,45072,1663],{},[2098,45074,42999],{"mathvariant":2304},[2098,45076,42219],{"mathvariant":2304},[2098,45078,37489],{},[2245,45080,36247],{},[2245,45082,129],{"stretchy":2311},[2098,45084,41903],{},[2098,45086,2854],{},[2245,45088,3205],{"stretchy":2311},[2724,45090,45091,45093],{},[2098,45092,44264],{},[2098,45094,8695],{},[2245,45096,2375],{"separator":2113},[2105,45098,45099],{"encoding":2107},"H(t)=\\frac{\\hbar\\omega_0}{2}\\sigma_z+\\hbar\\Omega\\cos(\\omega t)\\sigma_x,",[54,45101,45103,45130,45289],{"className":45102,"ariaHidden":2113},[2112],[54,45104,45106,45109,45112,45115,45118,45121,45124,45127],{"className":45105},[2117],[54,45107],{"className":45108,"style":2325},[2121],[54,45110,2694],{"className":45111,"style":2709},[2126,2167],[54,45113,129],{"className":45114},[2497],[54,45116,2854],{"className":45117},[2126,2167],[54,45119,3205],{"className":45120},[2336],[54,45122],{"className":45123,"style":2475},[2270],[54,45125,122],{"className":45126},[2479],[54,45128],{"className":45129,"style":2475},[2270],[54,45131,45133,45137,45240,45280,45283,45286],{"className":45132},[2117],[54,45134],{"className":45135,"style":45136},[2121],"height:2.0519em;vertical-align:-0.686em;",[54,45138,45140,45143,45237],{"className":45139},[2126],[54,45141],{"className":45142},[2497,9349],[54,45144,45146],{"className":45145},[9047],[54,45147,45149,45229],{"className":45148},[2139,2756],[54,45150,45152,45226],{"className":45151},[2143],[54,45153,45156,45167,45175],{"className":45154,"style":45155},[2147],"height:1.3659em;",[54,45157,45158,45161],{"style":11273},[54,45159],{"className":45160,"style":3866},[2155],[54,45162,45164],{"className":45163},[2126],[54,45165,133],{"className":45166},[2126],[54,45168,45169,45172],{"style":9434},[54,45170],{"className":45171,"style":3866},[2155],[54,45173],{"className":45174,"style":9442},[9441],[54,45176,45177,45180],{"style":9445},[54,45178],{"className":45179,"style":3866},[2155],[54,45181,45183,45186],{"className":45182},[2126],[54,45184,42999],{"className":45185},[2126],[54,45187,45189,45192],{"className":45188},[2126],[54,45190,41903],{"className":45191,"style":2332},[2126,2167],[54,45193,45195],{"className":45194},[2135],[54,45196,45198,45218],{"className":45197},[2139,2756],[54,45199,45201,45215],{"className":45200},[2143],[54,45202,45204],{"className":45203,"style":13837},[2147],[54,45205,45206,45209],{"style":4128},[54,45207],{"className":45208,"style":2156},[2155],[54,45210,45212],{"className":45211},[2160,2161,2162,2163],[54,45213,150],{"className":45214},[2126,2163],[54,45216,2781],{"className":45217},[2780],[54,45219,45221],{"className":45220},[2143],[54,45222,45224],{"className":45223,"style":2788},[2147],[54,45225],{},[54,45227,2781],{"className":45228},[2780],[54,45230,45232],{"className":45231},[2143],[54,45233,45235],{"className":45234,"style":11311},[2147],[54,45236],{},[54,45238],{"className":45239},[2336,9349],[54,45241,45243,45246],{"className":45242},[2126],[54,45244,44264],{"className":45245,"style":2332},[2126,2167],[54,45247,45249],{"className":45248},[2135],[54,45250,45252,45272],{"className":45251},[2139,2756],[54,45253,45255,45269],{"className":45254},[2143],[54,45256,45258],{"className":45257,"style":4063},[2147],[54,45259,45260,45263],{"style":4128},[54,45261],{"className":45262,"style":2156},[2155],[54,45264,45266],{"className":45265},[2160,2161,2162,2163],[54,45267,215],{"className":45268,"style":20138},[2126,2167,2163],[54,45270,2781],{"className":45271},[2780],[54,45273,45275],{"className":45274},[2143],[54,45276,45278],{"className":45277,"style":2788},[2147],[54,45279],{},[54,45281],{"className":45282,"style":2271},[2270],[54,45284,1663],{"className":45285},[2275],[54,45287],{"className":45288,"style":2271},[2270],[54,45290,45292,45295,45299,45302,45305,45308,45311,45314,45317,45357],{"className":45291},[2117],[54,45293],{"className":45294,"style":2325},[2121],[54,45296,45298],{"className":45297},[2126],"ℏΩ",[54,45300],{"className":45301,"style":2521},[2270],[54,45303,37489],{"className":45304},[7126],[54,45306,129],{"className":45307},[2497],[54,45309,41903],{"className":45310,"style":2332},[2126,2167],[54,45312,2854],{"className":45313},[2126,2167],[54,45315,3205],{"className":45316},[2336],[54,45318,45320,45323],{"className":45319},[2126],[54,45321,44264],{"className":45322,"style":2332},[2126,2167],[54,45324,45326],{"className":45325},[2135],[54,45327,45329,45349],{"className":45328},[2139,2756],[54,45330,45332,45346],{"className":45331},[2143],[54,45333,45335],{"className":45334,"style":4063},[2147],[54,45336,45337,45340],{"style":4128},[54,45338],{"className":45339,"style":2156},[2155],[54,45341,45343],{"className":45342},[2160,2161,2162,2163],[54,45344,8695],{"className":45345},[2126,2167,2163],[54,45347,2781],{"className":45348},[2780],[54,45350,45352],{"className":45351},[2143],[54,45353,45355],{"className":45354,"style":2788},[2147],[54,45356],{},[54,45358,2375],{"className":45359},[2517],[13,45361,28011,45362,45422,45423,45451,45452,45489],{},[54,45363,45365,45386],{"className":45364},[2079],[54,45366,45368],{"className":45367},[2083],[2085,45369,45370],{"xmlns":2087},[2089,45371,45372,45384],{},[2092,45373,45374,45376,45378,45380,45382],{},[2098,45375,41903],{},[2245,45377,122],{},[2249,45379,133],{},[2098,45381,8119],{},[2098,45383,21834],{},[2105,45385,42164],{"encoding":2107},[54,45387,45389,45407],{"className":45388,"ariaHidden":2113},[2112],[54,45390,45392,45395,45398,45401,45404],{"className":45391},[2117],[54,45393],{"className":45394,"style":2195},[2121],[54,45396,41903],{"className":45397,"style":2332},[2126,2167],[54,45399],{"className":45400,"style":2475},[2270],[54,45402,122],{"className":45403},[2479],[54,45405],{"className":45406,"style":2475},[2270],[54,45408,45410,45413,45416,45419],{"className":45409},[2117],[54,45411],{"className":45412,"style":6591},[2121],[54,45414,133],{"className":45415},[2126],[54,45417,8119],{"className":45418,"style":2332},[2126,2167],[54,45420,21834],{"className":45421,"style":26895},[2126,2167]," is the drive angular frequency and ",[54,45424,45426,45439],{"className":45425},[2079],[54,45427,45429],{"className":45428},[2083],[2085,45430,45431],{"xmlns":2087},[2089,45432,45433,45437],{},[2092,45434,45435],{},[2098,45436,42219],{"mathvariant":2304},[2105,45438,42222],{"encoding":2107},[54,45440,45442],{"className":45441,"ariaHidden":2113},[2112],[54,45443,45445,45448],{"className":45444},[2117],[54,45446],{"className":45447,"style":2122},[2121],[54,45449,42219],{"className":45450},[2126]," is the on-resonance Rabi angular frequency. The computational model specifies the drive strength using ",[54,45453,45455,45474],{"className":45454},[2079],[54,45456,45458],{"className":45457},[2083],[2085,45459,45460],{"xmlns":2087},[2089,45461,45462,45472],{},[2092,45463,45464,45466,45468,45470],{},[2098,45465,42219],{"mathvariant":2304},[2098,45467,194],{"mathvariant":2304},[2249,45469,133],{},[2098,45471,8119],{},[2105,45473,42264],{"encoding":2107},[54,45475,45477],{"className":45476,"ariaHidden":2113},[2112],[54,45478,45480,45483,45486],{"className":45479},[2117],[54,45481],{"className":45482,"style":2325},[2121],[54,45484,42277],{"className":45485},[2126],[54,45487,8119],{"className":45488,"style":2332},[2126,2167]," in MHz. Transforming into a frame rotating at the drive frequency and applying the rotating-wave approximation gives the effective time-independent Hamiltonian",[54,45491,45493],{"className":45492},[2340],[54,45494,45496,45555],{"className":45495},[2079],[54,45497,45499],{"className":45498},[2083],[2085,45500,45501],{"xmlns":2087,"display":2349},[2089,45502,45503,45552],{},[2092,45504,45505,45518,45520,45526,45550],{},[2724,45506,45507,45509],{},[2098,45508,2694],{},[2092,45510,45511,45513,45516],{},[2098,45512,19926],{"mathvariant":2304},[2098,45514,45515],{"mathvariant":2304},"W",[2098,45517,2214],{"mathvariant":2304},[2245,45519,122],{},[9047,45521,45522,45524],{},[2098,45523,42999],{"mathvariant":2304},[2249,45525,133],{},[2092,45527,45528,45530,45532,45538,45540,45542,45548],{},[2245,45529,129],{"fence":2113},[2098,45531,42302],{"mathvariant":2304},[2724,45533,45534,45536],{},[2098,45535,44264],{},[2098,45537,215],{},[2245,45539,1663],{},[2098,45541,42219],{"mathvariant":2304},[2724,45543,45544,45546],{},[2098,45545,44264],{},[2098,45547,8695],{},[2245,45549,3205],{"fence":2113},[2245,45551,2375],{"separator":2113},[2105,45553,45554],{"encoding":2107},"H_{\\mathrm{RWA}}=\\frac{\\hbar}{2}\\left(\\Delta\\sigma_z+\\Omega\\sigma_x\\right),",[54,45556,45558,45619],{"className":45557,"ariaHidden":2113},[2112],[54,45559,45561,45564,45610,45613,45616],{"className":45560},[2117],[54,45562],{"className":45563,"style":2743},[2121],[54,45565,45567,45570],{"className":45566},[2126],[54,45568,2694],{"className":45569,"style":2709},[2126,2167],[54,45571,45573],{"className":45572},[2135],[54,45574,45576,45602],{"className":45575},[2139,2756],[54,45577,45579,45599],{"className":45578},[2143],[54,45580,45582],{"className":45581,"style":2763},[2147],[54,45583,45584,45587],{"style":4066},[54,45585],{"className":45586,"style":2156},[2155],[54,45588,45590],{"className":45589},[2160,2161,2162,2163],[54,45591,45593],{"className":45592},[2126,2163],[54,45594,45596],{"className":45595},[2126,2163],[54,45597,42869],{"className":45598},[2126,43453,2163],[54,45600,2781],{"className":45601},[2780],[54,45603,45605],{"className":45604},[2143],[54,45606,45608],{"className":45607,"style":2788},[2147],[54,45609],{},[54,45611],{"className":45612,"style":2475},[2270],[54,45614,122],{"className":45615},[2479],[54,45617],{"className":45618,"style":2475},[2270],[54,45620,45622,45625,45687,45690,45794,45797],{"className":45621},[2117],[54,45623],{"className":45624,"style":45136},[2121],[54,45626,45628,45631,45684],{"className":45627},[2126],[54,45629],{"className":45630},[2497,9349],[54,45632,45634],{"className":45633},[9047],[54,45635,45637,45676],{"className":45636},[2139,2756],[54,45638,45640,45673],{"className":45639},[2143],[54,45641,45643,45654,45662],{"className":45642,"style":45155},[2147],[54,45644,45645,45648],{"style":11273},[54,45646],{"className":45647,"style":3866},[2155],[54,45649,45651],{"className":45650},[2126],[54,45652,133],{"className":45653},[2126],[54,45655,45656,45659],{"style":9434},[54,45657],{"className":45658,"style":3866},[2155],[54,45660],{"className":45661,"style":9442},[9441],[54,45663,45664,45667],{"style":9445},[54,45665],{"className":45666,"style":3866},[2155],[54,45668,45670],{"className":45669},[2126],[54,45671,42999],{"className":45672},[2126],[54,45674,2781],{"className":45675},[2780],[54,45677,45679],{"className":45678},[2143],[54,45680,45682],{"className":45681,"style":11311},[2147],[54,45683],{},[54,45685],{"className":45686},[2336,9349],[54,45688],{"className":45689,"style":2521},[2270],[54,45691,45693,45696,45699,45739,45742,45745,45748,45751,45791],{"className":45692},[2493],[54,45694,129],{"className":45695,"style":2499},[2497,2498],[54,45697,42302],{"className":45698},[2126],[54,45700,45702,45705],{"className":45701},[2126],[54,45703,44264],{"className":45704,"style":2332},[2126,2167],[54,45706,45708],{"className":45707},[2135],[54,45709,45711,45731],{"className":45710},[2139,2756],[54,45712,45714,45728],{"className":45713},[2143],[54,45715,45717],{"className":45716,"style":4063},[2147],[54,45718,45719,45722],{"style":4128},[54,45720],{"className":45721,"style":2156},[2155],[54,45723,45725],{"className":45724},[2160,2161,2162,2163],[54,45726,215],{"className":45727,"style":20138},[2126,2167,2163],[54,45729,2781],{"className":45730},[2780],[54,45732,45734],{"className":45733},[2143],[54,45735,45737],{"className":45736,"style":2788},[2147],[54,45738],{},[54,45740],{"className":45741,"style":2271},[2270],[54,45743,1663],{"className":45744},[2275],[54,45746],{"className":45747,"style":2271},[2270],[54,45749,42219],{"className":45750},[2126],[54,45752,45754,45757],{"className":45753},[2126],[54,45755,44264],{"className":45756,"style":2332},[2126,2167],[54,45758,45760],{"className":45759},[2135],[54,45761,45763,45783],{"className":45762},[2139,2756],[54,45764,45766,45780],{"className":45765},[2143],[54,45767,45769],{"className":45768,"style":4063},[2147],[54,45770,45771,45774],{"style":4128},[54,45772],{"className":45773,"style":2156},[2155],[54,45775,45777],{"className":45776},[2160,2161,2162,2163],[54,45778,8695],{"className":45779},[2126,2167,2163],[54,45781,2781],{"className":45782},[2780],[54,45784,45786],{"className":45785},[2143],[54,45787,45789],{"className":45788,"style":2788},[2147],[54,45790],{},[54,45792,3205],{"className":45793,"style":2499},[2336,2498],[54,45795],{"className":45796,"style":2521},[2270],[54,45798,2375],{"className":45799},[2517],[13,45801,45802],{},"with detuning",[54,45804,45806],{"className":45805},[2340],[54,45807,45809,45857],{"className":45808},[2079],[54,45810,45812],{"className":45811},[2083],[2085,45813,45814],{"xmlns":2087,"display":2349},[2089,45815,45816,45854],{},[2092,45817,45818,45820,45822,45824,45826,45832,45834,45836,45838,45840,45842,45844,45850,45852],{},[2098,45819,42302],{"mathvariant":2304},[2245,45821,122],{},[2098,45823,41903],{},[2245,45825,2247],{},[2724,45827,45828,45830],{},[2098,45829,41903],{},[2249,45831,150],{},[2245,45833,122],{},[2249,45835,133],{},[2098,45837,8119],{},[2245,45839,129],{"stretchy":2311},[2098,45841,21834],{},[2245,45843,2247],{},[2724,45845,45846,45848],{},[2098,45847,21834],{},[2249,45849,150],{},[2245,45851,3205],{"stretchy":2311},[2098,45853,29],{"mathvariant":2304},[2105,45855,45856],{"encoding":2107},"\\Delta=\\omega-\\omega_0=2\\pi(f-f_0).",[54,45858,45860,45878,45896,45951,45978],{"className":45859,"ariaHidden":2113},[2112],[54,45861,45863,45866,45869,45872,45875],{"className":45862},[2117],[54,45864],{"className":45865,"style":2122},[2121],[54,45867,42302],{"className":45868},[2126],[54,45870],{"className":45871,"style":2475},[2270],[54,45873,122],{"className":45874},[2479],[54,45876],{"className":45877,"style":2475},[2270],[54,45879,45881,45884,45887,45890,45893],{"className":45880},[2117],[54,45882],{"className":45883,"style":2263},[2121],[54,45885,41903],{"className":45886,"style":2332},[2126,2167],[54,45888],{"className":45889,"style":2271},[2270],[54,45891,2247],{"className":45892},[2275],[54,45894],{"className":45895,"style":2271},[2270],[54,45897,45899,45902,45942,45945,45948],{"className":45898},[2117],[54,45900],{"className":45901,"style":24753},[2121],[54,45903,45905,45908],{"className":45904},[2126],[54,45906,41903],{"className":45907,"style":2332},[2126,2167],[54,45909,45911],{"className":45910},[2135],[54,45912,45914,45934],{"className":45913},[2139,2756],[54,45915,45917,45931],{"className":45916},[2143],[54,45918,45920],{"className":45919,"style":13837},[2147],[54,45921,45922,45925],{"style":4128},[54,45923],{"className":45924,"style":2156},[2155],[54,45926,45928],{"className":45927},[2160,2161,2162,2163],[54,45929,150],{"className":45930},[2126,2163],[54,45932,2781],{"className":45933},[2780],[54,45935,45937],{"className":45936},[2143],[54,45938,45940],{"className":45939,"style":2788},[2147],[54,45941],{},[54,45943],{"className":45944,"style":2475},[2270],[54,45946,122],{"className":45947},[2479],[54,45949],{"className":45950,"style":2475},[2270],[54,45952,45954,45957,45960,45963,45966,45969,45972,45975],{"className":45953},[2117],[54,45955],{"className":45956,"style":2325},[2121],[54,45958,133],{"className":45959},[2126],[54,45961,8119],{"className":45962,"style":2332},[2126,2167],[54,45964,129],{"className":45965},[2497],[54,45967,21834],{"className":45968,"style":26895},[2126,2167],[54,45970],{"className":45971,"style":2271},[2270],[54,45973,2247],{"className":45974},[2275],[54,45976],{"className":45977,"style":2271},[2270],[54,45979,45981,45984,46024,46027],{"className":45980},[2117],[54,45982],{"className":45983,"style":2325},[2121],[54,45985,45987,45990],{"className":45986},[2126],[54,45988,21834],{"className":45989,"style":26895},[2126,2167],[54,45991,45993],{"className":45992},[2135],[54,45994,45996,46016],{"className":45995},[2139,2756],[54,45997,45999,46013],{"className":45998},[2143],[54,46000,46002],{"className":46001,"style":13837},[2147],[54,46003,46004,46007],{"style":41824},[54,46005],{"className":46006,"style":2156},[2155],[54,46008,46010],{"className":46009},[2160,2161,2162,2163],[54,46011,150],{"className":46012},[2126,2163],[54,46014,2781],{"className":46015},[2780],[54,46017,46019],{"className":46018},[2143],[54,46020,46022],{"className":46021,"style":2788},[2147],[54,46023],{},[54,46025,3205],{"className":46026},[2336],[54,46028,29],{"className":46029},[2126],[13,46031,46032,46033,10355,46061,46089,46090,46118,46119,46147],{},"This Hamiltonian describes precession around an effective control axis with components proportional to ",[54,46034,46036,46049],{"className":46035},[2079],[54,46037,46039],{"className":46038},[2083],[2085,46040,46041],{"xmlns":2087},[2089,46042,46043,46047],{},[2092,46044,46045],{},[2098,46046,42219],{"mathvariant":2304},[2105,46048,42222],{"encoding":2107},[54,46050,46052],{"className":46051,"ariaHidden":2113},[2112],[54,46053,46055,46058],{"className":46054},[2117],[54,46056],{"className":46057,"style":2122},[2121],[54,46059,42219],{"className":46060},[2126],[54,46062,46064,46077],{"className":46063},[2079],[54,46065,46067],{"className":46066},[2083],[2085,46068,46069],{"xmlns":2087},[2089,46070,46071,46075],{},[2092,46072,46073],{},[2098,46074,42302],{"mathvariant":2304},[2105,46076,42305],{"encoding":2107},[54,46078,46080],{"className":46079,"ariaHidden":2113},[2112],[54,46081,46083,46086],{"className":46082},[2117],[54,46084],{"className":46085,"style":2122},[2121],[54,46087,42302],{"className":46088},[2126],". The transverse component ",[54,46091,46093,46106],{"className":46092},[2079],[54,46094,46096],{"className":46095},[2083],[2085,46097,46098],{"xmlns":2087},[2089,46099,46100,46104],{},[2092,46101,46102],{},[2098,46103,42219],{"mathvariant":2304},[2105,46105,42222],{"encoding":2107},[54,46107,46109],{"className":46108,"ariaHidden":2113},[2112],[54,46110,46112,46115],{"className":46111},[2117],[54,46113],{"className":46114,"style":2122},[2121],[54,46116,42219],{"className":46117},[2126]," drives population transfer. A longitudinal component ",[54,46120,46122,46135],{"className":46121},[2079],[54,46123,46125],{"className":46124},[2083],[2085,46126,46127],{"xmlns":2087},[2089,46128,46129,46133],{},[2092,46130,46131],{},[2098,46132,42302],{"mathvariant":2304},[2105,46134,42305],{"encoding":2107},[54,46136,46138],{"className":46137,"ariaHidden":2113},[2112],[54,46139,46141,46144],{"className":46140},[2117],[54,46142],{"className":46143,"style":2122},[2121],[54,46145,42302],{"className":46146},[2126]," tilts the rotation axis away from the resonant direction.",[31,46149,46151],{"id":46150},"excited-state-probability","Excited-State Probability",[13,46153,46154,46155,46193,46194,46222],{},"The simulation assumes that the qubit begins in the ground state ",[54,46156,46158,46175],{"className":46157},[2079],[54,46159,46161],{"className":46160},[2083],[2085,46162,46163],{"xmlns":2087},[2089,46164,46165,46173],{},[2092,46166,46167,46169,46171],{},[2245,46168,2305],{"stretchy":2311},[2098,46170,37470],{},[2245,46172,2312],{"stretchy":2311},[2105,46174,41452],{"encoding":2107},[54,46176,46178],{"className":46177,"ariaHidden":2113},[2112],[54,46179,46181,46184,46187,46190],{"className":46180},[2117],[54,46182],{"className":46183,"style":2325},[2121],[54,46185,2305],{"className":46186},[2497],[54,46188,37470],{"className":46189,"style":2332},[2126,2167],[54,46191,2312],{"className":46192},[2336],". Evolving under the RWA Hamiltonian, the probability of occupying the excited state after pulse duration ",[54,46195,46197,46210],{"className":46196},[2079],[54,46198,46200],{"className":46199},[2083],[2085,46201,46202],{"xmlns":2087},[2089,46203,46204,46208],{},[2092,46205,46206],{},[2098,46207,41261],{},[2105,46209,41387],{"encoding":2107},[54,46211,46213],{"className":46212,"ariaHidden":2113},[2112],[54,46214,46216,46219],{"className":46215},[2117],[54,46217],{"className":46218,"style":2195},[2121],[54,46220,41261],{"className":46221,"style":41333},[2126,2167]," is",[54,46224,46226],{"className":46225},[2340],[54,46227,46229,46308],{"className":46228},[2079],[54,46230,46232],{"className":46231},[2083],[2085,46233,46234],{"xmlns":2087,"display":2349},[2089,46235,46236,46305],{},[2092,46237,46238,46244,46246,46248,46250,46252,46254,46256,46272,46283,46303],{},[2724,46239,46240,46242],{},[2098,46241,2730],{},[2098,46243,2842],{},[2245,46245,129],{"stretchy":2311},[2098,46247,21834],{},[2245,46249,2375],{"separator":2113},[2098,46251,41261],{},[2245,46253,3205],{"stretchy":2311},[2245,46255,122],{},[9047,46257,46258,46264],{},[2095,46259,46260,46262],{},[2098,46261,42219],{"mathvariant":2304},[2249,46263,133],{},[3331,46265,46266,46268,46270],{},[2098,46267,42219],{"mathvariant":2304},[2098,46269,19926],{},[2249,46271,133],{},[2095,46273,46274,46281],{},[2092,46275,46276,46279],{},[2098,46277,46278],{},"sin",[2245,46280,36247],{},[2249,46282,133],{},[2092,46284,46285,46287,46301],{},[2245,46286,129],{"fence":2113},[9047,46288,46289,46299],{},[2092,46290,46291,46297],{},[2724,46292,46293,46295],{},[2098,46294,42219],{"mathvariant":2304},[2098,46296,19926],{},[2098,46298,41261],{},[2249,46300,133],{},[2245,46302,3205],{"fence":2113},[2245,46304,2375],{"separator":2113},[2105,46306,46307],{"encoding":2107},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right),",[54,46309,46311,46384],{"className":46310,"ariaHidden":2113},[2112],[54,46312,46314,46317,46357,46360,46363,46366,46369,46372,46375,46378,46381],{"className":46313},[2117],[54,46315],{"className":46316,"style":2325},[2121],[54,46318,46320,46323],{"className":46319},[2126],[54,46321,2730],{"className":46322,"style":2776},[2126,2167],[54,46324,46326],{"className":46325},[2135],[54,46327,46329,46349],{"className":46328},[2139,2756],[54,46330,46332,46346],{"className":46331},[2143],[54,46333,46335],{"className":46334,"style":4063},[2147],[54,46336,46337,46340],{"style":7193},[54,46338],{"className":46339,"style":2156},[2155],[54,46341,46343],{"className":46342},[2160,2161,2162,2163],[54,46344,2842],{"className":46345},[2126,2167,2163],[54,46347,2781],{"className":46348},[2780],[54,46350,46352],{"className":46351},[2143],[54,46353,46355],{"className":46354,"style":2788},[2147],[54,46356],{},[54,46358,129],{"className":46359},[2497],[54,46361,21834],{"className":46362,"style":26895},[2126,2167],[54,46364,2375],{"className":46365},[2517],[54,46367],{"className":46368,"style":2521},[2270],[54,46370,41261],{"className":46371,"style":41333},[2126,2167],[54,46373,3205],{"className":46374},[2336],[54,46376],{"className":46377,"style":2475},[2270],[54,46379,122],{"className":46380},[2479],[54,46382],{"className":46383,"style":2475},[2270],[54,46385,46387,46391,46533,46536,46567,46570,46688,46691],{"className":46386},[2117],[54,46388],{"className":46389,"style":46390},[2121],"height:2.4706em;vertical-align:-0.9795em;",[54,46392,46394,46397,46530],{"className":46393},[2126],[54,46395],{"className":46396},[2497,9349],[54,46398,46400],{"className":46399},[9047],[54,46401,46403,46521],{"className":46402},[2139,2756],[54,46404,46406,46518],{"className":46405},[2143],[54,46407,46410,46473,46481],{"className":46408,"style":46409},[2147],"height:1.4911em;",[54,46411,46412,46415],{"style":11273},[54,46413],{"className":46414,"style":3866},[2155],[54,46416,46418],{"className":46417},[2126],[54,46419,46421,46424],{"className":46420},[2126],[54,46422,42219],{"className":46423},[2126],[54,46425,46427],{"className":46426},[2135],[54,46428,46430,46464],{"className":46429},[2139,2756],[54,46431,46433,46461],{"className":46432},[2143],[54,46434,46437,46449],{"className":46435,"style":46436},[2147],"height:0.7959em;",[54,46438,46440,46443],{"style":46439},"top:-2.4065em;margin-left:0em;margin-right:0.05em;",[54,46441],{"className":46442,"style":2156},[2155],[54,46444,46446],{"className":46445},[2160,2161,2162,2163],[54,46447,19926],{"className":46448,"style":19951},[2126,2167,2163],[54,46450,46452,46455],{"style":46451},"top:-3.0448em;margin-right:0.05em;",[54,46453],{"className":46454,"style":2156},[2155],[54,46456,46458],{"className":46457},[2160,2161,2162,2163],[54,46459,133],{"className":46460},[2126,2163],[54,46462,2781],{"className":46463},[2780],[54,46465,46467],{"className":46466},[2143],[54,46468,46471],{"className":46469,"style":46470},[2147],"height:0.2935em;",[54,46472],{},[54,46474,46475,46478],{"style":9434},[54,46476],{"className":46477,"style":3866},[2155],[54,46479],{"className":46480,"style":9442},[9441],[54,46482,46483,46486],{"style":9445},[54,46484],{"className":46485,"style":3866},[2155],[54,46487,46489],{"className":46488},[2126],[54,46490,46492,46495],{"className":46491},[2126],[54,46493,42219],{"className":46494},[2126],[54,46496,46498],{"className":46497},[2135],[54,46499,46501],{"className":46500},[2139],[54,46502,46504],{"className":46503},[2143],[54,46505,46507],{"className":46506,"style":12780},[2147],[54,46508,46509,46512],{"style":2151},[54,46510],{"className":46511,"style":2156},[2155],[54,46513,46515],{"className":46514},[2160,2161,2162,2163],[54,46516,133],{"className":46517},[2126,2163],[54,46519,2781],{"className":46520},[2780],[54,46522,46524],{"className":46523},[2143],[54,46525,46528],{"className":46526,"style":46527},[2147],"height:0.9795em;",[54,46529],{},[54,46531],{"className":46532},[2336,9349],[54,46534],{"className":46535,"style":2521},[2270],[54,46537,46539,46542],{"className":46538},[7126],[54,46540,46278],{"className":46541},[7126],[54,46543,46545],{"className":46544},[2135],[54,46546,46548],{"className":46547},[2139],[54,46549,46551],{"className":46550},[2143],[54,46552,46555],{"className":46553,"style":46554},[2147],"height:0.8719em;",[54,46556,46558,46561],{"style":46557},"top:-3.1208em;margin-right:0.05em;",[54,46559],{"className":46560,"style":2156},[2155],[54,46562,46564],{"className":46563},[2160,2161,2162,2163],[54,46565,133],{"className":46566},[2126,2163],[54,46568],{"className":46569,"style":2521},[2270],[54,46571,46573,46579,46682],{"className":46572},[2493],[54,46574,46576],{"className":46575,"style":2499},[2497,2498],[54,46577,129],{"className":46578},[2503,2162],[54,46580,46582,46585,46679],{"className":46581},[2126],[54,46583],{"className":46584},[2497,9349],[54,46586,46588],{"className":46587},[9047],[54,46589,46591,46671],{"className":46590},[2139,2756],[54,46592,46594,46668],{"className":46593},[2143],[54,46595,46598,46609,46617],{"className":46596,"style":46597},[2147],"height:1.3603em;",[54,46599,46600,46603],{"style":11273},[54,46601],{"className":46602,"style":3866},[2155],[54,46604,46606],{"className":46605},[2126],[54,46607,133],{"className":46608},[2126],[54,46610,46611,46614],{"style":9434},[54,46612],{"className":46613,"style":3866},[2155],[54,46615],{"className":46616,"style":9442},[9441],[54,46618,46619,46622],{"style":9445},[54,46620],{"className":46621,"style":3866},[2155],[54,46623,46625,46665],{"className":46624},[2126],[54,46626,46628,46631],{"className":46627},[2126],[54,46629,42219],{"className":46630},[2126],[54,46632,46634],{"className":46633},[2135],[54,46635,46637,46657],{"className":46636},[2139,2756],[54,46638,46640,46654],{"className":46639},[2143],[54,46641,46643],{"className":46642,"style":2763},[2147],[54,46644,46645,46648],{"style":4954},[54,46646],{"className":46647,"style":2156},[2155],[54,46649,46651],{"className":46650},[2160,2161,2162,2163],[54,46652,19926],{"className":46653,"style":19951},[2126,2167,2163],[54,46655,2781],{"className":46656},[2780],[54,46658,46660],{"className":46659},[2143],[54,46661,46663],{"className":46662,"style":2788},[2147],[54,46664],{},[54,46666,41261],{"className":46667,"style":41333},[2126,2167],[54,46669,2781],{"className":46670},[2780],[54,46672,46674],{"className":46673},[2143],[54,46675,46677],{"className":46676,"style":11311},[2147],[54,46678],{},[54,46680],{"className":46681},[2336,9349],[54,46683,46685],{"className":46684,"style":2499},[2336,2498],[54,46686,3205],{"className":46687},[2503,2162],[54,46689],{"className":46690,"style":2521},[2270],[54,46692,2375],{"className":46693},[2517],[13,46695,46696],{},"where",[54,46698,46700],{"className":46699},[2340],[54,46701,46703,46741],{"className":46702},[2079],[54,46704,46706],{"className":46705},[2083],[2085,46707,46708],{"xmlns":2087,"display":2349},[2089,46709,46710,46738],{},[2092,46711,46712,46718,46720],{},[2724,46713,46714,46716],{},[2098,46715,42219],{"mathvariant":2304},[2098,46717,19926],{},[2245,46719,122],{},[9052,46721,46722],{},[2092,46723,46724,46730,46732],{},[2095,46725,46726,46728],{},[2098,46727,42219],{"mathvariant":2304},[2249,46729,133],{},[2245,46731,1663],{},[2095,46733,46734,46736],{},[2098,46735,42302],{"mathvariant":2304},[2249,46737,133],{},[2105,46739,46740],{"encoding":2107},"\\Omega_R=\\sqrt{\\Omega^2+\\Delta^2}",[54,46742,46744,46799],{"className":46743,"ariaHidden":2113},[2112],[54,46745,46747,46750,46790,46793,46796],{"className":46746},[2117],[54,46748],{"className":46749,"style":2743},[2121],[54,46751,46753,46756],{"className":46752},[2126],[54,46754,42219],{"className":46755},[2126],[54,46757,46759],{"className":46758},[2135],[54,46760,46762,46782],{"className":46761},[2139,2756],[54,46763,46765,46779],{"className":46764},[2143],[54,46766,46768],{"className":46767,"style":2763},[2147],[54,46769,46770,46773],{"style":4954},[54,46771],{"className":46772,"style":2156},[2155],[54,46774,46776],{"className":46775},[2160,2161,2162,2163],[54,46777,19926],{"className":46778,"style":19951},[2126,2167,2163],[54,46780,2781],{"className":46781},[2780],[54,46783,46785],{"className":46784},[2143],[54,46786,46788],{"className":46787,"style":2788},[2147],[54,46789],{},[54,46791],{"className":46792,"style":2475},[2270],[54,46794,122],{"className":46795},[2479],[54,46797],{"className":46798,"style":2475},[2270],[54,46800,46802,46806],{"className":46801},[2117],[54,46803],{"className":46804,"style":46805},[2121],"height:1.24em;vertical-align:-0.1777em;",[54,46807,46809],{"className":46808},[2126,9375],[54,46810,46812,46918],{"className":46811},[2139,2756],[54,46813,46815,46915],{"className":46814},[2143],[54,46816,46819,46898],{"className":46817,"style":46818},[2147],"height:1.0623em;",[54,46820,46823,46827],{"className":46821,"style":46822},[9389],"top:-3.2em;",[54,46824],{"className":46825,"style":46826},[2155],"height:3.2em;",[54,46828,46831,46860,46863,46866,46869],{"className":46829,"style":46830},[2126],"padding-left:1em;",[54,46832,46834,46837],{"className":46833},[2126],[54,46835,42219],{"className":46836},[2126],[54,46838,46840],{"className":46839},[2135],[54,46841,46843],{"className":46842},[2139],[54,46844,46846],{"className":46845},[2143],[54,46847,46849],{"className":46848,"style":28966},[2147],[54,46850,46851,46854],{"style":28969},[54,46852],{"className":46853,"style":2156},[2155],[54,46855,46857],{"className":46856},[2160,2161,2162,2163],[54,46858,133],{"className":46859},[2126,2163],[54,46861],{"className":46862,"style":2271},[2270],[54,46864,1663],{"className":46865},[2275],[54,46867],{"className":46868,"style":2271},[2270],[54,46870,46872,46875],{"className":46871},[2126],[54,46873,42302],{"className":46874},[2126],[54,46876,46878],{"className":46877},[2135],[54,46879,46881],{"className":46880},[2139],[54,46882,46884],{"className":46883},[2143],[54,46885,46887],{"className":46886,"style":28966},[2147],[54,46888,46889,46892],{"style":28969},[54,46890],{"className":46891,"style":2156},[2155],[54,46893,46895],{"className":46894},[2160,2161,2162,2163],[54,46896,133],{"className":46897},[2126,2163],[54,46899,46901,46904],{"style":46900},"top:-3.0223em;",[54,46902],{"className":46903,"style":46826},[2155],[54,46905,46908],{"className":46906,"style":46907},[9409],"min-width:1.02em;height:1.28em;",[5837,46909,46912],{"xmlns":5839,"width":9413,"height":46910,"viewBox":46911,"preserveAspectRatio":9416},"1.28em","0 0 400000 1296",[5846,46913],{"d":46914},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[54,46916,2781],{"className":46917},[2780],[54,46919,46921],{"className":46920},[2143],[54,46922,46925],{"className":46923,"style":46924},[2147],"height:0.1777em;",[54,46926],{},[13,46928,46929],{},"is the generalized Rabi angular frequency. The amplitude coefficient",[54,46931,46933],{"className":46932},[2340],[54,46934,46936,46964],{"className":46935},[2079],[54,46937,46939],{"className":46938},[2083],[2085,46940,46941],{"xmlns":2087,"display":2349},[2089,46942,46943,46961],{},[2092,46944,46945],{},[9047,46946,46947,46953],{},[2095,46948,46949,46951],{},[2098,46950,42219],{"mathvariant":2304},[2249,46952,133],{},[3331,46954,46955,46957,46959],{},[2098,46956,42219],{"mathvariant":2304},[2098,46958,19926],{},[2249,46960,133],{},[2105,46962,46963],{"encoding":2107},"\\frac{\\Omega^2}{\\Omega_R^2}",[54,46965,46967],{"className":46966,"ariaHidden":2113},[2112],[54,46968,46970,46973],{"className":46969},[2117],[54,46971],{"className":46972,"style":46390},[2121],[54,46974,46976,46979,47106],{"className":46975},[2126],[54,46977],{"className":46978},[2497,9349],[54,46980,46982],{"className":46981},[9047],[54,46983,46985,47098],{"className":46984},[2139,2756],[54,46986,46988,47095],{"className":46987},[2143],[54,46989,46991,47050,47058],{"className":46990,"style":46409},[2147],[54,46992,46993,46996],{"style":11273},[54,46994],{"className":46995,"style":3866},[2155],[54,46997,46999],{"className":46998},[2126],[54,47000,47002,47005],{"className":47001},[2126],[54,47003,42219],{"className":47004},[2126],[54,47006,47008],{"className":47007},[2135],[54,47009,47011,47042],{"className":47010},[2139,2756],[54,47012,47014,47039],{"className":47013},[2143],[54,47015,47017,47028],{"className":47016,"style":46436},[2147],[54,47018,47019,47022],{"style":46439},[54,47020],{"className":47021,"style":2156},[2155],[54,47023,47025],{"className":47024},[2160,2161,2162,2163],[54,47026,19926],{"className":47027,"style":19951},[2126,2167,2163],[54,47029,47030,47033],{"style":46451},[54,47031],{"className":47032,"style":2156},[2155],[54,47034,47036],{"className":47035},[2160,2161,2162,2163],[54,47037,133],{"className":47038},[2126,2163],[54,47040,2781],{"className":47041},[2780],[54,47043,47045],{"className":47044},[2143],[54,47046,47048],{"className":47047,"style":46470},[2147],[54,47049],{},[54,47051,47052,47055],{"style":9434},[54,47053],{"className":47054,"style":3866},[2155],[54,47056],{"className":47057,"style":9442},[9441],[54,47059,47060,47063],{"style":9445},[54,47061],{"className":47062,"style":3866},[2155],[54,47064,47066],{"className":47065},[2126],[54,47067,47069,47072],{"className":47068},[2126],[54,47070,42219],{"className":47071},[2126],[54,47073,47075],{"className":47074},[2135],[54,47076,47078],{"className":47077},[2139],[54,47079,47081],{"className":47080},[2143],[54,47082,47084],{"className":47083,"style":12780},[2147],[54,47085,47086,47089],{"style":2151},[54,47087],{"className":47088,"style":2156},[2155],[54,47090,47092],{"className":47091},[2160,2161,2162,2163],[54,47093,133],{"className":47094},[2126,2163],[54,47096,2781],{"className":47097},[2780],[54,47099,47101],{"className":47100},[2143],[54,47102,47104],{"className":47103,"style":46527},[2147],[54,47105],{},[54,47107],{"className":47108},[2336,9349],[13,47110,47111,47112,47163],{},"sets the maximum possible population transfer. At exact resonance, ",[54,47113,47115,47133],{"className":47114},[2079],[54,47116,47118],{"className":47117},[2083],[2085,47119,47120],{"xmlns":2087},[2089,47121,47122,47130],{},[2092,47123,47124,47126,47128],{},[2098,47125,42302],{"mathvariant":2304},[2245,47127,122],{},[2249,47129,150],{},[2105,47131,47132],{"encoding":2107},"\\Delta=0",[54,47134,47136,47154],{"className":47135,"ariaHidden":2113},[2112],[54,47137,47139,47142,47145,47148,47151],{"className":47138},[2117],[54,47140],{"className":47141,"style":2122},[2121],[54,47143,42302],{"className":47144},[2126],[54,47146],{"className":47147,"style":2475},[2270],[54,47149,122],{"className":47150},[2479],[54,47152],{"className":47153,"style":2475},[2270],[54,47155,47157,47160],{"className":47156},[2117],[54,47158],{"className":47159,"style":2285},[2121],[54,47161,150],{"className":47162},[2126],", so",[54,47165,47167],{"className":47166},[2340],[54,47168,47170,47202],{"className":47169},[2079],[54,47171,47173],{"className":47172},[2083],[2085,47174,47175],{"xmlns":2087,"display":2349},[2089,47176,47177,47199],{},[2092,47178,47179,47195,47197],{},[9047,47180,47181,47187],{},[2095,47182,47183,47185],{},[2098,47184,42219],{"mathvariant":2304},[2249,47186,133],{},[3331,47188,47189,47191,47193],{},[2098,47190,42219],{"mathvariant":2304},[2098,47192,19926],{},[2249,47194,133],{},[2245,47196,122],{},[2249,47198,43900],{},[2105,47200,47201],{"encoding":2107},"\\frac{\\Omega^2}{\\Omega_R^2}=1.",[54,47203,47205,47356],{"className":47204,"ariaHidden":2113},[2112],[54,47206,47208,47211,47347,47350,47353],{"className":47207},[2117],[54,47209],{"className":47210,"style":46390},[2121],[54,47212,47214,47217,47344],{"className":47213},[2126],[54,47215],{"className":47216},[2497,9349],[54,47218,47220],{"className":47219},[9047],[54,47221,47223,47336],{"className":47222},[2139,2756],[54,47224,47226,47333],{"className":47225},[2143],[54,47227,47229,47288,47296],{"className":47228,"style":46409},[2147],[54,47230,47231,47234],{"style":11273},[54,47232],{"className":47233,"style":3866},[2155],[54,47235,47237],{"className":47236},[2126],[54,47238,47240,47243],{"className":47239},[2126],[54,47241,42219],{"className":47242},[2126],[54,47244,47246],{"className":47245},[2135],[54,47247,47249,47280],{"className":47248},[2139,2756],[54,47250,47252,47277],{"className":47251},[2143],[54,47253,47255,47266],{"className":47254,"style":46436},[2147],[54,47256,47257,47260],{"style":46439},[54,47258],{"className":47259,"style":2156},[2155],[54,47261,47263],{"className":47262},[2160,2161,2162,2163],[54,47264,19926],{"className":47265,"style":19951},[2126,2167,2163],[54,47267,47268,47271],{"style":46451},[54,47269],{"className":47270,"style":2156},[2155],[54,47272,47274],{"className":47273},[2160,2161,2162,2163],[54,47275,133],{"className":47276},[2126,2163],[54,47278,2781],{"className":47279},[2780],[54,47281,47283],{"className":47282},[2143],[54,47284,47286],{"className":47285,"style":46470},[2147],[54,47287],{},[54,47289,47290,47293],{"style":9434},[54,47291],{"className":47292,"style":3866},[2155],[54,47294],{"className":47295,"style":9442},[9441],[54,47297,47298,47301],{"style":9445},[54,47299],{"className":47300,"style":3866},[2155],[54,47302,47304],{"className":47303},[2126],[54,47305,47307,47310],{"className":47306},[2126],[54,47308,42219],{"className":47309},[2126],[54,47311,47313],{"className":47312},[2135],[54,47314,47316],{"className":47315},[2139],[54,47317,47319],{"className":47318},[2143],[54,47320,47322],{"className":47321,"style":12780},[2147],[54,47323,47324,47327],{"style":2151},[54,47325],{"className":47326,"style":2156},[2155],[54,47328,47330],{"className":47329},[2160,2161,2162,2163],[54,47331,133],{"className":47332},[2126,2163],[54,47334,2781],{"className":47335},[2780],[54,47337,47339],{"className":47338},[2143],[54,47340,47342],{"className":47341,"style":46527},[2147],[54,47343],{},[54,47345],{"className":47346},[2336,9349],[54,47348],{"className":47349,"style":2475},[2270],[54,47351,122],{"className":47352},[2479],[54,47354],{"className":47355,"style":2475},[2270],[54,47357,47359,47362],{"className":47358},[2117],[54,47360],{"className":47361,"style":2285},[2121],[54,47363,43900],{"className":47364},[2126],[13,47366,47367],{},"Full population inversion is achievable in the ideal resonant model. For any nonzero detuning, the coefficient falls below one, so the state generally cannot reach unit excited-state probability.",[13,47369,47370],{},"At resonance, the excited-state probability simplifies to",[54,47372,47374],{"className":47373},[2340],[54,47375,47377,47431],{"className":47376},[2079],[54,47378,47380],{"className":47379},[2083],[2085,47381,47382],{"xmlns":2087,"display":2349},[2089,47383,47384,47428],{},[2092,47385,47386,47392,47394,47396,47398,47400,47410,47426],{},[2724,47387,47388,47390],{},[2098,47389,2730],{},[2098,47391,2842],{},[2245,47393,129],{"stretchy":2311},[2098,47395,41261],{},[2245,47397,3205],{"stretchy":2311},[2245,47399,122],{},[2095,47401,47402,47408],{},[2092,47403,47404,47406],{},[2098,47405,46278],{},[2245,47407,36247],{},[2249,47409,133],{},[2092,47411,47412,47414,47424],{},[2245,47413,129],{"fence":2113},[9047,47415,47416,47422],{},[2092,47417,47418,47420],{},[2098,47419,42219],{"mathvariant":2304},[2098,47421,41261],{},[2249,47423,133],{},[2245,47425,3205],{"fence":2113},[2098,47427,29],{"mathvariant":2304},[2105,47429,47430],{"encoding":2107},"P_e(\\tau)=\\sin^2\\left(\\frac{\\Omega\\tau}{2}\\right).",[54,47432,47434,47498],{"className":47433,"ariaHidden":2113},[2112],[54,47435,47437,47440,47480,47483,47486,47489,47492,47495],{"className":47436},[2117],[54,47438],{"className":47439,"style":2325},[2121],[54,47441,47443,47446],{"className":47442},[2126],[54,47444,2730],{"className":47445,"style":2776},[2126,2167],[54,47447,47449],{"className":47448},[2135],[54,47450,47452,47472],{"className":47451},[2139,2756],[54,47453,47455,47469],{"className":47454},[2143],[54,47456,47458],{"className":47457,"style":4063},[2147],[54,47459,47460,47463],{"style":7193},[54,47461],{"className":47462,"style":2156},[2155],[54,47464,47466],{"className":47465},[2160,2161,2162,2163],[54,47467,2842],{"className":47468},[2126,2167,2163],[54,47470,2781],{"className":47471},[2780],[54,47473,47475],{"className":47474},[2143],[54,47476,47478],{"className":47477,"style":2788},[2147],[54,47479],{},[54,47481,129],{"className":47482},[2497],[54,47484,41261],{"className":47485,"style":41333},[2126,2167],[54,47487,3205],{"className":47488},[2336],[54,47490],{"className":47491,"style":2475},[2270],[54,47493,122],{"className":47494},[2479],[54,47496],{"className":47497,"style":2475},[2270],[54,47499,47501,47504,47533,47536,47616,47619],{"className":47500},[2117],[54,47502],{"className":47503,"style":37225},[2121],[54,47505,47507,47510],{"className":47506},[7126],[54,47508,46278],{"className":47509},[7126],[54,47511,47513],{"className":47512},[2135],[54,47514,47516],{"className":47515},[2139],[54,47517,47519],{"className":47518},[2143],[54,47520,47522],{"className":47521,"style":46554},[2147],[54,47523,47524,47527],{"style":46557},[54,47525],{"className":47526,"style":2156},[2155],[54,47528,47530],{"className":47529},[2160,2161,2162,2163],[54,47531,133],{"className":47532},[2126,2163],[54,47534],{"className":47535,"style":2521},[2270],[54,47537,47539,47545,47610],{"className":47538},[2493],[54,47540,47542],{"className":47541,"style":2499},[2497,2498],[54,47543,129],{"className":47544},[2503,2162],[54,47546,47548,47551,47607],{"className":47547},[2126],[54,47549],{"className":47550},[2497,9349],[54,47552,47554],{"className":47553},[9047],[54,47555,47557,47599],{"className":47556},[2139,2756],[54,47558,47560,47596],{"className":47559},[2143],[54,47561,47563,47574,47582],{"className":47562,"style":46597},[2147],[54,47564,47565,47568],{"style":11273},[54,47566],{"className":47567,"style":3866},[2155],[54,47569,47571],{"className":47570},[2126],[54,47572,133],{"className":47573},[2126],[54,47575,47576,47579],{"style":9434},[54,47577],{"className":47578,"style":3866},[2155],[54,47580],{"className":47581,"style":9442},[9441],[54,47583,47584,47587],{"style":9445},[54,47585],{"className":47586,"style":3866},[2155],[54,47588,47590,47593],{"className":47589},[2126],[54,47591,42219],{"className":47592},[2126],[54,47594,41261],{"className":47595,"style":41333},[2126,2167],[54,47597,2781],{"className":47598},[2780],[54,47600,47602],{"className":47601},[2143],[54,47603,47605],{"className":47604,"style":11311},[2147],[54,47606],{},[54,47608],{"className":47609},[2336,9349],[54,47611,47613],{"className":47612,"style":2499},[2336,2498],[54,47614,3205],{"className":47615},[2503,2162],[54,47617],{"className":47618,"style":2521},[2270],[54,47620,29],{"className":47621},[2126],[13,47623,47624,47625,47654],{},"The pulse duration required for ideal population inversion defines the standard ",[54,47626,47628,47642],{"className":47627},[2079],[54,47629,47631],{"className":47630},[2083],[2085,47632,47633],{"xmlns":2087},[2089,47634,47635,47639],{},[2092,47636,47637],{},[2098,47638,8119],{},[2105,47640,47641],{"encoding":2107},"\\pi",[54,47643,47645],{"className":47644,"ariaHidden":2113},[2112],[54,47646,47648,47651],{"className":47647},[2117],[54,47649],{"className":47650,"style":2195},[2121],[54,47652,8119],{"className":47653,"style":2332},[2126,2167],"-pulse time,",[54,47656,47658],{"className":47657},[2340],[54,47659,47661,47711],{"className":47660},[2079],[54,47662,47664],{"className":47663},[2083],[2085,47665,47666],{"xmlns":2087,"display":2349},[2089,47667,47668,47708],{},[2092,47669,47670,47676,47678,47684,47686,47706],{},[2724,47671,47672,47674],{},[2098,47673,41261],{},[2098,47675,8119],{},[2245,47677,122],{},[9047,47679,47680,47682],{},[2098,47681,8119],{},[2098,47683,42219],{"mathvariant":2304},[2245,47685,122],{},[9047,47687,47688,47690],{},[2249,47689,174],{},[2092,47691,47692,47694,47696,47698,47700,47702,47704],{},[2249,47693,133],{},[2245,47695,129],{"stretchy":2311},[2098,47697,42219],{"mathvariant":2304},[2098,47699,194],{"mathvariant":2304},[2249,47701,133],{},[2098,47703,8119],{},[2245,47705,3205],{"stretchy":2311},[2098,47707,29],{"mathvariant":2304},[2105,47709,47710],{"encoding":2107},"\\tau_\\pi=\\frac{\\pi}{\\Omega}=\\frac{1}{2(\\Omega\u002F2\\pi)}.",[54,47712,47714,47770,47849],{"className":47713,"ariaHidden":2113},[2112],[54,47715,47717,47720,47761,47764,47767],{"className":47716},[2117],[54,47718],{"className":47719,"style":24753},[2121],[54,47721,47723,47726],{"className":47722},[2126],[54,47724,41261],{"className":47725,"style":41333},[2126,2167],[54,47727,47729],{"className":47728},[2135],[54,47730,47732,47753],{"className":47731},[2139,2756],[54,47733,47735,47750],{"className":47734},[2143],[54,47736,47738],{"className":47737,"style":4063},[2147],[54,47739,47741,47744],{"style":47740},"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;",[54,47742],{"className":47743,"style":2156},[2155],[54,47745,47747],{"className":47746},[2160,2161,2162,2163],[54,47748,8119],{"className":47749,"style":2332},[2126,2167,2163],[54,47751,2781],{"className":47752},[2780],[54,47754,47756],{"className":47755},[2143],[54,47757,47759],{"className":47758,"style":2788},[2147],[54,47760],{},[54,47762],{"className":47763,"style":2475},[2270],[54,47765,122],{"className":47766},[2479],[54,47768],{"className":47769,"style":2475},[2270],[54,47771,47773,47777,47840,47843,47846],{"className":47772},[2117],[54,47774],{"className":47775,"style":47776},[2121],"height:1.7936em;vertical-align:-0.686em;",[54,47778,47780,47783,47837],{"className":47779},[2126],[54,47781],{"className":47782},[2497,9349],[54,47784,47786],{"className":47785},[9047],[54,47787,47789,47829],{"className":47788},[2139,2756],[54,47790,47792,47826],{"className":47791},[2143],[54,47793,47796,47807,47815],{"className":47794,"style":47795},[2147],"height:1.1076em;",[54,47797,47798,47801],{"style":11273},[54,47799],{"className":47800,"style":3866},[2155],[54,47802,47804],{"className":47803},[2126],[54,47805,42219],{"className":47806},[2126],[54,47808,47809,47812],{"style":9434},[54,47810],{"className":47811,"style":3866},[2155],[54,47813],{"className":47814,"style":9442},[9441],[54,47816,47817,47820],{"style":9445},[54,47818],{"className":47819,"style":3866},[2155],[54,47821,47823],{"className":47822},[2126],[54,47824,8119],{"className":47825,"style":2332},[2126,2167],[54,47827,2781],{"className":47828},[2780],[54,47830,47832],{"className":47831},[2143],[54,47833,47835],{"className":47834,"style":11311},[2147],[54,47836],{},[54,47838],{"className":47839},[2336,9349],[54,47841],{"className":47842,"style":2475},[2270],[54,47844,122],{"className":47845},[2479],[54,47847],{"className":47848,"style":2475},[2270],[54,47850,47852,47855,47929],{"className":47851},[2117],[54,47853],{"className":47854,"style":35395},[2121],[54,47856,47858,47861,47926],{"className":47857},[2126],[54,47859],{"className":47860},[2497,9349],[54,47862,47864],{"className":47863},[9047],[54,47865,47867,47918],{"className":47866},[2139,2756],[54,47868,47870,47915],{"className":47869},[2143],[54,47871,47873,47896,47904],{"className":47872,"style":9362},[2147],[54,47874,47875,47878],{"style":11273},[54,47876],{"className":47877,"style":3866},[2155],[54,47879,47881,47884,47887,47890,47893],{"className":47880},[2126],[54,47882,133],{"className":47883},[2126],[54,47885,129],{"className":47886},[2497],[54,47888,42277],{"className":47889},[2126],[54,47891,8119],{"className":47892,"style":2332},[2126,2167],[54,47894,3205],{"className":47895},[2336],[54,47897,47898,47901],{"style":9434},[54,47899],{"className":47900,"style":3866},[2155],[54,47902],{"className":47903,"style":9442},[9441],[54,47905,47906,47909],{"style":9445},[54,47907],{"className":47908,"style":3866},[2155],[54,47910,47912],{"className":47911},[2126],[54,47913,174],{"className":47914},[2126],[54,47916,2781],{"className":47917},[2780],[54,47919,47921],{"className":47920},[2143],[54,47922,47924],{"className":47923,"style":29113},[2147],[54,47925],{},[54,47927],{"className":47928},[2336,9349],[54,47930,29],{"className":47931},[2126],[13,47933,47934],{},"Given the default drive strength",[54,47936,47938],{"className":47937},[2340],[54,47939,47941,47981],{"className":47940},[2079],[54,47942,47944],{"className":47943},[2083],[2085,47945,47946],{"xmlns":2087,"display":2349},[2089,47947,47948,47978],{},[2092,47949,47950,47960,47962,47965,47967,47976],{},[9047,47951,47952,47954],{},[2098,47953,42219],{"mathvariant":2304},[2092,47955,47956,47958],{},[2249,47957,133],{},[2098,47959,8119],{},[2245,47961,122],{},[2249,47963,47964],{},"20",[5723,47966,5725],{},[2092,47968,47969,47972,47974],{},[2098,47970,47971],{"mathvariant":2304},"M",[2098,47973,2694],{"mathvariant":2304},[2098,47975,215],{"mathvariant":2304},[2245,47977,2375],{"separator":2113},[2105,47979,47980],{"encoding":2107},"\\frac{\\Omega}{2\\pi}=20~\\mathrm{MHz},",[54,47982,47984,48065],{"className":47983,"ariaHidden":2113},[2112],[54,47985,47987,47991,48056,48059,48062],{"className":47986},[2117],[54,47988],{"className":47989,"style":47990},[2121],"height:2.0463em;vertical-align:-0.686em;",[54,47992,47994,47997,48053],{"className":47993},[2126],[54,47995],{"className":47996},[2497,9349],[54,47998,48000],{"className":47999},[9047],[54,48001,48003,48045],{"className":48002},[2139,2756],[54,48004,48006,48042],{"className":48005},[2143],[54,48007,48009,48023,48031],{"className":48008,"style":46597},[2147],[54,48010,48011,48014],{"style":11273},[54,48012],{"className":48013,"style":3866},[2155],[54,48015,48017,48020],{"className":48016},[2126],[54,48018,133],{"className":48019},[2126],[54,48021,8119],{"className":48022,"style":2332},[2126,2167],[54,48024,48025,48028],{"style":9434},[54,48026],{"className":48027,"style":3866},[2155],[54,48029],{"className":48030,"style":9442},[9441],[54,48032,48033,48036],{"style":9445},[54,48034],{"className":48035,"style":3866},[2155],[54,48037,48039],{"className":48038},[2126],[54,48040,42219],{"className":48041},[2126],[54,48043,2781],{"className":48044},[2780],[54,48046,48048],{"className":48047},[2143],[54,48049,48051],{"className":48050,"style":11311},[2147],[54,48052],{},[54,48054],{"className":48055},[2336,9349],[54,48057],{"className":48058,"style":2475},[2270],[54,48060,122],{"className":48061},[2479],[54,48063],{"className":48064,"style":2475},[2270],[54,48066,48068,48072,48075,48078,48085],{"className":48067},[2117],[54,48069],{"className":48070,"style":48071},[2121],"height:0.8778em;vertical-align:-0.1944em;",[54,48073,47964],{"className":48074},[2126],[54,48076,5725],{"className":48077},[2270,43446],[54,48079,48081],{"className":48080},[2126],[54,48082,48084],{"className":48083},[2126,43453],"MHz",[54,48086,2375],{"className":48087},[2517],[13,48089,48090,48091,48119],{},"the ideal ",[54,48092,48094,48107],{"className":48093},[2079],[54,48095,48097],{"className":48096},[2083],[2085,48098,48099],{"xmlns":2087},[2089,48100,48101,48105],{},[2092,48102,48103],{},[2098,48104,8119],{},[2105,48106,47641],{"encoding":2107},[54,48108,48110],{"className":48109,"ariaHidden":2113},[2112],[54,48111,48113,48116],{"className":48112},[2117],[54,48114],{"className":48115,"style":2195},[2121],[54,48117,8119],{"className":48118,"style":2332},[2126,2167],"-pulse time is",[54,48121,48123],{"className":48122},[2340],[54,48124,48126,48159],{"className":48125},[2079],[54,48127,48129],{"className":48128},[2083],[2085,48130,48131],{"xmlns":2087,"display":2349},[2089,48132,48133,48156],{},[2092,48134,48135,48141,48143,48146,48148,48154],{},[2724,48136,48137,48139],{},[2098,48138,41261],{},[2098,48140,8119],{},[2245,48142,122],{},[2249,48144,48145],{},"25",[5723,48147,5725],{},[2092,48149,48150,48152],{},[2098,48151,3976],{"mathvariant":2304},[2098,48153,14089],{"mathvariant":2304},[2098,48155,29],{"mathvariant":2304},[2105,48157,48158],{"encoding":2107},"\\tau_\\pi=25~\\mathrm{ns}.",[54,48160,48162,48217],{"className":48161,"ariaHidden":2113},[2112],[54,48163,48165,48168,48208,48211,48214],{"className":48164},[2117],[54,48166],{"className":48167,"style":24753},[2121],[54,48169,48171,48174],{"className":48170},[2126],[54,48172,41261],{"className":48173,"style":41333},[2126,2167],[54,48175,48177],{"className":48176},[2135],[54,48178,48180,48200],{"className":48179},[2139,2756],[54,48181,48183,48197],{"className":48182},[2143],[54,48184,48186],{"className":48185,"style":4063},[2147],[54,48187,48188,48191],{"style":47740},[54,48189],{"className":48190,"style":2156},[2155],[54,48192,48194],{"className":48193},[2160,2161,2162,2163],[54,48195,8119],{"className":48196,"style":2332},[2126,2167,2163],[54,48198,2781],{"className":48199},[2780],[54,48201,48203],{"className":48202},[2143],[54,48204,48206],{"className":48205,"style":2788},[2147],[54,48207],{},[54,48209],{"className":48210,"style":2475},[2270],[54,48212,122],{"className":48213},[2479],[54,48215],{"className":48216,"style":2475},[2270],[54,48218,48220,48223,48226,48229,48236],{"className":48219},[2117],[54,48221],{"className":48222,"style":2285},[2121],[54,48224,48145],{"className":48225},[2126],[54,48227,5725],{"className":48228},[2270,43446],[54,48230,48232],{"className":48231},[2126],[54,48233,48235],{"className":48234},[2126,43453],"ns",[54,48237,29],{"className":48238},[2126],[13,48240,48241],{},"A full period of the resonant excited-state probability oscillation is",[54,48243,48245],{"className":48244},[2340],[54,48246,48248,48296],{"className":48247},[2079],[54,48249,48251],{"className":48250},[2083],[2085,48252,48253],{"xmlns":2087,"display":2349},[2089,48254,48255,48293],{},[2092,48256,48257,48263,48265,48275,48277,48291],{},[2724,48258,48259,48261],{},[2098,48260,36603],{},[2098,48262,19926],{},[2245,48264,122],{},[9047,48266,48267,48273],{},[2092,48268,48269,48271],{},[2249,48270,133],{},[2098,48272,8119],{},[2098,48274,42219],{"mathvariant":2304},[2245,48276,122],{},[9047,48278,48279,48281],{},[2249,48280,174],{},[2092,48282,48283,48285,48287,48289],{},[2098,48284,42219],{"mathvariant":2304},[2098,48286,194],{"mathvariant":2304},[2249,48288,133],{},[2098,48290,8119],{},[2098,48292,29],{"mathvariant":2304},[2105,48294,48295],{"encoding":2107},"T_R=\\frac{2\\pi}{\\Omega}=\\frac{1}{\\Omega\u002F2\\pi}.",[54,48297,48299,48354,48435],{"className":48298,"ariaHidden":2113},[2112],[54,48300,48302,48305,48345,48348,48351],{"className":48301},[2117],[54,48303],{"className":48304,"style":2743},[2121],[54,48306,48308,48311],{"className":48307},[2126],[54,48309,36603],{"className":48310,"style":2776},[2126,2167],[54,48312,48314],{"className":48313},[2135],[54,48315,48317,48337],{"className":48316},[2139,2756],[54,48318,48320,48334],{"className":48319},[2143],[54,48321,48323],{"className":48322,"style":2763},[2147],[54,48324,48325,48328],{"style":7193},[54,48326],{"className":48327,"style":2156},[2155],[54,48329,48331],{"className":48330},[2160,2161,2162,2163],[54,48332,19926],{"className":48333,"style":19951},[2126,2167,2163],[54,48335,2781],{"className":48336},[2780],[54,48338,48340],{"className":48339},[2143],[54,48341,48343],{"className":48342,"style":2788},[2147],[54,48344],{},[54,48346],{"className":48347,"style":2475},[2270],[54,48349,122],{"className":48350},[2479],[54,48352],{"className":48353,"style":2475},[2270],[54,48355,48357,48361,48426,48429,48432],{"className":48356},[2117],[54,48358],{"className":48359,"style":48360},[2121],"height:2.0074em;vertical-align:-0.686em;",[54,48362,48364,48367,48423],{"className":48363},[2126],[54,48365],{"className":48366},[2497,9349],[54,48368,48370],{"className":48369},[9047],[54,48371,48373,48415],{"className":48372},[2139,2756],[54,48374,48376,48412],{"className":48375},[2143],[54,48377,48379,48390,48398],{"className":48378,"style":9362},[2147],[54,48380,48381,48384],{"style":11273},[54,48382],{"className":48383,"style":3866},[2155],[54,48385,48387],{"className":48386},[2126],[54,48388,42219],{"className":48389},[2126],[54,48391,48392,48395],{"style":9434},[54,48393],{"className":48394,"style":3866},[2155],[54,48396],{"className":48397,"style":9442},[9441],[54,48399,48400,48403],{"style":9445},[54,48401],{"className":48402,"style":3866},[2155],[54,48404,48406,48409],{"className":48405},[2126],[54,48407,133],{"className":48408},[2126],[54,48410,8119],{"className":48411,"style":2332},[2126,2167],[54,48413,2781],{"className":48414},[2780],[54,48416,48418],{"className":48417},[2143],[54,48419,48421],{"className":48420,"style":11311},[2147],[54,48422],{},[54,48424],{"className":48425},[2336,9349],[54,48427],{"className":48428,"style":2475},[2270],[54,48430,122],{"className":48431},[2479],[54,48433],{"className":48434,"style":2475},[2270],[54,48436,48438,48441,48506],{"className":48437},[2117],[54,48439],{"className":48440,"style":35395},[2121],[54,48442,48444,48447,48503],{"className":48443},[2126],[54,48445],{"className":48446},[2497,9349],[54,48448,48450],{"className":48449},[9047],[54,48451,48453,48495],{"className":48452},[2139,2756],[54,48454,48456,48492],{"className":48455},[2143],[54,48457,48459,48473,48481],{"className":48458,"style":9362},[2147],[54,48460,48461,48464],{"style":11273},[54,48462],{"className":48463,"style":3866},[2155],[54,48465,48467,48470],{"className":48466},[2126],[54,48468,42277],{"className":48469},[2126],[54,48471,8119],{"className":48472,"style":2332},[2126,2167],[54,48474,48475,48478],{"style":9434},[54,48476],{"className":48477,"style":3866},[2155],[54,48479],{"className":48480,"style":9442},[9441],[54,48482,48483,48486],{"style":9445},[54,48484],{"className":48485,"style":3866},[2155],[54,48487,48489],{"className":48488},[2126],[54,48490,174],{"className":48491},[2126],[54,48493,2781],{"className":48494},[2780],[54,48496,48498],{"className":48497},[2143],[54,48499,48501],{"className":48500,"style":29113},[2147],[54,48502],{},[54,48504],{"className":48505},[2336,9349],[54,48507,29],{"className":48508},[2126],[13,48510,48511],{},"For the same default drive strength, the full temporal period is",[54,48513,48515],{"className":48514},[2340],[54,48516,48518,48551],{"className":48517},[2079],[54,48519,48521],{"className":48520},[2083],[2085,48522,48523],{"xmlns":2087,"display":2349},[2089,48524,48525,48548],{},[2092,48526,48527,48533,48535,48538,48540,48546],{},[2724,48528,48529,48531],{},[2098,48530,36603],{},[2098,48532,19926],{},[2245,48534,122],{},[2249,48536,48537],{},"50",[5723,48539,5725],{},[2092,48541,48542,48544],{},[2098,48543,3976],{"mathvariant":2304},[2098,48545,14089],{"mathvariant":2304},[2098,48547,29],{"mathvariant":2304},[2105,48549,48550],{"encoding":2107},"T_R=50~\\mathrm{ns}.",[54,48552,48554,48609],{"className":48553,"ariaHidden":2113},[2112],[54,48555,48557,48560,48600,48603,48606],{"className":48556},[2117],[54,48558],{"className":48559,"style":2743},[2121],[54,48561,48563,48566],{"className":48562},[2126],[54,48564,36603],{"className":48565,"style":2776},[2126,2167],[54,48567,48569],{"className":48568},[2135],[54,48570,48572,48592],{"className":48571},[2139,2756],[54,48573,48575,48589],{"className":48574},[2143],[54,48576,48578],{"className":48577,"style":2763},[2147],[54,48579,48580,48583],{"style":7193},[54,48581],{"className":48582,"style":2156},[2155],[54,48584,48586],{"className":48585},[2160,2161,2162,2163],[54,48587,19926],{"className":48588,"style":19951},[2126,2167,2163],[54,48590,2781],{"className":48591},[2780],[54,48593,48595],{"className":48594},[2143],[54,48596,48598],{"className":48597,"style":2788},[2147],[54,48599],{},[54,48601],{"className":48602,"style":2475},[2270],[54,48604,122],{"className":48605},[2479],[54,48607],{"className":48608,"style":2475},[2270],[54,48610,48612,48615,48618,48621,48627],{"className":48611},[2117],[54,48613],{"className":48614,"style":2285},[2121],[54,48616,48537],{"className":48617},[2126],[54,48619,5725],{"className":48620},[2270,43446],[54,48622,48624],{"className":48623},[2126],[54,48625,48235],{"className":48626},[2126,43453],[54,48628,29],{"className":48629},[2126],[31,48631,48633],{"id":48632},"phenomenological-dephasing","Phenomenological Dephasing",[13,48635,48636],{},"The computational model includes an optional phenomenological damping factor,",[54,48638,48640],{"className":48639},[2340],[54,48641,48643,48715],{"className":48642},[2079],[54,48644,48646],{"className":48645},[2083],[2085,48647,48648],{"xmlns":2087,"display":2349},[2089,48649,48650,48712],{},[2092,48651,48652,48658,48660,48662,48664,48666,48668,48670,48676,48678,48680,48682,48684,48686,48688,48690,48710],{},[2724,48653,48654,48656],{},[2098,48655,2730],{},[2098,48657,2842],{},[2245,48659,129],{"stretchy":2311},[2098,48661,21834],{},[2245,48663,2375],{"separator":2113},[2098,48665,41261],{},[2245,48667,3205],{"stretchy":2311},[2245,48669,14490],{},[2724,48671,48672,48674],{},[2098,48673,2730],{},[2098,48675,2842],{},[2245,48677,129],{"stretchy":2311},[2098,48679,21834],{},[2245,48681,2375],{"separator":2113},[2098,48683,41261],{},[2245,48685,3205],{"stretchy":2311},[2098,48687,23625],{},[2245,48689,36247],{},[2092,48691,48692,48694,48696,48708],{},[2245,48693,129],{"fence":2113},[2245,48695,2247],{},[9047,48697,48698,48700],{},[2098,48699,41261],{},[3331,48701,48702,48704,48706],{},[2098,48703,36603],{},[2249,48705,133],{},[2245,48707,20325],{},[2245,48709,3205],{"fence":2113},[2098,48711,29],{"mathvariant":2304},[2105,48713,48714],{"encoding":2107},"P_e(f,\\tau)\\rightarrow P_e(f,\\tau)\\exp\\left(-\\frac{\\tau}{T_2^\\ast}\\right).",[54,48716,48718,48791],{"className":48717,"ariaHidden":2113},[2112],[54,48719,48721,48724,48764,48767,48770,48773,48776,48779,48782,48785,48788],{"className":48720},[2117],[54,48722],{"className":48723,"style":2325},[2121],[54,48725,48727,48730],{"className":48726},[2126],[54,48728,2730],{"className":48729,"style":2776},[2126,2167],[54,48731,48733],{"className":48732},[2135],[54,48734,48736,48756],{"className":48735},[2139,2756],[54,48737,48739,48753],{"className":48738},[2143],[54,48740,48742],{"className":48741,"style":4063},[2147],[54,48743,48744,48747],{"style":7193},[54,48745],{"className":48746,"style":2156},[2155],[54,48748,48750],{"className":48749},[2160,2161,2162,2163],[54,48751,2842],{"className":48752},[2126,2167,2163],[54,48754,2781],{"className":48755},[2780],[54,48757,48759],{"className":48758},[2143],[54,48760,48762],{"className":48761,"style":2788},[2147],[54,48763],{},[54,48765,129],{"className":48766},[2497],[54,48768,21834],{"className":48769,"style":26895},[2126,2167],[54,48771,2375],{"className":48772},[2517],[54,48774],{"className":48775,"style":2521},[2270],[54,48777,41261],{"className":48778,"style":41333},[2126,2167],[54,48780,3205],{"className":48781},[2336],[54,48783],{"className":48784,"style":2475},[2270],[54,48786,14490],{"className":48787},[2479],[54,48789],{"className":48790,"style":2475},[2270],[54,48792,48794,48798,48838,48841,48844,48847,48850,48853,48856,48859,48862,48865,48997,49000],{"className":48793},[2117],[54,48795],{"className":48796,"style":48797},[2121],"height:2.4023em;vertical-align:-0.9523em;",[54,48799,48801,48804],{"className":48800},[2126],[54,48802,2730],{"className":48803,"style":2776},[2126,2167],[54,48805,48807],{"className":48806},[2135],[54,48808,48810,48830],{"className":48809},[2139,2756],[54,48811,48813,48827],{"className":48812},[2143],[54,48814,48816],{"className":48815,"style":4063},[2147],[54,48817,48818,48821],{"style":7193},[54,48819],{"className":48820,"style":2156},[2155],[54,48822,48824],{"className":48823},[2160,2161,2162,2163],[54,48825,2842],{"className":48826},[2126,2167,2163],[54,48828,2781],{"className":48829},[2780],[54,48831,48833],{"className":48832},[2143],[54,48834,48836],{"className":48835,"style":2788},[2147],[54,48837],{},[54,48839,129],{"className":48840},[2497],[54,48842,21834],{"className":48843,"style":26895},[2126,2167],[54,48845,2375],{"className":48846},[2517],[54,48848],{"className":48849,"style":2521},[2270],[54,48851,41261],{"className":48852,"style":41333},[2126,2167],[54,48854,3205],{"className":48855},[2336],[54,48857],{"className":48858,"style":2521},[2270],[54,48860,23625],{"className":48861},[7126],[54,48863],{"className":48864,"style":2521},[2270],[54,48866,48868,48874,48877,48991],{"className":48867},[2493],[54,48869,48871],{"className":48870,"style":2499},[2497,2498],[54,48872,129],{"className":48873},[2503,2162],[54,48875,2247],{"className":48876},[2126],[54,48878,48880,48883,48988],{"className":48879},[2126],[54,48881],{"className":48882},[2497,9349],[54,48884,48886],{"className":48885},[9047],[54,48887,48889,48979],{"className":48888},[2139,2756],[54,48890,48892,48976],{"className":48891},[2143],[54,48893,48895,48957,48965],{"className":48894,"style":47795},[2147],[54,48896,48897,48900],{"style":11273},[54,48898],{"className":48899,"style":3866},[2155],[54,48901,48903],{"className":48902},[2126],[54,48904,48906,48909],{"className":48905},[2126],[54,48907,36603],{"className":48908,"style":2776},[2126,2167],[54,48910,48912],{"className":48911},[2135],[54,48913,48915,48948],{"className":48914},[2139,2756],[54,48916,48918,48945],{"className":48917},[2143],[54,48919,48922,48934],{"className":48920,"style":48921},[2147],"height:0.6705em;",[54,48923,48925,48928],{"style":48924},"top:-2.4337em;margin-left:-0.1389em;margin-right:0.05em;",[54,48926],{"className":48927,"style":2156},[2155],[54,48929,48931],{"className":48930},[2160,2161,2162,2163],[54,48932,133],{"className":48933},[2126,2163],[54,48935,48936,48939],{"style":46451},[54,48937],{"className":48938,"style":2156},[2155],[54,48940,48942],{"className":48941},[2160,2161,2162,2163],[54,48943,20325],{"className":48944},[2275,2163],[54,48946,2781],{"className":48947},[2780],[54,48949,48951],{"className":48950},[2143],[54,48952,48955],{"className":48953,"style":48954},[2147],"height:0.2663em;",[54,48956],{},[54,48958,48959,48962],{"style":9434},[54,48960],{"className":48961,"style":3866},[2155],[54,48963],{"className":48964,"style":9442},[9441],[54,48966,48967,48970],{"style":9445},[54,48968],{"className":48969,"style":3866},[2155],[54,48971,48973],{"className":48972},[2126],[54,48974,41261],{"className":48975,"style":41333},[2126,2167],[54,48977,2781],{"className":48978},[2780],[54,48980,48982],{"className":48981},[2143],[54,48983,48986],{"className":48984,"style":48985},[2147],"height:0.9523em;",[54,48987],{},[54,48989],{"className":48990},[2336,9349],[54,48992,48994],{"className":48993,"style":2499},[2336,2498],[54,48995,3205],{"className":48996},[2503,2162],[54,48998],{"className":48999,"style":2521},[2270],[54,49001,29],{"className":49002},[2126],[13,49004,49005],{},"This exponential envelope attenuates the plotted Rabi fringes as pulse duration increases. The factor is a compact visualization device for contrast decay. A rigorous open-quantum-system treatment would require a density matrix model with relaxation channels, dephasing channels, finite temperature effects, shaped pulse envelopes, and stochastic dynamics. The default dephasing parameter is",[54,49007,49009],{"className":49008},[2340],[54,49010,49012,49047],{"className":49011},[2079],[54,49013,49015],{"className":49014},[2083],[2085,49016,49017],{"xmlns":2087,"display":2349},[2089,49018,49019,49044],{},[2092,49020,49021,49029,49031,49034,49036,49042],{},[3331,49022,49023,49025,49027],{},[2098,49024,36603],{},[2249,49026,133],{},[2245,49028,20325],{},[2245,49030,122],{},[2249,49032,49033],{},"500",[5723,49035,5725],{},[2092,49037,49038,49040],{},[2098,49039,3976],{"mathvariant":2304},[2098,49041,14089],{"mathvariant":2304},[2098,49043,29],{"mathvariant":2304},[2105,49045,49046],{"encoding":2107},"T_2^\\ast=500~\\mathrm{ns}.",[54,49048,49050,49118],{"className":49049,"ariaHidden":2113},[2112],[54,49051,49053,49057,49109,49112,49115],{"className":49052},[2117],[54,49054],{"className":49055,"style":49056},[2121],"height:0.9857em;vertical-align:-0.247em;",[54,49058,49060,49063],{"className":49059},[2126],[54,49061,36603],{"className":49062,"style":2776},[2126,2167],[54,49064,49066],{"className":49065},[2135],[54,49067,49069,49101],{"className":49068},[2139,2756],[54,49070,49072,49098],{"className":49071},[2143],[54,49073,49075,49087],{"className":49074,"style":36702},[2147],[54,49076,49078,49081],{"style":49077},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[54,49079],{"className":49080,"style":2156},[2155],[54,49082,49084],{"className":49083},[2160,2161,2162,2163],[54,49085,133],{"className":49086},[2126,2163],[54,49088,49089,49092],{"style":2462},[54,49090],{"className":49091,"style":2156},[2155],[54,49093,49095],{"className":49094},[2160,2161,2162,2163],[54,49096,20325],{"className":49097},[2275,2163],[54,49099,2781],{"className":49100},[2780],[54,49102,49104],{"className":49103},[2143],[54,49105,49107],{"className":49106,"style":3469},[2147],[54,49108],{},[54,49110],{"className":49111,"style":2475},[2270],[54,49113,122],{"className":49114},[2479],[54,49116],{"className":49117,"style":2475},[2270],[54,49119,49121,49124,49127,49130,49136],{"className":49120},[2117],[54,49122],{"className":49123,"style":2285},[2121],[54,49125,49033],{"className":49126},[2126],[54,49128,5725],{"className":49129},[2270,43446],[54,49131,49133],{"className":49132},[2126],[54,49134,48235],{"className":49135},[2126,43453],[54,49137,29],{"className":49138},[2126],[13,49140,49141,49142,49145],{},"Setting ",[51,49143,49144],{},"T2STAR_NS = None"," disables the phenomenological damping factor.",[31,49147,49149],{"id":49148},"detuning-and-generalized-rabi-frequency","Detuning and Generalized Rabi Frequency",[13,49151,49152],{},"Detuning quantifies the spectral mismatch between the applied drive and the natural qubit transition,",[54,49154,49156],{"className":49155},[2340],[54,49157,49159,49195],{"className":49158},[2079],[54,49160,49162],{"className":49161},[2083],[2085,49163,49164],{"xmlns":2087,"display":2349},[2089,49165,49166,49192],{},[2092,49167,49168,49170,49172,49174,49176,49178,49180,49182,49188,49190],{},[2098,49169,42302],{"mathvariant":2304},[2245,49171,122],{},[2249,49173,133],{},[2098,49175,8119],{},[2245,49177,129],{"stretchy":2311},[2098,49179,21834],{},[2245,49181,2247],{},[2724,49183,49184,49186],{},[2098,49185,21834],{},[2249,49187,150],{},[2245,49189,3205],{"stretchy":2311},[2098,49191,29],{"mathvariant":2304},[2105,49193,49194],{"encoding":2107},"\\Delta=2\\pi(f-f_0).",[54,49196,49198,49216,49243],{"className":49197,"ariaHidden":2113},[2112],[54,49199,49201,49204,49207,49210,49213],{"className":49200},[2117],[54,49202],{"className":49203,"style":2122},[2121],[54,49205,42302],{"className":49206},[2126],[54,49208],{"className":49209,"style":2475},[2270],[54,49211,122],{"className":49212},[2479],[54,49214],{"className":49215,"style":2475},[2270],[54,49217,49219,49222,49225,49228,49231,49234,49237,49240],{"className":49218},[2117],[54,49220],{"className":49221,"style":2325},[2121],[54,49223,133],{"className":49224},[2126],[54,49226,8119],{"className":49227,"style":2332},[2126,2167],[54,49229,129],{"className":49230},[2497],[54,49232,21834],{"className":49233,"style":26895},[2126,2167],[54,49235],{"className":49236,"style":2271},[2270],[54,49238,2247],{"className":49239},[2275],[54,49241],{"className":49242,"style":2271},[2270],[54,49244,49246,49249,49289,49292],{"className":49245},[2117],[54,49247],{"className":49248,"style":2325},[2121],[54,49250,49252,49255],{"className":49251},[2126],[54,49253,21834],{"className":49254,"style":26895},[2126,2167],[54,49256,49258],{"className":49257},[2135],[54,49259,49261,49281],{"className":49260},[2139,2756],[54,49262,49264,49278],{"className":49263},[2143],[54,49265,49267],{"className":49266,"style":13837},[2147],[54,49268,49269,49272],{"style":41824},[54,49270],{"className":49271,"style":2156},[2155],[54,49273,49275],{"className":49274},[2160,2161,2162,2163],[54,49276,150],{"className":49277},[2126,2163],[54,49279,2781],{"className":49280},[2780],[54,49282,49284],{"className":49283},[2143],[54,49285,49287],{"className":49286,"style":2788},[2147],[54,49288],{},[54,49290,3205],{"className":49291},[2336],[54,49293,29],{"className":49294},[2126],[13,49296,49297],{},"Expressed in ordinary frequency units, the generalized Rabi frequency is",[54,49299,49301],{"className":49300},[2340],[54,49302,49304,49392],{"className":49303},[2079],[54,49305,49307],{"className":49306},[2083],[2085,49308,49309],{"xmlns":2087,"display":2349},[2089,49310,49311,49389],{},[2092,49312,49313,49319,49321,49323,49325,49327,49341,49343,49387],{},[2724,49314,49315,49317],{},[2098,49316,42532],{},[2098,49318,19926],{},[2245,49320,129],{"stretchy":2311},[2098,49322,21834],{},[2245,49324,3205],{"stretchy":2311},[2245,49326,122],{},[9047,49328,49329,49335],{},[2724,49330,49331,49333],{},[2098,49332,42219],{"mathvariant":2304},[2098,49334,19926],{},[2092,49336,49337,49339],{},[2249,49338,133],{},[2098,49340,8119],{},[2245,49342,122],{},[9052,49344,49345],{},[2092,49346,49347,49367,49369,49371,49373,49375,49381],{},[2095,49348,49349,49365],{},[2092,49350,49351,49353,49363],{},[2245,49352,129],{"fence":2113},[9047,49354,49355,49357],{},[2098,49356,42219],{"mathvariant":2304},[2092,49358,49359,49361],{},[2249,49360,133],{},[2098,49362,8119],{},[2245,49364,3205],{"fence":2113},[2249,49366,133],{},[2245,49368,1663],{},[2245,49370,129],{"stretchy":2311},[2098,49372,21834],{},[2245,49374,2247],{},[2724,49376,49377,49379],{},[2098,49378,21834],{},[2249,49380,150],{},[2095,49382,49383,49385],{},[2245,49384,3205],{"stretchy":2311},[2249,49386,133],{},[2098,49388,29],{"mathvariant":2304},[2105,49390,49391],{"encoding":2107},"\\nu_R(f)=\\frac{\\Omega_R}{2\\pi}=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[54,49393,49395,49459,49576],{"className":49394,"ariaHidden":2113},[2112],[54,49396,49398,49401,49441,49444,49447,49450,49453,49456],{"className":49397},[2117],[54,49399],{"className":49400,"style":2325},[2121],[54,49402,49404,49407],{"className":49403},[2126],[54,49405,42532],{"className":49406,"style":42553},[2126,2167],[54,49408,49410],{"className":49409},[2135],[54,49411,49413,49433],{"className":49412},[2139,2756],[54,49414,49416,49430],{"className":49415},[2143],[54,49417,49419],{"className":49418,"style":2763},[2147],[54,49420,49421,49424],{"style":42568},[54,49422],{"className":49423,"style":2156},[2155],[54,49425,49427],{"className":49426},[2160,2161,2162,2163],[54,49428,19926],{"className":49429,"style":19951},[2126,2167,2163],[54,49431,2781],{"className":49432},[2780],[54,49434,49436],{"className":49435},[2143],[54,49437,49439],{"className":49438,"style":2788},[2147],[54,49440],{},[54,49442,129],{"className":49443},[2497],[54,49445,21834],{"className":49446,"style":26895},[2126,2167],[54,49448,3205],{"className":49449},[2336],[54,49451],{"className":49452,"style":2475},[2270],[54,49454,122],{"className":49455},[2479],[54,49457],{"className":49458,"style":2475},[2270],[54,49460,49462,49465,49567,49570,49573],{"className":49461},[2117],[54,49463],{"className":49464,"style":47990},[2121],[54,49466,49468,49471,49564],{"className":49467},[2126],[54,49469],{"className":49470},[2497,9349],[54,49472,49474],{"className":49473},[9047],[54,49475,49477,49556],{"className":49476},[2139,2756],[54,49478,49480,49553],{"className":49479},[2143],[54,49481,49483,49497,49505],{"className":49482,"style":46597},[2147],[54,49484,49485,49488],{"style":11273},[54,49486],{"className":49487,"style":3866},[2155],[54,49489,49491,49494],{"className":49490},[2126],[54,49492,133],{"className":49493},[2126],[54,49495,8119],{"className":49496,"style":2332},[2126,2167],[54,49498,49499,49502],{"style":9434},[54,49500],{"className":49501,"style":3866},[2155],[54,49503],{"className":49504,"style":9442},[9441],[54,49506,49507,49510],{"style":9445},[54,49508],{"className":49509,"style":3866},[2155],[54,49511,49513],{"className":49512},[2126],[54,49514,49516,49519],{"className":49515},[2126],[54,49517,42219],{"className":49518},[2126],[54,49520,49522],{"className":49521},[2135],[54,49523,49525,49545],{"className":49524},[2139,2756],[54,49526,49528,49542],{"className":49527},[2143],[54,49529,49531],{"className":49530,"style":2763},[2147],[54,49532,49533,49536],{"style":4954},[54,49534],{"className":49535,"style":2156},[2155],[54,49537,49539],{"className":49538},[2160,2161,2162,2163],[54,49540,19926],{"className":49541,"style":19951},[2126,2167,2163],[54,49543,2781],{"className":49544},[2780],[54,49546,49548],{"className":49547},[2143],[54,49549,49551],{"className":49550,"style":2788},[2147],[54,49552],{},[54,49554,2781],{"className":49555},[2780],[54,49557,49559],{"className":49558},[2143],[54,49560,49562],{"className":49561,"style":11311},[2147],[54,49563],{},[54,49565],{"className":49566},[2336,9349],[54,49568],{"className":49569,"style":2475},[2270],[54,49571,122],{"className":49572},[2479],[54,49574],{"className":49575,"style":2475},[2270],[54,49577,49579,49583,49837],{"className":49578},[2117],[54,49580],{"className":49581,"style":49582},[2121],"height:3.04em;vertical-align:-1.0541em;",[54,49584,49586],{"className":49585},[2126,9375],[54,49587,49589,49828],{"className":49588},[2139,2756],[54,49590,49592,49825],{"className":49591},[2143],[54,49593,49596,49808],{"className":49594,"style":49595},[2147],"height:1.9859em;",[54,49597,49600,49604],{"className":49598,"style":49599},[9389],"top:-5em;",[54,49601],{"className":49602,"style":49603},[2155],"height:5em;",[54,49605,49607,49715,49718,49721,49724,49727,49730,49733,49736,49739,49779],{"className":49606,"style":46830},[2126],[54,49608,49610,49690],{"className":49609},[2493],[54,49611,49613,49619,49684],{"className":49612},[2493],[54,49614,49616],{"className":49615,"style":2499},[2497,2498],[54,49617,129],{"className":49618},[2503,2162],[54,49620,49622,49625,49681],{"className":49621},[2126],[54,49623],{"className":49624},[2497,9349],[54,49626,49628],{"className":49627},[9047],[54,49629,49631,49673],{"className":49630},[2139,2756],[54,49632,49634,49670],{"className":49633},[2143],[54,49635,49637,49651,49659],{"className":49636,"style":46597},[2147],[54,49638,49639,49642],{"style":11273},[54,49640],{"className":49641,"style":3866},[2155],[54,49643,49645,49648],{"className":49644},[2126],[54,49646,133],{"className":49647},[2126],[54,49649,8119],{"className":49650,"style":2332},[2126,2167],[54,49652,49653,49656],{"style":9434},[54,49654],{"className":49655,"style":3866},[2155],[54,49657],{"className":49658,"style":9442},[9441],[54,49660,49661,49664],{"style":9445},[54,49662],{"className":49663,"style":3866},[2155],[54,49665,49667],{"className":49666},[2126],[54,49668,42219],{"className":49669},[2126],[54,49671,2781],{"className":49672},[2780],[54,49674,49676],{"className":49675},[2143],[54,49677,49679],{"className":49678,"style":11311},[2147],[54,49680],{},[54,49682],{"className":49683},[2336,9349],[54,49685,49687],{"className":49686,"style":2499},[2336,2498],[54,49688,3205],{"className":49689},[2503,2162],[54,49691,49693],{"className":49692},[2135],[54,49694,49696],{"className":49695},[2139],[54,49697,49699],{"className":49698},[2143],[54,49700,49703],{"className":49701,"style":49702},[2147],"height:1.654em;",[54,49704,49706,49709],{"style":49705},"top:-3.9029em;margin-right:0.05em;",[54,49707],{"className":49708,"style":2156},[2155],[54,49710,49712],{"className":49711},[2160,2161,2162,2163],[54,49713,133],{"className":49714},[2126,2163],[54,49716],{"className":49717,"style":2271},[2270],[54,49719,1663],{"className":49720},[2275],[54,49722],{"className":49723,"style":2271},[2270],[54,49725,129],{"className":49726},[2497],[54,49728,21834],{"className":49729,"style":26895},[2126,2167],[54,49731],{"className":49732,"style":2271},[2270],[54,49734,2247],{"className":49735},[2275],[54,49737],{"className":49738,"style":2271},[2270],[54,49740,49742,49745],{"className":49741},[2126],[54,49743,21834],{"className":49744,"style":26895},[2126,2167],[54,49746,49748],{"className":49747},[2135],[54,49749,49751,49771],{"className":49750},[2139,2756],[54,49752,49754,49768],{"className":49753},[2143],[54,49755,49757],{"className":49756,"style":13837},[2147],[54,49758,49759,49762],{"style":41824},[54,49760],{"className":49761,"style":2156},[2155],[54,49763,49765],{"className":49764},[2160,2161,2162,2163],[54,49766,150],{"className":49767},[2126,2163],[54,49769,2781],{"className":49770},[2780],[54,49772,49774],{"className":49773},[2143],[54,49775,49777],{"className":49776,"style":2788},[2147],[54,49778],{},[54,49780,49782,49785],{"className":49781},[2336],[54,49783,3205],{"className":49784},[2336],[54,49786,49788],{"className":49787},[2135],[54,49789,49791],{"className":49790},[2139],[54,49792,49794],{"className":49793},[2143],[54,49795,49797],{"className":49796,"style":28966},[2147],[54,49798,49799,49802],{"style":28969},[54,49800],{"className":49801,"style":2156},[2155],[54,49803,49805],{"className":49804},[2160,2161,2162,2163],[54,49806,133],{"className":49807},[2126,2163],[54,49809,49811,49814],{"style":49810},"top:-3.9459em;",[54,49812],{"className":49813,"style":49603},[2155],[54,49815,49818],{"className":49816,"style":49817},[9409],"min-width:1.02em;height:3.08em;",[5837,49819,49822],{"xmlns":5839,"width":9413,"height":49820,"viewBox":49821,"preserveAspectRatio":9416},"3.08em","0 0 400000 3240",[5846,49823],{"d":49824},"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z",[54,49826,2781],{"className":49827},[2780],[54,49829,49831],{"className":49830},[2143],[54,49832,49835],{"className":49833,"style":49834},[2147],"height:1.0541em;",[54,49836],{},[54,49838,29],{"className":49839},[2126],[13,49841,49842],{},"This expression explains the ridge visible in the Fourier-domain heatmap. At resonance, the generalized frequency equals the applied drive strength,",[54,49844,49846],{"className":49845},[2340],[54,49847,49849,49891],{"className":49848},[2079],[54,49850,49852],{"className":49851},[2083],[2085,49853,49854],{"xmlns":2087,"display":2349},[2089,49855,49856,49888],{},[2092,49857,49858,49864,49866,49872,49874,49876,49886],{},[2724,49859,49860,49862],{},[2098,49861,42532],{},[2098,49863,19926],{},[2245,49865,129],{"stretchy":2311},[2724,49867,49868,49870],{},[2098,49869,21834],{},[2249,49871,150],{},[2245,49873,3205],{"stretchy":2311},[2245,49875,122],{},[9047,49877,49878,49880],{},[2098,49879,42219],{"mathvariant":2304},[2092,49881,49882,49884],{},[2249,49883,133],{},[2098,49885,8119],{},[2098,49887,29],{"mathvariant":2304},[2105,49889,49890],{"encoding":2107},"\\nu_R(f_0)=\\frac{\\Omega}{2\\pi}.",[54,49892,49894,49995],{"className":49893,"ariaHidden":2113},[2112],[54,49895,49897,49900,49940,49943,49983,49986,49989,49992],{"className":49896},[2117],[54,49898],{"className":49899,"style":2325},[2121],[54,49901,49903,49906],{"className":49902},[2126],[54,49904,42532],{"className":49905,"style":42553},[2126,2167],[54,49907,49909],{"className":49908},[2135],[54,49910,49912,49932],{"className":49911},[2139,2756],[54,49913,49915,49929],{"className":49914},[2143],[54,49916,49918],{"className":49917,"style":2763},[2147],[54,49919,49920,49923],{"style":42568},[54,49921],{"className":49922,"style":2156},[2155],[54,49924,49926],{"className":49925},[2160,2161,2162,2163],[54,49927,19926],{"className":49928,"style":19951},[2126,2167,2163],[54,49930,2781],{"className":49931},[2780],[54,49933,49935],{"className":49934},[2143],[54,49936,49938],{"className":49937,"style":2788},[2147],[54,49939],{},[54,49941,129],{"className":49942},[2497],[54,49944,49946,49949],{"className":49945},[2126],[54,49947,21834],{"className":49948,"style":26895},[2126,2167],[54,49950,49952],{"className":49951},[2135],[54,49953,49955,49975],{"className":49954},[2139,2756],[54,49956,49958,49972],{"className":49957},[2143],[54,49959,49961],{"className":49960,"style":13837},[2147],[54,49962,49963,49966],{"style":41824},[54,49964],{"className":49965,"style":2156},[2155],[54,49967,49969],{"className":49968},[2160,2161,2162,2163],[54,49970,150],{"className":49971},[2126,2163],[54,49973,2781],{"className":49974},[2780],[54,49976,49978],{"className":49977},[2143],[54,49979,49981],{"className":49980,"style":2788},[2147],[54,49982],{},[54,49984,3205],{"className":49985},[2336],[54,49987],{"className":49988,"style":2475},[2270],[54,49990,122],{"className":49991},[2479],[54,49993],{"className":49994,"style":2475},[2270],[54,49996,49998,50001,50066],{"className":49997},[2117],[54,49999],{"className":50000,"style":47990},[2121],[54,50002,50004,50007,50063],{"className":50003},[2126],[54,50005],{"className":50006},[2497,9349],[54,50008,50010],{"className":50009},[9047],[54,50011,50013,50055],{"className":50012},[2139,2756],[54,50014,50016,50052],{"className":50015},[2143],[54,50017,50019,50033,50041],{"className":50018,"style":46597},[2147],[54,50020,50021,50024],{"style":11273},[54,50022],{"className":50023,"style":3866},[2155],[54,50025,50027,50030],{"className":50026},[2126],[54,50028,133],{"className":50029},[2126],[54,50031,8119],{"className":50032,"style":2332},[2126,2167],[54,50034,50035,50038],{"style":9434},[54,50036],{"className":50037,"style":3866},[2155],[54,50039],{"className":50040,"style":9442},[9441],[54,50042,50043,50046],{"style":9445},[54,50044],{"className":50045,"style":3866},[2155],[54,50047,50049],{"className":50048},[2126],[54,50050,42219],{"className":50051},[2126],[54,50053,2781],{"className":50054},[2780],[54,50056,50058],{"className":50057},[2143],[54,50059,50061],{"className":50060,"style":11311},[2147],[54,50062],{},[54,50064],{"className":50065},[2336,9349],[54,50067,29],{"className":50068},[2126],[13,50070,50071,50072,50141,50142,50244],{},"As the system moves away from resonance, ",[54,50073,50075,50092],{"className":50074},[2079],[54,50076,50078],{"className":50077},[2083],[2085,50079,50080],{"xmlns":2087},[2089,50081,50082,50090],{},[2092,50083,50084],{},[2724,50085,50086,50088],{},[2098,50087,42532],{},[2098,50089,19926],{},[2105,50091,42537],{"encoding":2107},[54,50093,50095],{"className":50094,"ariaHidden":2113},[2112],[54,50096,50098,50101],{"className":50097},[2117],[54,50099],{"className":50100,"style":24753},[2121],[54,50102,50104,50107],{"className":50103},[2126],[54,50105,42532],{"className":50106,"style":42553},[2126,2167],[54,50108,50110],{"className":50109},[2135],[54,50111,50113,50133],{"className":50112},[2139,2756],[54,50114,50116,50130],{"className":50115},[2143],[54,50117,50119],{"className":50118,"style":2763},[2147],[54,50120,50121,50124],{"style":42568},[54,50122],{"className":50123,"style":2156},[2155],[54,50125,50127],{"className":50126},[2160,2161,2162,2163],[54,50128,19926],{"className":50129,"style":19951},[2126,2167,2163],[54,50131,2781],{"className":50132},[2780],[54,50134,50136],{"className":50135},[2143],[54,50137,50139],{"className":50138,"style":2788},[2147],[54,50140],{}," increases symmetrically with ",[54,50143,50145,50171],{"className":50144},[2079],[54,50146,50148],{"className":50147},[2083],[2085,50149,50150],{"xmlns":2087},[2089,50151,50152,50168],{},[2092,50153,50154,50156,50158,50160,50166],{},[2245,50155,2305],{"stretchy":2311},[2098,50157,21834],{},[2245,50159,2247],{},[2724,50161,50162,50164],{},[2098,50163,21834],{},[2249,50165,150],{},[2245,50167,2305],{"stretchy":2311},[2105,50169,50170],{"encoding":2107},"\\lvert f-f_0\\rvert",[54,50172,50174,50195],{"className":50173,"ariaHidden":2113},[2112],[54,50175,50177,50180,50183,50186,50189,50192],{"className":50176},[2117],[54,50178],{"className":50179,"style":2325},[2121],[54,50181,2305],{"className":50182},[2497],[54,50184,21834],{"className":50185,"style":26895},[2126,2167],[54,50187],{"className":50188,"style":2271},[2270],[54,50190,2247],{"className":50191},[2275],[54,50193],{"className":50194,"style":2271},[2270],[54,50196,50198,50201,50241],{"className":50197},[2117],[54,50199],{"className":50200,"style":2325},[2121],[54,50202,50204,50207],{"className":50203},[2126],[54,50205,21834],{"className":50206,"style":26895},[2126,2167],[54,50208,50210],{"className":50209},[2135],[54,50211,50213,50233],{"className":50212},[2139,2756],[54,50214,50216,50230],{"className":50215},[2143],[54,50217,50219],{"className":50218,"style":13837},[2147],[54,50220,50221,50224],{"style":41824},[54,50222],{"className":50223,"style":2156},[2155],[54,50225,50227],{"className":50226},[2160,2161,2162,2163],[54,50228,150],{"className":50229},[2126,2163],[54,50231,2781],{"className":50232},[2780],[54,50234,50236],{"className":50235},[2143],[54,50237,50239],{"className":50238,"style":2788},[2147],[54,50240],{},[54,50242,2305],{"className":50243},[2336],". The oscillations accelerate along the pulse-duration axis, and the maximum achievable excited-state probability decreases. This tradeoff is the defining visual signature of detuned Rabi dynamics.",[31,50246,50248],{"id":50247},"quantum-information-interpretation","Quantum Information Interpretation",[13,50250,50251,50252,50322],{},"In quantum information language, a calibrated resonant Rabi pulse implements a single-qubit rotation. At exact resonance, the effective Hamiltonian is proportional to ",[54,50253,50255,50273],{"className":50254},[2079],[54,50256,50258],{"className":50257},[2083],[2085,50259,50260],{"xmlns":2087},[2089,50261,50262,50270],{},[2092,50263,50264],{},[2724,50265,50266,50268],{},[2098,50267,44264],{},[2098,50269,8695],{},[2105,50271,50272],{"encoding":2107},"\\sigma_x",[54,50274,50276],{"className":50275,"ariaHidden":2113},[2112],[54,50277,50279,50282],{"className":50278},[2117],[54,50280],{"className":50281,"style":24753},[2121],[54,50283,50285,50288],{"className":50284},[2126],[54,50286,44264],{"className":50287,"style":2332},[2126,2167],[54,50289,50291],{"className":50290},[2135],[54,50292,50294,50314],{"className":50293},[2139,2756],[54,50295,50297,50311],{"className":50296},[2143],[54,50298,50300],{"className":50299,"style":4063},[2147],[54,50301,50302,50305],{"style":4128},[54,50303],{"className":50304,"style":2156},[2155],[54,50306,50308],{"className":50307},[2160,2161,2162,2163],[54,50309,8695],{"className":50310},[2126,2167,2163],[54,50312,2781],{"className":50313},[2780],[54,50315,50317],{"className":50316},[2143],[54,50318,50320],{"className":50319,"style":2788},[2147],[54,50321],{},", so the microwave pulse implements",[54,50324,50326],{"className":50325},[2340],[54,50327,50329,50386],{"className":50328},[2079],[54,50330,50332],{"className":50331},[2083],[2085,50333,50334],{"xmlns":2087,"display":2349},[2089,50335,50336,50383],{},[2092,50337,50338,50344,50346,50349,50351,50353,50355,50357,50381],{},[2724,50339,50340,50342],{},[2098,50341,19926],{},[2098,50343,8695],{},[2245,50345,129],{"stretchy":2311},[2098,50347,50348],{},"θ",[2245,50350,3205],{"stretchy":2311},[2245,50352,122],{},[2098,50354,23625],{},[2245,50356,36247],{},[2092,50358,50359,50361,50363,50379],{},[2245,50360,129],{"fence":2113},[2245,50362,2247],{},[9047,50364,50365,50377],{},[2092,50366,50367,50369,50371],{},[2098,50368,2849],{},[2098,50370,50348],{},[2724,50372,50373,50375],{},[2098,50374,44264],{},[2098,50376,8695],{},[2249,50378,133],{},[2245,50380,3205],{"fence":2113},[2245,50382,2375],{"separator":2113},[2105,50384,50385],{"encoding":2107},"R_x(\\theta)=\\exp\\left(-\\frac{i\\theta\\sigma_x}{2}\\right),",[54,50387,50389,50453],{"className":50388,"ariaHidden":2113},[2112],[54,50390,50392,50395,50435,50438,50441,50444,50447,50450],{"className":50391},[2117],[54,50393],{"className":50394,"style":2325},[2121],[54,50396,50398,50401],{"className":50397},[2126],[54,50399,19926],{"className":50400,"style":19951},[2126,2167],[54,50402,50404],{"className":50403},[2135],[54,50405,50407,50427],{"className":50406},[2139,2756],[54,50408,50410,50424],{"className":50409},[2143],[54,50411,50413],{"className":50412,"style":4063},[2147],[54,50414,50415,50418],{"style":19966},[54,50416],{"className":50417,"style":2156},[2155],[54,50419,50421],{"className":50420},[2160,2161,2162,2163],[54,50422,8695],{"className":50423},[2126,2167,2163],[54,50425,2781],{"className":50426},[2780],[54,50428,50430],{"className":50429},[2143],[54,50431,50433],{"className":50432,"style":2788},[2147],[54,50434],{},[54,50436,129],{"className":50437},[2497],[54,50439,50348],{"className":50440,"style":2168},[2126,2167],[54,50442,3205],{"className":50443},[2336],[54,50445],{"className":50446,"style":2475},[2270],[54,50448,122],{"className":50449},[2479],[54,50451],{"className":50452,"style":2475},[2270],[54,50454,50456,50459,50462,50465,50588,50591],{"className":50455},[2117],[54,50457],{"className":50458,"style":37225},[2121],[54,50460,23625],{"className":50461},[7126],[54,50463],{"className":50464,"style":2521},[2270],[54,50466,50468,50474,50477,50582],{"className":50467},[2493],[54,50469,50471],{"className":50470,"style":2499},[2497,2498],[54,50472,129],{"className":50473},[2503,2162],[54,50475,2247],{"className":50476},[2126],[54,50478,50480,50483,50579],{"className":50479},[2126],[54,50481],{"className":50482},[2497,9349],[54,50484,50486],{"className":50485},[9047],[54,50487,50489,50571],{"className":50488},[2139,2756],[54,50490,50492,50568],{"className":50491},[2143],[54,50493,50495,50506,50514],{"className":50494,"style":37315},[2147],[54,50496,50497,50500],{"style":11273},[54,50498],{"className":50499,"style":3866},[2155],[54,50501,50503],{"className":50502},[2126],[54,50504,133],{"className":50505},[2126],[54,50507,50508,50511],{"style":9434},[54,50509],{"className":50510,"style":3866},[2155],[54,50512],{"className":50513,"style":9442},[9441],[54,50515,50516,50519],{"style":9445},[54,50517],{"className":50518,"style":3866},[2155],[54,50520,50522,50525,50528],{"className":50521},[2126],[54,50523,2849],{"className":50524},[2126,2167],[54,50526,50348],{"className":50527,"style":2168},[2126,2167],[54,50529,50531,50534],{"className":50530},[2126],[54,50532,44264],{"className":50533,"style":2332},[2126,2167],[54,50535,50537],{"className":50536},[2135],[54,50538,50540,50560],{"className":50539},[2139,2756],[54,50541,50543,50557],{"className":50542},[2143],[54,50544,50546],{"className":50545,"style":4063},[2147],[54,50547,50548,50551],{"style":4128},[54,50549],{"className":50550,"style":2156},[2155],[54,50552,50554],{"className":50553},[2160,2161,2162,2163],[54,50555,8695],{"className":50556},[2126,2167,2163],[54,50558,2781],{"className":50559},[2780],[54,50561,50563],{"className":50562},[2143],[54,50564,50566],{"className":50565,"style":2788},[2147],[54,50567],{},[54,50569,2781],{"className":50570},[2780],[54,50572,50574],{"className":50573},[2143],[54,50575,50577],{"className":50576,"style":11311},[2147],[54,50578],{},[54,50580],{"className":50581},[2336,9349],[54,50583,50585],{"className":50584,"style":2499},[2336,2498],[54,50586,3205],{"className":50587},[2503,2162],[54,50589],{"className":50590,"style":2521},[2270],[54,50592,2375],{"className":50593},[2517],[13,50595,46696],{},[54,50597,50599],{"className":50598},[2340],[54,50600,50602,50624],{"className":50601},[2079],[54,50603,50605],{"className":50604},[2083],[2085,50606,50607],{"xmlns":2087,"display":2349},[2089,50608,50609,50621],{},[2092,50610,50611,50613,50615,50617,50619],{},[2098,50612,50348],{},[2245,50614,122],{},[2098,50616,42219],{"mathvariant":2304},[2098,50618,41261],{},[2098,50620,29],{"mathvariant":2304},[2105,50622,50623],{"encoding":2107},"\\theta=\\Omega\\tau.",[54,50625,50627,50645],{"className":50626,"ariaHidden":2113},[2112],[54,50628,50630,50633,50636,50639,50642],{"className":50629},[2117],[54,50631],{"className":50632,"style":4990},[2121],[54,50634,50348],{"className":50635,"style":2168},[2126,2167],[54,50637],{"className":50638,"style":2475},[2270],[54,50640,122],{"className":50641},[2479],[54,50643],{"className":50644,"style":2475},[2270],[54,50646,50648,50651,50654,50657],{"className":50647},[2117],[54,50649],{"className":50650,"style":2122},[2121],[54,50652,42219],{"className":50653},[2126],[54,50655,41261],{"className":50656,"style":41333},[2126,2167],[54,50658,29],{"className":50659},[2126],[13,50661,50662,50663,50691],{},"A calibrated resonant ",[54,50664,50666,50679],{"className":50665},[2079],[54,50667,50669],{"className":50668},[2083],[2085,50670,50671],{"xmlns":2087},[2089,50672,50673,50677],{},[2092,50674,50675],{},[2098,50676,8119],{},[2105,50678,47641],{"encoding":2107},[54,50680,50682],{"className":50681,"ariaHidden":2113},[2112],[54,50683,50685,50688],{"className":50684},[2117],[54,50686],{"className":50687,"style":2195},[2121],[54,50689,8119],{"className":50690,"style":2332},[2126,2167],"-pulse enacts",[54,50693,50695],{"className":50694},[2340],[54,50696,50698,50736],{"className":50697},[2079],[54,50699,50701],{"className":50700},[2083],[2085,50702,50703],{"xmlns":2087,"display":2349},[2089,50704,50705,50733],{},[2092,50706,50707,50713,50715,50717,50719,50721,50723,50725,50731],{},[2724,50708,50709,50711],{},[2098,50710,19926],{},[2098,50712,8695],{},[2245,50714,129],{"stretchy":2311},[2098,50716,8119],{},[2245,50718,3205],{"stretchy":2311},[2245,50720,122],{},[2245,50722,2247],{},[2098,50724,2849],{},[2724,50726,50727,50729],{},[2098,50728,44264],{},[2098,50730,8695],{},[2245,50732,2375],{"separator":2113},[2105,50734,50735],{"encoding":2107},"R_x(\\pi)=-i\\sigma_x,",[54,50737,50739,50803],{"className":50738,"ariaHidden":2113},[2112],[54,50740,50742,50745,50785,50788,50791,50794,50797,50800],{"className":50741},[2117],[54,50743],{"className":50744,"style":2325},[2121],[54,50746,50748,50751],{"className":50747},[2126],[54,50749,19926],{"className":50750,"style":19951},[2126,2167],[54,50752,50754],{"className":50753},[2135],[54,50755,50757,50777],{"className":50756},[2139,2756],[54,50758,50760,50774],{"className":50759},[2143],[54,50761,50763],{"className":50762,"style":4063},[2147],[54,50764,50765,50768],{"style":19966},[54,50766],{"className":50767,"style":2156},[2155],[54,50769,50771],{"className":50770},[2160,2161,2162,2163],[54,50772,8695],{"className":50773},[2126,2167,2163],[54,50775,2781],{"className":50776},[2780],[54,50778,50780],{"className":50779},[2143],[54,50781,50783],{"className":50782,"style":2788},[2147],[54,50784],{},[54,50786,129],{"className":50787},[2497],[54,50789,8119],{"className":50790,"style":2332},[2126,2167],[54,50792,3205],{"className":50793},[2336],[54,50795],{"className":50796,"style":2475},[2270],[54,50798,122],{"className":50799},[2479],[54,50801],{"className":50802,"style":2475},[2270],[54,50804,50806,50809,50812,50815,50855],{"className":50805},[2117],[54,50807],{"className":50808,"style":27543},[2121],[54,50810,2247],{"className":50811},[2126],[54,50813,2849],{"className":50814},[2126,2167],[54,50816,50818,50821],{"className":50817},[2126],[54,50819,44264],{"className":50820,"style":2332},[2126,2167],[54,50822,50824],{"className":50823},[2135],[54,50825,50827,50847],{"className":50826},[2139,2756],[54,50828,50830,50844],{"className":50829},[2143],[54,50831,50833],{"className":50832,"style":4063},[2147],[54,50834,50835,50838],{"style":4128},[54,50836],{"className":50837,"style":2156},[2155],[54,50839,50841],{"className":50840},[2160,2161,2162,2163],[54,50842,8695],{"className":50843},[2126,2167,2163],[54,50845,2781],{"className":50846},[2780],[54,50848,50850],{"className":50849},[2143],[54,50851,50853],{"className":50852,"style":2788},[2147],[54,50854],{},[54,50856,2375],{"className":50857},[2517],[13,50859,50860,50861,50889,50890,50926],{},"which corresponds to a quantum ",[54,50862,50864,50877],{"className":50863},[2079],[54,50865,50867],{"className":50866},[2083],[2085,50868,50869],{"xmlns":2087},[2089,50870,50871,50875],{},[2092,50872,50873],{},[2098,50874,7287],{},[2105,50876,7287],{"encoding":2107},[54,50878,50880],{"className":50879,"ariaHidden":2113},[2112],[54,50881,50883,50886],{"className":50882},[2117],[54,50884],{"className":50885,"style":2122},[2121],[54,50887,7287],{"className":50888,"style":7432},[2126,2167]," gate up to a global phase. A resonant ",[54,50891,50893,50911],{"className":50892},[2079],[54,50894,50896],{"className":50895},[2083],[2085,50897,50898],{"xmlns":2087},[2089,50899,50900,50908],{},[2092,50901,50902,50904,50906],{},[2098,50903,8119],{},[2098,50905,194],{"mathvariant":2304},[2249,50907,133],{},[2105,50909,50910],{"encoding":2107},"\\pi\u002F2",[54,50912,50914],{"className":50913,"ariaHidden":2113},[2112],[54,50915,50917,50920,50923],{"className":50916},[2117],[54,50918],{"className":50919,"style":2325},[2121],[54,50921,8119],{"className":50922,"style":2332},[2126,2167],[54,50924,25677],{"className":50925},[2126],"-pulse prepares a balanced coherent superposition,",[54,50928,50930],{"className":50929},[2340],[54,50931,50933,50979],{"className":50932},[2079],[54,50934,50936],{"className":50935},[2083],[2085,50937,50938],{"xmlns":2087,"display":2349},[2089,50939,50940,50976],{},[2092,50941,50942,50944,50946,50948,50950,50974],{},[2245,50943,2305],{"stretchy":2311},[2098,50945,37470],{},[2245,50947,2312],{"stretchy":2311},[2245,50949,14490],{},[9047,50951,50952,50970],{},[2092,50953,50954,50956,50958,50960,50962,50964,50966,50968],{},[2245,50955,2305],{"stretchy":2311},[2098,50957,37470],{},[2245,50959,2312],{"stretchy":2311},[2245,50961,2247],{},[2098,50963,2849],{},[2245,50965,2305],{"stretchy":2311},[2098,50967,2842],{},[2245,50969,2312],{"stretchy":2311},[9052,50971,50972],{},[2249,50973,133],{},[2245,50975,2375],{"separator":2113},[2105,50977,50978],{"encoding":2107},"\\lvert g\\rangle\\rightarrow\\frac{\\lvert g\\rangle-i\\lvert e\\rangle}{\\sqrt{2}},",[54,50980,50982,51006],{"className":50981,"ariaHidden":2113},[2112],[54,50983,50985,50988,50991,50994,50997,51000,51003],{"className":50984},[2117],[54,50986],{"className":50987,"style":2325},[2121],[54,50989,2305],{"className":50990},[2497],[54,50992,37470],{"className":50993,"style":2332},[2126,2167],[54,50995,2312],{"className":50996},[2336],[54,50998],{"className":50999,"style":2475},[2270],[54,51001,14490],{"className":51002},[2479],[54,51004],{"className":51005,"style":2475},[2270],[54,51007,51009,51013,51146],{"className":51008},[2117],[54,51010],{"className":51011,"style":51012},[2121],"height:2.357em;vertical-align:-0.93em;",[54,51014,51016,51019,51143],{"className":51015},[2126],[54,51017],{"className":51018},[2497,9349],[54,51020,51022],{"className":51021},[9047],[54,51023,51025,51135],{"className":51024},[2139,2756],[54,51026,51028,51132],{"className":51027},[2143],[54,51029,51031,51086,51094],{"className":51030,"style":28921},[2147],[54,51032,51033,51036],{"style":9365},[54,51034],{"className":51035,"style":3866},[2155],[54,51037,51039],{"className":51038},[2126],[54,51040,51042],{"className":51041},[2126,9375],[54,51043,51045,51078],{"className":51044},[2139,2756],[54,51046,51048,51075],{"className":51047},[2143],[54,51049,51051,51063],{"className":51050,"style":9385},[2147],[54,51052,51054,51057],{"className":51053,"style":3862},[9389],[54,51055],{"className":51056,"style":3866},[2155],[54,51058,51060],{"className":51059,"style":9396},[2126],[54,51061,133],{"className":51062},[2126],[54,51064,51065,51068],{"style":9402},[54,51066],{"className":51067,"style":3866},[2155],[54,51069,51071],{"className":51070,"style":9410},[9409],[5837,51072,51073],{"xmlns":5839,"width":9413,"height":9414,"viewBox":9415,"preserveAspectRatio":9416},[5846,51074],{"d":9419},[54,51076,2781],{"className":51077},[2780],[54,51079,51081],{"className":51080},[2143],[54,51082,51084],{"className":51083,"style":9429},[2147],[54,51085],{},[54,51087,51088,51091],{"style":9434},[54,51089],{"className":51090,"style":3866},[2155],[54,51092],{"className":51093,"style":9442},[9441],[54,51095,51096,51099],{"style":9445},[54,51097],{"className":51098,"style":3866},[2155],[54,51100,51102,51105,51108,51111,51114,51117,51120,51123,51126,51129],{"className":51101},[2126],[54,51103,2305],{"className":51104},[2497],[54,51106,37470],{"className":51107,"style":2332},[2126,2167],[54,51109,2312],{"className":51110},[2336],[54,51112],{"className":51113,"style":2271},[2270],[54,51115,2247],{"className":51116},[2275],[54,51118],{"className":51119,"style":2271},[2270],[54,51121,2849],{"className":51122},[2126,2167],[54,51124,2305],{"className":51125},[2497],[54,51127,2842],{"className":51128},[2126,2167],[54,51130,2312],{"className":51131},[2336],[54,51133,2781],{"className":51134},[2780],[54,51136,51138],{"className":51137},[2143],[54,51139,51141],{"className":51140,"style":9464},[2147],[54,51142],{},[54,51144],{"className":51145},[2336,9349],[54,51147,2375],{"className":51148},[2517],[13,51150,51151],{},"with a phase convention set by the microwave drive phase.",[13,51153,51154],{},"Detuning changes the rotation axis so it is no longer purely transverse. The unitary can be written as",[54,51156,51158],{"className":51157},[2340],[54,51159,51161,51231],{"className":51160},[2079],[54,51162,51164],{"className":51163},[2083],[2085,51165,51166],{"xmlns":2087,"display":2349},[2089,51167,51168,51228],{},[2092,51169,51170,51181,51183,51185,51187,51189,51191,51193,51226],{},[2724,51171,51172,51174],{},[2098,51173,19926],{},[3827,51175,51176,51178],{"accent":2113},[2098,51177,3976],{},[2245,51179,51180],{},"^",[2245,51182,129],{"stretchy":2311},[2098,51184,50348],{},[2245,51186,3205],{"stretchy":2311},[2245,51188,122],{},[2098,51190,23625],{},[2245,51192,36247],{},[2092,51194,51195,51197,51199,51209,51215,51218,51224],{},[2245,51196,129],{"fence":2113},[2245,51198,2247],{},[9047,51200,51201,51207],{},[2092,51202,51203,51205],{},[2098,51204,2849],{},[2098,51206,50348],{},[2249,51208,133],{},[3827,51210,51211,51213],{"accent":2113},[2098,51212,3976],{},[2245,51214,51180],{},[2245,51216,51217],{},"⋅",[3827,51219,51220,51222],{"accent":2113},[2098,51221,44264],{},[2245,51223,5733],{},[2245,51225,3205],{"fence":2113},[2245,51227,2375],{"separator":2113},[2105,51229,51230],{"encoding":2107},"R_{\\hat{n}}(\\theta)=\\exp\\left(-\\frac{i\\theta}{2}\\hat{n}\\cdot\\vec{\\sigma}\\right),",[54,51232,51234,51331],{"className":51233,"ariaHidden":2113},[2112],[54,51235,51237,51240,51313,51316,51319,51322,51325,51328],{"className":51236},[2117],[54,51238],{"className":51239,"style":2325},[2121],[54,51241,51243,51246],{"className":51242},[2126],[54,51244,19926],{"className":51245,"style":19951},[2126,2167],[54,51247,51249],{"className":51248},[2135],[54,51250,51252,51305],{"className":51251},[2139,2756],[54,51253,51255,51302],{"className":51254},[2143],[54,51256,51258],{"className":51257,"style":10799},[2147],[54,51259,51260,51263],{"style":19966},[54,51261],{"className":51262,"style":2156},[2155],[54,51264,51266],{"className":51265},[2160,2161,2162,2163],[54,51267,51269],{"className":51268},[2126,2163],[54,51270,51272],{"className":51271},[2126,3850,2163],[54,51273,51275],{"className":51274},[2139],[54,51276,51278],{"className":51277},[2143],[54,51279,51281,51290],{"className":51280,"style":4990},[2147],[54,51282,51284,51287],{"style":51283},"top:-2.7em;",[54,51285],{"className":51286,"style":2156},[2155],[54,51288,3976],{"className":51289},[2126,2167,2163],[54,51291,51292,51295],{"style":51283},[54,51293],{"className":51294,"style":2156},[2155],[54,51296,51299],{"className":51297,"style":51298},[3879],"left:-0.25em;",[54,51300,51180],{"className":51301},[2126,2163],[54,51303,2781],{"className":51304},[2780],[54,51306,51308],{"className":51307},[2143],[54,51309,51311],{"className":51310,"style":2788},[2147],[54,51312],{},[54,51314,129],{"className":51315},[2497],[54,51317,50348],{"className":51318,"style":2168},[2126,2167],[54,51320,3205],{"className":51321},[2336],[54,51323],{"className":51324,"style":2475},[2270],[54,51326,122],{"className":51327},[2479],[54,51329],{"className":51330,"style":2475},[2270],[54,51332,51334,51337,51340,51343,51502,51505],{"className":51333},[2117],[54,51335],{"className":51336,"style":37225},[2121],[54,51338,23625],{"className":51339},[7126],[54,51341],{"className":51342,"style":2521},[2270],[54,51344,51346,51352,51355,51420,51451,51454,51457,51460,51496],{"className":51345},[2493],[54,51347,51349],{"className":51348,"style":2499},[2497,2498],[54,51350,129],{"className":51351},[2503,2162],[54,51353,2247],{"className":51354},[2126],[54,51356,51358,51361,51417],{"className":51357},[2126],[54,51359],{"className":51360},[2497,9349],[54,51362,51364],{"className":51363},[9047],[54,51365,51367,51409],{"className":51366},[2139,2756],[54,51368,51370,51406],{"className":51369},[2143],[54,51371,51373,51384,51392],{"className":51372,"style":37315},[2147],[54,51374,51375,51378],{"style":11273},[54,51376],{"className":51377,"style":3866},[2155],[54,51379,51381],{"className":51380},[2126],[54,51382,133],{"className":51383},[2126],[54,51385,51386,51389],{"style":9434},[54,51387],{"className":51388,"style":3866},[2155],[54,51390],{"className":51391,"style":9442},[9441],[54,51393,51394,51397],{"style":9445},[54,51395],{"className":51396,"style":3866},[2155],[54,51398,51400,51403],{"className":51399},[2126],[54,51401,2849],{"className":51402},[2126,2167],[54,51404,50348],{"className":51405,"style":2168},[2126,2167],[54,51407,2781],{"className":51408},[2780],[54,51410,51412],{"className":51411},[2143],[54,51413,51415],{"className":51414,"style":11311},[2147],[54,51416],{},[54,51418],{"className":51419},[2336,9349],[54,51421,51423],{"className":51422},[2126,3850],[54,51424,51426],{"className":51425},[2139],[54,51427,51429],{"className":51428},[2143],[54,51430,51432,51440],{"className":51431,"style":4990},[2147],[54,51433,51434,51437],{"style":3862},[54,51435],{"className":51436,"style":3866},[2155],[54,51438,3976],{"className":51439},[2126,2167],[54,51441,51442,51445],{"style":3862},[54,51443],{"className":51444,"style":3866},[2155],[54,51446,51448],{"className":51447,"style":51298},[3879],[54,51449,51180],{"className":51450},[2126],[54,51452],{"className":51453,"style":2271},[2270],[54,51455,51217],{"className":51456},[2275],[54,51458],{"className":51459,"style":2271},[2270],[54,51461,51463],{"className":51462},[2126,3850],[54,51464,51466],{"className":51465},[2139],[54,51467,51469],{"className":51468},[2143],[54,51470,51472,51480],{"className":51471,"style":5813},[2147],[54,51473,51474,51477],{"style":3862},[54,51475],{"className":51476,"style":3866},[2155],[54,51478,44264],{"className":51479,"style":2332},[2126,2167],[54,51481,51482,51485],{"style":3862},[54,51483],{"className":51484,"style":3866},[2155],[54,51486,51489],{"className":51487,"style":51488},[3879],"left:-0.2355em;",[54,51490,51492],{"className":51491,"style":5835},[5834],[5837,51493,51494],{"xmlns":5839,"width":5840,"height":5841,"style":5842,"viewBox":5843,"preserveAspectRatio":5844},[5846,51495],{"d":5848},[54,51497,51499],{"className":51498,"style":2499},[2336,2498],[54,51500,3205],{"className":51501},[2503,2162],[54,51503],{"className":51504,"style":2521},[2270],[54,51506,2375],{"className":51507},[2517],[13,51509,46696],{},[54,51511,51513],{"className":51512},[2340],[54,51514,51516,51578],{"className":51515},[2079],[54,51517,51519],{"className":51518},[2083],[2085,51520,51521],{"xmlns":2087,"display":2349},[2089,51522,51523,51575],{},[2092,51524,51525,51531,51533,51543,51545,51547,51549,51551,51553,51555,51557,51559,51561,51563,51565,51571,51573],{},[3827,51526,51527,51529],{"accent":2113},[2098,51528,3976],{},[2245,51530,51180],{},[2245,51532,122],{},[9047,51534,51535,51537],{},[2249,51536,174],{},[2724,51538,51539,51541],{},[2098,51540,42219],{"mathvariant":2304},[2098,51542,19926],{},[2245,51544,129],{"stretchy":2311},[2098,51546,42219],{"mathvariant":2304},[2245,51548,2375],{"separator":2113},[2249,51550,150],{},[2245,51552,2375],{"separator":2113},[2098,51554,42302],{"mathvariant":2304},[2245,51556,3205],{"stretchy":2311},[2245,51558,2375],{"separator":2113},[2270,51560],{"width":17567},[2098,51562,50348],{},[2245,51564,122],{},[2724,51566,51567,51569],{},[2098,51568,42219],{"mathvariant":2304},[2098,51570,19926],{},[2098,51572,41261],{},[2098,51574,29],{"mathvariant":2304},[2105,51576,51577],{"encoding":2107},"\\hat{n}=\\frac{1}{\\Omega_R}(\\Omega,0,\\Delta), \\qquad \\theta=\\Omega_R\\tau.",[54,51579,51581,51627,51782],{"className":51580,"ariaHidden":2113},[2112],[54,51582,51584,51587,51618,51621,51624],{"className":51583},[2117],[54,51585],{"className":51586,"style":4990},[2121],[54,51588,51590],{"className":51589},[2126,3850],[54,51591,51593],{"className":51592},[2139],[54,51594,51596],{"className":51595},[2143],[54,51597,51599,51607],{"className":51598,"style":4990},[2147],[54,51600,51601,51604],{"style":3862},[54,51602],{"className":51603,"style":3866},[2155],[54,51605,3976],{"className":51606},[2126,2167],[54,51608,51609,51612],{"style":3862},[54,51610],{"className":51611,"style":3866},[2155],[54,51613,51615],{"className":51614,"style":51298},[3879],[54,51616,51180],{"className":51617},[2126],[54,51619],{"className":51620,"style":2475},[2270],[54,51622,122],{"className":51623},[2479],[54,51625],{"className":51626,"style":2475},[2270],[54,51628,51630,51634,51734,51737,51740,51743,51746,51749,51752,51755,51758,51761,51764,51767,51770,51773,51776,51779],{"className":51629},[2117],[54,51631],{"className":51632,"style":51633},[2121],"height:2.1574em;vertical-align:-0.836em;",[54,51635,51637,51640,51731],{"className":51636},[2126],[54,51638],{"className":51639},[2497,9349],[54,51641,51643],{"className":51642},[9047],[54,51644,51646,51722],{"className":51645},[2139,2756],[54,51647,51649,51719],{"className":51648},[2143],[54,51650,51652,51700,51708],{"className":51651,"style":9362},[2147],[54,51653,51654,51657],{"style":11273},[54,51655],{"className":51656,"style":3866},[2155],[54,51658,51660],{"className":51659},[2126],[54,51661,51663,51666],{"className":51662},[2126],[54,51664,42219],{"className":51665},[2126],[54,51667,51669],{"className":51668},[2135],[54,51670,51672,51692],{"className":51671},[2139,2756],[54,51673,51675,51689],{"className":51674},[2143],[54,51676,51678],{"className":51677,"style":2763},[2147],[54,51679,51680,51683],{"style":4954},[54,51681],{"className":51682,"style":2156},[2155],[54,51684,51686],{"className":51685},[2160,2161,2162,2163],[54,51687,19926],{"className":51688,"style":19951},[2126,2167,2163],[54,51690,2781],{"className":51691},[2780],[54,51693,51695],{"className":51694},[2143],[54,51696,51698],{"className":51697,"style":2788},[2147],[54,51699],{},[54,51701,51702,51705],{"style":9434},[54,51703],{"className":51704,"style":3866},[2155],[54,51706],{"className":51707,"style":9442},[9441],[54,51709,51710,51713],{"style":9445},[54,51711],{"className":51712,"style":3866},[2155],[54,51714,51716],{"className":51715},[2126],[54,51717,174],{"className":51718},[2126],[54,51720,2781],{"className":51721},[2780],[54,51723,51725],{"className":51724},[2143],[54,51726,51729],{"className":51727,"style":51728},[2147],"height:0.836em;",[54,51730],{},[54,51732],{"className":51733},[2336,9349],[54,51735,129],{"className":51736},[2497],[54,51738,42219],{"className":51739},[2126],[54,51741,2375],{"className":51742},[2517],[54,51744],{"className":51745,"style":2521},[2270],[54,51747,150],{"className":51748},[2126],[54,51750,2375],{"className":51751},[2517],[54,51753],{"className":51754,"style":2521},[2270],[54,51756,42302],{"className":51757},[2126],[54,51759,3205],{"className":51760},[2336],[54,51762,2375],{"className":51763},[2517],[54,51765],{"className":51766,"style":18185},[2270],[54,51768],{"className":51769,"style":2521},[2270],[54,51771,50348],{"className":51772,"style":2168},[2126,2167],[54,51774],{"className":51775,"style":2475},[2270],[54,51777,122],{"className":51778},[2479],[54,51780],{"className":51781,"style":2475},[2270],[54,51783,51785,51788,51828,51831],{"className":51784},[2117],[54,51786],{"className":51787,"style":2743},[2121],[54,51789,51791,51794],{"className":51790},[2126],[54,51792,42219],{"className":51793},[2126],[54,51795,51797],{"className":51796},[2135],[54,51798,51800,51820],{"className":51799},[2139,2756],[54,51801,51803,51817],{"className":51802},[2143],[54,51804,51806],{"className":51805,"style":2763},[2147],[54,51807,51808,51811],{"style":4954},[54,51809],{"className":51810,"style":2156},[2155],[54,51812,51814],{"className":51813},[2160,2161,2162,2163],[54,51815,19926],{"className":51816,"style":19951},[2126,2167,2163],[54,51818,2781],{"className":51819},[2780],[54,51821,51823],{"className":51822},[2143],[54,51824,51826],{"className":51825,"style":2788},[2147],[54,51827],{},[54,51829,41261],{"className":51830,"style":41333},[2126,2167],[54,51832,29],{"className":51833},[2126],[13,51835,51836],{},"This representation shows that detuning increases the rotation rate and tilts the control axis. Under this idealized model, the qubit still evolves coherently, although the trajectory generally misses the pure excited-state pole.",[31,51838,51840],{"id":51839},"bloch-sphere-interpretation","Bloch-Sphere Interpretation",[13,51842,51843],{},"A Bloch sphere maps any pure qubit state to a unit vector in three-dimensional space. Ground and excited states occupy opposite poles. Equal-amplitude coherent superpositions lie on the equator.",[13,51845,51846,51847,51885,51886,51914,51915,51950],{},"A resonant microwave drive rotates the Bloch vector around a transverse axis. Starting from ",[54,51848,51850,51867],{"className":51849},[2079],[54,51851,51853],{"className":51852},[2083],[2085,51854,51855],{"xmlns":2087},[2089,51856,51857,51865],{},[2092,51858,51859,51861,51863],{},[2245,51860,2305],{"stretchy":2311},[2098,51862,37470],{},[2245,51864,2312],{"stretchy":2311},[2105,51866,41452],{"encoding":2107},[54,51868,51870],{"className":51869,"ariaHidden":2113},[2112],[54,51871,51873,51876,51879,51882],{"className":51872},[2117],[54,51874],{"className":51875,"style":2325},[2121],[54,51877,2305],{"className":51878},[2497],[54,51880,37470],{"className":51881,"style":2332},[2126,2167],[54,51883,2312],{"className":51884},[2336],", a properly timed ",[54,51887,51889,51902],{"className":51888},[2079],[54,51890,51892],{"className":51891},[2083],[2085,51893,51894],{"xmlns":2087},[2089,51895,51896,51900],{},[2092,51897,51898],{},[2098,51899,8119],{},[2105,51901,47641],{"encoding":2107},[54,51903,51905],{"className":51904,"ariaHidden":2113},[2112],[54,51906,51908,51911],{"className":51907},[2117],[54,51909],{"className":51910,"style":2195},[2121],[54,51912,8119],{"className":51913,"style":2332},[2126,2167],"-pulse moves the state to the excited-state pole. A ",[54,51916,51918,51935],{"className":51917},[2079],[54,51919,51921],{"className":51920},[2083],[2085,51922,51923],{"xmlns":2087},[2089,51924,51925,51933],{},[2092,51926,51927,51929,51931],{},[2098,51928,8119],{},[2098,51930,194],{"mathvariant":2304},[2249,51932,133],{},[2105,51934,50910],{"encoding":2107},[54,51936,51938],{"className":51937,"ariaHidden":2113},[2112],[54,51939,51941,51944,51947],{"className":51940},[2117],[54,51942],{"className":51943,"style":2325},[2121],[54,51945,8119],{"className":51946,"style":2332},[2126,2167],[54,51948,25677],{"className":51949},[2126],"-pulse moves the state to the equator, where the ground-state and excited-state measurement probabilities are equal.",[13,51952,51953,51954,51982],{},"Detuning tilts the rotation axis toward the ",[54,51955,51957,51970],{"className":51956},[2079],[54,51958,51960],{"className":51959},[2083],[2085,51961,51962],{"xmlns":2087},[2089,51963,51964,51968],{},[2092,51965,51966],{},[2098,51967,215],{},[2105,51969,215],{"encoding":2107},[54,51971,51973],{"className":51972,"ariaHidden":2113},[2112],[54,51974,51976,51979],{"className":51975},[2117],[54,51977],{"className":51978,"style":2195},[2121],[54,51980,215],{"className":51981,"style":20138},[2126,2167],"-axis. The state vector then precesses around the tilted axis and generally fails to reach the excited-state pole. This geometric picture explains the amplitude coefficient",[54,51984,51986],{"className":51985},[2340],[54,51987,51989,52019],{"className":51988},[2079],[54,51990,51992],{"className":51991},[2083],[2085,51993,51994],{"xmlns":2087,"display":2349},[2089,51995,51996,52016],{},[2092,51997,51998,52014],{},[9047,51999,52000,52006],{},[2095,52001,52002,52004],{},[2098,52003,42219],{"mathvariant":2304},[2249,52005,133],{},[3331,52007,52008,52010,52012],{},[2098,52009,42219],{"mathvariant":2304},[2098,52011,19926],{},[2249,52013,133],{},[2098,52015,29],{"mathvariant":2304},[2105,52017,52018],{"encoding":2107},"\\frac{\\Omega^2}{\\Omega_R^2}.",[54,52020,52022],{"className":52021,"ariaHidden":2113},[2112],[54,52023,52025,52028,52164],{"className":52024},[2117],[54,52026],{"className":52027,"style":46390},[2121],[54,52029,52031,52034,52161],{"className":52030},[2126],[54,52032],{"className":52033},[2497,9349],[54,52035,52037],{"className":52036},[9047],[54,52038,52040,52153],{"className":52039},[2139,2756],[54,52041,52043,52150],{"className":52042},[2143],[54,52044,52046,52105,52113],{"className":52045,"style":46409},[2147],[54,52047,52048,52051],{"style":11273},[54,52049],{"className":52050,"style":3866},[2155],[54,52052,52054],{"className":52053},[2126],[54,52055,52057,52060],{"className":52056},[2126],[54,52058,42219],{"className":52059},[2126],[54,52061,52063],{"className":52062},[2135],[54,52064,52066,52097],{"className":52065},[2139,2756],[54,52067,52069,52094],{"className":52068},[2143],[54,52070,52072,52083],{"className":52071,"style":46436},[2147],[54,52073,52074,52077],{"style":46439},[54,52075],{"className":52076,"style":2156},[2155],[54,52078,52080],{"className":52079},[2160,2161,2162,2163],[54,52081,19926],{"className":52082,"style":19951},[2126,2167,2163],[54,52084,52085,52088],{"style":46451},[54,52086],{"className":52087,"style":2156},[2155],[54,52089,52091],{"className":52090},[2160,2161,2162,2163],[54,52092,133],{"className":52093},[2126,2163],[54,52095,2781],{"className":52096},[2780],[54,52098,52100],{"className":52099},[2143],[54,52101,52103],{"className":52102,"style":46470},[2147],[54,52104],{},[54,52106,52107,52110],{"style":9434},[54,52108],{"className":52109,"style":3866},[2155],[54,52111],{"className":52112,"style":9442},[9441],[54,52114,52115,52118],{"style":9445},[54,52116],{"className":52117,"style":3866},[2155],[54,52119,52121],{"className":52120},[2126],[54,52122,52124,52127],{"className":52123},[2126],[54,52125,42219],{"className":52126},[2126],[54,52128,52130],{"className":52129},[2135],[54,52131,52133],{"className":52132},[2139],[54,52134,52136],{"className":52135},[2143],[54,52137,52139],{"className":52138,"style":12780},[2147],[54,52140,52141,52144],{"style":2151},[54,52142],{"className":52143,"style":2156},[2155],[54,52145,52147],{"className":52146},[2160,2161,2162,2163],[54,52148,133],{"className":52149},[2126,2163],[54,52151,2781],{"className":52152},[2780],[54,52154,52156],{"className":52155},[2143],[54,52157,52159],{"className":52158,"style":46527},[2147],[54,52160],{},[54,52162],{"className":52163},[2336,9349],[54,52165,29],{"className":52166},[2126],[13,52168,52169,52170,52209],{},"As ",[54,52171,52173,52191],{"className":52172},[2079],[54,52174,52176],{"className":52175},[2083],[2085,52177,52178],{"xmlns":2087},[2089,52179,52180,52188],{},[2092,52181,52182,52184,52186],{},[2245,52183,2305],{"stretchy":2311},[2098,52185,42302],{"mathvariant":2304},[2245,52187,2305],{"stretchy":2311},[2105,52189,52190],{"encoding":2107},"\\lvert\\Delta\\rvert",[54,52192,52194],{"className":52193,"ariaHidden":2113},[2112],[54,52195,52197,52200,52203,52206],{"className":52196},[2117],[54,52198],{"className":52199,"style":2325},[2121],[54,52201,2305],{"className":52202},[2497],[54,52204,42302],{"className":52205},[2126],[54,52207,2305],{"className":52208},[2336]," grows, the rotation axis becomes more longitudinal, the oscillation frequency increases, and the maximum excited-state population decreases.",[31,52211,52213],{"id":52212},"fourier-analysis-dynamics","Fourier Analysis Dynamics",[13,52215,52216,52217,52245,52246,52344],{},"The spectral analysis computes a real Fast Fourier transform along the pulse-duration axis. For each sampled drive frequency ",[54,52218,52220,52233],{"className":52219},[2079],[54,52221,52223],{"className":52222},[2083],[2085,52224,52225],{"xmlns":2087},[2089,52226,52227,52231],{},[2092,52228,52229],{},[2098,52230,21834],{},[2105,52232,21834],{"encoding":2107},[54,52234,52236],{"className":52235,"ariaHidden":2113},[2112],[54,52237,52239,52242],{"className":52238},[2117],[54,52240],{"className":52241,"style":6591},[2121],[54,52243,21834],{"className":52244,"style":26895},[2126,2167],", the trace ",[54,52247,52249,52277],{"className":52248},[2079],[54,52250,52252],{"className":52251},[2083],[2085,52253,52254],{"xmlns":2087},[2089,52255,52256,52274],{},[2092,52257,52258,52264,52266,52268,52270,52272],{},[2724,52259,52260,52262],{},[2098,52261,2730],{},[2098,52263,2842],{},[2245,52265,129],{"stretchy":2311},[2098,52267,41261],{},[2245,52269,2375],{"separator":2113},[2098,52271,21834],{},[2245,52273,3205],{"stretchy":2311},[2105,52275,52276],{"encoding":2107},"P_e(\\tau,f)",[54,52278,52280],{"className":52279,"ariaHidden":2113},[2112],[54,52281,52283,52286,52326,52329,52332,52335,52338,52341],{"className":52282},[2117],[54,52284],{"className":52285,"style":2325},[2121],[54,52287,52289,52292],{"className":52288},[2126],[54,52290,2730],{"className":52291,"style":2776},[2126,2167],[54,52293,52295],{"className":52294},[2135],[54,52296,52298,52318],{"className":52297},[2139,2756],[54,52299,52301,52315],{"className":52300},[2143],[54,52302,52304],{"className":52303,"style":4063},[2147],[54,52305,52306,52309],{"style":7193},[54,52307],{"className":52308,"style":2156},[2155],[54,52310,52312],{"className":52311},[2160,2161,2162,2163],[54,52313,2842],{"className":52314},[2126,2167,2163],[54,52316,2781],{"className":52317},[2780],[54,52319,52321],{"className":52320},[2143],[54,52322,52324],{"className":52323,"style":2788},[2147],[54,52325],{},[54,52327,129],{"className":52328},[2497],[54,52330,41261],{"className":52331,"style":41333},[2126,2167],[54,52333,2375],{"className":52334},[2517],[54,52336],{"className":52337,"style":2521},[2270],[54,52339,21834],{"className":52340,"style":26895},[2126,2167],[54,52342,3205],{"className":52343},[2336]," is treated as a time-domain signal. The spectral amplitude is",[54,52346,52348],{"className":52347},[2340],[54,52349,52351,52414],{"className":52350},[2079],[54,52352,52354],{"className":52353},[2083],[2085,52355,52356],{"xmlns":2087,"display":2349},[2089,52357,52358,52411],{},[2092,52359,52360,52362,52364,52366,52368,52370,52372,52374,52409],{},[2098,52361,5413],{},[2245,52363,129],{"stretchy":2311},[2098,52365,42532],{},[2245,52367,2375],{"separator":2113},[2098,52369,21834],{},[2245,52371,3205],{"stretchy":2311},[2245,52373,122],{},[2092,52375,52376,52378,52385,52407],{},[2245,52377,2305],{"fence":2113},[2724,52379,52380,52383],{},[2098,52381,52382],{"mathvariant":2100},"F",[2098,52384,41261],{},[2092,52386,52387,52389,52395,52397,52399,52401,52403,52405],{},[2245,52388,2366],{"fence":2113},[2724,52390,52391,52393],{},[2098,52392,2730],{},[2098,52394,2842],{},[2245,52396,129],{"stretchy":2311},[2098,52398,41261],{},[2245,52400,2375],{"separator":2113},[2098,52402,21834],{},[2245,52404,3205],{"stretchy":2311},[2245,52406,2428],{"fence":2113},[2245,52408,2305],{"fence":2113},[2098,52410,29],{"mathvariant":2304},[2105,52412,52413],{"encoding":2107},"S(\\nu,f)=\\left|\\mathcal{F}_{\\tau}\\left\\{P_e(\\tau,f)\\right\\}\\right|.",[54,52415,52417,52453],{"className":52416,"ariaHidden":2113},[2112],[54,52418,52420,52423,52426,52429,52432,52435,52438,52441,52444,52447,52450],{"className":52419},[2117],[54,52421],{"className":52422,"style":2325},[2121],[54,52424,5413],{"className":52425,"style":5448},[2126,2167],[54,52427,129],{"className":52428},[2497],[54,52430,42532],{"className":52431,"style":42553},[2126,2167],[54,52433,2375],{"className":52434},[2517],[54,52436],{"className":52437,"style":2521},[2270],[54,52439,21834],{"className":52440,"style":26895},[2126,2167],[54,52442,3205],{"className":52443},[2336],[54,52445],{"className":52446,"style":2475},[2270],[54,52448,122],{"className":52449},[2479],[54,52451],{"className":52452,"style":2475},[2270],[54,52454,52456,52459,52583,52586],{"className":52455},[2117],[54,52457],{"className":52458,"style":2325},[2121],[54,52460,52462,52465,52510,52513,52580],{"className":52461},[2493],[54,52463,2305],{"className":52464,"style":2499},[2497,2498],[54,52466,52468,52472],{"className":52467},[2126],[54,52469,52382],{"className":52470,"style":52471},[2126,2130],"margin-right:0.0993em;",[54,52473,52475],{"className":52474},[2135],[54,52476,52478,52502],{"className":52477},[2139,2756],[54,52479,52481,52499],{"className":52480},[2143],[54,52482,52484],{"className":52483,"style":4063},[2147],[54,52485,52487,52490],{"style":52486},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[54,52488],{"className":52489,"style":2156},[2155],[54,52491,52493],{"className":52492},[2160,2161,2162,2163],[54,52494,52496],{"className":52495},[2126,2163],[54,52497,41261],{"className":52498,"style":41333},[2126,2167,2163],[54,52500,2781],{"className":52501},[2780],[54,52503,52505],{"className":52504},[2143],[54,52506,52508],{"className":52507,"style":2788},[2147],[54,52509],{},[54,52511],{"className":52512,"style":2521},[2270],[54,52514,52516,52519,52559,52562,52565,52568,52571,52574,52577],{"className":52515},[2493],[54,52517,2366],{"className":52518,"style":2499},[2497,2498],[54,52520,52522,52525],{"className":52521},[2126],[54,52523,2730],{"className":52524,"style":2776},[2126,2167],[54,52526,52528],{"className":52527},[2135],[54,52529,52531,52551],{"className":52530},[2139,2756],[54,52532,52534,52548],{"className":52533},[2143],[54,52535,52537],{"className":52536,"style":4063},[2147],[54,52538,52539,52542],{"style":7193},[54,52540],{"className":52541,"style":2156},[2155],[54,52543,52545],{"className":52544},[2160,2161,2162,2163],[54,52546,2842],{"className":52547},[2126,2167,2163],[54,52549,2781],{"className":52550},[2780],[54,52552,52554],{"className":52553},[2143],[54,52555,52557],{"className":52556,"style":2788},[2147],[54,52558],{},[54,52560,129],{"className":52561},[2497],[54,52563,41261],{"className":52564,"style":41333},[2126,2167],[54,52566,2375],{"className":52567},[2517],[54,52569],{"className":52570,"style":2521},[2270],[54,52572,21834],{"className":52573,"style":26895},[2126,2167],[54,52575,3205],{"className":52576},[2336],[54,52578,2428],{"className":52579,"style":2499},[2336,2498],[54,52581,2305],{"className":52582,"style":2499},[2336,2498],[54,52584],{"className":52585,"style":2521},[2270],[54,52587,29],{"className":52588},[2126],[13,52590,52591],{},"This probability expression contains a squared sinusoid. Using the identity",[54,52593,52595],{"className":52594},[2340],[54,52596,52598,52674],{"className":52597},[2079],[54,52599,52601],{"className":52600},[2083],[2085,52602,52603],{"xmlns":2087,"display":2349},[2089,52604,52605,52671],{},[2092,52606,52607,52617,52637,52639,52645],{},[2095,52608,52609,52615],{},[2092,52610,52611,52613],{},[2098,52612,46278],{},[2245,52614,36247],{},[2249,52616,133],{},[2092,52618,52619,52621,52635],{},[2245,52620,129],{"fence":2113},[9047,52622,52623,52633],{},[2092,52624,52625,52631],{},[2724,52626,52627,52629],{},[2098,52628,42219],{"mathvariant":2304},[2098,52630,19926],{},[2098,52632,41261],{},[2249,52634,133],{},[2245,52636,3205],{"fence":2113},[2245,52638,122],{},[9047,52640,52641,52643],{},[2249,52642,174],{},[2249,52644,133],{},[2092,52646,52647,52649,52651,52653,52655,52657,52659,52665,52667,52669],{},[2245,52648,11804],{"fence":2113},[2249,52650,174],{},[2245,52652,2247],{},[2098,52654,37489],{},[2245,52656,36247],{},[2245,52658,129],{"stretchy":2311},[2724,52660,52661,52663],{},[2098,52662,42219],{"mathvariant":2304},[2098,52664,19926],{},[2098,52666,41261],{},[2245,52668,3205],{"stretchy":2311},[2245,52670,11829],{"fence":2113},[2105,52672,52673],{"encoding":2107},"\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right)=\\frac{1}{2}\\left[1-\\cos(\\Omega_R\\tau)\\right]",[54,52675,52677,52841],{"className":52676,"ariaHidden":2113},[2112],[54,52678,52680,52683,52712,52715,52832,52835,52838],{"className":52679},[2117],[54,52681],{"className":52682,"style":37225},[2121],[54,52684,52686,52689],{"className":52685},[7126],[54,52687,46278],{"className":52688},[7126],[54,52690,52692],{"className":52691},[2135],[54,52693,52695],{"className":52694},[2139],[54,52696,52698],{"className":52697},[2143],[54,52699,52701],{"className":52700,"style":46554},[2147],[54,52702,52703,52706],{"style":46557},[54,52704],{"className":52705,"style":2156},[2155],[54,52707,52709],{"className":52708},[2160,2161,2162,2163],[54,52710,133],{"className":52711},[2126,2163],[54,52713],{"className":52714,"style":2521},[2270],[54,52716,52718,52724,52826],{"className":52717},[2493],[54,52719,52721],{"className":52720,"style":2499},[2497,2498],[54,52722,129],{"className":52723},[2503,2162],[54,52725,52727,52730,52823],{"className":52726},[2126],[54,52728],{"className":52729},[2497,9349],[54,52731,52733],{"className":52732},[9047],[54,52734,52736,52815],{"className":52735},[2139,2756],[54,52737,52739,52812],{"className":52738},[2143],[54,52740,52742,52753,52761],{"className":52741,"style":46597},[2147],[54,52743,52744,52747],{"style":11273},[54,52745],{"className":52746,"style":3866},[2155],[54,52748,52750],{"className":52749},[2126],[54,52751,133],{"className":52752},[2126],[54,52754,52755,52758],{"style":9434},[54,52756],{"className":52757,"style":3866},[2155],[54,52759],{"className":52760,"style":9442},[9441],[54,52762,52763,52766],{"style":9445},[54,52764],{"className":52765,"style":3866},[2155],[54,52767,52769,52809],{"className":52768},[2126],[54,52770,52772,52775],{"className":52771},[2126],[54,52773,42219],{"className":52774},[2126],[54,52776,52778],{"className":52777},[2135],[54,52779,52781,52801],{"className":52780},[2139,2756],[54,52782,52784,52798],{"className":52783},[2143],[54,52785,52787],{"className":52786,"style":2763},[2147],[54,52788,52789,52792],{"style":4954},[54,52790],{"className":52791,"style":2156},[2155],[54,52793,52795],{"className":52794},[2160,2161,2162,2163],[54,52796,19926],{"className":52797,"style":19951},[2126,2167,2163],[54,52799,2781],{"className":52800},[2780],[54,52802,52804],{"className":52803},[2143],[54,52805,52807],{"className":52806,"style":2788},[2147],[54,52808],{},[54,52810,41261],{"className":52811,"style":41333},[2126,2167],[54,52813,2781],{"className":52814},[2780],[54,52816,52818],{"className":52817},[2143],[54,52819,52821],{"className":52820,"style":11311},[2147],[54,52822],{},[54,52824],{"className":52825},[2336,9349],[54,52827,52829],{"className":52828,"style":2499},[2336,2498],[54,52830,3205],{"className":52831},[2503,2162],[54,52833],{"className":52834,"style":2475},[2270],[54,52836,122],{"className":52837},[2479],[54,52839],{"className":52840,"style":2475},[2270],[54,52842,52844,52847,52909,52912],{"className":52843},[2117],[54,52845],{"className":52846,"style":48360},[2121],[54,52848,52850,52853,52906],{"className":52849},[2126],[54,52851],{"className":52852},[2497,9349],[54,52854,52856],{"className":52855},[9047],[54,52857,52859,52898],{"className":52858},[2139,2756],[54,52860,52862,52895],{"className":52861},[2143],[54,52863,52865,52876,52884],{"className":52864,"style":9362},[2147],[54,52866,52867,52870],{"style":11273},[54,52868],{"className":52869,"style":3866},[2155],[54,52871,52873],{"className":52872},[2126],[54,52874,133],{"className":52875},[2126],[54,52877,52878,52881],{"style":9434},[54,52879],{"className":52880,"style":3866},[2155],[54,52882],{"className":52883,"style":9442},[9441],[54,52885,52886,52889],{"style":9445},[54,52887],{"className":52888,"style":3866},[2155],[54,52890,52892],{"className":52891},[2126],[54,52893,174],{"className":52894},[2126],[54,52896,2781],{"className":52897},[2780],[54,52899,52901],{"className":52900},[2143],[54,52902,52904],{"className":52903,"style":11311},[2147],[54,52905],{},[54,52907],{"className":52908},[2336,9349],[54,52910],{"className":52911,"style":2521},[2270],[54,52913,52915,52918,52921,52924,52927,52930,52933,52936,52976,52979,52982],{"className":52914},[2493],[54,52916,11804],{"className":52917,"style":2499},[2497,2498],[54,52919,174],{"className":52920},[2126],[54,52922],{"className":52923,"style":2271},[2270],[54,52925,2247],{"className":52926},[2275],[54,52928],{"className":52929,"style":2271},[2270],[54,52931,37489],{"className":52932},[7126],[54,52934,129],{"className":52935},[2497],[54,52937,52939,52942],{"className":52938},[2126],[54,52940,42219],{"className":52941},[2126],[54,52943,52945],{"className":52944},[2135],[54,52946,52948,52968],{"className":52947},[2139,2756],[54,52949,52951,52965],{"className":52950},[2143],[54,52952,52954],{"className":52953,"style":2763},[2147],[54,52955,52956,52959],{"style":4954},[54,52957],{"className":52958,"style":2156},[2155],[54,52960,52962],{"className":52961},[2160,2161,2162,2163],[54,52963,19926],{"className":52964,"style":19951},[2126,2167,2163],[54,52966,2781],{"className":52967},[2780],[54,52969,52971],{"className":52970},[2143],[54,52972,52974],{"className":52973,"style":2788},[2147],[54,52975],{},[54,52977,41261],{"className":52978,"style":41333},[2126,2167],[54,52980,3205],{"className":52981},[2336],[54,52983,11829],{"className":52984,"style":2499},[2336,2498],[13,52986,52987],{},"shows that the dominant oscillatory spectral component appears at the generalized frequency",[54,52989,52991],{"className":52990},[2340],[54,52992,52994,53030],{"className":52993},[2079],[54,52995,52997],{"className":52996},[2083],[2085,52998,52999],{"xmlns":2087,"display":2349},[2089,53000,53001,53027],{},[2092,53002,53003,53009,53011,53025],{},[2724,53004,53005,53007],{},[2098,53006,42532],{},[2098,53008,19926],{},[2245,53010,122],{},[9047,53012,53013,53019],{},[2724,53014,53015,53017],{},[2098,53016,42219],{"mathvariant":2304},[2098,53018,19926],{},[2092,53020,53021,53023],{},[2249,53022,133],{},[2098,53024,8119],{},[2098,53026,29],{"mathvariant":2304},[2105,53028,53029],{"encoding":2107},"\\nu_R=\\frac{\\Omega_R}{2\\pi}.",[54,53031,53033,53088],{"className":53032,"ariaHidden":2113},[2112],[54,53034,53036,53039,53079,53082,53085],{"className":53035},[2117],[54,53037],{"className":53038,"style":24753},[2121],[54,53040,53042,53045],{"className":53041},[2126],[54,53043,42532],{"className":53044,"style":42553},[2126,2167],[54,53046,53048],{"className":53047},[2135],[54,53049,53051,53071],{"className":53050},[2139,2756],[54,53052,53054,53068],{"className":53053},[2143],[54,53055,53057],{"className":53056,"style":2763},[2147],[54,53058,53059,53062],{"style":42568},[54,53060],{"className":53061,"style":2156},[2155],[54,53063,53065],{"className":53064},[2160,2161,2162,2163],[54,53066,19926],{"className":53067,"style":19951},[2126,2167,2163],[54,53069,2781],{"className":53070},[2780],[54,53072,53074],{"className":53073},[2143],[54,53075,53077],{"className":53076,"style":2788},[2147],[54,53078],{},[54,53080],{"className":53081,"style":2475},[2270],[54,53083,122],{"className":53084},[2479],[54,53086],{"className":53087,"style":2475},[2270],[54,53089,53091,53094,53196],{"className":53090},[2117],[54,53092],{"className":53093,"style":47990},[2121],[54,53095,53097,53100,53193],{"className":53096},[2126],[54,53098],{"className":53099},[2497,9349],[54,53101,53103],{"className":53102},[9047],[54,53104,53106,53185],{"className":53105},[2139,2756],[54,53107,53109,53182],{"className":53108},[2143],[54,53110,53112,53126,53134],{"className":53111,"style":46597},[2147],[54,53113,53114,53117],{"style":11273},[54,53115],{"className":53116,"style":3866},[2155],[54,53118,53120,53123],{"className":53119},[2126],[54,53121,133],{"className":53122},[2126],[54,53124,8119],{"className":53125,"style":2332},[2126,2167],[54,53127,53128,53131],{"style":9434},[54,53129],{"className":53130,"style":3866},[2155],[54,53132],{"className":53133,"style":9442},[9441],[54,53135,53136,53139],{"style":9445},[54,53137],{"className":53138,"style":3866},[2155],[54,53140,53142],{"className":53141},[2126],[54,53143,53145,53148],{"className":53144},[2126],[54,53146,42219],{"className":53147},[2126],[54,53149,53151],{"className":53150},[2135],[54,53152,53154,53174],{"className":53153},[2139,2756],[54,53155,53157,53171],{"className":53156},[2143],[54,53158,53160],{"className":53159,"style":2763},[2147],[54,53161,53162,53165],{"style":4954},[54,53163],{"className":53164,"style":2156},[2155],[54,53166,53168],{"className":53167},[2160,2161,2162,2163],[54,53169,19926],{"className":53170,"style":19951},[2126,2167,2163],[54,53172,2781],{"className":53173},[2780],[54,53175,53177],{"className":53176},[2143],[54,53178,53180],{"className":53179,"style":2788},[2147],[54,53181],{},[54,53183,2781],{"className":53184},[2780],[54,53186,53188],{"className":53187},[2143],[54,53189,53191],{"className":53190,"style":11311},[2147],[54,53192],{},[54,53194],{"className":53195},[2336,9349],[54,53197,29],{"className":53198},[2126],[13,53200,53201],{},"The computational workflow subtracts the mean signal along the pulse-duration axis, applies a Hann window to reduce spectral leakage, uses zero-padding to smooth the displayed Fourier grid, and overlays the exact generalized Rabi ridge on the Fourier heatmap. Zero-padding improves visual interpolation of the plotted spectrum, although the physical frequency resolution remains set by the sampled temporal window.",[31,53203,53205],{"id":53204},"computational-methodology","Computational Methodology",[13,53207,53208,53209,53237,53238,53266,53267,10355,53295,53323,53324,53458,53459,53678],{},"The notebook uses a deterministic closed-form workflow, so it does not require numerical time stepping of the Schrödinger equation. It defines a drive-frequency grid ",[54,53210,53212,53225],{"className":53211},[2079],[54,53213,53215],{"className":53214},[2083],[2085,53216,53217],{"xmlns":2087},[2089,53218,53219,53223],{},[2092,53220,53221],{},[2098,53222,21834],{},[2105,53224,21834],{"encoding":2107},[54,53226,53228],{"className":53227,"ariaHidden":2113},[2112],[54,53229,53231,53234],{"className":53230},[2117],[54,53232],{"className":53233,"style":6591},[2121],[54,53235,21834],{"className":53236,"style":26895},[2126,2167]," in GHz and a pulse-duration grid ",[54,53239,53241,53254],{"className":53240},[2079],[54,53242,53244],{"className":53243},[2083],[2085,53245,53246],{"xmlns":2087},[2089,53247,53248,53252],{},[2092,53249,53250],{},[2098,53251,41261],{},[2105,53253,41387],{"encoding":2107},[54,53255,53257],{"className":53256,"ariaHidden":2113},[2112],[54,53258,53260,53263],{"className":53259},[2117],[54,53261],{"className":53262,"style":2195},[2121],[54,53264,41261],{"className":53265,"style":41333},[2126,2167]," in ns, then converts both to SI units. From a dense two-dimensional mesh over ",[54,53268,53270,53283],{"className":53269},[2079],[54,53271,53273],{"className":53272},[2083],[2085,53274,53275],{"xmlns":2087},[2089,53276,53277,53281],{},[2092,53278,53279],{},[2098,53280,21834],{},[2105,53282,21834],{"encoding":2107},[54,53284,53286],{"className":53285,"ariaHidden":2113},[2112],[54,53287,53289,53292],{"className":53288},[2117],[54,53290],{"className":53291,"style":6591},[2121],[54,53293,21834],{"className":53294,"style":26895},[2126,2167],[54,53296,53298,53311],{"className":53297},[2079],[54,53299,53301],{"className":53300},[2083],[2085,53302,53303],{"xmlns":2087},[2089,53304,53305,53309],{},[2092,53306,53307],{},[2098,53308,41261],{},[2105,53310,41387],{"encoding":2107},[54,53312,53314],{"className":53313,"ariaHidden":2113},[2112],[54,53315,53317,53320],{"className":53316},[2117],[54,53318],{"className":53319,"style":2195},[2121],[54,53321,41261],{"className":53322,"style":41333},[2126,2167],", the workflow computes the detuning ",[54,53325,53327,53361],{"className":53326},[2079],[54,53328,53330],{"className":53329},[2083],[2085,53331,53332],{"xmlns":2087},[2089,53333,53334,53358],{},[2092,53335,53336,53338,53340,53342,53344,53346,53348,53350,53356],{},[2098,53337,42302],{"mathvariant":2304},[2245,53339,122],{},[2249,53341,133],{},[2098,53343,8119],{},[2245,53345,129],{"stretchy":2311},[2098,53347,21834],{},[2245,53349,2247],{},[2724,53351,53352,53354],{},[2098,53353,21834],{},[2249,53355,150],{},[2245,53357,3205],{"stretchy":2311},[2105,53359,53360],{"encoding":2107},"\\Delta=2\\pi(f-f_0)",[54,53362,53364,53382,53409],{"className":53363,"ariaHidden":2113},[2112],[54,53365,53367,53370,53373,53376,53379],{"className":53366},[2117],[54,53368],{"className":53369,"style":2122},[2121],[54,53371,42302],{"className":53372},[2126],[54,53374],{"className":53375,"style":2475},[2270],[54,53377,122],{"className":53378},[2479],[54,53380],{"className":53381,"style":2475},[2270],[54,53383,53385,53388,53391,53394,53397,53400,53403,53406],{"className":53384},[2117],[54,53386],{"className":53387,"style":2325},[2121],[54,53389,133],{"className":53390},[2126],[54,53392,8119],{"className":53393,"style":2332},[2126,2167],[54,53395,129],{"className":53396},[2497],[54,53398,21834],{"className":53399,"style":26895},[2126,2167],[54,53401],{"className":53402,"style":2271},[2270],[54,53404,2247],{"className":53405},[2275],[54,53407],{"className":53408,"style":2271},[2270],[54,53410,53412,53415,53455],{"className":53411},[2117],[54,53413],{"className":53414,"style":2325},[2121],[54,53416,53418,53421],{"className":53417},[2126],[54,53419,21834],{"className":53420,"style":26895},[2126,2167],[54,53422,53424],{"className":53423},[2135],[54,53425,53427,53447],{"className":53426},[2139,2756],[54,53428,53430,53444],{"className":53429},[2143],[54,53431,53433],{"className":53432,"style":13837},[2147],[54,53434,53435,53438],{"style":41824},[54,53436],{"className":53437,"style":2156},[2155],[54,53439,53441],{"className":53440},[2160,2161,2162,2163],[54,53442,150],{"className":53443},[2126,2163],[54,53445,2781],{"className":53446},[2780],[54,53448,53450],{"className":53449},[2143],[54,53451,53453],{"className":53452,"style":2788},[2147],[54,53454],{},[54,53456,3205],{"className":53457},[2336],", evaluates the generalized Rabi angular frequency ",[54,53460,53462,53499],{"className":53461},[2079],[54,53463,53465],{"className":53464},[2083],[2085,53466,53467],{"xmlns":2087},[2089,53468,53469,53497],{},[2092,53470,53471,53477,53479],{},[2724,53472,53473,53475],{},[2098,53474,42219],{"mathvariant":2304},[2098,53476,19926],{},[2245,53478,122],{},[9052,53480,53481],{},[2092,53482,53483,53489,53491],{},[2095,53484,53485,53487],{},[2098,53486,42219],{"mathvariant":2304},[2249,53488,133],{},[2245,53490,1663],{},[2095,53492,53493,53495],{},[2098,53494,42302],{"mathvariant":2304},[2249,53496,133],{},[2105,53498,46740],{"encoding":2107},[54,53500,53502,53557],{"className":53501,"ariaHidden":2113},[2112],[54,53503,53505,53508,53548,53551,53554],{"className":53504},[2117],[54,53506],{"className":53507,"style":2743},[2121],[54,53509,53511,53514],{"className":53510},[2126],[54,53512,42219],{"className":53513},[2126],[54,53515,53517],{"className":53516},[2135],[54,53518,53520,53540],{"className":53519},[2139,2756],[54,53521,53523,53537],{"className":53522},[2143],[54,53524,53526],{"className":53525,"style":2763},[2147],[54,53527,53528,53531],{"style":4954},[54,53529],{"className":53530,"style":2156},[2155],[54,53532,53534],{"className":53533},[2160,2161,2162,2163],[54,53535,19926],{"className":53536,"style":19951},[2126,2167,2163],[54,53538,2781],{"className":53539},[2780],[54,53541,53543],{"className":53542},[2143],[54,53544,53546],{"className":53545,"style":2788},[2147],[54,53547],{},[54,53549],{"className":53550,"style":2475},[2270],[54,53552,122],{"className":53553},[2479],[54,53555],{"className":53556,"style":2475},[2270],[54,53558,53560,53564],{"className":53559},[2117],[54,53561],{"className":53562,"style":53563},[2121],"height:1.04em;vertical-align:-0.1266em;",[54,53565,53567],{"className":53566},[2126,9375],[54,53568,53570,53669],{"className":53569},[2139,2756],[54,53571,53573,53666],{"className":53572},[2143],[54,53574,53577,53653],{"className":53575,"style":53576},[2147],"height:0.9134em;",[54,53578,53580,53583],{"className":53579,"style":3862},[9389],[54,53581],{"className":53582,"style":3866},[2155],[54,53584,53586,53615,53618,53621,53624],{"className":53585,"style":9396},[2126],[54,53587,53589,53592],{"className":53588},[2126],[54,53590,42219],{"className":53591},[2126],[54,53593,53595],{"className":53594},[2135],[54,53596,53598],{"className":53597},[2139],[54,53599,53601],{"className":53600},[2143],[54,53602,53604],{"className":53603,"style":28966},[2147],[54,53605,53606,53609],{"style":28969},[54,53607],{"className":53608,"style":2156},[2155],[54,53610,53612],{"className":53611},[2160,2161,2162,2163],[54,53613,133],{"className":53614},[2126,2163],[54,53616],{"className":53617,"style":2271},[2270],[54,53619,1663],{"className":53620},[2275],[54,53622],{"className":53623,"style":2271},[2270],[54,53625,53627,53630],{"className":53626},[2126],[54,53628,42302],{"className":53629},[2126],[54,53631,53633],{"className":53632},[2135],[54,53634,53636],{"className":53635},[2139],[54,53637,53639],{"className":53638},[2143],[54,53640,53642],{"className":53641,"style":28966},[2147],[54,53643,53644,53647],{"style":28969},[54,53645],{"className":53646,"style":2156},[2155],[54,53648,53650],{"className":53649},[2160,2161,2162,2163],[54,53651,133],{"className":53652},[2126,2163],[54,53654,53656,53659],{"style":53655},"top:-2.8734em;",[54,53657],{"className":53658,"style":3866},[2155],[54,53660,53662],{"className":53661,"style":9410},[9409],[5837,53663,53664],{"xmlns":5839,"width":9413,"height":9414,"viewBox":9415,"preserveAspectRatio":9416},[5846,53665],{"d":9419},[54,53667,2781],{"className":53668},[2780],[54,53670,53672],{"className":53671},[2143],[54,53673,53676],{"className":53674,"style":53675},[2147],"height:0.1266em;",[54,53677],{},", and calculates the analytic excited-state probability",[54,53680,53682],{"className":53681},[2340],[54,53683,53685,53763],{"className":53684},[2079],[54,53686,53688],{"className":53687},[2083],[2085,53689,53690],{"xmlns":2087,"display":2349},[2089,53691,53692,53760],{},[2092,53693,53694,53700,53702,53704,53706,53708,53710,53712,53728,53738,53758],{},[2724,53695,53696,53698],{},[2098,53697,2730],{},[2098,53699,2842],{},[2245,53701,129],{"stretchy":2311},[2098,53703,21834],{},[2245,53705,2375],{"separator":2113},[2098,53707,41261],{},[2245,53709,3205],{"stretchy":2311},[2245,53711,122],{},[9047,53713,53714,53720],{},[2095,53715,53716,53718],{},[2098,53717,42219],{"mathvariant":2304},[2249,53719,133],{},[3331,53721,53722,53724,53726],{},[2098,53723,42219],{"mathvariant":2304},[2098,53725,19926],{},[2249,53727,133],{},[2095,53729,53730,53736],{},[2092,53731,53732,53734],{},[2098,53733,46278],{},[2245,53735,36247],{},[2249,53737,133],{},[2092,53739,53740,53742,53756],{},[2245,53741,129],{"fence":2113},[9047,53743,53744,53754],{},[2092,53745,53746,53752],{},[2724,53747,53748,53750],{},[2098,53749,42219],{"mathvariant":2304},[2098,53751,19926],{},[2098,53753,41261],{},[2249,53755,133],{},[2245,53757,3205],{"fence":2113},[2098,53759,29],{"mathvariant":2304},[2105,53761,53762],{"encoding":2107},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right).",[54,53764,53766,53839],{"className":53765,"ariaHidden":2113},[2112],[54,53767,53769,53772,53812,53815,53818,53821,53824,53827,53830,53833,53836],{"className":53768},[2117],[54,53770],{"className":53771,"style":2325},[2121],[54,53773,53775,53778],{"className":53774},[2126],[54,53776,2730],{"className":53777,"style":2776},[2126,2167],[54,53779,53781],{"className":53780},[2135],[54,53782,53784,53804],{"className":53783},[2139,2756],[54,53785,53787,53801],{"className":53786},[2143],[54,53788,53790],{"className":53789,"style":4063},[2147],[54,53791,53792,53795],{"style":7193},[54,53793],{"className":53794,"style":2156},[2155],[54,53796,53798],{"className":53797},[2160,2161,2162,2163],[54,53799,2842],{"className":53800},[2126,2167,2163],[54,53802,2781],{"className":53803},[2780],[54,53805,53807],{"className":53806},[2143],[54,53808,53810],{"className":53809,"style":2788},[2147],[54,53811],{},[54,53813,129],{"className":53814},[2497],[54,53816,21834],{"className":53817,"style":26895},[2126,2167],[54,53819,2375],{"className":53820},[2517],[54,53822],{"className":53823,"style":2521},[2270],[54,53825,41261],{"className":53826,"style":41333},[2126,2167],[54,53828,3205],{"className":53829},[2336],[54,53831],{"className":53832,"style":2475},[2270],[54,53834,122],{"className":53835},[2479],[54,53837],{"className":53838,"style":2475},[2270],[54,53840,53842,53845,53981,53984,54013,54016,54133,54136],{"className":53841},[2117],[54,53843],{"className":53844,"style":46390},[2121],[54,53846,53848,53851,53978],{"className":53847},[2126],[54,53849],{"className":53850},[2497,9349],[54,53852,53854],{"className":53853},[9047],[54,53855,53857,53970],{"className":53856},[2139,2756],[54,53858,53860,53967],{"className":53859},[2143],[54,53861,53863,53922,53930],{"className":53862,"style":46409},[2147],[54,53864,53865,53868],{"style":11273},[54,53866],{"className":53867,"style":3866},[2155],[54,53869,53871],{"className":53870},[2126],[54,53872,53874,53877],{"className":53873},[2126],[54,53875,42219],{"className":53876},[2126],[54,53878,53880],{"className":53879},[2135],[54,53881,53883,53914],{"className":53882},[2139,2756],[54,53884,53886,53911],{"className":53885},[2143],[54,53887,53889,53900],{"className":53888,"style":46436},[2147],[54,53890,53891,53894],{"style":46439},[54,53892],{"className":53893,"style":2156},[2155],[54,53895,53897],{"className":53896},[2160,2161,2162,2163],[54,53898,19926],{"className":53899,"style":19951},[2126,2167,2163],[54,53901,53902,53905],{"style":46451},[54,53903],{"className":53904,"style":2156},[2155],[54,53906,53908],{"className":53907},[2160,2161,2162,2163],[54,53909,133],{"className":53910},[2126,2163],[54,53912,2781],{"className":53913},[2780],[54,53915,53917],{"className":53916},[2143],[54,53918,53920],{"className":53919,"style":46470},[2147],[54,53921],{},[54,53923,53924,53927],{"style":9434},[54,53925],{"className":53926,"style":3866},[2155],[54,53928],{"className":53929,"style":9442},[9441],[54,53931,53932,53935],{"style":9445},[54,53933],{"className":53934,"style":3866},[2155],[54,53936,53938],{"className":53937},[2126],[54,53939,53941,53944],{"className":53940},[2126],[54,53942,42219],{"className":53943},[2126],[54,53945,53947],{"className":53946},[2135],[54,53948,53950],{"className":53949},[2139],[54,53951,53953],{"className":53952},[2143],[54,53954,53956],{"className":53955,"style":12780},[2147],[54,53957,53958,53961],{"style":2151},[54,53959],{"className":53960,"style":2156},[2155],[54,53962,53964],{"className":53963},[2160,2161,2162,2163],[54,53965,133],{"className":53966},[2126,2163],[54,53968,2781],{"className":53969},[2780],[54,53971,53973],{"className":53972},[2143],[54,53974,53976],{"className":53975,"style":46527},[2147],[54,53977],{},[54,53979],{"className":53980},[2336,9349],[54,53982],{"className":53983,"style":2521},[2270],[54,53985,53987,53990],{"className":53986},[7126],[54,53988,46278],{"className":53989},[7126],[54,53991,53993],{"className":53992},[2135],[54,53994,53996],{"className":53995},[2139],[54,53997,53999],{"className":53998},[2143],[54,54000,54002],{"className":54001,"style":46554},[2147],[54,54003,54004,54007],{"style":46557},[54,54005],{"className":54006,"style":2156},[2155],[54,54008,54010],{"className":54009},[2160,2161,2162,2163],[54,54011,133],{"className":54012},[2126,2163],[54,54014],{"className":54015,"style":2521},[2270],[54,54017,54019,54025,54127],{"className":54018},[2493],[54,54020,54022],{"className":54021,"style":2499},[2497,2498],[54,54023,129],{"className":54024},[2503,2162],[54,54026,54028,54031,54124],{"className":54027},[2126],[54,54029],{"className":54030},[2497,9349],[54,54032,54034],{"className":54033},[9047],[54,54035,54037,54116],{"className":54036},[2139,2756],[54,54038,54040,54113],{"className":54039},[2143],[54,54041,54043,54054,54062],{"className":54042,"style":46597},[2147],[54,54044,54045,54048],{"style":11273},[54,54046],{"className":54047,"style":3866},[2155],[54,54049,54051],{"className":54050},[2126],[54,54052,133],{"className":54053},[2126],[54,54055,54056,54059],{"style":9434},[54,54057],{"className":54058,"style":3866},[2155],[54,54060],{"className":54061,"style":9442},[9441],[54,54063,54064,54067],{"style":9445},[54,54065],{"className":54066,"style":3866},[2155],[54,54068,54070,54110],{"className":54069},[2126],[54,54071,54073,54076],{"className":54072},[2126],[54,54074,42219],{"className":54075},[2126],[54,54077,54079],{"className":54078},[2135],[54,54080,54082,54102],{"className":54081},[2139,2756],[54,54083,54085,54099],{"className":54084},[2143],[54,54086,54088],{"className":54087,"style":2763},[2147],[54,54089,54090,54093],{"style":4954},[54,54091],{"className":54092,"style":2156},[2155],[54,54094,54096],{"className":54095},[2160,2161,2162,2163],[54,54097,19926],{"className":54098,"style":19951},[2126,2167,2163],[54,54100,2781],{"className":54101},[2780],[54,54103,54105],{"className":54104},[2143],[54,54106,54108],{"className":54107,"style":2788},[2147],[54,54109],{},[54,54111,41261],{"className":54112,"style":41333},[2126,2167],[54,54114,2781],{"className":54115},[2780],[54,54117,54119],{"className":54118},[2143],[54,54120,54122],{"className":54121,"style":11311},[2147],[54,54123],{},[54,54125],{"className":54126},[2336,9349],[54,54128,54130],{"className":54129,"style":2499},[2336,2498],[54,54131,3205],{"className":54132},[2503,2162],[54,54134],{"className":54135,"style":2521},[2270],[54,54137,29],{"className":54138},[2126],[13,54140,54141,54142,54257],{},"The routine optionally applies the damping envelope ",[54,54143,54145,54179],{"className":54144},[2079],[54,54146,54148],{"className":54147},[2083],[2085,54149,54150],{"xmlns":2087},[2089,54151,54152,54176],{},[2092,54153,54154,54156,54158,54160,54162,54164,54166,54174],{},[2098,54155,23625],{},[2245,54157,36247],{},[2245,54159,129],{"stretchy":2311},[2245,54161,2247],{},[2098,54163,41261],{},[2098,54165,194],{"mathvariant":2304},[3331,54167,54168,54170,54172],{},[2098,54169,36603],{},[2249,54171,133],{},[2245,54173,20325],{},[2245,54175,3205],{"stretchy":2311},[2105,54177,54178],{"encoding":2107},"\\exp(-\\tau\u002FT_2^\\ast)",[54,54180,54182],{"className":54181,"ariaHidden":2113},[2112],[54,54183,54185,54188,54191,54194,54197,54200,54203,54254],{"className":54184},[2117],[54,54186],{"className":54187,"style":2325},[2121],[54,54189,23625],{"className":54190},[7126],[54,54192,129],{"className":54193},[2497],[54,54195,2247],{"className":54196},[2126],[54,54198,41261],{"className":54199,"style":41333},[2126,2167],[54,54201,194],{"className":54202},[2126],[54,54204,54206,54209],{"className":54205},[2126],[54,54207,36603],{"className":54208,"style":2776},[2126,2167],[54,54210,54212],{"className":54211},[2135],[54,54213,54215,54246],{"className":54214},[2139,2756],[54,54216,54218,54243],{"className":54217},[2143],[54,54219,54221,54232],{"className":54220,"style":36139},[2147],[54,54222,54223,54226],{"style":42829},[54,54224],{"className":54225,"style":2156},[2155],[54,54227,54229],{"className":54228},[2160,2161,2162,2163],[54,54230,133],{"className":54231},[2126,2163],[54,54233,54234,54237],{"style":2151},[54,54235],{"className":54236,"style":2156},[2155],[54,54238,54240],{"className":54239},[2160,2161,2162,2163],[54,54241,20325],{"className":54242},[2275,2163],[54,54244,2781],{"className":54245},[2780],[54,54247,54249],{"className":54248},[2143],[54,54250,54252],{"className":54251,"style":42859},[2147],[54,54253],{},[54,54255,3205],{"className":54256},[2336]," and clips the result to the physical interval",[54,54259,54261],{"className":54260},[2340],[54,54262,54264,54290],{"className":54263},[2079],[54,54265,54267],{"className":54266},[2083],[2085,54268,54269],{"xmlns":2087,"display":2349},[2089,54270,54271,54287],{},[2092,54272,54273,54275,54277,54283,54285],{},[2249,54274,150],{},[2245,54276,27066],{},[2724,54278,54279,54281],{},[2098,54280,2730],{},[2098,54282,2842],{},[2245,54284,27066],{},[2249,54286,43900],{},[2105,54288,54289],{"encoding":2107},"0\\le P_e\\le 1.",[54,54291,54293,54312,54367],{"className":54292,"ariaHidden":2113},[2112],[54,54294,54296,54300,54303,54306,54309],{"className":54295},[2117],[54,54297],{"className":54298,"style":54299},[2121],"height:0.7804em;vertical-align:-0.136em;",[54,54301,150],{"className":54302},[2126],[54,54304],{"className":54305,"style":2475},[2270],[54,54307,27066],{"className":54308},[2479],[54,54310],{"className":54311,"style":2475},[2270],[54,54313,54315,54318,54358,54361,54364],{"className":54314},[2117],[54,54316],{"className":54317,"style":2743},[2121],[54,54319,54321,54324],{"className":54320},[2126],[54,54322,2730],{"className":54323,"style":2776},[2126,2167],[54,54325,54327],{"className":54326},[2135],[54,54328,54330,54350],{"className":54329},[2139,2756],[54,54331,54333,54347],{"className":54332},[2143],[54,54334,54336],{"className":54335,"style":4063},[2147],[54,54337,54338,54341],{"style":7193},[54,54339],{"className":54340,"style":2156},[2155],[54,54342,54344],{"className":54343},[2160,2161,2162,2163],[54,54345,2842],{"className":54346},[2126,2167,2163],[54,54348,2781],{"className":54349},[2780],[54,54351,54353],{"className":54352},[2143],[54,54354,54356],{"className":54355,"style":2788},[2147],[54,54357],{},[54,54359],{"className":54360,"style":2475},[2270],[54,54362,27066],{"className":54363},[2479],[54,54365],{"className":54366,"style":2475},[2270],[54,54368,54370,54373],{"className":54369},[2117],[54,54371],{"className":54372,"style":2285},[2121],[54,54374,43900],{"className":54375},[2126],[13,54377,54378],{},"Finally, the output is rendered as heatmaps, three-dimensional surfaces, one-dimensional slices, and Fourier-domain spectra. Standard scientific Python libraries provide the numerical arrays, visualization routines, and optional interactive rendering tools.",[31,54380,54382],{"id":54381},"graphical-interpretation","Graphical Interpretation",[13,54384,54385,54386,54484],{},"The ",[54,54387,54389,54417],{"className":54388},[2079],[54,54390,54392],{"className":54391},[2083],[2085,54393,54394],{"xmlns":2087},[2089,54395,54396,54414],{},[2092,54397,54398,54404,54406,54408,54410,54412],{},[2724,54399,54400,54402],{},[2098,54401,2730],{},[2098,54403,2842],{},[2245,54405,129],{"stretchy":2311},[2098,54407,21834],{},[2245,54409,2375],{"separator":2113},[2098,54411,41261],{},[2245,54413,3205],{"stretchy":2311},[2105,54415,54416],{"encoding":2107},"P_e(f,\\tau)",[54,54418,54420],{"className":54419,"ariaHidden":2113},[2112],[54,54421,54423,54426,54466,54469,54472,54475,54478,54481],{"className":54422},[2117],[54,54424],{"className":54425,"style":2325},[2121],[54,54427,54429,54432],{"className":54428},[2126],[54,54430,2730],{"className":54431,"style":2776},[2126,2167],[54,54433,54435],{"className":54434},[2135],[54,54436,54438,54458],{"className":54437},[2139,2756],[54,54439,54441,54455],{"className":54440},[2143],[54,54442,54444],{"className":54443,"style":4063},[2147],[54,54445,54446,54449],{"style":7193},[54,54447],{"className":54448,"style":2156},[2155],[54,54450,54452],{"className":54451},[2160,2161,2162,2163],[54,54453,2842],{"className":54454},[2126,2167,2163],[54,54456,2781],{"className":54457},[2780],[54,54459,54461],{"className":54460},[2143],[54,54462,54464],{"className":54463,"style":2788},[2147],[54,54465],{},[54,54467,129],{"className":54468},[2497],[54,54470,21834],{"className":54471,"style":26895},[2126,2167],[54,54473,2375],{"className":54474},[2517],[54,54476],{"className":54477,"style":2521},[2270],[54,54479,41261],{"className":54480,"style":41333},[2126,2167],[54,54482,3205],{"className":54483},[2336]," heatmap shows the predicted probability of detecting the qubit in the excited state after applying a control pulse. Its horizontal axis gives the applied drive frequency. The vertical axis gives the pulse duration. Brighter regions indicate higher excited-state probability.",[13,54486,54487],{},"The resonant interaction region appears near",[54,54489,54491],{"className":54490},[2340],[54,54492,54494,54517],{"className":54493},[2079],[54,54495,54497],{"className":54496},[2083],[2085,54498,54499],{"xmlns":2087,"display":2349},[2089,54500,54501,54515],{},[2092,54502,54503,54505,54507,54513],{},[2098,54504,21834],{},[2245,54506,122],{},[2724,54508,54509,54511],{},[2098,54510,21834],{},[2249,54512,150],{},[2098,54514,29],{"mathvariant":2304},[2105,54516,43596],{"encoding":2107},[54,54518,54520,54538],{"className":54519,"ariaHidden":2113},[2112],[54,54521,54523,54526,54529,54532,54535],{"className":54522},[2117],[54,54524],{"className":54525,"style":6591},[2121],[54,54527,21834],{"className":54528,"style":26895},[2126,2167],[54,54530],{"className":54531,"style":2475},[2270],[54,54533,122],{"className":54534},[2479],[54,54536],{"className":54537,"style":2475},[2270],[54,54539,54541,54544,54584],{"className":54540},[2117],[54,54542],{"className":54543,"style":6591},[2121],[54,54545,54547,54550],{"className":54546},[2126],[54,54548,21834],{"className":54549,"style":26895},[2126,2167],[54,54551,54553],{"className":54552},[2135],[54,54554,54556,54576],{"className":54555},[2139,2756],[54,54557,54559,54573],{"className":54558},[2143],[54,54560,54562],{"className":54561,"style":13837},[2147],[54,54563,54564,54567],{"style":41824},[54,54565],{"className":54566,"style":2156},[2155],[54,54568,54570],{"className":54569},[2160,2161,2162,2163],[54,54571,150],{"className":54572},[2126,2163],[54,54574,2781],{"className":54575},[2780],[54,54577,54579],{"className":54578},[2143],[54,54580,54582],{"className":54581,"style":2788},[2147],[54,54583],{},[54,54585,29],{"className":54586},[2126],[13,54588,54589,54590,54669],{},"At resonance, contrast is maximal and the pulse duration alone sets the rotation angle. With the default drive strength ",[54,54591,54593,54627],{"className":54592},[2079],[54,54594,54596],{"className":54595},[2083],[2085,54597,54598],{"xmlns":2087},[2089,54599,54600,54624],{},[2092,54601,54602,54604,54606,54608,54610,54612,54614,54616],{},[2098,54603,42219],{"mathvariant":2304},[2098,54605,194],{"mathvariant":2304},[2249,54607,133],{},[2098,54609,8119],{},[2245,54611,122],{},[2249,54613,47964],{},[5723,54615,5725],{},[2092,54617,54618,54620,54622],{},[2098,54619,47971],{"mathvariant":2304},[2098,54621,2694],{"mathvariant":2304},[2098,54623,215],{"mathvariant":2304},[2105,54625,54626],{"encoding":2107},"\\Omega\u002F2\\pi=20~\\mathrm{MHz}",[54,54628,54630,54651],{"className":54629,"ariaHidden":2113},[2112],[54,54631,54633,54636,54639,54642,54645,54648],{"className":54632},[2117],[54,54634],{"className":54635,"style":2325},[2121],[54,54637,42277],{"className":54638},[2126],[54,54640,8119],{"className":54641,"style":2332},[2126,2167],[54,54643],{"className":54644,"style":2475},[2270],[54,54646,122],{"className":54647},[2479],[54,54649],{"className":54650,"style":2475},[2270],[54,54652,54654,54657,54660,54663],{"className":54653},[2117],[54,54655],{"className":54656,"style":2122},[2121],[54,54658,47964],{"className":54659},[2126],[54,54661,5725],{"className":54662},[2270,43446],[54,54664,54666],{"className":54665},[2126],[54,54667,48084],{"className":54668},[2126,43453],", the ideal resonant maxima appear near",[54,54671,54673],{"className":54672},[2340],[54,54674,54676,54749],{"className":54675},[2079],[54,54677,54679],{"className":54678},[2083],[2085,54680,54681],{"xmlns":2087,"display":2349},[2089,54682,54683,54746],{},[2092,54684,54685,54687,54689,54691,54693,54699,54701,54703,54706,54708,54714,54716,54718,54721,54723,54729,54731,54733,54736,54738,54744],{},[2098,54686,41261],{},[2245,54688,122],{},[2249,54690,48145],{},[5723,54692,5725],{},[2092,54694,54695,54697],{},[2098,54696,3976],{"mathvariant":2304},[2098,54698,14089],{"mathvariant":2304},[2245,54700,2375],{"separator":2113},[5723,54702,5725],{},[2249,54704,54705],{},"75",[5723,54707,5725],{},[2092,54709,54710,54712],{},[2098,54711,3976],{"mathvariant":2304},[2098,54713,14089],{"mathvariant":2304},[2245,54715,2375],{"separator":2113},[5723,54717,5725],{},[2249,54719,54720],{},"125",[5723,54722,5725],{},[2092,54724,54725,54727],{},[2098,54726,3976],{"mathvariant":2304},[2098,54728,14089],{"mathvariant":2304},[2245,54730,2375],{"separator":2113},[5723,54732,5725],{},[2249,54734,54735],{},"175",[5723,54737,5725],{},[2092,54739,54740,54742],{},[2098,54741,3976],{"mathvariant":2304},[2098,54743,14089],{"mathvariant":2304},[2245,54745,2375],{"separator":2113},[2105,54747,54748],{"encoding":2107},"\\tau=25~\\mathrm{ns},\\ 75~\\mathrm{ns},\\ 125~\\mathrm{ns},\\ 175~\\mathrm{ns},",[54,54750,54752,54770],{"className":54751,"ariaHidden":2113},[2112],[54,54753,54755,54758,54761,54764,54767],{"className":54754},[2117],[54,54756],{"className":54757,"style":2195},[2121],[54,54759,41261],{"className":54760,"style":41333},[2126,2167],[54,54762],{"className":54763,"style":2475},[2270],[54,54765,122],{"className":54766},[2479],[54,54768],{"className":54769,"style":2475},[2270],[54,54771,54773,54777,54780,54783,54789,54792,54795,54798,54801,54804,54810,54813,54816,54819,54822,54825,54831,54834,54837,54840,54843,54846,54852],{"className":54772},[2117],[54,54774],{"className":54775,"style":54776},[2121],"height:0.8389em;vertical-align:-0.1944em;",[54,54778,48145],{"className":54779},[2126],[54,54781,5725],{"className":54782},[2270,43446],[54,54784,54786],{"className":54785},[2126],[54,54787,48235],{"className":54788},[2126,43453],[54,54790,2375],{"className":54791},[2517],[54,54793,5725],{"className":54794},[2270],[54,54796],{"className":54797,"style":2521},[2270],[54,54799,54705],{"className":54800},[2126],[54,54802,5725],{"className":54803},[2270,43446],[54,54805,54807],{"className":54806},[2126],[54,54808,48235],{"className":54809},[2126,43453],[54,54811,2375],{"className":54812},[2517],[54,54814,5725],{"className":54815},[2270],[54,54817],{"className":54818,"style":2521},[2270],[54,54820,54720],{"className":54821},[2126],[54,54823,5725],{"className":54824},[2270,43446],[54,54826,54828],{"className":54827},[2126],[54,54829,48235],{"className":54830},[2126,43453],[54,54832,2375],{"className":54833},[2517],[54,54835,5725],{"className":54836},[2270],[54,54838],{"className":54839,"style":2521},[2270],[54,54841,54735],{"className":54842},[2126],[54,54844,5725],{"className":54845},[2270,43446],[54,54847,54849],{"className":54848},[2126],[54,54850,48235],{"className":54851},[2126,43453],[54,54853,2375],{"className":54854},[2517],[13,54856,54857],{},"before the optional phenomenological damping factor is applied.",[13,54859,54860,54861,54930,54931,55057],{},"Off resonance, the fringes show lower contrast and faster temporal oscillations. This behavior follows from the generalized Rabi frequency ",[54,54862,54864,54881],{"className":54863},[2079],[54,54865,54867],{"className":54866},[2083],[2085,54868,54869],{"xmlns":2087},[2089,54870,54871,54879],{},[2092,54872,54873],{},[2724,54874,54875,54877],{},[2098,54876,42219],{"mathvariant":2304},[2098,54878,19926],{},[2105,54880,42459],{"encoding":2107},[54,54882,54884],{"className":54883,"ariaHidden":2113},[2112],[54,54885,54887,54890],{"className":54886},[2117],[54,54888],{"className":54889,"style":2743},[2121],[54,54891,54893,54896],{"className":54892},[2126],[54,54894,42219],{"className":54895},[2126],[54,54897,54899],{"className":54898},[2135],[54,54900,54902,54922],{"className":54901},[2139,2756],[54,54903,54905,54919],{"className":54904},[2143],[54,54906,54908],{"className":54907,"style":2763},[2147],[54,54909,54910,54913],{"style":4954},[54,54911],{"className":54912,"style":2156},[2155],[54,54914,54916],{"className":54915},[2160,2161,2162,2163],[54,54917,19926],{"className":54918,"style":19951},[2126,2167,2163],[54,54920,2781],{"className":54921},[2780],[54,54923,54925],{"className":54924},[2143],[54,54926,54928],{"className":54927,"style":2788},[2147],[54,54929],{},", which increases with detuning, and from the amplitude coefficient ",[54,54932,54934,54962],{"className":54933},[2079],[54,54935,54937],{"className":54936},[2083],[2085,54938,54939],{"xmlns":2087},[2089,54940,54941,54959],{},[2092,54942,54943,54949,54951],{},[2095,54944,54945,54947],{},[2098,54946,42219],{"mathvariant":2304},[2249,54948,133],{},[2098,54950,194],{"mathvariant":2304},[3331,54952,54953,54955,54957],{},[2098,54954,42219],{"mathvariant":2304},[2098,54956,19926],{},[2249,54958,133],{},[2105,54960,54961],{"encoding":2107},"\\Omega^2\u002F\\Omega_R^2",[54,54963,54965],{"className":54964,"ariaHidden":2113},[2112],[54,54966,54968,54972,55001,55004],{"className":54967},[2117],[54,54969],{"className":54970,"style":54971},[2121],"height:1.0894em;vertical-align:-0.2753em;",[54,54973,54975,54978],{"className":54974},[2126],[54,54976,42219],{"className":54977},[2126],[54,54979,54981],{"className":54980},[2135],[54,54982,54984],{"className":54983},[2139],[54,54985,54987],{"className":54986},[2143],[54,54988,54990],{"className":54989,"style":12780},[2147],[54,54991,54992,54995],{"style":2151},[54,54993],{"className":54994,"style":2156},[2155],[54,54996,54998],{"className":54997},[2160,2161,2162,2163],[54,54999,133],{"className":55000},[2126,2163],[54,55002,194],{"className":55003},[2126],[54,55005,55007,55010],{"className":55006},[2126],[54,55008,42219],{"className":55009},[2126],[54,55011,55013],{"className":55012},[2135],[54,55014,55016,55048],{"className":55015},[2139,2756],[54,55017,55019,55045],{"className":55018},[2143],[54,55020,55022,55034],{"className":55021,"style":12780},[2147],[54,55023,55025,55028],{"style":55024},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[54,55026],{"className":55027,"style":2156},[2155],[54,55029,55031],{"className":55030},[2160,2161,2162,2163],[54,55032,19926],{"className":55033,"style":19951},[2126,2167,2163],[54,55035,55036,55039],{"style":2151},[54,55037],{"className":55038,"style":2156},[2155],[54,55040,55042],{"className":55041},[2160,2161,2162,2163],[54,55043,133],{"className":55044},[2126,2163],[54,55046,2781],{"className":55047},[2780],[54,55049,55051],{"className":55050},[2143],[54,55052,55055],{"className":55053,"style":55054},[2147],"height:0.2753em;",[54,55056],{},", which decreases with detuning.",[13,55059,55060],{},"The three-dimensional surface plot represents the same probability data as a height field. This view emphasizes the central resonant ridge and the suppressed off-resonant oscillations. Cross-section plots isolate one-dimensional behavior. A fixed-frequency slice shows how probability evolves with pulse duration. At a fixed duration, a complementary slice shows how probability changes across the drive-frequency sweep.",[13,55062,55063,55064,55161],{},"The FFT heatmap characterizes the oscillation-frequency content of ",[54,55065,55067,55094],{"className":55066},[2079],[54,55068,55070],{"className":55069},[2083],[2085,55071,55072],{"xmlns":2087},[2089,55073,55074,55092],{},[2092,55075,55076,55082,55084,55086,55088,55090],{},[2724,55077,55078,55080],{},[2098,55079,2730],{},[2098,55081,2842],{},[2245,55083,129],{"stretchy":2311},[2098,55085,41261],{},[2245,55087,2375],{"separator":2113},[2098,55089,21834],{},[2245,55091,3205],{"stretchy":2311},[2105,55093,52276],{"encoding":2107},[54,55095,55097],{"className":55096,"ariaHidden":2113},[2112],[54,55098,55100,55103,55143,55146,55149,55152,55155,55158],{"className":55099},[2117],[54,55101],{"className":55102,"style":2325},[2121],[54,55104,55106,55109],{"className":55105},[2126],[54,55107,2730],{"className":55108,"style":2776},[2126,2167],[54,55110,55112],{"className":55111},[2135],[54,55113,55115,55135],{"className":55114},[2139,2756],[54,55116,55118,55132],{"className":55117},[2143],[54,55119,55121],{"className":55120,"style":4063},[2147],[54,55122,55123,55126],{"style":7193},[54,55124],{"className":55125,"style":2156},[2155],[54,55127,55129],{"className":55128},[2160,2161,2162,2163],[54,55130,2842],{"className":55131},[2126,2167,2163],[54,55133,2781],{"className":55134},[2780],[54,55136,55138],{"className":55137},[2143],[54,55139,55141],{"className":55140,"style":2788},[2147],[54,55142],{},[54,55144,129],{"className":55145},[2497],[54,55147,41261],{"className":55148,"style":41333},[2126,2167],[54,55150,2375],{"className":55151},[2517],[54,55153],{"className":55154,"style":2521},[2270],[54,55156,21834],{"className":55157,"style":26895},[2126,2167],[54,55159,3205],{"className":55160},[2336],". An overlaid theoretical curve follows",[54,55163,55165],{"className":55164},[2340],[54,55166,55168,55240],{"className":55167},[2079],[54,55169,55171],{"className":55170},[2083],[2085,55172,55173],{"xmlns":2087,"display":2349},[2089,55174,55175,55237],{},[2092,55176,55177,55183,55185,55187,55189,55191,55235],{},[2724,55178,55179,55181],{},[2098,55180,42532],{},[2098,55182,19926],{},[2245,55184,129],{"stretchy":2311},[2098,55186,21834],{},[2245,55188,3205],{"stretchy":2311},[2245,55190,122],{},[9052,55192,55193],{},[2092,55194,55195,55215,55217,55219,55221,55223,55229],{},[2095,55196,55197,55213],{},[2092,55198,55199,55201,55211],{},[2245,55200,129],{"fence":2113},[9047,55202,55203,55205],{},[2098,55204,42219],{"mathvariant":2304},[2092,55206,55207,55209],{},[2249,55208,133],{},[2098,55210,8119],{},[2245,55212,3205],{"fence":2113},[2249,55214,133],{},[2245,55216,1663],{},[2245,55218,129],{"stretchy":2311},[2098,55220,21834],{},[2245,55222,2247],{},[2724,55224,55225,55227],{},[2098,55226,21834],{},[2249,55228,150],{},[2095,55230,55231,55233],{},[2245,55232,3205],{"stretchy":2311},[2249,55234,133],{},[2098,55236,29],{"mathvariant":2304},[2105,55238,55239],{"encoding":2107},"\\nu_R(f)=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[54,55241,55243,55307],{"className":55242,"ariaHidden":2113},[2112],[54,55244,55246,55249,55289,55292,55295,55298,55301,55304],{"className":55245},[2117],[54,55247],{"className":55248,"style":2325},[2121],[54,55250,55252,55255],{"className":55251},[2126],[54,55253,42532],{"className":55254,"style":42553},[2126,2167],[54,55256,55258],{"className":55257},[2135],[54,55259,55261,55281],{"className":55260},[2139,2756],[54,55262,55264,55278],{"className":55263},[2143],[54,55265,55267],{"className":55266,"style":2763},[2147],[54,55268,55269,55272],{"style":42568},[54,55270],{"className":55271,"style":2156},[2155],[54,55273,55275],{"className":55274},[2160,2161,2162,2163],[54,55276,19926],{"className":55277,"style":19951},[2126,2167,2163],[54,55279,2781],{"className":55280},[2780],[54,55282,55284],{"className":55283},[2143],[54,55285,55287],{"className":55286,"style":2788},[2147],[54,55288],{},[54,55290,129],{"className":55291},[2497],[54,55293,21834],{"className":55294,"style":26895},[2126,2167],[54,55296,3205],{"className":55297},[2336],[54,55299],{"className":55300,"style":2475},[2270],[54,55302,122],{"className":55303},[2479],[54,55305],{"className":55306,"style":2475},[2270],[54,55308,55310,55313,55556],{"className":55309},[2117],[54,55311],{"className":55312,"style":49582},[2121],[54,55314,55316],{"className":55315},[2126,9375],[54,55317,55319,55548],{"className":55318},[2139,2756],[54,55320,55322,55545],{"className":55321},[2143],[54,55323,55325,55533],{"className":55324,"style":49595},[2147],[54,55326,55328,55331],{"className":55327,"style":49599},[9389],[54,55329],{"className":55330,"style":49603},[2155],[54,55332,55334,55440,55443,55446,55449,55452,55455,55458,55461,55464,55504],{"className":55333,"style":46830},[2126],[54,55335,55337,55417],{"className":55336},[2493],[54,55338,55340,55346,55411],{"className":55339},[2493],[54,55341,55343],{"className":55342,"style":2499},[2497,2498],[54,55344,129],{"className":55345},[2503,2162],[54,55347,55349,55352,55408],{"className":55348},[2126],[54,55350],{"className":55351},[2497,9349],[54,55353,55355],{"className":55354},[9047],[54,55356,55358,55400],{"className":55357},[2139,2756],[54,55359,55361,55397],{"className":55360},[2143],[54,55362,55364,55378,55386],{"className":55363,"style":46597},[2147],[54,55365,55366,55369],{"style":11273},[54,55367],{"className":55368,"style":3866},[2155],[54,55370,55372,55375],{"className":55371},[2126],[54,55373,133],{"className":55374},[2126],[54,55376,8119],{"className":55377,"style":2332},[2126,2167],[54,55379,55380,55383],{"style":9434},[54,55381],{"className":55382,"style":3866},[2155],[54,55384],{"className":55385,"style":9442},[9441],[54,55387,55388,55391],{"style":9445},[54,55389],{"className":55390,"style":3866},[2155],[54,55392,55394],{"className":55393},[2126],[54,55395,42219],{"className":55396},[2126],[54,55398,2781],{"className":55399},[2780],[54,55401,55403],{"className":55402},[2143],[54,55404,55406],{"className":55405,"style":11311},[2147],[54,55407],{},[54,55409],{"className":55410},[2336,9349],[54,55412,55414],{"className":55413,"style":2499},[2336,2498],[54,55415,3205],{"className":55416},[2503,2162],[54,55418,55420],{"className":55419},[2135],[54,55421,55423],{"className":55422},[2139],[54,55424,55426],{"className":55425},[2143],[54,55427,55429],{"className":55428,"style":49702},[2147],[54,55430,55431,55434],{"style":49705},[54,55432],{"className":55433,"style":2156},[2155],[54,55435,55437],{"className":55436},[2160,2161,2162,2163],[54,55438,133],{"className":55439},[2126,2163],[54,55441],{"className":55442,"style":2271},[2270],[54,55444,1663],{"className":55445},[2275],[54,55447],{"className":55448,"style":2271},[2270],[54,55450,129],{"className":55451},[2497],[54,55453,21834],{"className":55454,"style":26895},[2126,2167],[54,55456],{"className":55457,"style":2271},[2270],[54,55459,2247],{"className":55460},[2275],[54,55462],{"className":55463,"style":2271},[2270],[54,55465,55467,55470],{"className":55466},[2126],[54,55468,21834],{"className":55469,"style":26895},[2126,2167],[54,55471,55473],{"className":55472},[2135],[54,55474,55476,55496],{"className":55475},[2139,2756],[54,55477,55479,55493],{"className":55478},[2143],[54,55480,55482],{"className":55481,"style":13837},[2147],[54,55483,55484,55487],{"style":41824},[54,55485],{"className":55486,"style":2156},[2155],[54,55488,55490],{"className":55489},[2160,2161,2162,2163],[54,55491,150],{"className":55492},[2126,2163],[54,55494,2781],{"className":55495},[2780],[54,55497,55499],{"className":55498},[2143],[54,55500,55502],{"className":55501,"style":2788},[2147],[54,55503],{},[54,55505,55507,55510],{"className":55506},[2336],[54,55508,3205],{"className":55509},[2336],[54,55511,55513],{"className":55512},[2135],[54,55514,55516],{"className":55515},[2139],[54,55517,55519],{"className":55518},[2143],[54,55520,55522],{"className":55521,"style":28966},[2147],[54,55523,55524,55527],{"style":28969},[54,55525],{"className":55526,"style":2156},[2155],[54,55528,55530],{"className":55529},[2160,2161,2162,2163],[54,55531,133],{"className":55532},[2126,2163],[54,55534,55535,55538],{"style":49810},[54,55536],{"className":55537,"style":49603},[2155],[54,55539,55541],{"className":55540,"style":49817},[9409],[5837,55542,55543],{"xmlns":5839,"width":9413,"height":49820,"viewBox":49821,"preserveAspectRatio":9416},[5846,55544],{"d":49824},[54,55546,2781],{"className":55547},[2780],[54,55549,55551],{"className":55550},[2143],[54,55552,55554],{"className":55553,"style":49834},[2147],[54,55555],{},[54,55557,29],{"className":55558},[2126],[31,55560,55562],{"id":55561},"primary-modeling-assumptions","Primary Modeling Assumptions",[13,55564,55565],{},"This notebook is an idealized mathematical physics visualization designed for conceptual clarity and calibration intuition.",[41409,55567,55568,55578],{},[41412,55569,55570],{},[41415,55571,55572,55575],{},[41418,55573,55574],{},"Assumption",[41418,55576,55577],{},"Implication",[41425,55579,55580,55664,55672,55679,55687,55724,55814,55971,56019,56027,56035,56043],{},[41415,55581,55582,55585],{},[41430,55583,55584],{},"Two-level truncation",[41430,55586,55587,55588,10355,55626,29],{},"The simulated model includes only ",[54,55589,55591,55608],{"className":55590},[2079],[54,55592,55594],{"className":55593},[2083],[2085,55595,55596],{"xmlns":2087},[2089,55597,55598,55606],{},[2092,55599,55600,55602,55604],{},[2245,55601,2305],{"stretchy":2311},[2098,55603,37470],{},[2245,55605,2312],{"stretchy":2311},[2105,55607,41452],{"encoding":2107},[54,55609,55611],{"className":55610,"ariaHidden":2113},[2112],[54,55612,55614,55617,55620,55623],{"className":55613},[2117],[54,55615],{"className":55616,"style":2325},[2121],[54,55618,2305],{"className":55619},[2497],[54,55621,37470],{"className":55622,"style":2332},[2126,2167],[54,55624,2312],{"className":55625},[2336],[54,55627,55629,55646],{"className":55628},[2079],[54,55630,55632],{"className":55631},[2083],[2085,55633,55634],{"xmlns":2087},[2089,55635,55636,55644],{},[2092,55637,55638,55640,55642],{},[2245,55639,2305],{"stretchy":2311},[2098,55641,2842],{},[2245,55643,2312],{"stretchy":2311},[2105,55645,41498],{"encoding":2107},[54,55647,55649],{"className":55648,"ariaHidden":2113},[2112],[54,55650,55652,55655,55658,55661],{"className":55651},[2117],[54,55653],{"className":55654,"style":2325},[2121],[54,55656,2305],{"className":55657},[2497],[54,55659,2842],{"className":55660},[2126,2167],[54,55662,2312],{"className":55663},[2336],[41415,55665,55666,55669],{},[41430,55667,55668],{},"Classical drive",[41430,55670,55671],{},"The microwave field is prescribed externally instead of quantized as a field mode.",[41415,55673,55674,55676],{},[41430,55675,42872],{},[41430,55677,55678],{},"Fast counter-rotating terms are omitted.",[41415,55680,55681,55684],{},[41430,55682,55683],{},"Square pulse",[41430,55685,55686],{},"The drive is assumed to turn on and off instantaneously.",[41415,55688,55689,55692],{},[41430,55690,55691],{},"Uniform drive amplitude",[41430,55693,55694,55695,55723],{},"The parameter ",[54,55696,55698,55711],{"className":55697},[2079],[54,55699,55701],{"className":55700},[2083],[2085,55702,55703],{"xmlns":2087},[2089,55704,55705,55709],{},[2092,55706,55707],{},[2098,55708,42219],{"mathvariant":2304},[2105,55710,42222],{"encoding":2107},[54,55712,55714],{"className":55713,"ariaHidden":2113},[2112],[54,55715,55717,55720],{"className":55716},[2117],[54,55718],{"className":55719,"style":2122},[2121],[54,55721,42219],{"className":55722},[2126]," remains constant across the frequency sweep.",[41415,55725,55726,55729],{},[41430,55727,55728],{},"Optional damping envelope",[41430,55730,55694,55731,55813],{},[54,55732,55734,55753],{"className":55733},[2079],[54,55735,55737],{"className":55736},[2083],[2085,55738,55739],{"xmlns":2087},[2089,55740,55741,55751],{},[2092,55742,55743],{},[3331,55744,55745,55747,55749],{},[2098,55746,36603],{},[2249,55748,133],{},[2245,55750,20325],{},[2105,55752,42798],{"encoding":2107},[54,55754,55756],{"className":55755,"ariaHidden":2113},[2112],[54,55757,55759,55762],{"className":55758},[2117],[54,55760],{"className":55761,"style":42808},[2121],[54,55763,55765,55768],{"className":55764},[2126],[54,55766,36603],{"className":55767,"style":2776},[2126,2167],[54,55769,55771],{"className":55770},[2135],[54,55772,55774,55805],{"className":55773},[2139,2756],[54,55775,55777,55802],{"className":55776},[2143],[54,55778,55780,55791],{"className":55779,"style":36139},[2147],[54,55781,55782,55785],{"style":42829},[54,55783],{"className":55784,"style":2156},[2155],[54,55786,55788],{"className":55787},[2160,2161,2162,2163],[54,55789,133],{"className":55790},[2126,2163],[54,55792,55793,55796],{"style":2151},[54,55794],{"className":55795,"style":2156},[2155],[54,55797,55799],{"className":55798},[2160,2161,2162,2163],[54,55800,20325],{"className":55801},[2275,2163],[54,55803,2781],{"className":55804},[2780],[54,55806,55808],{"className":55807},[2143],[54,55809,55811],{"className":55810,"style":42859},[2147],[54,55812],{}," enters as a simple exponential contrast envelope.",[41415,55815,55816,55890],{},[41430,55817,55818,55819,55889],{},"No explicit ",[54,55820,55822,55840],{"className":55821},[2079],[54,55823,55825],{"className":55824},[2083],[2085,55826,55827],{"xmlns":2087},[2089,55828,55829,55837],{},[2092,55830,55831],{},[2724,55832,55833,55835],{},[2098,55834,36603],{},[2249,55836,174],{},[2105,55838,55839],{"encoding":2107},"T_1",[54,55841,55843],{"className":55842,"ariaHidden":2113},[2112],[54,55844,55846,55849],{"className":55845},[2117],[54,55847],{"className":55848,"style":2743},[2121],[54,55850,55852,55855],{"className":55851},[2126],[54,55853,36603],{"className":55854,"style":2776},[2126,2167],[54,55856,55858],{"className":55857},[2135],[54,55859,55861,55881],{"className":55860},[2139,2756],[54,55862,55864,55878],{"className":55863},[2143],[54,55865,55867],{"className":55866,"style":13837},[2147],[54,55868,55869,55872],{"style":7193},[54,55870],{"className":55871,"style":2156},[2155],[54,55873,55875],{"className":55874},[2160,2161,2162,2163],[54,55876,174],{"className":55877},[2126,2163],[54,55879,2781],{"className":55880},[2780],[54,55882,55884],{"className":55883},[2143],[54,55885,55887],{"className":55886,"style":2788},[2147],[54,55888],{}," relaxation",[41430,55891,55892,55893,55931,55932,55970],{},"Energy decay from ",[54,55894,55896,55913],{"className":55895},[2079],[54,55897,55899],{"className":55898},[2083],[2085,55900,55901],{"xmlns":2087},[2089,55902,55903,55911],{},[2092,55904,55905,55907,55909],{},[2245,55906,2305],{"stretchy":2311},[2098,55908,2842],{},[2245,55910,2312],{"stretchy":2311},[2105,55912,41498],{"encoding":2107},[54,55914,55916],{"className":55915,"ariaHidden":2113},[2112],[54,55917,55919,55922,55925,55928],{"className":55918},[2117],[54,55920],{"className":55921,"style":2325},[2121],[54,55923,2305],{"className":55924},[2497],[54,55926,2842],{"className":55927},[2126,2167],[54,55929,2312],{"className":55930},[2336]," to ",[54,55933,55935,55952],{"className":55934},[2079],[54,55936,55938],{"className":55937},[2083],[2085,55939,55940],{"xmlns":2087},[2089,55941,55942,55950],{},[2092,55943,55944,55946,55948],{},[2245,55945,2305],{"stretchy":2311},[2098,55947,37470],{},[2245,55949,2312],{"stretchy":2311},[2105,55951,41452],{"encoding":2107},[54,55953,55955],{"className":55954,"ariaHidden":2113},[2112],[54,55956,55958,55961,55964,55967],{"className":55957},[2117],[54,55959],{"className":55960,"style":2325},[2121],[54,55962,2305],{"className":55963},[2497],[54,55965,37470],{"className":55966,"style":2332},[2126,2167],[54,55968,2312],{"className":55969},[2336]," is omitted.",[41415,55972,55973,55976],{},[41430,55974,55975],{},"No higher-state leakage",[41430,55977,55978,55979,56018],{},"Higher levels, including a transmon ",[54,55980,55982,56000],{"className":55981},[2079],[54,55983,55985],{"className":55984},[2083],[2085,55986,55987],{"xmlns":2087},[2089,55988,55989,55997],{},[2092,55990,55991,55993,55995],{},[2245,55992,2305],{"stretchy":2311},[2098,55994,21834],{},[2245,55996,2312],{"stretchy":2311},[2105,55998,55999],{"encoding":2107},"\\lvert f\\rangle",[54,56001,56003],{"className":56002,"ariaHidden":2113},[2112],[54,56004,56006,56009,56012,56015],{"className":56005},[2117],[54,56007],{"className":56008,"style":2325},[2121],[54,56010,2305],{"className":56011},[2497],[54,56013,21834],{"className":56014,"style":26895},[2126,2167],[54,56016,2312],{"className":56017},[2336]," state, are excluded.",[41415,56020,56021,56024],{},[41430,56022,56023],{},"No AC Stark shift",[41430,56025,56026],{},"Drive-induced transition-frequency shifts are excluded.",[41415,56028,56029,56032],{},[41430,56030,56031],{},"No Bloch-Siegert shift",[41430,56033,56034],{},"Counter-rotating corrections are excluded.",[41415,56036,56037,56040],{},[41430,56038,56039],{},"No experimental readout model",[41430,56041,56042],{},"Assignment errors and finite signal-to-noise effects are omitted.",[41415,56044,56045,56048],{},[41430,56046,56047],{},"No pulse-envelope shaping",[41430,56049,56050],{},"Gaussian, DRAG, cosine, and hardware-specific envelopes are omitted.",[13,56052,56053],{},"A hardware-accurate superconducting-qubit simulation requires additional structure, including anharmonic multilevel dynamics, calibrated microwave pulse envelopes, drive phase control, amplitude-dependent frequency shifts, finite thermal populations, leakage channels, and measurement infidelity modeled with density matrix dynamics.",[31,56055,56057],{"id":56056},"notebook-control-knobs","Notebook Control Knobs",[13,56059,56060],{},"The first code section centralizes the main physical and numerical parameters for direct adjustment.",[41409,56062,56063,56076],{},[41412,56064,56065],{},[41415,56066,56067,56070,56074],{},[41418,56068,56069],{},"Parameter",[41418,56071,56073],{"align":56072},"right","Default",[41418,56075,41423],{},[41425,56077,56078,56162,56215,56230,56245,56260,56275,56290,56304],{},[41415,56079,56080,56085,56089],{},[41430,56081,56082],{},[51,56083,56084],{},"F0_GHZ",[41430,56086,56087],{"align":56072},[51,56088,43359],{},[41430,56090,56091,56092,56161],{},"Qubit transition frequency ",[54,56093,56095,56112],{"className":56094},[2079],[54,56096,56098],{"className":56097},[2083],[2085,56099,56100],{"xmlns":2087},[2089,56101,56102,56110],{},[2092,56103,56104],{},[2724,56105,56106,56108],{},[2098,56107,21834],{},[2249,56109,150],{},[2105,56111,41794],{"encoding":2107},[54,56113,56115],{"className":56114,"ariaHidden":2113},[2112],[54,56116,56118,56121],{"className":56117},[2117],[54,56119],{"className":56120,"style":6591},[2121],[54,56122,56124,56127],{"className":56123},[2126],[54,56125,21834],{"className":56126,"style":26895},[2126,2167],[54,56128,56130],{"className":56129},[2135],[54,56131,56133,56153],{"className":56132},[2139,2756],[54,56134,56136,56150],{"className":56135},[2143],[54,56137,56139],{"className":56138,"style":13837},[2147],[54,56140,56141,56144],{"style":41824},[54,56142],{"className":56143,"style":2156},[2155],[54,56145,56147],{"className":56146},[2160,2161,2162,2163],[54,56148,150],{"className":56149},[2126,2163],[54,56151,2781],{"className":56152},[2780],[54,56154,56156],{"className":56155},[2143],[54,56157,56159],{"className":56158,"style":2788},[2147],[54,56160],{}," in GHz",[41415,56163,56164,56169,56174],{},[41430,56165,56166],{},[51,56167,56168],{},"OMEGA_ONRESONANCE_MHZ",[41430,56170,56171],{"align":56072},[51,56172,56173],{},"20.0",[41430,56175,56176,56177,56214],{},"On-resonance Rabi rate ",[54,56178,56180,56199],{"className":56179},[2079],[54,56181,56183],{"className":56182},[2083],[2085,56184,56185],{"xmlns":2087},[2089,56186,56187,56197],{},[2092,56188,56189,56191,56193,56195],{},[2098,56190,42219],{"mathvariant":2304},[2098,56192,194],{"mathvariant":2304},[2249,56194,133],{},[2098,56196,8119],{},[2105,56198,42264],{"encoding":2107},[54,56200,56202],{"className":56201,"ariaHidden":2113},[2112],[54,56203,56205,56208,56211],{"className":56204},[2117],[54,56206],{"className":56207,"style":2325},[2121],[54,56209,42277],{"className":56210},[2126],[54,56212,8119],{"className":56213,"style":2332},[2126,2167]," in MHz",[41415,56216,56217,56222,56227],{},[41430,56218,56219],{},[51,56220,56221],{},"FREQ_MIN_GHZ",[41430,56223,56224],{"align":56072},[51,56225,56226],{},"4.90",[41430,56228,56229],{},"Minimum swept drive frequency in GHz",[41415,56231,56232,56237,56242],{},[41430,56233,56234],{},[51,56235,56236],{},"FREQ_MAX_GHZ",[41430,56238,56239],{"align":56072},[51,56240,56241],{},"5.10",[41430,56243,56244],{},"Maximum swept drive frequency in GHz",[41415,56246,56247,56252,56257],{},[41430,56248,56249],{},[51,56250,56251],{},"N_FREQ",[41430,56253,56254],{"align":56072},[51,56255,56256],{},"401",[41430,56258,56259],{},"Number of drive-frequency samples",[41415,56261,56262,56267,56272],{},[41430,56263,56264],{},[51,56265,56266],{},"DUR_MIN_NS",[41430,56268,56269],{"align":56072},[51,56270,56271],{},"0.0",[41430,56273,56274],{},"Minimum pulse duration in ns",[41415,56276,56277,56282,56287],{},[41430,56278,56279],{},[51,56280,56281],{},"DUR_MAX_NS",[41430,56283,56284],{"align":56072},[51,56285,56286],{},"200.0",[41430,56288,56289],{},"Maximum pulse duration in ns",[41415,56291,56292,56297,56301],{},[41430,56293,56294],{},[51,56295,56296],{},"N_DUR",[41430,56298,56299],{"align":56072},[51,56300,56256],{},[41430,56302,56303],{},"Number of pulse-duration samples",[41415,56305,56306,56311,56316],{},[41430,56307,56308],{},[51,56309,56310],{},"T2STAR_NS",[41430,56312,56313],{"align":56072},[51,56314,56315],{},"500.0",[41430,56317,56318,56319,56322],{},"Optional damping time in ns, with ",[51,56320,56321],{},"None"," disabling damping",[13,56324,56325],{},"Additional cross-section plotting cells expose localized slice controls.",[41409,56327,56328,56338],{},[41412,56329,56330],{},[41415,56331,56332,56334,56336],{},[41418,56333,56069],{},[41418,56335,56073],{"align":56072},[41418,56337,41423],{},[41425,56339,56340,56439],{},[41415,56341,56342,56347,56351],{},[41430,56343,56344],{},[51,56345,56346],{},"F_CROSS_GHZ",[41430,56348,56349],{"align":56072},[51,56350,43359],{},[41430,56352,56353,56354],{},"Drive-frequency slice used to plot ",[54,56355,56357,56381],{"className":56356},[2079],[54,56358,56360],{"className":56359},[2083],[2085,56361,56362],{"xmlns":2087},[2089,56363,56364,56378],{},[2092,56365,56366,56372,56374,56376],{},[2724,56367,56368,56370],{},[2098,56369,2730],{},[2098,56371,2842],{},[2245,56373,129],{"stretchy":2311},[2098,56375,41261],{},[2245,56377,3205],{"stretchy":2311},[2105,56379,56380],{"encoding":2107},"P_e(\\tau)",[54,56382,56384],{"className":56383,"ariaHidden":2113},[2112],[54,56385,56387,56390,56430,56433,56436],{"className":56386},[2117],[54,56388],{"className":56389,"style":2325},[2121],[54,56391,56393,56396],{"className":56392},[2126],[54,56394,2730],{"className":56395,"style":2776},[2126,2167],[54,56397,56399],{"className":56398},[2135],[54,56400,56402,56422],{"className":56401},[2139,2756],[54,56403,56405,56419],{"className":56404},[2143],[54,56406,56408],{"className":56407,"style":4063},[2147],[54,56409,56410,56413],{"style":7193},[54,56411],{"className":56412,"style":2156},[2155],[54,56414,56416],{"className":56415},[2160,2161,2162,2163],[54,56417,2842],{"className":56418},[2126,2167,2163],[54,56420,2781],{"className":56421},[2780],[54,56423,56425],{"className":56424},[2143],[54,56426,56428],{"className":56427,"style":2788},[2147],[54,56429],{},[54,56431,129],{"className":56432},[2497],[54,56434,41261],{"className":56435,"style":41333},[2126,2167],[54,56437,3205],{"className":56438},[2336],[41415,56440,56441,56446,56451],{},[41430,56442,56443],{},[51,56444,56445],{},"T_CROSS_NS",[41430,56447,56448],{"align":56072},[51,56449,56450],{},"100.0",[41430,56452,56453,56454],{},"Pulse-duration slice used to plot ",[54,56455,56457,56481],{"className":56456},[2079],[54,56458,56460],{"className":56459},[2083],[2085,56461,56462],{"xmlns":2087},[2089,56463,56464,56478],{},[2092,56465,56466,56472,56474,56476],{},[2724,56467,56468,56470],{},[2098,56469,2730],{},[2098,56471,2842],{},[2245,56473,129],{"stretchy":2311},[2098,56475,21834],{},[2245,56477,3205],{"stretchy":2311},[2105,56479,56480],{"encoding":2107},"P_e(f)",[54,56482,56484],{"className":56483,"ariaHidden":2113},[2112],[54,56485,56487,56490,56530,56533,56536],{"className":56486},[2117],[54,56488],{"className":56489,"style":2325},[2121],[54,56491,56493,56496],{"className":56492},[2126],[54,56494,2730],{"className":56495,"style":2776},[2126,2167],[54,56497,56499],{"className":56498},[2135],[54,56500,56502,56522],{"className":56501},[2139,2756],[54,56503,56505,56519],{"className":56504},[2143],[54,56506,56508],{"className":56507,"style":4063},[2147],[54,56509,56510,56513],{"style":7193},[54,56511],{"className":56512,"style":2156},[2155],[54,56514,56516],{"className":56515},[2160,2161,2162,2163],[54,56517,2842],{"className":56518},[2126,2167,2163],[54,56520,2781],{"className":56521},[2780],[54,56523,56525],{"className":56524},[2143],[54,56526,56528],{"className":56527,"style":2788},[2147],[54,56529],{},[54,56531,129],{"className":56532},[2497],[54,56534,21834],{"className":56535,"style":26895},[2126,2167],[54,56537,3205],{"className":56538},[2336],[13,56540,56541],{},"Separate FFT processing controls support spectral analysis.",[41409,56543,56544,56554],{},[41412,56545,56546],{},[41415,56547,56548,56550,56552],{},[41418,56549,56069],{},[41418,56551,56073],{"align":56072},[41418,56553,41423],{},[41425,56555,56556,56570,56585,56599,56614,56629],{},[41415,56557,56558,56563,56567],{},[41430,56559,56560],{},[51,56561,56562],{},"FFT_PAD",[41430,56564,56565],{"align":56072},[51,56566,35297],{},[41430,56568,56569],{},"Zero-padding factor used in the Fourier transform",[41415,56571,56572,56577,56582],{},[41430,56573,56574],{},[51,56575,56576],{},"WINDOW",[41430,56578,56579],{"align":56072},[51,56580,56581],{},"\"hann\"",[41430,56583,56584],{},"Window function applied along the pulse-duration axis",[41415,56586,56587,56592,56596],{},[41430,56588,56589],{},[51,56590,56591],{},"DETREND_MEAN",[41430,56593,56594],{"align":56072},[51,56595,13434],{},[41430,56597,56598],{},"Flag that subtracts the mean before the FFT",[41415,56600,56601,56606,56611],{},[41430,56602,56603],{},[51,56604,56605],{},"AMP_MODE",[41430,56607,56608],{"align":56072},[51,56609,56610],{},"\"magnitude\"",[41430,56612,56613],{},"Display mode for magnitude, power, or dB-scaled amplitude",[41415,56615,56616,56621,56626],{},[41430,56617,56618],{},[51,56619,56620],{},"NORM_MODE",[41430,56622,56623],{"align":56072},[51,56624,56625],{},"\"global\"",[41430,56627,56628],{},"Normalization mode applied globally or per drive-frequency column",[41415,56630,56631,56636,56641],{},[41430,56632,56633],{},[51,56634,56635],{},"FREQ_UNITS",[41430,56637,56638],{"align":56072},[51,56639,56640],{},"\"MHz\"",[41430,56642,56643],{},"Frequency units used on the Fourier-frequency axis",[13,56645,56646],{},"Three-dimensional visualization cells add rendering controls for upsampling, colormaps, face-count limits, and viewing angles. These settings change the appearance of the rendered figures while preserving the analytic probability model.",[44,56648,56650],{"className":46,"code":56649,"language":48,"meta":49,"style":49},"# @title Install dependencies\nimport sys\n%pip -q install pyvista pythreejs trame\n",[51,56651,56652,56657,56664],{"__ignoreMap":49},[54,56653,56654],{"class":56,"line":57},[54,56655,56656],{"class":111},"# @title Install dependencies\n",[54,56658,56659,56661],{"class":56,"line":74},[54,56660,61],{"class":60},[54,56662,56663],{"class":64}," sys\n",[54,56665,56666,56668,56671,56673],{"class":56,"line":88},[54,56667,8325],{"class":121},[54,56669,56670],{"class":64},"pip ",[54,56672,1923],{"class":121},[54,56674,56675],{"class":64},"q install pyvista pythreejs trame\n",[44,56677,56679],{"className":46,"code":56678,"language":48,"meta":49,"style":49},"# @title Controls & imports\nimport math\nfrom typing import Optional, Tuple\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n\ntry:\n import pyvista as pv # optional; only for 3D interactive\n PV_AVAILABLE = True\nexcept Exception:\n PV_AVAILABLE = False\n\nplt.rcParams['figure.dpi'] = 250\nplt.rcParams['axes.grid'] = False\n\n# ---------- Control knobs ----------\nF0_GHZ: float = 5.000 # Qubit transition freq f0 [GHz]\nOMEGA_ONRESONANCE_MHZ: float = 20.0 # Ω\u002F2π on-resonance [MHz]\nFREQ_MIN_GHZ: float = 4.90 # Sweep min f [GHz]\nFREQ_MAX_GHZ: float = 5.10 # Sweep max f [GHz]\nN_FREQ: int = 401 # # of frequency points\nDUR_MIN_NS: float = 0.0 # Sweep min duration τ [ns]\nDUR_MAX_NS: float = 200.0 # Sweep max duration τ [ns]\nN_DUR: int = 401 # # of duration points\nT2STAR_NS: Optional[float] = 500.0 # None to disable exp(-τ\u002FT2*)\n\nprint('Controls loaded.')\n",[51,56680,56681,56686,56693,56704,56714,56725,56737,56741,56748,56764,56774,56782,56791,56795,56810,56823,56827,56832,56848,56864,56880,56896,56912,56928,56944,56959,56978,56982],{"__ignoreMap":49},[54,56682,56683],{"class":56,"line":57},[54,56684,56685],{"class":111},"# @title Controls & imports\n",[54,56687,56688,56690],{"class":56,"line":74},[54,56689,61],{"class":60},[54,56691,56692],{"class":64}," math\n",[54,56694,56695,56697,56699,56701],{"class":56,"line":88},[54,56696,77],{"class":60},[54,56698,722],{"class":64},[54,56700,61],{"class":60},[54,56702,56703],{"class":64}," Optional, Tuple\n",[54,56705,56706,56708,56710,56712],{"class":56,"line":101},[54,56707,61],{"class":60},[54,56709,65],{"class":64},[54,56711,68],{"class":60},[54,56713,71],{"class":64},[54,56715,56716,56718,56721,56723],{"class":56,"line":108},[54,56717,61],{"class":60},[54,56719,56720],{"class":64}," matplotlib.pyplot ",[54,56722,68],{"class":60},[54,56724,715],{"class":64},[54,56726,56727,56729,56732,56734],{"class":56,"line":115},[54,56728,61],{"class":60},[54,56730,56731],{"class":64}," pandas ",[54,56733,68],{"class":60},[54,56735,56736],{"class":64}," pd\n",[54,56738,56739],{"class":56,"line":139},[54,56740,105],{"emptyLinePlaceholder":104},[54,56742,56743,56746],{"class":56,"line":159},[54,56744,56745],{"class":60},"try",[54,56747,1284],{"class":64},[54,56749,56750,56753,56756,56758,56761],{"class":56,"line":183},[54,56751,56752],{"class":60}," import",[54,56754,56755],{"class":64}," pyvista ",[54,56757,68],{"class":60},[54,56759,56760],{"class":64}," pv ",[54,56762,56763],{"class":111},"# optional; only for 3D interactive\n",[54,56765,56766,56769,56771],{"class":56,"line":210},[54,56767,56768],{"class":132}," PV_AVAILABLE",[54,56770,979],{"class":121},[54,56772,56773],{"class":132}," True\n",[54,56775,56776,56779],{"class":56,"line":227},[54,56777,56778],{"class":60},"except",[54,56780,56781],{"class":64}," Exception:\n",[54,56783,56784,56786,56788],{"class":56,"line":232},[54,56785,56768],{"class":132},[54,56787,979],{"class":121},[54,56789,56790],{"class":132}," False\n",[54,56792,56793],{"class":56,"line":246},[54,56794,105],{"emptyLinePlaceholder":104},[54,56796,56797,56800,56803,56805,56807],{"class":56,"line":259},[54,56798,56799],{"class":64},"plt.rcParams[",[54,56801,56802],{"class":279},"'figure.dpi'",[54,56804,7762],{"class":64},[54,56806,122],{"class":121},[54,56808,56809],{"class":132}," 250\n",[54,56811,56812,56814,56817,56819,56821],{"class":56,"line":264},[54,56813,56799],{"class":64},[54,56815,56816],{"class":279},"'axes.grid'",[54,56818,7762],{"class":64},[54,56820,122],{"class":121},[54,56822,56790],{"class":132},[54,56824,56825],{"class":56,"line":270},[54,56826,105],{"emptyLinePlaceholder":104},[54,56828,56829],{"class":56,"line":285},[54,56830,56831],{"class":111},"# ---------- Control knobs ----------\n",[54,56833,56834,56836,56838,56840,56842,56845],{"class":56,"line":297},[54,56835,56084],{"class":132},[54,56837,959],{"class":64},[54,56839,962],{"class":121},[54,56841,979],{"class":121},[54,56843,56844],{"class":132}," 5.000",[54,56846,56847],{"class":111}," # Qubit transition freq f0 [GHz]\n",[54,56849,56850,56852,56854,56856,56858,56861],{"class":56,"line":310},[54,56851,56168],{"class":132},[54,56853,959],{"class":64},[54,56855,962],{"class":121},[54,56857,979],{"class":121},[54,56859,56860],{"class":132}," 20.0",[54,56862,56863],{"class":111}," # Ω\u002F2π on-resonance [MHz]\n",[54,56865,56866,56868,56870,56872,56874,56877],{"class":56,"line":324},[54,56867,56221],{"class":132},[54,56869,959],{"class":64},[54,56871,962],{"class":121},[54,56873,979],{"class":121},[54,56875,56876],{"class":132}," 4.90",[54,56878,56879],{"class":111}," # Sweep min f [GHz]\n",[54,56881,56882,56884,56886,56888,56890,56893],{"class":56,"line":333},[54,56883,56236],{"class":132},[54,56885,959],{"class":64},[54,56887,962],{"class":121},[54,56889,979],{"class":121},[54,56891,56892],{"class":132}," 5.10",[54,56894,56895],{"class":111}," # Sweep max f [GHz]\n",[54,56897,56898,56900,56902,56904,56906,56909],{"class":56,"line":345},[54,56899,56251],{"class":132},[54,56901,959],{"class":64},[54,56903,976],{"class":121},[54,56905,979],{"class":121},[54,56907,56908],{"class":132}," 401",[54,56910,56911],{"class":111}," # # of frequency points\n",[54,56913,56914,56916,56918,56920,56922,56925],{"class":56,"line":358},[54,56915,56266],{"class":132},[54,56917,959],{"class":64},[54,56919,962],{"class":121},[54,56921,979],{"class":121},[54,56923,56924],{"class":132}," 0.0",[54,56926,56927],{"class":111}," # Sweep min duration τ [ns]\n",[54,56929,56930,56932,56934,56936,56938,56941],{"class":56,"line":908},[54,56931,56281],{"class":132},[54,56933,959],{"class":64},[54,56935,962],{"class":121},[54,56937,979],{"class":121},[54,56939,56940],{"class":132}," 200.0",[54,56942,56943],{"class":111}," # Sweep max duration τ [ns]\n",[54,56945,56946,56948,56950,56952,56954,56956],{"class":56,"line":913},[54,56947,56296],{"class":132},[54,56949,959],{"class":64},[54,56951,976],{"class":121},[54,56953,979],{"class":121},[54,56955,56908],{"class":132},[54,56957,56958],{"class":111}," # # of duration points\n",[54,56960,56961,56963,56966,56968,56970,56972,56975],{"class":56,"line":918},[54,56962,56310],{"class":132},[54,56964,56965],{"class":64},": Optional[",[54,56967,962],{"class":121},[54,56969,7762],{"class":64},[54,56971,122],{"class":121},[54,56973,56974],{"class":132}," 500.0",[54,56976,56977],{"class":111}," # None to disable exp(-τ\u002FT2*)\n",[54,56979,56980],{"class":56,"line":923},[54,56981,105],{"emptyLinePlaceholder":104},[54,56983,56984,56986,56988,56991],{"class":56,"line":935},[54,56985,8081],{"class":121},[54,56987,129],{"class":64},[54,56989,56990],{"class":279},"'Controls loaded.'",[54,56992,136],{"class":64},[44,56994,56997],{"className":56995,"code":56996,"language":6595,"meta":49},[8087],"Controls loaded.\n",[51,56998,56996],{"__ignoreMap":49},[44,57000,57002],{"className":46,"code":57001,"language":48,"meta":49,"style":49},"def rabi_probability(\n f_ghz: np.ndarray,\n tau_ns: np.ndarray,\n f0_ghz: float,\n omega_onres_mhz: float,\n t2star_ns: Optional[float] = None,\n) -> np.ndarray:\n \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n\n Args:\n f_ghz: 1D array of drive frequencies [GHz].\n tau_ns: 1D array of pulse durations [ns].\n f0_ghz: Qubit frequency on resonance [GHz].\n omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n\n Returns:\n 2D array P_e with shape (len(tau_ns), len(f_ghz)).\n \"\"\"\n two_pi = 2.0 * math.pi\n f = f_ghz * 1e9\n f0 = f0_ghz * 1e9\n tau = tau_ns * 1e-9\n omega = two_pi * omega_onres_mhz * 1e6 # rad\u002Fs\n\n F, TAU = np.meshgrid(f, tau, indexing='xy')\n Delta = two_pi * (F - f0) # rad\u002Fs\n Omega_R = np.sqrt(omega**2 + Delta**2)\n\n with np.errstate(divide='ignore', invalid='ignore'):\n amp = (omega \u002F Omega_R) ** 2\n phase = 0.5 * Omega_R * TAU\n Pe = amp * np.sin(phase) ** 2\n\n if t2star_ns is not None and t2star_ns > 0.0:\n decay = np.exp(-TAU \u002F (t2star_ns * 1e-9))\n Pe = Pe * decay\n\n return np.clip(Pe, 0.0, 1.0)\n",[51,57003,57004,57013,57020,57027,57038,57049,57067,57072,57077,57081,57085,57090,57095,57100,57105,57110,57114,57118,57123,57127,57142,57157,57171,57186,57209,57213,57241,57263,57292,57296,57327,57347,57367,57390,57394,57418,57448,57462,57466],{"__ignoreMap":49},[54,57005,57006,57008,57011],{"class":56,"line":57},[54,57007,926],{"class":60},[54,57009,57010],{"class":125}," rabi_probability",[54,57012,932],{"class":64},[54,57014,57015,57018],{"class":56,"line":74},[54,57016,57017],{"class":938}," f_ghz",[54,57019,942],{"class":64},[54,57021,57022,57025],{"class":56,"line":88},[54,57023,57024],{"class":938}," tau_ns",[54,57026,942],{"class":64},[54,57028,57029,57032,57034,57036],{"class":56,"line":101},[54,57030,57031],{"class":938}," f0_ghz",[54,57033,959],{"class":64},[54,57035,962],{"class":121},[54,57037,965],{"class":64},[54,57039,57040,57043,57045,57047],{"class":56,"line":108},[54,57041,57042],{"class":938}," omega_onres_mhz",[54,57044,959],{"class":64},[54,57046,962],{"class":121},[54,57048,965],{"class":64},[54,57050,57051,57054,57056,57058,57060,57062,57065],{"class":56,"line":115},[54,57052,57053],{"class":938}," t2star_ns",[54,57055,56965],{"class":64},[54,57057,962],{"class":121},[54,57059,7762],{"class":64},[54,57061,122],{"class":121},[54,57063,57064],{"class":132}," None",[54,57066,965],{"class":64},[54,57068,57069],{"class":56,"line":139},[54,57070,57071],{"class":64},") -> np.ndarray:\n",[54,57073,57074],{"class":56,"line":159},[54,57075,57076],{"class":279}," \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n",[54,57078,57079],{"class":56,"line":183},[54,57080,105],{"emptyLinePlaceholder":104},[54,57082,57083],{"class":56,"line":210},[54,57084,1041],{"class":279},[54,57086,57087],{"class":56,"line":227},[54,57088,57089],{"class":279}," f_ghz: 1D array of drive frequencies [GHz].\n",[54,57091,57092],{"class":56,"line":232},[54,57093,57094],{"class":279}," tau_ns: 1D array of pulse durations [ns].\n",[54,57096,57097],{"class":56,"line":246},[54,57098,57099],{"class":279}," f0_ghz: Qubit frequency on resonance [GHz].\n",[54,57101,57102],{"class":56,"line":259},[54,57103,57104],{"class":279}," omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n",[54,57106,57107],{"class":56,"line":264},[54,57108,57109],{"class":279}," t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n",[54,57111,57112],{"class":56,"line":270},[54,57113,105],{"emptyLinePlaceholder":104},[54,57115,57116],{"class":56,"line":285},[54,57117,1112],{"class":279},[54,57119,57120],{"class":56,"line":297},[54,57121,57122],{"class":279}," 2D array P_e with shape (len(tau_ns), len(f_ghz)).\n",[54,57124,57125],{"class":56,"line":310},[54,57126,1024],{"class":279},[54,57128,57129,57132,57134,57137,57139],{"class":56,"line":324},[54,57130,57131],{"class":64}," two_pi ",[54,57133,122],{"class":121},[54,57135,57136],{"class":132}," 2.0",[54,57138,7785],{"class":121},[54,57140,57141],{"class":64}," math.pi\n",[54,57143,57144,57147,57149,57152,57154],{"class":56,"line":333},[54,57145,57146],{"class":64}," f ",[54,57148,122],{"class":121},[54,57150,57151],{"class":64}," f_ghz ",[54,57153,1947],{"class":121},[54,57155,57156],{"class":132}," 1e9\n",[54,57158,57159,57162,57164,57167,57169],{"class":56,"line":345},[54,57160,57161],{"class":64}," f0 ",[54,57163,122],{"class":121},[54,57165,57166],{"class":64}," f0_ghz ",[54,57168,1947],{"class":121},[54,57170,57156],{"class":132},[54,57172,57173,57176,57178,57181,57183],{"class":56,"line":358},[54,57174,57175],{"class":64}," tau ",[54,57177,122],{"class":121},[54,57179,57180],{"class":64}," tau_ns ",[54,57182,1947],{"class":121},[54,57184,57185],{"class":132}," 1e-9\n",[54,57187,57188,57191,57193,57196,57198,57201,57203,57206],{"class":56,"line":908},[54,57189,57190],{"class":64}," omega ",[54,57192,122],{"class":121},[54,57194,57195],{"class":64}," two_pi ",[54,57197,1947],{"class":121},[54,57199,57200],{"class":64}," omega_onres_mhz ",[54,57202,1947],{"class":121},[54,57204,57205],{"class":132}," 1e6",[54,57207,57208],{"class":111}," # rad\u002Fs\n",[54,57210,57211],{"class":56,"line":913},[54,57212,105],{"emptyLinePlaceholder":104},[54,57214,57215,57218,57221,57223,57225,57228,57231,57234,57236,57239],{"class":56,"line":918},[54,57216,57217],{"class":64}," F, ",[54,57219,57220],{"class":132},"TAU",[54,57222,979],{"class":121},[54,57224,1172],{"class":64},[54,57226,57227],{"class":125},"meshgrid",[54,57229,57230],{"class":64},"(f, tau, ",[54,57232,57233],{"class":1711},"indexing",[54,57235,122],{"class":121},[54,57237,57238],{"class":279},"'xy'",[54,57240,136],{"class":64},[54,57242,57243,57246,57248,57250,57252,57255,57257,57260],{"class":56,"line":923},[54,57244,57245],{"class":64}," Delta ",[54,57247,122],{"class":121},[54,57249,57195],{"class":64},[54,57251,1947],{"class":121},[54,57253,57254],{"class":64}," (F ",[54,57256,1923],{"class":121},[54,57258,57259],{"class":64}," f0) ",[54,57261,57262],{"class":111},"# rad\u002Fs\n",[54,57264,57265,57268,57270,57272,57274,57277,57279,57281,57283,57286,57288,57290],{"class":56,"line":935},[54,57266,57267],{"class":64}," Omega_R ",[54,57269,122],{"class":121},[54,57271,1172],{"class":64},[54,57273,9375],{"class":125},[54,57275,57276],{"class":64},"(omega",[54,57278,1933],{"class":121},[54,57280,133],{"class":132},[54,57282,7869],{"class":121},[54,57284,57285],{"class":64}," Delta",[54,57287,1933],{"class":121},[54,57289,133],{"class":132},[54,57291,136],{"class":64},[54,57293,57294],{"class":56,"line":945},[54,57295,105],{"emptyLinePlaceholder":104},[54,57297,57298,57301,57303,57306,57308,57311,57313,57316,57318,57321,57323,57325],{"class":56,"line":953},[54,57299,57300],{"class":60}," with",[54,57302,1172],{"class":64},[54,57304,57305],{"class":125},"errstate",[54,57307,129],{"class":64},[54,57309,57310],{"class":1711},"divide",[54,57312,122],{"class":121},[54,57314,57315],{"class":279},"'ignore'",[54,57317,171],{"class":64},[54,57319,57320],{"class":1711},"invalid",[54,57322,122],{"class":121},[54,57324,57315],{"class":279},[54,57326,1439],{"class":64},[54,57328,57329,57332,57334,57337,57339,57342,57344],{"class":56,"line":968},[54,57330,57331],{"class":64}," amp ",[54,57333,122],{"class":121},[54,57335,57336],{"class":64}," (omega ",[54,57338,194],{"class":121},[54,57340,57341],{"class":64}," Omega_R) ",[54,57343,1933],{"class":121},[54,57345,57346],{"class":132}," 2\n",[54,57348,57349,57352,57354,57357,57359,57362,57364],{"class":56,"line":987},[54,57350,57351],{"class":64}," phase ",[54,57353,122],{"class":121},[54,57355,57356],{"class":132}," 0.5",[54,57358,7785],{"class":121},[54,57360,57361],{"class":64}," Omega_R ",[54,57363,1947],{"class":121},[54,57365,57366],{"class":132}," TAU\n",[54,57368,57369,57372,57374,57377,57379,57381,57383,57386,57388],{"class":56,"line":1005},[54,57370,57371],{"class":64}," Pe ",[54,57373,122],{"class":121},[54,57375,57376],{"class":64}," amp ",[54,57378,1947],{"class":121},[54,57380,1172],{"class":64},[54,57382,46278],{"class":125},[54,57384,57385],{"class":64},"(phase) ",[54,57387,1933],{"class":121},[54,57389,57346],{"class":132},[54,57391,57392],{"class":56,"line":1021},[54,57393,105],{"emptyLinePlaceholder":104},[54,57395,57396,57398,57401,57403,57405,57407,57410,57412,57414,57416],{"class":56,"line":1027},[54,57397,1273],{"class":60},[54,57399,57400],{"class":64}," t2star_ns ",[54,57402,36565],{"class":60},[54,57404,21536],{"class":60},[54,57406,57064],{"class":132},[54,57408,57409],{"class":60}," and",[54,57411,57400],{"class":64},[54,57413,1207],{"class":121},[54,57415,56924],{"class":132},[54,57417,1284],{"class":64},[54,57419,57420,57423,57425,57427,57429,57431,57433,57435,57438,57441,57443,57446],{"class":56,"line":1033},[54,57421,57422],{"class":64}," decay ",[54,57424,122],{"class":121},[54,57426,1172],{"class":64},[54,57428,23625],{"class":125},[54,57430,129],{"class":64},[54,57432,1923],{"class":121},[54,57434,57220],{"class":132},[54,57436,57437],{"class":121}," \u002F",[54,57439,57440],{"class":64}," (t2star_ns ",[54,57442,1947],{"class":121},[54,57444,57445],{"class":132}," 1e-9",[54,57447,22096],{"class":64},[54,57449,57450,57452,57454,57457,57459],{"class":56,"line":1038},[54,57451,57371],{"class":64},[54,57453,122],{"class":121},[54,57455,57456],{"class":64}," Pe ",[54,57458,1947],{"class":121},[54,57460,57461],{"class":64}," decay\n",[54,57463,57464],{"class":56,"line":1044},[54,57465,105],{"emptyLinePlaceholder":104},[54,57467,57468,57470,57472,57475,57478,57480,57482,57485],{"class":56,"line":1050},[54,57469,2060],{"class":60},[54,57471,1172],{"class":64},[54,57473,57474],{"class":125},"clip",[54,57476,57477],{"class":64},"(Pe, ",[54,57479,56271],{"class":132},[54,57481,171],{"class":64},[54,57483,57484],{"class":132},"1.0",[54,57486,136],{"class":64},[44,57488,57490],{"className":46,"code":57489,"language":48,"meta":49,"style":49},"freq_ghz = np.linspace(FREQ_MIN_GHZ, FREQ_MAX_GHZ, N_FREQ)\ndur_ns = np.linspace(DUR_MIN_NS, DUR_MAX_NS, N_DUR)\nPe = rabi_probability(freq_ghz, dur_ns, F0_GHZ, OMEGA_ONRESONANCE_MHZ, T2STAR_NS)\nprint('Grid shape (τ × f):', Pe.shape)\n",[51,57491,57492,57518,57543,57567],{"__ignoreMap":49},[54,57493,57494,57497,57499,57501,57504,57506,57508,57510,57512,57514,57516],{"class":56,"line":57},[54,57495,57496],{"class":64},"freq_ghz ",[54,57498,122],{"class":121},[54,57500,1172],{"class":64},[54,57502,57503],{"class":125},"linspace",[54,57505,129],{"class":64},[54,57507,56221],{"class":132},[54,57509,171],{"class":64},[54,57511,56236],{"class":132},[54,57513,171],{"class":64},[54,57515,56251],{"class":132},[54,57517,136],{"class":64},[54,57519,57520,57523,57525,57527,57529,57531,57533,57535,57537,57539,57541],{"class":56,"line":74},[54,57521,57522],{"class":64},"dur_ns ",[54,57524,122],{"class":121},[54,57526,1172],{"class":64},[54,57528,57503],{"class":125},[54,57530,129],{"class":64},[54,57532,56266],{"class":132},[54,57534,171],{"class":64},[54,57536,56281],{"class":132},[54,57538,171],{"class":64},[54,57540,56296],{"class":132},[54,57542,136],{"class":64},[54,57544,57545,57548,57550,57552,57555,57557,57559,57561,57563,57565],{"class":56,"line":88},[54,57546,57547],{"class":64},"Pe ",[54,57549,122],{"class":121},[54,57551,57010],{"class":125},[54,57553,57554],{"class":64},"(freq_ghz, dur_ns, ",[54,57556,56084],{"class":132},[54,57558,171],{"class":64},[54,57560,56168],{"class":132},[54,57562,171],{"class":64},[54,57564,56310],{"class":132},[54,57566,136],{"class":64},[54,57568,57569,57571,57573,57576],{"class":56,"line":101},[54,57570,8081],{"class":121},[54,57572,129],{"class":64},[54,57574,57575],{"class":279},"'Grid shape (τ × f):'",[54,57577,57578],{"class":64},", Pe.shape)\n",[44,57580,57583],{"className":57581,"code":57582,"language":6595,"meta":49},[8087],"Grid shape (τ × f): (401, 401)\n",[51,57584,57582],{"__ignoreMap":49},[44,57586,57588],{"className":46,"code":57587,"language":48,"meta":49,"style":49},"controls_table = pd.DataFrame(\n [\n ['f0 (GHz)', F0_GHZ],\n ['Ω\u002F2π on-res (MHz)', OMEGA_ONRESONANCE_MHZ],\n ['f sweep min (GHz)', FREQ_MIN_GHZ],\n ['f sweep max (GHz)', FREQ_MAX_GHZ],\n ['N_f', N_FREQ],\n ['τ sweep min (ns)', DUR_MIN_NS],\n ['τ sweep max (ns)', DUR_MAX_NS],\n ['N_τ', N_DUR],\n ['T2* (ns; None=off)', T2STAR_NS],\n ],\n columns=['Parameter', 'Value']\n)\ncontrols_table\n",[51,57589,57590,57605,57610,57625,57638,57651,57664,57677,57690,57703,57716,57729,57734,57753,57757],{"__ignoreMap":49},[54,57591,57592,57595,57597,57600,57603],{"class":56,"line":57},[54,57593,57594],{"class":64},"controls_table ",[54,57596,122],{"class":121},[54,57598,57599],{"class":64}," pd.",[54,57601,57602],{"class":125},"DataFrame",[54,57604,932],{"class":64},[54,57606,57607],{"class":56,"line":74},[54,57608,57609],{"class":64}," [\n",[54,57611,57612,57615,57618,57620,57622],{"class":56,"line":88},[54,57613,57614],{"class":64}," [",[54,57616,57617],{"class":279},"'f0 (GHz)'",[54,57619,171],{"class":64},[54,57621,56084],{"class":132},[54,57623,57624],{"class":64},"],\n",[54,57626,57627,57629,57632,57634,57636],{"class":56,"line":101},[54,57628,57614],{"class":64},[54,57630,57631],{"class":279},"'Ω\u002F2π on-res (MHz)'",[54,57633,171],{"class":64},[54,57635,56168],{"class":132},[54,57637,57624],{"class":64},[54,57639,57640,57642,57645,57647,57649],{"class":56,"line":108},[54,57641,57614],{"class":64},[54,57643,57644],{"class":279},"'f sweep min (GHz)'",[54,57646,171],{"class":64},[54,57648,56221],{"class":132},[54,57650,57624],{"class":64},[54,57652,57653,57655,57658,57660,57662],{"class":56,"line":115},[54,57654,57614],{"class":64},[54,57656,57657],{"class":279},"'f sweep max (GHz)'",[54,57659,171],{"class":64},[54,57661,56236],{"class":132},[54,57663,57624],{"class":64},[54,57665,57666,57668,57671,57673,57675],{"class":56,"line":139},[54,57667,57614],{"class":64},[54,57669,57670],{"class":279},"'N_f'",[54,57672,171],{"class":64},[54,57674,56251],{"class":132},[54,57676,57624],{"class":64},[54,57678,57679,57681,57684,57686,57688],{"class":56,"line":159},[54,57680,57614],{"class":64},[54,57682,57683],{"class":279},"'τ sweep min (ns)'",[54,57685,171],{"class":64},[54,57687,56266],{"class":132},[54,57689,57624],{"class":64},[54,57691,57692,57694,57697,57699,57701],{"class":56,"line":183},[54,57693,57614],{"class":64},[54,57695,57696],{"class":279},"'τ sweep max (ns)'",[54,57698,171],{"class":64},[54,57700,56281],{"class":132},[54,57702,57624],{"class":64},[54,57704,57705,57707,57710,57712,57714],{"class":56,"line":210},[54,57706,57614],{"class":64},[54,57708,57709],{"class":279},"'N_τ'",[54,57711,171],{"class":64},[54,57713,56296],{"class":132},[54,57715,57624],{"class":64},[54,57717,57718,57720,57723,57725,57727],{"class":56,"line":227},[54,57719,57614],{"class":64},[54,57721,57722],{"class":279},"'T2* (ns; None=off)'",[54,57724,171],{"class":64},[54,57726,56310],{"class":132},[54,57728,57624],{"class":64},[54,57730,57731],{"class":56,"line":232},[54,57732,57733],{"class":64}," ],\n",[54,57735,57736,57739,57741,57743,57746,57748,57751],{"class":56,"line":246},[54,57737,57738],{"class":1711}," columns",[54,57740,122],{"class":121},[54,57742,11804],{"class":64},[54,57744,57745],{"class":279},"'Parameter'",[54,57747,171],{"class":64},[54,57749,57750],{"class":279},"'Value'",[54,57752,1350],{"class":64},[54,57754,57755],{"class":56,"line":259},[54,57756,136],{"class":64},[54,57758,57759],{"class":56,"line":264},[54,57760,57761],{"class":64},"controls_table\n",[44,57763,57766],{"className":57764,"code":57765,"language":6595,"meta":49},[8087],"Parameter Value\n0 f0 (GHz) 5.0\n1 Ω\u002F2π on-res (MHz) 20.0\n2 f sweep min (GHz) 4.9\n3 f sweep max (GHz) 5.1\n4 N_f 401.0\n5 τ sweep min (ns) 0.0\n6 τ sweep max (ns) 200.0\n7 N_τ 401.0\n8 T2* (ns; None=off) 500.0\n",[51,57767,57765],{"__ignoreMap":49},[44,57769,57771],{"className":46,"code":57770,"language":48,"meta":49,"style":49},"fig, ax = plt.subplots(figsize=(7, 5))\nc = ax.pcolormesh(freq_ghz, dur_ns, Pe, shading='auto')\ncb = fig.colorbar(c, ax=ax, label=r'$P_e$')\nax.set_xlabel('Drive frequency f (GHz)')\nax.set_ylabel('Pulse duration τ (ns)')\nax.set_title('Rabi excited-state probability $P_e(f,\\\\,\\\\tau)$')\nfig.tight_layout()\nplt.show()\n",[51,57772,57773,57805,57831,57867,57882,57896,57920,57930],{"__ignoreMap":49},[54,57774,57775,57778,57780,57783,57786,57788,57791,57793,57795,57798,57800,57803],{"class":56,"line":57},[54,57776,57777],{"class":64},"fig, ax ",[54,57779,122],{"class":121},[54,57781,57782],{"class":64}," plt.",[54,57784,57785],{"class":125},"subplots",[54,57787,129],{"class":64},[54,57789,57790],{"class":1711},"figsize",[54,57792,122],{"class":121},[54,57794,129],{"class":64},[54,57796,57797],{"class":132},"7",[54,57799,171],{"class":64},[54,57801,57802],{"class":132},"5",[54,57804,22096],{"class":64},[54,57806,57807,57810,57812,57815,57818,57821,57824,57826,57829],{"class":56,"line":74},[54,57808,57809],{"class":64},"c ",[54,57811,122],{"class":121},[54,57813,57814],{"class":64}," ax.",[54,57816,57817],{"class":125},"pcolormesh",[54,57819,57820],{"class":64},"(freq_ghz, dur_ns, Pe, ",[54,57822,57823],{"class":1711},"shading",[54,57825,122],{"class":121},[54,57827,57828],{"class":279},"'auto'",[54,57830,136],{"class":64},[54,57832,57833,57836,57838,57841,57844,57847,57850,57852,57855,57857,57859,57861,57865],{"class":56,"line":88},[54,57834,57835],{"class":64},"cb ",[54,57837,122],{"class":121},[54,57839,57840],{"class":64}," fig.",[54,57842,57843],{"class":125},"colorbar",[54,57845,57846],{"class":64},"(c, ",[54,57848,57849],{"class":1711},"ax",[54,57851,122],{"class":121},[54,57853,57854],{"class":64},"ax, ",[54,57856,20568],{"class":1711},[54,57858,122],{"class":121},[54,57860,1772],{"class":60},[54,57862,57864],{"class":57863},"sVyAn","'$P_e$'",[54,57866,136],{"class":64},[54,57868,57869,57872,57875,57877,57880],{"class":56,"line":101},[54,57870,57871],{"class":64},"ax.",[54,57873,57874],{"class":125},"set_xlabel",[54,57876,129],{"class":64},[54,57878,57879],{"class":279},"'Drive frequency f (GHz)'",[54,57881,136],{"class":64},[54,57883,57884,57886,57889,57891,57894],{"class":56,"line":108},[54,57885,57871],{"class":64},[54,57887,57888],{"class":125},"set_ylabel",[54,57890,129],{"class":64},[54,57892,57893],{"class":279},"'Pulse duration τ (ns)'",[54,57895,136],{"class":64},[54,57897,57898,57900,57903,57905,57908,57911,57913,57915,57918],{"class":56,"line":115},[54,57899,57871],{"class":64},[54,57901,57902],{"class":125},"set_title",[54,57904,129],{"class":64},[54,57906,57907],{"class":279},"'Rabi excited-state probability $P_e(f,",[54,57909,57910],{"class":121},"\\\\",[54,57912,2375],{"class":279},[54,57914,57910],{"class":121},[54,57916,57917],{"class":279},"tau)$'",[54,57919,136],{"class":64},[54,57921,57922,57925,57928],{"class":56,"line":139},[54,57923,57924],{"class":64},"fig.",[54,57926,57927],{"class":125},"tight_layout",[54,57929,330],{"class":64},[54,57931,57932,57934,57936],{"class":56,"line":159},[54,57933,24932],{"class":64},[54,57935,25169],{"class":125},[54,57937,330],{"class":64},[380,57939],{"alt":57940,"src":57941},"Output 1 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.webp",[44,57943,57945],{"className":46,"code":57944,"language":48,"meta":49,"style":49},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n\nimport importlib.util\nimport subprocess\nimport sys\n\n# -------------------------------------------------------------------------\n# Control knobs\n# -------------------------------------------------------------------------\nINSTALL_BLOCH_DEPENDENCIES = True\n\nBLOCH_REQUIRED_PACKAGES = {\n \"numpy\": \"numpy\",\n \"matplotlib\": \"matplotlib\",\n \"scipy\": \"scipy\",\n \"qutip\": \"qutip\",\n}\n\nBLOCH_F0_GHZ = globals().get(\"F0_GHZ\", 5.000)\nBLOCH_OMEGA_ONRESONANCE_MHZ = globals().get(\n \"OMEGA_ONRESONANCE_MHZ\",\n 20.0,\n)\n\nBLOCH_RESONANT_DETUNING_MHZ = 0.0\nBLOCH_DETUNED_DETUNING_MHZ = 40.0\n\n# Use None to automatically plot one resonant pi pulse.\nBLOCH_DURATION_NS = None\n\nBLOCH_NUM_TIME_POINTS = 300\nBLOCH_FIGSIZE = (8.8, 7.8)\nBLOCH_DPI = 250\nBLOCH_VIEW = [-60, 25]\n\nBLOCH_SHOW_CONTROL_AXIS_ARROWS = True\nBLOCH_SHOW_ARROW_TIP_LABELS = True\nBLOCH_SHOW_ENDPOINT_MARKERS = True\nBLOCH_SHOW_LEGEND = True\nBLOCH_PRINT_SUMMARY = True\n\nCOLOR_RESONANT_TRAJECTORY = \"tab:blue\"\nCOLOR_DETUNED_TRAJECTORY = \"tab:orange\"\nCOLOR_RESONANT_AXIS = \"tab:green\"\nCOLOR_DETUNED_AXIS = \"tab:red\"\nCOLOR_INITIAL_STATE = \"black\"\n\nTRAJECTORY_LINEWIDTH = 2.4\nCONTROL_AXIS_LINEWIDTH = 2.2\nENDPOINT_MARKER_SIZE = 45\nARROW_LENGTH_RATIO = 0.12\nARROW_LABEL_SCALE = 1.16\n\nLEGEND_LOCATION = \"upper left\"\nLEGEND_BBOX_TO_ANCHOR = (1.02, 1.02)\n\n\ndef package_is_available(package_name: str) -> bool:\n \"\"\"Return True when a package can be imported.\"\"\"\n return importlib.util.find_spec(package_name) is not None\n\n\ndef install_missing_packages(\n required_packages: dict[str, str],\n) -> None:\n \"\"\"Install missing notebook packages using uv pip with a pip fallback.\"\"\"\n if not INSTALL_BLOCH_DEPENDENCIES:\n return\n\n missing_package_names = [\n pip_name\n for import_name, pip_name in required_packages.items()\n if not package_is_available(import_name)\n ]\n\n if not missing_package_names:\n return\n\n try:\n subprocess.run(\n [sys.executable, \"-m\", \"pip\", \"install\", \"-q\", \"uv\"],\n check=True,\n )\n subprocess.run(\n [\n sys.executable,\n \"-m\",\n \"uv\",\n \"pip\",\n \"install\",\n \"--system\",\n \"-q\",\n *missing_package_names,\n ],\n check=True,\n )\n except (subprocess.CalledProcessError, FileNotFoundError):\n subprocess.run(\n [\n sys.executable,\n \"-m\",\n \"pip\",\n \"install\",\n \"-q\",\n *missing_package_names,\n ],\n check=True,\n )\n\n\ninstall_missing_packages(BLOCH_REQUIRED_PACKAGES)\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport qutip as qt\nfrom matplotlib.lines import Line2D\n\ntry:\n from IPython import get_ipython\n\n ipython = get_ipython()\n if ipython is not None:\n ipython.run_line_magic(\"matplotlib\", \"inline\")\nexcept ImportError:\n pass\n\nplt.rcParams.update(\n {\n \"figure.dpi\": BLOCH_DPI,\n }\n)\n\n\ndef get_pi_pulse_duration_ns(omega_onresonance_mhz: float) -> float:\n \"\"\"Return the resonant pi-pulse duration in nanoseconds.\"\"\"\n return 1.0e3 \u002F (2.0 * omega_onresonance_mhz)\n\n\ndef solve_rwa_bloch_trajectory(\n detuning_mhz: float,\n duration_ns: float,\n num_time_points: int,\n) -> dict[str, np.ndarray | float]:\n \"\"\"Solve the RWA two-level trajectory and return Bloch observables.\"\"\"\n omega_rad_s = 2.0 * np.pi * BLOCH_OMEGA_ONRESONANCE_MHZ * 1.0e6\n delta_rad_s = 2.0 * np.pi * detuning_mhz * 1.0e6\n omega_generalized_rad_s = np.hypot(omega_rad_s, delta_rad_s)\n\n time_s = np.linspace(0.0, duration_ns * 1.0e-9, num_time_points)\n\n ground_state = qt.basis(2, 0)\n excited_state = qt.basis(2, 1)\n excited_projector = excited_state * excited_state.dag()\n\n hamiltonian = 0.5 * (\n delta_rad_s * qt.sigmaz()\n + omega_rad_s * qt.sigmax()\n )\n\n result = qt.sesolve(\n hamiltonian,\n ground_state,\n time_s,\n )\n\n x_expectation = np.real(np.asarray(qt.expect(qt.sigmax(), result.states)))\n y_expectation = np.real(np.asarray(qt.expect(qt.sigmay(), result.states)))\n z_expectation = np.real(np.asarray(qt.expect(qt.sigmaz(), result.states)))\n\n excited_probability = np.real(\n np.asarray(qt.expect(excited_projector, result.states))\n )\n\n bloch_vectors = np.vstack(\n [\n x_expectation,\n y_expectation,\n z_expectation,\n ]\n )\n\n control_axis = np.array(\n [\n omega_rad_s \u002F omega_generalized_rad_s,\n 0.0,\n delta_rad_s \u002F omega_generalized_rad_s,\n ]\n )\n\n return {\n \"time_ns\": time_s * 1.0e9,\n \"vectors\": bloch_vectors,\n \"excited_probability\": excited_probability,\n \"control_axis\": control_axis,\n \"generalized_rabi_mhz\": (\n omega_generalized_rad_s \u002F (2.0 * np.pi * 1.0e6)\n ),\n }\n\n\ndef render_empty_bloch_sphere(\n bloch_sphere: qt.Bloch,\n fallback_figure: plt.Figure,\n):\n \"\"\"Render an empty QuTiP Bloch sphere and return its Matplotlib axis.\"\"\"\n if hasattr(bloch_sphere, \"render\"):\n bloch_sphere.render()\n elif hasattr(bloch_sphere, \"make_sphere\"):\n bloch_sphere.make_sphere()\n else:\n bloch_sphere.show()\n\n axis = getattr(bloch_sphere, \"axes\", None)\n\n if axis is None and fallback_figure.axes:\n axis = fallback_figure.axes[-1]\n\n if axis is None:\n raise RuntimeError(\"Could not access the Matplotlib Bloch axis.\")\n\n return axis\n\n\ndef draw_bloch_trajectory(\n axis,\n vectors: np.ndarray,\n color: str,\n label: str,\n) -> None:\n \"\"\"Draw a Bloch-vector trajectory on an existing 3D axis.\"\"\"\n axis.plot(\n vectors[0],\n vectors[1],\n vectors[2],\n color=color,\n linewidth=TRAJECTORY_LINEWIDTH,\n label=label,\n )\n\n\ndef draw_endpoint_marker(\n axis,\n vector: np.ndarray,\n color: str,\n label: str,\n) -> None:\n \"\"\"Draw a marker at the final point of a Bloch trajectory.\"\"\"\n axis.scatter(\n vector[0],\n vector[1],\n vector[2],\n color=color,\n s=ENDPOINT_MARKER_SIZE,\n depthshade=True,\n label=label,\n )\n\n\ndef draw_control_axis_arrow(\n axis,\n vector: np.ndarray,\n color: str,\n label: str,\n) -> None:\n \"\"\"Draw a normalized effective-control-axis arrow.\"\"\"\n axis.quiver(\n 0.0,\n 0.0,\n 0.0,\n vector[0],\n vector[1],\n vector[2],\n color=color,\n linewidth=CONTROL_AXIS_LINEWIDTH,\n arrow_length_ratio=ARROW_LENGTH_RATIO,\n normalize=False,\n label=label,\n )\n\n\ndef add_arrow_tip_label(\n axis,\n vector: np.ndarray,\n text: str,\n color: str,\n) -> None:\n \"\"\"Place a text label slightly beyond the arrow tip.\"\"\"\n label_position = ARROW_LABEL_SCALE * vector\n\n axis.text(\n label_position[0],\n label_position[1],\n label_position[2],\n text,\n color=color,\n fontsize=10,\n ha=\"center\",\n va=\"center\",\n )\n\n\ndef add_initial_state_label(axis) -> None:\n \"\"\"Label the initialized ground-state pole.\"\"\"\n axis.scatter(\n 0.0,\n 0.0,\n 1.0,\n color=COLOR_INITIAL_STATE,\n s=ENDPOINT_MARKER_SIZE,\n depthshade=True,\n )\n axis.text(\n 0.0,\n 0.0,\n 1.16,\n r\"initial $|g\\rangle$\",\n color=COLOR_INITIAL_STATE,\n fontsize=9,\n ha=\"center\",\n va=\"center\",\n )\n\n\nif BLOCH_DURATION_NS is None:\n bloch_duration_ns = get_pi_pulse_duration_ns(\n BLOCH_OMEGA_ONRESONANCE_MHZ\n )\nelse:\n bloch_duration_ns = BLOCH_DURATION_NS\n\nresonant_trajectory = solve_rwa_bloch_trajectory(\n detuning_mhz=BLOCH_RESONANT_DETUNING_MHZ,\n duration_ns=bloch_duration_ns,\n num_time_points=BLOCH_NUM_TIME_POINTS,\n)\n\ndetuned_trajectory = solve_rwa_bloch_trajectory(\n detuning_mhz=BLOCH_DETUNED_DETUNING_MHZ,\n duration_ns=bloch_duration_ns,\n num_time_points=BLOCH_NUM_TIME_POINTS,\n)\n\nresonant_drive_ghz = (\n BLOCH_F0_GHZ\n + BLOCH_RESONANT_DETUNING_MHZ * 1.0e-3\n)\ndetuned_drive_ghz = (\n BLOCH_F0_GHZ\n + BLOCH_DETUNED_DETUNING_MHZ * 1.0e-3\n)\n\nfigure = plt.figure(figsize=BLOCH_FIGSIZE)\n\nbloch = qt.Bloch(fig=figure)\nbloch.view = BLOCH_VIEW\nbloch.zlabel = [r\"$|g\\rangle$\", r\"$|e\\rangle$\"]\nbloch.title = (\n \"Bloch-sphere Rabi trajectories\\n\"\n rf\"resonant $\\Delta\u002F2\\pi={BLOCH_RESONANT_DETUNING_MHZ:.1f}$ MHz, \"\n rf\"detuned $\\Delta\u002F2\\pi={BLOCH_DETUNED_DETUNING_MHZ:.1f}$ MHz\"\n)\n\nbloch_axis = render_empty_bloch_sphere(\n bloch_sphere=bloch,\n fallback_figure=figure,\n)\n\ndraw_bloch_trajectory(\n axis=bloch_axis,\n vectors=resonant_trajectory[\"vectors\"],\n color=COLOR_RESONANT_TRAJECTORY,\n label=\"Resonant state trajectory\",\n)\ndraw_bloch_trajectory(\n axis=bloch_axis,\n vectors=detuned_trajectory[\"vectors\"],\n color=COLOR_DETUNED_TRAJECTORY,\n label=\"Detuned state trajectory\",\n)\n\nif BLOCH_SHOW_ENDPOINT_MARKERS:\n draw_endpoint_marker(\n axis=bloch_axis,\n vector=resonant_trajectory[\"vectors\"][:, -1],\n color=COLOR_RESONANT_TRAJECTORY,\n label=\"Resonant final state\",\n )\n draw_endpoint_marker(\n axis=bloch_axis,\n vector=detuned_trajectory[\"vectors\"][:, -1],\n color=COLOR_DETUNED_TRAJECTORY,\n label=\"Detuned final state\",\n )\n add_initial_state_label(bloch_axis)\n\nif BLOCH_SHOW_CONTROL_AXIS_ARROWS:\n draw_control_axis_arrow(\n axis=bloch_axis,\n vector=resonant_trajectory[\"control_axis\"],\n color=COLOR_RESONANT_AXIS,\n label=r\"Resonant control axis $\\hat{n}_{\\mathrm{res}}$\",\n )\n draw_control_axis_arrow(\n axis=bloch_axis,\n vector=detuned_trajectory[\"control_axis\"],\n color=COLOR_DETUNED_AXIS,\n label=r\"Detuned control axis $\\hat{n}_{\\mathrm{det}}$\",\n )\n\nif BLOCH_SHOW_ARROW_TIP_LABELS and BLOCH_SHOW_CONTROL_AXIS_ARROWS:\n add_arrow_tip_label(\n axis=bloch_axis,\n vector=resonant_trajectory[\"control_axis\"],\n text=r\"$\\hat{n}_{\\mathrm{res}}$\",\n color=COLOR_RESONANT_AXIS,\n )\n add_arrow_tip_label(\n axis=bloch_axis,\n vector=detuned_trajectory[\"control_axis\"],\n text=r\"$\\hat{n}_{\\mathrm{det}}$\",\n color=COLOR_DETUNED_AXIS,\n )\n\nif BLOCH_SHOW_LEGEND:\n legend_handles = [\n Line2D(\n [0],\n [0],\n color=COLOR_RESONANT_TRAJECTORY,\n linewidth=TRAJECTORY_LINEWIDTH,\n label=\"Resonant state trajectory\",\n ),\n Line2D(\n [0],\n [0],\n color=COLOR_DETUNED_TRAJECTORY,\n linewidth=TRAJECTORY_LINEWIDTH,\n label=\"Detuned state trajectory\",\n ),\n Line2D(\n [0],\n [0],\n color=COLOR_RESONANT_AXIS,\n linewidth=CONTROL_AXIS_LINEWIDTH,\n label=r\"Resonant control axis $\\hat{n}_{\\mathrm{res}}$\",\n ),\n Line2D(\n [0],\n [0],\n color=COLOR_DETUNED_AXIS,\n linewidth=CONTROL_AXIS_LINEWIDTH,\n label=r\"Detuned control axis $\\hat{n}_{\\mathrm{det}}$\",\n ),\n ]\n\n bloch_axis.legend(\n handles=legend_handles,\n loc=LEGEND_LOCATION,\n bbox_to_anchor=LEGEND_BBOX_TO_ANCHOR,\n frameon=True,\n borderaxespad=0.0,\n )\n\nplt.show()\n\nif BLOCH_PRINT_SUMMARY:\n resonant_axis = resonant_trajectory[\"control_axis\"]\n detuned_axis = detuned_trajectory[\"control_axis\"]\n\n print(\"Bloch-sphere trajectory summary\")\n print(f\"Natural transition frequency f0: {BLOCH_F0_GHZ:.6f} GHz\")\n print(f\"Resonant drive frequency: {resonant_drive_ghz:.6f} GHz\")\n print(f\"Detuned drive frequency: {detuned_drive_ghz:.6f} GHz\")\n print(f\"On-resonance Rabi rate: {BLOCH_OMEGA_ONRESONANCE_MHZ:.3f} MHz\")\n print(f\"Plotted duration: {bloch_duration_ns:.3f} ns\")\n print(\n \"Resonant control axis n_res: \"\n f\"({resonant_axis[0]:.6f}, \"\n f\"{resonant_axis[1]:.6f}, \"\n f\"{resonant_axis[2]:.6f})\"\n )\n print(\n \"Detuned control axis n_det: \"\n f\"({detuned_axis[0]:.6f}, \"\n f\"{detuned_axis[1]:.6f}, \"\n f\"{detuned_axis[2]:.6f})\"\n )\n print(\n \"Resonant final excited-state probability: \"\n f\"{resonant_trajectory['excited_probability'][-1]:.6f}\"\n )\n print(\n \"Detuned final excited-state probability: \"\n f\"{detuned_trajectory['excited_probability'][-1]:.6f}\"\n )\n print(\n \"Detuned generalized Rabi frequency: \"\n f\"{detuned_trajectory['generalized_rabi_mhz']:.6f} MHz\"\n )\n",[51,57946,57947,57952,57957,57961,57968,57975,57981,57985,57990,57995,57999,58008,58012,58021,58033,58045,58057,58069,58073,58077,58103,58118,58125,58132,58136,58140,58150,58160,58164,58169,58179,58183,58192,58212,58221,58241,58245,58254,58263,58272,58281,58290,58294,58304,58314,58324,58334,58344,58348,58358,58368,58378,58388,58398,58402,58412,58430,58434,58438,58462,58467,58486,58490,58494,58503,58519,58527,58532,58543,58548,58552,58561,58566,58582,58593,58598,58602,58611,58615,58619,58626,58635,58665,58676,58681,58689,58694,58699,58706,58713,58720,58727,58734,58741,58749,58754,58764,58768,58776,58784,58788,58792,58798,58804,58810,58816,58822,58826,58836,58840,58844,58848,58859,58863,58873,58883,58895,58907,58912,58919,58933,58938,58951,58967,58987,58995,59001,59006,59017,59023,59035,59041,59046,59051,59056,59079,59085,59105,59110,59115,59125,59137,59149,59161,59181,59187,59211,59234,59250,59255,59282,59287,59311,59333,59354,59359,59373,59388,59406,59411,59416,59431,59437,59443,59449,59454,59459,59491,59518,59544,59549,59563,59578,59583,59588,59602,59607,59613,59619,59625,59630,59635,59640,59654,59659,59670,59678,59688,59693,59698,59703,59710,59726,59735,59744,59753,59762,59785,59791,59796,59801,59806,59816,59825,59834,59839,59845,59861,59872,59887,59897,59904,59913,59918,59940,59945,59962,59979,59984,59997,60011,60016,60024,60029,60034,60044,60052,60060,60071,60082,60091,60097,60107,60117,60126,60135,60146,60158,60169,60174,60179,60184,60194,60201,60209,60220,60231,60240,60246,60256,60266,60275,60284,60293,60305,60317,60326,60331,60336,60341,60351,60358,60365,60376,60387,60396,60402,60412,60420,60427,60434,60443,60452,60461,60470,60481,60493,60505,60514,60519,60524,60529,60539,60546,60553,60565,60576,60585,60591,60607,60612,60621,60631,60640,60649,60655,60664,60676,60689,60701,60706,60711,60716,60735,60741,60750,60757,60764,60772,60783,60794,60805,60810,60819,60826,60833,60841,60862,60873,60885,60896,60907,60912,60917,60922,60936,60948,60954,60959,60967,60977,60982,60994,61005,61015,61026,61031,61036,61048,61059,61068,61079,61084,61089,61099,61105,61119,61124,61134,61139,61151,61156,61161,61184,61189,61212,61223,61262,61272,61284,61304,61322,61327,61332,61344,61354,61364,61369,61374,61382,61392,61407,61418,61430,61435,61442,61451,61465,61476,61488,61493,61498,61508,61516,61526,61547,61558,61570,61575,61582,61591,61610,61621,61633,61638,61647,61652,61662,61670,61679,61693,61704,61730,61735,61742,61751,61764,61775,61798,61803,61808,61822,61830,61839,61852,61874,61885,61890,61897,61906,61919,61940,61951,61956,61961,61971,61981,61989,61999,62008,62020,62032,62044,62049,62056,62065,62074,62085,62096,62107,62112,62119,62128,62137,62148,62159,62180,62185,62192,62201,62210,62221,62232,62253,62258,62263,62268,62278,62289,62301,62313,62325,62337,62342,62347,62356,62361,62371,62386,62401,62406,62418,62443,62468,62493,62518,62544,62551,62557,62582,62603,62625,62630,62637,62643,62665,62686,62707,62712,62719,62725,62753,62758,62765,62771,62798,62803,62810,62816,62839],{"__ignoreMap":49},[54,57948,57949],{"class":56,"line":57},[54,57950,57951],{"class":111},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n",[54,57953,57954],{"class":56,"line":74},[54,57955,57956],{"class":279},"\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n",[54,57958,57959],{"class":56,"line":88},[54,57960,105],{"emptyLinePlaceholder":104},[54,57962,57963,57965],{"class":56,"line":101},[54,57964,61],{"class":60},[54,57966,57967],{"class":64}," importlib.util\n",[54,57969,57970,57972],{"class":56,"line":108},[54,57971,61],{"class":60},[54,57973,57974],{"class":64}," subprocess\n",[54,57976,57977,57979],{"class":56,"line":115},[54,57978,61],{"class":60},[54,57980,56663],{"class":64},[54,57982,57983],{"class":56,"line":139},[54,57984,105],{"emptyLinePlaceholder":104},[54,57986,57987],{"class":56,"line":159},[54,57988,57989],{"class":111},"# -------------------------------------------------------------------------\n",[54,57991,57992],{"class":56,"line":183},[54,57993,57994],{"class":111},"# Control knobs\n",[54,57996,57997],{"class":56,"line":210},[54,57998,57989],{"class":111},[54,58000,58001,58004,58006],{"class":56,"line":227},[54,58002,58003],{"class":132},"INSTALL_BLOCH_DEPENDENCIES",[54,58005,979],{"class":121},[54,58007,56773],{"class":132},[54,58009,58010],{"class":56,"line":232},[54,58011,105],{"emptyLinePlaceholder":104},[54,58013,58014,58017,58019],{"class":56,"line":246},[54,58015,58016],{"class":132},"BLOCH_REQUIRED_PACKAGES",[54,58018,979],{"class":121},[54,58020,23279],{"class":64},[54,58022,58023,58026,58028,58031],{"class":56,"line":259},[54,58024,58025],{"class":279}," \"numpy\"",[54,58027,959],{"class":64},[54,58029,58030],{"class":279},"\"numpy\"",[54,58032,965],{"class":64},[54,58034,58035,58038,58040,58043],{"class":56,"line":264},[54,58036,58037],{"class":279}," \"matplotlib\"",[54,58039,959],{"class":64},[54,58041,58042],{"class":279},"\"matplotlib\"",[54,58044,965],{"class":64},[54,58046,58047,58050,58052,58055],{"class":56,"line":270},[54,58048,58049],{"class":279}," \"scipy\"",[54,58051,959],{"class":64},[54,58053,58054],{"class":279},"\"scipy\"",[54,58056,965],{"class":64},[54,58058,58059,58062,58064,58067],{"class":56,"line":285},[54,58060,58061],{"class":279}," \"qutip\"",[54,58063,959],{"class":64},[54,58065,58066],{"class":279},"\"qutip\"",[54,58068,965],{"class":64},[54,58070,58071],{"class":56,"line":297},[54,58072,23431],{"class":64},[54,58074,58075],{"class":56,"line":310},[54,58076,105],{"emptyLinePlaceholder":104},[54,58078,58079,58082,58084,58087,58089,58092,58094,58097,58099,58101],{"class":56,"line":324},[54,58080,58081],{"class":132},"BLOCH_F0_GHZ",[54,58083,979],{"class":121},[54,58085,58086],{"class":121}," globals",[54,58088,13539],{"class":64},[54,58090,58091],{"class":125},"get",[54,58093,129],{"class":64},[54,58095,58096],{"class":279},"\"F0_GHZ\"",[54,58098,171],{"class":64},[54,58100,43359],{"class":132},[54,58102,136],{"class":64},[54,58104,58105,58108,58110,58112,58114,58116],{"class":56,"line":333},[54,58106,58107],{"class":132},"BLOCH_OMEGA_ONRESONANCE_MHZ",[54,58109,979],{"class":121},[54,58111,58086],{"class":121},[54,58113,13539],{"class":64},[54,58115,58091],{"class":125},[54,58117,932],{"class":64},[54,58119,58120,58123],{"class":56,"line":345},[54,58121,58122],{"class":279}," \"OMEGA_ONRESONANCE_MHZ\"",[54,58124,965],{"class":64},[54,58126,58127,58130],{"class":56,"line":358},[54,58128,58129],{"class":132}," 20.0",[54,58131,965],{"class":64},[54,58133,58134],{"class":56,"line":908},[54,58135,136],{"class":64},[54,58137,58138],{"class":56,"line":913},[54,58139,105],{"emptyLinePlaceholder":104},[54,58141,58142,58145,58147],{"class":56,"line":918},[54,58143,58144],{"class":132},"BLOCH_RESONANT_DETUNING_MHZ",[54,58146,979],{"class":121},[54,58148,58149],{"class":132}," 0.0\n",[54,58151,58152,58155,58157],{"class":56,"line":923},[54,58153,58154],{"class":132},"BLOCH_DETUNED_DETUNING_MHZ",[54,58156,979],{"class":121},[54,58158,58159],{"class":132}," 40.0\n",[54,58161,58162],{"class":56,"line":935},[54,58163,105],{"emptyLinePlaceholder":104},[54,58165,58166],{"class":56,"line":945},[54,58167,58168],{"class":111},"# Use None to automatically plot one resonant pi pulse.\n",[54,58170,58171,58174,58176],{"class":56,"line":953},[54,58172,58173],{"class":132},"BLOCH_DURATION_NS",[54,58175,979],{"class":121},[54,58177,58178],{"class":132}," None\n",[54,58180,58181],{"class":56,"line":968},[54,58182,105],{"emptyLinePlaceholder":104},[54,58184,58185,58188,58190],{"class":56,"line":987},[54,58186,58187],{"class":132},"BLOCH_NUM_TIME_POINTS",[54,58189,979],{"class":121},[54,58191,23177],{"class":132},[54,58193,58194,58197,58199,58202,58205,58207,58210],{"class":56,"line":1005},[54,58195,58196],{"class":132},"BLOCH_FIGSIZE",[54,58198,979],{"class":121},[54,58200,58201],{"class":64}," (",[54,58203,58204],{"class":132},"8.8",[54,58206,171],{"class":64},[54,58208,58209],{"class":132},"7.8",[54,58211,136],{"class":64},[54,58213,58214,58217,58219],{"class":56,"line":1021},[54,58215,58216],{"class":132},"BLOCH_DPI",[54,58218,979],{"class":121},[54,58220,56809],{"class":132},[54,58222,58223,58226,58228,58230,58232,58235,58237,58239],{"class":56,"line":1027},[54,58224,58225],{"class":132},"BLOCH_VIEW",[54,58227,979],{"class":121},[54,58229,7983],{"class":64},[54,58231,1923],{"class":121},[54,58233,58234],{"class":132},"60",[54,58236,171],{"class":64},[54,58238,48145],{"class":132},[54,58240,1350],{"class":64},[54,58242,58243],{"class":56,"line":1033},[54,58244,105],{"emptyLinePlaceholder":104},[54,58246,58247,58250,58252],{"class":56,"line":1038},[54,58248,58249],{"class":132},"BLOCH_SHOW_CONTROL_AXIS_ARROWS",[54,58251,979],{"class":121},[54,58253,56773],{"class":132},[54,58255,58256,58259,58261],{"class":56,"line":1044},[54,58257,58258],{"class":132},"BLOCH_SHOW_ARROW_TIP_LABELS",[54,58260,979],{"class":121},[54,58262,56773],{"class":132},[54,58264,58265,58268,58270],{"class":56,"line":1050},[54,58266,58267],{"class":132},"BLOCH_SHOW_ENDPOINT_MARKERS",[54,58269,979],{"class":121},[54,58271,56773],{"class":132},[54,58273,58274,58277,58279],{"class":56,"line":1056},[54,58275,58276],{"class":132},"BLOCH_SHOW_LEGEND",[54,58278,979],{"class":121},[54,58280,56773],{"class":132},[54,58282,58283,58286,58288],{"class":56,"line":1062},[54,58284,58285],{"class":132},"BLOCH_PRINT_SUMMARY",[54,58287,979],{"class":121},[54,58289,56773],{"class":132},[54,58291,58292],{"class":56,"line":1068},[54,58293,105],{"emptyLinePlaceholder":104},[54,58295,58296,58299,58301],{"class":56,"line":1074},[54,58297,58298],{"class":132},"COLOR_RESONANT_TRAJECTORY",[54,58300,979],{"class":121},[54,58302,58303],{"class":279}," \"tab:blue\"\n",[54,58305,58306,58309,58311],{"class":56,"line":1080},[54,58307,58308],{"class":132},"COLOR_DETUNED_TRAJECTORY",[54,58310,979],{"class":121},[54,58312,58313],{"class":279}," \"tab:orange\"\n",[54,58315,58316,58319,58321],{"class":56,"line":1086},[54,58317,58318],{"class":132},"COLOR_RESONANT_AXIS",[54,58320,979],{"class":121},[54,58322,58323],{"class":279}," \"tab:green\"\n",[54,58325,58326,58329,58331],{"class":56,"line":1092},[54,58327,58328],{"class":132},"COLOR_DETUNED_AXIS",[54,58330,979],{"class":121},[54,58332,58333],{"class":279}," \"tab:red\"\n",[54,58335,58336,58339,58341],{"class":56,"line":1098},[54,58337,58338],{"class":132},"COLOR_INITIAL_STATE",[54,58340,979],{"class":121},[54,58342,58343],{"class":279}," \"black\"\n",[54,58345,58346],{"class":56,"line":1104},[54,58347,105],{"emptyLinePlaceholder":104},[54,58349,58350,58353,58355],{"class":56,"line":1109},[54,58351,58352],{"class":132},"TRAJECTORY_LINEWIDTH",[54,58354,979],{"class":121},[54,58356,58357],{"class":132}," 2.4\n",[54,58359,58360,58363,58365],{"class":56,"line":1115},[54,58361,58362],{"class":132},"CONTROL_AXIS_LINEWIDTH",[54,58364,979],{"class":121},[54,58366,58367],{"class":132}," 2.2\n",[54,58369,58370,58373,58375],{"class":56,"line":1121},[54,58371,58372],{"class":132},"ENDPOINT_MARKER_SIZE",[54,58374,979],{"class":121},[54,58376,58377],{"class":132}," 45\n",[54,58379,58380,58383,58385],{"class":56,"line":1126},[54,58381,58382],{"class":132},"ARROW_LENGTH_RATIO",[54,58384,979],{"class":121},[54,58386,58387],{"class":132}," 0.12\n",[54,58389,58390,58393,58395],{"class":56,"line":1131},[54,58391,58392],{"class":132},"ARROW_LABEL_SCALE",[54,58394,979],{"class":121},[54,58396,58397],{"class":132}," 1.16\n",[54,58399,58400],{"class":56,"line":1136},[54,58401,105],{"emptyLinePlaceholder":104},[54,58403,58404,58407,58409],{"class":56,"line":1153},[54,58405,58406],{"class":132},"LEGEND_LOCATION",[54,58408,979],{"class":121},[54,58410,58411],{"class":279}," \"upper left\"\n",[54,58413,58414,58417,58419,58421,58424,58426,58428],{"class":56,"line":1164},[54,58415,58416],{"class":132},"LEGEND_BBOX_TO_ANCHOR",[54,58418,979],{"class":121},[54,58420,58201],{"class":64},[54,58422,58423],{"class":132},"1.02",[54,58425,171],{"class":64},[54,58427,58423],{"class":132},[54,58429,136],{"class":64},[54,58431,58432],{"class":56,"line":1180},[54,58433,105],{"emptyLinePlaceholder":104},[54,58435,58436],{"class":56,"line":1213},[54,58437,105],{"emptyLinePlaceholder":104},[54,58439,58440,58442,58445,58447,58450,58452,58455,58458,58460],{"class":56,"line":1219},[54,58441,926],{"class":60},[54,58443,58444],{"class":125}," package_is_available",[54,58446,129],{"class":64},[54,58448,58449],{"class":938},"package_name",[54,58451,959],{"class":64},[54,58453,58454],{"class":121},"str",[54,58456,58457],{"class":64},") -> ",[54,58459,995],{"class":121},[54,58461,1284],{"class":64},[54,58463,58464],{"class":56,"line":1247},[54,58465,58466],{"class":279}," \"\"\"Return True when a package can be imported.\"\"\"\n",[54,58468,58469,58471,58474,58477,58480,58482,58484],{"class":56,"line":1270},[54,58470,2060],{"class":60},[54,58472,58473],{"class":64}," importlib.util.",[54,58475,58476],{"class":125},"find_spec",[54,58478,58479],{"class":64},"(package_name) ",[54,58481,36565],{"class":60},[54,58483,21536],{"class":60},[54,58485,58178],{"class":132},[54,58487,58488],{"class":56,"line":1287},[54,58489,105],{"emptyLinePlaceholder":104},[54,58491,58492],{"class":56,"line":1296},[54,58493,105],{"emptyLinePlaceholder":104},[54,58495,58496,58498,58501],{"class":56,"line":1325},[54,58497,926],{"class":60},[54,58499,58500],{"class":125}," install_missing_packages",[54,58502,932],{"class":64},[54,58504,58505,58508,58511,58513,58515,58517],{"class":56,"line":1333},[54,58506,58507],{"class":938}," required_packages",[54,58509,58510],{"class":64},": dict[",[54,58512,58454],{"class":121},[54,58514,171],{"class":64},[54,58516,58454],{"class":121},[54,58518,57624],{"class":64},[54,58520,58521,58523,58525],{"class":56,"line":1353},[54,58522,58457],{"class":64},[54,58524,56321],{"class":132},[54,58526,1284],{"class":64},[54,58528,58529],{"class":56,"line":1361},[54,58530,58531],{"class":279}," \"\"\"Install missing notebook packages using uv pip with a pip fallback.\"\"\"\n",[54,58533,58534,58536,58538,58541],{"class":56,"line":1368},[54,58535,1273],{"class":60},[54,58537,21536],{"class":60},[54,58539,58540],{"class":132}," INSTALL_BLOCH_DEPENDENCIES",[54,58542,1284],{"class":64},[54,58544,58545],{"class":56,"line":1388},[54,58546,58547],{"class":60}," return\n",[54,58549,58550],{"class":56,"line":1395},[54,58551,105],{"emptyLinePlaceholder":104},[54,58553,58554,58557,58559],{"class":56,"line":1411},[54,58555,58556],{"class":64}," missing_package_names ",[54,58558,122],{"class":121},[54,58560,1733],{"class":64},[54,58562,58563],{"class":56,"line":1416},[54,58564,58565],{"class":64}," pip_name\n",[54,58567,58568,58570,58573,58575,58578,58580],{"class":56,"line":1421},[54,58569,1823],{"class":60},[54,58571,58572],{"class":64}," import_name, pip_name ",[54,58574,1192],{"class":60},[54,58576,58577],{"class":64}," required_packages.",[54,58579,22163],{"class":125},[54,58581,330],{"class":64},[54,58583,58584,58586,58588,58590],{"class":56,"line":1442},[54,58585,1290],{"class":60},[54,58587,21536],{"class":60},[54,58589,58444],{"class":125},[54,58591,58592],{"class":64},"(import_name)\n",[54,58594,58595],{"class":56,"line":1447},[54,58596,58597],{"class":64}," ]\n",[54,58599,58600],{"class":56,"line":1453},[54,58601,105],{"emptyLinePlaceholder":104},[54,58603,58604,58606,58608],{"class":56,"line":1458},[54,58605,1273],{"class":60},[54,58607,21536],{"class":60},[54,58609,58610],{"class":64}," missing_package_names:\n",[54,58612,58613],{"class":56,"line":1469},[54,58614,58547],{"class":60},[54,58616,58617],{"class":56,"line":1489},[54,58618,105],{"emptyLinePlaceholder":104},[54,58620,58621,58624],{"class":56,"line":1529},[54,58622,58623],{"class":60}," try",[54,58625,1284],{"class":64},[54,58627,58628,58631,58633],{"class":56,"line":1534},[54,58629,58630],{"class":64}," subprocess.",[54,58632,22057],{"class":125},[54,58634,932],{"class":64},[54,58636,58637,58640,58643,58645,58648,58650,58653,58655,58658,58660,58663],{"class":56,"line":1544},[54,58638,58639],{"class":64}," [sys.executable, ",[54,58641,58642],{"class":279},"\"-m\"",[54,58644,171],{"class":64},[54,58646,58647],{"class":279},"\"pip\"",[54,58649,171],{"class":64},[54,58651,58652],{"class":279},"\"install\"",[54,58654,171],{"class":64},[54,58656,58657],{"class":279},"\"-q\"",[54,58659,171],{"class":64},[54,58661,58662],{"class":279},"\"uv\"",[54,58664,57624],{"class":64},[54,58666,58667,58670,58672,58674],{"class":56,"line":1559},[54,58668,58669],{"class":1711}," check",[54,58671,122],{"class":121},[54,58673,13434],{"class":132},[54,58675,965],{"class":64},[54,58677,58678],{"class":56,"line":1592},[54,58679,58680],{"class":64}," )\n",[54,58682,58683,58685,58687],{"class":56,"line":1597},[54,58684,58630],{"class":64},[54,58686,22057],{"class":125},[54,58688,932],{"class":64},[54,58690,58691],{"class":56,"line":1608},[54,58692,58693],{"class":64}," [\n",[54,58695,58696],{"class":56,"line":1613},[54,58697,58698],{"class":64}," sys.executable,\n",[54,58700,58701,58704],{"class":56,"line":1627},[54,58702,58703],{"class":279}," \"-m\"",[54,58705,965],{"class":64},[54,58707,58708,58711],{"class":56,"line":1632},[54,58709,58710],{"class":279}," \"uv\"",[54,58712,965],{"class":64},[54,58714,58715,58718],{"class":56,"line":1643},[54,58716,58717],{"class":279}," \"pip\"",[54,58719,965],{"class":64},[54,58721,58722,58725],{"class":56,"line":1671},[54,58723,58724],{"class":279}," \"install\"",[54,58726,965],{"class":64},[54,58728,58729,58732],{"class":56,"line":1690},[54,58730,58731],{"class":279}," \"--system\"",[54,58733,965],{"class":64},[54,58735,58736,58739],{"class":56,"line":1695},[54,58737,58738],{"class":279}," \"-q\"",[54,58740,965],{"class":64},[54,58742,58743,58746],{"class":56,"line":1720},[54,58744,58745],{"class":121}," *",[54,58747,58748],{"class":64},"missing_package_names,\n",[54,58750,58751],{"class":56,"line":1725},[54,58752,58753],{"class":64}," ],\n",[54,58755,58756,58758,58760,58762],{"class":56,"line":1736},[54,58757,58669],{"class":1711},[54,58759,122],{"class":121},[54,58761,13434],{"class":132},[54,58763,965],{"class":64},[54,58765,58766],{"class":56,"line":1780},[54,58767,58680],{"class":64},[54,58769,58770,58773],{"class":56,"line":1789},[54,58771,58772],{"class":60}," except",[54,58774,58775],{"class":64}," (subprocess.CalledProcessError, FileNotFoundError):\n",[54,58777,58778,58780,58782],{"class":56,"line":1794},[54,58779,58630],{"class":64},[54,58781,22057],{"class":125},[54,58783,932],{"class":64},[54,58785,58786],{"class":56,"line":1805},[54,58787,58693],{"class":64},[54,58789,58790],{"class":56,"line":1820},[54,58791,58698],{"class":64},[54,58793,58794,58796],{"class":56,"line":1835},[54,58795,58703],{"class":279},[54,58797,965],{"class":64},[54,58799,58800,58802],{"class":56,"line":1846},[54,58801,58717],{"class":279},[54,58803,965],{"class":64},[54,58805,58806,58808],{"class":56,"line":1851},[54,58807,58724],{"class":279},[54,58809,965],{"class":64},[54,58811,58812,58814],{"class":56,"line":1877},[54,58813,58738],{"class":279},[54,58815,965],{"class":64},[54,58817,58818,58820],{"class":56,"line":1893},[54,58819,58745],{"class":121},[54,58821,58748],{"class":64},[54,58823,58824],{"class":56,"line":1912},[54,58825,58753],{"class":64},[54,58827,58828,58830,58832,58834],{"class":56,"line":1952},[54,58829,58669],{"class":1711},[54,58831,122],{"class":121},[54,58833,13434],{"class":132},[54,58835,965],{"class":64},[54,58837,58838],{"class":56,"line":1975},[54,58839,58680],{"class":64},[54,58841,58842],{"class":56,"line":1981},[54,58843,105],{"emptyLinePlaceholder":104},[54,58845,58846],{"class":56,"line":1989},[54,58847,105],{"emptyLinePlaceholder":104},[54,58849,58850,58853,58855,58857],{"class":56,"line":1998},[54,58851,58852],{"class":125},"install_missing_packages",[54,58854,129],{"class":64},[54,58856,58016],{"class":132},[54,58858,136],{"class":64},[54,58860,58861],{"class":56,"line":2003},[54,58862,105],{"emptyLinePlaceholder":104},[54,58864,58865,58867,58869,58871],{"class":56,"line":2019},[54,58866,61],{"class":60},[54,58868,56720],{"class":64},[54,58870,68],{"class":60},[54,58872,715],{"class":64},[54,58874,58875,58877,58879,58881],{"class":56,"line":2024},[54,58876,61],{"class":60},[54,58878,65],{"class":64},[54,58880,68],{"class":60},[54,58882,71],{"class":64},[54,58884,58885,58887,58890,58892],{"class":56,"line":2044},[54,58886,61],{"class":60},[54,58888,58889],{"class":64}," qutip ",[54,58891,68],{"class":60},[54,58893,58894],{"class":64}," qt\n",[54,58896,58897,58899,58902,58904],{"class":56,"line":2057},[54,58898,77],{"class":60},[54,58900,58901],{"class":64}," matplotlib.lines ",[54,58903,61],{"class":60},[54,58905,58906],{"class":64}," Line2D\n",[54,58908,58910],{"class":56,"line":58909},118,[54,58911,105],{"emptyLinePlaceholder":104},[54,58913,58915,58917],{"class":56,"line":58914},119,[54,58916,56745],{"class":60},[54,58918,1284],{"class":64},[54,58920,58922,58925,58928,58930],{"class":56,"line":58921},120,[54,58923,58924],{"class":60}," from",[54,58926,58927],{"class":64}," IPython ",[54,58929,61],{"class":60},[54,58931,58932],{"class":64}," get_ipython\n",[54,58934,58936],{"class":56,"line":58935},121,[54,58937,105],{"emptyLinePlaceholder":104},[54,58939,58941,58944,58946,58949],{"class":56,"line":58940},122,[54,58942,58943],{"class":64}," ipython ",[54,58945,122],{"class":121},[54,58947,58948],{"class":125}," get_ipython",[54,58950,330],{"class":64},[54,58952,58954,58956,58959,58961,58963,58965],{"class":56,"line":58953},123,[54,58955,1273],{"class":60},[54,58957,58958],{"class":64}," ipython ",[54,58960,36565],{"class":60},[54,58962,21536],{"class":60},[54,58964,57064],{"class":132},[54,58966,1284],{"class":64},[54,58968,58970,58973,58976,58978,58980,58982,58985],{"class":56,"line":58969},124,[54,58971,58972],{"class":64}," ipython.",[54,58974,58975],{"class":125},"run_line_magic",[54,58977,129],{"class":64},[54,58979,58042],{"class":279},[54,58981,171],{"class":64},[54,58983,58984],{"class":279},"\"inline\"",[54,58986,136],{"class":64},[54,58988,58990,58992],{"class":56,"line":58989},125,[54,58991,56778],{"class":60},[54,58993,58994],{"class":64}," ImportError:\n",[54,58996,58998],{"class":56,"line":58997},126,[54,58999,59000],{"class":60}," pass\n",[54,59002,59004],{"class":56,"line":59003},127,[54,59005,105],{"emptyLinePlaceholder":104},[54,59007,59009,59012,59015],{"class":56,"line":59008},128,[54,59010,59011],{"class":64},"plt.rcParams.",[54,59013,59014],{"class":125},"update",[54,59016,932],{"class":64},[54,59018,59020],{"class":56,"line":59019},129,[54,59021,59022],{"class":64}," {\n",[54,59024,59026,59029,59031,59033],{"class":56,"line":59025},130,[54,59027,59028],{"class":279}," \"figure.dpi\"",[54,59030,959],{"class":64},[54,59032,58216],{"class":132},[54,59034,965],{"class":64},[54,59036,59038],{"class":56,"line":59037},131,[54,59039,59040],{"class":64}," }\n",[54,59042,59044],{"class":56,"line":59043},132,[54,59045,136],{"class":64},[54,59047,59049],{"class":56,"line":59048},133,[54,59050,105],{"emptyLinePlaceholder":104},[54,59052,59054],{"class":56,"line":59053},134,[54,59055,105],{"emptyLinePlaceholder":104},[54,59057,59059,59061,59064,59066,59069,59071,59073,59075,59077],{"class":56,"line":59058},135,[54,59060,926],{"class":60},[54,59062,59063],{"class":125}," get_pi_pulse_duration_ns",[54,59065,129],{"class":64},[54,59067,59068],{"class":938},"omega_onresonance_mhz",[54,59070,959],{"class":64},[54,59072,962],{"class":121},[54,59074,58457],{"class":64},[54,59076,962],{"class":121},[54,59078,1284],{"class":64},[54,59080,59082],{"class":56,"line":59081},136,[54,59083,59084],{"class":279}," \"\"\"Return the resonant pi-pulse duration in nanoseconds.\"\"\"\n",[54,59086,59088,59090,59093,59095,59097,59100,59102],{"class":56,"line":59087},137,[54,59089,2060],{"class":60},[54,59091,59092],{"class":132}," 1.0e3",[54,59094,57437],{"class":121},[54,59096,58201],{"class":64},[54,59098,59099],{"class":132},"2.0",[54,59101,7785],{"class":121},[54,59103,59104],{"class":64}," omega_onresonance_mhz)\n",[54,59106,59108],{"class":56,"line":59107},138,[54,59109,105],{"emptyLinePlaceholder":104},[54,59111,59113],{"class":56,"line":59112},139,[54,59114,105],{"emptyLinePlaceholder":104},[54,59116,59118,59120,59123],{"class":56,"line":59117},140,[54,59119,926],{"class":60},[54,59121,59122],{"class":125}," solve_rwa_bloch_trajectory",[54,59124,932],{"class":64},[54,59126,59128,59131,59133,59135],{"class":56,"line":59127},141,[54,59129,59130],{"class":938}," detuning_mhz",[54,59132,959],{"class":64},[54,59134,962],{"class":121},[54,59136,965],{"class":64},[54,59138,59140,59143,59145,59147],{"class":56,"line":59139},142,[54,59141,59142],{"class":938}," duration_ns",[54,59144,959],{"class":64},[54,59146,962],{"class":121},[54,59148,965],{"class":64},[54,59150,59152,59155,59157,59159],{"class":56,"line":59151},143,[54,59153,59154],{"class":938}," num_time_points",[54,59156,959],{"class":64},[54,59158,976],{"class":121},[54,59160,965],{"class":64},[54,59162,59164,59167,59169,59172,59175,59178],{"class":56,"line":59163},144,[54,59165,59166],{"class":64},") -> dict[",[54,59168,58454],{"class":121},[54,59170,59171],{"class":64},", np.ndarray ",[54,59173,59174],{"class":121},"|",[54,59176,59177],{"class":121}," float",[54,59179,59180],{"class":64},"]:\n",[54,59182,59184],{"class":56,"line":59183},145,[54,59185,59186],{"class":279}," \"\"\"Solve the RWA two-level trajectory and return Bloch observables.\"\"\"\n",[54,59188,59190,59193,59195,59197,59199,59201,59203,59206,59208],{"class":56,"line":59189},146,[54,59191,59192],{"class":64}," omega_rad_s ",[54,59194,122],{"class":121},[54,59196,57136],{"class":132},[54,59198,7785],{"class":121},[54,59200,8503],{"class":64},[54,59202,1947],{"class":121},[54,59204,59205],{"class":132}," BLOCH_OMEGA_ONRESONANCE_MHZ",[54,59207,7785],{"class":121},[54,59209,59210],{"class":132}," 1.0e6\n",[54,59212,59214,59217,59219,59221,59223,59225,59227,59230,59232],{"class":56,"line":59213},147,[54,59215,59216],{"class":64}," delta_rad_s ",[54,59218,122],{"class":121},[54,59220,57136],{"class":132},[54,59222,7785],{"class":121},[54,59224,8503],{"class":64},[54,59226,1947],{"class":121},[54,59228,59229],{"class":64}," detuning_mhz ",[54,59231,1947],{"class":121},[54,59233,59210],{"class":132},[54,59235,59237,59240,59242,59244,59247],{"class":56,"line":59236},148,[54,59238,59239],{"class":64}," omega_generalized_rad_s ",[54,59241,122],{"class":121},[54,59243,1172],{"class":64},[54,59245,59246],{"class":125},"hypot",[54,59248,59249],{"class":64},"(omega_rad_s, delta_rad_s)\n",[54,59251,59253],{"class":56,"line":59252},149,[54,59254,105],{"emptyLinePlaceholder":104},[54,59256,59258,59261,59263,59265,59267,59269,59271,59274,59276,59279],{"class":56,"line":59257},150,[54,59259,59260],{"class":64}," time_s ",[54,59262,122],{"class":121},[54,59264,1172],{"class":64},[54,59266,57503],{"class":125},[54,59268,129],{"class":64},[54,59270,56271],{"class":132},[54,59272,59273],{"class":64},", duration_ns ",[54,59275,1947],{"class":121},[54,59277,59278],{"class":132}," 1.0e-9",[54,59280,59281],{"class":64},", num_time_points)\n",[54,59283,59285],{"class":56,"line":59284},151,[54,59286,105],{"emptyLinePlaceholder":104},[54,59288,59290,59293,59295,59298,59301,59303,59305,59307,59309],{"class":56,"line":59289},152,[54,59291,59292],{"class":64}," ground_state ",[54,59294,122],{"class":121},[54,59296,59297],{"class":64}," qt.",[54,59299,59300],{"class":125},"basis",[54,59302,129],{"class":64},[54,59304,133],{"class":132},[54,59306,171],{"class":64},[54,59308,150],{"class":132},[54,59310,136],{"class":64},[54,59312,59314,59317,59319,59321,59323,59325,59327,59329,59331],{"class":56,"line":59313},153,[54,59315,59316],{"class":64}," excited_state ",[54,59318,122],{"class":121},[54,59320,59297],{"class":64},[54,59322,59300],{"class":125},[54,59324,129],{"class":64},[54,59326,133],{"class":132},[54,59328,171],{"class":64},[54,59330,174],{"class":132},[54,59332,136],{"class":64},[54,59334,59336,59339,59341,59344,59346,59349,59352],{"class":56,"line":59335},154,[54,59337,59338],{"class":64}," excited_projector ",[54,59340,122],{"class":121},[54,59342,59343],{"class":64}," excited_state ",[54,59345,1947],{"class":121},[54,59347,59348],{"class":64}," excited_state.",[54,59350,59351],{"class":125},"dag",[54,59353,330],{"class":64},[54,59355,59357],{"class":56,"line":59356},155,[54,59358,105],{"emptyLinePlaceholder":104},[54,59360,59362,59365,59367,59369,59371],{"class":56,"line":59361},156,[54,59363,59364],{"class":64}," hamiltonian ",[54,59366,122],{"class":121},[54,59368,57356],{"class":132},[54,59370,7785],{"class":121},[54,59372,890],{"class":64},[54,59374,59376,59379,59381,59383,59386],{"class":56,"line":59375},157,[54,59377,59378],{"class":64}," delta_rad_s ",[54,59380,1947],{"class":121},[54,59382,59297],{"class":64},[54,59384,59385],{"class":125},"sigmaz",[54,59387,330],{"class":64},[54,59389,59391,59394,59397,59399,59401,59404],{"class":56,"line":59390},158,[54,59392,59393],{"class":121}," +",[54,59395,59396],{"class":64}," omega_rad_s ",[54,59398,1947],{"class":121},[54,59400,59297],{"class":64},[54,59402,59403],{"class":125},"sigmax",[54,59405,330],{"class":64},[54,59407,59409],{"class":56,"line":59408},159,[54,59410,1216],{"class":64},[54,59412,59414],{"class":56,"line":59413},160,[54,59415,105],{"emptyLinePlaceholder":104},[54,59417,59419,59422,59424,59426,59429],{"class":56,"line":59418},161,[54,59420,59421],{"class":64}," result ",[54,59423,122],{"class":121},[54,59425,59297],{"class":64},[54,59427,59428],{"class":125},"sesolve",[54,59430,932],{"class":64},[54,59432,59434],{"class":56,"line":59433},162,[54,59435,59436],{"class":64}," hamiltonian,\n",[54,59438,59440],{"class":56,"line":59439},163,[54,59441,59442],{"class":64}," ground_state,\n",[54,59444,59446],{"class":56,"line":59445},164,[54,59447,59448],{"class":64}," time_s,\n",[54,59450,59452],{"class":56,"line":59451},165,[54,59453,1216],{"class":64},[54,59455,59457],{"class":56,"line":59456},166,[54,59458,105],{"emptyLinePlaceholder":104},[54,59460,59462,59465,59467,59469,59472,59475,59478,59481,59484,59486,59488],{"class":56,"line":59461},167,[54,59463,59464],{"class":64}," x_expectation ",[54,59466,122],{"class":121},[54,59468,1172],{"class":64},[54,59470,59471],{"class":125},"real",[54,59473,59474],{"class":64},"(np.",[54,59476,59477],{"class":125},"asarray",[54,59479,59480],{"class":64},"(qt.",[54,59482,59483],{"class":125},"expect",[54,59485,59480],{"class":64},[54,59487,59403],{"class":125},[54,59489,59490],{"class":64},"(), result.states)))\n",[54,59492,59494,59497,59499,59501,59503,59505,59507,59509,59511,59513,59516],{"class":56,"line":59493},168,[54,59495,59496],{"class":64}," y_expectation ",[54,59498,122],{"class":121},[54,59500,1172],{"class":64},[54,59502,59471],{"class":125},[54,59504,59474],{"class":64},[54,59506,59477],{"class":125},[54,59508,59480],{"class":64},[54,59510,59483],{"class":125},[54,59512,59480],{"class":64},[54,59514,59515],{"class":125},"sigmay",[54,59517,59490],{"class":64},[54,59519,59521,59524,59526,59528,59530,59532,59534,59536,59538,59540,59542],{"class":56,"line":59520},169,[54,59522,59523],{"class":64}," z_expectation ",[54,59525,122],{"class":121},[54,59527,1172],{"class":64},[54,59529,59471],{"class":125},[54,59531,59474],{"class":64},[54,59533,59477],{"class":125},[54,59535,59480],{"class":64},[54,59537,59483],{"class":125},[54,59539,59480],{"class":64},[54,59541,59385],{"class":125},[54,59543,59490],{"class":64},[54,59545,59547],{"class":56,"line":59546},170,[54,59548,105],{"emptyLinePlaceholder":104},[54,59550,59552,59555,59557,59559,59561],{"class":56,"line":59551},171,[54,59553,59554],{"class":64}," excited_probability ",[54,59556,122],{"class":121},[54,59558,1172],{"class":64},[54,59560,59471],{"class":125},[54,59562,932],{"class":64},[54,59564,59566,59569,59571,59573,59575],{"class":56,"line":59565},172,[54,59567,59568],{"class":64}," np.",[54,59570,59477],{"class":125},[54,59572,59480],{"class":64},[54,59574,59483],{"class":125},[54,59576,59577],{"class":64},"(excited_projector, result.states))\n",[54,59579,59581],{"class":56,"line":59580},173,[54,59582,1216],{"class":64},[54,59584,59586],{"class":56,"line":59585},174,[54,59587,105],{"emptyLinePlaceholder":104},[54,59589,59591,59594,59596,59598,59600],{"class":56,"line":59590},175,[54,59592,59593],{"class":64}," bloch_vectors ",[54,59595,122],{"class":121},[54,59597,1172],{"class":64},[54,59599,22629],{"class":125},[54,59601,932],{"class":64},[54,59603,59605],{"class":56,"line":59604},176,[54,59606,22137],{"class":64},[54,59608,59610],{"class":56,"line":59609},177,[54,59611,59612],{"class":64}," x_expectation,\n",[54,59614,59616],{"class":56,"line":59615},178,[54,59617,59618],{"class":64}," y_expectation,\n",[54,59620,59622],{"class":56,"line":59621},179,[54,59623,59624],{"class":64}," z_expectation,\n",[54,59626,59628],{"class":56,"line":59627},180,[54,59629,22183],{"class":64},[54,59631,59633],{"class":56,"line":59632},181,[54,59634,1216],{"class":64},[54,59636,59638],{"class":56,"line":59637},182,[54,59639,105],{"emptyLinePlaceholder":104},[54,59641,59643,59646,59648,59650,59652],{"class":56,"line":59642},183,[54,59644,59645],{"class":64}," control_axis ",[54,59647,122],{"class":121},[54,59649,1172],{"class":64},[54,59651,1175],{"class":125},[54,59653,932],{"class":64},[54,59655,59657],{"class":56,"line":59656},184,[54,59658,22137],{"class":64},[54,59660,59662,59665,59667],{"class":56,"line":59661},185,[54,59663,59664],{"class":64}," omega_rad_s ",[54,59666,194],{"class":121},[54,59668,59669],{"class":64}," omega_generalized_rad_s,\n",[54,59671,59673,59676],{"class":56,"line":59672},186,[54,59674,59675],{"class":132}," 0.0",[54,59677,965],{"class":64},[54,59679,59681,59684,59686],{"class":56,"line":59680},187,[54,59682,59683],{"class":64}," delta_rad_s ",[54,59685,194],{"class":121},[54,59687,59669],{"class":64},[54,59689,59691],{"class":56,"line":59690},188,[54,59692,22183],{"class":64},[54,59694,59696],{"class":56,"line":59695},189,[54,59697,1216],{"class":64},[54,59699,59701],{"class":56,"line":59700},190,[54,59702,105],{"emptyLinePlaceholder":104},[54,59704,59706,59708],{"class":56,"line":59705},191,[54,59707,2060],{"class":60},[54,59709,23279],{"class":64},[54,59711,59713,59716,59719,59721,59724],{"class":56,"line":59712},192,[54,59714,59715],{"class":279}," \"time_ns\"",[54,59717,59718],{"class":64},": time_s ",[54,59720,1947],{"class":121},[54,59722,59723],{"class":132}," 1.0e9",[54,59725,965],{"class":64},[54,59727,59729,59732],{"class":56,"line":59728},193,[54,59730,59731],{"class":279}," \"vectors\"",[54,59733,59734],{"class":64},": bloch_vectors,\n",[54,59736,59738,59741],{"class":56,"line":59737},194,[54,59739,59740],{"class":279}," \"excited_probability\"",[54,59742,59743],{"class":64},": excited_probability,\n",[54,59745,59747,59750],{"class":56,"line":59746},195,[54,59748,59749],{"class":279}," \"control_axis\"",[54,59751,59752],{"class":64},": control_axis,\n",[54,59754,59756,59759],{"class":56,"line":59755},196,[54,59757,59758],{"class":279}," \"generalized_rabi_mhz\"",[54,59760,59761],{"class":64},": (\n",[54,59763,59765,59768,59770,59772,59774,59776,59778,59780,59783],{"class":56,"line":59764},197,[54,59766,59767],{"class":64}," omega_generalized_rad_s ",[54,59769,194],{"class":121},[54,59771,58201],{"class":64},[54,59773,59099],{"class":132},[54,59775,7785],{"class":121},[54,59777,8503],{"class":64},[54,59779,1947],{"class":121},[54,59781,59782],{"class":132}," 1.0e6",[54,59784,136],{"class":64},[54,59786,59788],{"class":56,"line":59787},198,[54,59789,59790],{"class":64}," ),\n",[54,59792,59794],{"class":56,"line":59793},199,[54,59795,59040],{"class":64},[54,59797,59799],{"class":56,"line":59798},200,[54,59800,105],{"emptyLinePlaceholder":104},[54,59802,59804],{"class":56,"line":59803},201,[54,59805,105],{"emptyLinePlaceholder":104},[54,59807,59809,59811,59814],{"class":56,"line":59808},202,[54,59810,926],{"class":60},[54,59812,59813],{"class":125}," render_empty_bloch_sphere",[54,59815,932],{"class":64},[54,59817,59819,59822],{"class":56,"line":59818},203,[54,59820,59821],{"class":938}," bloch_sphere",[54,59823,59824],{"class":64},": qt.Bloch,\n",[54,59826,59828,59831],{"class":56,"line":59827},204,[54,59829,59830],{"class":938}," fallback_figure",[54,59832,59833],{"class":64},": plt.Figure,\n",[54,59835,59837],{"class":56,"line":59836},205,[54,59838,1439],{"class":64},[54,59840,59842],{"class":56,"line":59841},206,[54,59843,59844],{"class":279}," \"\"\"Render an empty QuTiP Bloch sphere and return its Matplotlib axis.\"\"\"\n",[54,59846,59848,59850,59853,59856,59859],{"class":56,"line":59847},207,[54,59849,1273],{"class":60},[54,59851,59852],{"class":121}," hasattr",[54,59854,59855],{"class":64},"(bloch_sphere, ",[54,59857,59858],{"class":279},"\"render\"",[54,59860,1439],{"class":64},[54,59862,59864,59867,59870],{"class":56,"line":59863},208,[54,59865,59866],{"class":64}," bloch_sphere.",[54,59868,59869],{"class":125},"render",[54,59871,330],{"class":64},[54,59873,59875,59878,59880,59882,59885],{"class":56,"line":59874},209,[54,59876,59877],{"class":60}," elif",[54,59879,59852],{"class":121},[54,59881,59855],{"class":64},[54,59883,59884],{"class":279},"\"make_sphere\"",[54,59886,1439],{"class":64},[54,59888,59890,59892,59895],{"class":56,"line":59889},210,[54,59891,59866],{"class":64},[54,59893,59894],{"class":125},"make_sphere",[54,59896,330],{"class":64},[54,59898,59900,59902],{"class":56,"line":59899},211,[54,59901,1356],{"class":60},[54,59903,1284],{"class":64},[54,59905,59907,59909,59911],{"class":56,"line":59906},212,[54,59908,59866],{"class":64},[54,59910,25169],{"class":125},[54,59912,330],{"class":64},[54,59914,59916],{"class":56,"line":59915},213,[54,59917,105],{"emptyLinePlaceholder":104},[54,59919,59921,59924,59926,59929,59931,59934,59936,59938],{"class":56,"line":59920},214,[54,59922,59923],{"class":64}," axis ",[54,59925,122],{"class":121},[54,59927,59928],{"class":121}," getattr",[54,59930,59855],{"class":64},[54,59932,59933],{"class":279},"\"axes\"",[54,59935,171],{"class":64},[54,59937,56321],{"class":132},[54,59939,136],{"class":64},[54,59941,59943],{"class":56,"line":59942},215,[54,59944,105],{"emptyLinePlaceholder":104},[54,59946,59948,59950,59953,59955,59957,59959],{"class":56,"line":59947},216,[54,59949,1273],{"class":60},[54,59951,59952],{"class":64}," axis ",[54,59954,36565],{"class":60},[54,59956,57064],{"class":132},[54,59958,57409],{"class":60},[54,59960,59961],{"class":64}," fallback_figure.axes:\n",[54,59963,59965,59968,59970,59973,59975,59977],{"class":56,"line":59964},217,[54,59966,59967],{"class":64}," axis ",[54,59969,122],{"class":121},[54,59971,59972],{"class":64}," fallback_figure.axes[",[54,59974,1923],{"class":121},[54,59976,174],{"class":132},[54,59978,1350],{"class":64},[54,59980,59982],{"class":56,"line":59981},218,[54,59983,105],{"emptyLinePlaceholder":104},[54,59985,59987,59989,59991,59993,59995],{"class":56,"line":59986},219,[54,59988,1273],{"class":60},[54,59990,59952],{"class":64},[54,59992,36565],{"class":60},[54,59994,57064],{"class":132},[54,59996,1284],{"class":64},[54,59998,60000,60003,60006,60009],{"class":56,"line":59999},220,[54,60001,60002],{"class":60}," raise",[54,60004,60005],{"class":64}," RuntimeError(",[54,60007,60008],{"class":279},"\"Could not access the Matplotlib Bloch axis.\"",[54,60010,136],{"class":64},[54,60012,60014],{"class":56,"line":60013},221,[54,60015,105],{"emptyLinePlaceholder":104},[54,60017,60019,60021],{"class":56,"line":60018},222,[54,60020,2060],{"class":60},[54,60022,60023],{"class":64}," axis\n",[54,60025,60027],{"class":56,"line":60026},223,[54,60028,105],{"emptyLinePlaceholder":104},[54,60030,60032],{"class":56,"line":60031},224,[54,60033,105],{"emptyLinePlaceholder":104},[54,60035,60037,60039,60042],{"class":56,"line":60036},225,[54,60038,926],{"class":60},[54,60040,60041],{"class":125}," draw_bloch_trajectory",[54,60043,932],{"class":64},[54,60045,60047,60050],{"class":56,"line":60046},226,[54,60048,60049],{"class":938}," axis",[54,60051,965],{"class":64},[54,60053,60055,60058],{"class":56,"line":60054},227,[54,60056,60057],{"class":938}," vectors",[54,60059,942],{"class":64},[54,60061,60063,60065,60067,60069],{"class":56,"line":60062},228,[54,60064,24964],{"class":938},[54,60066,959],{"class":64},[54,60068,58454],{"class":121},[54,60070,965],{"class":64},[54,60072,60074,60076,60078,60080],{"class":56,"line":60073},229,[54,60075,24988],{"class":938},[54,60077,959],{"class":64},[54,60079,58454],{"class":121},[54,60081,965],{"class":64},[54,60083,60085,60087,60089],{"class":56,"line":60084},230,[54,60086,58457],{"class":64},[54,60088,56321],{"class":132},[54,60090,1284],{"class":64},[54,60092,60094],{"class":56,"line":60093},231,[54,60095,60096],{"class":279}," \"\"\"Draw a Bloch-vector trajectory on an existing 3D axis.\"\"\"\n",[54,60098,60100,60103,60105],{"class":56,"line":60099},232,[54,60101,60102],{"class":64}," axis.",[54,60104,24935],{"class":125},[54,60106,932],{"class":64},[54,60108,60110,60113,60115],{"class":56,"line":60109},233,[54,60111,60112],{"class":64}," vectors[",[54,60114,150],{"class":132},[54,60116,57624],{"class":64},[54,60118,60120,60122,60124],{"class":56,"line":60119},234,[54,60121,60112],{"class":64},[54,60123,174],{"class":132},[54,60125,57624],{"class":64},[54,60127,60129,60131,60133],{"class":56,"line":60128},235,[54,60130,60112],{"class":64},[54,60132,133],{"class":132},[54,60134,57624],{"class":64},[54,60136,60138,60141,60143],{"class":56,"line":60137},236,[54,60139,60140],{"class":1711}," color",[54,60142,122],{"class":121},[54,60144,60145],{"class":64},"color,\n",[54,60147,60149,60152,60154,60156],{"class":56,"line":60148},237,[54,60150,60151],{"class":1711}," linewidth",[54,60153,122],{"class":121},[54,60155,58352],{"class":132},[54,60157,965],{"class":64},[54,60159,60161,60164,60166],{"class":56,"line":60160},238,[54,60162,60163],{"class":1711}," label",[54,60165,122],{"class":121},[54,60167,60168],{"class":64},"label,\n",[54,60170,60172],{"class":56,"line":60171},239,[54,60173,1216],{"class":64},[54,60175,60177],{"class":56,"line":60176},240,[54,60178,105],{"emptyLinePlaceholder":104},[54,60180,60182],{"class":56,"line":60181},241,[54,60183,105],{"emptyLinePlaceholder":104},[54,60185,60187,60189,60192],{"class":56,"line":60186},242,[54,60188,926],{"class":60},[54,60190,60191],{"class":125}," draw_endpoint_marker",[54,60193,932],{"class":64},[54,60195,60197,60199],{"class":56,"line":60196},243,[54,60198,60049],{"class":938},[54,60200,965],{"class":64},[54,60202,60204,60207],{"class":56,"line":60203},244,[54,60205,60206],{"class":938}," vector",[54,60208,942],{"class":64},[54,60210,60212,60214,60216,60218],{"class":56,"line":60211},245,[54,60213,24964],{"class":938},[54,60215,959],{"class":64},[54,60217,58454],{"class":121},[54,60219,965],{"class":64},[54,60221,60223,60225,60227,60229],{"class":56,"line":60222},246,[54,60224,24988],{"class":938},[54,60226,959],{"class":64},[54,60228,58454],{"class":121},[54,60230,965],{"class":64},[54,60232,60234,60236,60238],{"class":56,"line":60233},247,[54,60235,58457],{"class":64},[54,60237,56321],{"class":132},[54,60239,1284],{"class":64},[54,60241,60243],{"class":56,"line":60242},248,[54,60244,60245],{"class":279}," \"\"\"Draw a marker at the final point of a Bloch trajectory.\"\"\"\n",[54,60247,60249,60251,60254],{"class":56,"line":60248},249,[54,60250,60102],{"class":64},[54,60252,60253],{"class":125},"scatter",[54,60255,932],{"class":64},[54,60257,60259,60262,60264],{"class":56,"line":60258},250,[54,60260,60261],{"class":64}," vector[",[54,60263,150],{"class":132},[54,60265,57624],{"class":64},[54,60267,60269,60271,60273],{"class":56,"line":60268},251,[54,60270,60261],{"class":64},[54,60272,174],{"class":132},[54,60274,57624],{"class":64},[54,60276,60278,60280,60282],{"class":56,"line":60277},252,[54,60279,60261],{"class":64},[54,60281,133],{"class":132},[54,60283,57624],{"class":64},[54,60285,60287,60289,60291],{"class":56,"line":60286},253,[54,60288,60140],{"class":1711},[54,60290,122],{"class":121},[54,60292,60145],{"class":64},[54,60294,60296,60299,60301,60303],{"class":56,"line":60295},254,[54,60297,60298],{"class":1711}," s",[54,60300,122],{"class":121},[54,60302,58372],{"class":132},[54,60304,965],{"class":64},[54,60306,60308,60311,60313,60315],{"class":56,"line":60307},255,[54,60309,60310],{"class":1711}," depthshade",[54,60312,122],{"class":121},[54,60314,13434],{"class":132},[54,60316,965],{"class":64},[54,60318,60320,60322,60324],{"class":56,"line":60319},256,[54,60321,60163],{"class":1711},[54,60323,122],{"class":121},[54,60325,60168],{"class":64},[54,60327,60329],{"class":56,"line":60328},257,[54,60330,1216],{"class":64},[54,60332,60334],{"class":56,"line":60333},258,[54,60335,105],{"emptyLinePlaceholder":104},[54,60337,60339],{"class":56,"line":60338},259,[54,60340,105],{"emptyLinePlaceholder":104},[54,60342,60344,60346,60349],{"class":56,"line":60343},260,[54,60345,926],{"class":60},[54,60347,60348],{"class":125}," draw_control_axis_arrow",[54,60350,932],{"class":64},[54,60352,60354,60356],{"class":56,"line":60353},261,[54,60355,60049],{"class":938},[54,60357,965],{"class":64},[54,60359,60361,60363],{"class":56,"line":60360},262,[54,60362,60206],{"class":938},[54,60364,942],{"class":64},[54,60366,60368,60370,60372,60374],{"class":56,"line":60367},263,[54,60369,24964],{"class":938},[54,60371,959],{"class":64},[54,60373,58454],{"class":121},[54,60375,965],{"class":64},[54,60377,60379,60381,60383,60385],{"class":56,"line":60378},264,[54,60380,24988],{"class":938},[54,60382,959],{"class":64},[54,60384,58454],{"class":121},[54,60386,965],{"class":64},[54,60388,60390,60392,60394],{"class":56,"line":60389},265,[54,60391,58457],{"class":64},[54,60393,56321],{"class":132},[54,60395,1284],{"class":64},[54,60397,60399],{"class":56,"line":60398},266,[54,60400,60401],{"class":279}," \"\"\"Draw a normalized effective-control-axis arrow.\"\"\"\n",[54,60403,60405,60407,60410],{"class":56,"line":60404},267,[54,60406,60102],{"class":64},[54,60408,60409],{"class":125},"quiver",[54,60411,932],{"class":64},[54,60413,60415,60418],{"class":56,"line":60414},268,[54,60416,60417],{"class":132}," 0.0",[54,60419,965],{"class":64},[54,60421,60423,60425],{"class":56,"line":60422},269,[54,60424,60417],{"class":132},[54,60426,965],{"class":64},[54,60428,60430,60432],{"class":56,"line":60429},270,[54,60431,60417],{"class":132},[54,60433,965],{"class":64},[54,60435,60437,60439,60441],{"class":56,"line":60436},271,[54,60438,60261],{"class":64},[54,60440,150],{"class":132},[54,60442,57624],{"class":64},[54,60444,60446,60448,60450],{"class":56,"line":60445},272,[54,60447,60261],{"class":64},[54,60449,174],{"class":132},[54,60451,57624],{"class":64},[54,60453,60455,60457,60459],{"class":56,"line":60454},273,[54,60456,60261],{"class":64},[54,60458,133],{"class":132},[54,60460,57624],{"class":64},[54,60462,60464,60466,60468],{"class":56,"line":60463},274,[54,60465,60140],{"class":1711},[54,60467,122],{"class":121},[54,60469,60145],{"class":64},[54,60471,60473,60475,60477,60479],{"class":56,"line":60472},275,[54,60474,60151],{"class":1711},[54,60476,122],{"class":121},[54,60478,58362],{"class":132},[54,60480,965],{"class":64},[54,60482,60484,60487,60489,60491],{"class":56,"line":60483},276,[54,60485,60486],{"class":1711}," arrow_length_ratio",[54,60488,122],{"class":121},[54,60490,58382],{"class":132},[54,60492,965],{"class":64},[54,60494,60496,60499,60501,60503],{"class":56,"line":60495},277,[54,60497,60498],{"class":1711}," normalize",[54,60500,122],{"class":121},[54,60502,21520],{"class":132},[54,60504,965],{"class":64},[54,60506,60508,60510,60512],{"class":56,"line":60507},278,[54,60509,60163],{"class":1711},[54,60511,122],{"class":121},[54,60513,60168],{"class":64},[54,60515,60517],{"class":56,"line":60516},279,[54,60518,1216],{"class":64},[54,60520,60522],{"class":56,"line":60521},280,[54,60523,105],{"emptyLinePlaceholder":104},[54,60525,60527],{"class":56,"line":60526},281,[54,60528,105],{"emptyLinePlaceholder":104},[54,60530,60532,60534,60537],{"class":56,"line":60531},282,[54,60533,926],{"class":60},[54,60535,60536],{"class":125}," add_arrow_tip_label",[54,60538,932],{"class":64},[54,60540,60542,60544],{"class":56,"line":60541},283,[54,60543,60049],{"class":938},[54,60545,965],{"class":64},[54,60547,60549,60551],{"class":56,"line":60548},284,[54,60550,60206],{"class":938},[54,60552,942],{"class":64},[54,60554,60556,60559,60561,60563],{"class":56,"line":60555},285,[54,60557,60558],{"class":938}," text",[54,60560,959],{"class":64},[54,60562,58454],{"class":121},[54,60564,965],{"class":64},[54,60566,60568,60570,60572,60574],{"class":56,"line":60567},286,[54,60569,24964],{"class":938},[54,60571,959],{"class":64},[54,60573,58454],{"class":121},[54,60575,965],{"class":64},[54,60577,60579,60581,60583],{"class":56,"line":60578},287,[54,60580,58457],{"class":64},[54,60582,56321],{"class":132},[54,60584,1284],{"class":64},[54,60586,60588],{"class":56,"line":60587},288,[54,60589,60590],{"class":279}," \"\"\"Place a text label slightly beyond the arrow tip.\"\"\"\n",[54,60592,60594,60597,60599,60602,60604],{"class":56,"line":60593},289,[54,60595,60596],{"class":64}," label_position ",[54,60598,122],{"class":121},[54,60600,60601],{"class":132}," ARROW_LABEL_SCALE",[54,60603,7785],{"class":121},[54,60605,60606],{"class":64}," vector\n",[54,60608,60610],{"class":56,"line":60609},290,[54,60611,105],{"emptyLinePlaceholder":104},[54,60613,60615,60617,60619],{"class":56,"line":60614},291,[54,60616,60102],{"class":64},[54,60618,6595],{"class":125},[54,60620,932],{"class":64},[54,60622,60624,60627,60629],{"class":56,"line":60623},292,[54,60625,60626],{"class":64}," label_position[",[54,60628,150],{"class":132},[54,60630,57624],{"class":64},[54,60632,60634,60636,60638],{"class":56,"line":60633},293,[54,60635,60626],{"class":64},[54,60637,174],{"class":132},[54,60639,57624],{"class":64},[54,60641,60643,60645,60647],{"class":56,"line":60642},294,[54,60644,60626],{"class":64},[54,60646,133],{"class":132},[54,60648,57624],{"class":64},[54,60650,60652],{"class":56,"line":60651},295,[54,60653,60654],{"class":64}," text,\n",[54,60656,60658,60660,60662],{"class":56,"line":60657},296,[54,60659,60140],{"class":1711},[54,60661,122],{"class":121},[54,60663,60145],{"class":64},[54,60665,60667,60670,60672,60674],{"class":56,"line":60666},297,[54,60668,60669],{"class":1711}," fontsize",[54,60671,122],{"class":121},[54,60673,23361],{"class":132},[54,60675,965],{"class":64},[54,60677,60679,60682,60684,60687],{"class":56,"line":60678},298,[54,60680,60681],{"class":1711}," ha",[54,60683,122],{"class":121},[54,60685,60686],{"class":279},"\"center\"",[54,60688,965],{"class":64},[54,60690,60692,60695,60697,60699],{"class":56,"line":60691},299,[54,60693,60694],{"class":1711}," va",[54,60696,122],{"class":121},[54,60698,60686],{"class":279},[54,60700,965],{"class":64},[54,60702,60704],{"class":56,"line":60703},300,[54,60705,1216],{"class":64},[54,60707,60709],{"class":56,"line":60708},301,[54,60710,105],{"emptyLinePlaceholder":104},[54,60712,60714],{"class":56,"line":60713},302,[54,60715,105],{"emptyLinePlaceholder":104},[54,60717,60719,60721,60724,60726,60729,60731,60733],{"class":56,"line":60718},303,[54,60720,926],{"class":60},[54,60722,60723],{"class":125}," add_initial_state_label",[54,60725,129],{"class":64},[54,60727,60728],{"class":938},"axis",[54,60730,58457],{"class":64},[54,60732,56321],{"class":132},[54,60734,1284],{"class":64},[54,60736,60738],{"class":56,"line":60737},304,[54,60739,60740],{"class":279}," \"\"\"Label the initialized ground-state pole.\"\"\"\n",[54,60742,60744,60746,60748],{"class":56,"line":60743},305,[54,60745,60102],{"class":64},[54,60747,60253],{"class":125},[54,60749,932],{"class":64},[54,60751,60753,60755],{"class":56,"line":60752},306,[54,60754,60417],{"class":132},[54,60756,965],{"class":64},[54,60758,60760,60762],{"class":56,"line":60759},307,[54,60761,60417],{"class":132},[54,60763,965],{"class":64},[54,60765,60767,60770],{"class":56,"line":60766},308,[54,60768,60769],{"class":132}," 1.0",[54,60771,965],{"class":64},[54,60773,60775,60777,60779,60781],{"class":56,"line":60774},309,[54,60776,60140],{"class":1711},[54,60778,122],{"class":121},[54,60780,58338],{"class":132},[54,60782,965],{"class":64},[54,60784,60786,60788,60790,60792],{"class":56,"line":60785},310,[54,60787,60298],{"class":1711},[54,60789,122],{"class":121},[54,60791,58372],{"class":132},[54,60793,965],{"class":64},[54,60795,60797,60799,60801,60803],{"class":56,"line":60796},311,[54,60798,60310],{"class":1711},[54,60800,122],{"class":121},[54,60802,13434],{"class":132},[54,60804,965],{"class":64},[54,60806,60808],{"class":56,"line":60807},312,[54,60809,1216],{"class":64},[54,60811,60813,60815,60817],{"class":56,"line":60812},313,[54,60814,60102],{"class":64},[54,60816,6595],{"class":125},[54,60818,932],{"class":64},[54,60820,60822,60824],{"class":56,"line":60821},314,[54,60823,60417],{"class":132},[54,60825,965],{"class":64},[54,60827,60829,60831],{"class":56,"line":60828},315,[54,60830,60417],{"class":132},[54,60832,965],{"class":64},[54,60834,60836,60839],{"class":56,"line":60835},316,[54,60837,60838],{"class":132}," 1.16",[54,60840,965],{"class":64},[54,60842,60844,60847,60850,60852,60854,60857,60860],{"class":56,"line":60843},317,[54,60845,60846],{"class":60}," r",[54,60848,60849],{"class":57863},"\"initial $",[54,60851,59174],{"class":64},[54,60853,37470],{"class":57863},[54,60855,60856],{"class":121},"\\r",[54,60858,60859],{"class":57863},"angle$\"",[54,60861,965],{"class":64},[54,60863,60865,60867,60869,60871],{"class":56,"line":60864},318,[54,60866,60140],{"class":1711},[54,60868,122],{"class":121},[54,60870,58338],{"class":132},[54,60872,965],{"class":64},[54,60874,60876,60878,60880,60883],{"class":56,"line":60875},319,[54,60877,60669],{"class":1711},[54,60879,122],{"class":121},[54,60881,60882],{"class":132},"9",[54,60884,965],{"class":64},[54,60886,60888,60890,60892,60894],{"class":56,"line":60887},320,[54,60889,60681],{"class":1711},[54,60891,122],{"class":121},[54,60893,60686],{"class":279},[54,60895,965],{"class":64},[54,60897,60899,60901,60903,60905],{"class":56,"line":60898},321,[54,60900,60694],{"class":1711},[54,60902,122],{"class":121},[54,60904,60686],{"class":279},[54,60906,965],{"class":64},[54,60908,60910],{"class":56,"line":60909},322,[54,60911,1216],{"class":64},[54,60913,60915],{"class":56,"line":60914},323,[54,60916,105],{"emptyLinePlaceholder":104},[54,60918,60920],{"class":56,"line":60919},324,[54,60921,105],{"emptyLinePlaceholder":104},[54,60923,60925,60927,60930,60932,60934],{"class":56,"line":60924},325,[54,60926,1201],{"class":60},[54,60928,60929],{"class":132}," BLOCH_DURATION_NS",[54,60931,46222],{"class":60},[54,60933,57064],{"class":132},[54,60935,1284],{"class":64},[54,60937,60939,60942,60944,60946],{"class":56,"line":60938},326,[54,60940,60941],{"class":64}," bloch_duration_ns ",[54,60943,122],{"class":121},[54,60945,59063],{"class":125},[54,60947,932],{"class":64},[54,60949,60951],{"class":56,"line":60950},327,[54,60952,60953],{"class":132}," BLOCH_OMEGA_ONRESONANCE_MHZ\n",[54,60955,60957],{"class":56,"line":60956},328,[54,60958,1216],{"class":64},[54,60960,60962,60965],{"class":56,"line":60961},329,[54,60963,60964],{"class":60},"else",[54,60966,1284],{"class":64},[54,60968,60970,60972,60974],{"class":56,"line":60969},330,[54,60971,60941],{"class":64},[54,60973,122],{"class":121},[54,60975,60976],{"class":132}," BLOCH_DURATION_NS\n",[54,60978,60980],{"class":56,"line":60979},331,[54,60981,105],{"emptyLinePlaceholder":104},[54,60983,60985,60988,60990,60992],{"class":56,"line":60984},332,[54,60986,60987],{"class":64},"resonant_trajectory ",[54,60989,122],{"class":121},[54,60991,59122],{"class":125},[54,60993,932],{"class":64},[54,60995,60997,60999,61001,61003],{"class":56,"line":60996},333,[54,60998,59130],{"class":1711},[54,61000,122],{"class":121},[54,61002,58144],{"class":132},[54,61004,965],{"class":64},[54,61006,61008,61010,61012],{"class":56,"line":61007},334,[54,61009,59142],{"class":1711},[54,61011,122],{"class":121},[54,61013,61014],{"class":64},"bloch_duration_ns,\n",[54,61016,61018,61020,61022,61024],{"class":56,"line":61017},335,[54,61019,59154],{"class":1711},[54,61021,122],{"class":121},[54,61023,58187],{"class":132},[54,61025,965],{"class":64},[54,61027,61029],{"class":56,"line":61028},336,[54,61030,136],{"class":64},[54,61032,61034],{"class":56,"line":61033},337,[54,61035,105],{"emptyLinePlaceholder":104},[54,61037,61039,61042,61044,61046],{"class":56,"line":61038},338,[54,61040,61041],{"class":64},"detuned_trajectory ",[54,61043,122],{"class":121},[54,61045,59122],{"class":125},[54,61047,932],{"class":64},[54,61049,61051,61053,61055,61057],{"class":56,"line":61050},339,[54,61052,59130],{"class":1711},[54,61054,122],{"class":121},[54,61056,58154],{"class":132},[54,61058,965],{"class":64},[54,61060,61062,61064,61066],{"class":56,"line":61061},340,[54,61063,59142],{"class":1711},[54,61065,122],{"class":121},[54,61067,61014],{"class":64},[54,61069,61071,61073,61075,61077],{"class":56,"line":61070},341,[54,61072,59154],{"class":1711},[54,61074,122],{"class":121},[54,61076,58187],{"class":132},[54,61078,965],{"class":64},[54,61080,61082],{"class":56,"line":61081},342,[54,61083,136],{"class":64},[54,61085,61087],{"class":56,"line":61086},343,[54,61088,105],{"emptyLinePlaceholder":104},[54,61090,61092,61095,61097],{"class":56,"line":61091},344,[54,61093,61094],{"class":64},"resonant_drive_ghz ",[54,61096,122],{"class":121},[54,61098,890],{"class":64},[54,61100,61102],{"class":56,"line":61101},345,[54,61103,61104],{"class":132}," BLOCH_F0_GHZ\n",[54,61106,61108,61111,61114,61116],{"class":56,"line":61107},346,[54,61109,61110],{"class":121}," +",[54,61112,61113],{"class":132}," BLOCH_RESONANT_DETUNING_MHZ",[54,61115,7785],{"class":121},[54,61117,61118],{"class":132}," 1.0e-3\n",[54,61120,61122],{"class":56,"line":61121},347,[54,61123,136],{"class":64},[54,61125,61127,61130,61132],{"class":56,"line":61126},348,[54,61128,61129],{"class":64},"detuned_drive_ghz ",[54,61131,122],{"class":121},[54,61133,890],{"class":64},[54,61135,61137],{"class":56,"line":61136},349,[54,61138,61104],{"class":132},[54,61140,61142,61144,61147,61149],{"class":56,"line":61141},350,[54,61143,61110],{"class":121},[54,61145,61146],{"class":132}," BLOCH_DETUNED_DETUNING_MHZ",[54,61148,7785],{"class":121},[54,61150,61118],{"class":132},[54,61152,61154],{"class":56,"line":61153},351,[54,61155,136],{"class":64},[54,61157,61159],{"class":56,"line":61158},352,[54,61160,105],{"emptyLinePlaceholder":104},[54,61162,61164,61167,61169,61171,61174,61176,61178,61180,61182],{"class":56,"line":61163},353,[54,61165,61166],{"class":64},"figure ",[54,61168,122],{"class":121},[54,61170,57782],{"class":64},[54,61172,61173],{"class":125},"figure",[54,61175,129],{"class":64},[54,61177,57790],{"class":1711},[54,61179,122],{"class":121},[54,61181,58196],{"class":132},[54,61183,136],{"class":64},[54,61185,61187],{"class":56,"line":61186},354,[54,61188,105],{"emptyLinePlaceholder":104},[54,61190,61192,61195,61197,61199,61202,61204,61207,61209],{"class":56,"line":61191},355,[54,61193,61194],{"class":64},"bloch ",[54,61196,122],{"class":121},[54,61198,59297],{"class":64},[54,61200,61201],{"class":125},"Bloch",[54,61203,129],{"class":64},[54,61205,61206],{"class":1711},"fig",[54,61208,122],{"class":121},[54,61210,61211],{"class":64},"figure)\n",[54,61213,61215,61218,61220],{"class":56,"line":61214},356,[54,61216,61217],{"class":64},"bloch.view ",[54,61219,122],{"class":121},[54,61221,61222],{"class":132}," BLOCH_VIEW\n",[54,61224,61226,61229,61231,61233,61235,61238,61240,61242,61244,61246,61248,61250,61252,61254,61256,61258,61260],{"class":56,"line":61225},357,[54,61227,61228],{"class":64},"bloch.zlabel ",[54,61230,122],{"class":121},[54,61232,7983],{"class":64},[54,61234,1772],{"class":60},[54,61236,61237],{"class":57863},"\"$",[54,61239,59174],{"class":64},[54,61241,37470],{"class":57863},[54,61243,60856],{"class":121},[54,61245,60859],{"class":57863},[54,61247,171],{"class":64},[54,61249,1772],{"class":60},[54,61251,61237],{"class":57863},[54,61253,59174],{"class":64},[54,61255,2842],{"class":57863},[54,61257,60856],{"class":121},[54,61259,60859],{"class":57863},[54,61261,1350],{"class":64},[54,61263,61265,61268,61270],{"class":56,"line":61264},358,[54,61266,61267],{"class":64},"bloch.title ",[54,61269,122],{"class":121},[54,61271,890],{"class":64},[54,61273,61275,61278,61281],{"class":56,"line":61274},359,[54,61276,61277],{"class":279}," \"Bloch-sphere Rabi trajectories",[54,61279,61280],{"class":121},"\\n",[54,61282,61283],{"class":279},"\"\n",[54,61285,61287,61290,61293,61296,61299,61301],{"class":56,"line":61286},360,[54,61288,61289],{"class":60}," rf",[54,61291,61292],{"class":279},"\"resonant $\\Delta\u002F2\\pi=",[54,61294,61295],{"class":132},"{BLOCH_RESONANT_DETUNING_MHZ",[54,61297,61298],{"class":60},":.1f",[54,61300,2428],{"class":132},[54,61302,61303],{"class":279},"$ MHz, \"\n",[54,61305,61307,61309,61312,61315,61317,61319],{"class":56,"line":61306},361,[54,61308,61289],{"class":60},[54,61310,61311],{"class":279},"\"detuned $\\Delta\u002F2\\pi=",[54,61313,61314],{"class":132},"{BLOCH_DETUNED_DETUNING_MHZ",[54,61316,61298],{"class":60},[54,61318,2428],{"class":132},[54,61320,61321],{"class":279},"$ MHz\"\n",[54,61323,61325],{"class":56,"line":61324},362,[54,61326,136],{"class":64},[54,61328,61330],{"class":56,"line":61329},363,[54,61331,105],{"emptyLinePlaceholder":104},[54,61333,61335,61338,61340,61342],{"class":56,"line":61334},364,[54,61336,61337],{"class":64},"bloch_axis ",[54,61339,122],{"class":121},[54,61341,59813],{"class":125},[54,61343,932],{"class":64},[54,61345,61347,61349,61351],{"class":56,"line":61346},365,[54,61348,59821],{"class":1711},[54,61350,122],{"class":121},[54,61352,61353],{"class":64},"bloch,\n",[54,61355,61357,61359,61361],{"class":56,"line":61356},366,[54,61358,59830],{"class":1711},[54,61360,122],{"class":121},[54,61362,61363],{"class":64},"figure,\n",[54,61365,61367],{"class":56,"line":61366},367,[54,61368,136],{"class":64},[54,61370,61372],{"class":56,"line":61371},368,[54,61373,105],{"emptyLinePlaceholder":104},[54,61375,61377,61380],{"class":56,"line":61376},369,[54,61378,61379],{"class":125},"draw_bloch_trajectory",[54,61381,932],{"class":64},[54,61383,61385,61387,61389],{"class":56,"line":61384},370,[54,61386,60049],{"class":1711},[54,61388,122],{"class":121},[54,61390,61391],{"class":64},"bloch_axis,\n",[54,61393,61395,61397,61399,61402,61405],{"class":56,"line":61394},371,[54,61396,60057],{"class":1711},[54,61398,122],{"class":121},[54,61400,61401],{"class":64},"resonant_trajectory[",[54,61403,61404],{"class":279},"\"vectors\"",[54,61406,57624],{"class":64},[54,61408,61410,61412,61414,61416],{"class":56,"line":61409},372,[54,61411,24964],{"class":1711},[54,61413,122],{"class":121},[54,61415,58298],{"class":132},[54,61417,965],{"class":64},[54,61419,61421,61423,61425,61428],{"class":56,"line":61420},373,[54,61422,24988],{"class":1711},[54,61424,122],{"class":121},[54,61426,61427],{"class":279},"\"Resonant state trajectory\"",[54,61429,965],{"class":64},[54,61431,61433],{"class":56,"line":61432},374,[54,61434,136],{"class":64},[54,61436,61438,61440],{"class":56,"line":61437},375,[54,61439,61379],{"class":125},[54,61441,932],{"class":64},[54,61443,61445,61447,61449],{"class":56,"line":61444},376,[54,61446,60049],{"class":1711},[54,61448,122],{"class":121},[54,61450,61391],{"class":64},[54,61452,61454,61456,61458,61461,61463],{"class":56,"line":61453},377,[54,61455,60057],{"class":1711},[54,61457,122],{"class":121},[54,61459,61460],{"class":64},"detuned_trajectory[",[54,61462,61404],{"class":279},[54,61464,57624],{"class":64},[54,61466,61468,61470,61472,61474],{"class":56,"line":61467},378,[54,61469,24964],{"class":1711},[54,61471,122],{"class":121},[54,61473,58308],{"class":132},[54,61475,965],{"class":64},[54,61477,61479,61481,61483,61486],{"class":56,"line":61478},379,[54,61480,24988],{"class":1711},[54,61482,122],{"class":121},[54,61484,61485],{"class":279},"\"Detuned state trajectory\"",[54,61487,965],{"class":64},[54,61489,61491],{"class":56,"line":61490},380,[54,61492,136],{"class":64},[54,61494,61496],{"class":56,"line":61495},381,[54,61497,105],{"emptyLinePlaceholder":104},[54,61499,61501,61503,61506],{"class":56,"line":61500},382,[54,61502,1201],{"class":60},[54,61504,61505],{"class":132}," BLOCH_SHOW_ENDPOINT_MARKERS",[54,61507,1284],{"class":64},[54,61509,61511,61514],{"class":56,"line":61510},383,[54,61512,61513],{"class":125}," draw_endpoint_marker",[54,61515,932],{"class":64},[54,61517,61519,61522,61524],{"class":56,"line":61518},384,[54,61520,61521],{"class":1711}," axis",[54,61523,122],{"class":121},[54,61525,61391],{"class":64},[54,61527,61529,61532,61534,61536,61538,61541,61543,61545],{"class":56,"line":61528},385,[54,61530,61531],{"class":1711}," vector",[54,61533,122],{"class":121},[54,61535,61401],{"class":64},[54,61537,61404],{"class":279},[54,61539,61540],{"class":64},"][:, ",[54,61542,1923],{"class":121},[54,61544,174],{"class":132},[54,61546,57624],{"class":64},[54,61548,61550,61552,61554,61556],{"class":56,"line":61549},386,[54,61551,60140],{"class":1711},[54,61553,122],{"class":121},[54,61555,58298],{"class":132},[54,61557,965],{"class":64},[54,61559,61561,61563,61565,61568],{"class":56,"line":61560},387,[54,61562,60163],{"class":1711},[54,61564,122],{"class":121},[54,61566,61567],{"class":279},"\"Resonant final state\"",[54,61569,965],{"class":64},[54,61571,61573],{"class":56,"line":61572},388,[54,61574,1216],{"class":64},[54,61576,61578,61580],{"class":56,"line":61577},389,[54,61579,61513],{"class":125},[54,61581,932],{"class":64},[54,61583,61585,61587,61589],{"class":56,"line":61584},390,[54,61586,61521],{"class":1711},[54,61588,122],{"class":121},[54,61590,61391],{"class":64},[54,61592,61594,61596,61598,61600,61602,61604,61606,61608],{"class":56,"line":61593},391,[54,61595,61531],{"class":1711},[54,61597,122],{"class":121},[54,61599,61460],{"class":64},[54,61601,61404],{"class":279},[54,61603,61540],{"class":64},[54,61605,1923],{"class":121},[54,61607,174],{"class":132},[54,61609,57624],{"class":64},[54,61611,61613,61615,61617,61619],{"class":56,"line":61612},392,[54,61614,60140],{"class":1711},[54,61616,122],{"class":121},[54,61618,58308],{"class":132},[54,61620,965],{"class":64},[54,61622,61624,61626,61628,61631],{"class":56,"line":61623},393,[54,61625,60163],{"class":1711},[54,61627,122],{"class":121},[54,61629,61630],{"class":279},"\"Detuned final state\"",[54,61632,965],{"class":64},[54,61634,61636],{"class":56,"line":61635},394,[54,61637,1216],{"class":64},[54,61639,61641,61644],{"class":56,"line":61640},395,[54,61642,61643],{"class":125}," add_initial_state_label",[54,61645,61646],{"class":64},"(bloch_axis)\n",[54,61648,61650],{"class":56,"line":61649},396,[54,61651,105],{"emptyLinePlaceholder":104},[54,61653,61655,61657,61660],{"class":56,"line":61654},397,[54,61656,1201],{"class":60},[54,61658,61659],{"class":132}," BLOCH_SHOW_CONTROL_AXIS_ARROWS",[54,61661,1284],{"class":64},[54,61663,61665,61668],{"class":56,"line":61664},398,[54,61666,61667],{"class":125}," draw_control_axis_arrow",[54,61669,932],{"class":64},[54,61671,61673,61675,61677],{"class":56,"line":61672},399,[54,61674,61521],{"class":1711},[54,61676,122],{"class":121},[54,61678,61391],{"class":64},[54,61680,61682,61684,61686,61688,61691],{"class":56,"line":61681},400,[54,61683,61531],{"class":1711},[54,61685,122],{"class":121},[54,61687,61401],{"class":64},[54,61689,61690],{"class":279},"\"control_axis\"",[54,61692,57624],{"class":64},[54,61694,61696,61698,61700,61702],{"class":56,"line":61695},401,[54,61697,60140],{"class":1711},[54,61699,122],{"class":121},[54,61701,58318],{"class":132},[54,61703,965],{"class":64},[54,61705,61707,61709,61711,61713,61716,61719,61722,61725,61728],{"class":56,"line":61706},402,[54,61708,60163],{"class":1711},[54,61710,122],{"class":121},[54,61712,1772],{"class":60},[54,61714,61715],{"class":57863},"\"Resonant control axis $",[54,61717,61718],{"class":121},"\\h",[54,61720,61721],{"class":57863},"at{n}_{",[54,61723,61724],{"class":121},"\\m",[54,61726,61727],{"class":57863},"athrm{res}}$\"",[54,61729,965],{"class":64},[54,61731,61733],{"class":56,"line":61732},403,[54,61734,1216],{"class":64},[54,61736,61738,61740],{"class":56,"line":61737},404,[54,61739,61667],{"class":125},[54,61741,932],{"class":64},[54,61743,61745,61747,61749],{"class":56,"line":61744},405,[54,61746,61521],{"class":1711},[54,61748,122],{"class":121},[54,61750,61391],{"class":64},[54,61752,61754,61756,61758,61760,61762],{"class":56,"line":61753},406,[54,61755,61531],{"class":1711},[54,61757,122],{"class":121},[54,61759,61460],{"class":64},[54,61761,61690],{"class":279},[54,61763,57624],{"class":64},[54,61765,61767,61769,61771,61773],{"class":56,"line":61766},407,[54,61768,60140],{"class":1711},[54,61770,122],{"class":121},[54,61772,58328],{"class":132},[54,61774,965],{"class":64},[54,61776,61778,61780,61782,61784,61787,61789,61791,61793,61796],{"class":56,"line":61777},408,[54,61779,60163],{"class":1711},[54,61781,122],{"class":121},[54,61783,1772],{"class":60},[54,61785,61786],{"class":57863},"\"Detuned control axis $",[54,61788,61718],{"class":121},[54,61790,61721],{"class":57863},[54,61792,61724],{"class":121},[54,61794,61795],{"class":57863},"athrm{det}}$\"",[54,61797,965],{"class":64},[54,61799,61801],{"class":56,"line":61800},409,[54,61802,1216],{"class":64},[54,61804,61806],{"class":56,"line":61805},410,[54,61807,105],{"emptyLinePlaceholder":104},[54,61809,61811,61813,61816,61818,61820],{"class":56,"line":61810},411,[54,61812,1201],{"class":60},[54,61814,61815],{"class":132}," BLOCH_SHOW_ARROW_TIP_LABELS",[54,61817,57409],{"class":60},[54,61819,61659],{"class":132},[54,61821,1284],{"class":64},[54,61823,61825,61828],{"class":56,"line":61824},412,[54,61826,61827],{"class":125}," add_arrow_tip_label",[54,61829,932],{"class":64},[54,61831,61833,61835,61837],{"class":56,"line":61832},413,[54,61834,61521],{"class":1711},[54,61836,122],{"class":121},[54,61838,61391],{"class":64},[54,61840,61842,61844,61846,61848,61850],{"class":56,"line":61841},414,[54,61843,61531],{"class":1711},[54,61845,122],{"class":121},[54,61847,61401],{"class":64},[54,61849,61690],{"class":279},[54,61851,57624],{"class":64},[54,61853,61855,61858,61860,61862,61864,61866,61868,61870,61872],{"class":56,"line":61854},415,[54,61856,61857],{"class":1711}," text",[54,61859,122],{"class":121},[54,61861,1772],{"class":60},[54,61863,61237],{"class":57863},[54,61865,61718],{"class":121},[54,61867,61721],{"class":57863},[54,61869,61724],{"class":121},[54,61871,61727],{"class":57863},[54,61873,965],{"class":64},[54,61875,61877,61879,61881,61883],{"class":56,"line":61876},416,[54,61878,60140],{"class":1711},[54,61880,122],{"class":121},[54,61882,58318],{"class":132},[54,61884,965],{"class":64},[54,61886,61888],{"class":56,"line":61887},417,[54,61889,1216],{"class":64},[54,61891,61893,61895],{"class":56,"line":61892},418,[54,61894,61827],{"class":125},[54,61896,932],{"class":64},[54,61898,61900,61902,61904],{"class":56,"line":61899},419,[54,61901,61521],{"class":1711},[54,61903,122],{"class":121},[54,61905,61391],{"class":64},[54,61907,61909,61911,61913,61915,61917],{"class":56,"line":61908},420,[54,61910,61531],{"class":1711},[54,61912,122],{"class":121},[54,61914,61460],{"class":64},[54,61916,61690],{"class":279},[54,61918,57624],{"class":64},[54,61920,61922,61924,61926,61928,61930,61932,61934,61936,61938],{"class":56,"line":61921},421,[54,61923,61857],{"class":1711},[54,61925,122],{"class":121},[54,61927,1772],{"class":60},[54,61929,61237],{"class":57863},[54,61931,61718],{"class":121},[54,61933,61721],{"class":57863},[54,61935,61724],{"class":121},[54,61937,61795],{"class":57863},[54,61939,965],{"class":64},[54,61941,61943,61945,61947,61949],{"class":56,"line":61942},422,[54,61944,60140],{"class":1711},[54,61946,122],{"class":121},[54,61948,58328],{"class":132},[54,61950,965],{"class":64},[54,61952,61954],{"class":56,"line":61953},423,[54,61955,1216],{"class":64},[54,61957,61959],{"class":56,"line":61958},424,[54,61960,105],{"emptyLinePlaceholder":104},[54,61962,61964,61966,61969],{"class":56,"line":61963},425,[54,61965,1201],{"class":60},[54,61967,61968],{"class":132}," BLOCH_SHOW_LEGEND",[54,61970,1284],{"class":64},[54,61972,61974,61977,61979],{"class":56,"line":61973},426,[54,61975,61976],{"class":64}," legend_handles ",[54,61978,122],{"class":121},[54,61980,1733],{"class":64},[54,61982,61984,61987],{"class":56,"line":61983},427,[54,61985,61986],{"class":125}," Line2D",[54,61988,932],{"class":64},[54,61990,61992,61995,61997],{"class":56,"line":61991},428,[54,61993,61994],{"class":64}," [",[54,61996,150],{"class":132},[54,61998,57624],{"class":64},[54,62000,62002,62004,62006],{"class":56,"line":62001},429,[54,62003,61994],{"class":64},[54,62005,150],{"class":132},[54,62007,57624],{"class":64},[54,62009,62011,62014,62016,62018],{"class":56,"line":62010},430,[54,62012,62013],{"class":1711}," color",[54,62015,122],{"class":121},[54,62017,58298],{"class":132},[54,62019,965],{"class":64},[54,62021,62023,62026,62028,62030],{"class":56,"line":62022},431,[54,62024,62025],{"class":1711}," linewidth",[54,62027,122],{"class":121},[54,62029,58352],{"class":132},[54,62031,965],{"class":64},[54,62033,62035,62038,62040,62042],{"class":56,"line":62034},432,[54,62036,62037],{"class":1711}," label",[54,62039,122],{"class":121},[54,62041,61427],{"class":279},[54,62043,965],{"class":64},[54,62045,62047],{"class":56,"line":62046},433,[54,62048,59790],{"class":64},[54,62050,62052,62054],{"class":56,"line":62051},434,[54,62053,61986],{"class":125},[54,62055,932],{"class":64},[54,62057,62059,62061,62063],{"class":56,"line":62058},435,[54,62060,61994],{"class":64},[54,62062,150],{"class":132},[54,62064,57624],{"class":64},[54,62066,62068,62070,62072],{"class":56,"line":62067},436,[54,62069,61994],{"class":64},[54,62071,150],{"class":132},[54,62073,57624],{"class":64},[54,62075,62077,62079,62081,62083],{"class":56,"line":62076},437,[54,62078,62013],{"class":1711},[54,62080,122],{"class":121},[54,62082,58308],{"class":132},[54,62084,965],{"class":64},[54,62086,62088,62090,62092,62094],{"class":56,"line":62087},438,[54,62089,62025],{"class":1711},[54,62091,122],{"class":121},[54,62093,58352],{"class":132},[54,62095,965],{"class":64},[54,62097,62099,62101,62103,62105],{"class":56,"line":62098},439,[54,62100,62037],{"class":1711},[54,62102,122],{"class":121},[54,62104,61485],{"class":279},[54,62106,965],{"class":64},[54,62108,62110],{"class":56,"line":62109},440,[54,62111,59790],{"class":64},[54,62113,62115,62117],{"class":56,"line":62114},441,[54,62116,61986],{"class":125},[54,62118,932],{"class":64},[54,62120,62122,62124,62126],{"class":56,"line":62121},442,[54,62123,61994],{"class":64},[54,62125,150],{"class":132},[54,62127,57624],{"class":64},[54,62129,62131,62133,62135],{"class":56,"line":62130},443,[54,62132,61994],{"class":64},[54,62134,150],{"class":132},[54,62136,57624],{"class":64},[54,62138,62140,62142,62144,62146],{"class":56,"line":62139},444,[54,62141,62013],{"class":1711},[54,62143,122],{"class":121},[54,62145,58318],{"class":132},[54,62147,965],{"class":64},[54,62149,62151,62153,62155,62157],{"class":56,"line":62150},445,[54,62152,62025],{"class":1711},[54,62154,122],{"class":121},[54,62156,58362],{"class":132},[54,62158,965],{"class":64},[54,62160,62162,62164,62166,62168,62170,62172,62174,62176,62178],{"class":56,"line":62161},446,[54,62163,62037],{"class":1711},[54,62165,122],{"class":121},[54,62167,1772],{"class":60},[54,62169,61715],{"class":57863},[54,62171,61718],{"class":121},[54,62173,61721],{"class":57863},[54,62175,61724],{"class":121},[54,62177,61727],{"class":57863},[54,62179,965],{"class":64},[54,62181,62183],{"class":56,"line":62182},447,[54,62184,59790],{"class":64},[54,62186,62188,62190],{"class":56,"line":62187},448,[54,62189,61986],{"class":125},[54,62191,932],{"class":64},[54,62193,62195,62197,62199],{"class":56,"line":62194},449,[54,62196,61994],{"class":64},[54,62198,150],{"class":132},[54,62200,57624],{"class":64},[54,62202,62204,62206,62208],{"class":56,"line":62203},450,[54,62205,61994],{"class":64},[54,62207,150],{"class":132},[54,62209,57624],{"class":64},[54,62211,62213,62215,62217,62219],{"class":56,"line":62212},451,[54,62214,62013],{"class":1711},[54,62216,122],{"class":121},[54,62218,58328],{"class":132},[54,62220,965],{"class":64},[54,62222,62224,62226,62228,62230],{"class":56,"line":62223},452,[54,62225,62025],{"class":1711},[54,62227,122],{"class":121},[54,62229,58362],{"class":132},[54,62231,965],{"class":64},[54,62233,62235,62237,62239,62241,62243,62245,62247,62249,62251],{"class":56,"line":62234},453,[54,62236,62037],{"class":1711},[54,62238,122],{"class":121},[54,62240,1772],{"class":60},[54,62242,61786],{"class":57863},[54,62244,61718],{"class":121},[54,62246,61721],{"class":57863},[54,62248,61724],{"class":121},[54,62250,61795],{"class":57863},[54,62252,965],{"class":64},[54,62254,62256],{"class":56,"line":62255},454,[54,62257,59790],{"class":64},[54,62259,62261],{"class":56,"line":62260},455,[54,62262,58597],{"class":64},[54,62264,62266],{"class":56,"line":62265},456,[54,62267,105],{"emptyLinePlaceholder":104},[54,62269,62271,62274,62276],{"class":56,"line":62270},457,[54,62272,62273],{"class":64}," bloch_axis.",[54,62275,25114],{"class":125},[54,62277,932],{"class":64},[54,62279,62281,62284,62286],{"class":56,"line":62280},458,[54,62282,62283],{"class":1711}," handles",[54,62285,122],{"class":121},[54,62287,62288],{"class":64},"legend_handles,\n",[54,62290,62292,62295,62297,62299],{"class":56,"line":62291},459,[54,62293,62294],{"class":1711}," loc",[54,62296,122],{"class":121},[54,62298,58406],{"class":132},[54,62300,965],{"class":64},[54,62302,62304,62307,62309,62311],{"class":56,"line":62303},460,[54,62305,62306],{"class":1711}," bbox_to_anchor",[54,62308,122],{"class":121},[54,62310,58416],{"class":132},[54,62312,965],{"class":64},[54,62314,62316,62319,62321,62323],{"class":56,"line":62315},461,[54,62317,62318],{"class":1711}," frameon",[54,62320,122],{"class":121},[54,62322,13434],{"class":132},[54,62324,965],{"class":64},[54,62326,62328,62331,62333,62335],{"class":56,"line":62327},462,[54,62329,62330],{"class":1711}," borderaxespad",[54,62332,122],{"class":121},[54,62334,56271],{"class":132},[54,62336,965],{"class":64},[54,62338,62340],{"class":56,"line":62339},463,[54,62341,1216],{"class":64},[54,62343,62345],{"class":56,"line":62344},464,[54,62346,105],{"emptyLinePlaceholder":104},[54,62348,62350,62352,62354],{"class":56,"line":62349},465,[54,62351,24932],{"class":64},[54,62353,25169],{"class":125},[54,62355,330],{"class":64},[54,62357,62359],{"class":56,"line":62358},466,[54,62360,105],{"emptyLinePlaceholder":104},[54,62362,62364,62366,62369],{"class":56,"line":62363},467,[54,62365,1201],{"class":60},[54,62367,62368],{"class":132}," BLOCH_PRINT_SUMMARY",[54,62370,1284],{"class":64},[54,62372,62374,62377,62379,62382,62384],{"class":56,"line":62373},468,[54,62375,62376],{"class":64}," resonant_axis ",[54,62378,122],{"class":121},[54,62380,62381],{"class":64}," resonant_trajectory[",[54,62383,61690],{"class":279},[54,62385,1350],{"class":64},[54,62387,62389,62392,62394,62397,62399],{"class":56,"line":62388},469,[54,62390,62391],{"class":64}," detuned_axis ",[54,62393,122],{"class":121},[54,62395,62396],{"class":64}," detuned_trajectory[",[54,62398,61690],{"class":279},[54,62400,1350],{"class":64},[54,62402,62404],{"class":56,"line":62403},470,[54,62405,105],{"emptyLinePlaceholder":104},[54,62407,62409,62411,62413,62416],{"class":56,"line":62408},471,[54,62410,1616],{"class":121},[54,62412,129],{"class":64},[54,62414,62415],{"class":279},"\"Bloch-sphere trajectory summary\"",[54,62417,136],{"class":64},[54,62419,62421,62423,62425,62427,62430,62433,62436,62438,62441],{"class":56,"line":62420},472,[54,62422,1616],{"class":121},[54,62424,129],{"class":64},[54,62426,21834],{"class":60},[54,62428,62429],{"class":279},"\"Natural transition frequency f0: ",[54,62431,62432],{"class":132},"{BLOCH_F0_GHZ",[54,62434,62435],{"class":60},":.6f",[54,62437,2428],{"class":132},[54,62439,62440],{"class":279}," GHz\"",[54,62442,136],{"class":64},[54,62444,62446,62448,62450,62452,62455,62457,62460,62462,62464,62466],{"class":56,"line":62445},473,[54,62447,1616],{"class":121},[54,62449,129],{"class":64},[54,62451,21834],{"class":60},[54,62453,62454],{"class":279},"\"Resonant drive frequency: ",[54,62456,2366],{"class":132},[54,62458,62459],{"class":64},"resonant_drive_ghz",[54,62461,62435],{"class":60},[54,62463,2428],{"class":132},[54,62465,62440],{"class":279},[54,62467,136],{"class":64},[54,62469,62471,62473,62475,62477,62480,62482,62485,62487,62489,62491],{"class":56,"line":62470},474,[54,62472,1616],{"class":121},[54,62474,129],{"class":64},[54,62476,21834],{"class":60},[54,62478,62479],{"class":279},"\"Detuned drive frequency: ",[54,62481,2366],{"class":132},[54,62483,62484],{"class":64},"detuned_drive_ghz",[54,62486,62435],{"class":60},[54,62488,2428],{"class":132},[54,62490,62440],{"class":279},[54,62492,136],{"class":64},[54,62494,62496,62498,62500,62502,62505,62508,62511,62513,62516],{"class":56,"line":62495},475,[54,62497,1616],{"class":121},[54,62499,129],{"class":64},[54,62501,21834],{"class":60},[54,62503,62504],{"class":279},"\"On-resonance Rabi rate: ",[54,62506,62507],{"class":132},"{BLOCH_OMEGA_ONRESONANCE_MHZ",[54,62509,62510],{"class":60},":.3f",[54,62512,2428],{"class":132},[54,62514,62515],{"class":279}," MHz\"",[54,62517,136],{"class":64},[54,62519,62521,62523,62525,62527,62530,62532,62535,62537,62539,62542],{"class":56,"line":62520},476,[54,62522,1616],{"class":121},[54,62524,129],{"class":64},[54,62526,21834],{"class":60},[54,62528,62529],{"class":279},"\"Plotted duration: ",[54,62531,2366],{"class":132},[54,62533,62534],{"class":64},"bloch_duration_ns",[54,62536,62510],{"class":60},[54,62538,2428],{"class":132},[54,62540,62541],{"class":279}," ns\"",[54,62543,136],{"class":64},[54,62545,62547,62549],{"class":56,"line":62546},477,[54,62548,1616],{"class":121},[54,62550,932],{"class":64},[54,62552,62554],{"class":56,"line":62553},478,[54,62555,62556],{"class":279}," \"Resonant control axis n_res: \"\n",[54,62558,62560,62563,62566,62568,62571,62573,62575,62577,62579],{"class":56,"line":62559},479,[54,62561,62562],{"class":60}," f",[54,62564,62565],{"class":279},"\"(",[54,62567,2366],{"class":132},[54,62569,62570],{"class":64},"resonant_axis[",[54,62572,150],{"class":132},[54,62574,11829],{"class":64},[54,62576,62435],{"class":60},[54,62578,2428],{"class":132},[54,62580,62581],{"class":279},", \"\n",[54,62583,62585,62587,62589,62591,62593,62595,62597,62599,62601],{"class":56,"line":62584},480,[54,62586,62562],{"class":60},[54,62588,21837],{"class":279},[54,62590,2366],{"class":132},[54,62592,62570],{"class":64},[54,62594,174],{"class":132},[54,62596,11829],{"class":64},[54,62598,62435],{"class":60},[54,62600,2428],{"class":132},[54,62602,62581],{"class":279},[54,62604,62606,62608,62610,62612,62614,62616,62618,62620,62622],{"class":56,"line":62605},481,[54,62607,62562],{"class":60},[54,62609,21837],{"class":279},[54,62611,2366],{"class":132},[54,62613,62570],{"class":64},[54,62615,133],{"class":132},[54,62617,11829],{"class":64},[54,62619,62435],{"class":60},[54,62621,2428],{"class":132},[54,62623,62624],{"class":279},")\"\n",[54,62626,62628],{"class":56,"line":62627},482,[54,62629,1216],{"class":64},[54,62631,62633,62635],{"class":56,"line":62632},483,[54,62634,1616],{"class":121},[54,62636,932],{"class":64},[54,62638,62640],{"class":56,"line":62639},484,[54,62641,62642],{"class":279}," \"Detuned control axis n_det: \"\n",[54,62644,62646,62648,62650,62652,62655,62657,62659,62661,62663],{"class":56,"line":62645},485,[54,62647,62562],{"class":60},[54,62649,62565],{"class":279},[54,62651,2366],{"class":132},[54,62653,62654],{"class":64},"detuned_axis[",[54,62656,150],{"class":132},[54,62658,11829],{"class":64},[54,62660,62435],{"class":60},[54,62662,2428],{"class":132},[54,62664,62581],{"class":279},[54,62666,62668,62670,62672,62674,62676,62678,62680,62682,62684],{"class":56,"line":62667},486,[54,62669,62562],{"class":60},[54,62671,21837],{"class":279},[54,62673,2366],{"class":132},[54,62675,62654],{"class":64},[54,62677,174],{"class":132},[54,62679,11829],{"class":64},[54,62681,62435],{"class":60},[54,62683,2428],{"class":132},[54,62685,62581],{"class":279},[54,62687,62689,62691,62693,62695,62697,62699,62701,62703,62705],{"class":56,"line":62688},487,[54,62690,62562],{"class":60},[54,62692,21837],{"class":279},[54,62694,2366],{"class":132},[54,62696,62654],{"class":64},[54,62698,133],{"class":132},[54,62700,11829],{"class":64},[54,62702,62435],{"class":60},[54,62704,2428],{"class":132},[54,62706,62624],{"class":279},[54,62708,62710],{"class":56,"line":62709},488,[54,62711,1216],{"class":64},[54,62713,62715,62717],{"class":56,"line":62714},489,[54,62716,1616],{"class":121},[54,62718,932],{"class":64},[54,62720,62722],{"class":56,"line":62721},490,[54,62723,62724],{"class":279}," \"Resonant final excited-state probability: \"\n",[54,62726,62728,62730,62732,62734,62736,62739,62741,62743,62745,62747,62749,62751],{"class":56,"line":62727},491,[54,62729,62562],{"class":60},[54,62731,21837],{"class":279},[54,62733,2366],{"class":132},[54,62735,61401],{"class":64},[54,62737,62738],{"class":279},"'excited_probability'",[54,62740,1311],{"class":64},[54,62742,1923],{"class":121},[54,62744,174],{"class":132},[54,62746,11829],{"class":64},[54,62748,62435],{"class":60},[54,62750,2428],{"class":132},[54,62752,61283],{"class":279},[54,62754,62756],{"class":56,"line":62755},492,[54,62757,1216],{"class":64},[54,62759,62761,62763],{"class":56,"line":62760},493,[54,62762,1616],{"class":121},[54,62764,932],{"class":64},[54,62766,62768],{"class":56,"line":62767},494,[54,62769,62770],{"class":279}," \"Detuned final excited-state probability: \"\n",[54,62772,62774,62776,62778,62780,62782,62784,62786,62788,62790,62792,62794,62796],{"class":56,"line":62773},495,[54,62775,62562],{"class":60},[54,62777,21837],{"class":279},[54,62779,2366],{"class":132},[54,62781,61460],{"class":64},[54,62783,62738],{"class":279},[54,62785,1311],{"class":64},[54,62787,1923],{"class":121},[54,62789,174],{"class":132},[54,62791,11829],{"class":64},[54,62793,62435],{"class":60},[54,62795,2428],{"class":132},[54,62797,61283],{"class":279},[54,62799,62801],{"class":56,"line":62800},496,[54,62802,1216],{"class":64},[54,62804,62806,62808],{"class":56,"line":62805},497,[54,62807,1616],{"class":121},[54,62809,932],{"class":64},[54,62811,62813],{"class":56,"line":62812},498,[54,62814,62815],{"class":279}," \"Detuned generalized Rabi frequency: \"\n",[54,62817,62819,62821,62823,62825,62827,62830,62832,62834,62836],{"class":56,"line":62818},499,[54,62820,62562],{"class":60},[54,62822,21837],{"class":279},[54,62824,2366],{"class":132},[54,62826,61460],{"class":64},[54,62828,62829],{"class":279},"'generalized_rabi_mhz'",[54,62831,11829],{"class":64},[54,62833,62435],{"class":60},[54,62835,2428],{"class":132},[54,62837,62838],{"class":279}," MHz\"\n",[54,62840,62842],{"class":56,"line":62841},500,[54,62843,1216],{"class":64},[380,62845],{"alt":62846,"src":62847},"Output 2 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-02.webp",[44,62849,62852],{"className":62850,"code":62851,"language":6595,"meta":49},[8087],"Bloch-sphere trajectory summary\nNatural transition frequency f0: 5.000000 GHz\nResonant drive frequency: 5.000000 GHz\nDetuned drive frequency: 5.040000 GHz\nOn-resonance Rabi rate: 20.000 MHz\nPlotted duration: 25.000 ns\nResonant control axis n_res: (1.000000, 0.000000, 0.000000)\nDetuned control axis n_det: (0.447214, 0.000000, 0.894427)\nResonant final excited-state probability: 1.000000\nDetuned final excited-state probability: 0.026264\nDetuned generalized Rabi frequency: 44.721360 MHz\n",[51,62853,62851],{"__ignoreMap":49},[44,62855,62857],{"className":46,"code":62856,"language":48,"meta":49,"style":49},"# --- High-resolution 3D surface with colormap & legend ---\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# Controls for surface quality and appearance\nUPSAMPLE_FACTOR: int = 2 # 2=~4× more faces; try 3 for even smoother\nSURFACE_CMAP: str = 'inferno' # e.g., 'viridis', 'plasma', 'magma', 'inferno'\nZ_MIN, Z_MAX = 0.0, 1.0 # clamp\u002Fnormalize expected Pe range\n\n# --- Bilinear upsampling (SciPy-free) ---\n# 1) Upsample frequency axis for each fixed τ\nfreq_hi = np.linspace(freq_ghz.min(), freq_ghz.max(),\n UPSAMPLE_FACTOR * (len(freq_ghz) - 1) + 1)\nPe_tau_f = np.empty((len(dur_ns), len(freq_hi)), dtype=float)\nfor i in range(len(dur_ns)):\n Pe_tau_f[i, :] = np.interp(freq_hi, freq_ghz, Pe[i, :])\n\n# 2) Upsample τ axis for each fixed frequency\ndur_hi = np.linspace(dur_ns.min(), dur_ns.max(),\n UPSAMPLE_FACTOR * (len(dur_ns) - 1) + 1)\nPe_hi = np.empty((len(dur_hi), len(freq_hi)), dtype=float)\nfor j in range(len(freq_hi)):\n Pe_hi[:, j] = np.interp(dur_hi, dur_ns, Pe_tau_f[:, j])\n\n# Mesh for plotting\nF_hi, T_hi = np.meshgrid(freq_hi, dur_hi, indexing='xy')\n\n# --- Plot ---\nfig = plt.figure(figsize=(8, 6), dpi=200)\nax = fig.add_subplot(111, projection='3d')\n\nsurf = ax.plot_surface(\n F_hi, T_hi, Pe_hi,\n rstride=1, cstride=1, # use every row\u002Fcolumn (after upsampling)\n linewidth=0, antialiased=True,\n cmap=SURFACE_CMAP,\n vmin=Z_MIN, vmax=Z_MAX,\n)\n\nax.set_xlabel('f (GHz)')\nax.set_ylabel(r'$\\tau$ (ns)')\nax.set_zlabel(r'$P_e$')\nax.set_title('Rabi $P_e$ surface')\n\n# Colorbar (legend for the colormap)\ncb = fig.colorbar(surf, ax=ax, shrink=0.7, pad=0.08)\ncb.set_label(r'$P_e$')\n\nfig.tight_layout()\nplt.show()\n",[51,62858,62859,62864,62876,62886,62896,62900,62905,62921,62938,62959,62963,62968,62973,62997,63023,63055,63072,63087,63091,63096,63119,63145,63173,63190,63204,63208,63213,63235,63239,63244,63282,63311,63315,63329,63334,63358,63378,63389,63409,63413,63417,63430,63460,63475,63488,63492,63497,63536,63552,63556,63564],{"__ignoreMap":49},[54,62860,62861],{"class":56,"line":57},[54,62862,62863],{"class":111},"# --- High-resolution 3D surface with colormap & legend ---\n",[54,62865,62866,62868,62871,62873],{"class":56,"line":74},[54,62867,77],{"class":60},[54,62869,62870],{"class":64}," mpl_toolkits.mplot3d ",[54,62872,61],{"class":60},[54,62874,62875],{"class":64}," Axes3D\n",[54,62877,62878,62880,62882,62884],{"class":56,"line":88},[54,62879,61],{"class":60},[54,62881,65],{"class":64},[54,62883,68],{"class":60},[54,62885,71],{"class":64},[54,62887,62888,62890,62892,62894],{"class":56,"line":101},[54,62889,61],{"class":60},[54,62891,56720],{"class":64},[54,62893,68],{"class":60},[54,62895,715],{"class":64},[54,62897,62898],{"class":56,"line":108},[54,62899,105],{"emptyLinePlaceholder":104},[54,62901,62902],{"class":56,"line":115},[54,62903,62904],{"class":111},"# Controls for surface quality and appearance\n",[54,62906,62907,62910,62912,62914,62916,62918],{"class":56,"line":139},[54,62908,62909],{"class":132},"UPSAMPLE_FACTOR",[54,62911,959],{"class":64},[54,62913,976],{"class":121},[54,62915,979],{"class":121},[54,62917,1872],{"class":132},[54,62919,62920],{"class":111}," # 2=~4× more faces; try 3 for even smoother\n",[54,62922,62923,62926,62928,62930,62932,62935],{"class":56,"line":159},[54,62924,62925],{"class":132},"SURFACE_CMAP",[54,62927,959],{"class":64},[54,62929,58454],{"class":121},[54,62931,979],{"class":121},[54,62933,62934],{"class":279}," 'inferno'",[54,62936,62937],{"class":111}," # e.g., 'viridis', 'plasma', 'magma', 'inferno'\n",[54,62939,62940,62943,62945,62948,62950,62952,62954,62956],{"class":56,"line":183},[54,62941,62942],{"class":132},"Z_MIN",[54,62944,171],{"class":64},[54,62946,62947],{"class":132},"Z_MAX",[54,62949,979],{"class":121},[54,62951,56924],{"class":132},[54,62953,171],{"class":64},[54,62955,57484],{"class":132},[54,62957,62958],{"class":111}," # clamp\u002Fnormalize expected Pe range\n",[54,62960,62961],{"class":56,"line":210},[54,62962,105],{"emptyLinePlaceholder":104},[54,62964,62965],{"class":56,"line":227},[54,62966,62967],{"class":111},"# --- Bilinear upsampling (SciPy-free) ---\n",[54,62969,62970],{"class":56,"line":232},[54,62971,62972],{"class":111},"# 1) Upsample frequency axis for each fixed τ\n",[54,62974,62975,62978,62980,62982,62984,62987,62989,62992,62994],{"class":56,"line":246},[54,62976,62977],{"class":64},"freq_hi ",[54,62979,122],{"class":121},[54,62981,1172],{"class":64},[54,62983,57503],{"class":125},[54,62985,62986],{"class":64},"(freq_ghz.",[54,62988,24786],{"class":125},[54,62990,62991],{"class":64},"(), freq_ghz.",[54,62993,36269],{"class":125},[54,62995,62996],{"class":64},"(),\n",[54,62998,62999,63002,63004,63006,63008,63011,63013,63015,63017,63019,63021],{"class":56,"line":259},[54,63000,63001],{"class":132}," UPSAMPLE_FACTOR",[54,63003,7785],{"class":121},[54,63005,58201],{"class":64},[54,63007,1319],{"class":121},[54,63009,63010],{"class":64},"(freq_ghz) ",[54,63012,1923],{"class":121},[54,63014,982],{"class":132},[54,63016,204],{"class":64},[54,63018,1663],{"class":121},[54,63020,982],{"class":132},[54,63022,136],{"class":64},[54,63024,63025,63028,63030,63032,63035,63038,63040,63043,63045,63048,63050,63053],{"class":56,"line":264},[54,63026,63027],{"class":64},"Pe_tau_f ",[54,63029,122],{"class":121},[54,63031,1172],{"class":64},[54,63033,63034],{"class":125},"empty",[54,63036,63037],{"class":64},"((",[54,63039,1319],{"class":121},[54,63041,63042],{"class":64},"(dur_ns), ",[54,63044,1319],{"class":121},[54,63046,63047],{"class":64},"(freq_hi)), ",[54,63049,1712],{"class":1711},[54,63051,63052],{"class":121},"=float",[54,63054,136],{"class":64},[54,63056,63057,63059,63061,63063,63065,63067,63069],{"class":56,"line":270},[54,63058,1186],{"class":60},[54,63060,7802],{"class":64},[54,63062,1192],{"class":60},[54,63064,7777],{"class":121},[54,63066,129],{"class":64},[54,63068,1319],{"class":121},[54,63070,63071],{"class":64},"(dur_ns)):\n",[54,63073,63074,63077,63079,63081,63084],{"class":56,"line":285},[54,63075,63076],{"class":64}," Pe_tau_f[i, :] ",[54,63078,122],{"class":121},[54,63080,1172],{"class":64},[54,63082,63083],{"class":125},"interp",[54,63085,63086],{"class":64},"(freq_hi, freq_ghz, Pe[i, :])\n",[54,63088,63089],{"class":56,"line":297},[54,63090,105],{"emptyLinePlaceholder":104},[54,63092,63093],{"class":56,"line":310},[54,63094,63095],{"class":111},"# 2) Upsample τ axis for each fixed frequency\n",[54,63097,63098,63101,63103,63105,63107,63110,63112,63115,63117],{"class":56,"line":324},[54,63099,63100],{"class":64},"dur_hi ",[54,63102,122],{"class":121},[54,63104,1172],{"class":64},[54,63106,57503],{"class":125},[54,63108,63109],{"class":64},"(dur_ns.",[54,63111,24786],{"class":125},[54,63113,63114],{"class":64},"(), dur_ns.",[54,63116,36269],{"class":125},[54,63118,62996],{"class":64},[54,63120,63121,63124,63126,63128,63130,63133,63135,63137,63139,63141,63143],{"class":56,"line":333},[54,63122,63123],{"class":132}," UPSAMPLE_FACTOR",[54,63125,7785],{"class":121},[54,63127,58201],{"class":64},[54,63129,1319],{"class":121},[54,63131,63132],{"class":64},"(dur_ns) ",[54,63134,1923],{"class":121},[54,63136,982],{"class":132},[54,63138,204],{"class":64},[54,63140,1663],{"class":121},[54,63142,982],{"class":132},[54,63144,136],{"class":64},[54,63146,63147,63150,63152,63154,63156,63158,63160,63163,63165,63167,63169,63171],{"class":56,"line":345},[54,63148,63149],{"class":64},"Pe_hi ",[54,63151,122],{"class":121},[54,63153,1172],{"class":64},[54,63155,63034],{"class":125},[54,63157,63037],{"class":64},[54,63159,1319],{"class":121},[54,63161,63162],{"class":64},"(dur_hi), ",[54,63164,1319],{"class":121},[54,63166,63047],{"class":64},[54,63168,1712],{"class":1711},[54,63170,63052],{"class":121},[54,63172,136],{"class":64},[54,63174,63175,63177,63179,63181,63183,63185,63187],{"class":56,"line":358},[54,63176,1186],{"class":60},[54,63178,8236],{"class":64},[54,63180,1192],{"class":60},[54,63182,7777],{"class":121},[54,63184,129],{"class":64},[54,63186,1319],{"class":121},[54,63188,63189],{"class":64},"(freq_hi)):\n",[54,63191,63192,63195,63197,63199,63201],{"class":56,"line":908},[54,63193,63194],{"class":64}," Pe_hi[:, j] ",[54,63196,122],{"class":121},[54,63198,1172],{"class":64},[54,63200,63083],{"class":125},[54,63202,63203],{"class":64},"(dur_hi, dur_ns, Pe_tau_f[:, j])\n",[54,63205,63206],{"class":56,"line":913},[54,63207,105],{"emptyLinePlaceholder":104},[54,63209,63210],{"class":56,"line":918},[54,63211,63212],{"class":111},"# Mesh for plotting\n",[54,63214,63215,63218,63220,63222,63224,63227,63229,63231,63233],{"class":56,"line":923},[54,63216,63217],{"class":64},"F_hi, T_hi ",[54,63219,122],{"class":121},[54,63221,1172],{"class":64},[54,63223,57227],{"class":125},[54,63225,63226],{"class":64},"(freq_hi, dur_hi, ",[54,63228,57233],{"class":1711},[54,63230,122],{"class":121},[54,63232,57238],{"class":279},[54,63234,136],{"class":64},[54,63236,63237],{"class":56,"line":935},[54,63238,105],{"emptyLinePlaceholder":104},[54,63240,63241],{"class":56,"line":945},[54,63242,63243],{"class":111},"# --- Plot ---\n",[54,63245,63246,63249,63251,63253,63255,63257,63259,63261,63263,63265,63267,63270,63272,63275,63277,63280],{"class":56,"line":953},[54,63247,63248],{"class":64},"fig ",[54,63250,122],{"class":121},[54,63252,57782],{"class":64},[54,63254,61173],{"class":125},[54,63256,129],{"class":64},[54,63258,57790],{"class":1711},[54,63260,122],{"class":121},[54,63262,129],{"class":64},[54,63264,35297],{"class":132},[54,63266,171],{"class":64},[54,63268,63269],{"class":132},"6",[54,63271,1769],{"class":64},[54,63273,63274],{"class":1711},"dpi",[54,63276,122],{"class":121},[54,63278,63279],{"class":132},"200",[54,63281,136],{"class":64},[54,63283,63284,63287,63289,63291,63294,63296,63299,63301,63304,63306,63309],{"class":56,"line":968},[54,63285,63286],{"class":64},"ax ",[54,63288,122],{"class":121},[54,63290,57840],{"class":64},[54,63292,63293],{"class":125},"add_subplot",[54,63295,129],{"class":64},[54,63297,63298],{"class":132},"111",[54,63300,171],{"class":64},[54,63302,63303],{"class":1711},"projection",[54,63305,122],{"class":121},[54,63307,63308],{"class":279},"'3d'",[54,63310,136],{"class":64},[54,63312,63313],{"class":56,"line":987},[54,63314,105],{"emptyLinePlaceholder":104},[54,63316,63317,63320,63322,63324,63327],{"class":56,"line":1005},[54,63318,63319],{"class":64},"surf ",[54,63321,122],{"class":121},[54,63323,57814],{"class":64},[54,63325,63326],{"class":125},"plot_surface",[54,63328,932],{"class":64},[54,63330,63331],{"class":56,"line":1021},[54,63332,63333],{"class":64}," F_hi, T_hi, Pe_hi,\n",[54,63335,63336,63339,63341,63343,63345,63348,63350,63352,63355],{"class":56,"line":1027},[54,63337,63338],{"class":1711}," rstride",[54,63340,122],{"class":121},[54,63342,174],{"class":132},[54,63344,171],{"class":64},[54,63346,63347],{"class":1711},"cstride",[54,63349,122],{"class":121},[54,63351,174],{"class":132},[54,63353,63354],{"class":64},", ",[54,63356,63357],{"class":111},"# use every row\u002Fcolumn (after upsampling)\n",[54,63359,63360,63363,63365,63367,63369,63372,63374,63376],{"class":56,"line":1033},[54,63361,63362],{"class":1711}," linewidth",[54,63364,122],{"class":121},[54,63366,150],{"class":132},[54,63368,171],{"class":64},[54,63370,63371],{"class":1711},"antialiased",[54,63373,122],{"class":121},[54,63375,13434],{"class":132},[54,63377,965],{"class":64},[54,63379,63380,63383,63385,63387],{"class":56,"line":1038},[54,63381,63382],{"class":1711}," cmap",[54,63384,122],{"class":121},[54,63386,62925],{"class":132},[54,63388,965],{"class":64},[54,63390,63391,63394,63396,63398,63400,63403,63405,63407],{"class":56,"line":1044},[54,63392,63393],{"class":1711}," vmin",[54,63395,122],{"class":121},[54,63397,62942],{"class":132},[54,63399,171],{"class":64},[54,63401,63402],{"class":1711},"vmax",[54,63404,122],{"class":121},[54,63406,62947],{"class":132},[54,63408,965],{"class":64},[54,63410,63411],{"class":56,"line":1050},[54,63412,136],{"class":64},[54,63414,63415],{"class":56,"line":1056},[54,63416,105],{"emptyLinePlaceholder":104},[54,63418,63419,63421,63423,63425,63428],{"class":56,"line":1062},[54,63420,57871],{"class":64},[54,63422,57874],{"class":125},[54,63424,129],{"class":64},[54,63426,63427],{"class":279},"'f (GHz)'",[54,63429,136],{"class":64},[54,63431,63432,63434,63436,63438,63440,63443,63446,63449,63451,63453,63455,63458],{"class":56,"line":1068},[54,63433,57871],{"class":64},[54,63435,57888],{"class":125},[54,63437,129],{"class":64},[54,63439,1772],{"class":60},[54,63441,63442],{"class":57863},"'$",[54,63444,63445],{"class":121},"\\t",[54,63447,63448],{"class":57863},"au$ ",[54,63450,129],{"class":132},[54,63452,48235],{"class":57863},[54,63454,3205],{"class":132},[54,63456,63457],{"class":57863},"'",[54,63459,136],{"class":64},[54,63461,63462,63464,63467,63469,63471,63473],{"class":56,"line":1074},[54,63463,57871],{"class":64},[54,63465,63466],{"class":125},"set_zlabel",[54,63468,129],{"class":64},[54,63470,1772],{"class":60},[54,63472,57864],{"class":57863},[54,63474,136],{"class":64},[54,63476,63477,63479,63481,63483,63486],{"class":56,"line":1080},[54,63478,57871],{"class":64},[54,63480,57902],{"class":125},[54,63482,129],{"class":64},[54,63484,63485],{"class":279},"'Rabi $P_e$ surface'",[54,63487,136],{"class":64},[54,63489,63490],{"class":56,"line":1086},[54,63491,105],{"emptyLinePlaceholder":104},[54,63493,63494],{"class":56,"line":1092},[54,63495,63496],{"class":111},"# Colorbar (legend for the colormap)\n",[54,63498,63499,63501,63503,63505,63507,63510,63512,63514,63516,63519,63521,63524,63526,63529,63531,63534],{"class":56,"line":1098},[54,63500,57835],{"class":64},[54,63502,122],{"class":121},[54,63504,57840],{"class":64},[54,63506,57843],{"class":125},[54,63508,63509],{"class":64},"(surf, ",[54,63511,57849],{"class":1711},[54,63513,122],{"class":121},[54,63515,57854],{"class":64},[54,63517,63518],{"class":1711},"shrink",[54,63520,122],{"class":121},[54,63522,63523],{"class":132},"0.7",[54,63525,171],{"class":64},[54,63527,63528],{"class":1711},"pad",[54,63530,122],{"class":121},[54,63532,63533],{"class":132},"0.08",[54,63535,136],{"class":64},[54,63537,63538,63541,63544,63546,63548,63550],{"class":56,"line":1104},[54,63539,63540],{"class":64},"cb.",[54,63542,63543],{"class":125},"set_label",[54,63545,129],{"class":64},[54,63547,1772],{"class":60},[54,63549,57864],{"class":57863},[54,63551,136],{"class":64},[54,63553,63554],{"class":56,"line":1109},[54,63555,105],{"emptyLinePlaceholder":104},[54,63557,63558,63560,63562],{"class":56,"line":1115},[54,63559,57924],{"class":64},[54,63561,57927],{"class":125},[54,63563,330],{"class":64},[54,63565,63566,63568,63570],{"class":56,"line":1121},[54,63567,24932],{"class":64},[54,63569,25169],{"class":125},[54,63571,330],{"class":64},[380,63573],{"alt":63574,"src":63575},"Output 3 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-03.webp",[44,63577,63579],{"className":46,"code":63578,"language":48,"meta":49,"style":49},"# @title Cross-sections\n# Slicing controls\nF_CROSS_GHZ: float = 5.000 # drive frequency slice [GHz]\nT_CROSS_NS: float = 100.0 # duration slice [ns]\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef _nearest_index(vec: np.ndarray, value: float, name: str) -> int:\n \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n vmin, vmax = float(vec.min()), float(vec.max())\n if value \u003C vmin or value > vmax:\n print(f\"[Note] {name}={value} is outside sweep; clamping to [{vmin}, {vmax}].\")\n value = np.clip(value, vmin, vmax)\n return int(np.argmin(np.abs(vec - value)))\n\n# --- Slice at f = F_CROSS_GHZ ---\nj = _nearest_index(freq_ghz, F_CROSS_GHZ, \"F_CROSS_GHZ\")\nPe_vs_tau = Pe[:, j]\n\nfig, ax = plt.subplots(figsize=(6, 3.5))\nax.plot(dur_ns, Pe_vs_tau, color='blue')\nax.set_xlabel(r'$\\tau$ (ns)')\nax.set_ylabel(r'$P_e$')\n# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\nax.set_title(r'$P_e(\\tau)$ at f = ' + f'{freq_ghz[j]:.6f} GHz')\nax.set_ylim(0.0, 1.0)\nfig.tight_layout()\nplt.show()\n\n# --- Slice at τ = T_CROSS_NS ---\ni = _nearest_index(dur_ns, T_CROSS_NS, \"T_CROSS_NS\")\nPe_vs_f = Pe[i, :]\n\nfig, ax = plt.subplots(figsize=(6, 3.5))\nax.plot(freq_ghz, Pe_vs_f, color='blue')\nax.set_xlabel('f (GHz)')\nax.set_ylabel(r'$P_e$')\nax.set_title(r'$P_e(f)$ at $\\tau$ = ' + f'{dur_ns[i]:.3f} ns')\nax.set_ylim(0.0, 1.0)\nfig.tight_layout()\nplt.show()\n",[51,63580,63581,63586,63591,63606,63622,63626,63636,63646,63650,63686,63691,63716,63738,63787,63801,63825,63829,63834,63855,63865,63869,63896,63915,63941,63955,63960,64006,64023,64031,64039,64043,64048,64069,64079,64083,64109,64126,64138,64152,64198,64214,64222],{"__ignoreMap":49},[54,63582,63583],{"class":56,"line":57},[54,63584,63585],{"class":111},"# @title Cross-sections\n",[54,63587,63588],{"class":56,"line":74},[54,63589,63590],{"class":111},"# Slicing controls\n",[54,63592,63593,63595,63597,63599,63601,63603],{"class":56,"line":88},[54,63594,56346],{"class":132},[54,63596,959],{"class":64},[54,63598,962],{"class":121},[54,63600,979],{"class":121},[54,63602,56844],{"class":132},[54,63604,63605],{"class":111}," # drive frequency slice [GHz]\n",[54,63607,63608,63610,63612,63614,63616,63619],{"class":56,"line":101},[54,63609,56445],{"class":132},[54,63611,959],{"class":64},[54,63613,962],{"class":121},[54,63615,979],{"class":121},[54,63617,63618],{"class":132}," 100.0",[54,63620,63621],{"class":111}," # duration slice [ns]\n",[54,63623,63624],{"class":56,"line":108},[54,63625,105],{"emptyLinePlaceholder":104},[54,63627,63628,63630,63632,63634],{"class":56,"line":115},[54,63629,61],{"class":60},[54,63631,65],{"class":64},[54,63633,68],{"class":60},[54,63635,71],{"class":64},[54,63637,63638,63640,63642,63644],{"class":56,"line":139},[54,63639,61],{"class":60},[54,63641,56720],{"class":64},[54,63643,68],{"class":60},[54,63645,715],{"class":64},[54,63647,63648],{"class":56,"line":159},[54,63649,105],{"emptyLinePlaceholder":104},[54,63651,63652,63654,63657,63659,63662,63665,63668,63670,63672,63674,63676,63678,63680,63682,63684],{"class":56,"line":183},[54,63653,926],{"class":60},[54,63655,63656],{"class":125}," _nearest_index",[54,63658,129],{"class":64},[54,63660,63661],{"class":938},"vec",[54,63663,63664],{"class":64},": np.ndarray, ",[54,63666,63667],{"class":938},"value",[54,63669,959],{"class":64},[54,63671,962],{"class":121},[54,63673,171],{"class":64},[54,63675,21564],{"class":938},[54,63677,959],{"class":64},[54,63679,58454],{"class":121},[54,63681,58457],{"class":64},[54,63683,976],{"class":121},[54,63685,1284],{"class":64},[54,63687,63688],{"class":56,"line":210},[54,63689,63690],{"class":279}," \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n",[54,63692,63693,63696,63698,63700,63703,63705,63708,63710,63712,63714],{"class":56,"line":227},[54,63694,63695],{"class":64}," vmin, vmax ",[54,63697,122],{"class":121},[54,63699,59177],{"class":121},[54,63701,63702],{"class":64},"(vec.",[54,63704,24786],{"class":125},[54,63706,63707],{"class":64},"()), ",[54,63709,962],{"class":121},[54,63711,63702],{"class":64},[54,63713,36269],{"class":125},[54,63715,21643],{"class":64},[54,63717,63718,63720,63723,63725,63728,63731,63733,63735],{"class":56,"line":232},[54,63719,1273],{"class":60},[54,63721,63722],{"class":64}," value ",[54,63724,5233],{"class":121},[54,63726,63727],{"class":64}," vmin ",[54,63729,63730],{"class":60},"or",[54,63732,63722],{"class":64},[54,63734,1207],{"class":121},[54,63736,63737],{"class":64}," vmax:\n",[54,63739,63740,63743,63745,63747,63750,63752,63754,63756,63758,63760,63762,63764,63767,63769,63772,63774,63776,63778,63780,63782,63785],{"class":56,"line":246},[54,63741,63742],{"class":121}," print",[54,63744,129],{"class":64},[54,63746,21834],{"class":60},[54,63748,63749],{"class":279},"\"[Note] ",[54,63751,2366],{"class":132},[54,63753,21564],{"class":64},[54,63755,2428],{"class":132},[54,63757,122],{"class":279},[54,63759,2366],{"class":132},[54,63761,63667],{"class":64},[54,63763,2428],{"class":132},[54,63765,63766],{"class":279}," is outside sweep; clamping to [",[54,63768,2366],{"class":132},[54,63770,63771],{"class":64},"vmin",[54,63773,2428],{"class":132},[54,63775,171],{"class":279},[54,63777,2366],{"class":132},[54,63779,63402],{"class":64},[54,63781,2428],{"class":132},[54,63783,63784],{"class":279},"].\"",[54,63786,136],{"class":64},[54,63788,63789,63792,63794,63796,63798],{"class":56,"line":259},[54,63790,63791],{"class":64}," value ",[54,63793,122],{"class":121},[54,63795,1172],{"class":64},[54,63797,57474],{"class":125},[54,63799,63800],{"class":64},"(value, vmin, vmax)\n",[54,63802,63803,63805,63807,63809,63812,63814,63817,63820,63822],{"class":56,"line":264},[54,63804,2060],{"class":60},[54,63806,1651],{"class":121},[54,63808,59474],{"class":64},[54,63810,63811],{"class":125},"argmin",[54,63813,59474],{"class":64},[54,63815,63816],{"class":125},"abs",[54,63818,63819],{"class":64},"(vec ",[54,63821,1923],{"class":121},[54,63823,63824],{"class":64}," value)))\n",[54,63826,63827],{"class":56,"line":270},[54,63828,105],{"emptyLinePlaceholder":104},[54,63830,63831],{"class":56,"line":285},[54,63832,63833],{"class":111},"# --- Slice at f = F_CROSS_GHZ ---\n",[54,63835,63836,63839,63841,63843,63846,63848,63850,63853],{"class":56,"line":297},[54,63837,63838],{"class":64},"j ",[54,63840,122],{"class":121},[54,63842,63656],{"class":125},[54,63844,63845],{"class":64},"(freq_ghz, ",[54,63847,56346],{"class":132},[54,63849,171],{"class":64},[54,63851,63852],{"class":279},"\"F_CROSS_GHZ\"",[54,63854,136],{"class":64},[54,63856,63857,63860,63862],{"class":56,"line":310},[54,63858,63859],{"class":64},"Pe_vs_tau ",[54,63861,122],{"class":121},[54,63863,63864],{"class":64}," Pe[:, j]\n",[54,63866,63867],{"class":56,"line":324},[54,63868,105],{"emptyLinePlaceholder":104},[54,63870,63871,63873,63875,63877,63879,63881,63883,63885,63887,63889,63891,63894],{"class":56,"line":333},[54,63872,57777],{"class":64},[54,63874,122],{"class":121},[54,63876,57782],{"class":64},[54,63878,57785],{"class":125},[54,63880,129],{"class":64},[54,63882,57790],{"class":1711},[54,63884,122],{"class":121},[54,63886,129],{"class":64},[54,63888,63269],{"class":132},[54,63890,171],{"class":64},[54,63892,63893],{"class":132},"3.5",[54,63895,22096],{"class":64},[54,63897,63898,63900,63902,63905,63908,63910,63913],{"class":56,"line":345},[54,63899,57871],{"class":64},[54,63901,24935],{"class":125},[54,63903,63904],{"class":64},"(dur_ns, Pe_vs_tau, ",[54,63906,63907],{"class":1711},"color",[54,63909,122],{"class":121},[54,63911,63912],{"class":279},"'blue'",[54,63914,136],{"class":64},[54,63916,63917,63919,63921,63923,63925,63927,63929,63931,63933,63935,63937,63939],{"class":56,"line":358},[54,63918,57871],{"class":64},[54,63920,57874],{"class":125},[54,63922,129],{"class":64},[54,63924,1772],{"class":60},[54,63926,63442],{"class":57863},[54,63928,63445],{"class":121},[54,63930,63448],{"class":57863},[54,63932,129],{"class":132},[54,63934,48235],{"class":57863},[54,63936,3205],{"class":132},[54,63938,63457],{"class":57863},[54,63940,136],{"class":64},[54,63942,63943,63945,63947,63949,63951,63953],{"class":56,"line":908},[54,63944,57871],{"class":64},[54,63946,57888],{"class":125},[54,63948,129],{"class":64},[54,63950,1772],{"class":60},[54,63952,57864],{"class":57863},[54,63954,136],{"class":64},[54,63956,63957],{"class":56,"line":913},[54,63958,63959],{"class":111},"# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\n",[54,63961,63962,63964,63966,63968,63970,63973,63975,63977,63980,63982,63985,63987,63990,63992,63994,63997,63999,64001,64004],{"class":56,"line":918},[54,63963,57871],{"class":64},[54,63965,57902],{"class":125},[54,63967,129],{"class":64},[54,63969,1772],{"class":60},[54,63971,63972],{"class":57863},"'$P_e",[54,63974,129],{"class":132},[54,63976,63445],{"class":121},[54,63978,63979],{"class":57863},"au",[54,63981,3205],{"class":132},[54,63983,63984],{"class":57863},"$ at f = '",[54,63986,7869],{"class":121},[54,63988,63989],{"class":60}," f",[54,63991,63457],{"class":279},[54,63993,2366],{"class":132},[54,63995,63996],{"class":64},"freq_ghz[j]",[54,63998,62435],{"class":60},[54,64000,2428],{"class":132},[54,64002,64003],{"class":279}," GHz'",[54,64005,136],{"class":64},[54,64007,64008,64010,64013,64015,64017,64019,64021],{"class":56,"line":923},[54,64009,57871],{"class":64},[54,64011,64012],{"class":125},"set_ylim",[54,64014,129],{"class":64},[54,64016,56271],{"class":132},[54,64018,171],{"class":64},[54,64020,57484],{"class":132},[54,64022,136],{"class":64},[54,64024,64025,64027,64029],{"class":56,"line":935},[54,64026,57924],{"class":64},[54,64028,57927],{"class":125},[54,64030,330],{"class":64},[54,64032,64033,64035,64037],{"class":56,"line":945},[54,64034,24932],{"class":64},[54,64036,25169],{"class":125},[54,64038,330],{"class":64},[54,64040,64041],{"class":56,"line":953},[54,64042,105],{"emptyLinePlaceholder":104},[54,64044,64045],{"class":56,"line":968},[54,64046,64047],{"class":111},"# --- Slice at τ = T_CROSS_NS ---\n",[54,64049,64050,64053,64055,64057,64060,64062,64064,64067],{"class":56,"line":987},[54,64051,64052],{"class":64},"i ",[54,64054,122],{"class":121},[54,64056,63656],{"class":125},[54,64058,64059],{"class":64},"(dur_ns, ",[54,64061,56445],{"class":132},[54,64063,171],{"class":64},[54,64065,64066],{"class":279},"\"T_CROSS_NS\"",[54,64068,136],{"class":64},[54,64070,64071,64074,64076],{"class":56,"line":1005},[54,64072,64073],{"class":64},"Pe_vs_f ",[54,64075,122],{"class":121},[54,64077,64078],{"class":64}," Pe[i, :]\n",[54,64080,64081],{"class":56,"line":1021},[54,64082,105],{"emptyLinePlaceholder":104},[54,64084,64085,64087,64089,64091,64093,64095,64097,64099,64101,64103,64105,64107],{"class":56,"line":1027},[54,64086,57777],{"class":64},[54,64088,122],{"class":121},[54,64090,57782],{"class":64},[54,64092,57785],{"class":125},[54,64094,129],{"class":64},[54,64096,57790],{"class":1711},[54,64098,122],{"class":121},[54,64100,129],{"class":64},[54,64102,63269],{"class":132},[54,64104,171],{"class":64},[54,64106,63893],{"class":132},[54,64108,22096],{"class":64},[54,64110,64111,64113,64115,64118,64120,64122,64124],{"class":56,"line":1033},[54,64112,57871],{"class":64},[54,64114,24935],{"class":125},[54,64116,64117],{"class":64},"(freq_ghz, Pe_vs_f, ",[54,64119,63907],{"class":1711},[54,64121,122],{"class":121},[54,64123,63912],{"class":279},[54,64125,136],{"class":64},[54,64127,64128,64130,64132,64134,64136],{"class":56,"line":1038},[54,64129,57871],{"class":64},[54,64131,57874],{"class":125},[54,64133,129],{"class":64},[54,64135,63427],{"class":279},[54,64137,136],{"class":64},[54,64139,64140,64142,64144,64146,64148,64150],{"class":56,"line":1044},[54,64141,57871],{"class":64},[54,64143,57888],{"class":125},[54,64145,129],{"class":64},[54,64147,1772],{"class":60},[54,64149,57864],{"class":57863},[54,64151,136],{"class":64},[54,64153,64154,64156,64158,64160,64162,64164,64166,64168,64170,64173,64175,64178,64180,64182,64184,64186,64189,64191,64193,64196],{"class":56,"line":1050},[54,64155,57871],{"class":64},[54,64157,57902],{"class":125},[54,64159,129],{"class":64},[54,64161,1772],{"class":60},[54,64163,63972],{"class":57863},[54,64165,129],{"class":132},[54,64167,21834],{"class":57863},[54,64169,3205],{"class":132},[54,64171,64172],{"class":57863},"$ at $",[54,64174,63445],{"class":121},[54,64176,64177],{"class":57863},"au$ = '",[54,64179,7869],{"class":121},[54,64181,63989],{"class":60},[54,64183,63457],{"class":279},[54,64185,2366],{"class":132},[54,64187,64188],{"class":64},"dur_ns[i]",[54,64190,62510],{"class":60},[54,64192,2428],{"class":132},[54,64194,64195],{"class":279}," ns'",[54,64197,136],{"class":64},[54,64199,64200,64202,64204,64206,64208,64210,64212],{"class":56,"line":1056},[54,64201,57871],{"class":64},[54,64203,64012],{"class":125},[54,64205,129],{"class":64},[54,64207,56271],{"class":132},[54,64209,171],{"class":64},[54,64211,57484],{"class":132},[54,64213,136],{"class":64},[54,64215,64216,64218,64220],{"class":56,"line":1062},[54,64217,57924],{"class":64},[54,64219,57927],{"class":125},[54,64221,330],{"class":64},[54,64223,64224,64226,64228],{"class":56,"line":1068},[54,64225,24932],{"class":64},[54,64227,25169],{"class":125},[54,64229,330],{"class":64},[380,64231],{"alt":64232,"src":64233},"Output 4 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-04.webp",[380,64235],{"alt":64236,"src":64237},"Output 5 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-05.webp",[44,64239,64241],{"className":46,"code":64240,"language":48,"meta":49,"style":49},"# @title FFT of Rabi oscillations along tau (controls + computation)\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# -------- Control knobs (FFT & display) --------\nFFT_PAD: int = 8 # zero-padding factor (2, 4, 8, ...)\nWINDOW: str = \"hann\" # \"hann\", \"boxcar\"\nDETREND_MEAN: bool = True # subtract mean along τ before FFT\nAMP_MODE: str = \"magnitude\" # \"magnitude\" | \"power\" | \"db\"\nNORM_MODE: str = \"global\" # \"global\" | \"per_f\" (column-wise)\nFREQ_UNITS: str = \"MHz\" # \"MHz\" or \"GHz\" for Fourier axis\n\n# -------- Helpers --------\ndef _make_window(n: int, kind: str) -> np.ndarray:\n if kind.lower() == \"hann\":\n return np.hanning(n)\n return np.ones(n, dtype=float) # boxcar\n\ndef rfft_spectrogram(\n Pe: np.ndarray,\n dur_ns: np.ndarray,\n pad: int = 8,\n window: str = \"hann\",\n detrend_mean: bool = True,\n amp_mode: str = \"magnitude\",\n norm_mode: str = \"global\",\n freq_units: str = \"MHz\",\n) -> tuple[np.ndarray, np.ndarray]:\n \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n\n Args:\n Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n dur_ns: τ grid [ns], length N_tau, evenly spaced.\n pad: zero-padding factor (≥1).\n window: \"hann\" or \"boxcar\".\n detrend_mean: subtract column-wise mean before FFT.\n amp_mode: \"magnitude\", \"power\", or \"db\".\n norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n\n Returns:\n f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n S: 2D nonnegative spectrogram (N_w, N_f).\n \"\"\"\n Pe = np.asarray(Pe, float)\n dur_ns = np.asarray(dur_ns, float)\n assert Pe.shape[0] == dur_ns.size, \"Pe must be shaped (N_tau, N_f).\"\n\n # Uniform sampling assumption (true for np.linspace)\n dt = float(dur_ns[1] - dur_ns[0]) * 1e-9 # [s]\n n_tau = dur_ns.size\n nfft = int(2 ** int(np.ceil(np.log2(n_tau))) * max(1, pad))\n w = _make_window(n_tau, window)\n\n # Prepare output\n S_list = []\n X = Pe.copy()\n if detrend_mean:\n X = X - X.mean(axis=0, keepdims=True)\n X *= w[:, None]\n\n # Real FFT along τ (axis=0)\n Xf = np.fft.rfft(X, n=nfft, axis=0) # shape (N_w, N_f)\n f_fft_hz = np.fft.rfftfreq(nfft, d=dt) # [Hz]\n\n # Amplitude selection\n if amp_mode.lower() == \"power\":\n S = np.abs(Xf) ** 2\n else:\n S = np.abs(Xf)\n\n # Normalization\n if norm_mode.lower() == \"per_f\":\n col_max = np.maximum(S.max(axis=0, keepdims=True), 1e-15)\n S = S \u002F col_max\n else:\n S = S \u002F max(S.max(), 1e-15)\n\n if amp_mode.lower() == \"db\":\n S = 20.0 * np.log10(np.maximum(S, 1e-12)) # dB, clipped floor\n\n # Units\n if freq_units.upper() == \"GHz\":\n f_fft = f_fft_hz * 1e-9\n else:\n f_fft = f_fft_hz * 1e-6\n\n return f_fft, S\n\n# Compute spectrogram with current grid\nf_fft_axis, S_fft = rfft_spectrogram(\n Pe, dur_ns,\n pad=FFT_PAD,\n window=WINDOW,\n detrend_mean=DETREND_MEAN,\n amp_mode=AMP_MODE,\n norm_mode=NORM_MODE,\n freq_units=FREQ_UNITS,\n)\nprint(\"FFT grid:\", S_fft.shape, \"| Fourier freq range:\", f_fft_axis[0], \"→\", f_fft_axis[-1], FREQ_UNITS)\n",[51,64242,64243,64248,64258,64268,64272,64277,64293,64309,64325,64341,64357,64373,64377,64382,64408,64426,64439,64460,64464,64473,64480,64487,64502,64517,64532,64547,64562,64577,64582,64587,64591,64595,64600,64605,64610,64615,64620,64625,64630,64635,64639,64643,64648,64653,64657,64674,64691,64711,64715,64720,64753,64763,64806,64818,64822,64827,64836,64851,64858,64895,64909,64913,64918,64953,64978,64982,64987,65005,65023,65029,65042,65046,65051,65069,65109,65123,65129,65151,65155,65172,65203,65207,65212,65231,65245,65251,65264,65268,65275,65279,65284,65295,65300,65310,65320,65330,65340,65350,65360,65364],{"__ignoreMap":49},[54,64244,64245],{"class":56,"line":57},[54,64246,64247],{"class":111},"# @title FFT of Rabi oscillations along tau (controls + computation)\n",[54,64249,64250,64252,64254,64256],{"class":56,"line":74},[54,64251,61],{"class":60},[54,64253,65],{"class":64},[54,64255,68],{"class":60},[54,64257,71],{"class":64},[54,64259,64260,64262,64264,64266],{"class":56,"line":88},[54,64261,61],{"class":60},[54,64263,56720],{"class":64},[54,64265,68],{"class":60},[54,64267,715],{"class":64},[54,64269,64270],{"class":56,"line":101},[54,64271,105],{"emptyLinePlaceholder":104},[54,64273,64274],{"class":56,"line":108},[54,64275,64276],{"class":111},"# -------- Control knobs (FFT & display) --------\n",[54,64278,64279,64281,64283,64285,64287,64290],{"class":56,"line":115},[54,64280,56562],{"class":132},[54,64282,959],{"class":64},[54,64284,976],{"class":121},[54,64286,979],{"class":121},[54,64288,64289],{"class":132}," 8",[54,64291,64292],{"class":111}," # zero-padding factor (2, 4, 8, ...)\n",[54,64294,64295,64297,64299,64301,64303,64306],{"class":56,"line":139},[54,64296,56576],{"class":132},[54,64298,959],{"class":64},[54,64300,58454],{"class":121},[54,64302,979],{"class":121},[54,64304,64305],{"class":279}," \"hann\"",[54,64307,64308],{"class":111}," # \"hann\", \"boxcar\"\n",[54,64310,64311,64313,64315,64317,64319,64322],{"class":56,"line":159},[54,64312,56591],{"class":132},[54,64314,959],{"class":64},[54,64316,995],{"class":121},[54,64318,979],{"class":121},[54,64320,64321],{"class":132}," True",[54,64323,64324],{"class":111}," # subtract mean along τ before FFT\n",[54,64326,64327,64329,64331,64333,64335,64338],{"class":56,"line":183},[54,64328,56605],{"class":132},[54,64330,959],{"class":64},[54,64332,58454],{"class":121},[54,64334,979],{"class":121},[54,64336,64337],{"class":279}," \"magnitude\"",[54,64339,64340],{"class":111}," # \"magnitude\" | \"power\" | \"db\"\n",[54,64342,64343,64345,64347,64349,64351,64354],{"class":56,"line":210},[54,64344,56620],{"class":132},[54,64346,959],{"class":64},[54,64348,58454],{"class":121},[54,64350,979],{"class":121},[54,64352,64353],{"class":279}," \"global\"",[54,64355,64356],{"class":111}," # \"global\" | \"per_f\" (column-wise)\n",[54,64358,64359,64361,64363,64365,64367,64370],{"class":56,"line":227},[54,64360,56635],{"class":132},[54,64362,959],{"class":64},[54,64364,58454],{"class":121},[54,64366,979],{"class":121},[54,64368,64369],{"class":279}," \"MHz\"",[54,64371,64372],{"class":111}," # \"MHz\" or \"GHz\" for Fourier axis\n",[54,64374,64375],{"class":56,"line":232},[54,64376,105],{"emptyLinePlaceholder":104},[54,64378,64379],{"class":56,"line":246},[54,64380,64381],{"class":111},"# -------- Helpers --------\n",[54,64383,64384,64386,64389,64391,64393,64395,64397,64399,64402,64404,64406],{"class":56,"line":259},[54,64385,926],{"class":60},[54,64387,64388],{"class":125}," _make_window",[54,64390,129],{"class":64},[54,64392,3976],{"class":938},[54,64394,959],{"class":64},[54,64396,976],{"class":121},[54,64398,171],{"class":64},[54,64400,64401],{"class":938},"kind",[54,64403,959],{"class":64},[54,64405,58454],{"class":121},[54,64407,57071],{"class":64},[54,64409,64410,64412,64415,64418,64420,64422,64424],{"class":56,"line":264},[54,64411,1273],{"class":60},[54,64413,64414],{"class":64}," kind.",[54,64416,64417],{"class":125},"lower",[54,64419,1233],{"class":64},[54,64421,1279],{"class":121},[54,64423,64305],{"class":279},[54,64425,1284],{"class":64},[54,64427,64428,64431,64433,64436],{"class":56,"line":270},[54,64429,64430],{"class":60}," return",[54,64432,1172],{"class":64},[54,64434,64435],{"class":125},"hanning",[54,64437,64438],{"class":64},"(n)\n",[54,64440,64441,64443,64445,64448,64451,64453,64455,64457],{"class":56,"line":285},[54,64442,2060],{"class":60},[54,64444,1172],{"class":64},[54,64446,64447],{"class":125},"ones",[54,64449,64450],{"class":64},"(n, ",[54,64452,1712],{"class":1711},[54,64454,63052],{"class":121},[54,64456,22274],{"class":64},[54,64458,64459],{"class":111},"# boxcar\n",[54,64461,64462],{"class":56,"line":297},[54,64463,105],{"emptyLinePlaceholder":104},[54,64465,64466,64468,64471],{"class":56,"line":310},[54,64467,926],{"class":60},[54,64469,64470],{"class":125}," rfft_spectrogram",[54,64472,932],{"class":64},[54,64474,64475,64478],{"class":56,"line":324},[54,64476,64477],{"class":938}," Pe",[54,64479,942],{"class":64},[54,64481,64482,64485],{"class":56,"line":333},[54,64483,64484],{"class":938}," dur_ns",[54,64486,942],{"class":64},[54,64488,64489,64492,64494,64496,64498,64500],{"class":56,"line":345},[54,64490,64491],{"class":938}," pad",[54,64493,959],{"class":64},[54,64495,976],{"class":121},[54,64497,979],{"class":121},[54,64499,64289],{"class":132},[54,64501,965],{"class":64},[54,64503,64504,64507,64509,64511,64513,64515],{"class":56,"line":358},[54,64505,64506],{"class":938}," window",[54,64508,959],{"class":64},[54,64510,58454],{"class":121},[54,64512,979],{"class":121},[54,64514,64305],{"class":279},[54,64516,965],{"class":64},[54,64518,64519,64522,64524,64526,64528,64530],{"class":56,"line":908},[54,64520,64521],{"class":938}," detrend_mean",[54,64523,959],{"class":64},[54,64525,995],{"class":121},[54,64527,979],{"class":121},[54,64529,64321],{"class":132},[54,64531,965],{"class":64},[54,64533,64534,64537,64539,64541,64543,64545],{"class":56,"line":913},[54,64535,64536],{"class":938}," amp_mode",[54,64538,959],{"class":64},[54,64540,58454],{"class":121},[54,64542,979],{"class":121},[54,64544,64337],{"class":279},[54,64546,965],{"class":64},[54,64548,64549,64552,64554,64556,64558,64560],{"class":56,"line":918},[54,64550,64551],{"class":938}," norm_mode",[54,64553,959],{"class":64},[54,64555,58454],{"class":121},[54,64557,979],{"class":121},[54,64559,64353],{"class":279},[54,64561,965],{"class":64},[54,64563,64564,64567,64569,64571,64573,64575],{"class":56,"line":923},[54,64565,64566],{"class":938}," freq_units",[54,64568,959],{"class":64},[54,64570,58454],{"class":121},[54,64572,979],{"class":121},[54,64574,64369],{"class":279},[54,64576,965],{"class":64},[54,64578,64579],{"class":56,"line":935},[54,64580,64581],{"class":64},") -> tuple[np.ndarray, np.ndarray]:\n",[54,64583,64584],{"class":56,"line":945},[54,64585,64586],{"class":279}," \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n",[54,64588,64589],{"class":56,"line":953},[54,64590,105],{"emptyLinePlaceholder":104},[54,64592,64593],{"class":56,"line":968},[54,64594,1041],{"class":279},[54,64596,64597],{"class":56,"line":987},[54,64598,64599],{"class":279}," Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n",[54,64601,64602],{"class":56,"line":1005},[54,64603,64604],{"class":279}," dur_ns: τ grid [ns], length N_tau, evenly spaced.\n",[54,64606,64607],{"class":56,"line":1021},[54,64608,64609],{"class":279}," pad: zero-padding factor (≥1).\n",[54,64611,64612],{"class":56,"line":1027},[54,64613,64614],{"class":279}," window: \"hann\" or \"boxcar\".\n",[54,64616,64617],{"class":56,"line":1033},[54,64618,64619],{"class":279}," detrend_mean: subtract column-wise mean before FFT.\n",[54,64621,64622],{"class":56,"line":1038},[54,64623,64624],{"class":279}," amp_mode: \"magnitude\", \"power\", or \"db\".\n",[54,64626,64627],{"class":56,"line":1044},[54,64628,64629],{"class":279}," norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n",[54,64631,64632],{"class":56,"line":1050},[54,64633,64634],{"class":279}," freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n",[54,64636,64637],{"class":56,"line":1056},[54,64638,105],{"emptyLinePlaceholder":104},[54,64640,64641],{"class":56,"line":1062},[54,64642,1112],{"class":279},[54,64644,64645],{"class":56,"line":1068},[54,64646,64647],{"class":279}," f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n",[54,64649,64650],{"class":56,"line":1074},[54,64651,64652],{"class":279}," S: 2D nonnegative spectrogram (N_w, N_f).\n",[54,64654,64655],{"class":56,"line":1080},[54,64656,1024],{"class":279},[54,64658,64659,64662,64664,64666,64668,64670,64672],{"class":56,"line":1086},[54,64660,64661],{"class":64}," Pe ",[54,64663,122],{"class":121},[54,64665,1172],{"class":64},[54,64667,59477],{"class":125},[54,64669,57477],{"class":64},[54,64671,962],{"class":121},[54,64673,136],{"class":64},[54,64675,64676,64679,64681,64683,64685,64687,64689],{"class":56,"line":1092},[54,64677,64678],{"class":64}," dur_ns ",[54,64680,122],{"class":121},[54,64682,1172],{"class":64},[54,64684,59477],{"class":125},[54,64686,64059],{"class":64},[54,64688,962],{"class":121},[54,64690,136],{"class":64},[54,64692,64693,64696,64699,64701,64703,64705,64708],{"class":56,"line":1098},[54,64694,64695],{"class":60}," assert",[54,64697,64698],{"class":64}," Pe.shape[",[54,64700,150],{"class":132},[54,64702,7762],{"class":64},[54,64704,1279],{"class":121},[54,64706,64707],{"class":64}," dur_ns.size, ",[54,64709,64710],{"class":279},"\"Pe must be shaped (N_tau, N_f).\"\n",[54,64712,64713],{"class":56,"line":1104},[54,64714,105],{"emptyLinePlaceholder":104},[54,64716,64717],{"class":56,"line":1109},[54,64718,64719],{"class":111}," # Uniform sampling assumption (true for np.linspace)\n",[54,64721,64722,64725,64727,64729,64732,64734,64736,64738,64741,64743,64746,64748,64750],{"class":56,"line":1115},[54,64723,64724],{"class":64}," dt ",[54,64726,122],{"class":121},[54,64728,59177],{"class":121},[54,64730,64731],{"class":64},"(dur_ns[",[54,64733,174],{"class":132},[54,64735,7762],{"class":64},[54,64737,1923],{"class":121},[54,64739,64740],{"class":64}," dur_ns[",[54,64742,150],{"class":132},[54,64744,64745],{"class":64},"]) ",[54,64747,1947],{"class":121},[54,64749,57445],{"class":132},[54,64751,64752],{"class":111}," # [s]\n",[54,64754,64755,64758,64760],{"class":56,"line":1121},[54,64756,64757],{"class":64}," n_tau ",[54,64759,122],{"class":121},[54,64761,64762],{"class":64}," dur_ns.size\n",[54,64764,64765,64768,64770,64772,64774,64776,64779,64781,64783,64786,64788,64791,64794,64796,64799,64801,64803],{"class":56,"line":1126},[54,64766,64767],{"class":64}," nfft ",[54,64769,122],{"class":121},[54,64771,1651],{"class":121},[54,64773,129],{"class":64},[54,64775,133],{"class":132},[54,64777,64778],{"class":121}," **",[54,64780,1651],{"class":121},[54,64782,59474],{"class":64},[54,64784,64785],{"class":125},"ceil",[54,64787,59474],{"class":64},[54,64789,64790],{"class":125},"log2",[54,64792,64793],{"class":64},"(n_tau))) ",[54,64795,1947],{"class":121},[54,64797,64798],{"class":121}," max",[54,64800,129],{"class":64},[54,64802,174],{"class":132},[54,64804,64805],{"class":64},", pad))\n",[54,64807,64808,64811,64813,64815],{"class":56,"line":1131},[54,64809,64810],{"class":64}," w ",[54,64812,122],{"class":121},[54,64814,64388],{"class":125},[54,64816,64817],{"class":64},"(n_tau, window)\n",[54,64819,64820],{"class":56,"line":1136},[54,64821,105],{"emptyLinePlaceholder":104},[54,64823,64824],{"class":56,"line":1153},[54,64825,64826],{"class":111}," # Prepare output\n",[54,64828,64829,64832,64834],{"class":56,"line":1164},[54,64830,64831],{"class":64}," S_list ",[54,64833,122],{"class":121},[54,64835,1466],{"class":64},[54,64837,64838,64841,64843,64846,64849],{"class":56,"line":1180},[54,64839,64840],{"class":64}," X ",[54,64842,122],{"class":121},[54,64844,64845],{"class":64}," Pe.",[54,64847,64848],{"class":125},"copy",[54,64850,330],{"class":64},[54,64852,64853,64855],{"class":56,"line":1213},[54,64854,1273],{"class":60},[54,64856,64857],{"class":64}," detrend_mean:\n",[54,64859,64860,64863,64865,64868,64870,64873,64876,64878,64880,64882,64884,64886,64889,64891,64893],{"class":56,"line":1219},[54,64861,64862],{"class":64}," X ",[54,64864,122],{"class":121},[54,64866,64867],{"class":64}," X ",[54,64869,1923],{"class":121},[54,64871,64872],{"class":64}," X.",[54,64874,64875],{"class":125},"mean",[54,64877,129],{"class":64},[54,64879,60728],{"class":1711},[54,64881,122],{"class":121},[54,64883,150],{"class":132},[54,64885,171],{"class":64},[54,64887,64888],{"class":1711},"keepdims",[54,64890,122],{"class":121},[54,64892,13434],{"class":132},[54,64894,136],{"class":64},[54,64896,64897,64899,64902,64905,64907],{"class":56,"line":1247},[54,64898,64840],{"class":64},[54,64900,64901],{"class":121},"*=",[54,64903,64904],{"class":64}," w[:, ",[54,64906,56321],{"class":132},[54,64908,1350],{"class":64},[54,64910,64911],{"class":56,"line":1270},[54,64912,105],{"emptyLinePlaceholder":104},[54,64914,64915],{"class":56,"line":1287},[54,64916,64917],{"class":111}," # Real FFT along τ (axis=0)\n",[54,64919,64920,64923,64925,64928,64931,64934,64936,64938,64941,64943,64945,64947,64950],{"class":56,"line":1296},[54,64921,64922],{"class":64}," Xf ",[54,64924,122],{"class":121},[54,64926,64927],{"class":64}," np.fft.",[54,64929,64930],{"class":125},"rfft",[54,64932,64933],{"class":64},"(X, ",[54,64935,3976],{"class":1711},[54,64937,122],{"class":121},[54,64939,64940],{"class":64},"nfft, ",[54,64942,60728],{"class":1711},[54,64944,122],{"class":121},[54,64946,150],{"class":132},[54,64948,64949],{"class":64},") ",[54,64951,64952],{"class":111},"# shape (N_w, N_f)\n",[54,64954,64955,64958,64960,64962,64965,64968,64970,64972,64975],{"class":56,"line":1325},[54,64956,64957],{"class":64}," f_fft_hz ",[54,64959,122],{"class":121},[54,64961,64927],{"class":64},[54,64963,64964],{"class":125},"rfftfreq",[54,64966,64967],{"class":64},"(nfft, ",[54,64969,4515],{"class":1711},[54,64971,122],{"class":121},[54,64973,64974],{"class":64},"dt) ",[54,64976,64977],{"class":111},"# [Hz]\n",[54,64979,64980],{"class":56,"line":1333},[54,64981,105],{"emptyLinePlaceholder":104},[54,64983,64984],{"class":56,"line":1353},[54,64985,64986],{"class":111}," # Amplitude selection\n",[54,64988,64989,64991,64994,64996,64998,65000,65003],{"class":56,"line":1361},[54,64990,1273],{"class":60},[54,64992,64993],{"class":64}," amp_mode.",[54,64995,64417],{"class":125},[54,64997,1233],{"class":64},[54,64999,1279],{"class":121},[54,65001,65002],{"class":279}," \"power\"",[54,65004,1284],{"class":64},[54,65006,65007,65010,65012,65014,65016,65019,65021],{"class":56,"line":1368},[54,65008,65009],{"class":64}," S ",[54,65011,122],{"class":121},[54,65013,1172],{"class":64},[54,65015,63816],{"class":125},[54,65017,65018],{"class":64},"(Xf) ",[54,65020,1933],{"class":121},[54,65022,57346],{"class":132},[54,65024,65025,65027],{"class":56,"line":1388},[54,65026,1356],{"class":60},[54,65028,1284],{"class":64},[54,65030,65031,65033,65035,65037,65039],{"class":56,"line":1395},[54,65032,65009],{"class":64},[54,65034,122],{"class":121},[54,65036,1172],{"class":64},[54,65038,63816],{"class":125},[54,65040,65041],{"class":64},"(Xf)\n",[54,65043,65044],{"class":56,"line":1411},[54,65045,105],{"emptyLinePlaceholder":104},[54,65047,65048],{"class":56,"line":1416},[54,65049,65050],{"class":111}," # Normalization\n",[54,65052,65053,65055,65058,65060,65062,65064,65067],{"class":56,"line":1421},[54,65054,1273],{"class":60},[54,65056,65057],{"class":64}," norm_mode.",[54,65059,64417],{"class":125},[54,65061,1233],{"class":64},[54,65063,1279],{"class":121},[54,65065,65066],{"class":279}," \"per_f\"",[54,65068,1284],{"class":64},[54,65070,65071,65074,65076,65078,65081,65084,65086,65088,65090,65092,65094,65096,65098,65100,65102,65104,65107],{"class":56,"line":1442},[54,65072,65073],{"class":64}," col_max ",[54,65075,122],{"class":121},[54,65077,1172],{"class":64},[54,65079,65080],{"class":125},"maximum",[54,65082,65083],{"class":64},"(S.",[54,65085,36269],{"class":125},[54,65087,129],{"class":64},[54,65089,60728],{"class":1711},[54,65091,122],{"class":121},[54,65093,150],{"class":132},[54,65095,171],{"class":64},[54,65097,64888],{"class":1711},[54,65099,122],{"class":121},[54,65101,13434],{"class":132},[54,65103,1769],{"class":64},[54,65105,65106],{"class":132},"1e-15",[54,65108,136],{"class":64},[54,65110,65111,65113,65115,65118,65120],{"class":56,"line":1447},[54,65112,65009],{"class":64},[54,65114,122],{"class":121},[54,65116,65117],{"class":64}," S ",[54,65119,194],{"class":121},[54,65121,65122],{"class":64}," col_max\n",[54,65124,65125,65127],{"class":56,"line":1453},[54,65126,1356],{"class":60},[54,65128,1284],{"class":64},[54,65130,65131,65133,65135,65137,65139,65141,65143,65145,65147,65149],{"class":56,"line":1458},[54,65132,65009],{"class":64},[54,65134,122],{"class":121},[54,65136,65117],{"class":64},[54,65138,194],{"class":121},[54,65140,64798],{"class":121},[54,65142,65083],{"class":64},[54,65144,36269],{"class":125},[54,65146,13506],{"class":64},[54,65148,65106],{"class":132},[54,65150,136],{"class":64},[54,65152,65153],{"class":56,"line":1469},[54,65154,105],{"emptyLinePlaceholder":104},[54,65156,65157,65159,65161,65163,65165,65167,65170],{"class":56,"line":1489},[54,65158,1273],{"class":60},[54,65160,64993],{"class":64},[54,65162,64417],{"class":125},[54,65164,1233],{"class":64},[54,65166,1279],{"class":121},[54,65168,65169],{"class":279}," \"db\"",[54,65171,1284],{"class":64},[54,65173,65174,65176,65178,65180,65182,65184,65187,65189,65191,65194,65197,65200],{"class":56,"line":1529},[54,65175,65009],{"class":64},[54,65177,122],{"class":121},[54,65179,56860],{"class":132},[54,65181,7785],{"class":121},[54,65183,1172],{"class":64},[54,65185,65186],{"class":125},"log10",[54,65188,59474],{"class":64},[54,65190,65080],{"class":125},[54,65192,65193],{"class":64},"(S, ",[54,65195,65196],{"class":132},"1e-12",[54,65198,65199],{"class":64},")) ",[54,65201,65202],{"class":111},"# dB, clipped floor\n",[54,65204,65205],{"class":56,"line":1534},[54,65206,105],{"emptyLinePlaceholder":104},[54,65208,65209],{"class":56,"line":1544},[54,65210,65211],{"class":111}," # Units\n",[54,65213,65214,65216,65219,65222,65224,65226,65229],{"class":56,"line":1559},[54,65215,1273],{"class":60},[54,65217,65218],{"class":64}," freq_units.",[54,65220,65221],{"class":125},"upper",[54,65223,1233],{"class":64},[54,65225,1279],{"class":121},[54,65227,65228],{"class":279}," \"GHz\"",[54,65230,1284],{"class":64},[54,65232,65233,65236,65238,65241,65243],{"class":56,"line":1592},[54,65234,65235],{"class":64}," f_fft ",[54,65237,122],{"class":121},[54,65239,65240],{"class":64}," f_fft_hz ",[54,65242,1947],{"class":121},[54,65244,57185],{"class":132},[54,65246,65247,65249],{"class":56,"line":1597},[54,65248,1356],{"class":60},[54,65250,1284],{"class":64},[54,65252,65253,65255,65257,65259,65261],{"class":56,"line":1608},[54,65254,65235],{"class":64},[54,65256,122],{"class":121},[54,65258,65240],{"class":64},[54,65260,1947],{"class":121},[54,65262,65263],{"class":132}," 1e-6\n",[54,65265,65266],{"class":56,"line":1613},[54,65267,105],{"emptyLinePlaceholder":104},[54,65269,65270,65272],{"class":56,"line":1627},[54,65271,2060],{"class":60},[54,65273,65274],{"class":64}," f_fft, S\n",[54,65276,65277],{"class":56,"line":1632},[54,65278,105],{"emptyLinePlaceholder":104},[54,65280,65281],{"class":56,"line":1643},[54,65282,65283],{"class":111},"# Compute spectrogram with current grid\n",[54,65285,65286,65289,65291,65293],{"class":56,"line":1671},[54,65287,65288],{"class":64},"f_fft_axis, S_fft ",[54,65290,122],{"class":121},[54,65292,64470],{"class":125},[54,65294,932],{"class":64},[54,65296,65297],{"class":56,"line":1690},[54,65298,65299],{"class":64}," Pe, dur_ns,\n",[54,65301,65302,65304,65306,65308],{"class":56,"line":1695},[54,65303,64491],{"class":1711},[54,65305,122],{"class":121},[54,65307,56562],{"class":132},[54,65309,965],{"class":64},[54,65311,65312,65314,65316,65318],{"class":56,"line":1720},[54,65313,64506],{"class":1711},[54,65315,122],{"class":121},[54,65317,56576],{"class":132},[54,65319,965],{"class":64},[54,65321,65322,65324,65326,65328],{"class":56,"line":1725},[54,65323,64521],{"class":1711},[54,65325,122],{"class":121},[54,65327,56591],{"class":132},[54,65329,965],{"class":64},[54,65331,65332,65334,65336,65338],{"class":56,"line":1736},[54,65333,64536],{"class":1711},[54,65335,122],{"class":121},[54,65337,56605],{"class":132},[54,65339,965],{"class":64},[54,65341,65342,65344,65346,65348],{"class":56,"line":1780},[54,65343,64551],{"class":1711},[54,65345,122],{"class":121},[54,65347,56620],{"class":132},[54,65349,965],{"class":64},[54,65351,65352,65354,65356,65358],{"class":56,"line":1789},[54,65353,64566],{"class":1711},[54,65355,122],{"class":121},[54,65357,56635],{"class":132},[54,65359,965],{"class":64},[54,65361,65362],{"class":56,"line":1794},[54,65363,136],{"class":64},[54,65365,65366,65368,65370,65373,65376,65379,65382,65384,65386,65389,65391,65393,65395,65397,65399],{"class":56,"line":1805},[54,65367,8081],{"class":121},[54,65369,129],{"class":64},[54,65371,65372],{"class":279},"\"FFT grid:\"",[54,65374,65375],{"class":64},", S_fft.shape, ",[54,65377,65378],{"class":279},"\"| Fourier freq range:\"",[54,65380,65381],{"class":64},", f_fft_axis[",[54,65383,150],{"class":132},[54,65385,1316],{"class":64},[54,65387,65388],{"class":279},"\"→\"",[54,65390,65381],{"class":64},[54,65392,1923],{"class":121},[54,65394,174],{"class":132},[54,65396,1316],{"class":64},[54,65398,56635],{"class":132},[54,65400,136],{"class":64},[44,65402,65405],{"className":65403,"code":65404,"language":6595,"meta":49},[8087],"FFT grid: (2049, 401) | Fourier freq range: 0.0 → 999.9999999999999 MHz\n",[51,65406,65404],{"__ignoreMap":49},[44,65408,65410],{"className":46,"code":65409,"language":48,"meta":49,"style":49},"# @title Theoretical Rabi ridge\nimport math\ntwo_pi = 2.0 * math.pi\nf0 = F0_GHZ * 1e9\ndelta_hz = two_pi * (freq_ghz*1e9 - f0) # rad\u002Fs detuning\nomega_onres = two_pi * OMEGA_ONRESONANCE_MHZ * 1e6 # rad\u002Fs\nOmega_R = np.sqrt(omega_onres**2 + delta_hz**2) # rad\u002Fs\nridge_hz = Omega_R \u002F (2.0 * math.pi) # Hz\nif FREQ_UNITS.upper() == \"GHz\":\n ridge_axis = ridge_hz * 1e-9\nelse:\n ridge_axis = ridge_hz * 1e-6 # MHz\n",[51,65411,65412,65417,65423,65436,65450,65478,65498,65530,65553,65572,65586,65592],{"__ignoreMap":49},[54,65413,65414],{"class":56,"line":57},[54,65415,65416],{"class":111},"# @title Theoretical Rabi ridge\n",[54,65418,65419,65421],{"class":56,"line":74},[54,65420,61],{"class":60},[54,65422,56692],{"class":64},[54,65424,65425,65428,65430,65432,65434],{"class":56,"line":88},[54,65426,65427],{"class":64},"two_pi ",[54,65429,122],{"class":121},[54,65431,57136],{"class":132},[54,65433,7785],{"class":121},[54,65435,57141],{"class":64},[54,65437,65438,65441,65443,65446,65448],{"class":56,"line":101},[54,65439,65440],{"class":64},"f0 ",[54,65442,122],{"class":121},[54,65444,65445],{"class":132}," F0_GHZ",[54,65447,7785],{"class":121},[54,65449,57156],{"class":132},[54,65451,65452,65455,65457,65459,65461,65464,65466,65469,65472,65475],{"class":56,"line":108},[54,65453,65454],{"class":64},"delta_hz ",[54,65456,122],{"class":121},[54,65458,57195],{"class":64},[54,65460,1947],{"class":121},[54,65462,65463],{"class":64}," (freq_ghz",[54,65465,1947],{"class":121},[54,65467,65468],{"class":132},"1e9",[54,65470,65471],{"class":121}," -",[54,65473,65474],{"class":64}," f0) ",[54,65476,65477],{"class":111},"# rad\u002Fs detuning\n",[54,65479,65480,65483,65485,65487,65489,65492,65494,65496],{"class":56,"line":115},[54,65481,65482],{"class":64},"omega_onres ",[54,65484,122],{"class":121},[54,65486,57195],{"class":64},[54,65488,1947],{"class":121},[54,65490,65491],{"class":132}," OMEGA_ONRESONANCE_MHZ",[54,65493,7785],{"class":121},[54,65495,57205],{"class":132},[54,65497,57208],{"class":111},[54,65499,65500,65503,65505,65507,65509,65512,65514,65516,65518,65521,65523,65525,65528],{"class":56,"line":139},[54,65501,65502],{"class":64},"Omega_R ",[54,65504,122],{"class":121},[54,65506,1172],{"class":64},[54,65508,9375],{"class":125},[54,65510,65511],{"class":64},"(omega_onres",[54,65513,1933],{"class":121},[54,65515,133],{"class":132},[54,65517,7869],{"class":121},[54,65519,65520],{"class":64}," delta_hz",[54,65522,1933],{"class":121},[54,65524,133],{"class":132},[54,65526,65527],{"class":64},") ",[54,65529,57262],{"class":111},[54,65531,65532,65535,65537,65539,65541,65543,65545,65547,65550],{"class":56,"line":159},[54,65533,65534],{"class":64},"ridge_hz ",[54,65536,122],{"class":121},[54,65538,57361],{"class":64},[54,65540,194],{"class":121},[54,65542,58201],{"class":64},[54,65544,59099],{"class":132},[54,65546,7785],{"class":121},[54,65548,65549],{"class":64}," math.pi) ",[54,65551,65552],{"class":111},"# Hz\n",[54,65554,65555,65557,65560,65562,65564,65566,65568,65570],{"class":56,"line":183},[54,65556,1201],{"class":60},[54,65558,65559],{"class":132}," FREQ_UNITS",[54,65561,29],{"class":64},[54,65563,65221],{"class":125},[54,65565,1233],{"class":64},[54,65567,1279],{"class":121},[54,65569,65228],{"class":279},[54,65571,1284],{"class":64},[54,65573,65574,65577,65579,65582,65584],{"class":56,"line":210},[54,65575,65576],{"class":64}," ridge_axis ",[54,65578,122],{"class":121},[54,65580,65581],{"class":64}," ridge_hz ",[54,65583,1947],{"class":121},[54,65585,57185],{"class":132},[54,65587,65588,65590],{"class":56,"line":227},[54,65589,60964],{"class":60},[54,65591,1284],{"class":64},[54,65593,65594,65596,65598,65600,65602,65605],{"class":56,"line":232},[54,65595,65576],{"class":64},[54,65597,122],{"class":121},[54,65599,65581],{"class":64},[54,65601,1947],{"class":121},[54,65603,65604],{"class":132}," 1e-6",[54,65606,65607],{"class":111}," # MHz\n",[44,65609,65611],{"className":46,"code":65610,"language":48,"meta":49,"style":49},"# @title 2D heatmap\nfig, ax = plt.subplots(figsize=(7.5, 5.5))\nim = ax.pcolormesh(freq_ghz, f_fft_axis, S_fft, shading=\"auto\", cmap=\"inferno\")\ncb = fig.colorbar(im, ax=ax)\ncb.set_label(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\n\nax.plot(freq_ghz, ridge_axis, \"--\", lw=1.5, label=r\"$\\Omega_R\u002F2\\pi$\")\nax.set_xlabel(\"Drive frequency f (GHz)\")\nax.set_ylabel(f\"Fourier frequency ({FREQ_UNITS})\")\nax.set_title(\"Fourier transform along pulse length: $|\\\\mathcal{F}_\\\\tau\\\\{P_e\\\\}|$\")\nax.legend(loc=\"upper right\")\nfig.tight_layout()\nplt.show()\n",[51,65612,65613,65618,65646,65679,65699,65742,65746,65792,65805,65826,65864,65882,65890],{"__ignoreMap":49},[54,65614,65615],{"class":56,"line":57},[54,65616,65617],{"class":111},"# @title 2D heatmap\n",[54,65619,65620,65622,65624,65626,65628,65630,65632,65634,65636,65639,65641,65644],{"class":56,"line":74},[54,65621,57777],{"class":64},[54,65623,122],{"class":121},[54,65625,57782],{"class":64},[54,65627,57785],{"class":125},[54,65629,129],{"class":64},[54,65631,57790],{"class":1711},[54,65633,122],{"class":121},[54,65635,129],{"class":64},[54,65637,65638],{"class":132},"7.5",[54,65640,171],{"class":64},[54,65642,65643],{"class":132},"5.5",[54,65645,22096],{"class":64},[54,65647,65648,65651,65653,65655,65657,65660,65662,65664,65667,65669,65672,65674,65677],{"class":56,"line":88},[54,65649,65650],{"class":64},"im ",[54,65652,122],{"class":121},[54,65654,57814],{"class":64},[54,65656,57817],{"class":125},[54,65658,65659],{"class":64},"(freq_ghz, f_fft_axis, S_fft, ",[54,65661,57823],{"class":1711},[54,65663,122],{"class":121},[54,65665,65666],{"class":279},"\"auto\"",[54,65668,171],{"class":64},[54,65670,65671],{"class":1711},"cmap",[54,65673,122],{"class":121},[54,65675,65676],{"class":279},"\"inferno\"",[54,65678,136],{"class":64},[54,65680,65681,65683,65685,65687,65689,65692,65694,65696],{"class":56,"line":101},[54,65682,57835],{"class":64},[54,65684,122],{"class":121},[54,65686,57840],{"class":64},[54,65688,57843],{"class":125},[54,65690,65691],{"class":64},"(im, ",[54,65693,57849],{"class":1711},[54,65695,122],{"class":121},[54,65697,65698],{"class":64},"ax)\n",[54,65700,65701,65703,65705,65707,65710,65712,65714,65717,65720,65723,65725,65727,65730,65732,65735,65737,65740],{"class":56,"line":108},[54,65702,63540],{"class":64},[54,65704,63543],{"class":125},[54,65706,129],{"class":64},[54,65708,65709],{"class":279},"\"FFT amplitude\"",[54,65711,7869],{"class":121},[54,65713,58201],{"class":64},[54,65715,65716],{"class":279},"\" (dB)\"",[54,65718,65719],{"class":60}," if",[54,65721,65722],{"class":132}," AMP_MODE",[54,65724,29],{"class":64},[54,65726,64417],{"class":125},[54,65728,65729],{"class":64},"()",[54,65731,1279],{"class":121},[54,65733,65734],{"class":279},"\"db\"",[54,65736,8036],{"class":60},[54,65738,65739],{"class":279}," \" (arb.)\"",[54,65741,22096],{"class":64},[54,65743,65744],{"class":56,"line":115},[54,65745,105],{"emptyLinePlaceholder":104},[54,65747,65748,65750,65752,65755,65758,65760,65763,65765,65768,65770,65772,65774,65776,65778,65781,65784,65787,65790],{"class":56,"line":139},[54,65749,57871],{"class":64},[54,65751,24935],{"class":125},[54,65753,65754],{"class":64},"(freq_ghz, ridge_axis, ",[54,65756,65757],{"class":279},"\"--\"",[54,65759,171],{"class":64},[54,65761,65762],{"class":1711},"lw",[54,65764,122],{"class":121},[54,65766,65767],{"class":132},"1.5",[54,65769,171],{"class":64},[54,65771,20568],{"class":1711},[54,65773,122],{"class":121},[54,65775,1772],{"class":60},[54,65777,61237],{"class":57863},[54,65779,65780],{"class":121},"\\O",[54,65782,65783],{"class":57863},"mega_R\u002F2",[54,65785,65786],{"class":121},"\\p",[54,65788,65789],{"class":57863},"i$\"",[54,65791,136],{"class":64},[54,65793,65794,65796,65798,65800,65803],{"class":56,"line":159},[54,65795,57871],{"class":64},[54,65797,57874],{"class":125},[54,65799,129],{"class":64},[54,65801,65802],{"class":279},"\"Drive frequency f (GHz)\"",[54,65804,136],{"class":64},[54,65806,65807,65809,65811,65813,65815,65818,65821,65824],{"class":56,"line":183},[54,65808,57871],{"class":64},[54,65810,57888],{"class":125},[54,65812,129],{"class":64},[54,65814,21834],{"class":60},[54,65816,65817],{"class":279},"\"Fourier frequency (",[54,65819,65820],{"class":132},"{FREQ_UNITS}",[54,65822,65823],{"class":279},")\"",[54,65825,136],{"class":64},[54,65827,65828,65830,65832,65834,65837,65839,65841,65844,65847,65849,65852,65854,65857,65859,65862],{"class":56,"line":210},[54,65829,57871],{"class":64},[54,65831,57902],{"class":125},[54,65833,129],{"class":64},[54,65835,65836],{"class":279},"\"Fourier transform along pulse length: $|",[54,65838,57910],{"class":121},[54,65840,2130],{"class":279},[54,65842,65843],{"class":132},"{F}",[54,65845,65846],{"class":279},"_",[54,65848,57910],{"class":121},[54,65850,65851],{"class":279},"tau",[54,65853,57910],{"class":121},[54,65855,65856],{"class":279},"{P_e",[54,65858,57910],{"class":121},[54,65860,65861],{"class":279},"}|$\"",[54,65863,136],{"class":64},[54,65865,65866,65868,65870,65872,65875,65877,65880],{"class":56,"line":227},[54,65867,57871],{"class":64},[54,65869,25114],{"class":125},[54,65871,129],{"class":64},[54,65873,65874],{"class":1711},"loc",[54,65876,122],{"class":121},[54,65878,65879],{"class":279},"\"upper right\"",[54,65881,136],{"class":64},[54,65883,65884,65886,65888],{"class":56,"line":232},[54,65885,57924],{"class":64},[54,65887,57927],{"class":125},[54,65889,330],{"class":64},[54,65891,65892,65894,65896],{"class":56,"line":246},[54,65893,24932],{"class":64},[54,65895,25169],{"class":125},[54,65897,330],{"class":64},[380,65899],{"alt":65900,"src":65901},"Output 6 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-06.webp",[44,65903,65905],{"className":46,"code":65904,"language":48,"meta":49,"style":49},"# --- Adaptive high-res 3D surface ---\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# ------ Control knobs ------\nUPSAMPLE_F_FREQ: int = 3 # try 2–3; upsample the drive-frequency axis only\nMAX_FACES: int = 300_000 # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\nMIN_ROWS: int = 128 # keep at least this many Fourier-frequency rows\nMAX_ROWS: int = 1024 # hard cap for rows to keep plotting fast\n\n# Preserve global color limits so the legend stays consistent\nvmin = float(np.nanmin(S_fft))\nvmax = float(np.nanmax(S_fft))\n\n# Shapes\nn_rows, n_cols = S_fft.shape # rows=Fourier freq (ν), cols=drive freq (f)\n\n# 1) Upsample along the typically smaller axis (drive frequency) only\nn_cols_up = min(UPSAMPLE_F_FREQ * (n_cols - 1) + 1, 1200) # sensible upper bound\nfreq_hi = np.linspace(freq_ghz.min(), freq_ghz.max(), n_cols_up)\n\nS_up_f = np.empty((n_rows, n_cols_up), dtype=float)\nfor i in range(n_rows):\n S_up_f[i, :] = np.interp(freq_hi, freq_ghz, S_fft[i, :])\n\n# 2) Determine how many Fourier rows we can afford for MAX_FACES\n# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\nn_rows_cap = int(MAX_FACES \u002F max(n_cols_up - 1, 1)) + 1\nn_rows_plot = int(np.clip(n_rows_cap, MIN_ROWS, min(MAX_ROWS, n_rows)))\n\n# Down-sample ν-axis evenly to n_rows_plot\nrow_idx = np.linspace(0, n_rows - 1, n_rows_plot).astype(int)\nf_fft_plot = f_fft_axis[row_idx]\nS_final = S_up_f[row_idx, :]\n\n# Mesh for plotting\nF2_hi, FFOUR_hi = np.meshgrid(freq_hi, f_fft_plot, indexing=\"xy\")\n\n# 3) Plot (keeps 'inferno')\nfig = plt.figure(figsize=(8, 6), dpi=200)\nax = fig.add_subplot(111, projection=\"3d\")\nsurf = ax.plot_surface(\n F2_hi, FFOUR_hi, S_final,\n rstride=1, cstride=1,\n linewidth=0, antialiased=True,\n cmap=\"inferno\",\n vmin=vmin, vmax=vmax\n)\n\nax.set_xlabel(\"f (GHz)\")\nax.set_ylabel(f\"Fourier freq ({FREQ_UNITS})\")\nax.set_zlabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower() == \"db\" else \" (arb.)\"))\nax.set_title(\"Rabi FFT surface (adaptive resolution)\")\nfig.colorbar(surf, ax=ax, shrink=0.7, pad=0.08, label=\"Amplitude\")\n\n# A clearer default view\nax.view_init(elev=25, azim=-60)\n\nfig.tight_layout()\nplt.show()\n",[51,65906,65907,65912,65922,65932,65942,65946,65951,65968,65985,66002,66019,66023,66028,66045,66061,66065,66070,66083,66087,66092,66130,66151,66155,66175,66188,66202,66206,66211,66216,66251,66280,66284,66289,66323,66333,66343,66347,66351,66377,66381,66386,66420,66445,66457,66467,66485,66503,66513,66529,66533,66537,66550,66569,66605,66618,66657,66661,66666,66693,66697,66705],{"__ignoreMap":49},[54,65908,65909],{"class":56,"line":57},[54,65910,65911],{"class":111},"# --- Adaptive high-res 3D surface ---\n",[54,65913,65914,65916,65918,65920],{"class":56,"line":74},[54,65915,77],{"class":60},[54,65917,62870],{"class":64},[54,65919,61],{"class":60},[54,65921,62875],{"class":64},[54,65923,65924,65926,65928,65930],{"class":56,"line":88},[54,65925,61],{"class":60},[54,65927,65],{"class":64},[54,65929,68],{"class":60},[54,65931,71],{"class":64},[54,65933,65934,65936,65938,65940],{"class":56,"line":101},[54,65935,61],{"class":60},[54,65937,56720],{"class":64},[54,65939,68],{"class":60},[54,65941,715],{"class":64},[54,65943,65944],{"class":56,"line":108},[54,65945,105],{"emptyLinePlaceholder":104},[54,65947,65948],{"class":56,"line":115},[54,65949,65950],{"class":111},"# ------ Control knobs ------\n",[54,65952,65953,65956,65958,65960,65962,65965],{"class":56,"line":139},[54,65954,65955],{"class":132},"UPSAMPLE_F_FREQ",[54,65957,959],{"class":64},[54,65959,976],{"class":121},[54,65961,979],{"class":121},[54,65963,65964],{"class":132}," 3",[54,65966,65967],{"class":111}," # try 2–3; upsample the drive-frequency axis only\n",[54,65969,65970,65973,65975,65977,65979,65982],{"class":56,"line":159},[54,65971,65972],{"class":132},"MAX_FACES",[54,65974,959],{"class":64},[54,65976,976],{"class":121},[54,65978,979],{"class":121},[54,65980,65981],{"class":132}," 300_000",[54,65983,65984],{"class":111}," # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\n",[54,65986,65987,65990,65992,65994,65996,65999],{"class":56,"line":183},[54,65988,65989],{"class":132},"MIN_ROWS",[54,65991,959],{"class":64},[54,65993,976],{"class":121},[54,65995,979],{"class":121},[54,65997,65998],{"class":132}," 128",[54,66000,66001],{"class":111}," # keep at least this many Fourier-frequency rows\n",[54,66003,66004,66007,66009,66011,66013,66016],{"class":56,"line":210},[54,66005,66006],{"class":132},"MAX_ROWS",[54,66008,959],{"class":64},[54,66010,976],{"class":121},[54,66012,979],{"class":121},[54,66014,66015],{"class":132}," 1024",[54,66017,66018],{"class":111}," # hard cap for rows to keep plotting fast\n",[54,66020,66021],{"class":56,"line":227},[54,66022,105],{"emptyLinePlaceholder":104},[54,66024,66025],{"class":56,"line":232},[54,66026,66027],{"class":111},"# Preserve global color limits so the legend stays consistent\n",[54,66029,66030,66033,66035,66037,66039,66042],{"class":56,"line":246},[54,66031,66032],{"class":64},"vmin ",[54,66034,122],{"class":121},[54,66036,59177],{"class":121},[54,66038,59474],{"class":64},[54,66040,66041],{"class":125},"nanmin",[54,66043,66044],{"class":64},"(S_fft))\n",[54,66046,66047,66050,66052,66054,66056,66059],{"class":56,"line":259},[54,66048,66049],{"class":64},"vmax ",[54,66051,122],{"class":121},[54,66053,59177],{"class":121},[54,66055,59474],{"class":64},[54,66057,66058],{"class":125},"nanmax",[54,66060,66044],{"class":64},[54,66062,66063],{"class":56,"line":264},[54,66064,105],{"emptyLinePlaceholder":104},[54,66066,66067],{"class":56,"line":270},[54,66068,66069],{"class":111},"# Shapes\n",[54,66071,66072,66075,66077,66080],{"class":56,"line":285},[54,66073,66074],{"class":64},"n_rows, n_cols ",[54,66076,122],{"class":121},[54,66078,66079],{"class":64}," S_fft.shape ",[54,66081,66082],{"class":111},"# rows=Fourier freq (ν), cols=drive freq (f)\n",[54,66084,66085],{"class":56,"line":297},[54,66086,105],{"emptyLinePlaceholder":104},[54,66088,66089],{"class":56,"line":310},[54,66090,66091],{"class":111},"# 1) Upsample along the typically smaller axis (drive frequency) only\n",[54,66093,66094,66097,66099,66101,66103,66105,66107,66110,66112,66114,66116,66118,66120,66122,66125,66127],{"class":56,"line":324},[54,66095,66096],{"class":64},"n_cols_up ",[54,66098,122],{"class":121},[54,66100,2032],{"class":121},[54,66102,129],{"class":64},[54,66104,65955],{"class":132},[54,66106,7785],{"class":121},[54,66108,66109],{"class":64}," (n_cols ",[54,66111,1923],{"class":121},[54,66113,982],{"class":132},[54,66115,204],{"class":64},[54,66117,1663],{"class":121},[54,66119,982],{"class":132},[54,66121,171],{"class":64},[54,66123,66124],{"class":132},"1200",[54,66126,22274],{"class":64},[54,66128,66129],{"class":111},"# sensible upper bound\n",[54,66131,66132,66134,66136,66138,66140,66142,66144,66146,66148],{"class":56,"line":333},[54,66133,62977],{"class":64},[54,66135,122],{"class":121},[54,66137,1172],{"class":64},[54,66139,57503],{"class":125},[54,66141,62986],{"class":64},[54,66143,24786],{"class":125},[54,66145,62991],{"class":64},[54,66147,36269],{"class":125},[54,66149,66150],{"class":64},"(), n_cols_up)\n",[54,66152,66153],{"class":56,"line":345},[54,66154,105],{"emptyLinePlaceholder":104},[54,66156,66157,66160,66162,66164,66166,66169,66171,66173],{"class":56,"line":358},[54,66158,66159],{"class":64},"S_up_f ",[54,66161,122],{"class":121},[54,66163,1172],{"class":64},[54,66165,63034],{"class":125},[54,66167,66168],{"class":64},"((n_rows, n_cols_up), ",[54,66170,1712],{"class":1711},[54,66172,63052],{"class":121},[54,66174,136],{"class":64},[54,66176,66177,66179,66181,66183,66185],{"class":56,"line":908},[54,66178,1186],{"class":60},[54,66180,7802],{"class":64},[54,66182,1192],{"class":60},[54,66184,7777],{"class":121},[54,66186,66187],{"class":64},"(n_rows):\n",[54,66189,66190,66193,66195,66197,66199],{"class":56,"line":913},[54,66191,66192],{"class":64}," S_up_f[i, :] ",[54,66194,122],{"class":121},[54,66196,1172],{"class":64},[54,66198,63083],{"class":125},[54,66200,66201],{"class":64},"(freq_hi, freq_ghz, S_fft[i, :])\n",[54,66203,66204],{"class":56,"line":918},[54,66205,105],{"emptyLinePlaceholder":104},[54,66207,66208],{"class":56,"line":923},[54,66209,66210],{"class":111},"# 2) Determine how many Fourier rows we can afford for MAX_FACES\n",[54,66212,66213],{"class":56,"line":935},[54,66214,66215],{"class":111},"# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\n",[54,66217,66218,66221,66223,66225,66227,66229,66231,66233,66236,66238,66240,66242,66244,66247,66249],{"class":56,"line":945},[54,66219,66220],{"class":64},"n_rows_cap ",[54,66222,122],{"class":121},[54,66224,1651],{"class":121},[54,66226,129],{"class":64},[54,66228,65972],{"class":132},[54,66230,57437],{"class":121},[54,66232,64798],{"class":121},[54,66234,66235],{"class":64},"(n_cols_up ",[54,66237,1923],{"class":121},[54,66239,982],{"class":132},[54,66241,171],{"class":64},[54,66243,174],{"class":132},[54,66245,66246],{"class":64},")) ",[54,66248,1663],{"class":121},[54,66250,1640],{"class":132},[54,66252,66253,66256,66258,66260,66262,66264,66267,66269,66271,66273,66275,66277],{"class":56,"line":953},[54,66254,66255],{"class":64},"n_rows_plot ",[54,66257,122],{"class":121},[54,66259,1651],{"class":121},[54,66261,59474],{"class":64},[54,66263,57474],{"class":125},[54,66265,66266],{"class":64},"(n_rows_cap, ",[54,66268,65989],{"class":132},[54,66270,171],{"class":64},[54,66272,24786],{"class":121},[54,66274,129],{"class":64},[54,66276,66006],{"class":132},[54,66278,66279],{"class":64},", n_rows)))\n",[54,66281,66282],{"class":56,"line":968},[54,66283,105],{"emptyLinePlaceholder":104},[54,66285,66286],{"class":56,"line":987},[54,66287,66288],{"class":111},"# Down-sample ν-axis evenly to n_rows_plot\n",[54,66290,66291,66294,66296,66298,66300,66302,66304,66307,66309,66311,66314,66317,66319,66321],{"class":56,"line":1005},[54,66292,66293],{"class":64},"row_idx ",[54,66295,122],{"class":121},[54,66297,1172],{"class":64},[54,66299,57503],{"class":125},[54,66301,129],{"class":64},[54,66303,150],{"class":132},[54,66305,66306],{"class":64},", n_rows ",[54,66308,1923],{"class":121},[54,66310,982],{"class":132},[54,66312,66313],{"class":64},", n_rows_plot).",[54,66315,66316],{"class":125},"astype",[54,66318,129],{"class":64},[54,66320,976],{"class":121},[54,66322,136],{"class":64},[54,66324,66325,66328,66330],{"class":56,"line":1021},[54,66326,66327],{"class":64},"f_fft_plot ",[54,66329,122],{"class":121},[54,66331,66332],{"class":64}," f_fft_axis[row_idx]\n",[54,66334,66335,66338,66340],{"class":56,"line":1027},[54,66336,66337],{"class":64},"S_final ",[54,66339,122],{"class":121},[54,66341,66342],{"class":64}," S_up_f[row_idx, :]\n",[54,66344,66345],{"class":56,"line":1033},[54,66346,105],{"emptyLinePlaceholder":104},[54,66348,66349],{"class":56,"line":1038},[54,66350,63212],{"class":111},[54,66352,66353,66356,66359,66361,66363,66365,66368,66370,66372,66375],{"class":56,"line":1044},[54,66354,66355],{"class":64},"F2_hi, ",[54,66357,66358],{"class":132},"FFOUR_hi",[54,66360,979],{"class":121},[54,66362,1172],{"class":64},[54,66364,57227],{"class":125},[54,66366,66367],{"class":64},"(freq_hi, f_fft_plot, ",[54,66369,57233],{"class":1711},[54,66371,122],{"class":121},[54,66373,66374],{"class":279},"\"xy\"",[54,66376,136],{"class":64},[54,66378,66379],{"class":56,"line":1050},[54,66380,105],{"emptyLinePlaceholder":104},[54,66382,66383],{"class":56,"line":1056},[54,66384,66385],{"class":111},"# 3) Plot (keeps 'inferno')\n",[54,66387,66388,66390,66392,66394,66396,66398,66400,66402,66404,66406,66408,66410,66412,66414,66416,66418],{"class":56,"line":1062},[54,66389,63248],{"class":64},[54,66391,122],{"class":121},[54,66393,57782],{"class":64},[54,66395,61173],{"class":125},[54,66397,129],{"class":64},[54,66399,57790],{"class":1711},[54,66401,122],{"class":121},[54,66403,129],{"class":64},[54,66405,35297],{"class":132},[54,66407,171],{"class":64},[54,66409,63269],{"class":132},[54,66411,1769],{"class":64},[54,66413,63274],{"class":1711},[54,66415,122],{"class":121},[54,66417,63279],{"class":132},[54,66419,136],{"class":64},[54,66421,66422,66424,66426,66428,66430,66432,66434,66436,66438,66440,66443],{"class":56,"line":1068},[54,66423,63286],{"class":64},[54,66425,122],{"class":121},[54,66427,57840],{"class":64},[54,66429,63293],{"class":125},[54,66431,129],{"class":64},[54,66433,63298],{"class":132},[54,66435,171],{"class":64},[54,66437,63303],{"class":1711},[54,66439,122],{"class":121},[54,66441,66442],{"class":279},"\"3d\"",[54,66444,136],{"class":64},[54,66446,66447,66449,66451,66453,66455],{"class":56,"line":1074},[54,66448,63319],{"class":64},[54,66450,122],{"class":121},[54,66452,57814],{"class":64},[54,66454,63326],{"class":125},[54,66456,932],{"class":64},[54,66458,66459,66462,66464],{"class":56,"line":1080},[54,66460,66461],{"class":64}," F2_hi, ",[54,66463,66358],{"class":132},[54,66465,66466],{"class":64},", S_final,\n",[54,66468,66469,66471,66473,66475,66477,66479,66481,66483],{"class":56,"line":1086},[54,66470,63338],{"class":1711},[54,66472,122],{"class":121},[54,66474,174],{"class":132},[54,66476,171],{"class":64},[54,66478,63347],{"class":1711},[54,66480,122],{"class":121},[54,66482,174],{"class":132},[54,66484,965],{"class":64},[54,66486,66487,66489,66491,66493,66495,66497,66499,66501],{"class":56,"line":1092},[54,66488,63362],{"class":1711},[54,66490,122],{"class":121},[54,66492,150],{"class":132},[54,66494,171],{"class":64},[54,66496,63371],{"class":1711},[54,66498,122],{"class":121},[54,66500,13434],{"class":132},[54,66502,965],{"class":64},[54,66504,66505,66507,66509,66511],{"class":56,"line":1098},[54,66506,63382],{"class":1711},[54,66508,122],{"class":121},[54,66510,65676],{"class":279},[54,66512,965],{"class":64},[54,66514,66515,66517,66519,66522,66524,66526],{"class":56,"line":1104},[54,66516,63393],{"class":1711},[54,66518,122],{"class":121},[54,66520,66521],{"class":64},"vmin, ",[54,66523,63402],{"class":1711},[54,66525,122],{"class":121},[54,66527,66528],{"class":64},"vmax\n",[54,66530,66531],{"class":56,"line":1109},[54,66532,136],{"class":64},[54,66534,66535],{"class":56,"line":1115},[54,66536,105],{"emptyLinePlaceholder":104},[54,66538,66539,66541,66543,66545,66548],{"class":56,"line":1121},[54,66540,57871],{"class":64},[54,66542,57874],{"class":125},[54,66544,129],{"class":64},[54,66546,66547],{"class":279},"\"f (GHz)\"",[54,66549,136],{"class":64},[54,66551,66552,66554,66556,66558,66560,66563,66565,66567],{"class":56,"line":1126},[54,66553,57871],{"class":64},[54,66555,57888],{"class":125},[54,66557,129],{"class":64},[54,66559,21834],{"class":60},[54,66561,66562],{"class":279},"\"Fourier freq (",[54,66564,65820],{"class":132},[54,66566,65823],{"class":279},[54,66568,136],{"class":64},[54,66570,66571,66573,66575,66577,66579,66581,66583,66585,66587,66589,66591,66593,66595,66597,66599,66601,66603],{"class":56,"line":1131},[54,66572,57871],{"class":64},[54,66574,63466],{"class":125},[54,66576,129],{"class":64},[54,66578,65709],{"class":279},[54,66580,7869],{"class":121},[54,66582,58201],{"class":64},[54,66584,65716],{"class":279},[54,66586,65719],{"class":60},[54,66588,65722],{"class":132},[54,66590,29],{"class":64},[54,66592,64417],{"class":125},[54,66594,1233],{"class":64},[54,66596,1279],{"class":121},[54,66598,65169],{"class":279},[54,66600,8036],{"class":60},[54,66602,65739],{"class":279},[54,66604,22096],{"class":64},[54,66606,66607,66609,66611,66613,66616],{"class":56,"line":1136},[54,66608,57871],{"class":64},[54,66610,57902],{"class":125},[54,66612,129],{"class":64},[54,66614,66615],{"class":279},"\"Rabi FFT surface (adaptive resolution)\"",[54,66617,136],{"class":64},[54,66619,66620,66622,66624,66626,66628,66630,66632,66634,66636,66638,66640,66642,66644,66646,66648,66650,66652,66655],{"class":56,"line":1153},[54,66621,57924],{"class":64},[54,66623,57843],{"class":125},[54,66625,63509],{"class":64},[54,66627,57849],{"class":1711},[54,66629,122],{"class":121},[54,66631,57854],{"class":64},[54,66633,63518],{"class":1711},[54,66635,122],{"class":121},[54,66637,63523],{"class":132},[54,66639,171],{"class":64},[54,66641,63528],{"class":1711},[54,66643,122],{"class":121},[54,66645,63533],{"class":132},[54,66647,171],{"class":64},[54,66649,20568],{"class":1711},[54,66651,122],{"class":121},[54,66653,66654],{"class":279},"\"Amplitude\"",[54,66656,136],{"class":64},[54,66658,66659],{"class":56,"line":1164},[54,66660,105],{"emptyLinePlaceholder":104},[54,66662,66663],{"class":56,"line":1180},[54,66664,66665],{"class":111},"# A clearer default view\n",[54,66667,66668,66670,66673,66675,66678,66680,66682,66684,66687,66689,66691],{"class":56,"line":1213},[54,66669,57871],{"class":64},[54,66671,66672],{"class":125},"view_init",[54,66674,129],{"class":64},[54,66676,66677],{"class":1711},"elev",[54,66679,122],{"class":121},[54,66681,48145],{"class":132},[54,66683,171],{"class":64},[54,66685,66686],{"class":1711},"azim",[54,66688,13671],{"class":121},[54,66690,58234],{"class":132},[54,66692,136],{"class":64},[54,66694,66695],{"class":56,"line":1219},[54,66696,105],{"emptyLinePlaceholder":104},[54,66698,66699,66701,66703],{"class":56,"line":1247},[54,66700,57924],{"class":64},[54,66702,57927],{"class":125},[54,66704,330],{"class":64},[54,66706,66707,66709,66711],{"class":56,"line":1270},[54,66708,24932],{"class":64},[54,66710,25169],{"class":125},[54,66712,330],{"class":64},[380,66714],{"alt":66715,"src":66716},"Output 7 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-07.webp",[44,66718,66720],{"className":46,"code":66719,"language":48,"meta":49,"style":49},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\nF_STAR_GHZ: float = float(F0_GHZ) # slice near resonance\nNU_STAR = float(ridge_axis[np.argmin(np.abs(freq_ghz - F_STAR_GHZ))]) # near Ω_R\u002F2π\n\n# Slice at f = F_STAR_GHZ\nj = int(np.argmin(np.abs(freq_ghz - F_STAR_GHZ)))\nfig, ax = plt.subplots(figsize=(6.0, 3.8))\nax.plot(f_fft_axis, S_fft[:, j], color='blue')\nax.axvline(ridge_axis[j], ls=\"--\", lw=1.0, label=r\"$\\Omega_R\u002F2\\pi$\")\nax.set_xlabel(f\"Fourier frequency ({FREQ_UNITS})\")\nax.set_ylabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\nax.set_title(fr\"Slice at $f={freq_ghz[j]:.6f}$ GHz\")\nax.legend()\nfig.tight_layout(); plt.show()\n\n# Slice at ν = NU_STAR\ni = int(np.argmin(np.abs(f_fft_axis - NU_STAR)))\nfig, ax = plt.subplots(figsize=(6.0, 3.8))\nax.plot(freq_ghz, S_fft[i, :], color='blue')\nax.set_xlabel(\"Drive frequency f (GHz)\")\nax.set_ylabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\nax.set_title(fr\"Slice at Fourier freq ≈ {f_fft_axis[i]:.3f} {FREQ_UNITS}\")\nfig.tight_layout(); plt.show()\n",[51,66721,66722,66727,66750,66782,66786,66791,66815,66843,66860,66905,66923,66959,66986,66994,67007,67011,67016,67042,67068,67085,67097,67133,67162],{"__ignoreMap":49},[54,66723,66724],{"class":56,"line":57},[54,66725,66726],{"class":111},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\n",[54,66728,66729,66732,66734,66736,66738,66740,66742,66744,66747],{"class":56,"line":74},[54,66730,66731],{"class":132},"F_STAR_GHZ",[54,66733,959],{"class":64},[54,66735,962],{"class":121},[54,66737,979],{"class":121},[54,66739,59177],{"class":121},[54,66741,129],{"class":64},[54,66743,56084],{"class":132},[54,66745,66746],{"class":64},") ",[54,66748,66749],{"class":111},"# slice near resonance\n",[54,66751,66752,66755,66757,66759,66762,66764,66766,66768,66771,66773,66776,66779],{"class":56,"line":88},[54,66753,66754],{"class":132},"NU_STAR",[54,66756,979],{"class":121},[54,66758,59177],{"class":121},[54,66760,66761],{"class":64},"(ridge_axis[np.",[54,66763,63811],{"class":125},[54,66765,59474],{"class":64},[54,66767,63816],{"class":125},[54,66769,66770],{"class":64},"(freq_ghz ",[54,66772,1923],{"class":121},[54,66774,66775],{"class":132}," F_STAR_GHZ",[54,66777,66778],{"class":64},"))]) ",[54,66780,66781],{"class":111},"# near Ω_R\u002F2π\n",[54,66783,66784],{"class":56,"line":101},[54,66785,105],{"emptyLinePlaceholder":104},[54,66787,66788],{"class":56,"line":108},[54,66789,66790],{"class":111},"# Slice at f = F_STAR_GHZ\n",[54,66792,66793,66795,66797,66799,66801,66803,66805,66807,66809,66811,66813],{"class":56,"line":115},[54,66794,63838],{"class":64},[54,66796,122],{"class":121},[54,66798,1651],{"class":121},[54,66800,59474],{"class":64},[54,66802,63811],{"class":125},[54,66804,59474],{"class":64},[54,66806,63816],{"class":125},[54,66808,66770],{"class":64},[54,66810,1923],{"class":121},[54,66812,66775],{"class":132},[54,66814,13622],{"class":64},[54,66816,66817,66819,66821,66823,66825,66827,66829,66831,66833,66836,66838,66841],{"class":56,"line":139},[54,66818,57777],{"class":64},[54,66820,122],{"class":121},[54,66822,57782],{"class":64},[54,66824,57785],{"class":125},[54,66826,129],{"class":64},[54,66828,57790],{"class":1711},[54,66830,122],{"class":121},[54,66832,129],{"class":64},[54,66834,66835],{"class":132},"6.0",[54,66837,171],{"class":64},[54,66839,66840],{"class":132},"3.8",[54,66842,22096],{"class":64},[54,66844,66845,66847,66849,66852,66854,66856,66858],{"class":56,"line":159},[54,66846,57871],{"class":64},[54,66848,24935],{"class":125},[54,66850,66851],{"class":64},"(f_fft_axis, S_fft[:, j], ",[54,66853,63907],{"class":1711},[54,66855,122],{"class":121},[54,66857,63912],{"class":279},[54,66859,136],{"class":64},[54,66861,66862,66864,66867,66870,66873,66875,66877,66879,66881,66883,66885,66887,66889,66891,66893,66895,66897,66899,66901,66903],{"class":56,"line":183},[54,66863,57871],{"class":64},[54,66865,66866],{"class":125},"axvline",[54,66868,66869],{"class":64},"(ridge_axis[j], ",[54,66871,66872],{"class":1711},"ls",[54,66874,122],{"class":121},[54,66876,65757],{"class":279},[54,66878,171],{"class":64},[54,66880,65762],{"class":1711},[54,66882,122],{"class":121},[54,66884,57484],{"class":132},[54,66886,171],{"class":64},[54,66888,20568],{"class":1711},[54,66890,122],{"class":121},[54,66892,1772],{"class":60},[54,66894,61237],{"class":57863},[54,66896,65780],{"class":121},[54,66898,65783],{"class":57863},[54,66900,65786],{"class":121},[54,66902,65789],{"class":57863},[54,66904,136],{"class":64},[54,66906,66907,66909,66911,66913,66915,66917,66919,66921],{"class":56,"line":210},[54,66908,57871],{"class":64},[54,66910,57874],{"class":125},[54,66912,129],{"class":64},[54,66914,21834],{"class":60},[54,66916,65817],{"class":279},[54,66918,65820],{"class":132},[54,66920,65823],{"class":279},[54,66922,136],{"class":64},[54,66924,66925,66927,66929,66931,66933,66935,66937,66939,66941,66943,66945,66947,66949,66951,66953,66955,66957],{"class":56,"line":227},[54,66926,57871],{"class":64},[54,66928,57888],{"class":125},[54,66930,129],{"class":64},[54,66932,65709],{"class":279},[54,66934,7869],{"class":121},[54,66936,58201],{"class":64},[54,66938,65716],{"class":279},[54,66940,65719],{"class":60},[54,66942,65722],{"class":132},[54,66944,29],{"class":64},[54,66946,64417],{"class":125},[54,66948,65729],{"class":64},[54,66950,1279],{"class":121},[54,66952,65734],{"class":279},[54,66954,8036],{"class":60},[54,66956,65739],{"class":279},[54,66958,22096],{"class":64},[54,66960,66961,66963,66965,66967,66970,66973,66975,66977,66979,66981,66984],{"class":56,"line":232},[54,66962,57871],{"class":64},[54,66964,57902],{"class":125},[54,66966,129],{"class":64},[54,66968,66969],{"class":60},"fr",[54,66971,66972],{"class":279},"\"Slice at $f=",[54,66974,2366],{"class":132},[54,66976,63996],{"class":64},[54,66978,62435],{"class":60},[54,66980,2428],{"class":132},[54,66982,66983],{"class":279},"$ GHz\"",[54,66985,136],{"class":64},[54,66987,66988,66990,66992],{"class":56,"line":246},[54,66989,57871],{"class":64},[54,66991,25114],{"class":125},[54,66993,330],{"class":64},[54,66995,66996,66998,67000,67003,67005],{"class":56,"line":259},[54,66997,57924],{"class":64},[54,66999,57927],{"class":125},[54,67001,67002],{"class":64},"(); plt.",[54,67004,25169],{"class":125},[54,67006,330],{"class":64},[54,67008,67009],{"class":56,"line":264},[54,67010,105],{"emptyLinePlaceholder":104},[54,67012,67013],{"class":56,"line":270},[54,67014,67015],{"class":111},"# Slice at ν = NU_STAR\n",[54,67017,67018,67020,67022,67024,67026,67028,67030,67032,67035,67037,67040],{"class":56,"line":285},[54,67019,64052],{"class":64},[54,67021,122],{"class":121},[54,67023,1651],{"class":121},[54,67025,59474],{"class":64},[54,67027,63811],{"class":125},[54,67029,59474],{"class":64},[54,67031,63816],{"class":125},[54,67033,67034],{"class":64},"(f_fft_axis ",[54,67036,1923],{"class":121},[54,67038,67039],{"class":132}," NU_STAR",[54,67041,13622],{"class":64},[54,67043,67044,67046,67048,67050,67052,67054,67056,67058,67060,67062,67064,67066],{"class":56,"line":297},[54,67045,57777],{"class":64},[54,67047,122],{"class":121},[54,67049,57782],{"class":64},[54,67051,57785],{"class":125},[54,67053,129],{"class":64},[54,67055,57790],{"class":1711},[54,67057,122],{"class":121},[54,67059,129],{"class":64},[54,67061,66835],{"class":132},[54,67063,171],{"class":64},[54,67065,66840],{"class":132},[54,67067,22096],{"class":64},[54,67069,67070,67072,67074,67077,67079,67081,67083],{"class":56,"line":310},[54,67071,57871],{"class":64},[54,67073,24935],{"class":125},[54,67075,67076],{"class":64},"(freq_ghz, S_fft[i, :], ",[54,67078,63907],{"class":1711},[54,67080,122],{"class":121},[54,67082,63912],{"class":279},[54,67084,136],{"class":64},[54,67086,67087,67089,67091,67093,67095],{"class":56,"line":324},[54,67088,57871],{"class":64},[54,67090,57874],{"class":125},[54,67092,129],{"class":64},[54,67094,65802],{"class":279},[54,67096,136],{"class":64},[54,67098,67099,67101,67103,67105,67107,67109,67111,67113,67115,67117,67119,67121,67123,67125,67127,67129,67131],{"class":56,"line":333},[54,67100,57871],{"class":64},[54,67102,57888],{"class":125},[54,67104,129],{"class":64},[54,67106,65709],{"class":279},[54,67108,7869],{"class":121},[54,67110,58201],{"class":64},[54,67112,65716],{"class":279},[54,67114,65719],{"class":60},[54,67116,65722],{"class":132},[54,67118,29],{"class":64},[54,67120,64417],{"class":125},[54,67122,65729],{"class":64},[54,67124,1279],{"class":121},[54,67126,65734],{"class":279},[54,67128,8036],{"class":60},[54,67130,65739],{"class":279},[54,67132,22096],{"class":64},[54,67134,67135,67137,67139,67141,67143,67146,67148,67151,67153,67155,67158,67160],{"class":56,"line":345},[54,67136,57871],{"class":64},[54,67138,57902],{"class":125},[54,67140,129],{"class":64},[54,67142,66969],{"class":60},[54,67144,67145],{"class":279},"\"Slice at Fourier freq ≈ ",[54,67147,2366],{"class":132},[54,67149,67150],{"class":64},"f_fft_axis[i]",[54,67152,62510],{"class":60},[54,67154,2428],{"class":132},[54,67156,67157],{"class":132}," {FREQ_UNITS}",[54,67159,21837],{"class":279},[54,67161,136],{"class":64},[54,67163,67164,67166,67168,67170,67172],{"class":56,"line":358},[54,67165,57924],{"class":64},[54,67167,57927],{"class":125},[54,67169,67002],{"class":64},[54,67171,25169],{"class":125},[54,67173,330],{"class":64},[380,67175],{"alt":67176,"src":67177},"Output 8 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-08.webp",[380,67179],{"alt":67180,"src":67181},"Output 9 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-09.webp",[31,67183,489],{"id":488},[13,67185,41155,67186,496,67188,29],{},[20,67187,41227],{"href":41226},[20,67189,67191],{"href":67190},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FRabi_Oscillation_Visualization_for_Excited_State_Probability.ipynb","open it in Colab",[502,67193,67194],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}",{"title":49,"searchDepth":74,"depth":74,"links":67196},[67197,67198,67199,67200,67201,67202,67203,67204,67205,67206,67207,67208,67209,67210,67211],{"id":41403,"depth":74,"text":41404},{"id":42883,"depth":74,"text":42884},{"id":43672,"depth":74,"text":43673},{"id":45021,"depth":74,"text":45022},{"id":46150,"depth":74,"text":46151},{"id":48632,"depth":74,"text":48633},{"id":49148,"depth":74,"text":49149},{"id":50247,"depth":74,"text":50248},{"id":51839,"depth":74,"text":51840},{"id":52212,"depth":74,"text":52213},{"id":53204,"depth":74,"text":53205},{"id":54381,"depth":74,"text":54382},{"id":55561,"depth":74,"text":55562},{"id":56056,"depth":74,"text":56057},{"id":488,"depth":74,"text":489},[518,519,520,67213],"Rabi oscillations",[67215],{"username":8,"name":524,"role":525,"bio":526,"links":67216},[67217,67218],{"label":529,"href":530},{"label":532,"href":41226},{"username":8,"name":524,"role":534},"A primer on the quantum dynamics of a driven two-level system, visualizing excited-state probability under the rotating-wave approximation (RWA).","Deep dive · Hardware",{},"\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Frabi-oscillation-visualization",[],{"title":41216,"description":67220},"blog\u002Fexpert-notes\u002Frabi-oscillation-visualization",[67229,572],"hardware","cLyZBkTx0Kr_9zY5k_7c4mZTRRqgWxllLNGfFbxk58w",{"id":67232,"title":67233,"authors":67234,"body":67235,"breadcrumb":71822,"builders":71824,"byline":71829,"challenge":535,"courseAuthor":535,"courseLead":535,"dek":535,"description":71830,"draft":538,"extension":539,"eyebrow":67221,"finish":535,"fork":535,"hero":535,"heroAlt":535,"heroCta":535,"heroImage":535,"kind":541,"lessonCount":535,"meta":71831,"navigation":104,"newsItems":535,"next":535,"ogImage":71832,"order":535,"outcomes":535,"path":71833,"publishDate":545,"readingTime":71834,"related":71835,"relatedProjects":535,"seo":71836,"stem":71837,"tags":71838,"track":535,"trackName":535,"__hash__":71839},"blog\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications.md","A Sampling-Budget & Bit-String Analogy for Qubit Classifications",[8],{"type":10,"value":67236,"toc":71818},[67237,67244,71659,71665,71668,71671,71677,71683,71686,71688,71806,71808,71815],[13,67238,67239],{},[585,67240,587,67241,41228],{},[20,67242,41227],{"href":67243},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[44,67245,67247],{"className":46,"code":67246,"language":48,"meta":49,"style":49},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n# generates tables and figures, so you can iterate further.\n#\n# Contents created in the current directory:\n# - Sampling_Budget_Qubit_Bitstrings.ipynb (Colab notebook)\n# - summary.csv (summary table)\n# - bitstrings_\u003CMODEL>.csv (raw bit-strings per model)\n# - fig_ones_fraction.png (bar chart with Wilson CI)\n# - fig_required_shots.png (required shots for ±ε)\n# - simulation_config.json (run parameters)\n\nimport json\nimport math\nfrom dataclasses import dataclass\nfrom typing import List, Dict, Tuple\n\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom IPython.display import display, Markdown\nimport nbformat as nbf\n\nplt.rcParams['figure.dpi'] = 200\n\n# ---------- Helper math ----------\n\ndef wilson_ci(k: int, n: int, z: float = 1.96) -> Tuple[float, float]:\n \"\"\"\n Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n Returns (low, high). Works for edge cases k=0 or k=n.\n \"\"\"\n if n == 0:\n return (0.0, 1.0)\n p_hat = k \u002F n\n denom = 1.0 + (z ** 2) \u002F n\n center = (p_hat + (z ** 2) \u002F (2 * n)) \u002F denom\n span = (z * math.sqrt((p_hat * (1 - p_hat) \u002F n) + (z ** 2) \u002F (4 * n ** 2))) \u002F denom\n return (max(0.0, center - span), min(1.0, center + span))\n\n\ndef shots_for_epsilon(p: float, epsilon: float = 0.01, z: float = 1.96) -> int:\n \"\"\"\n Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n \"\"\"\n p = min(max(p, 0.0), 1.0)\n var = p * (1.0 - p)\n if var == 0.0:\n return 1\n n = (z ** 2) * var \u002F (epsilon ** 2)\n return max(1, int(math.ceil(n)))\n\n\n# ---------- Domain model ----------\n\n@dataclass(frozen=True)\nclass QubitModel:\n name: str\n satisfaction_prob: float # Probability a single shot meets the analytical criterion (bit = 1)\n description: str\n\n\nDEFAULT_MODELS: List[QubitModel] = [\n QubitModel(\n \"Toy working qubit\",\n 0.50,\n \"Simple demonstrator; about half shots meet the criterion.\"\n ),\n QubitModel(\n \"NISQ qubit\",\n 0.60,\n \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n ),\n QubitModel(\n \"Fault-tolerant (logical)\",\n 0.90,\n \"Error-corrected logical qubit; high single-shot success probability.\"\n ),\n QubitModel(\n \"Topologically protected\",\n 0.98,\n \"Intrinsic protection; very high single-shot success probability.\"\n ),\n QubitModel(\n \"Ideal qubit\",\n 1.00,\n \"Theoretical perfect qubit; always successful in a single shot.\"\n ),\n]\n\n# ---------- Simulation configuration ----------\n\nCONFIG: Dict = {\n \"random_seed\": 123456,\n \"shots_per_model\": 10000,\n \"epsilons\": [0.02, 0.01, 0.005], # ±2%, ±1%, ±0.5% targets\n \"cost_model\": {\n \"default_cost_per_shot_usd\": 0.00005,\n \"fixed_overhead_usd\": 0.10\n },\n \"timing\": {\n \"time_per_shot_seconds\": 2e-6\n }\n}\n\nrng = np.random.default_rng(CONFIG[\"random_seed\"])\n\n\n# ---------- Core simulation ----------\n\ndef simulate_bitstring(p: float, shots: int, rng_: np.random.Generator) -> np.ndarray:\n \"\"\"\n Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n \"\"\"\n return rng_.binomial(1, p, size=shots).astype(int)\n\n\ndef summarize_model(model: QubitModel, bits: np.ndarray, epsilons: List[float]) -> Dict:\n \"\"\"\n Calculate summary statistics for a given model and its simulated bitstring.\n \"\"\"\n n = int(bits.size)\n ones = int(bits.sum())\n zeros = int(n - ones)\n frac = ones \u002F n if n > 0 else float(\"nan\")\n lo, hi = wilson_ci(ones, n)\n\n row = {\n \"Model\": model.name,\n \"Description\": model.description,\n \"Configured p (target)\": model.satisfaction_prob,\n \"Shots\": n,\n \"Ones (successes)\": ones,\n \"Zeros (failures)\": zeros,\n \"Ones fraction (observed)\": frac,\n \"Wilson CI 95% low\": lo,\n \"Wilson CI 95% high\": hi,\n \"CI width (95%)\": hi - lo,\n }\n for eps in epsilons:\n row[f\"Required shots for ±{int(eps*100)}%\"] = shots_for_epsilon(model.satisfaction_prob, eps)\n return row\n\n\ndef apply_costs(shots: int, cost_per_shot: float, fixed_overhead: float) -> float:\n \"\"\"\n Calculate illustrative cost based on shots and cost model.\n \"\"\"\n return fixed_overhead + shots * cost_per_shot\n\n\n# ---------- Run the simulation ----------\n\nall_bitstrings: Dict[str, np.ndarray] = {}\nsummary_rows: List[Dict] = []\n\nprint(\"Running simulation...\")\nfor m in DEFAULT_MODELS:\n bits = simulate_bitstring(m.satisfaction_prob, CONFIG[\"shots_per_model\"], rng)\n all_bitstrings[m.name] = bits\n summary_rows.append(summarize_model(m, bits, CONFIG[\"epsilons\"]))\n\nsummary_df = pd.DataFrame(summary_rows)\n\n# Add cost estimates\ncps = CONFIG[\"cost_model\"][\"default_cost_per_shot_usd\"]\nover = CONFIG[\"cost_model\"][\"fixed_overhead_usd\"]\nif cps > 0 or over > 0:\n summary_df[\"Illustrative cost (USD)\"] = [\n apply_costs(int(row[\"Shots\"]), cps, over)\n for _, row in summary_df.iterrows()\n ]\n\n# ---------- Save artifacts ----------\n\n# Save raw bitstrings and summary CSV\nfor name, arr in all_bitstrings.items():\n out_path = f\"bitstrings_{name.replace(' ', '_')}.csv\"\n pd.DataFrame({\"bit\": arr}).to_csv(out_path, index=False)\n\nsummary_csv_path = \"summary.csv\"\nsummary_df.to_csv(summary_csv_path, index=False)\n\n# Save config JSON\nwith open(\"simulation_config.json\", \"w\") as f:\n json.dump(CONFIG, f, indent=2)\n\nprint(\"Saved data to CSV and JSON files.\")\n\n# ---------- Visualizations ----------\n\n# Figure 1: Observed ones fraction with Wilson CI\nplt.style.use('seaborn-v0_8-whitegrid')\nplt.figure(figsize=(10, 6))\nx = np.arange(len(summary_df))\ny = summary_df[\"Ones fraction (observed)\"].to_numpy()\n\n# --- FIX APPLIED HERE ---\n# Take the absolute value to prevent tiny negative numbers from floating-point\n# inaccuracies from causing a ValueError in the errorbar plot.\nlower_error = y - summary_df[\"Wilson CI 95% low\"].to_numpy()\nupper_error = summary_df[\"Wilson CI 95% high\"].to_numpy() - y\nyerr = np.vstack([np.abs(lower_error), np.abs(upper_error)])\n# --- END OF FIX ---\n\nplt.bar(x, y, color='skyblue', edgecolor='black')\nplt.errorbar(x, y, yerr=yerr, fmt=\"none\", capsize=5, color=\"black\", elinewidth=1.5)\nplt.xticks(x, summary_df[\"Model\"].tolist(), rotation=25, ha=\"right\")\nplt.ylabel(\"Observed Ones Fraction\")\nplt.title(\"Observed Single-Shot Success Rates with 95% Wilson CI\", fontsize=14)\nplt.tight_layout()\nfig1_path = \"fig_ones_fraction.png\"\nplt.savefig(fig1_path, dpi=200, bbox_inches='tight')\nplt.show()\n\n# Figure 2: Required shots at several ±epsilon targets\nplt.figure(figsize=(10, 6))\nwidth = 0.2\nx = np.arange(len(DEFAULT_MODELS))\ncolors = ['#ff9999','#66b3ff','#99ff99']\n\nfor i, eps in enumerate(CONFIG[\"epsilons\"]):\n req = [shots_for_epsilon(m.satisfaction_prob, eps) for m in DEFAULT_MODELS]\n plt.bar(x + i * width, req, width, label=f\"±{eps:.1%}\", color=colors[i], edgecolor='black')\n\nplt.xticks(x + width, [m.name for m in DEFAULT_MODELS], rotation=25, ha=\"right\")\nplt.ylabel(\"Required Shots (Log Scale)\")\nplt.title(\"Approximate Shots to Estimate Success Rate within ±ε\", fontsize=14)\nplt.yscale('log')\nplt.legend(title=\"Error Tolerance\")\nplt.tight_layout()\nfig2_path = \"fig_required_shots.png\"\nplt.savefig(fig2_path, dpi=200, bbox_inches='tight')\nplt.show()\n\nprint(\"Generated and saved plots.\")\n\n# ---------- Display interactive table ----------\n\ndisplay(Markdown(\"## Qubit Sampling Budget - Summary\"))\ndisplay(summary_df.style.format({\n \"Configured p (target)\": \"{:.2f}\",\n \"Ones fraction (observed)\": \"{:.4f}\",\n \"Wilson CI 95% low\": \"{:.4f}\",\n \"Wilson CI 95% high\": \"{:.4f}\",\n \"CI width (95%)\": \"{:.4f}\",\n \"Illustrative cost (USD)\": \"${:.2f}\"\n}))\n\n\n# ---------- Build a ready-to-run Colab notebook ----------\n\nmd_intro = r\"\"\"\n# Sampling Budget & Bit-String Analogy for Qubit Classes\n\n**High school level (concise):** We treat each measurement (“single shot”) as a bit: `1` if the outcome matches the analytical prediction, `0` if it does not. Different qubit types have different chances of landing a `1`. We simulate many shots, show how often we get `1`s, and estimate how many shots we need for a target accuracy.\n\n**Graduate level (concise):** We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ (noisy intermediate-scale quantum), fault-tolerant (logical), topologically protected, and ideal. We report Wilson score intervals (95%) for observed proportions and estimate the required shot budget $n \\approx z^2\\,p(1-p)\u002F\\epsilon^2$ to bound absolute error by ±$\\epsilon$ at ~95% confidence.\n\"\"\"\n\ncode_config = r\"\"\"\n# @title Configuration\nfrom dataclasses import dataclass\nfrom typing import List, Dict\nimport numpy as np\n\nRANDOM_SEED = 579345 # @param {type:\"number\"}\nSHOTS_PER_MODEL = 10000 # @param {type:\"number\"}\nEPSILONS = [0.02, 0.01, 0.005] # @param\nCOST_PER_SHOT_USD = 0.00005 # @param {type:\"number\"}\nFIXED_OVERHEAD_USD = 0.10 # @param {type:\"number\"}\n\n@dataclass(frozen=True)\nclass QubitModel:\n name: str\n satisfaction_prob: float\n description: str\n\nMODELS: List[QubitModel] = [\n QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n]\n\nrng = np.random.default_rng(RANDOM_SEED)\n\"\"\"\n\ncode_lib_and_run = r\"\"\"\n# @title Run Simulation & Display Results\nimport math\nfrom typing import Tuple, Dict\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom IPython.display import display, Markdown\n\ndef wilson_ci(k: int, n: int, z: float = 1.96) -> Tuple[float, float]:\n if n == 0: return (0.0, 1.0)\n p_hat, z2_n = k \u002F n, z**2 \u002F n\n denom = 1.0 + z2_n\n center = (p_hat + z2_n \u002F 2) \u002F denom\n span = (z * math.sqrt(p_hat * (1 - p_hat) \u002F n + z**2 \u002F (4 * n**2))) \u002F denom\n return (max(0.0, center - span), min(1.0, center + span))\n\ndef shots_for_epsilon(p: float, epsilon: float = 0.01, z: float = 1.96) -> int:\n if p * (1-p) == 0: return 1\n return max(1, int(math.ceil((z**2 * p * (1-p)) \u002F (epsilon**2))))\n\ndef summarize_model(model, bits: np.ndarray, epsilons) -> Dict:\n n = len(bits)\n ones = sum(bits)\n lo, hi = wilson_ci(ones, n)\n row = {\n \"Model\": model.name, \"p (target)\": model.satisfaction_prob, \"Shots\": n,\n \"Ones\": ones, \"p (observed)\": ones \u002F n if n > 0 else 0,\n \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n }\n for eps in epsilons:\n row[f\"Shots for ±{eps:.1%}\"] = shots_for_epsilon(model.satisfaction_prob, eps)\n return row\n\n# --- Run Simulation ---\nsummary_rows = [summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\nsummary_df = pd.DataFrame(summary_rows)\nif COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n summary_df[\"Cost (USD)\"] = FIXED_OVERHEAD_USD + summary_df[\"Shots\"] * COST_PER_SHOT_USD\n\ndisplay(Markdown(\"### Simulation Summary\"))\ndisplay(summary_df.style.format(precision=4).background_gradient(cmap='viridis', subset=['p (observed)', 'CI width']))\n\n# --- Plotting ---\nplt.style.use('seaborn-v0_8-whitegrid')\nfig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 6))\n\n# Plot 1\ny = summary_df[\"p (observed)\"].to_numpy()\nlower_error = y - summary_df[\"CI 95% low\"]\nupper_error = summary_df[\"CI 95% high\"] - y\nyerr = np.vstack([np.abs(lower_error), np.abs(upper_error)])\n\nax1.bar(summary_df[\"Model\"], y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black')\nax1.set_title(\"Observed Success Rate (95% Wilson CI)\", fontsize=14)\nax1.tick_params(axis='x', rotation=30, labelsize=10)\n\n# Plot 2\nwidth = 0.2\nx = np.arange(len(MODELS))\ncolors = ['#ff9999','#66b3ff','#99ff99']\nfor i, eps in enumerate(EPSILONS):\n req = summary_df[f\"Shots for ±{eps:.1%}\"]\n ax2.bar(x + i * width, req, width, label=f\"±{eps:.1%}\", color=colors[i], edgecolor='black')\n\nax2.set_xticks(x + width, summary_df[\"Model\"])\nax2.set_ylabel(\"Required Shots (Log Scale)\")\nax2.set_yscale('log')\nax2.set_title(\"Shots Needed for ±ε Tolerance\", fontsize=14)\nax2.tick_params(axis='x', rotation=30, labelsize=10)\nax2.legend(title=\"Tolerance\")\n\nplt.tight_layout()\nplt.show()\n\"\"\"\n\nnb = nbf.v4.new_notebook()\nnb[\"cells\"] = [\n nbf.v4.new_markdown_cell(md_intro),\n nbf.v4.new_code_cell(code_config),\n nbf.v4.new_code_cell(code_lib_and_run),\n]\n\nnb_path = \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\nwith open(nb_path, \"w\", encoding=\"utf-8\") as f:\n nbf.write(nb, f)\n\n# ---------- Final confirmation ----------\nprint(\"\\n\" + \"=\"*50)\nprint(\"SCRIPT COMPLETE\")\nprint(\"=\"*50)\nprint(f\"Google Colab notebook saved to: {nb_path}\")\nprint(f\"Summary data saved to: {summary_csv_path}\")\nprint(f\"Plots saved to: {fig1_path}, {fig2_path}\")\n",[51,67248,67249,67254,67259,67264,67269,67274,67279,67284,67289,67294,67299,67303,67310,67316,67328,67339,67343,67353,67363,67373,67385,67397,67401,67414,67418,67423,67427,67474,67478,67483,67488,67492,67505,67519,67533,67558,67594,67660,67693,67697,67701,67748,67752,67757,67762,67766,67790,67811,67824,67830,67860,67882,67886,67890,67895,67899,67915,67926,67934,67944,67951,67955,67959,67971,67978,67985,67992,67997,68001,68007,68014,68021,68026,68030,68036,68043,68050,68055,68059,68065,68072,68079,68084,68088,68094,68101,68108,68113,68117,68121,68125,68130,68134,68146,68158,68170,68196,68203,68215,68225,68229,68236,68246,68250,68254,68258,68283,68287,68291,68296,68300,68332,68336,68341,68345,68378,68382,68386,68417,68421,68426,68430,68441,68458,68475,68508,68520,68524,68533,68541,68549,68557,68565,68573,68581,68589,68603,68616,68629,68633,68645,68682,68689,68693,68697,68736,68740,68745,68749,68766,68770,68774,68779,68783,68797,68806,68810,68821,68835,68857,68867,68892,68896,68910,68914,68919,68941,68961,68983,68997,69015,69032,69036,69040,69045,69049,69054,69070,69107,69138,69142,69152,69170,69174,69179,69204,69228,69232,69243,69247,69252,69256,69261,69276,69298,69317,69337,69341,69346,69351,69356,69383,69409,69433,69438,69442,69471,69525,69562,69575,69598,69606,69616,69644,69652,69656,69661,69683,69693,69713,69737,69741,69763,69788,69845,69849,69888,69901,69922,69936,69953,69961,69971,69996,70004,70008,70019,70023,70028,70032,70049,70062,70082,70102,70125,70147,70166,70183,70188,70192,70196,70201,70205,70218,70223,70227,70258,70262,70343,70347,70351,70362,70367,70372,70377,70382,70386,70394,70401,70412,70419,70426,70430,70441,70446,70451,70456,70461,70465,70479,70484,70489,70494,70499,70504,70508,70512,70524,70528,70532,70543,70548,70553,70558,70563,70567,70572,70577,70581,70601,70613,70623,70633,70653,70713,70744,70748,70763,70779,70830,70834,70849,70860,70871,70883,70888,70903,70918,70923,70927,70932,70950,70955,70959,70964,70972,70984,70989,71013,71017,71032,71059,71063,71068,71079,71106,71110,71115,71128,71136,71147,71155,71159,71177,71199,71211,71215,71220,71225,71241,71249,71263,71271,71297,71301,71318,71339,71350,71362,71373,71385,71389,71396,71403,71407,71411,71426,71440,71451,71461,71470,71474,71478,71488,71515,71526,71530,71535,71558,71569,71584,71606,71628],{"__ignoreMap":49},[54,67250,67251],{"class":56,"line":57},[54,67252,67253],{"class":111},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n",[54,67255,67256],{"class":56,"line":74},[54,67257,67258],{"class":111},"# generates tables and figures, so you can iterate further.\n",[54,67260,67261],{"class":56,"line":88},[54,67262,67263],{"class":111},"#\n",[54,67265,67266],{"class":56,"line":101},[54,67267,67268],{"class":111},"# Contents created in the current directory:\n",[54,67270,67271],{"class":56,"line":108},[54,67272,67273],{"class":111},"# - Sampling_Budget_Qubit_Bitstrings.ipynb (Colab notebook)\n",[54,67275,67276],{"class":56,"line":115},[54,67277,67278],{"class":111},"# - summary.csv (summary table)\n",[54,67280,67281],{"class":56,"line":139},[54,67282,67283],{"class":111},"# - bitstrings_\u003CMODEL>.csv (raw bit-strings per model)\n",[54,67285,67286],{"class":56,"line":159},[54,67287,67288],{"class":111},"# - fig_ones_fraction.png (bar chart with Wilson CI)\n",[54,67290,67291],{"class":56,"line":183},[54,67292,67293],{"class":111},"# - fig_required_shots.png (required shots for ±ε)\n",[54,67295,67296],{"class":56,"line":210},[54,67297,67298],{"class":111},"# - simulation_config.json (run parameters)\n",[54,67300,67301],{"class":56,"line":227},[54,67302,105],{"emptyLinePlaceholder":104},[54,67304,67305,67307],{"class":56,"line":232},[54,67306,61],{"class":60},[54,67308,67309],{"class":64}," json\n",[54,67311,67312,67314],{"class":56,"line":246},[54,67313,61],{"class":60},[54,67315,56692],{"class":64},[54,67317,67318,67320,67323,67325],{"class":56,"line":259},[54,67319,77],{"class":60},[54,67321,67322],{"class":64}," dataclasses ",[54,67324,61],{"class":60},[54,67326,67327],{"class":64}," dataclass\n",[54,67329,67330,67332,67334,67336],{"class":56,"line":264},[54,67331,77],{"class":60},[54,67333,722],{"class":64},[54,67335,61],{"class":60},[54,67337,67338],{"class":64}," List, Dict, Tuple\n",[54,67340,67341],{"class":56,"line":270},[54,67342,105],{"emptyLinePlaceholder":104},[54,67344,67345,67347,67349,67351],{"class":56,"line":285},[54,67346,61],{"class":60},[54,67348,65],{"class":64},[54,67350,68],{"class":60},[54,67352,71],{"class":64},[54,67354,67355,67357,67359,67361],{"class":56,"line":297},[54,67356,61],{"class":60},[54,67358,56731],{"class":64},[54,67360,68],{"class":60},[54,67362,56736],{"class":64},[54,67364,67365,67367,67369,67371],{"class":56,"line":310},[54,67366,61],{"class":60},[54,67368,56720],{"class":64},[54,67370,68],{"class":60},[54,67372,715],{"class":64},[54,67374,67375,67377,67380,67382],{"class":56,"line":324},[54,67376,77],{"class":60},[54,67378,67379],{"class":64}," IPython.display ",[54,67381,61],{"class":60},[54,67383,67384],{"class":64}," display, Markdown\n",[54,67386,67387,67389,67392,67394],{"class":56,"line":333},[54,67388,61],{"class":60},[54,67390,67391],{"class":64}," nbformat ",[54,67393,68],{"class":60},[54,67395,67396],{"class":64}," nbf\n",[54,67398,67399],{"class":56,"line":345},[54,67400,105],{"emptyLinePlaceholder":104},[54,67402,67403,67405,67407,67409,67411],{"class":56,"line":358},[54,67404,56799],{"class":64},[54,67406,56802],{"class":279},[54,67408,7762],{"class":64},[54,67410,122],{"class":121},[54,67412,67413],{"class":132}," 200\n",[54,67415,67416],{"class":56,"line":908},[54,67417,105],{"emptyLinePlaceholder":104},[54,67419,67420],{"class":56,"line":913},[54,67421,67422],{"class":111},"# ---------- Helper math ----------\n",[54,67424,67425],{"class":56,"line":918},[54,67426,105],{"emptyLinePlaceholder":104},[54,67428,67429,67431,67434,67436,67438,67440,67442,67444,67446,67448,67450,67452,67454,67456,67458,67460,67463,67466,67468,67470,67472],{"class":56,"line":923},[54,67430,926],{"class":60},[54,67432,67433],{"class":125}," wilson_ci",[54,67435,129],{"class":64},[54,67437,3182],{"class":938},[54,67439,959],{"class":64},[54,67441,976],{"class":121},[54,67443,171],{"class":64},[54,67445,3976],{"class":938},[54,67447,959],{"class":64},[54,67449,976],{"class":121},[54,67451,171],{"class":64},[54,67453,215],{"class":938},[54,67455,959],{"class":64},[54,67457,962],{"class":121},[54,67459,979],{"class":121},[54,67461,67462],{"class":132}," 1.96",[54,67464,67465],{"class":64},") -> Tuple[",[54,67467,962],{"class":121},[54,67469,171],{"class":64},[54,67471,962],{"class":121},[54,67473,59180],{"class":64},[54,67475,67476],{"class":56,"line":935},[54,67477,1024],{"class":279},[54,67479,67480],{"class":56,"line":945},[54,67481,67482],{"class":279}," Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n",[54,67484,67485],{"class":56,"line":953},[54,67486,67487],{"class":279}," Returns (low, high). Works for edge cases k=0 or k=n.\n",[54,67489,67490],{"class":56,"line":968},[54,67491,1024],{"class":279},[54,67493,67494,67496,67499,67501,67503],{"class":56,"line":987},[54,67495,1273],{"class":60},[54,67497,67498],{"class":64}," n ",[54,67500,1279],{"class":121},[54,67502,20814],{"class":132},[54,67504,1284],{"class":64},[54,67506,67507,67509,67511,67513,67515,67517],{"class":56,"line":1005},[54,67508,64430],{"class":60},[54,67510,58201],{"class":64},[54,67512,56271],{"class":132},[54,67514,171],{"class":64},[54,67516,57484],{"class":132},[54,67518,136],{"class":64},[54,67520,67521,67524,67526,67528,67530],{"class":56,"line":1021},[54,67522,67523],{"class":64}," p_hat ",[54,67525,122],{"class":121},[54,67527,1276],{"class":64},[54,67529,194],{"class":121},[54,67531,67532],{"class":64}," n\n",[54,67534,67535,67538,67540,67543,67545,67548,67550,67552,67554,67556],{"class":56,"line":1027},[54,67536,67537],{"class":64}," denom ",[54,67539,122],{"class":121},[54,67541,67542],{"class":132}," 1.0",[54,67544,7869],{"class":121},[54,67546,67547],{"class":64}," (z ",[54,67549,1933],{"class":121},[54,67551,1872],{"class":132},[54,67553,204],{"class":64},[54,67555,194],{"class":121},[54,67557,67532],{"class":64},[54,67559,67560,67563,67565,67568,67570,67572,67574,67576,67578,67580,67582,67584,67586,67589,67591],{"class":56,"line":1033},[54,67561,67562],{"class":64}," center ",[54,67564,122],{"class":121},[54,67566,67567],{"class":64}," (p_hat ",[54,67569,1663],{"class":121},[54,67571,67547],{"class":64},[54,67573,1933],{"class":121},[54,67575,1872],{"class":132},[54,67577,204],{"class":64},[54,67579,194],{"class":121},[54,67581,58201],{"class":64},[54,67583,133],{"class":132},[54,67585,7785],{"class":121},[54,67587,67588],{"class":64}," n)) ",[54,67590,194],{"class":121},[54,67592,67593],{"class":64}," denom\n",[54,67595,67596,67599,67601,67603,67605,67608,67610,67613,67615,67617,67619,67621,67624,67626,67629,67631,67633,67635,67637,67639,67641,67643,67645,67647,67649,67651,67653,67656,67658],{"class":56,"line":1038},[54,67597,67598],{"class":64}," span ",[54,67600,122],{"class":121},[54,67602,67547],{"class":64},[54,67604,1947],{"class":121},[54,67606,67607],{"class":64}," math.",[54,67609,9375],{"class":125},[54,67611,67612],{"class":64},"((p_hat ",[54,67614,1947],{"class":121},[54,67616,58201],{"class":64},[54,67618,174],{"class":132},[54,67620,65471],{"class":121},[54,67622,67623],{"class":64}," p_hat) ",[54,67625,194],{"class":121},[54,67627,67628],{"class":64}," n) ",[54,67630,1663],{"class":121},[54,67632,67547],{"class":64},[54,67634,1933],{"class":121},[54,67636,1872],{"class":132},[54,67638,204],{"class":64},[54,67640,194],{"class":121},[54,67642,58201],{"class":64},[54,67644,11708],{"class":132},[54,67646,7785],{"class":121},[54,67648,67498],{"class":64},[54,67650,1933],{"class":121},[54,67652,1872],{"class":132},[54,67654,67655],{"class":64},"))) ",[54,67657,194],{"class":121},[54,67659,67593],{"class":64},[54,67661,67662,67664,67666,67668,67670,67672,67675,67677,67680,67682,67684,67686,67688,67690],{"class":56,"line":1044},[54,67663,2060],{"class":60},[54,67665,58201],{"class":64},[54,67667,36269],{"class":121},[54,67669,129],{"class":64},[54,67671,56271],{"class":132},[54,67673,67674],{"class":64},", center ",[54,67676,1923],{"class":121},[54,67678,67679],{"class":64}," span), ",[54,67681,24786],{"class":121},[54,67683,129],{"class":64},[54,67685,57484],{"class":132},[54,67687,67674],{"class":64},[54,67689,1663],{"class":121},[54,67691,67692],{"class":64}," span))\n",[54,67694,67695],{"class":56,"line":1050},[54,67696,105],{"emptyLinePlaceholder":104},[54,67698,67699],{"class":56,"line":1056},[54,67700,105],{"emptyLinePlaceholder":104},[54,67702,67703,67705,67708,67710,67712,67714,67716,67718,67721,67723,67725,67727,67730,67732,67734,67736,67738,67740,67742,67744,67746],{"class":56,"line":1062},[54,67704,926],{"class":60},[54,67706,67707],{"class":125}," shots_for_epsilon",[54,67709,129],{"class":64},[54,67711,13],{"class":938},[54,67713,959],{"class":64},[54,67715,962],{"class":121},[54,67717,171],{"class":64},[54,67719,67720],{"class":938},"epsilon",[54,67722,959],{"class":64},[54,67724,962],{"class":121},[54,67726,979],{"class":121},[54,67728,67729],{"class":132}," 0.01",[54,67731,171],{"class":64},[54,67733,215],{"class":938},[54,67735,959],{"class":64},[54,67737,962],{"class":121},[54,67739,979],{"class":121},[54,67741,67462],{"class":132},[54,67743,58457],{"class":64},[54,67745,976],{"class":121},[54,67747,1284],{"class":64},[54,67749,67750],{"class":56,"line":1068},[54,67751,1024],{"class":279},[54,67753,67754],{"class":56,"line":1074},[54,67755,67756],{"class":279}," Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n",[54,67758,67759],{"class":56,"line":1080},[54,67760,67761],{"class":279}," at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n",[54,67763,67764],{"class":56,"line":1086},[54,67765,1024],{"class":279},[54,67767,67768,67771,67773,67775,67777,67779,67782,67784,67786,67788],{"class":56,"line":1092},[54,67769,67770],{"class":64}," p ",[54,67772,122],{"class":121},[54,67774,2032],{"class":121},[54,67776,129],{"class":64},[54,67778,36269],{"class":121},[54,67780,67781],{"class":64},"(p, ",[54,67783,56271],{"class":132},[54,67785,1769],{"class":64},[54,67787,57484],{"class":132},[54,67789,136],{"class":64},[54,67791,67792,67795,67797,67800,67802,67804,67806,67808],{"class":56,"line":1098},[54,67793,67794],{"class":64}," var ",[54,67796,122],{"class":121},[54,67798,67799],{"class":64}," p ",[54,67801,1947],{"class":121},[54,67803,58201],{"class":64},[54,67805,57484],{"class":132},[54,67807,65471],{"class":121},[54,67809,67810],{"class":64}," p)\n",[54,67812,67813,67815,67818,67820,67822],{"class":56,"line":1104},[54,67814,1273],{"class":60},[54,67816,67817],{"class":64}," var ",[54,67819,1279],{"class":121},[54,67821,56924],{"class":132},[54,67823,1284],{"class":64},[54,67825,67826,67828],{"class":56,"line":1109},[54,67827,64430],{"class":60},[54,67829,1640],{"class":132},[54,67831,67832,67835,67837,67839,67841,67843,67845,67847,67849,67851,67854,67856,67858],{"class":56,"line":1115},[54,67833,67834],{"class":64}," n ",[54,67836,122],{"class":121},[54,67838,67547],{"class":64},[54,67840,1933],{"class":121},[54,67842,1872],{"class":132},[54,67844,204],{"class":64},[54,67846,1947],{"class":121},[54,67848,67817],{"class":64},[54,67850,194],{"class":121},[54,67852,67853],{"class":64}," (epsilon ",[54,67855,1933],{"class":121},[54,67857,1872],{"class":132},[54,67859,136],{"class":64},[54,67861,67862,67864,67866,67868,67870,67872,67874,67877,67879],{"class":56,"line":1121},[54,67863,2060],{"class":60},[54,67865,64798],{"class":121},[54,67867,129],{"class":64},[54,67869,174],{"class":132},[54,67871,171],{"class":64},[54,67873,976],{"class":121},[54,67875,67876],{"class":64},"(math.",[54,67878,64785],{"class":125},[54,67880,67881],{"class":64},"(n)))\n",[54,67883,67884],{"class":56,"line":1126},[54,67885,105],{"emptyLinePlaceholder":104},[54,67887,67888],{"class":56,"line":1131},[54,67889,105],{"emptyLinePlaceholder":104},[54,67891,67892],{"class":56,"line":1136},[54,67893,67894],{"class":111},"# ---------- Domain model ----------\n",[54,67896,67897],{"class":56,"line":1153},[54,67898,105],{"emptyLinePlaceholder":104},[54,67900,67901,67904,67906,67909,67911,67913],{"class":56,"line":1164},[54,67902,67903],{"class":125},"@dataclass",[54,67905,129],{"class":64},[54,67907,67908],{"class":1711},"frozen",[54,67910,122],{"class":121},[54,67912,13434],{"class":132},[54,67914,136],{"class":64},[54,67916,67917,67920,67924],{"class":56,"line":1180},[54,67918,67919],{"class":60},"class",[54,67921,67923],{"class":67922},"sU0A5"," QubitModel",[54,67925,1284],{"class":64},[54,67927,67928,67931],{"class":56,"line":1213},[54,67929,67930],{"class":64}," name: ",[54,67932,67933],{"class":121},"str\n",[54,67935,67936,67939,67941],{"class":56,"line":1219},[54,67937,67938],{"class":64}," satisfaction_prob: ",[54,67940,962],{"class":121},[54,67942,67943],{"class":111}," # Probability a single shot meets the analytical criterion (bit = 1)\n",[54,67945,67946,67949],{"class":56,"line":1247},[54,67947,67948],{"class":64}," description: ",[54,67950,67933],{"class":121},[54,67952,67953],{"class":56,"line":1270},[54,67954,105],{"emptyLinePlaceholder":104},[54,67956,67957],{"class":56,"line":1287},[54,67958,105],{"emptyLinePlaceholder":104},[54,67960,67961,67964,67967,67969],{"class":56,"line":1296},[54,67962,67963],{"class":132},"DEFAULT_MODELS",[54,67965,67966],{"class":64},": List[QubitModel] ",[54,67968,122],{"class":121},[54,67970,1733],{"class":64},[54,67972,67973,67976],{"class":56,"line":1325},[54,67974,67975],{"class":125}," QubitModel",[54,67977,932],{"class":64},[54,67979,67980,67983],{"class":56,"line":1333},[54,67981,67982],{"class":279}," \"Toy working qubit\"",[54,67984,965],{"class":64},[54,67986,67987,67990],{"class":56,"line":1353},[54,67988,67989],{"class":132}," 0.50",[54,67991,965],{"class":64},[54,67993,67994],{"class":56,"line":1361},[54,67995,67996],{"class":279}," \"Simple demonstrator; about half shots meet the criterion.\"\n",[54,67998,67999],{"class":56,"line":1368},[54,68000,22188],{"class":64},[54,68002,68003,68005],{"class":56,"line":1388},[54,68004,67975],{"class":125},[54,68006,932],{"class":64},[54,68008,68009,68012],{"class":56,"line":1395},[54,68010,68011],{"class":279}," \"NISQ qubit\"",[54,68013,965],{"class":64},[54,68015,68016,68019],{"class":56,"line":1411},[54,68017,68018],{"class":132}," 0.60",[54,68020,965],{"class":64},[54,68022,68023],{"class":56,"line":1416},[54,68024,68025],{"class":279}," \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n",[54,68027,68028],{"class":56,"line":1421},[54,68029,22188],{"class":64},[54,68031,68032,68034],{"class":56,"line":1442},[54,68033,67975],{"class":125},[54,68035,932],{"class":64},[54,68037,68038,68041],{"class":56,"line":1447},[54,68039,68040],{"class":279}," \"Fault-tolerant (logical)\"",[54,68042,965],{"class":64},[54,68044,68045,68048],{"class":56,"line":1453},[54,68046,68047],{"class":132}," 0.90",[54,68049,965],{"class":64},[54,68051,68052],{"class":56,"line":1458},[54,68053,68054],{"class":279}," \"Error-corrected logical qubit; high single-shot success probability.\"\n",[54,68056,68057],{"class":56,"line":1469},[54,68058,22188],{"class":64},[54,68060,68061,68063],{"class":56,"line":1489},[54,68062,67975],{"class":125},[54,68064,932],{"class":64},[54,68066,68067,68070],{"class":56,"line":1529},[54,68068,68069],{"class":279}," \"Topologically protected\"",[54,68071,965],{"class":64},[54,68073,68074,68077],{"class":56,"line":1534},[54,68075,68076],{"class":132}," 0.98",[54,68078,965],{"class":64},[54,68080,68081],{"class":56,"line":1544},[54,68082,68083],{"class":279}," \"Intrinsic protection; very high single-shot success probability.\"\n",[54,68085,68086],{"class":56,"line":1559},[54,68087,22188],{"class":64},[54,68089,68090,68092],{"class":56,"line":1592},[54,68091,67975],{"class":125},[54,68093,932],{"class":64},[54,68095,68096,68099],{"class":56,"line":1597},[54,68097,68098],{"class":279}," \"Ideal qubit\"",[54,68100,965],{"class":64},[54,68102,68103,68106],{"class":56,"line":1608},[54,68104,68105],{"class":132}," 1.00",[54,68107,965],{"class":64},[54,68109,68110],{"class":56,"line":1613},[54,68111,68112],{"class":279}," \"Theoretical perfect qubit; always successful in a single shot.\"\n",[54,68114,68115],{"class":56,"line":1627},[54,68116,22188],{"class":64},[54,68118,68119],{"class":56,"line":1632},[54,68120,1350],{"class":64},[54,68122,68123],{"class":56,"line":1643},[54,68124,105],{"emptyLinePlaceholder":104},[54,68126,68127],{"class":56,"line":1671},[54,68128,68129],{"class":111},"# ---------- Simulation configuration ----------\n",[54,68131,68132],{"class":56,"line":1690},[54,68133,105],{"emptyLinePlaceholder":104},[54,68135,68136,68139,68142,68144],{"class":56,"line":1695},[54,68137,68138],{"class":132},"CONFIG",[54,68140,68141],{"class":64},": Dict ",[54,68143,122],{"class":121},[54,68145,23279],{"class":64},[54,68147,68148,68151,68153,68156],{"class":56,"line":1720},[54,68149,68150],{"class":279}," \"random_seed\"",[54,68152,959],{"class":64},[54,68154,68155],{"class":132},"123456",[54,68157,965],{"class":64},[54,68159,68160,68163,68165,68168],{"class":56,"line":1725},[54,68161,68162],{"class":279}," \"shots_per_model\"",[54,68164,959],{"class":64},[54,68166,68167],{"class":132},"10000",[54,68169,965],{"class":64},[54,68171,68172,68175,68177,68180,68182,68185,68187,68190,68193],{"class":56,"line":1736},[54,68173,68174],{"class":279}," \"epsilons\"",[54,68176,23413],{"class":64},[54,68178,68179],{"class":132},"0.02",[54,68181,171],{"class":64},[54,68183,68184],{"class":132},"0.01",[54,68186,171],{"class":64},[54,68188,68189],{"class":132},"0.005",[54,68191,68192],{"class":64},"], ",[54,68194,68195],{"class":111},"# ±2%, ±1%, ±0.5% targets\n",[54,68197,68198,68201],{"class":56,"line":1780},[54,68199,68200],{"class":279}," \"cost_model\"",[54,68202,23298],{"class":64},[54,68204,68205,68208,68210,68213],{"class":56,"line":1789},[54,68206,68207],{"class":279}," \"default_cost_per_shot_usd\"",[54,68209,959],{"class":64},[54,68211,68212],{"class":132},"0.00005",[54,68214,965],{"class":64},[54,68216,68217,68220,68222],{"class":56,"line":1794},[54,68218,68219],{"class":279}," \"fixed_overhead_usd\"",[54,68221,959],{"class":64},[54,68223,68224],{"class":132},"0.10\n",[54,68226,68227],{"class":56,"line":1805},[54,68228,23319],{"class":64},[54,68230,68231,68234],{"class":56,"line":1820},[54,68232,68233],{"class":279}," \"timing\"",[54,68235,23298],{"class":64},[54,68237,68238,68241,68243],{"class":56,"line":1835},[54,68239,68240],{"class":279}," \"time_per_shot_seconds\"",[54,68242,959],{"class":64},[54,68244,68245],{"class":132},"2e-6\n",[54,68247,68248],{"class":56,"line":1846},[54,68249,59040],{"class":64},[54,68251,68252],{"class":56,"line":1851},[54,68253,23431],{"class":64},[54,68255,68256],{"class":56,"line":1877},[54,68257,105],{"emptyLinePlaceholder":104},[54,68259,68260,68263,68265,68268,68271,68273,68275,68277,68280],{"class":56,"line":1893},[54,68261,68262],{"class":64},"rng ",[54,68264,122],{"class":121},[54,68266,68267],{"class":64}," np.random.",[54,68269,68270],{"class":125},"default_rng",[54,68272,129],{"class":64},[54,68274,68138],{"class":132},[54,68276,11804],{"class":64},[54,68278,68279],{"class":279},"\"random_seed\"",[54,68281,68282],{"class":64},"])\n",[54,68284,68285],{"class":56,"line":1912},[54,68286,105],{"emptyLinePlaceholder":104},[54,68288,68289],{"class":56,"line":1952},[54,68290,105],{"emptyLinePlaceholder":104},[54,68292,68293],{"class":56,"line":1975},[54,68294,68295],{"class":111},"# ---------- Core simulation ----------\n",[54,68297,68298],{"class":56,"line":1981},[54,68299,105],{"emptyLinePlaceholder":104},[54,68301,68302,68304,68307,68309,68311,68313,68315,68317,68320,68322,68324,68326,68329],{"class":56,"line":1989},[54,68303,926],{"class":60},[54,68305,68306],{"class":125}," simulate_bitstring",[54,68308,129],{"class":64},[54,68310,13],{"class":938},[54,68312,959],{"class":64},[54,68314,962],{"class":121},[54,68316,171],{"class":64},[54,68318,68319],{"class":938},"shots",[54,68321,959],{"class":64},[54,68323,976],{"class":121},[54,68325,171],{"class":64},[54,68327,68328],{"class":938},"rng_",[54,68330,68331],{"class":64},": np.random.Generator) -> np.ndarray:\n",[54,68333,68334],{"class":56,"line":1998},[54,68335,1024],{"class":279},[54,68337,68338],{"class":56,"line":2003},[54,68339,68340],{"class":279}," Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n",[54,68342,68343],{"class":56,"line":2019},[54,68344,1024],{"class":279},[54,68346,68347,68349,68352,68355,68357,68359,68362,68365,68367,68370,68372,68374,68376],{"class":56,"line":2024},[54,68348,2060],{"class":60},[54,68350,68351],{"class":64}," rng_.",[54,68353,68354],{"class":125},"binomial",[54,68356,129],{"class":64},[54,68358,174],{"class":132},[54,68360,68361],{"class":64},", p, ",[54,68363,68364],{"class":1711},"size",[54,68366,122],{"class":121},[54,68368,68369],{"class":64},"shots).",[54,68371,66316],{"class":125},[54,68373,129],{"class":64},[54,68375,976],{"class":121},[54,68377,136],{"class":64},[54,68379,68380],{"class":56,"line":2044},[54,68381,105],{"emptyLinePlaceholder":104},[54,68383,68384],{"class":56,"line":2057},[54,68385,105],{"emptyLinePlaceholder":104},[54,68387,68388,68390,68393,68395,68398,68401,68404,68406,68409,68412,68414],{"class":56,"line":58909},[54,68389,926],{"class":60},[54,68391,68392],{"class":125}," summarize_model",[54,68394,129],{"class":64},[54,68396,68397],{"class":938},"model",[54,68399,68400],{"class":64},": QubitModel, ",[54,68402,68403],{"class":938},"bits",[54,68405,63664],{"class":64},[54,68407,68408],{"class":938},"epsilons",[54,68410,68411],{"class":64},": List[",[54,68413,962],{"class":121},[54,68415,68416],{"class":64},"]) -> Dict:\n",[54,68418,68419],{"class":56,"line":58914},[54,68420,1024],{"class":279},[54,68422,68423],{"class":56,"line":58921},[54,68424,68425],{"class":279}," Calculate summary statistics for a given model and its simulated bitstring.\n",[54,68427,68428],{"class":56,"line":58935},[54,68429,1024],{"class":279},[54,68431,68432,68434,68436,68438],{"class":56,"line":58940},[54,68433,67834],{"class":64},[54,68435,122],{"class":121},[54,68437,1651],{"class":121},[54,68439,68440],{"class":64},"(bits.size)\n",[54,68442,68443,68446,68448,68450,68453,68456],{"class":56,"line":58953},[54,68444,68445],{"class":64}," ones ",[54,68447,122],{"class":121},[54,68449,1651],{"class":121},[54,68451,68452],{"class":64},"(bits.",[54,68454,68455],{"class":125},"sum",[54,68457,21643],{"class":64},[54,68459,68460,68463,68465,68467,68470,68472],{"class":56,"line":58969},[54,68461,68462],{"class":64}," zeros ",[54,68464,122],{"class":121},[54,68466,1651],{"class":121},[54,68468,68469],{"class":64},"(n ",[54,68471,1923],{"class":121},[54,68473,68474],{"class":64}," ones)\n",[54,68476,68477,68480,68482,68485,68487,68489,68491,68493,68495,68497,68499,68501,68503,68506],{"class":56,"line":58989},[54,68478,68479],{"class":64}," frac ",[54,68481,122],{"class":121},[54,68483,68484],{"class":64}," ones ",[54,68486,194],{"class":121},[54,68488,67498],{"class":64},[54,68490,1201],{"class":60},[54,68492,67498],{"class":64},[54,68494,1207],{"class":121},[54,68496,20814],{"class":132},[54,68498,8036],{"class":60},[54,68500,59177],{"class":121},[54,68502,129],{"class":64},[54,68504,68505],{"class":279},"\"nan\"",[54,68507,136],{"class":64},[54,68509,68510,68513,68515,68517],{"class":56,"line":58997},[54,68511,68512],{"class":64}," lo, hi ",[54,68514,122],{"class":121},[54,68516,67433],{"class":125},[54,68518,68519],{"class":64},"(ones, n)\n",[54,68521,68522],{"class":56,"line":59003},[54,68523,105],{"emptyLinePlaceholder":104},[54,68525,68526,68529,68531],{"class":56,"line":59008},[54,68527,68528],{"class":64}," row ",[54,68530,122],{"class":121},[54,68532,23279],{"class":64},[54,68534,68535,68538],{"class":56,"line":59019},[54,68536,68537],{"class":279}," \"Model\"",[54,68539,68540],{"class":64},": model.name,\n",[54,68542,68543,68546],{"class":56,"line":59025},[54,68544,68545],{"class":279}," \"Description\"",[54,68547,68548],{"class":64},": model.description,\n",[54,68550,68551,68554],{"class":56,"line":59037},[54,68552,68553],{"class":279}," \"Configured p (target)\"",[54,68555,68556],{"class":64},": model.satisfaction_prob,\n",[54,68558,68559,68562],{"class":56,"line":59043},[54,68560,68561],{"class":279}," \"Shots\"",[54,68563,68564],{"class":64},": n,\n",[54,68566,68567,68570],{"class":56,"line":59048},[54,68568,68569],{"class":279}," \"Ones (successes)\"",[54,68571,68572],{"class":64},": ones,\n",[54,68574,68575,68578],{"class":56,"line":59053},[54,68576,68577],{"class":279}," \"Zeros (failures)\"",[54,68579,68580],{"class":64},": zeros,\n",[54,68582,68583,68586],{"class":56,"line":59058},[54,68584,68585],{"class":279}," \"Ones fraction (observed)\"",[54,68587,68588],{"class":64},": frac,\n",[54,68590,68591,68594,68597,68600],{"class":56,"line":59081},[54,68592,68593],{"class":279}," \"Wilson CI 95",[54,68595,68596],{"class":132},"% lo",[54,68598,68599],{"class":279},"w\"",[54,68601,68602],{"class":64},": lo,\n",[54,68604,68605,68607,68610,68613],{"class":56,"line":59087},[54,68606,68593],{"class":279},[54,68608,68609],{"class":132},"% hi",[54,68611,68612],{"class":279},"gh\"",[54,68614,68615],{"class":64},": hi,\n",[54,68617,68618,68621,68624,68626],{"class":56,"line":59107},[54,68619,68620],{"class":279}," \"CI width (95%)\"",[54,68622,68623],{"class":64},": hi ",[54,68625,1923],{"class":121},[54,68627,68628],{"class":64}," lo,\n",[54,68630,68631],{"class":56,"line":59112},[54,68632,59040],{"class":64},[54,68634,68635,68637,68640,68642],{"class":56,"line":59117},[54,68636,1472],{"class":60},[54,68638,68639],{"class":64}," eps ",[54,68641,1192],{"class":60},[54,68643,68644],{"class":64}," epsilons:\n",[54,68646,68647,68650,68652,68655,68657,68659,68662,68664,68666,68668,68670,68673,68675,68677,68679],{"class":56,"line":59127},[54,68648,68649],{"class":64}," row[",[54,68651,21834],{"class":60},[54,68653,68654],{"class":279},"\"Required shots for ±",[54,68656,2366],{"class":132},[54,68658,976],{"class":121},[54,68660,68661],{"class":64},"(eps",[54,68663,1947],{"class":121},[54,68665,5238],{"class":132},[54,68667,3205],{"class":64},[54,68669,2428],{"class":132},[54,68671,68672],{"class":279},"%\"",[54,68674,7762],{"class":64},[54,68676,122],{"class":121},[54,68678,67707],{"class":125},[54,68680,68681],{"class":64},"(model.satisfaction_prob, eps)\n",[54,68683,68684,68686],{"class":56,"line":59139},[54,68685,2060],{"class":60},[54,68687,68688],{"class":64}," row\n",[54,68690,68691],{"class":56,"line":59151},[54,68692,105],{"emptyLinePlaceholder":104},[54,68694,68695],{"class":56,"line":59163},[54,68696,105],{"emptyLinePlaceholder":104},[54,68698,68699,68701,68704,68706,68708,68710,68712,68714,68717,68719,68721,68723,68726,68728,68730,68732,68734],{"class":56,"line":59183},[54,68700,926],{"class":60},[54,68702,68703],{"class":125}," apply_costs",[54,68705,129],{"class":64},[54,68707,68319],{"class":938},[54,68709,959],{"class":64},[54,68711,976],{"class":121},[54,68713,171],{"class":64},[54,68715,68716],{"class":938},"cost_per_shot",[54,68718,959],{"class":64},[54,68720,962],{"class":121},[54,68722,171],{"class":64},[54,68724,68725],{"class":938},"fixed_overhead",[54,68727,959],{"class":64},[54,68729,962],{"class":121},[54,68731,58457],{"class":64},[54,68733,962],{"class":121},[54,68735,1284],{"class":64},[54,68737,68738],{"class":56,"line":59189},[54,68739,1024],{"class":279},[54,68741,68742],{"class":56,"line":59213},[54,68743,68744],{"class":279}," Calculate illustrative cost based on shots and cost model.\n",[54,68746,68747],{"class":56,"line":59236},[54,68748,1024],{"class":279},[54,68750,68751,68753,68756,68758,68761,68763],{"class":56,"line":59252},[54,68752,2060],{"class":60},[54,68754,68755],{"class":64}," fixed_overhead ",[54,68757,1663],{"class":121},[54,68759,68760],{"class":64}," shots ",[54,68762,1947],{"class":121},[54,68764,68765],{"class":64}," cost_per_shot\n",[54,68767,68768],{"class":56,"line":59257},[54,68769,105],{"emptyLinePlaceholder":104},[54,68771,68772],{"class":56,"line":59284},[54,68773,105],{"emptyLinePlaceholder":104},[54,68775,68776],{"class":56,"line":59289},[54,68777,68778],{"class":111},"# ---------- Run the simulation ----------\n",[54,68780,68781],{"class":56,"line":59313},[54,68782,105],{"emptyLinePlaceholder":104},[54,68784,68785,68788,68790,68793,68795],{"class":56,"line":59335},[54,68786,68787],{"class":64},"all_bitstrings: Dict[",[54,68789,58454],{"class":121},[54,68791,68792],{"class":64},", np.ndarray] ",[54,68794,122],{"class":121},[54,68796,23264],{"class":64},[54,68798,68799,68802,68804],{"class":56,"line":59356},[54,68800,68801],{"class":64},"summary_rows: List[Dict] ",[54,68803,122],{"class":121},[54,68805,1466],{"class":64},[54,68807,68808],{"class":56,"line":59361},[54,68809,105],{"emptyLinePlaceholder":104},[54,68811,68812,68814,68816,68819],{"class":56,"line":59375},[54,68813,8081],{"class":121},[54,68815,129],{"class":64},[54,68817,68818],{"class":279},"\"Running simulation...\"",[54,68820,136],{"class":64},[54,68822,68823,68825,68828,68830,68833],{"class":56,"line":59390},[54,68824,1186],{"class":60},[54,68826,68827],{"class":64}," m ",[54,68829,1192],{"class":60},[54,68831,68832],{"class":132}," DEFAULT_MODELS",[54,68834,1284],{"class":64},[54,68836,68837,68840,68842,68844,68847,68849,68851,68854],{"class":56,"line":59408},[54,68838,68839],{"class":64}," bits ",[54,68841,122],{"class":121},[54,68843,68306],{"class":125},[54,68845,68846],{"class":64},"(m.satisfaction_prob, ",[54,68848,68138],{"class":132},[54,68850,11804],{"class":64},[54,68852,68853],{"class":279},"\"shots_per_model\"",[54,68855,68856],{"class":64},"], rng)\n",[54,68858,68859,68862,68864],{"class":56,"line":59413},[54,68860,68861],{"class":64}," all_bitstrings[m.name] ",[54,68863,122],{"class":121},[54,68865,68866],{"class":64}," bits\n",[54,68868,68869,68872,68874,68876,68879,68882,68884,68886,68889],{"class":56,"line":59418},[54,68870,68871],{"class":64}," summary_rows.",[54,68873,1495],{"class":125},[54,68875,129],{"class":64},[54,68877,68878],{"class":125},"summarize_model",[54,68880,68881],{"class":64},"(m, bits, ",[54,68883,68138],{"class":132},[54,68885,11804],{"class":64},[54,68887,68888],{"class":279},"\"epsilons\"",[54,68890,68891],{"class":64},"]))\n",[54,68893,68894],{"class":56,"line":59433},[54,68895,105],{"emptyLinePlaceholder":104},[54,68897,68898,68901,68903,68905,68907],{"class":56,"line":59439},[54,68899,68900],{"class":64},"summary_df ",[54,68902,122],{"class":121},[54,68904,57599],{"class":64},[54,68906,57602],{"class":125},[54,68908,68909],{"class":64},"(summary_rows)\n",[54,68911,68912],{"class":56,"line":59445},[54,68913,105],{"emptyLinePlaceholder":104},[54,68915,68916],{"class":56,"line":59451},[54,68917,68918],{"class":111},"# Add cost estimates\n",[54,68920,68921,68924,68926,68929,68931,68934,68936,68939],{"class":56,"line":59456},[54,68922,68923],{"class":64},"cps ",[54,68925,122],{"class":121},[54,68927,68928],{"class":132}," CONFIG",[54,68930,11804],{"class":64},[54,68932,68933],{"class":279},"\"cost_model\"",[54,68935,1311],{"class":64},[54,68937,68938],{"class":279},"\"default_cost_per_shot_usd\"",[54,68940,1350],{"class":64},[54,68942,68943,68946,68948,68950,68952,68954,68956,68959],{"class":56,"line":59461},[54,68944,68945],{"class":64},"over ",[54,68947,122],{"class":121},[54,68949,68928],{"class":132},[54,68951,11804],{"class":64},[54,68953,68933],{"class":279},[54,68955,1311],{"class":64},[54,68957,68958],{"class":279},"\"fixed_overhead_usd\"",[54,68960,1350],{"class":64},[54,68962,68963,68965,68968,68970,68972,68974,68977,68979,68981],{"class":56,"line":59493},[54,68964,1201],{"class":60},[54,68966,68967],{"class":64}," cps ",[54,68969,1207],{"class":121},[54,68971,20814],{"class":132},[54,68973,21723],{"class":60},[54,68975,68976],{"class":64}," over ",[54,68978,1207],{"class":121},[54,68980,20814],{"class":132},[54,68982,1284],{"class":64},[54,68984,68985,68988,68991,68993,68995],{"class":56,"line":59520},[54,68986,68987],{"class":64}," summary_df[",[54,68989,68990],{"class":279},"\"Illustrative cost (USD)\"",[54,68992,7762],{"class":64},[54,68994,122],{"class":121},[54,68996,1733],{"class":64},[54,68998,68999,69002,69004,69006,69009,69012],{"class":56,"line":59546},[54,69000,69001],{"class":125}," apply_costs",[54,69003,129],{"class":64},[54,69005,976],{"class":121},[54,69007,69008],{"class":64},"(row[",[54,69010,69011],{"class":279},"\"Shots\"",[54,69013,69014],{"class":64},"]), cps, over)\n",[54,69016,69017,69019,69022,69024,69027,69030],{"class":56,"line":59551},[54,69018,1823],{"class":60},[54,69020,69021],{"class":64}," _, row ",[54,69023,1192],{"class":60},[54,69025,69026],{"class":64}," summary_df.",[54,69028,69029],{"class":125},"iterrows",[54,69031,330],{"class":64},[54,69033,69034],{"class":56,"line":59565},[54,69035,58597],{"class":64},[54,69037,69038],{"class":56,"line":59580},[54,69039,105],{"emptyLinePlaceholder":104},[54,69041,69042],{"class":56,"line":59585},[54,69043,69044],{"class":111},"# ---------- Save artifacts ----------\n",[54,69046,69047],{"class":56,"line":59590},[54,69048,105],{"emptyLinePlaceholder":104},[54,69050,69051],{"class":56,"line":59604},[54,69052,69053],{"class":111},"# Save raw bitstrings and summary CSV\n",[54,69055,69056,69058,69061,69063,69066,69068],{"class":56,"line":59609},[54,69057,1186],{"class":60},[54,69059,69060],{"class":64}," name, arr ",[54,69062,1192],{"class":60},[54,69064,69065],{"class":64}," all_bitstrings.",[54,69067,22163],{"class":125},[54,69069,1486],{"class":64},[54,69071,69072,69075,69077,69079,69082,69084,69087,69090,69092,69095,69097,69100,69102,69104],{"class":56,"line":59615},[54,69073,69074],{"class":64}," out_path ",[54,69076,122],{"class":121},[54,69078,63989],{"class":60},[54,69080,69081],{"class":279},"\"bitstrings_",[54,69083,2366],{"class":132},[54,69085,69086],{"class":64},"name.",[54,69088,69089],{"class":125},"replace",[54,69091,129],{"class":64},[54,69093,69094],{"class":279},"' '",[54,69096,171],{"class":64},[54,69098,69099],{"class":279},"'_'",[54,69101,3205],{"class":64},[54,69103,2428],{"class":132},[54,69105,69106],{"class":279},".csv\"\n",[54,69108,69109,69112,69114,69117,69120,69123,69126,69129,69132,69134,69136],{"class":56,"line":59621},[54,69110,69111],{"class":64}," pd.",[54,69113,57602],{"class":125},[54,69115,69116],{"class":64},"({",[54,69118,69119],{"class":279},"\"bit\"",[54,69121,69122],{"class":64},": arr}).",[54,69124,69125],{"class":125},"to_csv",[54,69127,69128],{"class":64},"(out_path, ",[54,69130,69131],{"class":1711},"index",[54,69133,122],{"class":121},[54,69135,21520],{"class":132},[54,69137,136],{"class":64},[54,69139,69140],{"class":56,"line":59627},[54,69141,105],{"emptyLinePlaceholder":104},[54,69143,69144,69147,69149],{"class":56,"line":59632},[54,69145,69146],{"class":64},"summary_csv_path ",[54,69148,122],{"class":121},[54,69150,69151],{"class":279}," \"summary.csv\"\n",[54,69153,69154,69157,69159,69162,69164,69166,69168],{"class":56,"line":59637},[54,69155,69156],{"class":64},"summary_df.",[54,69158,69125],{"class":125},[54,69160,69161],{"class":64},"(summary_csv_path, ",[54,69163,69131],{"class":1711},[54,69165,122],{"class":121},[54,69167,21520],{"class":132},[54,69169,136],{"class":64},[54,69171,69172],{"class":56,"line":59642},[54,69173,105],{"emptyLinePlaceholder":104},[54,69175,69176],{"class":56,"line":59656},[54,69177,69178],{"class":111},"# Save config JSON\n",[54,69180,69181,69184,69187,69189,69192,69194,69197,69199,69201],{"class":56,"line":59661},[54,69182,69183],{"class":60},"with",[54,69185,69186],{"class":121}," open",[54,69188,129],{"class":64},[54,69190,69191],{"class":279},"\"simulation_config.json\"",[54,69193,171],{"class":64},[54,69195,69196],{"class":279},"\"w\"",[54,69198,204],{"class":64},[54,69200,68],{"class":60},[54,69202,69203],{"class":64}," f:\n",[54,69205,69206,69209,69212,69214,69216,69219,69222,69224,69226],{"class":56,"line":59672},[54,69207,69208],{"class":64}," json.",[54,69210,69211],{"class":125},"dump",[54,69213,129],{"class":64},[54,69215,68138],{"class":132},[54,69217,69218],{"class":64},", f, ",[54,69220,69221],{"class":1711},"indent",[54,69223,122],{"class":121},[54,69225,133],{"class":132},[54,69227,136],{"class":64},[54,69229,69230],{"class":56,"line":59680},[54,69231,105],{"emptyLinePlaceholder":104},[54,69233,69234,69236,69238,69241],{"class":56,"line":59690},[54,69235,8081],{"class":121},[54,69237,129],{"class":64},[54,69239,69240],{"class":279},"\"Saved data to CSV and JSON files.\"",[54,69242,136],{"class":64},[54,69244,69245],{"class":56,"line":59695},[54,69246,105],{"emptyLinePlaceholder":104},[54,69248,69249],{"class":56,"line":59700},[54,69250,69251],{"class":111},"# ---------- Visualizations ----------\n",[54,69253,69254],{"class":56,"line":59705},[54,69255,105],{"emptyLinePlaceholder":104},[54,69257,69258],{"class":56,"line":59712},[54,69259,69260],{"class":111},"# Figure 1: Observed ones fraction with Wilson CI\n",[54,69262,69263,69266,69269,69271,69274],{"class":56,"line":59728},[54,69264,69265],{"class":64},"plt.style.",[54,69267,69268],{"class":125},"use",[54,69270,129],{"class":64},[54,69272,69273],{"class":279},"'seaborn-v0_8-whitegrid'",[54,69275,136],{"class":64},[54,69277,69278,69280,69282,69284,69286,69288,69290,69292,69294,69296],{"class":56,"line":59737},[54,69279,24932],{"class":64},[54,69281,61173],{"class":125},[54,69283,129],{"class":64},[54,69285,57790],{"class":1711},[54,69287,122],{"class":121},[54,69289,129],{"class":64},[54,69291,23361],{"class":132},[54,69293,171],{"class":64},[54,69295,63269],{"class":132},[54,69297,22096],{"class":64},[54,69299,69300,69303,69305,69307,69310,69312,69314],{"class":56,"line":59746},[54,69301,69302],{"class":64},"x ",[54,69304,122],{"class":121},[54,69306,1172],{"class":64},[54,69308,69309],{"class":125},"arange",[54,69311,129],{"class":64},[54,69313,1319],{"class":121},[54,69315,69316],{"class":64},"(summary_df))\n",[54,69318,69319,69322,69324,69327,69330,69332,69335],{"class":56,"line":59755},[54,69320,69321],{"class":64},"y ",[54,69323,122],{"class":121},[54,69325,69326],{"class":64}," summary_df[",[54,69328,69329],{"class":279},"\"Ones fraction (observed)\"",[54,69331,22461],{"class":64},[54,69333,69334],{"class":125},"to_numpy",[54,69336,330],{"class":64},[54,69338,69339],{"class":56,"line":59764},[54,69340,105],{"emptyLinePlaceholder":104},[54,69342,69343],{"class":56,"line":59787},[54,69344,69345],{"class":111},"# --- FIX APPLIED HERE ---\n",[54,69347,69348],{"class":56,"line":59793},[54,69349,69350],{"class":111},"# Take the absolute value to prevent tiny negative numbers from floating-point\n",[54,69352,69353],{"class":56,"line":59798},[54,69354,69355],{"class":111},"# inaccuracies from causing a ValueError in the errorbar plot.\n",[54,69357,69358,69361,69363,69366,69368,69370,69373,69375,69377,69379,69381],{"class":56,"line":59803},[54,69359,69360],{"class":64},"lower_error ",[54,69362,122],{"class":121},[54,69364,69365],{"class":64}," y ",[54,69367,1923],{"class":121},[54,69369,69326],{"class":64},[54,69371,69372],{"class":279},"\"Wilson CI 95",[54,69374,68596],{"class":132},[54,69376,68599],{"class":279},[54,69378,22461],{"class":64},[54,69380,69334],{"class":125},[54,69382,330],{"class":64},[54,69384,69385,69388,69390,69392,69394,69396,69398,69400,69402,69404,69406],{"class":56,"line":59808},[54,69386,69387],{"class":64},"upper_error ",[54,69389,122],{"class":121},[54,69391,69326],{"class":64},[54,69393,69372],{"class":279},[54,69395,68609],{"class":132},[54,69397,68612],{"class":279},[54,69399,22461],{"class":64},[54,69401,69334],{"class":125},[54,69403,1233],{"class":64},[54,69405,1923],{"class":121},[54,69407,69408],{"class":64}," y\n",[54,69410,69411,69414,69416,69418,69420,69423,69425,69428,69430],{"class":56,"line":59818},[54,69412,69413],{"class":64},"yerr ",[54,69415,122],{"class":121},[54,69417,1172],{"class":64},[54,69419,22629],{"class":125},[54,69421,69422],{"class":64},"([np.",[54,69424,63816],{"class":125},[54,69426,69427],{"class":64},"(lower_error), np.",[54,69429,63816],{"class":125},[54,69431,69432],{"class":64},"(upper_error)])\n",[54,69434,69435],{"class":56,"line":59827},[54,69436,69437],{"class":111},"# --- END OF FIX ---\n",[54,69439,69440],{"class":56,"line":59836},[54,69441,105],{"emptyLinePlaceholder":104},[54,69443,69444,69446,69449,69452,69454,69456,69459,69461,69464,69466,69469],{"class":56,"line":59841},[54,69445,24932],{"class":64},[54,69447,69448],{"class":125},"bar",[54,69450,69451],{"class":64},"(x, y, ",[54,69453,63907],{"class":1711},[54,69455,122],{"class":121},[54,69457,69458],{"class":279},"'skyblue'",[54,69460,171],{"class":64},[54,69462,69463],{"class":1711},"edgecolor",[54,69465,122],{"class":121},[54,69467,69468],{"class":279},"'black'",[54,69470,136],{"class":64},[54,69472,69473,69475,69478,69480,69483,69485,69488,69491,69493,69496,69498,69501,69503,69505,69507,69509,69511,69514,69516,69519,69521,69523],{"class":56,"line":59847},[54,69474,24932],{"class":64},[54,69476,69477],{"class":125},"errorbar",[54,69479,69451],{"class":64},[54,69481,69482],{"class":1711},"yerr",[54,69484,122],{"class":121},[54,69486,69487],{"class":64},"yerr, ",[54,69489,69490],{"class":1711},"fmt",[54,69492,122],{"class":121},[54,69494,69495],{"class":279},"\"none\"",[54,69497,171],{"class":64},[54,69499,69500],{"class":1711},"capsize",[54,69502,122],{"class":121},[54,69504,57802],{"class":132},[54,69506,171],{"class":64},[54,69508,63907],{"class":1711},[54,69510,122],{"class":121},[54,69512,69513],{"class":279},"\"black\"",[54,69515,171],{"class":64},[54,69517,69518],{"class":1711},"elinewidth",[54,69520,122],{"class":121},[54,69522,65767],{"class":132},[54,69524,136],{"class":64},[54,69526,69527,69529,69531,69534,69537,69539,69541,69543,69546,69548,69550,69552,69555,69557,69560],{"class":56,"line":59863},[54,69528,24932],{"class":64},[54,69530,25077],{"class":125},[54,69532,69533],{"class":64},"(x, summary_df[",[54,69535,69536],{"class":279},"\"Model\"",[54,69538,22461],{"class":64},[54,69540,23926],{"class":125},[54,69542,13506],{"class":64},[54,69544,69545],{"class":1711},"rotation",[54,69547,122],{"class":121},[54,69549,48145],{"class":132},[54,69551,171],{"class":64},[54,69553,69554],{"class":1711},"ha",[54,69556,122],{"class":121},[54,69558,69559],{"class":279},"\"right\"",[54,69561,136],{"class":64},[54,69563,69564,69566,69568,69570,69573],{"class":56,"line":59874},[54,69565,24932],{"class":64},[54,69567,25137],{"class":125},[54,69569,129],{"class":64},[54,69571,69572],{"class":279},"\"Observed Ones Fraction\"",[54,69574,136],{"class":64},[54,69576,69577,69579,69581,69583,69586,69588,69591,69593,69596],{"class":56,"line":59889},[54,69578,24932],{"class":64},[54,69580,25151],{"class":125},[54,69582,129],{"class":64},[54,69584,69585],{"class":279},"\"Observed Single-Shot Success Rates with 95% Wilson CI\"",[54,69587,171],{"class":64},[54,69589,69590],{"class":1711},"fontsize",[54,69592,122],{"class":121},[54,69594,69595],{"class":132},"14",[54,69597,136],{"class":64},[54,69599,69600,69602,69604],{"class":56,"line":59899},[54,69601,24932],{"class":64},[54,69603,57927],{"class":125},[54,69605,330],{"class":64},[54,69607,69608,69611,69613],{"class":56,"line":59906},[54,69609,69610],{"class":64},"fig1_path ",[54,69612,122],{"class":121},[54,69614,69615],{"class":279}," \"fig_ones_fraction.png\"\n",[54,69617,69618,69620,69623,69626,69628,69630,69632,69634,69637,69639,69642],{"class":56,"line":59915},[54,69619,24932],{"class":64},[54,69621,69622],{"class":125},"savefig",[54,69624,69625],{"class":64},"(fig1_path, ",[54,69627,63274],{"class":1711},[54,69629,122],{"class":121},[54,69631,63279],{"class":132},[54,69633,171],{"class":64},[54,69635,69636],{"class":1711},"bbox_inches",[54,69638,122],{"class":121},[54,69640,69641],{"class":279},"'tight'",[54,69643,136],{"class":64},[54,69645,69646,69648,69650],{"class":56,"line":59920},[54,69647,24932],{"class":64},[54,69649,25169],{"class":125},[54,69651,330],{"class":64},[54,69653,69654],{"class":56,"line":59942},[54,69655,105],{"emptyLinePlaceholder":104},[54,69657,69658],{"class":56,"line":59947},[54,69659,69660],{"class":111},"# Figure 2: Required shots at several ±epsilon targets\n",[54,69662,69663,69665,69667,69669,69671,69673,69675,69677,69679,69681],{"class":56,"line":59964},[54,69664,24932],{"class":64},[54,69666,61173],{"class":125},[54,69668,129],{"class":64},[54,69670,57790],{"class":1711},[54,69672,122],{"class":121},[54,69674,129],{"class":64},[54,69676,23361],{"class":132},[54,69678,171],{"class":64},[54,69680,63269],{"class":132},[54,69682,22096],{"class":64},[54,69684,69685,69688,69690],{"class":56,"line":59981},[54,69686,69687],{"class":64},"width ",[54,69689,122],{"class":121},[54,69691,69692],{"class":132}," 0.2\n",[54,69694,69695,69697,69699,69701,69703,69705,69707,69709,69711],{"class":56,"line":59986},[54,69696,69302],{"class":64},[54,69698,122],{"class":121},[54,69700,1172],{"class":64},[54,69702,69309],{"class":125},[54,69704,129],{"class":64},[54,69706,1319],{"class":121},[54,69708,129],{"class":64},[54,69710,67963],{"class":132},[54,69712,22096],{"class":64},[54,69714,69715,69718,69720,69722,69725,69727,69730,69732,69735],{"class":56,"line":59999},[54,69716,69717],{"class":64},"colors ",[54,69719,122],{"class":121},[54,69721,7983],{"class":64},[54,69723,69724],{"class":279},"'#ff9999'",[54,69726,2375],{"class":64},[54,69728,69729],{"class":279},"'#66b3ff'",[54,69731,2375],{"class":64},[54,69733,69734],{"class":279},"'#99ff99'",[54,69736,1350],{"class":64},[54,69738,69739],{"class":56,"line":60013},[54,69740,105],{"emptyLinePlaceholder":104},[54,69742,69743,69745,69748,69750,69752,69754,69756,69758,69760],{"class":56,"line":60018},[54,69744,1186],{"class":60},[54,69746,69747],{"class":64}," i, eps ",[54,69749,1192],{"class":60},[54,69751,1513],{"class":121},[54,69753,129],{"class":64},[54,69755,68138],{"class":132},[54,69757,11804],{"class":64},[54,69759,68888],{"class":279},[54,69761,69762],{"class":64},"]):\n",[54,69764,69765,69768,69770,69772,69775,69778,69780,69782,69784,69786],{"class":56,"line":60026},[54,69766,69767],{"class":64}," req ",[54,69769,122],{"class":121},[54,69771,7983],{"class":64},[54,69773,69774],{"class":125},"shots_for_epsilon",[54,69776,69777],{"class":64},"(m.satisfaction_prob, eps) ",[54,69779,1186],{"class":60},[54,69781,68827],{"class":64},[54,69783,1192],{"class":60},[54,69785,68832],{"class":132},[54,69787,1350],{"class":64},[54,69789,69790,69793,69795,69798,69800,69802,69804,69807,69809,69811,69813,69816,69818,69821,69824,69826,69828,69830,69832,69834,69837,69839,69841,69843],{"class":56,"line":60031},[54,69791,69792],{"class":64}," plt.",[54,69794,69448],{"class":125},[54,69796,69797],{"class":64},"(x ",[54,69799,1663],{"class":121},[54,69801,7802],{"class":64},[54,69803,1947],{"class":121},[54,69805,69806],{"class":64}," width, req, width, ",[54,69808,20568],{"class":1711},[54,69810,122],{"class":121},[54,69812,21834],{"class":60},[54,69814,69815],{"class":279},"\"±",[54,69817,2366],{"class":132},[54,69819,69820],{"class":64},"eps",[54,69822,69823],{"class":60},":.1%",[54,69825,2428],{"class":132},[54,69827,21837],{"class":279},[54,69829,171],{"class":64},[54,69831,63907],{"class":1711},[54,69833,122],{"class":121},[54,69835,69836],{"class":64},"colors[i], ",[54,69838,69463],{"class":1711},[54,69840,122],{"class":121},[54,69842,69468],{"class":279},[54,69844,136],{"class":64},[54,69846,69847],{"class":56,"line":60036},[54,69848,105],{"emptyLinePlaceholder":104},[54,69850,69851,69853,69855,69857,69859,69862,69864,69866,69868,69870,69872,69874,69876,69878,69880,69882,69884,69886],{"class":56,"line":60046},[54,69852,24932],{"class":64},[54,69854,25077],{"class":125},[54,69856,69797],{"class":64},[54,69858,1663],{"class":121},[54,69860,69861],{"class":64}," width, [m.name ",[54,69863,1186],{"class":60},[54,69865,68827],{"class":64},[54,69867,1192],{"class":60},[54,69869,68832],{"class":132},[54,69871,1316],{"class":64},[54,69873,69545],{"class":1711},[54,69875,122],{"class":121},[54,69877,48145],{"class":132},[54,69879,171],{"class":64},[54,69881,69554],{"class":1711},[54,69883,122],{"class":121},[54,69885,69559],{"class":279},[54,69887,136],{"class":64},[54,69889,69890,69892,69894,69896,69899],{"class":56,"line":60054},[54,69891,24932],{"class":64},[54,69893,25137],{"class":125},[54,69895,129],{"class":64},[54,69897,69898],{"class":279},"\"Required Shots (Log Scale)\"",[54,69900,136],{"class":64},[54,69902,69903,69905,69907,69909,69912,69914,69916,69918,69920],{"class":56,"line":60062},[54,69904,24932],{"class":64},[54,69906,25151],{"class":125},[54,69908,129],{"class":64},[54,69910,69911],{"class":279},"\"Approximate Shots to Estimate Success Rate within ±ε\"",[54,69913,171],{"class":64},[54,69915,69590],{"class":1711},[54,69917,122],{"class":121},[54,69919,69595],{"class":132},[54,69921,136],{"class":64},[54,69923,69924,69926,69929,69931,69934],{"class":56,"line":60073},[54,69925,24932],{"class":64},[54,69927,69928],{"class":125},"yscale",[54,69930,129],{"class":64},[54,69932,69933],{"class":279},"'log'",[54,69935,136],{"class":64},[54,69937,69938,69940,69942,69944,69946,69948,69951],{"class":56,"line":60084},[54,69939,24932],{"class":64},[54,69941,25114],{"class":125},[54,69943,129],{"class":64},[54,69945,25151],{"class":1711},[54,69947,122],{"class":121},[54,69949,69950],{"class":279},"\"Error Tolerance\"",[54,69952,136],{"class":64},[54,69954,69955,69957,69959],{"class":56,"line":60093},[54,69956,24932],{"class":64},[54,69958,57927],{"class":125},[54,69960,330],{"class":64},[54,69962,69963,69966,69968],{"class":56,"line":60099},[54,69964,69965],{"class":64},"fig2_path ",[54,69967,122],{"class":121},[54,69969,69970],{"class":279}," \"fig_required_shots.png\"\n",[54,69972,69973,69975,69977,69980,69982,69984,69986,69988,69990,69992,69994],{"class":56,"line":60109},[54,69974,24932],{"class":64},[54,69976,69622],{"class":125},[54,69978,69979],{"class":64},"(fig2_path, ",[54,69981,63274],{"class":1711},[54,69983,122],{"class":121},[54,69985,63279],{"class":132},[54,69987,171],{"class":64},[54,69989,69636],{"class":1711},[54,69991,122],{"class":121},[54,69993,69641],{"class":279},[54,69995,136],{"class":64},[54,69997,69998,70000,70002],{"class":56,"line":60119},[54,69999,24932],{"class":64},[54,70001,25169],{"class":125},[54,70003,330],{"class":64},[54,70005,70006],{"class":56,"line":60128},[54,70007,105],{"emptyLinePlaceholder":104},[54,70009,70010,70012,70014,70017],{"class":56,"line":60137},[54,70011,8081],{"class":121},[54,70013,129],{"class":64},[54,70015,70016],{"class":279},"\"Generated and saved plots.\"",[54,70018,136],{"class":64},[54,70020,70021],{"class":56,"line":60148},[54,70022,105],{"emptyLinePlaceholder":104},[54,70024,70025],{"class":56,"line":60160},[54,70026,70027],{"class":111},"# ---------- Display interactive table ----------\n",[54,70029,70030],{"class":56,"line":60171},[54,70031,105],{"emptyLinePlaceholder":104},[54,70033,70034,70037,70039,70042,70044,70047],{"class":56,"line":60176},[54,70035,70036],{"class":125},"display",[54,70038,129],{"class":64},[54,70040,70041],{"class":125},"Markdown",[54,70043,129],{"class":64},[54,70045,70046],{"class":279},"\"## Qubit Sampling Budget - Summary\"",[54,70048,22096],{"class":64},[54,70050,70051,70053,70056,70059],{"class":56,"line":60181},[54,70052,70036],{"class":125},[54,70054,70055],{"class":64},"(summary_df.style.",[54,70057,70058],{"class":125},"format",[54,70060,70061],{"class":64},"({\n",[54,70063,70064,70067,70069,70071,70073,70076,70078,70080],{"class":56,"line":60186},[54,70065,70066],{"class":279}," \"Configured p (target)\"",[54,70068,959],{"class":64},[54,70070,21837],{"class":279},[54,70072,2366],{"class":132},[54,70074,70075],{"class":60},":.2f",[54,70077,2428],{"class":132},[54,70079,21837],{"class":279},[54,70081,965],{"class":64},[54,70083,70084,70087,70089,70091,70093,70096,70098,70100],{"class":56,"line":60196},[54,70085,70086],{"class":279}," \"Ones fraction (observed)\"",[54,70088,959],{"class":64},[54,70090,21837],{"class":279},[54,70092,2366],{"class":132},[54,70094,70095],{"class":60},":.4f",[54,70097,2428],{"class":132},[54,70099,21837],{"class":279},[54,70101,965],{"class":64},[54,70103,70104,70107,70109,70111,70113,70115,70117,70119,70121,70123],{"class":56,"line":60203},[54,70105,70106],{"class":279}," \"Wilson CI 95",[54,70108,68596],{"class":132},[54,70110,68599],{"class":279},[54,70112,959],{"class":64},[54,70114,21837],{"class":279},[54,70116,2366],{"class":132},[54,70118,70095],{"class":60},[54,70120,2428],{"class":132},[54,70122,21837],{"class":279},[54,70124,965],{"class":64},[54,70126,70127,70129,70131,70133,70135,70137,70139,70141,70143,70145],{"class":56,"line":60211},[54,70128,70106],{"class":279},[54,70130,68609],{"class":132},[54,70132,68612],{"class":279},[54,70134,959],{"class":64},[54,70136,21837],{"class":279},[54,70138,2366],{"class":132},[54,70140,70095],{"class":60},[54,70142,2428],{"class":132},[54,70144,21837],{"class":279},[54,70146,965],{"class":64},[54,70148,70149,70152,70154,70156,70158,70160,70162,70164],{"class":56,"line":60222},[54,70150,70151],{"class":279}," \"CI width (95%)\"",[54,70153,959],{"class":64},[54,70155,21837],{"class":279},[54,70157,2366],{"class":132},[54,70159,70095],{"class":60},[54,70161,2428],{"class":132},[54,70163,21837],{"class":279},[54,70165,965],{"class":64},[54,70167,70168,70171,70173,70175,70177,70179,70181],{"class":56,"line":60233},[54,70169,70170],{"class":279}," \"Illustrative cost (USD)\"",[54,70172,959],{"class":64},[54,70174,61237],{"class":279},[54,70176,2366],{"class":132},[54,70178,70075],{"class":60},[54,70180,2428],{"class":132},[54,70182,61283],{"class":279},[54,70184,70185],{"class":56,"line":60242},[54,70186,70187],{"class":64},"}))\n",[54,70189,70190],{"class":56,"line":60248},[54,70191,105],{"emptyLinePlaceholder":104},[54,70193,70194],{"class":56,"line":60258},[54,70195,105],{"emptyLinePlaceholder":104},[54,70197,70198],{"class":56,"line":60268},[54,70199,70200],{"class":111},"# ---------- Build a ready-to-run Colab notebook ----------\n",[54,70202,70203],{"class":56,"line":60277},[54,70204,105],{"emptyLinePlaceholder":104},[54,70206,70207,70210,70212,70215],{"class":56,"line":60286},[54,70208,70209],{"class":64},"md_intro ",[54,70211,122],{"class":121},[54,70213,70214],{"class":60}," r",[54,70216,70217],{"class":57863},"\"\"\"\n",[54,70219,70220],{"class":56,"line":60295},[54,70221,70222],{"class":111},"# Sampling Budget & Bit-String Analogy for Qubit Classes\n",[54,70224,70225],{"class":56,"line":60307},[54,70226,105],{"emptyLinePlaceholder":104},[54,70228,70229,70231,70234,70236,70239,70241,70243,70245,70248,70250,70253,70255],{"class":56,"line":60319},[54,70230,1933],{"class":132},[54,70232,70233],{"class":57863},"High school level ",[54,70235,129],{"class":132},[54,70237,70238],{"class":57863},"concise",[54,70240,3205],{"class":132},[54,70242,24541],{"class":57863},[54,70244,1933],{"class":132},[54,70246,70247],{"class":57863}," We treat each measurement ",[54,70249,129],{"class":132},[54,70251,70252],{"class":57863},"“single shot”",[54,70254,3205],{"class":132},[54,70256,70257],{"class":57863}," as a bit: `1` if the outcome matches the analytical prediction, `0` if it does not. Different qubit types have different chances of landing a `1`. We simulate many shots, show how often we get `1`s, and estimate how many shots we need for a target accuracy.\n",[54,70259,70260],{"class":56,"line":60328},[54,70261,105],{"emptyLinePlaceholder":104},[54,70263,70264,70266,70269,70271,70273,70275,70277,70279,70282,70284,70287,70289,70292,70294,70297,70299,70302,70304,70307,70309,70312,70315,70318,70321,70323,70325,70328,70330,70332,70335,70338,70340],{"class":56,"line":60333},[54,70265,1933],{"class":132},[54,70267,70268],{"class":57863},"Graduate level ",[54,70270,129],{"class":132},[54,70272,70238],{"class":57863},[54,70274,3205],{"class":132},[54,70276,24541],{"class":57863},[54,70278,1933],{"class":132},[54,70280,70281],{"class":57863}," We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ ",[54,70283,129],{"class":132},[54,70285,70286],{"class":57863},"noisy intermediate-scale quantum",[54,70288,3205],{"class":132},[54,70290,70291],{"class":57863},", fault-tolerant ",[54,70293,129],{"class":132},[54,70295,70296],{"class":57863},"logical",[54,70298,3205],{"class":132},[54,70300,70301],{"class":57863},", topologically protected, and ideal. We report Wilson score intervals ",[54,70303,129],{"class":132},[54,70305,70306],{"class":57863},"95%",[54,70308,3205],{"class":132},[54,70310,70311],{"class":57863}," for observed proportions and estimate the required shot budget $n ",[54,70313,70314],{"class":121},"\\a",[54,70316,70317],{"class":57863},"pprox z^2",[54,70319,70320],{"class":121},"\\,",[54,70322,13],{"class":57863},[54,70324,129],{"class":132},[54,70326,70327],{"class":57863},"1-p",[54,70329,3205],{"class":132},[54,70331,194],{"class":57863},[54,70333,70334],{"class":121},"\\e",[54,70336,70337],{"class":57863},"psilon^2$ to bound absolute error by ±$",[54,70339,70334],{"class":121},[54,70341,70342],{"class":57863},"psilon$ at ~95% confidence.\n",[54,70344,70345],{"class":56,"line":60338},[54,70346,70217],{"class":57863},[54,70348,70349],{"class":56,"line":60343},[54,70350,105],{"emptyLinePlaceholder":104},[54,70352,70353,70356,70358,70360],{"class":56,"line":60353},[54,70354,70355],{"class":64},"code_config ",[54,70357,122],{"class":121},[54,70359,70214],{"class":60},[54,70361,70217],{"class":57863},[54,70363,70364],{"class":56,"line":60360},[54,70365,70366],{"class":111},"# @title Configuration\n",[54,70368,70369],{"class":56,"line":60367},[54,70370,70371],{"class":57863},"from dataclasses import dataclass\n",[54,70373,70374],{"class":56,"line":60378},[54,70375,70376],{"class":57863},"from typing import List, Dict\n",[54,70378,70379],{"class":56,"line":60389},[54,70380,70381],{"class":57863},"import numpy as np\n",[54,70383,70384],{"class":56,"line":60398},[54,70385,105],{"emptyLinePlaceholder":104},[54,70387,70388,70391],{"class":56,"line":60404},[54,70389,70390],{"class":57863},"RANDOM_SEED = 579345 ",[54,70392,70393],{"class":111},"# @param {type:\"number\"}\n",[54,70395,70396,70399],{"class":56,"line":60414},[54,70397,70398],{"class":57863},"SHOTS_PER_MODEL = 10000 ",[54,70400,70393],{"class":111},[54,70402,70403,70406,70409],{"class":56,"line":60422},[54,70404,70405],{"class":57863},"EPSILONS = ",[54,70407,70408],{"class":132},"[0.02, 0.01, 0.005]",[54,70410,70411],{"class":111}," # @param\n",[54,70413,70414,70417],{"class":56,"line":60429},[54,70415,70416],{"class":57863},"COST_PER_SHOT_USD = 0.00005 ",[54,70418,70393],{"class":111},[54,70420,70421,70424],{"class":56,"line":60436},[54,70422,70423],{"class":57863},"FIXED_OVERHEAD_USD = 0.10 ",[54,70425,70393],{"class":111},[54,70427,70428],{"class":56,"line":60445},[54,70429,105],{"emptyLinePlaceholder":104},[54,70431,70432,70434,70436,70439],{"class":56,"line":60454},[54,70433,67903],{"class":57863},[54,70435,129],{"class":132},[54,70437,70438],{"class":57863},"frozen=True",[54,70440,136],{"class":132},[54,70442,70443],{"class":56,"line":60463},[54,70444,70445],{"class":57863},"class QubitModel:\n",[54,70447,70448],{"class":56,"line":60472},[54,70449,70450],{"class":57863}," name: str\n",[54,70452,70453],{"class":56,"line":60483},[54,70454,70455],{"class":57863}," satisfaction_prob: float\n",[54,70457,70458],{"class":56,"line":60495},[54,70459,70460],{"class":57863}," description: str\n",[54,70462,70463],{"class":56,"line":60507},[54,70464,105],{"emptyLinePlaceholder":104},[54,70466,70467,70470,70473,70476],{"class":56,"line":60516},[54,70468,70469],{"class":57863},"MODELS: List",[54,70471,70472],{"class":132},"[QubitModel]",[54,70474,70475],{"class":57863}," = ",[54,70477,70478],{"class":132},"[\n",[54,70480,70481],{"class":56,"line":60521},[54,70482,70483],{"class":132}," QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n",[54,70485,70486],{"class":56,"line":60526},[54,70487,70488],{"class":132}," QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n",[54,70490,70491],{"class":56,"line":60531},[54,70492,70493],{"class":132}," QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n",[54,70495,70496],{"class":56,"line":60541},[54,70497,70498],{"class":132}," QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n",[54,70500,70501],{"class":56,"line":60548},[54,70502,70503],{"class":132}," QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n",[54,70505,70506],{"class":56,"line":60555},[54,70507,1350],{"class":132},[54,70509,70510],{"class":56,"line":60567},[54,70511,105],{"emptyLinePlaceholder":104},[54,70513,70514,70517,70519,70522],{"class":56,"line":60578},[54,70515,70516],{"class":57863},"rng = np.random.default_rng",[54,70518,129],{"class":132},[54,70520,70521],{"class":57863},"RANDOM_SEED",[54,70523,136],{"class":132},[54,70525,70526],{"class":56,"line":60587},[54,70527,70217],{"class":57863},[54,70529,70530],{"class":56,"line":60593},[54,70531,105],{"emptyLinePlaceholder":104},[54,70533,70534,70537,70539,70541],{"class":56,"line":60609},[54,70535,70536],{"class":64},"code_lib_and_run ",[54,70538,122],{"class":121},[54,70540,70214],{"class":60},[54,70542,70217],{"class":57863},[54,70544,70545],{"class":56,"line":60614},[54,70546,70547],{"class":111},"# @title Run Simulation & Display Results\n",[54,70549,70550],{"class":56,"line":60623},[54,70551,70552],{"class":57863},"import math\n",[54,70554,70555],{"class":56,"line":60633},[54,70556,70557],{"class":57863},"from typing import Tuple, Dict\n",[54,70559,70560],{"class":56,"line":60642},[54,70561,70562],{"class":57863},"import pandas as pd\n",[54,70564,70565],{"class":56,"line":60651},[54,70566,70381],{"class":57863},[54,70568,70569],{"class":56,"line":60657},[54,70570,70571],{"class":57863},"import matplotlib.pyplot as plt\n",[54,70573,70574],{"class":56,"line":60666},[54,70575,70576],{"class":57863},"from IPython.display import display, Markdown\n",[54,70578,70579],{"class":56,"line":60678},[54,70580,105],{"emptyLinePlaceholder":104},[54,70582,70583,70586,70588,70591,70593,70596,70599],{"class":56,"line":60691},[54,70584,70585],{"class":57863},"def wilson_ci",[54,70587,129],{"class":132},[54,70589,70590],{"class":57863},"k: int, n: int, z: float = 1.96",[54,70592,3205],{"class":132},[54,70594,70595],{"class":57863}," -> Tuple",[54,70597,70598],{"class":132},"[float, float]",[54,70600,1284],{"class":57863},[54,70602,70603,70606,70608,70611],{"class":56,"line":60703},[54,70604,70605],{"class":57863}," if n == 0: return ",[54,70607,129],{"class":132},[54,70609,70610],{"class":57863},"0.0, 1.0",[54,70612,136],{"class":132},[54,70614,70615,70618,70620],{"class":56,"line":60708},[54,70616,70617],{"class":57863}," p_hat, z2_n = k \u002F n, z",[54,70619,1933],{"class":132},[54,70621,70622],{"class":57863},"2 \u002F n\n",[54,70624,70625,70628,70630],{"class":56,"line":60713},[54,70626,70627],{"class":57863}," denom = 1.0 ",[54,70629,1663],{"class":132},[54,70631,70632],{"class":57863}," z2_n\n",[54,70634,70635,70638,70640,70643,70645,70648,70650],{"class":56,"line":60718},[54,70636,70637],{"class":57863}," center = ",[54,70639,129],{"class":132},[54,70641,70642],{"class":57863},"p_hat ",[54,70644,1663],{"class":132},[54,70646,70647],{"class":57863}," z2_n \u002F 2",[54,70649,3205],{"class":132},[54,70651,70652],{"class":57863}," \u002F denom\n",[54,70654,70655,70658,70660,70663,70665,70668,70670,70672,70674,70676,70679,70681,70684,70686,70689,70691,70694,70696,70699,70701,70704,70706,70708,70711],{"class":56,"line":60737},[54,70656,70657],{"class":57863}," span = ",[54,70659,129],{"class":132},[54,70661,70662],{"class":57863},"z ",[54,70664,1947],{"class":132},[54,70666,70667],{"class":57863}," math.sqrt",[54,70669,129],{"class":132},[54,70671,70642],{"class":57863},[54,70673,1947],{"class":132},[54,70675,58201],{"class":132},[54,70677,70678],{"class":57863},"1 - p_hat",[54,70680,3205],{"class":132},[54,70682,70683],{"class":57863}," \u002F n ",[54,70685,1663],{"class":132},[54,70687,70688],{"class":57863}," z",[54,70690,1933],{"class":132},[54,70692,70693],{"class":57863},"2 \u002F ",[54,70695,129],{"class":132},[54,70697,70698],{"class":57863},"4 ",[54,70700,1947],{"class":132},[54,70702,70703],{"class":57863}," n",[54,70705,1933],{"class":132},[54,70707,133],{"class":57863},[54,70709,70710],{"class":132},")))",[54,70712,70652],{"class":57863},[54,70714,70715,70718,70720,70722,70724,70727,70729,70732,70734,70737,70739,70742],{"class":56,"line":60743},[54,70716,70717],{"class":57863}," return ",[54,70719,129],{"class":132},[54,70721,36269],{"class":57863},[54,70723,129],{"class":132},[54,70725,70726],{"class":57863},"0.0, center - span",[54,70728,3205],{"class":132},[54,70730,70731],{"class":57863},", min",[54,70733,129],{"class":132},[54,70735,70736],{"class":57863},"1.0, center ",[54,70738,1663],{"class":132},[54,70740,70741],{"class":57863}," span",[54,70743,22096],{"class":132},[54,70745,70746],{"class":56,"line":60752},[54,70747,105],{"emptyLinePlaceholder":104},[54,70749,70750,70753,70755,70758,70760],{"class":56,"line":60759},[54,70751,70752],{"class":57863},"def shots_for_epsilon",[54,70754,129],{"class":132},[54,70756,70757],{"class":57863},"p: float, epsilon: float = 0.01, z: float = 1.96",[54,70759,3205],{"class":132},[54,70761,70762],{"class":57863}," -> int:\n",[54,70764,70765,70768,70770,70772,70774,70776],{"class":56,"line":60766},[54,70766,70767],{"class":57863}," if p ",[54,70769,1947],{"class":132},[54,70771,58201],{"class":132},[54,70773,70327],{"class":57863},[54,70775,3205],{"class":132},[54,70777,70778],{"class":57863}," == 0: return 1\n",[54,70780,70781,70784,70786,70789,70791,70794,70796,70798,70800,70803,70805,70807,70809,70811,70813,70816,70819,70821,70823,70825,70827],{"class":56,"line":60774},[54,70782,70783],{"class":57863}," return max",[54,70785,129],{"class":132},[54,70787,70788],{"class":57863},"1, int",[54,70790,129],{"class":132},[54,70792,70793],{"class":57863},"math.ceil",[54,70795,63037],{"class":132},[54,70797,215],{"class":57863},[54,70799,1933],{"class":132},[54,70801,70802],{"class":57863},"2 ",[54,70804,1947],{"class":132},[54,70806,67799],{"class":57863},[54,70808,1947],{"class":132},[54,70810,58201],{"class":132},[54,70812,70327],{"class":57863},[54,70814,70815],{"class":132},"))",[54,70817,70818],{"class":57863}," \u002F ",[54,70820,129],{"class":132},[54,70822,67720],{"class":57863},[54,70824,1933],{"class":132},[54,70826,133],{"class":57863},[54,70828,70829],{"class":132},"))))\n",[54,70831,70832],{"class":56,"line":60785},[54,70833,105],{"emptyLinePlaceholder":104},[54,70835,70836,70839,70841,70844,70846],{"class":56,"line":60796},[54,70837,70838],{"class":57863},"def summarize_model",[54,70840,129],{"class":132},[54,70842,70843],{"class":57863},"model, bits: np.ndarray, epsilons",[54,70845,3205],{"class":132},[54,70847,70848],{"class":57863}," -> Dict:\n",[54,70850,70851,70854,70856,70858],{"class":56,"line":60807},[54,70852,70853],{"class":57863}," n = len",[54,70855,129],{"class":132},[54,70857,68403],{"class":57863},[54,70859,136],{"class":132},[54,70861,70862,70865,70867,70869],{"class":56,"line":60812},[54,70863,70864],{"class":57863}," ones = sum",[54,70866,129],{"class":132},[54,70868,68403],{"class":57863},[54,70870,136],{"class":132},[54,70872,70873,70876,70878,70881],{"class":56,"line":60821},[54,70874,70875],{"class":57863}," lo, hi = wilson_ci",[54,70877,129],{"class":132},[54,70879,70880],{"class":57863},"ones, n",[54,70882,136],{"class":132},[54,70884,70885],{"class":56,"line":60828},[54,70886,70887],{"class":57863}," row = {\n",[54,70889,70890,70893,70895,70898,70900],{"class":56,"line":60835},[54,70891,70892],{"class":57863}," \"Model\": model.name, \"p ",[54,70894,129],{"class":132},[54,70896,70897],{"class":57863},"target",[54,70899,3205],{"class":132},[54,70901,70902],{"class":57863},"\": model.satisfaction_prob, \"Shots\": n,\n",[54,70904,70905,70908,70910,70913,70915],{"class":56,"line":60843},[54,70906,70907],{"class":57863}," \"Ones\": ones, \"p ",[54,70909,129],{"class":132},[54,70911,70912],{"class":57863},"observed",[54,70914,3205],{"class":132},[54,70916,70917],{"class":57863},"\": ones \u002F n if n > 0 else 0,\n",[54,70919,70920],{"class":56,"line":60864},[54,70921,70922],{"class":57863}," \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n",[54,70924,70925],{"class":56,"line":60875},[54,70926,59040],{"class":57863},[54,70928,70929],{"class":56,"line":60887},[54,70930,70931],{"class":57863}," for eps in epsilons:\n",[54,70933,70934,70937,70940,70943,70945,70948],{"class":56,"line":60898},[54,70935,70936],{"class":57863}," row",[54,70938,70939],{"class":132},"[f\"Shots for ±{eps:.1%}\"]",[54,70941,70942],{"class":57863}," = shots_for_epsilon",[54,70944,129],{"class":132},[54,70946,70947],{"class":57863},"model.satisfaction_prob, eps",[54,70949,136],{"class":132},[54,70951,70952],{"class":56,"line":60909},[54,70953,70954],{"class":57863}," return row\n",[54,70956,70957],{"class":56,"line":60914},[54,70958,105],{"emptyLinePlaceholder":104},[54,70960,70961],{"class":56,"line":60919},[54,70962,70963],{"class":111},"# --- Run Simulation ---\n",[54,70965,70966,70969],{"class":56,"line":60924},[54,70967,70968],{"class":57863},"summary_rows = ",[54,70970,70971],{"class":132},"[summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\n",[54,70973,70974,70977,70979,70982],{"class":56,"line":60938},[54,70975,70976],{"class":57863},"summary_df = pd.DataFrame",[54,70978,129],{"class":132},[54,70980,70981],{"class":57863},"summary_rows",[54,70983,136],{"class":132},[54,70985,70986],{"class":56,"line":60950},[54,70987,70988],{"class":57863},"if COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n",[54,70990,70991,70994,70997,71000,71002,71005,71008,71010],{"class":56,"line":60956},[54,70992,70993],{"class":57863}," summary_df",[54,70995,70996],{"class":132},"[\"Cost (USD)\"]",[54,70998,70999],{"class":57863}," = FIXED_OVERHEAD_USD ",[54,71001,1663],{"class":132},[54,71003,71004],{"class":57863}," summary_df",[54,71006,71007],{"class":132},"[\"Shots\"]",[54,71009,7785],{"class":132},[54,71011,71012],{"class":57863}," COST_PER_SHOT_USD\n",[54,71014,71015],{"class":56,"line":60961},[54,71016,105],{"emptyLinePlaceholder":104},[54,71018,71019,71021,71023,71025,71027,71029],{"class":56,"line":60969},[54,71020,70036],{"class":57863},[54,71022,129],{"class":132},[54,71024,70041],{"class":57863},[54,71026,129],{"class":132},[54,71028,21837],{"class":57863},[54,71030,71031],{"class":111},"### Simulation Summary\"))\n",[54,71033,71034,71036,71038,71041,71043,71046,71048,71051,71053,71056],{"class":56,"line":60979},[54,71035,70036],{"class":57863},[54,71037,129],{"class":132},[54,71039,71040],{"class":57863},"summary_df.style.format",[54,71042,129],{"class":132},[54,71044,71045],{"class":57863},"precision=4",[54,71047,3205],{"class":132},[54,71049,71050],{"class":57863},".background_gradient",[54,71052,129],{"class":132},[54,71054,71055],{"class":57863},"cmap='viridis', subset=",[54,71057,71058],{"class":132},"['p (observed)', 'CI width']))\n",[54,71060,71061],{"class":56,"line":60984},[54,71062,105],{"emptyLinePlaceholder":104},[54,71064,71065],{"class":56,"line":60996},[54,71066,71067],{"class":111},"# --- Plotting ---\n",[54,71069,71070,71073,71075,71077],{"class":56,"line":61007},[54,71071,71072],{"class":57863},"plt.style.use",[54,71074,129],{"class":132},[54,71076,69273],{"class":57863},[54,71078,136],{"class":132},[54,71080,71081,71084,71086,71089,71091,71094,71096,71099,71101,71104],{"class":56,"line":61017},[54,71082,71083],{"class":57863},"fig, ",[54,71085,129],{"class":132},[54,71087,71088],{"class":57863},"ax1, ax2",[54,71090,3205],{"class":132},[54,71092,71093],{"class":57863}," = plt.subplots",[54,71095,129],{"class":132},[54,71097,71098],{"class":57863},"1, 2, figsize=",[54,71100,129],{"class":132},[54,71102,71103],{"class":57863},"16, 6",[54,71105,22096],{"class":132},[54,71107,71108],{"class":56,"line":61028},[54,71109,105],{"emptyLinePlaceholder":104},[54,71111,71112],{"class":56,"line":61033},[54,71113,71114],{"class":111},"# Plot 1\n",[54,71116,71117,71120,71123,71126],{"class":56,"line":61038},[54,71118,71119],{"class":57863},"y = summary_df",[54,71121,71122],{"class":132},"[\"p (observed)\"]",[54,71124,71125],{"class":57863},".to_numpy",[54,71127,330],{"class":132},[54,71129,71130,71133],{"class":56,"line":61050},[54,71131,71132],{"class":57863},"lower_error = y - summary_df",[54,71134,71135],{"class":132},"[\"CI 95% low\"]\n",[54,71137,71138,71141,71144],{"class":56,"line":61061},[54,71139,71140],{"class":57863},"upper_error = summary_df",[54,71142,71143],{"class":132},"[\"CI 95% high\"]",[54,71145,71146],{"class":57863}," - y\n",[54,71148,71149,71152],{"class":56,"line":61070},[54,71150,71151],{"class":57863},"yerr = np.vstack",[54,71153,71154],{"class":132},"([np.abs(lower_error), np.abs(upper_error)])\n",[54,71156,71157],{"class":56,"line":61081},[54,71158,105],{"emptyLinePlaceholder":104},[54,71160,71161,71164,71166,71169,71172,71175],{"class":56,"line":61086},[54,71162,71163],{"class":57863},"ax1.bar",[54,71165,129],{"class":132},[54,71167,71168],{"class":57863},"summary_df",[54,71170,71171],{"class":132},"[\"Model\"]",[54,71173,71174],{"class":57863},", y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black'",[54,71176,136],{"class":132},[54,71178,71179,71182,71184,71187,71189,71192,71194,71197],{"class":56,"line":61091},[54,71180,71181],{"class":57863},"ax1.set_title",[54,71183,129],{"class":132},[54,71185,71186],{"class":57863},"\"Observed Success Rate ",[54,71188,129],{"class":132},[54,71190,71191],{"class":57863},"95% Wilson CI",[54,71193,3205],{"class":132},[54,71195,71196],{"class":57863},"\", fontsize=14",[54,71198,136],{"class":132},[54,71200,71201,71204,71206,71209],{"class":56,"line":61101},[54,71202,71203],{"class":57863},"ax1.tick_params",[54,71205,129],{"class":132},[54,71207,71208],{"class":57863},"axis='x', rotation=30, labelsize=10",[54,71210,136],{"class":132},[54,71212,71213],{"class":56,"line":61107},[54,71214,105],{"emptyLinePlaceholder":104},[54,71216,71217],{"class":56,"line":61121},[54,71218,71219],{"class":111},"# Plot 2\n",[54,71221,71222],{"class":56,"line":61126},[54,71223,71224],{"class":57863},"width = 0.2\n",[54,71226,71227,71230,71232,71234,71236,71239],{"class":56,"line":61136},[54,71228,71229],{"class":57863},"x = np.arange",[54,71231,129],{"class":132},[54,71233,1319],{"class":57863},[54,71235,129],{"class":132},[54,71237,71238],{"class":57863},"MODELS",[54,71240,22096],{"class":132},[54,71242,71243,71246],{"class":56,"line":61141},[54,71244,71245],{"class":57863},"colors = ",[54,71247,71248],{"class":132},"['#ff9999','#66b3ff','#99ff99']\n",[54,71250,71251,71254,71256,71259,71261],{"class":56,"line":61153},[54,71252,71253],{"class":57863},"for i, eps in enumerate",[54,71255,129],{"class":132},[54,71257,71258],{"class":57863},"EPSILONS",[54,71260,3205],{"class":132},[54,71262,1284],{"class":57863},[54,71264,71265,71268],{"class":56,"line":61158},[54,71266,71267],{"class":57863}," req = summary_df",[54,71269,71270],{"class":132},"[f\"Shots for ±{eps:.1%}\"]\n",[54,71272,71273,71276,71278,71280,71282,71284,71286,71289,71292,71295],{"class":56,"line":61163},[54,71274,71275],{"class":57863}," ax2.bar",[54,71277,129],{"class":132},[54,71279,69302],{"class":57863},[54,71281,1663],{"class":132},[54,71283,7802],{"class":57863},[54,71285,1947],{"class":132},[54,71287,71288],{"class":57863}," width, req, width, label=f\"±{eps:.1%}\", color=colors",[54,71290,71291],{"class":132},"[i]",[54,71293,71294],{"class":57863},", edgecolor='black'",[54,71296,136],{"class":132},[54,71298,71299],{"class":56,"line":61186},[54,71300,105],{"emptyLinePlaceholder":104},[54,71302,71303,71306,71308,71310,71312,71315],{"class":56,"line":61191},[54,71304,71305],{"class":57863},"ax2.set_xticks",[54,71307,129],{"class":132},[54,71309,69302],{"class":57863},[54,71311,1663],{"class":132},[54,71313,71314],{"class":57863}," width, summary_df",[54,71316,71317],{"class":132},"[\"Model\"])\n",[54,71319,71320,71323,71325,71328,71330,71333,71335,71337],{"class":56,"line":61214},[54,71321,71322],{"class":57863},"ax2.set_ylabel",[54,71324,129],{"class":132},[54,71326,71327],{"class":57863},"\"Required Shots ",[54,71329,129],{"class":132},[54,71331,71332],{"class":57863},"Log Scale",[54,71334,3205],{"class":132},[54,71336,21837],{"class":57863},[54,71338,136],{"class":132},[54,71340,71341,71344,71346,71348],{"class":56,"line":61225},[54,71342,71343],{"class":57863},"ax2.set_yscale",[54,71345,129],{"class":132},[54,71347,69933],{"class":57863},[54,71349,136],{"class":132},[54,71351,71352,71355,71357,71360],{"class":56,"line":61264},[54,71353,71354],{"class":57863},"ax2.set_title",[54,71356,129],{"class":132},[54,71358,71359],{"class":57863},"\"Shots Needed for ±ε Tolerance\", fontsize=14",[54,71361,136],{"class":132},[54,71363,71364,71367,71369,71371],{"class":56,"line":61274},[54,71365,71366],{"class":57863},"ax2.tick_params",[54,71368,129],{"class":132},[54,71370,71208],{"class":57863},[54,71372,136],{"class":132},[54,71374,71375,71378,71380,71383],{"class":56,"line":61286},[54,71376,71377],{"class":57863},"ax2.legend",[54,71379,129],{"class":132},[54,71381,71382],{"class":57863},"title=\"Tolerance\"",[54,71384,136],{"class":132},[54,71386,71387],{"class":56,"line":61306},[54,71388,105],{"emptyLinePlaceholder":104},[54,71390,71391,71394],{"class":56,"line":61324},[54,71392,71393],{"class":57863},"plt.tight_layout",[54,71395,330],{"class":132},[54,71397,71398,71401],{"class":56,"line":61329},[54,71399,71400],{"class":57863},"plt.show",[54,71402,330],{"class":132},[54,71404,71405],{"class":56,"line":61334},[54,71406,70217],{"class":57863},[54,71408,71409],{"class":56,"line":61346},[54,71410,105],{"emptyLinePlaceholder":104},[54,71412,71413,71416,71418,71421,71424],{"class":56,"line":61356},[54,71414,71415],{"class":64},"nb ",[54,71417,122],{"class":121},[54,71419,71420],{"class":64}," nbf.v4.",[54,71422,71423],{"class":125},"new_notebook",[54,71425,330],{"class":64},[54,71427,71428,71431,71434,71436,71438],{"class":56,"line":61366},[54,71429,71430],{"class":64},"nb[",[54,71432,71433],{"class":279},"\"cells\"",[54,71435,7762],{"class":64},[54,71437,122],{"class":121},[54,71439,1733],{"class":64},[54,71441,71442,71445,71448],{"class":56,"line":61371},[54,71443,71444],{"class":64}," nbf.v4.",[54,71446,71447],{"class":125},"new_markdown_cell",[54,71449,71450],{"class":64},"(md_intro),\n",[54,71452,71453,71455,71458],{"class":56,"line":61376},[54,71454,71444],{"class":64},[54,71456,71457],{"class":125},"new_code_cell",[54,71459,71460],{"class":64},"(code_config),\n",[54,71462,71463,71465,71467],{"class":56,"line":61384},[54,71464,71444],{"class":64},[54,71466,71457],{"class":125},[54,71468,71469],{"class":64},"(code_lib_and_run),\n",[54,71471,71472],{"class":56,"line":61394},[54,71473,1350],{"class":64},[54,71475,71476],{"class":56,"line":61409},[54,71477,105],{"emptyLinePlaceholder":104},[54,71479,71480,71483,71485],{"class":56,"line":61420},[54,71481,71482],{"class":64},"nb_path ",[54,71484,122],{"class":121},[54,71486,71487],{"class":279}," \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\n",[54,71489,71490,71492,71494,71497,71499,71501,71504,71506,71509,71511,71513],{"class":56,"line":61432},[54,71491,69183],{"class":60},[54,71493,69186],{"class":121},[54,71495,71496],{"class":64},"(nb_path, ",[54,71498,69196],{"class":279},[54,71500,171],{"class":64},[54,71502,71503],{"class":1711},"encoding",[54,71505,122],{"class":121},[54,71507,71508],{"class":279},"\"utf-8\"",[54,71510,204],{"class":64},[54,71512,68],{"class":60},[54,71514,69203],{"class":64},[54,71516,71517,71520,71523],{"class":56,"line":61437},[54,71518,71519],{"class":64}," nbf.",[54,71521,71522],{"class":125},"write",[54,71524,71525],{"class":64},"(nb, f)\n",[54,71527,71528],{"class":56,"line":61444},[54,71529,105],{"emptyLinePlaceholder":104},[54,71531,71532],{"class":56,"line":61453},[54,71533,71534],{"class":111},"# ---------- Final confirmation ----------\n",[54,71536,71537,71539,71541,71543,71545,71547,71549,71552,71554,71556],{"class":56,"line":61467},[54,71538,8081],{"class":121},[54,71540,129],{"class":64},[54,71542,21837],{"class":279},[54,71544,61280],{"class":121},[54,71546,21837],{"class":279},[54,71548,7869],{"class":121},[54,71550,71551],{"class":279}," \"=\"",[54,71553,1947],{"class":121},[54,71555,48537],{"class":132},[54,71557,136],{"class":64},[54,71559,71560,71562,71564,71567],{"class":56,"line":61478},[54,71561,8081],{"class":121},[54,71563,129],{"class":64},[54,71565,71566],{"class":279},"\"SCRIPT COMPLETE\"",[54,71568,136],{"class":64},[54,71570,71571,71573,71575,71578,71580,71582],{"class":56,"line":61490},[54,71572,8081],{"class":121},[54,71574,129],{"class":64},[54,71576,71577],{"class":279},"\"=\"",[54,71579,1947],{"class":121},[54,71581,48537],{"class":132},[54,71583,136],{"class":64},[54,71585,71586,71588,71590,71592,71595,71597,71600,71602,71604],{"class":56,"line":61495},[54,71587,8081],{"class":121},[54,71589,129],{"class":64},[54,71591,21834],{"class":60},[54,71593,71594],{"class":279},"\"Google Colab notebook saved to: ",[54,71596,2366],{"class":132},[54,71598,71599],{"class":64},"nb_path",[54,71601,2428],{"class":132},[54,71603,21837],{"class":279},[54,71605,136],{"class":64},[54,71607,71608,71610,71612,71614,71617,71619,71622,71624,71626],{"class":56,"line":61500},[54,71609,8081],{"class":121},[54,71611,129],{"class":64},[54,71613,21834],{"class":60},[54,71615,71616],{"class":279},"\"Summary data saved to: ",[54,71618,2366],{"class":132},[54,71620,71621],{"class":64},"summary_csv_path",[54,71623,2428],{"class":132},[54,71625,21837],{"class":279},[54,71627,136],{"class":64},[54,71629,71630,71632,71634,71636,71639,71641,71644,71646,71648,71650,71653,71655,71657],{"class":56,"line":61510},[54,71631,8081],{"class":121},[54,71633,129],{"class":64},[54,71635,21834],{"class":60},[54,71637,71638],{"class":279},"\"Plots saved to: ",[54,71640,2366],{"class":132},[54,71642,71643],{"class":64},"fig1_path",[54,71645,2428],{"class":132},[54,71647,171],{"class":279},[54,71649,2366],{"class":132},[54,71651,71652],{"class":64},"fig2_path",[54,71654,2428],{"class":132},[54,71656,21837],{"class":279},[54,71658,136],{"class":64},[44,71660,71663],{"className":71661,"code":71662,"language":6595,"meta":49},[8087],"Running simulation...\nSaved data to CSV and JSON files.\n",[51,71664,71662],{"__ignoreMap":49},[380,71666],{"alt":57940,"src":71667},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.webp",[380,71669],{"alt":62846,"src":71670},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-02.webp",[44,71672,71675],{"className":71673,"code":71674,"language":6595,"meta":49},[8087],"Generated and saved plots.\n",[51,71676,71674],{"__ignoreMap":49},[44,71678,71681],{"className":71679,"code":71680,"language":6595,"meta":49},[8087],"==================================================\nSCRIPT COMPLETE\n==================================================\nGoogle Colab notebook saved to: Sampling_Budget_Qubit_Bitstrings.ipynb\nSummary data saved to: summary.csv\nPlots saved to: fig_ones_fraction.png, fig_required_shots.png\n",[51,71682,71680],{"__ignoreMap":49},[71684,71685],"hr",{},[31,71687,41113],{"id":41112},[26927,71689,71690,71697,71704,71711,71718,71725,71732,71739,71746,71753,71760,71767,71774,71785,71792,71799],{},[659,71691,71692,71693,71696],{},"Agresti, A. and Coull, B.A. (1998) 'Approximate is better than “exact” for interval estimation of binomial proportions', ",[585,71694,71695],{},"The American Statistician",", 52(2), pp. 119-126.",[659,71698,71699,71700,71703],{},"Brown, L.D., Cai, T.T. and DasGupta, A. (2001) 'Interval Estimation for a Binomial Proportion', ",[585,71701,71702],{},"Statistical Science",", 16(2), pp. 101-133.",[659,71705,71706,71707,71710],{},"Cochran, W.G. (1977) ",[585,71708,71709],{},"Sampling Techniques",". 3rd edn. New York: John Wiley & Sons.",[659,71712,71713,71714,71717],{},"Gottesman, D. (2009) 'An introduction to quantum error correction and fault-tolerant quantum computation', in ",[585,71715,71716],{},"Quantum Information Science and Its Contributions to Mathematics, Proceedings of Symposia in Applied Mathematics",", 72, pp. 13-58.",[659,71719,71720,71721,71724],{},"Harris, C.R., Millman, K.J., van der Walt, S.J., Gommers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N.J., Kern, R., Picus, M., Hoyer, S., van Kerkwijk, M.H., Brett, M., Haldane, A., del Río, J.F., Wiebe, M., Peterson, P., Gérard-Marchant, P., Sheppard, K., Reddy, T., Weckesser, W., Abbasi, H., Gohlke, C. and Oliphant, T.E. (2020) 'Array programming with NumPy', ",[585,71722,71723],{},"Nature",", 585, pp. 357–362.",[659,71726,71727,71728,71731],{},"Hunter, J.D. (2007) 'Matplotlib: A 2D Graphics Environment', ",[585,71729,71730],{},"Computing in Science & Engineering",", 9(3), pp. 90-95.",[659,71733,71734,71735,71738],{},"Kitaev, A.Y. (2003) 'Fault-tolerant quantum computation by anyons', ",[585,71736,71737],{},"Annals of Physics",", 303(1), pp. 2-30.",[659,71740,71741,71742,71745],{},"McKinney, W. (2010) 'Data Structures for Statistical Computing in Python', in ",[585,71743,71744],{},"Proceedings of the 9th Python in Science Conference",", pp. 56-61.",[659,71747,71748,71749,71752],{},"Nayak, C., Simon, S.H., Stern, A., Freedman, M. and Das Sarma, S. (2008) 'Non-Abelian anyons and topological quantum computation', ",[585,71750,71751],{},"Reviews of Modern Physics",", 80(3), pp. 1083-1159.",[659,71754,71755,71756,71759],{},"Newman, M. (2013) ",[585,71757,71758],{},"Computational Physics",". CreateSpace Independent Publishing Platform.",[659,71761,71762,71763,71766],{},"Nielsen, M.A. and Chuang, I.L. (2010) ",[585,71764,71765],{},"Quantum Computation and Quantum Information: 10th Anniversary Edition",". Cambridge: Cambridge University Press.",[659,71768,71769,71770,71773],{},"Preskill, J. (2018) 'Quantum Computing in the NISQ era and beyond', ",[585,71771,71772],{},"Quantum",", 2, p. 79.",[659,71775,71776,71777,71780,71781,29],{},"The pandas development team (2020) ",[585,71778,71779],{},"pandas-dev\u002Fpandas: Pandas",". Zenodo. Available at: ",[20,71782,71784],{"href":71783},"https:\u002F\u002Fwww.google.com\u002Fsearch?q=https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134","https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134",[659,71786,71787,71788,71791],{},"Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Millman, K.J., Mayorov, N., Nelson, A.R.J., Jones, E., Kern, R., Larson, E., Carey, C.J., Polat, İ., Feng, Y., Moore, E.W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E.A., Harris, C.R., Archibald, A.M., Ribeiro, A.H., Pedregosa, F., van Mulbregt, P. and SciPy 1.0 Contributors (2020) 'SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python', ",[585,71789,71790],{},"Nature Methods",", 17, pp. 261–272.",[659,71793,71794,71795,71798],{},"Wilde, M.M. (2017) ",[585,71796,71797],{},"Quantum Information Theory",". 2nd edn. Cambridge: Cambridge University Press.",[659,71800,71801,71802,71805],{},"Wilson, E.B. (1927) 'Probable inference, the law of succession, and statistical inference', ",[585,71803,71804],{},"Journal of the American Statistical Association",", 22(158), pp. 209-212.",[31,71807,489],{"id":488},[13,71809,41155,71810,496,71812,29],{},[20,71811,41227],{"href":67243},[20,71813,67191],{"href":71814},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[502,71816,71817],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .sU0A5, html code.shiki .sU0A5{--shiki-default:#E5C07B}html pre.shiki code .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":49,"searchDepth":74,"depth":74,"links":71819},[71820,71821],{"id":41112,"depth":74,"text":41113},{"id":488,"depth":74,"text":489},[518,519,520,71823],"Sampling budget",[71825],{"username":8,"name":524,"role":525,"bio":526,"links":71826},[71827,71828],{"label":529,"href":530},{"label":532,"href":67243},{"username":8,"name":524,"role":534},"A Sampling-Budget & Bit-String Analogy for Qubit Classifications. A notebook by Onri Jay Benally, republished with permission.",{},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications","4 min read",[],{"title":67233,"description":71830},"blog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications",[67229,573],"HETemA9RSbsiqyryZFFLOYbrY0iNodc-iLIt3qtwByA",1785631553474]