[{"data":1,"prerenderedAt":72180},["ShallowReactive",2],{"blog-post-expert-notes\u002F2-qubit-state-visualization":3,"sibling-dives-expert-notes\u002F2-qubit-state-visualization":574,"learn-track-expert-notes\u002F2-qubit-state-visualization":72179},{"id":4,"title":5,"authors":6,"body":8,"breadcrumb":516,"builders":521,"byline":532,"challenge":534,"courseAuthor":534,"courseLead":534,"dek":535,"description":536,"draft":537,"extension":538,"eyebrow":539,"finish":534,"fork":534,"hero":534,"heroAlt":534,"heroCta":534,"heroImage":534,"kind":540,"lessonCount":534,"meta":541,"navigation":103,"newsItems":534,"next":534,"ogImage":542,"order":534,"outcomes":534,"path":543,"publishDate":544,"readingTime":545,"related":546,"relatedProjects":547,"seo":567,"stem":569,"tags":570,"track":534,"trackName":534,"__hash__":573},"blog\u002Fblog\u002Fexpert-notes\u002F2-qubit-state-visualization.md","2-qubit states, visualized 8 ways",[7],"onri-jay-benally",{"type":9,"value":10,"toc":504},"minimark",[11,15,29,34,37,368,371,375,378,385,388,394,398,401,407,411,414,420,424,427,433,437,440,444,447,453,457,460,466,470,473,479,485,489,500],[12,13,14],"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.",[12,16,17,18,23,24,28],{},"Everything below comes from Onri's runnable notebook, shared openly on ",[19,20,22],"a",{"href":21},"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 ",[19,25,27],{"href":26},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002F2_Qubit_State_Visualization_in_Python.ipynb","Google Colab",".",[30,31,33],"h2",{"id":32},"one-circuit-two-states","One circuit, two states",[12,35,36],{},"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:",[38,39,42],"code-block",{"name":40,"tag":41},"2_qubit_state_visualization.ipynb","Python",[43,44,49],"pre",{"className":45,"code":46,"language":47,"meta":48,"style":48},"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","",[50,51,52,71,85,98,105,112,136,156,180,207,224,229,243,256,261,267,282,294,307,321,330,342,355],"code",{"__ignoreMap":48},[53,54,57,61,65,68],"span",{"class":55,"line":56},"line",1,[53,58,60],{"class":59},"seHd6","import",[53,62,64],{"class":63},"sn6KH"," numpy ",[53,66,67],{"class":59},"as",[53,69,70],{"class":63}," np\n",[53,72,74,77,80,82],{"class":55,"line":73},2,[53,75,76],{"class":59},"from",[53,78,79],{"class":63}," qiskit ",[53,81,60],{"class":59},[53,83,84],{"class":63}," QuantumCircuit\n",[53,86,88,90,93,95],{"class":55,"line":87},3,[53,89,76],{"class":59},[53,91,92],{"class":63}," qiskit.quantum_info ",[53,94,60],{"class":59},[53,96,97],{"class":63}," Statevector, DensityMatrix\n",[53,99,101],{"class":55,"line":100},4,[53,102,104],{"emptyLinePlaceholder":103},true,"\n",[53,106,108],{"class":55,"line":107},5,[53,109,111],{"class":110},"sV9Aq","# Create a circuit with 2 qubits\n",[53,113,115,118,122,126,129,133],{"class":55,"line":114},6,[53,116,117],{"class":63},"qc ",[53,119,121],{"class":120},"sjrmR","=",[53,123,125],{"class":124},"sVbv2"," QuantumCircuit",[53,127,128],{"class":63},"(",[53,130,132],{"class":131},"sVC51","2",[53,134,135],{"class":63},")\n",[53,137,139,142,145,147,150,153],{"class":55,"line":138},7,[53,140,141],{"class":63},"qc.",[53,143,144],{"class":124},"h",[53,146,128],{"class":63},[53,148,149],{"class":131},"0",[53,151,152],{"class":63},") ",[53,154,155],{"class":110},"# Hadamard on q0\n",[53,157,159,161,164,166,168,171,174,177],{"class":55,"line":158},8,[53,160,141],{"class":63},[53,162,163],{"class":124},"cx",[53,165,128],{"class":63},[53,167,149],{"class":131},[53,169,170],{"class":63},", ",[53,172,173],{"class":131},"1",[53,175,176],{"class":63},") ",[53,178,179],{"class":110},"# CNOT with q0 as control, q1 as target\n",[53,181,183,185,188,191,194,197,199,201,204],{"class":55,"line":182},9,[53,184,141],{"class":63},[53,186,187],{"class":124},"ry",[53,189,190],{"class":63},"(np.pi ",[53,192,193],{"class":120},"\u002F",[53,195,196],{"class":131}," 4",[53,198,170],{"class":63},[53,200,173],{"class":131},[53,202,203],{"class":63},") ",[53,205,206],{"class":110},"# Y-rotation of pi\u002F4 on q1\n",[53,208,210,212,215,217,219,221],{"class":55,"line":209},10,[53,211,141],{"class":63},[53,213,214],{"class":124},"z",[53,216,128],{"class":63},[53,218,173],{"class":131},[53,220,152],{"class":63},[53,222,223],{"class":110},"# Phase flip (Z gate) on q1\n",[53,225,227],{"class":55,"line":226},11,[53,228,104],{"emptyLinePlaceholder":103},[53,230,232,235,237,240],{"class":55,"line":231},12,[53,233,234],{"class":63},"final_statevector ",[53,236,121],{"class":120},[53,238,239],{"class":124}," Statevector",[53,241,242],{"class":63},"(qc)\n",[53,244,246,249,251,254],{"class":55,"line":245},13,[53,247,248],{"class":63},"final_density_matrix ",[53,250,121],{"class":120},[53,252,253],{"class":124}," DensityMatrix",[53,255,242],{"class":63},[53,257,259],{"class":55,"line":258},14,[53,260,104],{"emptyLinePlaceholder":103},[53,262,264],{"class":55,"line":263},15,[53,265,266],{"class":110},"# Generate all 8 plots for the final state\n",[53,268,270,273,276,280],{"class":55,"line":269},16,[53,271,272],{"class":124},"plot_bloch_multivector_viz",[53,274,275],{"class":63},"(final_density_matrix, ",[53,277,279],{"class":278},"subq3","\"Final Entangled State\"",[53,281,135],{"class":63},[53,283,285,288,290,292],{"class":55,"line":284},17,[53,286,287],{"class":124},"plot_q_sphere_viz",[53,289,275],{"class":63},[53,291,279],{"class":278},[53,293,135],{"class":63},[53,295,297,300,302,305],{"class":55,"line":296},18,[53,298,299],{"class":124},"plot_steering_ellipsoid",[53,301,275],{"class":63},[53,303,304],{"class":278},"\"Steering Ellipsoid for Final State\"",[53,306,135],{"class":63},[53,308,310,313,316,319],{"class":55,"line":309},19,[53,311,312],{"class":124},"visualize_schmidt_decomposition",[53,314,315],{"class":63},"(final_statevector, ",[53,317,318],{"class":278},"\"Schmidt Decomposition for Final State\"",[53,320,135],{"class":63},[53,322,324,327],{"class":55,"line":323},20,[53,325,326],{"class":124},"explain_toroidal_geometry",[53,328,329],{"class":63},"()\n",[53,331,333,336,338,340],{"class":55,"line":332},21,[53,334,335],{"class":124},"plot_correlation_invariants",[53,337,275],{"class":63},[53,339,279],{"class":278},[53,341,135],{"class":63},[53,343,345,348,350,353],{"class":55,"line":344},22,[53,346,347],{"class":124},"plot_wigner_function",[53,349,275],{"class":63},[53,351,352],{"class":278},"\"Wigner Functions for Final State\"",[53,354,135],{"class":63},[53,356,358,361,363,366],{"class":55,"line":357},23,[53,359,360],{"class":124},"plot_density_matrix",[53,362,275],{"class":63},[53,364,365],{"class":278},"\"Density Matrix for Final State\"",[53,367,135],{"class":63},[12,369,370],{},"Every method below shows the entangled state; where the contrast with |00⟩ is the whole point, both are shown.",[30,372,374],{"id":373},"_1-two-bloch-spheres-one-giveaway","1. Two Bloch spheres, one giveaway",[12,376,377],{},"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.",[379,380],"blog-figure",{"alt":381,"caption":382,"no":383,"src":384},"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",[12,386,387],{},"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.",[379,389],{"alt":390,"caption":391,"no":392,"src":393},"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",[30,395,397],{"id":396},"_2-the-q-sphere","2. The Q-sphere",[12,399,400],{},"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.",[379,402],{"alt":403,"caption":404,"no":405,"src":406},"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",[30,408,410],{"id":409},"_3-the-steering-ellipsoid","3. The steering ellipsoid",[12,412,413],{},"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.",[379,415],{"alt":416,"caption":417,"no":418,"src":419},"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",[30,421,423],{"id":422},"_4-schmidt-decomposition","4. Schmidt decomposition",[12,425,426],{},"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.",[379,428],{"alt":429,"caption":430,"no":431,"src":432},"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",[30,434,436],{"id":435},"_5-toroidal-geometry","5. Toroidal geometry",[12,438,439],{},"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.",[30,441,443],{"id":442},"_6-correlation-invariants","6. Correlation invariants",[12,445,446],{},"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.",[379,448],{"alt":449,"caption":450,"no":451,"src":452},"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",[30,454,456],{"id":455},"_7-the-wigner-function","7. The Wigner function",[12,458,459],{},"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.",[379,461],{"alt":462,"caption":463,"no":464,"src":465},"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",[30,467,469],{"id":468},"_8-the-density-matrix","8. The density matrix",[12,471,472],{},"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.",[379,474],{"alt":475,"caption":476,"no":477,"src":478},"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",[379,480],{"alt":481,"caption":482,"no":483,"src":484},"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",[30,486,488],{"id":487},"run-it-yourself","Run it yourself",[12,490,491,492,495,496,499],{},"Every figure above comes out of one script you can run top to bottom: ",[19,493,494],{"href":21},"open the notebook on GitHub"," or ",[19,497,498],{"href":26},"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.",[501,502,503],"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":48,"searchDepth":73,"depth":73,"links":505},[506,507,508,509,510,511,512,513,514,515],{"id":32,"depth":73,"text":33},{"id":373,"depth":73,"text":374},{"id":396,"depth":73,"text":397},{"id":409,"depth":73,"text":410},{"id":422,"depth":73,"text":423},{"id":435,"depth":73,"text":436},{"id":442,"depth":73,"text":443},{"id":455,"depth":73,"text":456},{"id":468,"depth":73,"text":469},{"id":487,"depth":73,"text":488},[517,518,519,520],"Qollab","Blog","Expert notes","2-qubit states",[522],{"username":7,"name":523,"role":524,"bio":525,"links":526},"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.",[527,530],{"label":528,"href":529},"GitHub ↗","https:\u002F\u002Fgithub.com\u002FOJB-Quantum",{"label":531,"href":21},"Original notebook ↗",{"username":7,"name":523,"role":533},"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",[],[548,555,562],{"username":549,"project":550,"title":551,"category":552,"thumb":553,"to":554},"q-inho","qave","QAVE","Visualization","\u002F_content\u002Fimages\u002Fqave\u002Fthumbnail.webp","\u002Fexplore\u002Fqave",{"username":556,"project":557,"title":558,"category":559,"thumb":560,"to":561},"AmberPincar","quantum-garden","Quantum Garden","Art","\u002F_content\u002Fimages\u002Fquantum-garden\u002Fthumbnail.webp","\u002Fexplore\u002Fquantum-garden",{"username":563,"project":564,"title":565,"category":559,"thumb":566},"xinyi","quantum-butterfly-field","Quantum Butterfly Field","\u002F_content\u002Fimages\u002Fquantum-butterfly-field\u002Fthumbnail.webp",{"description":568,"title":5},"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",[571,572,47],"visualization","quantum","hbpIuxS-SqH0GmOc1n1ds2B-L8OsTlp8DNHQ4vd3Ylw",[575,914,41553,67570],{"id":4,"title":5,"authors":576,"body":577,"breadcrumb":899,"builders":900,"byline":905,"challenge":534,"courseAuthor":534,"courseLead":534,"dek":535,"description":536,"draft":537,"extension":538,"eyebrow":539,"finish":534,"fork":534,"hero":534,"heroAlt":534,"heroCta":534,"heroImage":534,"kind":540,"lessonCount":534,"meta":906,"navigation":103,"newsItems":534,"next":534,"ogImage":542,"order":534,"outcomes":534,"path":543,"publishDate":544,"readingTime":545,"related":907,"relatedProjects":908,"seo":912,"stem":569,"tags":913,"track":534,"trackName":534,"__hash__":573},[7],{"type":9,"value":578,"toc":887},[579,581,587,589,591,823,825,827,829,831,833,835,837,839,841,843,845,847,849,851,853,855,857,859,861,863,865,867,869,871,873,875,877,879,885],[12,580,14],{},[12,582,17,583,23,585,28],{},[19,584,22],{"href":21},[19,586,27],{"href":26},[30,588,33],{"id":32},[12,590,36],{},[38,592,593],{"name":40,"tag":41},[43,594,595],{"className":45,"code":46,"language":47,"meta":48,"style":48},[50,596,597,607,617,627,631,635,649,663,681,701,715,719,729,739,743,747,757,767,777,787,793,803,813],{"__ignoreMap":48},[53,598,599,601,603,605],{"class":55,"line":56},[53,600,60],{"class":59},[53,602,64],{"class":63},[53,604,67],{"class":59},[53,606,70],{"class":63},[53,608,609,611,613,615],{"class":55,"line":73},[53,610,76],{"class":59},[53,612,79],{"class":63},[53,614,60],{"class":59},[53,616,84],{"class":63},[53,618,619,621,623,625],{"class":55,"line":87},[53,620,76],{"class":59},[53,622,92],{"class":63},[53,624,60],{"class":59},[53,626,97],{"class":63},[53,628,629],{"class":55,"line":100},[53,630,104],{"emptyLinePlaceholder":103},[53,632,633],{"class":55,"line":107},[53,634,111],{"class":110},[53,636,637,639,641,643,645,647],{"class":55,"line":114},[53,638,117],{"class":63},[53,640,121],{"class":120},[53,642,125],{"class":124},[53,644,128],{"class":63},[53,646,132],{"class":131},[53,648,135],{"class":63},[53,650,651,653,655,657,659,661],{"class":55,"line":138},[53,652,141],{"class":63},[53,654,144],{"class":124},[53,656,128],{"class":63},[53,658,149],{"class":131},[53,660,152],{"class":63},[53,662,155],{"class":110},[53,664,665,667,669,671,673,675,677,679],{"class":55,"line":158},[53,666,141],{"class":63},[53,668,163],{"class":124},[53,670,128],{"class":63},[53,672,149],{"class":131},[53,674,170],{"class":63},[53,676,173],{"class":131},[53,678,176],{"class":63},[53,680,179],{"class":110},[53,682,683,685,687,689,691,693,695,697,699],{"class":55,"line":182},[53,684,141],{"class":63},[53,686,187],{"class":124},[53,688,190],{"class":63},[53,690,193],{"class":120},[53,692,196],{"class":131},[53,694,170],{"class":63},[53,696,173],{"class":131},[53,698,203],{"class":63},[53,700,206],{"class":110},[53,702,703,705,707,709,711,713],{"class":55,"line":209},[53,704,141],{"class":63},[53,706,214],{"class":124},[53,708,128],{"class":63},[53,710,173],{"class":131},[53,712,152],{"class":63},[53,714,223],{"class":110},[53,716,717],{"class":55,"line":226},[53,718,104],{"emptyLinePlaceholder":103},[53,720,721,723,725,727],{"class":55,"line":231},[53,722,234],{"class":63},[53,724,121],{"class":120},[53,726,239],{"class":124},[53,728,242],{"class":63},[53,730,731,733,735,737],{"class":55,"line":245},[53,732,248],{"class":63},[53,734,121],{"class":120},[53,736,253],{"class":124},[53,738,242],{"class":63},[53,740,741],{"class":55,"line":258},[53,742,104],{"emptyLinePlaceholder":103},[53,744,745],{"class":55,"line":263},[53,746,266],{"class":110},[53,748,749,751,753,755],{"class":55,"line":269},[53,750,272],{"class":124},[53,752,275],{"class":63},[53,754,279],{"class":278},[53,756,135],{"class":63},[53,758,759,761,763,765],{"class":55,"line":284},[53,760,287],{"class":124},[53,762,275],{"class":63},[53,764,279],{"class":278},[53,766,135],{"class":63},[53,768,769,771,773,775],{"class":55,"line":296},[53,770,299],{"class":124},[53,772,275],{"class":63},[53,774,304],{"class":278},[53,776,135],{"class":63},[53,778,779,781,783,785],{"class":55,"line":309},[53,780,312],{"class":124},[53,782,315],{"class":63},[53,784,318],{"class":278},[53,786,135],{"class":63},[53,788,789,791],{"class":55,"line":323},[53,790,326],{"class":124},[53,792,329],{"class":63},[53,794,795,797,799,801],{"class":55,"line":332},[53,796,335],{"class":124},[53,798,275],{"class":63},[53,800,279],{"class":278},[53,802,135],{"class":63},[53,804,805,807,809,811],{"class":55,"line":344},[53,806,347],{"class":124},[53,808,275],{"class":63},[53,810,352],{"class":278},[53,812,135],{"class":63},[53,814,815,817,819,821],{"class":55,"line":357},[53,816,360],{"class":124},[53,818,275],{"class":63},[53,820,365],{"class":278},[53,822,135],{"class":63},[12,824,370],{},[30,826,374],{"id":373},[12,828,377],{},[379,830],{"alt":381,"caption":382,"no":383,"src":384},[12,832,387],{},[379,834],{"alt":390,"caption":391,"no":392,"src":393},[30,8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Quantum Diagonalization of Lattice Hamiltonians",[918],"mirko-amico",{"type":9,"value":920,"toc":41500},[921,932,937,952,956,959,987,991,994,1012,1016,2403,2407,2412,2676,2987,3654,3974,3977,3981,4285,4546,4817,5021,5800,6037,6040,6278,6600,6607,7554,7559,8045,8423,8430,8434,8532,8868,8874,8877,8925,8929,9011,9077,9081,9085,9193,9317,9323,9327,9333,10604,11433,11983,11986,12974,13366,13613,14023,14029,14032,14035,14102,14108,14112,14116,14119,14125,14769,17536,17744,18963,20197,20242,20246,20794,21226,21229,21233,21323,21326,21451,21454,21512,21737,21740,21795,21801,21804,21808,21811,21918,21922,21925,22314,22317,22532,22538,22542,23010,23014,23069,23399,23405,23497,23870,23874,23899,23903,23943,24033,24064,24390,24403,24466,24805,24817,24881,25053,25137,25230,25236,25239,25256,25262,25511,25515,25518,25892,26018,26614,26646,27111,27193,27199,27265,27982,28085,28348,28602,28933,29116,29458,30198,30512,32127,32159,33655,34132,34826,35609,35943,36270,36332,36341,36345,36544,36902,36905,37433,37436,37736,37792,38056,38549,38808,39750,40754,41194,41449,41453,41462,41467,41472,41477,41481,41484,41490,41492,41497],[12,922,923],{},[924,925,926,927,931],"em",{},"Republished with the author's permission from ",[19,928,930],{"href":929},"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.",[12,933,934],{},[924,935,936],{},"Usage estimate: 20 minutes on a Heron r2 (NOTE: This is an estimate only. Your runtime might vary.)",[43,938,940],{"className":45,"code":939,"language":47,"meta":48,"style":48},"# This cell is hidden from users – it disables some lint rules\n# ruff: noqa: E402 E722 F601\n",[50,941,942,947],{"__ignoreMap":48},[53,943,944],{"class":55,"line":56},[53,945,946],{"class":110},"# This cell is hidden from users – it disables some lint rules\n",[53,948,949],{"class":55,"line":73},[53,950,951],{"class":110},"# ruff: noqa: E402 E722 F601\n",[30,953,955],{"id":954},"background","Background",[12,957,958],{},"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.",[12,960,961,962,966,967,971,972,976,977,981,982,986],{},"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, ",[19,963,965],{"href":964},"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, ",[19,968,970],{"href":969},"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 ",[19,973,975],{"href":974},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2312.00178","review paper","), the Krylov quantum diagonalization (KQD) algorithm. This algorithm performs well at large scale ",[19,978,980],{"href":979},"#references","[1]"," on existing quantum hardware, shares similar ",[19,983,985],{"href":984},"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.",[30,988,990],{"id":989},"requirements","Requirements",[12,992,993],{},"Before starting this tutorial, be sure you have the following installed:",[995,996,997,1005],"ul",{},[998,999,1000,1001,1004],"li",{},"Qiskit SDK v2.0 or later, with ",[19,1002,571],{"href":1003},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fvisualization"," support",[998,1006,1007,1008,1011],{},"Qiskit Runtime v0.22 or later ( ",[50,1009,1010],{},"pip install qiskit-ibm-runtime"," )",[30,1013,1015],{"id":1014},"setup","Setup",[43,1017,1019],{"className":45,"code":1018,"language":47,"meta":48,"style":48},"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",[50,1020,1021,1031,1043,1055,1067,1079,1086,1098,1105,1109,1124,1128,1139,1151,1162,1174,1186,1198,1210,1214,1218,1230,1235,1245,1250,1255,1260,1272,1282,1290,1305,1324,1342,1358,1364,1370,1375,1381,1387,1393,1399,1405,1411,1417,1423,1429,1435,1441,1446,1452,1458,1463,1468,1473,1490,1501,1517,1550,1556,1584,1607,1624,1633,1662,1670,1690,1698,1705,1725,1732,1748,1753,1758,1779,1784,1790,1795,1806,1826,1866,1871,1881,1896,1929,1934,1945,1950,1964,1969,1980,2008,2027,2032,2057,2062,2073,2117,2126,2131,2142,2157,2172,2183,2188,2214,2230,2249,2289,2312,2318,2326,2335,2340,2356,2361,2381,2394],{"__ignoreMap":48},[53,1022,1023,1025,1027,1029],{"class":55,"line":56},[53,1024,60],{"class":59},[53,1026,64],{"class":63},[53,1028,67],{"class":59},[53,1030,70],{"class":63},[53,1032,1033,1035,1038,1040],{"class":55,"line":73},[53,1034,60],{"class":59},[53,1036,1037],{"class":63}," scipy ",[53,1039,67],{"class":59},[53,1041,1042],{"class":63}," sp\n",[53,1044,1045,1047,1050,1052],{"class":55,"line":87},[53,1046,60],{"class":59},[53,1048,1049],{"class":63}," matplotlib.pylab ",[53,1051,67],{"class":59},[53,1053,1054],{"class":63}," plt\n",[53,1056,1057,1059,1062,1064],{"class":55,"line":100},[53,1058,76],{"class":59},[53,1060,1061],{"class":63}," typing ",[53,1063,60],{"class":59},[53,1065,1066],{"class":63}," Union, List\n",[53,1068,1069,1071,1074,1076],{"class":55,"line":107},[53,1070,60],{"class":59},[53,1072,1073],{"class":63}," itertools ",[53,1075,67],{"class":59},[53,1077,1078],{"class":63}," it\n",[53,1080,1081,1083],{"class":55,"line":114},[53,1082,60],{"class":59},[53,1084,1085],{"class":63}," copy\n",[53,1087,1088,1090,1093,1095],{"class":55,"line":138},[53,1089,76],{"class":59},[53,1091,1092],{"class":63}," sympy ",[53,1094,60],{"class":59},[53,1096,1097],{"class":63}," Matrix\n",[53,1099,1100,1102],{"class":55,"line":158},[53,1101,60],{"class":59},[53,1103,1104],{"class":63}," warnings\n",[53,1106,1107],{"class":55,"line":182},[53,1108,104],{"emptyLinePlaceholder":103},[53,1110,1111,1114,1117,1119,1122],{"class":55,"line":209},[53,1112,1113],{"class":63},"warnings.",[53,1115,1116],{"class":124},"filterwarnings",[53,1118,128],{"class":63},[53,1120,1121],{"class":278},"\"ignore\"",[53,1123,135],{"class":63},[53,1125,1126],{"class":55,"line":226},[53,1127,104],{"emptyLinePlaceholder":103},[53,1129,1130,1132,1134,1136],{"class":55,"line":231},[53,1131,76],{"class":59},[53,1133,92],{"class":63},[53,1135,60],{"class":59},[53,1137,1138],{"class":63}," SparsePauliOp, Pauli, StabilizerState\n",[53,1140,1141,1143,1146,1148],{"class":55,"line":245},[53,1142,76],{"class":59},[53,1144,1145],{"class":63}," qiskit.circuit ",[53,1147,60],{"class":59},[53,1149,1150],{"class":63}," Parameter, IfElseOp\n",[53,1152,1153,1155,1157,1159],{"class":55,"line":258},[53,1154,76],{"class":59},[53,1156,79],{"class":63},[53,1158,60],{"class":59},[53,1160,1161],{"class":63}," 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generate_preset_pass_manager\n",[53,1211,1212],{"class":55,"line":309},[53,1213,104],{"emptyLinePlaceholder":103},[53,1215,1216],{"class":55,"line":323},[53,1217,104],{"emptyLinePlaceholder":103},[53,1219,1220,1222,1225,1227],{"class":55,"line":332},[53,1221,76],{"class":59},[53,1223,1224],{"class":63}," qiskit_ibm_runtime ",[53,1226,60],{"class":59},[53,1228,1229],{"class":63}," (\n",[53,1231,1232],{"class":55,"line":344},[53,1233,1234],{"class":63}," QiskitRuntimeService,\n",[53,1236,1237,1240,1242],{"class":55,"line":357},[53,1238,1239],{"class":63}," EstimatorV2 ",[53,1241,67],{"class":59},[53,1243,1244],{"class":63}," Estimator,\n",[53,1246,1248],{"class":55,"line":1247},24,[53,1249,135],{"class":63},[53,1251,1253],{"class":55,"line":1252},25,[53,1254,104],{"emptyLinePlaceholder":103},[53,1256,1258],{"class":55,"line":1257},26,[53,1259,104],{"emptyLinePlaceholder":103},[53,1261,1263,1266,1269],{"class":55,"line":1262},27,[53,1264,1265],{"class":59},"def",[53,1267,1268],{"class":124}," solve_regularized_gen_eig",[53,1270,1271],{"class":63},"(\n",[53,1273,1275,1279],{"class":55,"line":1274},28,[53,1276,1278],{"class":1277},"sb9H8"," h",[53,1280,1281],{"class":63},": np.ndarray,\n",[53,1283,1285,1288],{"class":55,"line":1284},29,[53,1286,1287],{"class":1277}," s",[53,1289,1281],{"class":63},[53,1291,1293,1296,1299,1302],{"class":55,"line":1292},30,[53,1294,1295],{"class":1277}," threshold",[53,1297,1298],{"class":63},": ",[53,1300,1301],{"class":120},"float",[53,1303,1304],{"class":63},",\n",[53,1306,1308,1311,1313,1316,1319,1322],{"class":55,"line":1307},31,[53,1309,1310],{"class":1277}," k",[53,1312,1298],{"class":63},[53,1314,1315],{"class":120},"int",[53,1317,1318],{"class":120}," =",[53,1320,1321],{"class":131}," 1",[53,1323,1304],{"class":63},[53,1325,1327,1330,1332,1335,1337,1340],{"class":55,"line":1326},32,[53,1328,1329],{"class":1277}," return_dimn",[53,1331,1298],{"class":63},[53,1333,1334],{"class":120},"bool",[53,1336,1318],{"class":120},[53,1338,1339],{"class":131}," False",[53,1341,1304],{"class":63},[53,1343,1345,1348,1350,1353,1355],{"class":55,"line":1344},33,[53,1346,1347],{"class":63},") -> Union[",[53,1349,1301],{"class":120},[53,1351,1352],{"class":63},", List[",[53,1354,1301],{"class":120},[53,1356,1357],{"class":63},"]]:\n",[53,1359,1361],{"class":55,"line":1360},34,[53,1362,1363],{"class":278}," \"\"\"\n",[53,1365,1367],{"class":55,"line":1366},35,[53,1368,1369],{"class":278}," Method for solving the generalized eigenvalue problem with regularization\n",[53,1371,1373],{"class":55,"line":1372},36,[53,1374,104],{"emptyLinePlaceholder":103},[53,1376,1378],{"class":55,"line":1377},37,[53,1379,1380],{"class":278}," Args:\n",[53,1382,1384],{"class":55,"line":1383},38,[53,1385,1386],{"class":278}," h (numpy.ndarray):\n",[53,1388,1390],{"class":55,"line":1389},39,[53,1391,1392],{"class":278}," The effective representation of the matrix in the Krylov subspace\n",[53,1394,1396],{"class":55,"line":1395},40,[53,1397,1398],{"class":278}," s (numpy.ndarray):\n",[53,1400,1402],{"class":55,"line":1401},41,[53,1403,1404],{"class":278}," The matrix of overlaps between vectors of the Krylov subspace\n",[53,1406,1408],{"class":55,"line":1407},42,[53,1409,1410],{"class":278}," threshold (float):\n",[53,1412,1414],{"class":55,"line":1413},43,[53,1415,1416],{"class":278}," Cut-off value for the eigenvalue of s\n",[53,1418,1420],{"class":55,"line":1419},44,[53,1421,1422],{"class":278}," k (int):\n",[53,1424,1426],{"class":55,"line":1425},45,[53,1427,1428],{"class":278}," Number of eigenvalues to return\n",[53,1430,1432],{"class":55,"line":1431},46,[53,1433,1434],{"class":278}," return_dimn (bool):\n",[53,1436,1438],{"class":55,"line":1437},47,[53,1439,1440],{"class":278}," Whether to return the size of the regularized subspace\n",[53,1442,1444],{"class":55,"line":1443},48,[53,1445,104],{"emptyLinePlaceholder":103},[53,1447,1449],{"class":55,"line":1448},49,[53,1450,1451],{"class":278}," Returns:\n",[53,1453,1455],{"class":55,"line":1454},50,[53,1456,1457],{"class":278}," lowest k-eigenvalue(s) that are the solution of the regularized generalized eigenvalue problem\n",[53,1459,1461],{"class":55,"line":1460},51,[53,1462,104],{"emptyLinePlaceholder":103},[53,1464,1466],{"class":55,"line":1465},52,[53,1467,104],{"emptyLinePlaceholder":103},[53,1469,1471],{"class":55,"line":1470},53,[53,1472,1363],{"class":278},[53,1474,1476,1479,1481,1484,1487],{"class":55,"line":1475},54,[53,1477,1478],{"class":63}," s_vals, s_vecs ",[53,1480,121],{"class":120},[53,1482,1483],{"class":63}," sp.linalg.",[53,1485,1486],{"class":124},"eigh",[53,1488,1489],{"class":63},"(s)\n",[53,1491,1493,1496,1498],{"class":55,"line":1492},55,[53,1494,1495],{"class":63}," s_vecs ",[53,1497,121],{"class":120},[53,1499,1500],{"class":63}," s_vecs.T\n",[53,1502,1504,1507,1509,1512,1515],{"class":55,"line":1503},56,[53,1505,1506],{"class":63}," good_vecs ",[53,1508,121],{"class":120},[53,1510,1511],{"class":63}," np.",[53,1513,1514],{"class":124},"array",[53,1516,1271],{"class":63},[53,1518,1520,1523,1526,1529,1532,1535,1538,1541,1544,1547],{"class":55,"line":1519},57,[53,1521,1522],{"class":63}," [vec ",[53,1524,1525],{"class":59},"for",[53,1527,1528],{"class":63}," val, vec ",[53,1530,1531],{"class":59},"in",[53,1533,1534],{"class":120}," zip",[53,1536,1537],{"class":63},"(s_vals, s_vecs) ",[53,1539,1540],{"class":59},"if",[53,1542,1543],{"class":63}," val ",[53,1545,1546],{"class":120},">",[53,1548,1549],{"class":63}," threshold]\n",[53,1551,1553],{"class":55,"line":1552},58,[53,1554,1555],{"class":63}," )\n",[53,1557,1559,1562,1564,1567,1570,1573,1576,1579,1581],{"class":55,"line":1558},59,[53,1560,1561],{"class":63}," h_reg ",[53,1563,121],{"class":120},[53,1565,1566],{"class":63}," good_vecs.",[53,1568,1569],{"class":124},"conj",[53,1571,1572],{"class":63},"() ",[53,1574,1575],{"class":120},"@",[53,1577,1578],{"class":63}," h ",[53,1580,1575],{"class":120},[53,1582,1583],{"class":63}," good_vecs.T\n",[53,1585,1587,1590,1592,1594,1596,1598,1600,1603,1605],{"class":55,"line":1586},60,[53,1588,1589],{"class":63}," s_reg ",[53,1591,121],{"class":120},[53,1593,1566],{"class":63},[53,1595,1569],{"class":124},[53,1597,1572],{"class":63},[53,1599,1575],{"class":120},[53,1601,1602],{"class":63}," s ",[53,1604,1575],{"class":120},[53,1606,1583],{"class":63},[53,1608,1610,1613,1616,1619,1621],{"class":55,"line":1609},61,[53,1611,1612],{"class":59}," if",[53,1614,1615],{"class":63}," k ",[53,1617,1618],{"class":120},"==",[53,1620,1321],{"class":131},[53,1622,1623],{"class":63},":\n",[53,1625,1627,1630],{"class":55,"line":1626},62,[53,1628,1629],{"class":59}," if",[53,1631,1632],{"class":63}," return_dimn:\n",[53,1634,1636,1639,1641,1643,1646,1648,1651,1653,1656,1659],{"class":55,"line":1635},63,[53,1637,1638],{"class":59}," return",[53,1640,1483],{"class":63},[53,1642,1486],{"class":124},[53,1644,1645],{"class":63},"(h_reg, 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else",[53,1697,1623],{"class":63},[53,1699,1701,1703],{"class":55,"line":1700},67,[53,1702,1629],{"class":59},[53,1704,1632],{"class":63},[53,1706,1708,1710,1712,1714,1716,1718,1721,1723],{"class":55,"line":1707},68,[53,1709,1638],{"class":59},[53,1711,1483],{"class":63},[53,1713,1486],{"class":124},[53,1715,1645],{"class":63},[53,1717,149],{"class":131},[53,1719,1720],{"class":63},"][:k], 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",[53,2001,2002],{"class":120},"+",[53,2004,1321],{"class":131},[53,2006,2007],{"class":63},", n_exc))\n",[53,2009,2011,2013,2015,2018,2021,2024],{"class":55,"line":2010},91,[53,2012,1955],{"class":120},[53,2014,128],{"class":63},[53,2016,2017],{"class":278},"\"n_exc\"",[53,2019,2020],{"class":63},", n_exc, ",[53,2022,2023],{"class":278},"\", subspace dimension\"",[53,2025,2026],{"class":63},", sub_dimn)\n",[53,2028,2030],{"class":55,"line":2029},92,[53,2031,104],{"emptyLinePlaceholder":103},[53,2033,2035,2038,2040,2042,2045,2048,2052,2055],{"class":55,"line":2034},93,[53,2036,2037],{"class":63}," few_particle_H ",[53,2039,121],{"class":120},[53,2041,1511],{"class":63},[53,2043,2044],{"class":124},"zeros",[53,2046,2047],{"class":63},"((sub_dimn, sub_dimn), 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",[53,2110,2111],{"class":2050},"r",[53,2113,121],{"class":120},[53,2115,2116],{"class":63},"n_exc)\n",[53,2118,2120,2123],{"class":55,"line":2119},97,[53,2121,2122],{"class":63}," ] ",[53,2124,2125],{"class":110},"# list all of the possible sets of n_exc indices of 1s in n_exc-particle states\n",[53,2127,2129],{"class":55,"line":2128},98,[53,2130,104],{"emptyLinePlaceholder":103},[53,2132,2134,2137,2139],{"class":55,"line":2133},99,[53,2135,2136],{"class":63}," m ",[53,2138,121],{"class":120},[53,2140,2141],{"class":131}," 0\n",[53,2143,2145,2147,2150,2152,2154],{"class":55,"line":2144},100,[53,2146,1811],{"class":59},[53,2148,2149],{"class":63}," i, i_set ",[53,2151,1531],{"class":59},[53,2153,1852],{"class":120},[53,2155,2156],{"class":63},"(sparse_vecs):\n",[53,2158,2160,2163,2166,2168,2170],{"class":55,"line":2159},101,[53,2161,2162],{"class":59}," for",[53,2164,2165],{"class":63}," j, j_set ",[53,2167,1531],{"class":59},[53,2169,1852],{"class":120},[53,2171,2156],{"class":63},[53,2173,2175,2178,2181],{"class":55,"line":2174},102,[53,2176,2177],{"class":63}," m ",[53,2179,2180],{"class":120},"+=",[53,2182,1979],{"class":131},[53,2184,2186],{"class":55,"line":2185},103,[53,2187,104],{"emptyLinePlaceholder":103},[53,2189,2191,2194,2197,2200,2203,2206,2209,2212],{"class":55,"line":2190},104,[53,2192,2193],{"class":59}," if",[53,2195,2196],{"class":120}," len",[53,2198,2199],{"class":63},"(i_set.",[53,2201,2202],{"class":124},"symmetric_difference",[53,2204,2205],{"class":63},"(j_set)) ",[53,2207,2208],{"class":120},"\u003C=",[53,2210,2211],{"class":131}," 2",[53,2213,1623],{"class":63},[53,2215,2217,2220,2223,2225,2227],{"class":55,"line":2216},105,[53,2218,2219],{"class":59}," for",[53,2221,2222],{"class":63}," p_x, p_z, coeff ",[53,2224,1531],{"class":59},[53,2226,1534],{"class":120},[53,2228,2229],{"class":63},"(H_x, H_z, H_c):\n",[53,2231,2233,2236,2239,2241,2244,2246],{"class":55,"line":2232},106,[53,2234,2235],{"class":59}," if",[53,2237,2238],{"class":63}," i_set.",[53,2240,2202],{"class":124},[53,2242,2243],{"class":63},"(j_set) ",[53,2245,1618],{"class":120},[53,2247,2248],{"class":63}," p_x:\n",[53,2250,2252,2255,2257,2260,2263,2265,2268,2270,2273,2275,2278,2281,2284,2287],{"class":55,"line":2251},107,[53,2253,2254],{"class":63}," sgn ",[53,2256,121],{"class":120},[53,2258,2259],{"class":63}," ((",[53,2261,2262],{"class":120},"-",[53,2264,173],{"class":131},[53,2266,2267],{"class":59},"j",[53,2269,203],{"class":63},[53,2271,2272],{"class":120},"**",[53,2274,2196],{"class":120},[53,2276,2277],{"class":63},"(p_x.",[53,2279,2280],{"class":124},"intersection",[53,2282,2283],{"class":63},"(p_z))) ",[53,2285,2286],{"class":120},"*",[53,2288,1229],{"class":63},[53,2290,2292,2295,2297,2299,2301,2303,2305,2307,2309],{"class":55,"line":2291},108,[53,2293,2294],{"class":63}," (",[53,2296,2262],{"class":120},[53,2298,173],{"class":131},[53,2300,203],{"class":63},[53,2302,2272],{"class":120},[53,2304,2196],{"class":120},[53,2306,2199],{"class":63},[53,2308,2280],{"class":124},[53,2310,2311],{"class":63},"(p_z))\n",[53,2313,2315],{"class":55,"line":2314},109,[53,2316,2317],{"class":63}," )\n",[53,2319,2321,2324],{"class":55,"line":2320},110,[53,2322,2323],{"class":59}," else",[53,2325,1623],{"class":63},[53,2327,2329,2331,2333],{"class":55,"line":2328},111,[53,2330,2254],{"class":63},[53,2332,121],{"class":120},[53,2334,2141],{"class":131},[53,2336,2338],{"class":55,"line":2337},112,[53,2339,104],{"emptyLinePlaceholder":103},[53,2341,2343,2346,2348,2351,2353],{"class":55,"line":2342},113,[53,2344,2345],{"class":63}," few_particle_H[i, j] ",[53,2347,2180],{"class":120},[53,2349,2350],{"class":63}," sgn ",[53,2352,2286],{"class":120},[53,2354,2355],{"class":63}," coeff\n",[53,2357,2359],{"class":55,"line":2358},114,[53,2360,104],{"emptyLinePlaceholder":103},[53,2362,2364,2367,2369,2372,2375,2378],{"class":55,"line":2363},115,[53,2365,2366],{"class":63}," gs_en ",[53,2368,121],{"class":120},[53,2370,2371],{"class":120}," min",[53,2373,2374],{"class":63},"(np.linalg.",[53,2376,2377],{"class":124},"eigvalsh",[53,2379,2380],{"class":63},"(few_particle_H))\n",[53,2382,2384,2386,2388,2391],{"class":55,"line":2383},116,[53,2385,1955],{"class":120},[53,2387,128],{"class":63},[53,2389,2390],{"class":278},"\"single particle ground state energy: \"",[53,2392,2393],{"class":63},", gs_en)\n",[53,2395,2397,2400],{"class":55,"line":2396},117,[53,2398,2399],{"class":59}," return",[53,2401,2402],{"class":63}," gs_en\n",[30,2404,2406],{"id":2405},"step-1-map-classical-inputs-to-a-quantum-problem","Step 1: Map classical inputs to a quantum problem",[2408,2409,2411],"h3",{"id":2410},"the-krylov-space","The Krylov space",[12,2413,2414,2415,2508,2509,2538,2539,2568,2569,2628,2629,28],{},"The Krylov space ",[53,2416,2419,2448],{"className":2417},[2418],"katex",[53,2420,2423],{"className":2421},[2422],"katex-mathml",[2424,2425,2427],"math",{"xmlns":2426},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[2428,2429,2430,2443],"semantics",{},[2431,2432,2433],"mrow",{},[2434,2435,2436,2441],"msup",{},[2437,2438,2440],"mi",{"mathvariant":2439},"script","K",[2437,2442,2111],{},[2444,2445,2447],"annotation",{"encoding":2446},"application\u002Fx-tex","\\mathcal{K}^r",[53,2449,2453],{"className":2450,"ariaHidden":2452},[2451],"katex-html","true",[53,2454,2457,2462],{"className":2455},[2456],"base",[53,2458],{"className":2459,"style":2461},[2460],"strut","height:0.6833em;",[53,2463,2466,2471],{"className":2464},[2465],"mord",[53,2467,2440],{"className":2468,"style":2470},[2465,2469],"mathcal","margin-right:0.0144em;",[53,2472,2475],{"className":2473},[2474],"msupsub",[53,2476,2479],{"className":2477},[2478],"vlist-t",[53,2480,2483],{"className":2481},[2482],"vlist-r",[53,2484,2488],{"className":2485,"style":2487},[2486],"vlist","height:0.6644em;",[53,2489,2491,2496],{"style":2490},"top:-3.063em;margin-right:0.05em;",[53,2492],{"className":2493,"style":2495},[2494],"pstrut","height:2.7em;",[53,2497,2503],{"className":2498},[2499,2500,2501,2502],"sizing","reset-size6","size3","mtight",[53,2504,2111],{"className":2505,"style":2507},[2465,2506,2502],"mathnormal","margin-right:0.0278em;"," of order ",[53,2510,2512,2525],{"className":2511},[2418],[53,2513,2515],{"className":2514},[2422],[2424,2516,2517],{"xmlns":2426},[2428,2518,2519,2523],{},[2431,2520,2521],{},[2437,2522,2111],{},[2444,2524,2111],{"encoding":2446},[53,2526,2528],{"className":2527,"ariaHidden":2452},[2451],[53,2529,2531,2535],{"className":2530},[2456],[53,2532],{"className":2533,"style":2534},[2460],"height:0.4306em;",[53,2536,2111],{"className":2537,"style":2507},[2465,2506]," is the space spanned by vectors obtained by multiplying higher powers of a matrix ",[53,2540,2542,2556],{"className":2541},[2418],[53,2543,2545],{"className":2544},[2422],[2424,2546,2547],{"xmlns":2426},[2428,2548,2549,2554],{},[2431,2550,2551],{},[2437,2552,2553],{},"A",[2444,2555,2553],{"encoding":2446},[53,2557,2559],{"className":2558,"ariaHidden":2452},[2451],[53,2560,2562,2565],{"className":2561},[2456],[53,2563],{"className":2564,"style":2461},[2460],[53,2566,2553],{"className":2567},[2465,2506],", up to ",[53,2570,2572,2593],{"className":2571},[2418],[53,2573,2575],{"className":2574},[2422],[2424,2576,2577],{"xmlns":2426},[2428,2578,2579,2590],{},[2431,2580,2581,2583,2587],{},[2437,2582,2111],{},[2584,2585,2586],"mo",{},"−",[2588,2589,173],"mn",{},[2444,2591,2592],{"encoding":2446},"r-1",[53,2594,2596,2618],{"className":2595,"ariaHidden":2452},[2451],[53,2597,2599,2603,2606,2611,2615],{"className":2598},[2456],[53,2600],{"className":2601,"style":2602},[2460],"height:0.6667em;vertical-align:-0.0833em;",[53,2604,2111],{"className":2605,"style":2507},[2465,2506],[53,2607],{"className":2608,"style":2610},[2609],"mspace","margin-right:0.2222em;",[53,2612,2586],{"className":2613},[2614],"mbin",[53,2616],{"className":2617,"style":2610},[2609],[53,2619,2621,2625],{"className":2620},[2456],[53,2622],{"className":2623,"style":2624},[2460],"height:0.6444em;",[53,2626,173],{"className":2627},[2465],", with a reference vector ",[53,2630,2632,2655],{"className":2631},[2418],[53,2633,2635],{"className":2634},[2422],[2424,2636,2637],{"xmlns":2426},[2428,2638,2639,2652],{},[2431,2640,2641,2645,2648],{},[2437,2642,2644],{"mathvariant":2643},"normal","∣",[2437,2646,2647],{},"v",[2584,2649,2651],{"stretchy":2650},"false","⟩",[2444,2653,2654],{"encoding":2446},"\\vert v \\rangle",[53,2656,2658],{"className":2657,"ariaHidden":2452},[2451],[53,2659,2661,2665,2668,2672],{"className":2660},[2456],[53,2662],{"className":2663,"style":2664},[2460],"height:1em;vertical-align:-0.25em;",[53,2666,2644],{"className":2667},[2465],[53,2669,2647],{"className":2670,"style":2671},[2465,2506],"margin-right:0.0359em;",[53,2673,2651],{"className":2674},[2675],"mclose",[53,2677,2680],{"className":2678},[2679],"katex-display",[53,2681,2683,2771],{"className":2682},[2418],[53,2684,2686],{"className":2685},[2422],[2424,2687,2689],{"xmlns":2426,"display":2688},"block",[2428,2690,2691,2768],{},[2431,2692,2693,2699,2701],{},[2434,2694,2695,2697],{},[2437,2696,2440],{"mathvariant":2439},[2437,2698,2111],{},[2584,2700,121],{},[2431,2702,2703,2706,2708,2710,2712,2715,2717,2719,2721,2723,2725,2731,2733,2735,2737,2739,2741,2743,2745,2747,2759,2761,2763,2765],{},[2584,2704,2705],{"fence":2452},"{",[2437,2707,2644],{"mathvariant":2643},[2437,2709,2647],{},[2584,2711,2651],{"stretchy":2650},[2584,2713,2714],{"separator":2452},",",[2437,2716,2553],{},[2437,2718,2644],{"mathvariant":2643},[2437,2720,2647],{},[2584,2722,2651],{"stretchy":2650},[2584,2724,2714],{"separator":2452},[2434,2726,2727,2729],{},[2437,2728,2553],{},[2588,2730,132],{},[2437,2732,2644],{"mathvariant":2643},[2437,2734,2647],{},[2584,2736,2651],{"stretchy":2650},[2584,2738,2714],{"separator":2452},[2437,2740,28],{"mathvariant":2643},[2437,2742,28],{"mathvariant":2643},[2437,2744,28],{"mathvariant":2643},[2584,2746,2714],{"separator":2452},[2434,2748,2749,2751],{},[2437,2750,2553],{},[2431,2752,2753,2755,2757],{},[2437,2754,2111],{},[2584,2756,2586],{},[2588,2758,173],{},[2437,2760,2644],{"mathvariant":2643},[2437,2762,2647],{},[2584,2764,2651],{"stretchy":2650},[2584,2766,2767],{"fence":2452},"}",[2444,2769,2770],{"encoding":2446},"\\mathcal{K}^r = \\left\\{ \\vert v \\rangle, A \\vert v \\rangle, A^2 \\vert v \\rangle, ..., A^{r-1} \\vert v \\rangle \\right\\}",[53,2772,2774,2822],{"className":2773,"ariaHidden":2452},[2451],[53,2775,2777,2781,2811,2815,2819],{"className":2776},[2456],[53,2778],{"className":2779,"style":2780},[2460],"height:0.7144em;",[53,2782,2784,2787],{"className":2783},[2465],[53,2785,2440],{"className":2786,"style":2470},[2465,2469],[53,2788,2790],{"className":2789},[2474],[53,2791,2793],{"className":2792},[2478],[53,2794,2796],{"className":2795},[2482],[53,2797,2799],{"className":2798,"style":2780},[2486],[53,2800,2802,2805],{"style":2801},"top:-3.113em;margin-right:0.05em;",[53,2803],{"className":2804,"style":2495},[2494],[53,2806,2808],{"className":2807},[2499,2500,2501,2502],[53,2809,2111],{"className":2810,"style":2507},[2465,2506,2502],[53,2812],{"className":2813,"style":2814},[2609],"margin-right:0.2778em;",[53,2816,121],{"className":2817},[2818],"mrel",[53,2820],{"className":2821,"style":2814},[2609],[53,2823,2825,2829],{"className":2824},[2456],[53,2826],{"className":2827,"style":2828},[2460],"height:1.2141em;vertical-align:-0.35em;",[53,2830,2833,2844,2847,2850,2853,2857,2861,2864,2867,2870,2873,2876,2879,2909,2912,2915,2918,2921,2924,2928,2931,2934,2972,2975,2978,2981],{"className":2831},[2832],"minner",[53,2834,2839],{"className":2835,"style":2838},[2836,2837],"mopen","delimcenter","top:0em;",[53,2840,2705],{"className":2841},[2842,2843],"delimsizing","size1",[53,2845,2644],{"className":2846},[2465],[53,2848,2647],{"className":2849,"style":2671},[2465,2506],[53,2851,2651],{"className":2852},[2675],[53,2854,2714],{"className":2855},[2856],"mpunct",[53,2858],{"className":2859,"style":2860},[2609],"margin-right:0.1667em;",[53,2862,2553],{"className":2863},[2465,2506],[53,2865,2644],{"className":2866},[2465],[53,2868,2647],{"className":2869,"style":2671},[2465,2506],[53,2871,2651],{"className":2872},[2675],[53,2874,2714],{"className":2875},[2856],[53,2877],{"className":2878,"style":2860},[2609],[53,2880,2882,2885],{"className":2881},[2465],[53,2883,2553],{"className":2884},[2465,2506],[53,2886,2888],{"className":2887},[2474],[53,2889,2891],{"className":2890},[2478],[53,2892,2894],{"className":2893},[2482],[53,2895,2898],{"className":2896,"style":2897},[2486],"height:0.8641em;",[53,2899,2900,2903],{"style":2801},[53,2901],{"className":2902,"style":2495},[2494],[53,2904,2906],{"className":2905},[2499,2500,2501,2502],[53,2907,132],{"className":2908},[2465,2502],[53,2910,2644],{"className":2911},[2465],[53,2913,2647],{"className":2914,"style":2671},[2465,2506],[53,2916,2651],{"className":2917},[2675],[53,2919,2714],{"className":2920},[2856],[53,2922],{"className":2923,"style":2860},[2609],[53,2925,2927],{"className":2926},[2465],"...",[53,2929,2714],{"className":2930},[2856],[53,2932],{"className":2933,"style":2860},[2609],[53,2935,2937,2940],{"className":2936},[2465],[53,2938,2553],{"className":2939},[2465,2506],[53,2941,2943],{"className":2942},[2474],[53,2944,2946],{"className":2945},[2478],[53,2947,2949],{"className":2948},[2482],[53,2950,2952],{"className":2951,"style":2897},[2486],[53,2953,2954,2957],{"style":2801},[53,2955],{"className":2956,"style":2495},[2494],[53,2958,2960],{"className":2959},[2499,2500,2501,2502],[53,2961,2963,2966,2969],{"className":2962},[2465,2502],[53,2964,2111],{"className":2965,"style":2507},[2465,2506,2502],[53,2967,2586],{"className":2968},[2614,2502],[53,2970,173],{"className":2971},[2465,2502],[53,2973,2644],{"className":2974},[2465],[53,2976,2647],{"className":2977,"style":2671},[2465,2506],[53,2979,2651],{"className":2980},[2675],[53,2982,2984],{"className":2983,"style":2838},[2675,2837],[53,2985,2767],{"className":2986},[2842,2843],[12,2988,2989,2990,3018,3019,3049,3050,3130,3131,3159,3160,3267,3268,3338,3339,3367,3368,3469,3470,3473,3474,3502,3503,28],{},"If the matrix ",[53,2991,2993,3006],{"className":2992},[2418],[53,2994,2996],{"className":2995},[2422],[2424,2997,2998],{"xmlns":2426},[2428,2999,3000,3004],{},[2431,3001,3002],{},[2437,3003,2553],{},[2444,3005,2553],{"encoding":2446},[53,3007,3009],{"className":3008,"ariaHidden":2452},[2451],[53,3010,3012,3015],{"className":3011},[2456],[53,3013],{"className":3014,"style":2461},[2460],[53,3016,2553],{"className":3017},[2465,2506]," is the Hamiltonian ",[53,3020,3022,3036],{"className":3021},[2418],[53,3023,3025],{"className":3024},[2422],[2424,3026,3027],{"xmlns":2426},[2428,3028,3029,3034],{},[2431,3030,3031],{},[2437,3032,3033],{},"H",[2444,3035,3033],{"encoding":2446},[53,3037,3039],{"className":3038,"ariaHidden":2452},[2451],[53,3040,3042,3045],{"className":3041},[2456],[53,3043],{"className":3044,"style":2461},[2460],[53,3046,3033],{"className":3047,"style":3048},[2465,2506],"margin-right:0.0813em;",", we'll refer to the corresponding space as the power Krylov space ",[53,3051,3053,3073],{"className":3052},[2418],[53,3054,3056],{"className":3055},[2422],[2424,3057,3058],{"xmlns":2426},[2428,3059,3060,3070],{},[2431,3061,3062],{},[3063,3064,3065,3067],"msub",{},[2437,3066,2440],{"mathvariant":2439},[2437,3068,3069],{},"P",[2444,3071,3072],{"encoding":2446},"\\mathcal{K}_P",[53,3074,3076],{"className":3075,"ariaHidden":2452},[2451],[53,3077,3079,3083],{"className":3078},[2456],[53,3080],{"className":3081,"style":3082},[2460],"height:0.8333em;vertical-align:-0.15em;",[53,3084,3086,3089],{"className":3085},[2465],[53,3087,2440],{"className":3088,"style":2470},[2465,2469],[53,3090,3092],{"className":3091},[2474],[53,3093,3096,3121],{"className":3094},[2478,3095],"vlist-t2",[53,3097,3099,3116],{"className":3098},[2482],[53,3100,3103],{"className":3101,"style":3102},[2486],"height:0.3283em;",[53,3104,3106,3109],{"style":3105},"top:-2.55em;margin-left:-0.0144em;margin-right:0.05em;",[53,3107],{"className":3108,"style":2495},[2494],[53,3110,3112],{"className":3111},[2499,2500,2501,2502],[53,3113,3069],{"className":3114,"style":3115},[2465,2506,2502],"margin-right:0.1389em;",[53,3117,3120],{"className":3118},[3119],"vlist-s","​",[53,3122,3124],{"className":3123},[2482],[53,3125,3128],{"className":3126,"style":3127},[2486],"height:0.15em;",[53,3129],{},". In the case where ",[53,3132,3134,3147],{"className":3133},[2418],[53,3135,3137],{"className":3136},[2422],[2424,3138,3139],{"xmlns":2426},[2428,3140,3141,3145],{},[2431,3142,3143],{},[2437,3144,2553],{},[2444,3146,2553],{"encoding":2446},[53,3148,3150],{"className":3149,"ariaHidden":2452},[2451],[53,3151,3153,3156],{"className":3152},[2456],[53,3154],{"className":3155,"style":2461},[2460],[53,3157,2553],{"className":3158},[2465,2506]," is the time-evolution operator generated by the Hamiltonian ",[53,3161,3163,3197],{"className":3162},[2418],[53,3164,3166],{"className":3165},[2422],[2424,3167,3168],{"xmlns":2426},[2428,3169,3170,3194],{},[2431,3171,3172,3175,3177],{},[2437,3173,3174],{},"U",[2584,3176,121],{},[2434,3178,3179,3182],{},[2437,3180,3181],{},"e",[2431,3183,3184,3186,3189,3191],{},[2584,3185,2586],{},[2437,3187,3188],{},"i",[2437,3190,3033],{},[2437,3192,3193],{},"t",[2444,3195,3196],{"encoding":2446},"U=e^{-iHt}",[53,3198,3200,3219],{"className":3199,"ariaHidden":2452},[2451],[53,3201,3203,3206,3210,3213,3216],{"className":3202},[2456],[53,3204],{"className":3205,"style":2461},[2460],[53,3207,3174],{"className":3208,"style":3209},[2465,2506],"margin-right:0.109em;",[53,3211],{"className":3212,"style":2814},[2609],[53,3214,121],{"className":3215},[2818],[53,3217],{"className":3218,"style":2814},[2609],[53,3220,3222,3226],{"className":3221},[2456],[53,3223],{"className":3224,"style":3225},[2460],"height:0.8413em;",[53,3227,3229,3232],{"className":3228},[2465],[53,3230,3181],{"className":3231},[2465,2506],[53,3233,3235],{"className":3234},[2474],[53,3236,3238],{"className":3237},[2478],[53,3239,3241],{"className":3240},[2482],[53,3242,3244],{"className":3243,"style":3225},[2486],[53,3245,3246,3249],{"style":2490},[53,3247],{"className":3248,"style":2495},[2494],[53,3250,3252],{"className":3251},[2499,2500,2501,2502],[53,3253,3255,3258,3261,3264],{"className":3254},[2465,2502],[53,3256,2586],{"className":3257},[2465,2502],[53,3259,3188],{"className":3260},[2465,2506,2502],[53,3262,3033],{"className":3263,"style":3048},[2465,2506,2502],[53,3265,3193],{"className":3266},[2465,2506,2502],", we'll refer to the space as the unitary Krylov space ",[53,3269,3271,3289],{"className":3270},[2418],[53,3272,3274],{"className":3273},[2422],[2424,3275,3276],{"xmlns":2426},[2428,3277,3278,3286],{},[2431,3279,3280],{},[3063,3281,3282,3284],{},[2437,3283,2440],{"mathvariant":2439},[2437,3285,3174],{},[2444,3287,3288],{"encoding":2446},"\\mathcal{K}_U",[53,3290,3292],{"className":3291,"ariaHidden":2452},[2451],[53,3293,3295,3298],{"className":3294},[2456],[53,3296],{"className":3297,"style":3082},[2460],[53,3299,3301,3304],{"className":3300},[2465],[53,3302,2440],{"className":3303,"style":2470},[2465,2469],[53,3305,3307],{"className":3306},[2474],[53,3308,3310,3330],{"className":3309},[2478,3095],[53,3311,3313,3327],{"className":3312},[2482],[53,3314,3316],{"className":3315,"style":3102},[2486],[53,3317,3318,3321],{"style":3105},[53,3319],{"className":3320,"style":2495},[2494],[53,3322,3324],{"className":3323},[2499,2500,2501,2502],[53,3325,3174],{"className":3326,"style":3209},[2465,2506,2502],[53,3328,3120],{"className":3329},[3119],[53,3331,3333],{"className":3332},[2482],[53,3334,3336],{"className":3335,"style":3127},[2486],[53,3337],{},". The power Krylov subspace that we use classically cannot be generated directly on a quantum computer as ",[53,3340,3342,3355],{"className":3341},[2418],[53,3343,3345],{"className":3344},[2422],[2424,3346,3347],{"xmlns":2426},[2428,3348,3349,3353],{},[2431,3350,3351],{},[2437,3352,3033],{},[2444,3354,3033],{"encoding":2446},[53,3356,3358],{"className":3357,"ariaHidden":2452},[2451],[53,3359,3361,3364],{"className":3360},[2456],[53,3362],{"className":3363,"style":2461},[2460],[53,3365,3033],{"className":3366,"style":3048},[2465,2506]," is not a unitary operator. Instead, we can use the time-evolution operator ",[53,3369,3371,3401],{"className":3370},[2418],[53,3372,3374],{"className":3373},[2422],[2424,3375,3376],{"xmlns":2426},[2428,3377,3378,3398],{},[2431,3379,3380,3382,3384],{},[2437,3381,3174],{},[2584,3383,121],{},[2434,3385,3386,3388],{},[2437,3387,3181],{},[2431,3389,3390,3392,3394,3396],{},[2584,3391,2586],{},[2437,3393,3188],{},[2437,3395,3033],{},[2437,3397,3193],{},[2444,3399,3400],{"encoding":2446},"U = e^{-iHt}",[53,3402,3404,3422],{"className":3403,"ariaHidden":2452},[2451],[53,3405,3407,3410,3413,3416,3419],{"className":3406},[2456],[53,3408],{"className":3409,"style":2461},[2460],[53,3411,3174],{"className":3412,"style":3209},[2465,2506],[53,3414],{"className":3415,"style":2814},[2609],[53,3417,121],{"className":3418},[2818],[53,3420],{"className":3421,"style":2814},[2609],[53,3423,3425,3428],{"className":3424},[2456],[53,3426],{"className":3427,"style":3225},[2460],[53,3429,3431,3434],{"className":3430},[2465],[53,3432,3181],{"className":3433},[2465,2506],[53,3435,3437],{"className":3436},[2474],[53,3438,3440],{"className":3439},[2478],[53,3441,3443],{"className":3442},[2482],[53,3444,3446],{"className":3445,"style":3225},[2486],[53,3447,3448,3451],{"style":2490},[53,3449],{"className":3450,"style":2495},[2494],[53,3452,3454],{"className":3453},[2499,2500,2501,2502],[53,3455,3457,3460,3463,3466],{"className":3456},[2465,2502],[53,3458,2586],{"className":3459},[2465,2502],[53,3461,3188],{"className":3462},[2465,2506,2502],[53,3464,3033],{"className":3465,"style":3048},[2465,2506,2502],[53,3467,3193],{"className":3468},[2465,2506,2502]," which can be shown to give similar ",[19,3471,3472],{"href":984},"convergence guarantees"," as the power method. Powers of ",[53,3475,3477,3490],{"className":3476},[2418],[53,3478,3480],{"className":3479},[2422],[2424,3481,3482],{"xmlns":2426},[2428,3483,3484,3488],{},[2431,3485,3486],{},[2437,3487,3174],{},[2444,3489,3174],{"encoding":2446},[53,3491,3493],{"className":3492,"ariaHidden":2452},[2451],[53,3494,3496,3499],{"className":3495},[2456],[53,3497],{"className":3498,"style":2461},[2460],[53,3500,3174],{"className":3501,"style":3209},[2465,2506]," then become different time steps ",[53,3504,3506,3548],{"className":3505},[2418],[53,3507,3509],{"className":3508},[2422],[2424,3510,3511],{"xmlns":2426},[2428,3512,3513,3545],{},[2431,3514,3515,3522,3524],{},[2434,3516,3517,3519],{},[2437,3518,3174],{},[2437,3520,3521],{},"k",[2584,3523,121],{},[2434,3525,3526,3528],{},[2437,3527,3181],{},[2431,3529,3530,3532,3534,3536,3538,3540,3542],{},[2584,3531,2586],{},[2437,3533,3188],{},[2437,3535,3033],{},[2584,3537,128],{"stretchy":2650},[2437,3539,3521],{},[2437,3541,3193],{},[2584,3543,3544],{"stretchy":2650},")",[2444,3546,3547],{"encoding":2446},"U^k = e^{-iH(kt)}",[53,3549,3551,3597],{"className":3550,"ariaHidden":2452},[2451],[53,3552,3554,3558,3588,3591,3594],{"className":3553},[2456],[53,3555],{"className":3556,"style":3557},[2460],"height:0.8491em;",[53,3559,3561,3564],{"className":3560},[2465],[53,3562,3174],{"className":3563,"style":3209},[2465,2506],[53,3565,3567],{"className":3566},[2474],[53,3568,3570],{"className":3569},[2478],[53,3571,3573],{"className":3572},[2482],[53,3574,3576],{"className":3575,"style":3557},[2486],[53,3577,3578,3581],{"style":2490},[53,3579],{"className":3580,"style":2495},[2494],[53,3582,3584],{"className":3583},[2499,2500,2501,2502],[53,3585,3521],{"className":3586,"style":3587},[2465,2506,2502],"margin-right:0.0315em;",[53,3589],{"className":3590,"style":2814},[2609],[53,3592,121],{"className":3593},[2818],[53,3595],{"className":3596,"style":2814},[2609],[53,3598,3600,3604],{"className":3599},[2456],[53,3601],{"className":3602,"style":3603},[2460],"height:0.888em;",[53,3605,3607,3610],{"className":3606},[2465],[53,3608,3181],{"className":3609},[2465,2506],[53,3611,3613],{"className":3612},[2474],[53,3614,3616],{"className":3615},[2478],[53,3617,3619],{"className":3618},[2482],[53,3620,3622],{"className":3621,"style":3603},[2486],[53,3623,3624,3627],{"style":2490},[53,3625],{"className":3626,"style":2495},[2494],[53,3628,3630],{"className":3629},[2499,2500,2501,2502],[53,3631,3633,3636,3639,3642,3645,3648,3651],{"className":3632},[2465,2502],[53,3634,2586],{"className":3635},[2465,2502],[53,3637,3188],{"className":3638},[2465,2506,2502],[53,3640,3033],{"className":3641,"style":3048},[2465,2506,2502],[53,3643,128],{"className":3644},[2836,2502],[53,3646,3521],{"className":3647,"style":3587},[2465,2506,2502],[53,3649,3193],{"className":3650},[2465,2506,2502],[53,3652,3544],{"className":3653},[2675,2502],[53,3655,3657],{"className":3656},[2679],[53,3658,3660,3748],{"className":3659},[2418],[53,3661,3663],{"className":3662},[2422],[2424,3664,3665],{"xmlns":2426,"display":2688},[2428,3666,3667,3745],{},[2431,3668,3669,3678,3680],{},[3670,3671,3672,3674,3676],"msubsup",{},[2437,3673,2440],{"mathvariant":2439},[2437,3675,3174],{},[2437,3677,2111],{},[2584,3679,121],{},[2431,3681,3682,3684,3686,3689,3691,3693,3695,3697,3699,3701,3703,3709,3711,3713,3715,3717,3719,3721,3723,3725,3737,3739,3741,3743],{},[2584,3683,2705],{"fence":2452},[2437,3685,2644],{"mathvariant":2643},[2437,3687,3688],{},"ψ",[2584,3690,2651],{"stretchy":2650},[2584,3692,2714],{"separator":2452},[2437,3694,3174],{},[2437,3696,2644],{"mathvariant":2643},[2437,3698,3688],{},[2584,3700,2651],{"stretchy":2650},[2584,3702,2714],{"separator":2452},[2434,3704,3705,3707],{},[2437,3706,3174],{},[2588,3708,132],{},[2437,3710,2644],{"mathvariant":2643},[2437,3712,3688],{},[2584,3714,2651],{"stretchy":2650},[2584,3716,2714],{"separator":2452},[2437,3718,28],{"mathvariant":2643},[2437,3720,28],{"mathvariant":2643},[2437,3722,28],{"mathvariant":2643},[2584,3724,2714],{"separator":2452},[2434,3726,3727,3729],{},[2437,3728,3174],{},[2431,3730,3731,3733,3735],{},[2437,3732,2111],{},[2584,3734,2586],{},[2588,3736,173],{},[2437,3738,2644],{"mathvariant":2643},[2437,3740,3688],{},[2584,3742,2651],{"stretchy":2650},[2584,3744,2767],{"fence":2452},[2444,3746,3747],{"encoding":2446},"\\mathcal{K}_U^r = \\left\\{ \\vert \\psi \\rangle, U \\vert \\psi \\rangle, U^2 \\vert \\psi \\rangle, ..., U^{r-1} \\vert \\psi \\rangle \\right\\}",[53,3749,3751,3820],{"className":3750,"ariaHidden":2452},[2451],[53,3752,3754,3758,3811,3814,3817],{"className":3753},[2456],[53,3755],{"className":3756,"style":3757},[2460],"height:0.9614em;vertical-align:-0.247em;",[53,3759,3761,3764],{"className":3760},[2465],[53,3762,2440],{"className":3763,"style":2470},[2465,2469],[53,3765,3767],{"className":3766},[2474],[53,3768,3770,3802],{"className":3769},[2478,3095],[53,3771,3773,3799],{"className":3772},[2482],[53,3774,3776,3788],{"className":3775,"style":2780},[2486],[53,3777,3779,3782],{"style":3778},"top:-2.453em;margin-left:-0.0144em;margin-right:0.05em;",[53,3780],{"className":3781,"style":2495},[2494],[53,3783,3785],{"className":3784},[2499,2500,2501,2502],[53,3786,3174],{"className":3787,"style":3209},[2465,2506,2502],[53,3789,3790,3793],{"style":2801},[53,3791],{"className":3792,"style":2495},[2494],[53,3794,3796],{"className":3795},[2499,2500,2501,2502],[53,3797,2111],{"className":3798,"style":2507},[2465,2506,2502],[53,3800,3120],{"className":3801},[3119],[53,3803,3805],{"className":3804},[2482],[53,3806,3809],{"className":3807,"style":3808},[2486],"height:0.247em;",[53,3810],{},[53,3812],{"className":3813,"style":2814},[2609],[53,3815,121],{"className":3816},[2818],[53,3818],{"className":3819,"style":2814},[2609],[53,3821,3823,3826],{"className":3822},[2456],[53,3824],{"className":3825,"style":2828},[2460],[53,3827,3829,3835,3838,3841,3844,3847,3850,3853,3856,3859,3862,3865,3868,3897,3900,3903,3906,3909,3912,3915,3918,3921,3959,3962,3965,3968],{"className":3828},[2832],[53,3830,3832],{"className":3831,"style":2838},[2836,2837],[53,3833,2705],{"className":3834},[2842,2843],[53,3836,2644],{"className":3837},[2465],[53,3839,3688],{"className":3840,"style":2671},[2465,2506],[53,3842,2651],{"className":3843},[2675],[53,3845,2714],{"className":3846},[2856],[53,3848],{"className":3849,"style":2860},[2609],[53,3851,3174],{"className":3852,"style":3209},[2465,2506],[53,3854,2644],{"className":3855},[2465],[53,3857,3688],{"className":3858,"style":2671},[2465,2506],[53,3860,2651],{"className":3861},[2675],[53,3863,2714],{"className":3864},[2856],[53,3866],{"className":3867,"style":2860},[2609],[53,3869,3871,3874],{"className":3870},[2465],[53,3872,3174],{"className":3873,"style":3209},[2465,2506],[53,3875,3877],{"className":3876},[2474],[53,3878,3880],{"className":3879},[2478],[53,3881,3883],{"className":3882},[2482],[53,3884,3886],{"className":3885,"style":2897},[2486],[53,3887,3888,3891],{"style":2801},[53,3889],{"className":3890,"style":2495},[2494],[53,3892,3894],{"className":3893},[2499,2500,2501,2502],[53,3895,132],{"className":3896},[2465,2502],[53,3898,2644],{"className":3899},[2465],[53,3901,3688],{"className":3902,"style":2671},[2465,2506],[53,3904,2651],{"className":3905},[2675],[53,3907,2714],{"className":3908},[2856],[53,3910],{"className":3911,"style":2860},[2609],[53,3913,2927],{"className":3914},[2465],[53,3916,2714],{"className":3917},[2856],[53,3919],{"className":3920,"style":2860},[2609],[53,3922,3924,3927],{"className":3923},[2465],[53,3925,3174],{"className":3926,"style":3209},[2465,2506],[53,3928,3930],{"className":3929},[2474],[53,3931,3933],{"className":3932},[2478],[53,3934,3936],{"className":3935},[2482],[53,3937,3939],{"className":3938,"style":2897},[2486],[53,3940,3941,3944],{"style":2801},[53,3942],{"className":3943,"style":2495},[2494],[53,3945,3947],{"className":3946},[2499,2500,2501,2502],[53,3948,3950,3953,3956],{"className":3949},[2465,2502],[53,3951,2111],{"className":3952,"style":2507},[2465,2506,2502],[53,3954,2586],{"className":3955},[2614,2502],[53,3957,173],{"className":3958},[2465,2502],[53,3960,2644],{"className":3961},[2465],[53,3963,3688],{"className":3964,"style":2671},[2465,2506],[53,3966,2651],{"className":3967},[2675],[53,3969,3971],{"className":3970,"style":2838},[2675,2837],[53,3972,2767],{"className":3973},[2842,2843],[12,3975,3976],{},"See the Appendix for a detailed derivation of how the unitary Krylov space allows to represents low-energy eigenstates accurately.",[2408,3978,3980],{"id":3979},"krylov-quantum-diagonalization-algorithm","Krylov quantum diagonalization algorithm",[12,3982,3983,3984,4012,4013,4082,4083,4152,4153,4223,4224,4284],{},"Given an Hamiltonian ",[53,3985,3987,4000],{"className":3986},[2418],[53,3988,3990],{"className":3989},[2422],[2424,3991,3992],{"xmlns":2426},[2428,3993,3994,3998],{},[2431,3995,3996],{},[2437,3997,3033],{},[2444,3999,3033],{"encoding":2446},[53,4001,4003],{"className":4002,"ariaHidden":2452},[2451],[53,4004,4006,4009],{"className":4005},[2456],[53,4007],{"className":4008,"style":2461},[2460],[53,4010,3033],{"className":4011,"style":3048},[2465,2506]," that we wish to diagonalize, first we consider the corresponding unitary Krylov space ",[53,4014,4016,4033],{"className":4015},[2418],[53,4017,4019],{"className":4018},[2422],[2424,4020,4021],{"xmlns":2426},[2428,4022,4023,4031],{},[2431,4024,4025],{},[3063,4026,4027,4029],{},[2437,4028,2440],{"mathvariant":2439},[2437,4030,3174],{},[2444,4032,3288],{"encoding":2446},[53,4034,4036],{"className":4035,"ariaHidden":2452},[2451],[53,4037,4039,4042],{"className":4038},[2456],[53,4040],{"className":4041,"style":3082},[2460],[53,4043,4045,4048],{"className":4044},[2465],[53,4046,2440],{"className":4047,"style":2470},[2465,2469],[53,4049,4051],{"className":4050},[2474],[53,4052,4054,4074],{"className":4053},[2478,3095],[53,4055,4057,4071],{"className":4056},[2482],[53,4058,4060],{"className":4059,"style":3102},[2486],[53,4061,4062,4065],{"style":3105},[53,4063],{"className":4064,"style":2495},[2494],[53,4066,4068],{"className":4067},[2499,2500,2501,2502],[53,4069,3174],{"className":4070,"style":3209},[2465,2506,2502],[53,4072,3120],{"className":4073},[3119],[53,4075,4077],{"className":4076},[2482],[53,4078,4080],{"className":4079,"style":3127},[2486],[53,4081],{},". The goal is to find a compact representation of the Hamiltonian in ",[53,4084,4086,4103],{"className":4085},[2418],[53,4087,4089],{"className":4088},[2422],[2424,4090,4091],{"xmlns":2426},[2428,4092,4093,4101],{},[2431,4094,4095],{},[3063,4096,4097,4099],{},[2437,4098,2440],{"mathvariant":2439},[2437,4100,3174],{},[2444,4102,3288],{"encoding":2446},[53,4104,4106],{"className":4105,"ariaHidden":2452},[2451],[53,4107,4109,4112],{"className":4108},[2456],[53,4110],{"className":4111,"style":3082},[2460],[53,4113,4115,4118],{"className":4114},[2465],[53,4116,2440],{"className":4117,"style":2470},[2465,2469],[53,4119,4121],{"className":4120},[2474],[53,4122,4124,4144],{"className":4123},[2478,3095],[53,4125,4127,4141],{"className":4126},[2482],[53,4128,4130],{"className":4129,"style":3102},[2486],[53,4131,4132,4135],{"style":3105},[53,4133],{"className":4134,"style":2495},[2494],[53,4136,4138],{"className":4137},[2499,2500,2501,2502],[53,4139,3174],{"className":4140,"style":3209},[2465,2506,2502],[53,4142,3120],{"className":4143},[3119],[53,4145,4147],{"className":4146},[2482],[53,4148,4150],{"className":4149,"style":3127},[2486],[53,4151],{},", which we'll refer to as ",[53,4154,4156,4176],{"className":4155},[2418],[53,4157,4159],{"className":4158},[2422],[2424,4160,4161],{"xmlns":2426},[2428,4162,4163,4173],{},[2431,4164,4165],{},[4166,4167,4168,4170],"mover",{"accent":2452},[2437,4169,3033],{},[2584,4171,4172],{},"~",[2444,4174,4175],{"encoding":2446},"\\tilde{H}",[53,4177,4179],{"className":4178,"ariaHidden":2452},[2451],[53,4180,4182,4186],{"className":4181},[2456],[53,4183],{"className":4184,"style":4185},[2460],"height:0.9202em;",[53,4187,4190],{"className":4188},[2465,4189],"accent",[53,4191,4193],{"className":4192},[2478],[53,4194,4196],{"className":4195},[2482],[53,4197,4199,4209],{"className":4198,"style":4185},[2486],[53,4200,4202,4206],{"style":4201},"top:-3em;",[53,4203],{"className":4204,"style":4205},[2494],"height:3em;",[53,4207,3033],{"className":4208,"style":3048},[2465,2506],[53,4210,4212,4215],{"style":4211},"top:-3.6023em;",[53,4213],{"className":4214,"style":4205},[2494],[53,4216,4220],{"className":4217,"style":4219},[4218],"accent-body","left:-0.1944em;",[53,4221,4172],{"className":4222},[2465],". The matrix elements of ",[53,4225,4227,4244],{"className":4226},[2418],[53,4228,4230],{"className":4229},[2422],[2424,4231,4232],{"xmlns":2426},[2428,4233,4234,4242],{},[2431,4235,4236],{},[4166,4237,4238,4240],{"accent":2452},[2437,4239,3033],{},[2584,4241,4172],{},[2444,4243,4175],{"encoding":2446},[53,4245,4247],{"className":4246,"ariaHidden":2452},[2451],[53,4248,4250,4253],{"className":4249},[2456],[53,4251],{"className":4252,"style":4185},[2460],[53,4254,4256],{"className":4255},[2465,4189],[53,4257,4259],{"className":4258},[2478],[53,4260,4262],{"className":4261},[2482],[53,4263,4265,4273],{"className":4264,"style":4185},[2486],[53,4266,4267,4270],{"style":4201},[53,4268],{"className":4269,"style":4205},[2494],[53,4271,3033],{"className":4272,"style":3048},[2465,2506],[53,4274,4275,4278],{"style":4211},[53,4276],{"className":4277,"style":4205},[2494],[53,4279,4281],{"className":4280,"style":4219},[4218],[53,4282,4172],{"className":4283},[2465],", the projection of the Hamiltonian in the Krylov space, can be calculated by calculating the following expectation values",[53,4286,4288],{"className":4287},[2679],[53,4289,4291,4346],{"className":4290},[2418],[53,4292,4294],{"className":4293},[2422],[2424,4295,4296],{"xmlns":2426,"display":2688},[2428,4297,4298,4343],{},[2431,4299,4300,4316,4318,4321,4327,4329,4331,4333,4339,4341],{},[3063,4301,4302,4308],{},[4166,4303,4304,4306],{"accent":2452},[2437,4305,3033],{},[2584,4307,4172],{},[2431,4309,4310,4313],{},[2437,4311,4312],{},"m",[2437,4314,4315],{},"n",[2584,4317,121],{},[2584,4319,4320],{"stretchy":2650},"⟨",[3063,4322,4323,4325],{},[2437,4324,3688],{},[2437,4326,4312],{},[2437,4328,2644],{"mathvariant":2643},[2437,4330,3033],{},[2437,4332,2644],{"mathvariant":2643},[3063,4334,4335,4337],{},[2437,4336,3688],{},[2437,4338,4315],{},[2584,4340,2651],{"stretchy":2650},[2584,4342,121],{},[2444,4344,4345],{"encoding":2446},"\\tilde{H}_{mn} = \\langle \\psi_m \\vert H \\vert \\psi_n \\rangle =",[53,4347,4349,4438],{"className":4348,"ariaHidden":2452},[2451],[53,4350,4352,4356,4429,4432,4435],{"className":4351},[2456],[53,4353],{"className":4354,"style":4355},[2460],"height:1.0702em;vertical-align:-0.15em;",[53,4357,4359,4390],{"className":4358},[2465],[53,4360,4362],{"className":4361},[2465,4189],[53,4363,4365],{"className":4364},[2478],[53,4366,4368],{"className":4367},[2482],[53,4369,4371,4379],{"className":4370,"style":4185},[2486],[53,4372,4373,4376],{"style":4201},[53,4374],{"className":4375,"style":4205},[2494],[53,4377,3033],{"className":4378,"style":3048},[2465,2506],[53,4380,4381,4384],{"style":4211},[53,4382],{"className":4383,"style":4205},[2494],[53,4385,4387],{"className":4386,"style":4219},[4218],[53,4388,4172],{"className":4389},[2465],[53,4391,4393],{"className":4392},[2474],[53,4394,4396,4421],{"className":4395},[2478,3095],[53,4397,4399,4418],{"className":4398},[2482],[53,4400,4403],{"className":4401,"style":4402},[2486],"height:0.1514em;",[53,4404,4406,4409],{"style":4405},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[53,4407],{"className":4408,"style":2495},[2494],[53,4410,4412],{"className":4411},[2499,2500,2501,2502],[53,4413,4415],{"className":4414},[2465,2502],[53,4416,2588],{"className":4417},[2465,2506,2502],[53,4419,3120],{"className":4420},[3119],[53,4422,4424],{"className":4423},[2482],[53,4425,4427],{"className":4426,"style":3127},[2486],[53,4428],{},[53,4430],{"className":4431,"style":2814},[2609],[53,4433,121],{"className":4434},[2818],[53,4436],{"className":4437,"style":2814},[2609],[53,4439,4441,4444,4447,4488,4491,4494,4497,4537,4540,4543],{"className":4440},[2456],[53,4442],{"className":4443,"style":2664},[2460],[53,4445,4320],{"className":4446},[2836],[53,4448,4450,4453],{"className":4449},[2465],[53,4451,3688],{"className":4452,"style":2671},[2465,2506],[53,4454,4456],{"className":4455},[2474],[53,4457,4459,4480],{"className":4458},[2478,3095],[53,4460,4462,4477],{"className":4461},[2482],[53,4463,4465],{"className":4464,"style":4402},[2486],[53,4466,4468,4471],{"style":4467},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[53,4469],{"className":4470,"style":2495},[2494],[53,4472,4474],{"className":4473},[2499,2500,2501,2502],[53,4475,4312],{"className":4476},[2465,2506,2502],[53,4478,3120],{"className":4479},[3119],[53,4481,4483],{"className":4482},[2482],[53,4484,4486],{"className":4485,"style":3127},[2486],[53,4487],{},[53,4489,2644],{"className":4490},[2465],[53,4492,3033],{"className":4493,"style":3048},[2465,2506],[53,4495,2644],{"className":4496},[2465],[53,4498,4500,4503],{"className":4499},[2465],[53,4501,3688],{"className":4502,"style":2671},[2465,2506],[53,4504,4506],{"className":4505},[2474],[53,4507,4509,4529],{"className":4508},[2478,3095],[53,4510,4512,4526],{"className":4511},[2482],[53,4513,4515],{"className":4514,"style":4402},[2486],[53,4516,4517,4520],{"style":4467},[53,4518],{"className":4519,"style":2495},[2494],[53,4521,4523],{"className":4522},[2499,2500,2501,2502],[53,4524,4315],{"className":4525},[2465,2506,2502],[53,4527,3120],{"className":4528},[3119],[53,4530,4532],{"className":4531},[2482],[53,4533,4535],{"className":4534,"style":3127},[2486],[53,4536],{},[53,4538,2651],{"className":4539},[2675],[53,4541],{"className":4542,"style":2814},[2609],[53,4544,121],{"className":4545},[2818],[53,4547,4549],{"className":4548},[2679],[53,4550,4552,4614],{"className":4551},[2418],[53,4553,4555],{"className":4554},[2422],[2424,4556,4557],{"xmlns":2426,"display":2688},[2428,4558,4559,4611],{},[2431,4560,4561,4563,4565,4567,4569,4585,4587,4605,4607,4609],{},[2584,4562,121],{},[2584,4564,4320],{"stretchy":2650},[2437,4566,3688],{},[2437,4568,2644],{"mathvariant":2643},[2434,4570,4571,4573],{},[2437,4572,3181],{},[2431,4574,4575,4577,4579],{},[2437,4576,3188],{},[2437,4578,3033],{},[3063,4580,4581,4583],{},[2437,4582,3193],{},[2437,4584,4312],{},[2437,4586,3033],{},[2434,4588,4589,4591],{},[2437,4590,3181],{},[2431,4592,4593,4595,4597,4599],{},[2584,4594,2586],{},[2437,4596,3188],{},[2437,4598,3033],{},[3063,4600,4601,4603],{},[2437,4602,3193],{},[2437,4604,4315],{},[2437,4606,2644],{"mathvariant":2643},[2437,4608,3688],{},[2584,4610,2651],{"stretchy":2650},[2444,4612,4613],{"encoding":2446},"= \\langle \\psi \\vert e^{i H t_m} H e^{-i H t_n} \\vert \\psi \\rangle",[53,4615,4617,4630],{"className":4616,"ariaHidden":2452},[2451],[53,4618,4620,4624,4627],{"className":4619},[2456],[53,4621],{"className":4622,"style":4623},[2460],"height:0.3669em;",[53,4625,121],{"className":4626},[2818],[53,4628],{"className":4629,"style":2814},[2609],[53,4631,4633,4637,4640,4643,4646,4727,4730,4808,4811,4814],{"className":4632},[2456],[53,4634],{"className":4635,"style":4636},[2460],"height:1.1413em;vertical-align:-0.25em;",[53,4638,4320],{"className":4639},[2836],[53,4641,3688],{"className":4642,"style":2671},[2465,2506],[53,4644,2644],{"className":4645},[2465],[53,4647,4649,4652],{"className":4648},[2465],[53,4650,3181],{"className":4651},[2465,2506],[53,4653,4655],{"className":4654},[2474],[53,4656,4658],{"className":4657},[2478],[53,4659,4661],{"className":4660},[2482],[53,4662,4665],{"className":4663,"style":4664},[2486],"height:0.8913em;",[53,4666,4667,4670],{"style":2801},[53,4668],{"className":4669,"style":2495},[2494],[53,4671,4673],{"className":4672},[2499,2500,2501,2502],[53,4674,4676,4679,4682],{"className":4675},[2465,2502],[53,4677,3188],{"className":4678},[2465,2506,2502],[53,4680,3033],{"className":4681,"style":3048},[2465,2506,2502],[53,4683,4685,4688],{"className":4684},[2465,2502],[53,4686,3193],{"className":4687},[2465,2506,2502],[53,4689,4691],{"className":4690},[2474],[53,4692,4694,4718],{"className":4693},[2478,3095],[53,4695,4697,4715],{"className":4696},[2482],[53,4698,4701],{"className":4699,"style":4700},[2486],"height:0.1645em;",[53,4702,4704,4708],{"style":4703},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[53,4705],{"className":4706,"style":4707},[2494],"height:2.5em;",[53,4709,4712],{"className":4710},[2499,4711,2843,2502],"reset-size3",[53,4713,4312],{"className":4714},[2465,2506,2502],[53,4716,3120],{"className":4717},[3119],[53,4719,4721],{"className":4720},[2482],[53,4722,4725],{"className":4723,"style":4724},[2486],"height:0.143em;",[53,4726],{},[53,4728,3033],{"className":4729,"style":3048},[2465,2506],[53,4731,4733,4736],{"className":4732},[2465],[53,4734,3181],{"className":4735},[2465,2506],[53,4737,4739],{"className":4738},[2474],[53,4740,4742],{"className":4741},[2478],[53,4743,4745],{"className":4744},[2482],[53,4746,4748],{"className":4747,"style":4664},[2486],[53,4749,4750,4753],{"style":2801},[53,4751],{"className":4752,"style":2495},[2494],[53,4754,4756],{"className":4755},[2499,2500,2501,2502],[53,4757,4759,4762,4765,4768],{"className":4758},[2465,2502],[53,4760,2586],{"className":4761},[2465,2502],[53,4763,3188],{"className":4764},[2465,2506,2502],[53,4766,3033],{"className":4767,"style":3048},[2465,2506,2502],[53,4769,4771,4774],{"className":4770},[2465,2502],[53,4772,3193],{"className":4773},[2465,2506,2502],[53,4775,4777],{"className":4776},[2474],[53,4778,4780,4800],{"className":4779},[2478,3095],[53,4781,4783,4797],{"className":4782},[2482],[53,4784,4786],{"className":4785,"style":4700},[2486],[53,4787,4788,4791],{"style":4703},[53,4789],{"className":4790,"style":4707},[2494],[53,4792,4794],{"className":4793},[2499,4711,2843,2502],[53,4795,4315],{"className":4796},[2465,2506,2502],[53,4798,3120],{"className":4799},[3119],[53,4801,4803],{"className":4802},[2482],[53,4804,4806],{"className":4805,"style":4724},[2486],[53,4807],{},[53,4809,2644],{"className":4810},[2465],[53,4812,3688],{"className":4813,"style":2671},[2465,2506],[53,4815,2651],{"className":4816},[2675],[53,4818,4820],{"className":4819},[2679],[53,4821,4823,4886],{"className":4822},[2418],[53,4824,4826],{"className":4825},[2422],[2424,4827,4828],{"xmlns":2426,"display":2688},[2428,4829,4830,4883],{},[2431,4831,4832,4834,4836,4838,4840,4857,4859,4877,4879,4881],{},[2584,4833,121],{},[2584,4835,4320],{"stretchy":2650},[2437,4837,3688],{},[2437,4839,2644],{"mathvariant":2643},[2434,4841,4842,4844],{},[2437,4843,3181],{},[2431,4845,4846,4848,4850,4852,4855],{},[2437,4847,3188],{},[2437,4849,3033],{},[2437,4851,4312],{},[2437,4853,4854],{},"d",[2437,4856,3193],{},[2437,4858,3033],{},[2434,4860,4861,4863],{},[2437,4862,3181],{},[2431,4864,4865,4867,4869,4871,4873,4875],{},[2584,4866,2586],{},[2437,4868,3188],{},[2437,4870,3033],{},[2437,4872,4315],{},[2437,4874,4854],{},[2437,4876,3193],{},[2437,4878,2644],{"mathvariant":2643},[2437,4880,3688],{},[2584,4882,2651],{"stretchy":2650},[2444,4884,4885],{"encoding":2446},"= \\langle \\psi \\vert e^{i H m dt} H e^{-i H n dt} \\vert \\psi \\rangle",[53,4887,4889,4901],{"className":4888,"ariaHidden":2452},[2451],[53,4890,4892,4895,4898],{"className":4891},[2456],[53,4893],{"className":4894,"style":4623},[2460],[53,4896,121],{"className":4897},[2818],[53,4899],{"className":4900,"style":2814},[2609],[53,4902,4904,4908,4911,4914,4917,4962,4965,5012,5015,5018],{"className":4903},[2456],[53,4905],{"className":4906,"style":4907},[2460],"height:1.1491em;vertical-align:-0.25em;",[53,4909,4320],{"className":4910},[2836],[53,4912,3688],{"className":4913,"style":2671},[2465,2506],[53,4915,2644],{"className":4916},[2465],[53,4918,4920,4923],{"className":4919},[2465],[53,4921,3181],{"className":4922},[2465,2506],[53,4924,4926],{"className":4925},[2474],[53,4927,4929],{"className":4928},[2478],[53,4930,4932],{"className":4931},[2482],[53,4933,4936],{"className":4934,"style":4935},[2486],"height:0.8991em;",[53,4937,4938,4941],{"style":2801},[53,4939],{"className":4940,"style":2495},[2494],[53,4942,4944],{"className":4943},[2499,2500,2501,2502],[53,4945,4947,4950,4953,4956,4959],{"className":4946},[2465,2502],[53,4948,3188],{"className":4949},[2465,2506,2502],[53,4951,3033],{"className":4952,"style":3048},[2465,2506,2502],[53,4954,4312],{"className":4955},[2465,2506,2502],[53,4957,4854],{"className":4958},[2465,2506,2502],[53,4960,3193],{"className":4961},[2465,2506,2502],[53,4963,3033],{"className":4964,"style":3048},[2465,2506],[53,4966,4968,4971],{"className":4967},[2465],[53,4969,3181],{"className":4970},[2465,2506],[53,4972,4974],{"className":4973},[2474],[53,4975,4977],{"className":4976},[2478],[53,4978,4980],{"className":4979},[2482],[53,4981,4983],{"className":4982,"style":4935},[2486],[53,4984,4985,4988],{"style":2801},[53,4986],{"className":4987,"style":2495},[2494],[53,4989,4991],{"className":4990},[2499,2500,2501,2502],[53,4992,4994,4997,5000,5003,5006,5009],{"className":4993},[2465,2502],[53,4995,2586],{"className":4996},[2465,2502],[53,4998,3188],{"className":4999},[2465,2506,2502],[53,5001,3033],{"className":5002,"style":3048},[2465,2506,2502],[53,5004,4315],{"className":5005},[2465,2506,2502],[53,5007,4854],{"className":5008},[2465,2506,2502],[53,5010,3193],{"className":5011},[2465,2506,2502],[53,5013,2644],{"className":5014},[2465],[53,5016,3688],{"className":5017,"style":2671},[2465,2506],[53,5019,2651],{"className":5020},[2675],[12,5022,5023,5024,5233,5234,5339,5340,5374,5375,5444,5445,5473,5474,5526,5527,5555,5556,5613,5614,5674,5675,5735,5736],{},"Where ",[53,5025,5027,5075],{"className":5026},[2418],[53,5028,5030],{"className":5029},[2422],[2424,5031,5032],{"xmlns":2426},[2428,5033,5034,5072],{},[2431,5035,5036,5038,5044,5046,5048,5066,5068,5070],{},[2437,5037,2644],{"mathvariant":2643},[3063,5039,5040,5042],{},[2437,5041,3688],{},[2437,5043,4315],{},[2584,5045,2651],{"stretchy":2650},[2584,5047,121],{},[2434,5049,5050,5052],{},[2437,5051,3181],{},[2431,5053,5054,5056,5058,5060],{},[2584,5055,2586],{},[2437,5057,3188],{},[2437,5059,3033],{},[3063,5061,5062,5064],{},[2437,5063,3193],{},[2437,5065,4315],{},[2437,5067,2644],{"mathvariant":2643},[2437,5069,3688],{},[2584,5071,2651],{"stretchy":2650},[2444,5073,5074],{"encoding":2446},"\\vert \\psi_n \\rangle = e^{-i H t_n} \\vert \\psi \\rangle",[53,5076,5078,5139],{"className":5077,"ariaHidden":2452},[2451],[53,5079,5081,5084,5087,5127,5130,5133,5136],{"className":5080},[2456],[53,5082],{"className":5083,"style":2664},[2460],[53,5085,2644],{"className":5086},[2465],[53,5088,5090,5093],{"className":5089},[2465],[53,5091,3688],{"className":5092,"style":2671},[2465,2506],[53,5094,5096],{"className":5095},[2474],[53,5097,5099,5119],{"className":5098},[2478,3095],[53,5100,5102,5116],{"className":5101},[2482],[53,5103,5105],{"className":5104,"style":4402},[2486],[53,5106,5107,5110],{"style":4467},[53,5108],{"className":5109,"style":2495},[2494],[53,5111,5113],{"className":5112},[2499,2500,2501,2502],[53,5114,4315],{"className":5115},[2465,2506,2502],[53,5117,3120],{"className":5118},[3119],[53,5120,5122],{"className":5121},[2482],[53,5123,5125],{"className":5124,"style":3127},[2486],[53,5126],{},[53,5128,2651],{"className":5129},[2675],[53,5131],{"className":5132,"style":2814},[2609],[53,5134,121],{"className":5135},[2818],[53,5137],{"className":5138,"style":2814},[2609],[53,5140,5142,5146,5224,5227,5230],{"className":5141},[2456],[53,5143],{"className":5144,"style":5145},[2460],"height:1.0913em;vertical-align:-0.25em;",[53,5147,5149,5152],{"className":5148},[2465],[53,5150,3181],{"className":5151},[2465,2506],[53,5153,5155],{"className":5154},[2474],[53,5156,5158],{"className":5157},[2478],[53,5159,5161],{"className":5160},[2482],[53,5162,5164],{"className":5163,"style":3225},[2486],[53,5165,5166,5169],{"style":2490},[53,5167],{"className":5168,"style":2495},[2494],[53,5170,5172],{"className":5171},[2499,2500,2501,2502],[53,5173,5175,5178,5181,5184],{"className":5174},[2465,2502],[53,5176,2586],{"className":5177},[2465,2502],[53,5179,3188],{"className":5180},[2465,2506,2502],[53,5182,3033],{"className":5183,"style":3048},[2465,2506,2502],[53,5185,5187,5190],{"className":5186},[2465,2502],[53,5188,3193],{"className":5189},[2465,2506,2502],[53,5191,5193],{"className":5192},[2474],[53,5194,5196,5216],{"className":5195},[2478,3095],[53,5197,5199,5213],{"className":5198},[2482],[53,5200,5202],{"className":5201,"style":4700},[2486],[53,5203,5204,5207],{"style":4703},[53,5205],{"className":5206,"style":4707},[2494],[53,5208,5210],{"className":5209},[2499,4711,2843,2502],[53,5211,4315],{"className":5212},[2465,2506,2502],[53,5214,3120],{"className":5215},[3119],[53,5217,5219],{"className":5218},[2482],[53,5220,5222],{"className":5221,"style":4724},[2486],[53,5223],{},[53,5225,2644],{"className":5226},[2465],[53,5228,3688],{"className":5229,"style":2671},[2465,2506],[53,5231,2651],{"className":5232},[2675]," are the vectors of the unitary Krylov space and ",[53,5235,5237,5263],{"className":5236},[2418],[53,5238,5240],{"className":5239},[2422],[2424,5241,5242],{"xmlns":2426},[2428,5243,5244,5260],{},[2431,5245,5246,5252,5254,5256,5258],{},[3063,5247,5248,5250],{},[2437,5249,3193],{},[2437,5251,4315],{},[2584,5253,121],{},[2437,5255,4315],{},[2437,5257,4854],{},[2437,5259,3193],{},[2444,5261,5262],{"encoding":2446},"t_n = n dt",[53,5264,5266,5323],{"className":5265,"ariaHidden":2452},[2451],[53,5267,5269,5273,5314,5317,5320],{"className":5268},[2456],[53,5270],{"className":5271,"style":5272},[2460],"height:0.7651em;vertical-align:-0.15em;",[53,5274,5276,5279],{"className":5275},[2465],[53,5277,3193],{"className":5278},[2465,2506],[53,5280,5282],{"className":5281},[2474],[53,5283,5285,5306],{"className":5284},[2478,3095],[53,5286,5288,5303],{"className":5287},[2482],[53,5289,5291],{"className":5290,"style":4402},[2486],[53,5292,5294,5297],{"style":5293},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[53,5295],{"className":5296,"style":2495},[2494],[53,5298,5300],{"className":5299},[2499,2500,2501,2502],[53,5301,4315],{"className":5302},[2465,2506,2502],[53,5304,3120],{"className":5305},[3119],[53,5307,5309],{"className":5308},[2482],[53,5310,5312],{"className":5311,"style":3127},[2486],[53,5313],{},[53,5315],{"className":5316,"style":2814},[2609],[53,5318,121],{"className":5319},[2818],[53,5321],{"className":5322,"style":2814},[2609],[53,5324,5326,5330,5333,5336],{"className":5325},[2456],[53,5327],{"className":5328,"style":5329},[2460],"height:0.6944em;",[53,5331,4315],{"className":5332},[2465,2506],[53,5334,4854],{"className":5335},[2465,2506],[53,5337,3193],{"className":5338},[2465,2506]," are the multiples of the time step ",[53,5341,5343,5359],{"className":5342},[2418],[53,5344,5346],{"className":5345},[2422],[2424,5347,5348],{"xmlns":2426},[2428,5349,5350,5356],{},[2431,5351,5352,5354],{},[2437,5353,4854],{},[2437,5355,3193],{},[2444,5357,5358],{"encoding":2446},"dt",[53,5360,5362],{"className":5361,"ariaHidden":2452},[2451],[53,5363,5365,5368,5371],{"className":5364},[2456],[53,5366],{"className":5367,"style":5329},[2460],[53,5369,4854],{"className":5370},[2465,2506],[53,5372,3193],{"className":5373},[2465,2506]," 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 ",[53,5376,5378,5395],{"className":5377},[2418],[53,5379,5381],{"className":5380},[2422],[2424,5382,5383],{"xmlns":2426},[2428,5384,5385,5393],{},[2431,5386,5387],{},[3063,5388,5389,5391],{},[2437,5390,2440],{"mathvariant":2439},[2437,5392,3174],{},[2444,5394,3288],{"encoding":2446},[53,5396,5398],{"className":5397,"ariaHidden":2452},[2451],[53,5399,5401,5404],{"className":5400},[2456],[53,5402],{"className":5403,"style":3082},[2460],[53,5405,5407,5410],{"className":5406},[2465],[53,5408,2440],{"className":5409,"style":2470},[2465,2469],[53,5411,5413],{"className":5412},[2474],[53,5414,5416,5436],{"className":5415},[2478,3095],[53,5417,5419,5433],{"className":5418},[2482],[53,5420,5422],{"className":5421,"style":3102},[2486],[53,5423,5424,5427],{"style":3105},[53,5425],{"className":5426,"style":2495},[2494],[53,5428,5430],{"className":5429},[2499,2500,2501,2502],[53,5431,3174],{"className":5432,"style":3209},[2465,2506,2502],[53,5434,3120],{"className":5435},[3119],[53,5437,5439],{"className":5438},[2482],[53,5440,5442],{"className":5441,"style":3127},[2486],[53,5443],{}," has dimension ",[53,5446,5448,5461],{"className":5447},[2418],[53,5449,5451],{"className":5450},[2422],[2424,5452,5453],{"xmlns":2426},[2428,5454,5455,5459],{},[2431,5456,5457],{},[2437,5458,2111],{},[2444,5460,2111],{"encoding":2446},[53,5462,5464],{"className":5463,"ariaHidden":2452},[2451],[53,5465,5467,5470],{"className":5466},[2456],[53,5468],{"className":5469,"style":2534},[2460],[53,5471,2111],{"className":5472,"style":2507},[2465,2506],", the Hamiltonian projected into the subspace will have dimensions ",[53,5475,5477,5496],{"className":5476},[2418],[53,5478,5480],{"className":5479},[2422],[2424,5481,5482],{"xmlns":2426},[2428,5483,5484,5493],{},[2431,5485,5486,5488,5491],{},[2437,5487,2111],{},[2584,5489,5490],{},"×",[2437,5492,2111],{},[2444,5494,5495],{"encoding":2446},"r \\times r",[53,5497,5499,5517],{"className":5498,"ariaHidden":2452},[2451],[53,5500,5502,5505,5508,5511,5514],{"className":5501},[2456],[53,5503],{"className":5504,"style":2602},[2460],[53,5506,2111],{"className":5507,"style":2507},[2465,2506],[53,5509],{"className":5510,"style":2610},[2609],[53,5512,5490],{"className":5513},[2614],[53,5515],{"className":5516,"style":2610},[2609],[53,5518,5520,5523],{"className":5519},[2456],[53,5521],{"className":5522,"style":2534},[2460],[53,5524,2111],{"className":5525,"style":2507},[2465,2506],". With ",[53,5528,5530,5543],{"className":5529},[2418],[53,5531,5533],{"className":5532},[2422],[2424,5534,5535],{"xmlns":2426},[2428,5536,5537,5541],{},[2431,5538,5539],{},[2437,5540,2111],{},[2444,5542,2111],{"encoding":2446},[53,5544,5546],{"className":5545,"ariaHidden":2452},[2451],[53,5547,5549,5552],{"className":5548},[2456],[53,5550],{"className":5551,"style":2534},[2460],[53,5553,2111],{"className":5554,"style":2507},[2465,2506]," small enough (generally ",[53,5557,5559,5581],{"className":5558},[2418],[53,5560,5562],{"className":5561},[2422],[2424,5563,5564],{"xmlns":2426},[2428,5565,5566,5578],{},[2431,5567,5568,5570,5573,5575],{},[2437,5569,2111],{},[2584,5571,5572],{},"\u003C",[2584,5574,5572],{},[2588,5576,5577],{},"100",[2444,5579,5580],{"encoding":2446},"r\u003C\u003C100",[53,5582,5584,5604],{"className":5583,"ariaHidden":2452},[2451],[53,5585,5587,5591,5594,5597,5601],{"className":5586},[2456],[53,5588],{"className":5589,"style":5590},[2460],"height:0.5782em;vertical-align:-0.0391em;",[53,5592,2111],{"className":5593,"style":2507},[2465,2506],[53,5595],{"className":5596,"style":2814},[2609],[53,5598,5600],{"className":5599},[2818],"\u003C\u003C",[53,5602],{"className":5603,"style":2814},[2609],[53,5605,5607,5610],{"className":5606},[2456],[53,5608],{"className":5609,"style":2624},[2460],[53,5611,5577],{"className":5612},[2465]," is sufficient to obtain convergence of estimates of eigenenergies) we can then easily diagonalize the projected Hamiltonian ",[53,5615,5617,5634],{"className":5616},[2418],[53,5618,5620],{"className":5619},[2422],[2424,5621,5622],{"xmlns":2426},[2428,5623,5624,5632],{},[2431,5625,5626],{},[4166,5627,5628,5630],{"accent":2452},[2437,5629,3033],{},[2584,5631,4172],{},[2444,5633,4175],{"encoding":2446},[53,5635,5637],{"className":5636,"ariaHidden":2452},[2451],[53,5638,5640,5643],{"className":5639},[2456],[53,5641],{"className":5642,"style":4185},[2460],[53,5644,5646],{"className":5645},[2465,4189],[53,5647,5649],{"className":5648},[2478],[53,5650,5652],{"className":5651},[2482],[53,5653,5655,5663],{"className":5654,"style":4185},[2486],[53,5656,5657,5660],{"style":4201},[53,5658],{"className":5659,"style":4205},[2494],[53,5661,3033],{"className":5662,"style":3048},[2465,2506],[53,5664,5665,5668],{"style":4211},[53,5666],{"className":5667,"style":4205},[2494],[53,5669,5671],{"className":5670,"style":4219},[4218],[53,5672,4172],{"className":5673},[2465],". However, we cannot directly diagonalize ",[53,5676,5678,5695],{"className":5677},[2418],[53,5679,5681],{"className":5680},[2422],[2424,5682,5683],{"xmlns":2426},[2428,5684,5685,5693],{},[2431,5686,5687],{},[4166,5688,5689,5691],{"accent":2452},[2437,5690,3033],{},[2584,5692,4172],{},[2444,5694,4175],{"encoding":2446},[53,5696,5698],{"className":5697,"ariaHidden":2452},[2451],[53,5699,5701,5704],{"className":5700},[2456],[53,5702],{"className":5703,"style":4185},[2460],[53,5705,5707],{"className":5706},[2465,4189],[53,5708,5710],{"className":5709},[2478],[53,5711,5713],{"className":5712},[2482],[53,5714,5716,5724],{"className":5715,"style":4185},[2486],[53,5717,5718,5721],{"style":4201},[53,5719],{"className":5720,"style":4205},[2494],[53,5722,3033],{"className":5723,"style":3048},[2465,2506],[53,5725,5726,5729],{"style":4211},[53,5727],{"className":5728,"style":4205},[2494],[53,5730,5732],{"className":5731,"style":4219},[4218],[53,5733,4172],{"className":5734},[2465]," because of the non-orthogonality of the Krylov space vectors. We'll have to measure their overlaps and construct a matrix ",[53,5737,5739,5758],{"className":5738},[2418],[53,5740,5742],{"className":5741},[2422],[2424,5743,5744],{"xmlns":2426},[2428,5745,5746,5755],{},[2431,5747,5748],{},[4166,5749,5750,5753],{"accent":2452},[2437,5751,5752],{},"S",[2584,5754,4172],{},[2444,5756,5757],{"encoding":2446},"\\tilde{S}",[53,5759,5761],{"className":5760,"ariaHidden":2452},[2451],[53,5762,5764,5767],{"className":5763},[2456],[53,5765],{"className":5766,"style":4185},[2460],[53,5768,5770],{"className":5769},[2465,4189],[53,5771,5773],{"className":5772},[2478],[53,5774,5776],{"className":5775},[2482],[53,5777,5779,5788],{"className":5778,"style":4185},[2486],[53,5780,5781,5784],{"style":4201},[53,5782],{"className":5783,"style":4205},[2494],[53,5785,5752],{"className":5786,"style":5787},[2465,2506],"margin-right:0.0576em;",[53,5789,5790,5793],{"style":4211},[53,5791],{"className":5792,"style":4205},[2494],[53,5794,5797],{"className":5795,"style":5796},[4218],"left:-0.1667em;",[53,5798,4172],{"className":5799},[2465],[53,5801,5803],{"className":5802},[2679],[53,5804,5806,5852],{"className":5805},[2418],[53,5807,5809],{"className":5808},[2422],[2424,5810,5811],{"xmlns":2426,"display":2688},[2428,5812,5813,5849],{},[2431,5814,5815,5829,5831,5833,5839,5841,5847],{},[3063,5816,5817,5823],{},[4166,5818,5819,5821],{"accent":2452},[2437,5820,5752],{},[2584,5822,4172],{},[2431,5824,5825,5827],{},[2437,5826,4312],{},[2437,5828,4315],{},[2584,5830,121],{},[2584,5832,4320],{"stretchy":2650},[3063,5834,5835,5837],{},[2437,5836,3688],{},[2437,5838,4312],{},[2437,5840,2644],{"mathvariant":2643},[3063,5842,5843,5845],{},[2437,5844,3688],{},[2437,5846,4315],{},[2584,5848,2651],{"stretchy":2650},[2444,5850,5851],{"encoding":2446},"\\tilde{S}_{mn} = \\langle \\psi_m \\vert \\psi_n \\rangle",[53,5853,5855,5942],{"className":5854,"ariaHidden":2452},[2451],[53,5856,5858,5861,5933,5936,5939],{"className":5857},[2456],[53,5859],{"className":5860,"style":4355},[2460],[53,5862,5864,5895],{"className":5863},[2465],[53,5865,5867],{"className":5866},[2465,4189],[53,5868,5870],{"className":5869},[2478],[53,5871,5873],{"className":5872},[2482],[53,5874,5876,5884],{"className":5875,"style":4185},[2486],[53,5877,5878,5881],{"style":4201},[53,5879],{"className":5880,"style":4205},[2494],[53,5882,5752],{"className":5883,"style":5787},[2465,2506],[53,5885,5886,5889],{"style":4211},[53,5887],{"className":5888,"style":4205},[2494],[53,5890,5892],{"className":5891,"style":5796},[4218],[53,5893,4172],{"className":5894},[2465],[53,5896,5898],{"className":5897},[2474],[53,5899,5901,5925],{"className":5900},[2478,3095],[53,5902,5904,5922],{"className":5903},[2482],[53,5905,5907],{"className":5906,"style":4402},[2486],[53,5908,5910,5913],{"style":5909},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[53,5911],{"className":5912,"style":2495},[2494],[53,5914,5916],{"className":5915},[2499,2500,2501,2502],[53,5917,5919],{"className":5918},[2465,2502],[53,5920,2588],{"className":5921},[2465,2506,2502],[53,5923,3120],{"className":5924},[3119],[53,5926,5928],{"className":5927},[2482],[53,5929,5931],{"className":5930,"style":3127},[2486],[53,5932],{},[53,5934],{"className":5935,"style":2814},[2609],[53,5937,121],{"className":5938},[2818],[53,5940],{"className":5941,"style":2814},[2609],[53,5943,5945,5948,5951,5991,5994,6034],{"className":5944},[2456],[53,5946],{"className":5947,"style":2664},[2460],[53,5949,4320],{"className":5950},[2836],[53,5952,5954,5957],{"className":5953},[2465],[53,5955,3688],{"className":5956,"style":2671},[2465,2506],[53,5958,5960],{"className":5959},[2474],[53,5961,5963,5983],{"className":5962},[2478,3095],[53,5964,5966,5980],{"className":5965},[2482],[53,5967,5969],{"className":5968,"style":4402},[2486],[53,5970,5971,5974],{"style":4467},[53,5972],{"className":5973,"style":2495},[2494],[53,5975,5977],{"className":5976},[2499,2500,2501,2502],[53,5978,4312],{"className":5979},[2465,2506,2502],[53,5981,3120],{"className":5982},[3119],[53,5984,5986],{"className":5985},[2482],[53,5987,5989],{"className":5988,"style":3127},[2486],[53,5990],{},[53,5992,2644],{"className":5993},[2465],[53,5995,5997,6000],{"className":5996},[2465],[53,5998,3688],{"className":5999,"style":2671},[2465,2506],[53,6001,6003],{"className":6002},[2474],[53,6004,6006,6026],{"className":6005},[2478,3095],[53,6007,6009,6023],{"className":6008},[2482],[53,6010,6012],{"className":6011,"style":4402},[2486],[53,6013,6014,6017],{"style":4467},[53,6015],{"className":6016,"style":2495},[2494],[53,6018,6020],{"className":6019},[2499,2500,2501,2502],[53,6021,4315],{"className":6022},[2465,2506,2502],[53,6024,3120],{"className":6025},[3119],[53,6027,6029],{"className":6028},[2482],[53,6030,6032],{"className":6031,"style":3127},[2486],[53,6033],{},[53,6035,2651],{"className":6036},[2675],[12,6038,6039],{},"This allows us to solve the eigenvalue problem in a non-orthogonal space (also called generalized eigenvalue problem)",[53,6041,6043],{"className":6042},[2679],[53,6044,6046,6097],{"className":6045},[2418],[53,6047,6049],{"className":6048},[2422],[2424,6050,6051],{"xmlns":2426,"display":2688},[2428,6052,6053,6094],{},[2431,6054,6055,6061,6065,6073,6075,6078,6080,6086,6088],{},[4166,6056,6057,6059],{"accent":2452},[2437,6058,3033],{},[2584,6060,4172],{},[6062,6063,6064],"mtext",{}," ",[4166,6066,6067,6070],{"accent":2452},[2437,6068,6069],{},"c",[2584,6071,6072],{},"⃗",[2584,6074,121],{},[2437,6076,6077],{},"E",[6062,6079,6064],{},[4166,6081,6082,6084],{"accent":2452},[2437,6083,5752],{},[2584,6085,4172],{},[6062,6087,6064],{},[4166,6089,6090,6092],{"accent":2452},[2437,6091,6069],{},[2584,6093,6072],{},[2444,6095,6096],{"encoding":2446},"\\tilde{H} \\ \\vec{c} = E \\ \\tilde{S} \\ 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can then obtain estimates of the eigenvalues and eigenstates of ",[53,6282,6284,6297],{"className":6283},[2418],[53,6285,6287],{"className":6286},[2422],[2424,6288,6289],{"xmlns":2426},[2428,6290,6291,6295],{},[2431,6292,6293],{},[2437,6294,3033],{},[2444,6296,3033],{"encoding":2446},[53,6298,6300],{"className":6299,"ariaHidden":2452},[2451],[53,6301,6303,6306],{"className":6302},[2456],[53,6304],{"className":6305,"style":2461},[2460],[53,6307,3033],{"className":6308,"style":3048},[2465,2506]," by looking at the ones of 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For example, the estimate of the ground state energy is obtained by taking the smallest eigenvalue ",[53,6372,6374,6387],{"className":6373},[2418],[53,6375,6377],{"className":6376},[2422],[2424,6378,6379],{"xmlns":2426},[2428,6380,6381,6385],{},[2431,6382,6383],{},[2437,6384,6069],{},[2444,6386,6069],{"encoding":2446},[53,6388,6390],{"className":6389,"ariaHidden":2452},[2451],[53,6391,6393,6396],{"className":6392},[2456],[53,6394],{"className":6395,"style":2534},[2460],[53,6397,6069],{"className":6398},[2465,2506]," and the ground state from the corresponding eigenvector ",[53,6401,6403,6421],{"className":6402},[2418],[53,6404,6406],{"className":6405},[2422],[2424,6407,6408],{"xmlns":2426},[2428,6409,6410,6418],{},[2431,6411,6412],{},[4166,6413,6414,6416],{"accent":2452},[2437,6415,6069],{},[2584,6417,6072],{},[2444,6419,6420],{"encoding":2446},"\\vec{c}",[53,6422,6424],{"className":6423,"ariaHidden":2452},[2451],[53,6425,6427,6430],{"className":6426},[2456],[53,6428],{"className":6429,"style":6152},[2460],[53,6431,6433],{"className":6432},[2465,4189],[53,6434,6436],{"className":6435},[2478],[53,6437,6439],{"className":6438},[2482],[53,6440,6442,6450],{"className":6441,"style":6152},[2486],[53,6443,6444,6447],{"style":4201},[53,6445],{"className":6446,"style":4205},[2494],[53,6448,6069],{"className":6449},[2465,2506],[53,6451,6452,6455],{"style":4201},[53,6453],{"className":6454,"style":4205},[2494],[53,6456,6458],{"className":6457,"style":6169},[4218],[53,6459,6461],{"className":6460,"style":6174},[6173],[6176,6462,6463],{"xmlns":6178,"width":6179,"height":6180,"style":6181,"viewBox":6182,"preserveAspectRatio":6183},[6185,6464],{"d":6187},". The coefficients in ",[53,6467,6469,6486],{"className":6468},[2418],[53,6470,6472],{"className":6471},[2422],[2424,6473,6474],{"xmlns":2426},[2428,6475,6476,6484],{},[2431,6477,6478],{},[4166,6479,6480,6482],{"accent":2452},[2437,6481,6069],{},[2584,6483,6072],{},[2444,6485,6420],{"encoding":2446},[53,6487,6489],{"className":6488,"ariaHidden":2452},[2451],[53,6490,6492,6495],{"className":6491},[2456],[53,6493],{"className":6494,"style":6152},[2460],[53,6496,6498],{"className":6497},[2465,4189],[53,6499,6501],{"className":6500},[2478],[53,6502,6504],{"className":6503},[2482],[53,6505,6507,6515],{"className":6506,"style":6152},[2486],[53,6508,6509,6512],{"style":4201},[53,6510],{"className":6511,"style":4205},[2494],[53,6513,6069],{"className":6514},[2465,2506],[53,6516,6517,6520],{"style":4201},[53,6518],{"className":6519,"style":4205},[2494],[53,6521,6523],{"className":6522,"style":6169},[4218],[53,6524,6526],{"className":6525,"style":6174},[6173],[6176,6527,6528],{"xmlns":6178,"width":6179,"height":6180,"style":6181,"viewBox":6182,"preserveAspectRatio":6183},[6185,6529],{"d":6187}," determines the contribution of the different vectors that span ",[53,6532,6534,6551],{"className":6533},[2418],[53,6535,6537],{"className":6536},[2422],[2424,6538,6539],{"xmlns":2426},[2428,6540,6541,6549],{},[2431,6542,6543],{},[3063,6544,6545,6547],{},[2437,6546,2440],{"mathvariant":2439},[2437,6548,3174],{},[2444,6550,3288],{"encoding":2446},[53,6552,6554],{"className":6553,"ariaHidden":2452},[2451],[53,6555,6557,6560],{"className":6556},[2456],[53,6558],{"className":6559,"style":3082},[2460],[53,6561,6563,6566],{"className":6562},[2465],[53,6564,2440],{"className":6565,"style":2470},[2465,2469],[53,6567,6569],{"className":6568},[2474],[53,6570,6572,6592],{"className":6571},[2478,3095],[53,6573,6575,6589],{"className":6574},[2482],[53,6576,6578],{"className":6577,"style":3102},[2486],[53,6579,6580,6583],{"style":3105},[53,6581],{"className":6582,"style":2495},[2494],[53,6584,6586],{"className":6585},[2499,2500,2501,2502],[53,6587,3174],{"className":6588,"style":3209},[2465,2506,2502],[53,6590,3120],{"className":6591},[3119],[53,6593,6595],{"className":6594},[2482],[53,6596,6598],{"className":6597,"style":3127},[2486],[53,6599],{},[12,6601,6602],{},[6603,6604],"img",{"alt":6605,"src":6606},"fig1.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-06.avif",[12,6608,6609,6610,6731,6732,170,6812,6893,6894,170,6981,7065,7066,7126,7127,7209,7210,7289,7290,7372,7373,7401,7402,7553],{},"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 ",[53,6611,6613,6641],{"className":6612},[2418],[53,6614,6616],{"className":6615},[2422],[2424,6617,6618],{"xmlns":2426},[2428,6619,6620,6638],{},[2431,6621,6622],{},[3063,6623,6624,6630],{},[4166,6625,6626,6628],{"accent":2452},[2437,6627,3033],{},[2584,6629,4172],{},[2431,6631,6632,6634,6636],{},[2437,6633,3188],{},[2584,6635,2714],{"separator":2452},[2437,6637,2267],{},[2444,6639,6640],{"encoding":2446},"\\tilde{H}_{i,j}",[53,6642,6644],{"className":6643,"ariaHidden":2452},[2451],[53,6645,6647,6651],{"className":6646},[2456],[53,6648],{"className":6649,"style":6650},[2460],"height:1.2063em;vertical-align:-0.2861em;",[53,6652,6654,6685],{"className":6653},[2465],[53,6655,6657],{"className":6656},[2465,4189],[53,6658,6660],{"className":6659},[2478],[53,6661,6663],{"className":6662},[2482],[53,6664,6666,6674],{"className":6665,"style":4185},[2486],[53,6667,6668,6671],{"style":4201},[53,6669],{"className":6670,"style":4205},[2494],[53,6672,3033],{"className":6673,"style":3048},[2465,2506],[53,6675,6676,6679],{"style":4211},[53,6677],{"className":6678,"style":4205},[2494],[53,6680,6682],{"className":6681,"style":4219},[4218],[53,6683,4172],{"className":6684},[2465],[53,6686,6688],{"className":6687},[2474],[53,6689,6691,6722],{"className":6690},[2478,3095],[53,6692,6694,6719],{"className":6693},[2482],[53,6695,6698],{"className":6696,"style":6697},[2486],"height:0.3117em;",[53,6699,6700,6703],{"style":4405},[53,6701],{"className":6702,"style":2495},[2494],[53,6704,6706],{"className":6705},[2499,2500,2501,2502],[53,6707,6709,6712,6715],{"className":6708},[2465,2502],[53,6710,3188],{"className":6711},[2465,2506,2502],[53,6713,2714],{"className":6714},[2856,2502],[53,6716,2267],{"className":6717,"style":6718},[2465,2506,2502],"margin-right:0.0572em;",[53,6720,3120],{"className":6721},[3119],[53,6723,6725],{"className":6724},[2482],[53,6726,6729],{"className":6727,"style":6728},[2486],"height:0.2861em;",[53,6730],{},", a Hadamard test between the state ",[53,6733,6735,6757],{"className":6734},[2418],[53,6736,6738],{"className":6737},[2422],[2424,6739,6740],{"xmlns":2426},[2428,6741,6742,6754],{},[2431,6743,6744,6746,6752],{},[2437,6745,2644],{"mathvariant":2643},[3063,6747,6748,6750],{},[2437,6749,3688],{},[2437,6751,3188],{},[2584,6753,2651],{"stretchy":2650},[2444,6755,6756],{"encoding":2446},"\\vert \\psi_i \\rangle",[53,6758,6760],{"className":6759,"ariaHidden":2452},[2451],[53,6761,6763,6766,6769,6809],{"className":6762},[2456],[53,6764],{"className":6765,"style":2664},[2460],[53,6767,2644],{"className":6768},[2465],[53,6770,6772,6775],{"className":6771},[2465],[53,6773,3688],{"className":6774,"style":2671},[2465,2506],[53,6776,6778],{"className":6777},[2474],[53,6779,6781,6801],{"className":6780},[2478,3095],[53,6782,6784,6798],{"className":6783},[2482],[53,6785,6787],{"className":6786,"style":6697},[2486],[53,6788,6789,6792],{"style":4467},[53,6790],{"className":6791,"style":2495},[2494],[53,6793,6795],{"className":6794},[2499,2500,2501,2502],[53,6796,3188],{"className":6797},[2465,2506,2502],[53,6799,3120],{"className":6800},[3119],[53,6802,6804],{"className":6803},[2482],[53,6805,6807],{"className":6806,"style":3127},[2486],[53,6808],{},[53,6810,2651],{"className":6811},[2675],[53,6813,6815,6837],{"className":6814},[2418],[53,6816,6818],{"className":6817},[2422],[2424,6819,6820],{"xmlns":2426},[2428,6821,6822,6834],{},[2431,6823,6824,6826,6832],{},[2437,6825,2644],{"mathvariant":2643},[3063,6827,6828,6830],{},[2437,6829,3688],{},[2437,6831,2267],{},[2584,6833,2651],{"stretchy":2650},[2444,6835,6836],{"encoding":2446},"\\vert \\psi_j \\rangle",[53,6838,6840],{"className":6839,"ariaHidden":2452},[2451],[53,6841,6843,6847,6850,6890],{"className":6842},[2456],[53,6844],{"className":6845,"style":6846},[2460],"height:1.0361em;vertical-align:-0.2861em;",[53,6848,2644],{"className":6849},[2465],[53,6851,6853,6856],{"className":6852},[2465],[53,6854,3688],{"className":6855,"style":2671},[2465,2506],[53,6857,6859],{"className":6858},[2474],[53,6860,6862,6882],{"className":6861},[2478,3095],[53,6863,6865,6879],{"className":6864},[2482],[53,6866,6868],{"className":6867,"style":6697},[2486],[53,6869,6870,6873],{"style":4467},[53,6871],{"className":6872,"style":2495},[2494],[53,6874,6876],{"className":6875},[2499,2500,2501,2502],[53,6877,2267],{"className":6878,"style":6718},[2465,2506,2502],[53,6880,3120],{"className":6881},[3119],[53,6883,6885],{"className":6884},[2482],[53,6886,6888],{"className":6887,"style":6728},[2486],[53,6889],{},[53,6891,2651],{"className":6892},[2675]," is carried out. This is highlighted in the figure by the color scheme for the matrix elements and the corresponding ",[53,6895,6897,6921],{"className":6896},[2418],[53,6898,6900],{"className":6899},[2422],[2424,6901,6902],{"xmlns":2426},[2428,6903,6904,6918],{},[2431,6905,6906,6909,6912],{},[6062,6907,6908],{},"Prep",[6062,6910,6911],{},"  ",[3063,6913,6914,6916],{},[2437,6915,3688],{},[2437,6917,3188],{},[2444,6919,6920],{"encoding":2446},"\\text{Prep} \\; \\psi_i",[53,6922,6924],{"className":6923,"ariaHidden":2452},[2451],[53,6925,6927,6931,6938,6941],{"className":6926},[2456],[53,6928],{"className":6929,"style":6930},[2460],"height:0.8889em;vertical-align:-0.1944em;",[53,6932,6935],{"className":6933},[2465,6934],"text",[53,6936,6908],{"className":6937},[2465],[53,6939],{"className":6940,"style":2814},[2609],[53,6942,6944,6947],{"className":6943},[2465],[53,6945,3688],{"className":6946,"style":2671},[2465,2506],[53,6948,6950],{"className":6949},[2474],[53,6951,6953,6973],{"className":6952},[2478,3095],[53,6954,6956,6970],{"className":6955},[2482],[53,6957,6959],{"className":6958,"style":6697},[2486],[53,6960,6961,6964],{"style":4467},[53,6962],{"className":6963,"style":2495},[2494],[53,6965,6967],{"className":6966},[2499,2500,2501,2502],[53,6968,3188],{"className":6969},[2465,2506,2502],[53,6971,3120],{"className":6972},[3119],[53,6974,6976],{"className":6975},[2482],[53,6977,6979],{"className":6978,"style":3127},[2486],[53,6980],{},[53,6982,6984,7006],{"className":6983},[2418],[53,6985,6987],{"className":6986},[2422],[2424,6988,6989],{"xmlns":2426},[2428,6990,6991,7003],{},[2431,6992,6993,6995,6997],{},[6062,6994,6908],{},[6062,6996,6911],{},[3063,6998,6999,7001],{},[2437,7000,3688],{},[2437,7002,2267],{},[2444,7004,7005],{"encoding":2446},"\\text{Prep} \\; \\psi_j",[53,7007,7009],{"className":7008,"ariaHidden":2452},[2451],[53,7010,7012,7016,7022,7025],{"className":7011},[2456],[53,7013],{"className":7014,"style":7015},[2460],"height:0.9805em;vertical-align:-0.2861em;",[53,7017,7019],{"className":7018},[2465,6934],[53,7020,6908],{"className":7021},[2465],[53,7023],{"className":7024,"style":2814},[2609],[53,7026,7028,7031],{"className":7027},[2465],[53,7029,3688],{"className":7030,"style":2671},[2465,2506],[53,7032,7034],{"className":7033},[2474],[53,7035,7037,7057],{"className":7036},[2478,3095],[53,7038,7040,7054],{"className":7039},[2482],[53,7041,7043],{"className":7042,"style":6697},[2486],[53,7044,7045,7048],{"style":4467},[53,7046],{"className":7047,"style":2495},[2494],[53,7049,7051],{"className":7050},[2499,2500,2501,2502],[53,7052,2267],{"className":7053,"style":6718},[2465,2506,2502],[53,7055,3120],{"className":7056},[3119],[53,7058,7060],{"className":7059},[2482],[53,7061,7063],{"className":7062,"style":6728},[2486],[53,7064],{}," 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 ",[53,7067,7069,7086],{"className":7068},[2418],[53,7070,7072],{"className":7071},[2422],[2424,7073,7074],{"xmlns":2426},[2428,7075,7076,7084],{},[2431,7077,7078],{},[4166,7079,7080,7082],{"accent":2452},[2437,7081,3033],{},[2584,7083,4172],{},[2444,7085,4175],{"encoding":2446},[53,7087,7089],{"className":7088,"ariaHidden":2452},[2451],[53,7090,7092,7095],{"className":7091},[2456],[53,7093],{"className":7094,"style":4185},[2460],[53,7096,7098],{"className":7097},[2465,4189],[53,7099,7101],{"className":7100},[2478],[53,7102,7104],{"className":7103},[2482],[53,7105,7107,7115],{"className":7106,"style":4185},[2486],[53,7108,7109,7112],{"style":4201},[53,7110],{"className":7111,"style":4205},[2494],[53,7113,3033],{"className":7114,"style":3048},[2465,2506],[53,7116,7117,7120],{"style":4211},[53,7118],{"className":7119,"style":4205},[2494],[53,7121,7123],{"className":7122,"style":4219},[4218],[53,7124,4172],{"className":7125},[2465],". 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 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operation prepares the system qubit in the state ",[53,7211,7213,7234],{"className":7212},[2418],[53,7214,7216],{"className":7215},[2422],[2424,7217,7218],{"xmlns":2426},[2428,7219,7220,7232],{},[2431,7221,7222,7224,7230],{},[2437,7223,2644],{"mathvariant":2643},[3063,7225,7226,7228],{},[2437,7227,3688],{},[2437,7229,3188],{},[2584,7231,2651],{"stretchy":2650},[2444,7233,6756],{"encoding":2446},[53,7235,7237],{"className":7236,"ariaHidden":2452},[2451],[53,7238,7240,7243,7246,7286],{"className":7239},[2456],[53,7241],{"className":7242,"style":2664},[2460],[53,7244,2644],{"className":7245},[2465],[53,7247,7249,7252],{"className":7248},[2465],[53,7250,3688],{"className":7251,"style":2671},[2465,2506],[53,7253,7255],{"className":7254},[2474],[53,7256,7258,7278],{"className":7257},[2478,3095],[53,7259,7261,7275],{"className":7260},[2482],[53,7262,7264],{"className":7263,"style":6697},[2486],[53,7265,7266,7269],{"style":4467},[53,7267],{"className":7268,"style":2495},[2494],[53,7270,7272],{"className":7271},[2499,2500,2501,2502],[53,7273,3188],{"className":7274},[2465,2506,2502],[53,7276,3120],{"className":7277},[3119],[53,7279,7281],{"className":7280},[2482],[53,7282,7284],{"className":7283,"style":3127},[2486],[53,7285],{},[53,7287,2651],{"className":7288},[2675]," controlled by the state of the ancilla qubit (similarly for ",[53,7291,7293,7314],{"className":7292},[2418],[53,7294,7296],{"className":7295},[2422],[2424,7297,7298],{"xmlns":2426},[2428,7299,7300,7312],{},[2431,7301,7302,7304,7306],{},[6062,7303,6908],{},[6062,7305,6911],{},[3063,7307,7308,7310],{},[2437,7309,3688],{},[2437,7311,2267],{},[2444,7313,7005],{"encoding":2446},[53,7315,7317],{"className":7316,"ariaHidden":2452},[2451],[53,7318,7320,7323,7329,7332],{"className":7319},[2456],[53,7321],{"className":7322,"style":7015},[2460],[53,7324,7326],{"className":7325},[2465,6934],[53,7327,6908],{"className":7328},[2465],[53,7330],{"className":7331,"style":2814},[2609],[53,7333,7335,7338],{"className":7334},[2465],[53,7336,3688],{"className":7337,"style":2671},[2465,2506],[53,7339,7341],{"className":7340},[2474],[53,7342,7344,7364],{"className":7343},[2478,3095],[53,7345,7347,7361],{"className":7346},[2482],[53,7348,7350],{"className":7349,"style":6697},[2486],[53,7351,7352,7355],{"style":4467},[53,7353],{"className":7354,"style":2495},[2494],[53,7356,7358],{"className":7357},[2499,2500,2501,2502],[53,7359,2267],{"className":7360,"style":6718},[2465,2506,2502],[53,7362,3120],{"className":7363},[3119],[53,7365,7367],{"className":7366},[2482],[53,7368,7370],{"className":7369,"style":6728},[2486],[53,7371],{},") and the operation ",[53,7374,7376,7389],{"className":7375},[2418],[53,7377,7379],{"className":7378},[2422],[2424,7380,7381],{"xmlns":2426},[2428,7382,7383,7387],{},[2431,7384,7385],{},[2437,7386,3069],{},[2444,7388,3069],{"encoding":2446},[53,7390,7392],{"className":7391,"ariaHidden":2452},[2451],[53,7393,7395,7398],{"className":7394},[2456],[53,7396],{"className":7397,"style":2461},[2460],[53,7399,3069],{"className":7400,"style":3115},[2465,2506]," represents Pauli decomposition of the system Hamiltonian ",[53,7403,7405,7434],{"className":7404},[2418],[53,7406,7408],{"className":7407},[2422],[2424,7409,7410],{"xmlns":2426},[2428,7411,7412,7431],{},[2431,7413,7414,7416,7418,7425],{},[2437,7415,3033],{},[2584,7417,121],{},[3063,7419,7420,7423],{},[2584,7421,7422],{},"∑",[2437,7424,3188],{},[3063,7426,7427,7429],{},[2437,7428,3069],{},[2437,7430,3188],{},[2444,7432,7433],{"encoding":2446},"H = \\sum_i P_i",[53,7435,7437,7455],{"className":7436,"ariaHidden":2452},[2451],[53,7438,7440,7443,7446,7449,7452],{"className":7439},[2456],[53,7441],{"className":7442,"style":2461},[2460],[53,7444,3033],{"className":7445,"style":3048},[2465,2506],[53,7447],{"className":7448,"style":2814},[2609],[53,7450,121],{"className":7451},[2818],[53,7453],{"className":7454,"style":2814},[2609],[53,7456,7458,7462,7509,7512],{"className":7457},[2456],[53,7459],{"className":7460,"style":7461},[2460],"height:1.0497em;vertical-align:-0.2997em;",[53,7463,7466,7472],{"className":7464},[7465],"mop",[53,7467,7422],{"className":7468,"style":7471},[7465,7469,7470],"op-symbol","small-op","position:relative;top:0em;",[53,7473,7475],{"className":7474},[2474],[53,7476,7478,7500],{"className":7477},[2478,3095],[53,7479,7481,7497],{"className":7480},[2482],[53,7482,7485],{"className":7483,"style":7484},[2486],"height:0.162em;",[53,7486,7488,7491],{"style":7487},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[53,7489],{"className":7490,"style":2495},[2494],[53,7492,7494],{"className":7493},[2499,2500,2501,2502],[53,7495,3188],{"className":7496},[2465,2506,2502],[53,7498,3120],{"className":7499},[3119],[53,7501,7503],{"className":7502},[2482],[53,7504,7507],{"className":7505,"style":7506},[2486],"height:0.2997em;",[53,7508],{},[53,7510],{"className":7511,"style":2860},[2609],[53,7513,7515,7518],{"className":7514},[2465],[53,7516,3069],{"className":7517,"style":3115},[2465,2506],[53,7519,7521],{"className":7520},[2474],[53,7522,7524,7545],{"className":7523},[2478,3095],[53,7525,7527,7542],{"className":7526},[2482],[53,7528,7530],{"className":7529,"style":6697},[2486],[53,7531,7533,7536],{"style":7532},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[53,7534],{"className":7535,"style":2495},[2494],[53,7537,7539],{"className":7538},[2499,2500,2501,2502],[53,7540,3188],{"className":7541},[2465,2506,2502],[53,7543,3120],{"className":7544},[3119],[53,7546,7548],{"className":7547},[2482],[53,7549,7551],{"className":7550,"style":3127},[2486],[53,7552],{},". A more detailed derivation of the operations calculated by the Hadamard test is given below.",[7555,7556,7558],"h4",{"id":7557},"define-hamiltonian","Define Hamiltonian",[12,7560,7561,7562,7591,7592],{},"Let's consider the Heisenberg Hamiltonian for ",[53,7563,7565,7579],{"className":7564},[2418],[53,7566,7568],{"className":7567},[2422],[2424,7569,7570],{"xmlns":2426},[2428,7571,7572,7577],{},[2431,7573,7574],{},[2437,7575,7576],{},"N",[2444,7578,7576],{"encoding":2446},[53,7580,7582],{"className":7581,"ariaHidden":2452},[2451],[53,7583,7585,7588],{"className":7584},[2456],[53,7586],{"className":7587,"style":2461},[2460],[53,7589,7576],{"className":7590,"style":3209},[2465,2506]," qubits on a linear chain: ",[53,7593,7595,7671],{"className":7594},[2418],[53,7596,7598],{"className":7597},[2422],[2424,7599,7600],{"xmlns":2426},[2428,7601,7602,7668],{},[2431,7603,7604,7606,7608,7622,7629,7635,7637,7644,7650,7652,7655,7662],{},[2437,7605,3033],{},[2584,7607,121],{},[3670,7609,7610,7612,7620],{},[2584,7611,7422],{},[2431,7613,7614,7616,7618],{},[2437,7615,3188],{},[2584,7617,2714],{"separator":2452},[2437,7619,2267],{},[2437,7621,7576],{},[3063,7623,7624,7627],{},[2437,7625,7626],{},"X",[2437,7628,3188],{},[3063,7630,7631,7633],{},[2437,7632,7626],{},[2437,7634,2267],{},[2584,7636,2002],{},[3063,7638,7639,7642],{},[2437,7640,7641],{},"Y",[2437,7643,3188],{},[3063,7645,7646,7648],{},[2437,7647,7641],{},[2437,7649,2267],{},[2584,7651,2586],{},[2437,7653,7654],{},"J",[3063,7656,7657,7660],{},[2437,7658,7659],{},"Z",[2437,7661,3188],{},[3063,7663,7664,7666],{},[2437,7665,7659],{},[2437,7667,2267],{},[2444,7669,7670],{"encoding":2446},"H= \\sum_{i,j}^N X_i X_j + Y_i Y_j - J Z_i Z_j",[53,7672,7674,7692,7856,7953],{"className":7673,"ariaHidden":2452},[2451],[53,7675,7677,7680,7683,7686,7689],{"className":7676},[2456],[53,7678],{"className":7679,"style":2461},[2460],[53,7681,3033],{"className":7682,"style":3048},[2465,2506],[53,7684],{"className":7685,"style":2814},[2609],[53,7687,121],{"className":7688},[2818],[53,7690],{"className":7691,"style":2814},[2609],[53,7693,7695,7699,7762,7765,7807,7847,7850,7853],{"className":7694},[2456],[53,7696],{"className":7697,"style":7698},[2460],"height:1.417em;vertical-align:-0.4358em;",[53,7700,7702,7705],{"className":7701},[7465],[53,7703,7422],{"className":7704,"style":7471},[7465,7469,7470],[53,7706,7708],{"className":7707},[2474],[53,7709,7711,7753],{"className":7710},[2478,3095],[53,7712,7714,7750],{"className":7713},[2482],[53,7715,7718,7738],{"className":7716,"style":7717},[2486],"height:0.9812em;",[53,7719,7720,7723],{"style":7487},[53,7721],{"className":7722,"style":2495},[2494],[53,7724,7726],{"className":7725},[2499,2500,2501,2502],[53,7727,7729,7732,7735],{"className":7728},[2465,2502],[53,7730,3188],{"className":7731},[2465,2506,2502],[53,7733,2714],{"className":7734},[2856,2502],[53,7736,2267],{"className":7737,"style":6718},[2465,2506,2502],[53,7739,7741,7744],{"style":7740},"top:-3.2029em;margin-right:0.05em;",[53,7742],{"className":7743,"style":2495},[2494],[53,7745,7747],{"className":7746},[2499,2500,2501,2502],[53,7748,7576],{"className":7749,"style":3209},[2465,2506,2502],[53,7751,3120],{"className":7752},[3119],[53,7754,7756],{"className":7755},[2482],[53,7757,7760],{"className":7758,"style":7759},[2486],"height:0.4358em;",[53,7761],{},[53,7763],{"className":7764,"style":2860},[2609],[53,7766,7768,7772],{"className":7767},[2465],[53,7769,7626],{"className":7770,"style":7771},[2465,2506],"margin-right:0.0785em;",[53,7773,7775],{"className":7774},[2474],[53,7776,7778,7799],{"className":7777},[2478,3095],[53,7779,7781,7796],{"className":7780},[2482],[53,7782,7784],{"className":7783,"style":6697},[2486],[53,7785,7787,7790],{"style":7786},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[53,7788],{"className":7789,"style":2495},[2494],[53,7791,7793],{"className":7792},[2499,2500,2501,2502],[53,7794,3188],{"className":7795},[2465,2506,2502],[53,7797,3120],{"className":7798},[3119],[53,7800,7802],{"className":7801},[2482],[53,7803,7805],{"className":7804,"style":3127},[2486],[53,7806],{},[53,7808,7810,7813],{"className":7809},[2465],[53,7811,7626],{"className":7812,"style":7771},[2465,2506],[53,7814,7816],{"className":7815},[2474],[53,7817,7819,7839],{"className":7818},[2478,3095],[53,7820,7822,7836],{"className":7821},[2482],[53,7823,7825],{"className":7824,"style":6697},[2486],[53,7826,7827,7830],{"style":7786},[53,7828],{"className":7829,"style":2495},[2494],[53,7831,7833],{"className":7832},[2499,2500,2501,2502],[53,7834,2267],{"className":7835,"style":6718},[2465,2506,2502],[53,7837,3120],{"className":7838},[3119],[53,7840,7842],{"className":7841},[2482],[53,7843,7845],{"className":7844,"style":6728},[2486],[53,7846],{},[53,7848],{"className":7849,"style":2610},[2609],[53,7851,2002],{"className":7852},[2614],[53,7854],{"className":7855,"style":2610},[2609],[53,7857,7859,7863,7904,7944,7947,7950],{"className":7858},[2456],[53,7860],{"className":7861,"style":7862},[2460],"height:0.9694em;vertical-align:-0.2861em;",[53,7864,7866,7869],{"className":7865},[2465],[53,7867,7641],{"className":7868,"style":2610},[2465,2506],[53,7870,7872],{"className":7871},[2474],[53,7873,7875,7896],{"className":7874},[2478,3095],[53,7876,7878,7893],{"className":7877},[2482],[53,7879,7881],{"className":7880,"style":6697},[2486],[53,7882,7884,7887],{"style":7883},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[53,7885],{"className":7886,"style":2495},[2494],[53,7888,7890],{"className":7889},[2499,2500,2501,2502],[53,7891,3188],{"className":7892},[2465,2506,2502],[53,7894,3120],{"className":7895},[3119],[53,7897,7899],{"className":7898},[2482],[53,7900,7902],{"className":7901,"style":3127},[2486],[53,7903],{},[53,7905,7907,7910],{"className":7906},[2465],[53,7908,7641],{"className":7909,"style":2610},[2465,2506],[53,7911,7913],{"className":7912},[2474],[53,7914,7916,7936],{"className":7915},[2478,3095],[53,7917,7919,7933],{"className":7918},[2482],[53,7920,7922],{"className":7921,"style":6697},[2486],[53,7923,7924,7927],{"style":7883},[53,7925],{"className":7926,"style":2495},[2494],[53,7928,7930],{"className":7929},[2499,2500,2501,2502],[53,7931,2267],{"className":7932,"style":6718},[2465,2506,2502],[53,7934,3120],{"className":7935},[3119],[53,7937,7939],{"className":7938},[2482],[53,7940,7942],{"className":7941,"style":6728},[2486],[53,7943],{},[53,7945],{"className":7946,"style":2610},[2609],[53,7948,2586],{"className":7949},[2614],[53,7951],{"className":7952,"style":2610},[2609],[53,7954,7956,7959,7963,8005],{"className":7955},[2456],[53,7957],{"className":7958,"style":7862},[2460],[53,7960,7654],{"className":7961,"style":7962},[2465,2506],"margin-right:0.0962em;",[53,7964,7966,7970],{"className":7965},[2465],[53,7967,7659],{"className":7968,"style":7969},[2465,2506],"margin-right:0.0715em;",[53,7971,7973],{"className":7972},[2474],[53,7974,7976,7997],{"className":7975},[2478,3095],[53,7977,7979,7994],{"className":7978},[2482],[53,7980,7982],{"className":7981,"style":6697},[2486],[53,7983,7985,7988],{"style":7984},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[53,7986],{"className":7987,"style":2495},[2494],[53,7989,7991],{"className":7990},[2499,2500,2501,2502],[53,7992,3188],{"className":7993},[2465,2506,2502],[53,7995,3120],{"className":7996},[3119],[53,7998,8000],{"className":7999},[2482],[53,8001,8003],{"className":8002,"style":3127},[2486],[53,8004],{},[53,8006,8008,8011],{"className":8007},[2465],[53,8009,7659],{"className":8010,"style":7969},[2465,2506],[53,8012,8014],{"className":8013},[2474],[53,8015,8017,8037],{"className":8016},[2478,3095],[53,8018,8020,8034],{"className":8019},[2482],[53,8021,8023],{"className":8022,"style":6697},[2486],[53,8024,8025,8028],{"style":7984},[53,8026],{"className":8027,"style":2495},[2494],[53,8029,8031],{"className":8030},[2499,2500,2501,2502],[53,8032,2267],{"className":8033,"style":6718},[2465,2506,2502],[53,8035,3120],{"className":8036},[3119],[53,8038,8040],{"className":8039},[2482],[53,8041,8043],{"className":8042,"style":6728},[2486],[53,8044],{},[43,8046,8048],{"className":45,"code":8047,"language":47,"meta":48,"style":48},"# 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",[50,8049,8050,8055,8065,8077,8081,8086,8135,8154,8164,8179,8197,8217,8240,8258,8284,8314,8344,8391,8395,8400,8416],{"__ignoreMap":48},[53,8051,8052],{"class":55,"line":56},[53,8053,8054],{"class":110},"# Define problem Hamiltonian.\n",[53,8056,8057,8060,8062],{"class":55,"line":73},[53,8058,8059],{"class":63},"n_qubits ",[53,8061,121],{"class":120},[53,8063,8064],{"class":131}," 30\n",[53,8066,8067,8070,8072,8074],{"class":55,"line":87},[53,8068,8069],{"class":63},"J ",[53,8071,121],{"class":120},[53,8073,1321],{"class":131},[53,8075,8076],{"class":110}," # coupling strength for ZZ interaction\n",[53,8078,8079],{"class":55,"line":100},[53,8080,104],{"emptyLinePlaceholder":103},[53,8082,8083],{"class":55,"line":107},[53,8084,8085],{"class":110},"# Define the Hamiltonian:\n",[53,8087,8088,8091,8093,8096,8099,8102,8104,8107,8109,8112,8114,8117,8119,8122,8125,8128,8130,8132],{"class":55,"line":114},[53,8089,8090],{"class":63},"H_int ",[53,8092,121],{"class":120},[53,8094,8095],{"class":63}," [[",[53,8097,8098],{"class":278},"\"I\"",[53,8100,8101],{"class":63},"] ",[53,8103,2286],{"class":120},[53,8105,8106],{"class":63}," n_qubits ",[53,8108,1525],{"class":59},[53,8110,8111],{"class":63}," _ ",[53,8113,1531],{"class":59},[53,8115,8116],{"class":120}," range",[53,8118,128],{"class":63},[53,8120,8121],{"class":131},"3",[53,8123,8124],{"class":120}," *",[53,8126,8127],{"class":63}," (n_qubits ",[53,8129,2262],{"class":120},[53,8131,1321],{"class":131},[53,8133,8134],{"class":63},"))]\n",[53,8136,8137,8139,8142,8144,8146,8148,8150,8152],{"class":55,"line":138},[53,8138,1525],{"class":59},[53,8140,8141],{"class":63}," i ",[53,8143,1531],{"class":59},[53,8145,8116],{"class":120},[53,8147,1999],{"class":63},[53,8149,2262],{"class":120},[53,8151,1321],{"class":131},[53,8153,1778],{"class":63},[53,8155,8156,8159,8161],{"class":55,"line":158},[53,8157,8158],{"class":63}," H_int[i][i] ",[53,8160,121],{"class":120},[53,8162,8163],{"class":278}," \"Z\"\n",[53,8165,8166,8169,8171,8173,8175,8177],{"class":55,"line":182},[53,8167,8168],{"class":63}," H_int[i][i ",[53,8170,2002],{"class":120},[53,8172,1321],{"class":131},[53,8174,8101],{"class":63},[53,8176,121],{"class":120},[53,8178,8163],{"class":278},[53,8180,8181,8183,8185,8187,8189,8191,8193,8195],{"class":55,"line":209},[53,8182,1525],{"class":59},[53,8184,8141],{"class":63},[53,8186,1531],{"class":59},[53,8188,8116],{"class":120},[53,8190,1999],{"class":63},[53,8192,2262],{"class":120},[53,8194,1321],{"class":131},[53,8196,1778],{"class":63},[53,8198,8199,8202,8204,8206,8209,8212,8214],{"class":55,"line":226},[53,8200,8201],{"class":63}," H_int[n_qubits ",[53,8203,2262],{"class":120},[53,8205,1321],{"class":131},[53,8207,8208],{"class":120}," +",[53,8210,8211],{"class":63}," i][i] ",[53,8213,121],{"class":120},[53,8215,8216],{"class":278}," \"X\"\n",[53,8218,8219,8221,8223,8225,8227,8230,8232,8234,8236,8238],{"class":55,"line":231},[53,8220,8201],{"class":63},[53,8222,2262],{"class":120},[53,8224,1321],{"class":131},[53,8226,8208],{"class":120},[53,8228,8229],{"class":63}," i][i ",[53,8231,2002],{"class":120},[53,8233,1321],{"class":131},[53,8235,8101],{"class":63},[53,8237,121],{"class":120},[53,8239,8216],{"class":278},[53,8241,8242,8244,8246,8248,8250,8252,8254,8256],{"class":55,"line":245},[53,8243,1525],{"class":59},[53,8245,8141],{"class":63},[53,8247,1531],{"class":59},[53,8249,8116],{"class":120},[53,8251,1999],{"class":63},[53,8253,2262],{"class":120},[53,8255,1321],{"class":131},[53,8257,1778],{"class":63},[53,8259,8260,8263,8265,8267,8269,8271,8273,8275,8277,8279,8281],{"class":55,"line":258},[53,8261,8262],{"class":63}," H_int[",[53,8264,132],{"class":131},[53,8266,8124],{"class":120},[53,8268,8127],{"class":63},[53,8270,2262],{"class":120},[53,8272,1321],{"class":131},[53,8274,203],{"class":63},[53,8276,2002],{"class":120},[53,8278,8211],{"class":63},[53,8280,121],{"class":120},[53,8282,8283],{"class":278}," \"Y\"\n",[53,8285,8286,8288,8290,8292,8294,8296,8298,8300,8302,8304,8306,8308,8310,8312],{"class":55,"line":263},[53,8287,8262],{"class":63},[53,8289,132],{"class":131},[53,8291,8124],{"class":120},[53,8293,8127],{"class":63},[53,8295,2262],{"class":120},[53,8297,1321],{"class":131},[53,8299,203],{"class":63},[53,8301,2002],{"class":120},[53,8303,8229],{"class":63},[53,8305,2002],{"class":120},[53,8307,1321],{"class":131},[53,8309,8101],{"class":63},[53,8311,121],{"class":120},[53,8313,8283],{"class":278},[53,8315,8316,8318,8320,8323,8326,8328,8331,8334,8336,8339,8341],{"class":55,"line":269},[53,8317,8090],{"class":63},[53,8319,121],{"class":120},[53,8321,8322],{"class":63}," [",[53,8324,8325],{"class":278},"\"\"",[53,8327,28],{"class":63},[53,8329,8330],{"class":124},"join",[53,8332,8333],{"class":63},"(term) ",[53,8335,1525],{"class":59},[53,8337,8338],{"class":63}," term ",[53,8340,1531],{"class":59},[53,8342,8343],{"class":63}," H_int]\n",[53,8345,8346,8349,8351,8354,8356,8359,8362,8364,8367,8369,8371,8373,8376,8379,8381,8383,8385,8387,8389],{"class":55,"line":284},[53,8347,8348],{"class":63},"H_tot ",[53,8350,121],{"class":120},[53,8352,8353],{"class":63}," [(term, J) ",[53,8355,1540],{"class":59},[53,8357,8358],{"class":63}," term.",[53,8360,8361],{"class":124},"count",[53,8363,128],{"class":63},[53,8365,8366],{"class":278},"\"Z\"",[53,8368,203],{"class":63},[53,8370,1618],{"class":120},[53,8372,2211],{"class":131},[53,8374,8375],{"class":59}," else",[53,8377,8378],{"class":63}," (term, ",[53,8380,173],{"class":131},[53,8382,203],{"class":63},[53,8384,1525],{"class":59},[53,8386,8338],{"class":63},[53,8388,1531],{"class":59},[53,8390,8343],{"class":63},[53,8392,8393],{"class":55,"line":296},[53,8394,104],{"emptyLinePlaceholder":103},[53,8396,8397],{"class":55,"line":309},[53,8398,8399],{"class":110},"# Get operator\n",[53,8401,8402,8405,8407,8410,8413],{"class":55,"line":323},[53,8403,8404],{"class":63},"H_op ",[53,8406,121],{"class":120},[53,8408,8409],{"class":63}," SparsePauliOp.",[53,8411,8412],{"class":124},"from_list",[53,8414,8415],{"class":63},"(H_tot)\n",[53,8417,8418,8421],{"class":55,"line":332},[53,8419,8420],{"class":120},"print",[53,8422,8415],{"class":63},[43,8424,8428],{"className":8425,"code":8427,"language":6934,"meta":48},[8426],"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",[50,8429,8427],{"__ignoreMap":48},[7555,8431,8433],{"id":8432},"set-parameters-for-the-algorithm","Set parameters for the algorithm",[12,8435,8436,8437,8439,8440,8443,8444,8497,8498,8531],{},"We heuristically choose a value for the time-step ",[50,8438,5358],{}," (based on upper bounds on the Hamiltonian norm). Ref ",[19,8441,8442],{"href":979},"[2]"," showed that a sufficiently small timestep is ",[53,8445,8447,8474],{"className":8446},[2418],[53,8448,8450],{"className":8449},[2422],[2424,8451,8452],{"xmlns":2426},[2428,8453,8454,8471],{},[2431,8455,8456,8459,8461,8463,8465,8467,8469],{},[2437,8457,8458],{},"π",[2437,8460,193],{"mathvariant":2643},[2437,8462,2644],{"mathvariant":2643},[2437,8464,2644],{"mathvariant":2643},[2437,8466,3033],{},[2437,8468,2644],{"mathvariant":2643},[2437,8470,2644],{"mathvariant":2643},[2444,8472,8473],{"encoding":2446},"\\pi\u002F\\vert \\vert H \\vert \\vert",[53,8475,8477],{"className":8476,"ariaHidden":2452},[2451],[53,8478,8480,8483,8486,8490,8493],{"className":8479},[2456],[53,8481],{"className":8482,"style":2664},[2460],[53,8484,8458],{"className":8485,"style":2671},[2465,2506],[53,8487,8489],{"className":8488},[2465],"\u002F∣∣",[53,8491,3033],{"className":8492,"style":3048},[2465,2506],[53,8494,8496],{"className":8495},[2465],"∣∣",", 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 ",[53,8499,8501,8516],{"className":8500},[2418],[53,8502,8504],{"className":8503},[2422],[2424,8505,8506],{"xmlns":2426},[2428,8507,8508,8514],{},[2431,8509,8510,8512],{},[2437,8511,4854],{},[2437,8513,3193],{},[2444,8515,5358],{"encoding":2446},[53,8517,8519],{"className":8518,"ariaHidden":2452},[2451],[53,8520,8522,8525,8528],{"className":8521},[2456],[53,8523],{"className":8524,"style":5329},[2460],[53,8526,4854],{"className":8527},[2465,2506],[53,8529,3193],{"className":8530},[2465,2506]," to be too small leads to worse conditioning of the Krylov subspace, since the Krylov basis vectors differ less from timestep to timestep.",[43,8533,8535],{"className":45,"code":8534,"language":47,"meta":48,"style":48},"# 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",[50,8536,8537,8542,8556,8569,8589,8603,8616,8628,8637,8679,8684,8693,8731,8751,8758,8766,8779,8791,8810,8824,8828,8833,8863],{"__ignoreMap":48},[53,8538,8539],{"class":55,"line":56},[53,8540,8541],{"class":110},"# Get Hamiltonian restricted to single-particle states\n",[53,8543,8544,8547,8549,8551,8553],{"class":55,"line":73},[53,8545,8546],{"class":63},"single_particle_H ",[53,8548,121],{"class":120},[53,8550,1511],{"class":63},[53,8552,2044],{"class":124},[53,8554,8555],{"class":63},"((n_qubits, n_qubits))\n",[53,8557,8558,8560,8562,8564,8566],{"class":55,"line":87},[53,8559,1525],{"class":59},[53,8561,8141],{"class":63},[53,8563,1531],{"class":59},[53,8565,8116],{"class":120},[53,8567,8568],{"class":63},"(n_qubits):\n",[53,8570,8571,8573,8576,8578,8580,8583,8585,8587],{"class":55,"line":100},[53,8572,1811],{"class":59},[53,8574,8575],{"class":63}," j ",[53,8577,1531],{"class":59},[53,8579,8116],{"class":120},[53,8581,8582],{"class":63},"(i ",[53,8584,2002],{"class":120},[53,8586,1321],{"class":131},[53,8588,1778],{"class":63},[53,8590,8591,8593,8595,8597,8599,8601],{"class":55,"line":107},[53,8592,2162],{"class":59},[53,8594,1814],{"class":63},[53,8596,1531],{"class":59},[53,8598,1819],{"class":63},[53,8600,1822],{"class":124},[53,8602,1825],{"class":63},[53,8604,8605,8608,8610,8613],{"class":55,"line":114},[53,8606,8607],{"class":63}," p_x ",[53,8609,121],{"class":120},[53,8611,8612],{"class":124}," Pauli",[53,8614,8615],{"class":63},"(p).x\n",[53,8617,8618,8621,8623,8625],{"class":55,"line":138},[53,8619,8620],{"class":63}," p_z ",[53,8622,121],{"class":120},[53,8624,8612],{"class":124},[53,8626,8627],{"class":63},"(p).z\n",[53,8629,8630,8632,8635],{"class":55,"line":158},[53,8631,2193],{"class":59},[53,8633,8634],{"class":120}," all",[53,8636,1271],{"class":63},[53,8638,8639,8642,8644,8647,8649,8652,8654,8657,8659,8662,8665,8667,8670,8672,8674,8676],{"class":55,"line":182},[53,8640,8641],{"class":63}," p_x[k] ",[53,8643,1618],{"class":120},[53,8645,8646],{"class":63}," ((i ",[53,8648,1618],{"class":120},[53,8650,8651],{"class":63}," k) ",[53,8653,2002],{"class":120},[53,8655,8656],{"class":63}," (j ",[53,8658,1618],{"class":120},[53,8660,8661],{"class":63}," k)) ",[53,8663,8664],{"class":120},"%",[53,8666,2211],{"class":131},[53,8668,8669],{"class":59}," for",[53,8671,1615],{"class":63},[53,8673,1531],{"class":59},[53,8675,8116],{"class":120},[53,8677,8678],{"class":63},"(n_qubits)\n",[53,8680,8681],{"class":55,"line":209},[53,8682,8683],{"class":63}," ):\n",[53,8685,8686,8689,8691],{"class":55,"line":226},[53,8687,8688],{"class":63}," sgn ",[53,8690,121],{"class":120},[53,8692,1229],{"class":63},[53,8694,8695,8698,8700,8702,8704,8706,8708,8711,8714,8717,8720,8722,8724,8726,8728],{"class":55,"line":231},[53,8696,8697],{"class":63}," (",[53,8699,2262],{"class":120},[53,8701,173],{"class":131},[53,8703,2267],{"class":59},[53,8705,203],{"class":63},[53,8707,2272],{"class":120},[53,8709,8710],{"class":120}," sum",[53,8712,8713],{"class":63},"(p_z[k] ",[53,8715,8716],{"class":59},"and",[53,8718,8719],{"class":63}," p_x[k] ",[53,8721,1525],{"class":59},[53,8723,1615],{"class":63},[53,8725,1531],{"class":59},[53,8727,8116],{"class":120},[53,8729,8730],{"class":63},"(n_qubits))\n",[53,8732,8733,8736,8738,8740,8742,8744,8746,8748],{"class":55,"line":245},[53,8734,8735],{"class":63}," ) ",[53,8737,2286],{"class":120},[53,8739,2259],{"class":63},[53,8741,2262],{"class":120},[53,8743,173],{"class":131},[53,8745,203],{"class":63},[53,8747,2272],{"class":120},[53,8749,8750],{"class":63}," p_z[i])\n",[53,8752,8753,8756],{"class":55,"line":258},[53,8754,8755],{"class":59}," else",[53,8757,1623],{"class":63},[53,8759,8760,8762,8764],{"class":55,"line":263},[53,8761,8688],{"class":63},[53,8763,121],{"class":120},[53,8765,2141],{"class":131},[53,8767,8768,8771,8773,8775,8777],{"class":55,"line":269},[53,8769,8770],{"class":63}," single_particle_H[i, j] ",[53,8772,2180],{"class":120},[53,8774,2350],{"class":63},[53,8776,2286],{"class":120},[53,8778,2355],{"class":63},[53,8780,8781,8783,8785,8787,8789],{"class":55,"line":284},[53,8782,1525],{"class":59},[53,8784,8141],{"class":63},[53,8786,1531],{"class":59},[53,8788,8116],{"class":120},[53,8790,8568],{"class":63},[53,8792,8793,8795,8797,8799,8801,8803,8805,8807],{"class":55,"line":296},[53,8794,1811],{"class":59},[53,8796,8575],{"class":63},[53,8798,1531],{"class":59},[53,8800,8116],{"class":120},[53,8802,8582],{"class":63},[53,8804,2002],{"class":120},[53,8806,1321],{"class":131},[53,8808,8809],{"class":63},", n_qubits):\n",[53,8811,8812,8815,8817,8819,8821],{"class":55,"line":309},[53,8813,8814],{"class":63}," single_particle_H[i, j] ",[53,8816,121],{"class":120},[53,8818,1511],{"class":63},[53,8820,1569],{"class":124},[53,8822,8823],{"class":63},"(single_particle_H[j, i])\n",[53,8825,8826],{"class":55,"line":323},[53,8827,104],{"emptyLinePlaceholder":103},[53,8829,8830],{"class":55,"line":332},[53,8831,8832],{"class":110},"# Set dt according to spectral norm\n",[53,8834,8835,8838,8840,8843,8845,8848,8851,8854,8857,8859,8861],{"class":55,"line":344},[53,8836,8837],{"class":63},"dt ",[53,8839,121],{"class":120},[53,8841,8842],{"class":63}," np.pi ",[53,8844,193],{"class":120},[53,8846,8847],{"class":63}," np.linalg.",[53,8849,8850],{"class":124},"norm",[53,8852,8853],{"class":63},"(single_particle_H, ",[53,8855,8856],{"class":2050},"ord",[53,8858,121],{"class":120},[53,8860,132],{"class":131},[53,8862,135],{"class":63},[53,8864,8865],{"class":55,"line":357},[53,8866,8867],{"class":63},"dt\n",[43,8869,8872],{"className":8870,"code":8871,"language":6934,"meta":48},[8426],"np.float64(0.10833078115826875)\n",[50,8873,8871],{"__ignoreMap":48},[12,8875,8876],{},"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.",[43,8878,8880],{"className":45,"code":8879,"language":47,"meta":48,"style":48},"# 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",[50,8881,8882,8887,8900,8910],{"__ignoreMap":48},[53,8883,8884],{"class":55,"line":56},[53,8885,8886],{"class":110},"# Set parameters for quantum Krylov algorithm\n",[53,8888,8889,8892,8894,8897],{"class":55,"line":73},[53,8890,8891],{"class":63},"krylov_dim ",[53,8893,121],{"class":120},[53,8895,8896],{"class":131}," 5",[53,8898,8899],{"class":110}," # size of Krylov subspace\n",[53,8901,8902,8905,8907],{"class":55,"line":87},[53,8903,8904],{"class":63},"num_trotter_steps ",[53,8906,121],{"class":120},[53,8908,8909],{"class":131}," 6\n",[53,8911,8912,8915,8917,8920,8922],{"class":55,"line":100},[53,8913,8914],{"class":63},"dt_circ ",[53,8916,121],{"class":120},[53,8918,8919],{"class":63}," dt ",[53,8921,193],{"class":120},[53,8923,8924],{"class":63}," num_trotter_steps\n",[7555,8926,8928],{"id":8927},"state-preparation","State preparation",[12,8930,8931,8932,8971,8972,9010],{},"Pick a reference state ",[53,8933,8935,8953],{"className":8934},[2418],[53,8936,8938],{"className":8937},[2422],[2424,8939,8940],{"xmlns":2426},[2428,8941,8942,8950],{},[2431,8943,8944,8946,8948],{},[2437,8945,2644],{"mathvariant":2643},[2437,8947,3688],{},[2584,8949,2651],{"stretchy":2650},[2444,8951,8952],{"encoding":2446},"\\vert \\psi \\rangle",[53,8954,8956],{"className":8955,"ariaHidden":2452},[2451],[53,8957,8959,8962,8965,8968],{"className":8958},[2456],[53,8960],{"className":8961,"style":2664},[2460],[53,8963,2644],{"className":8964},[2465],[53,8966,3688],{"className":8967,"style":2671},[2465,2506],[53,8969,2651],{"className":8970},[2675]," that has some overlap with the ground state. For this Hamiltonian, We use the a state with an excitation in the middle qubit ",[53,8973,8975,8994],{"className":8974},[2418],[53,8976,8978],{"className":8977},[2422],[2424,8979,8980],{"xmlns":2426},[2428,8981,8982,8991],{},[2431,8983,8984,8986,8989],{},[2437,8985,2644],{"mathvariant":2643},[2588,8987,8988],{},"00..010...00",[2584,8990,2651],{"stretchy":2650},[2444,8992,8993],{"encoding":2446},"\\vert 00..010...00 \\rangle",[53,8995,8997],{"className":8996,"ariaHidden":2452},[2451],[53,8998,9000,9003,9007],{"className":8999},[2456],[53,9001],{"className":9002,"style":2664},[2460],[53,9004,9006],{"className":9005},[2465],"∣00..010...00",[53,9008,2651],{"className":9009},[2675]," as our reference state.",[43,9012,9014],{"className":45,"code":9013,"language":47,"meta":48,"style":48},"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",[50,9015,9016,9027,9053],{"__ignoreMap":48},[53,9017,9018,9021,9023,9025],{"class":55,"line":56},[53,9019,9020],{"class":63},"qc_state_prep ",[53,9022,121],{"class":120},[53,9024,125],{"class":124},[53,9026,8678],{"class":63},[53,9028,9029,9032,9035,9037,9039,9041,9043,9045,9047,9049,9051],{"class":55,"line":73},[53,9030,9031],{"class":63},"qc_state_prep.",[53,9033,9034],{"class":124},"x",[53,9036,128],{"class":63},[53,9038,1315],{"class":120},[53,9040,1999],{"class":63},[53,9042,193],{"class":120},[53,9044,2211],{"class":131},[53,9046,203],{"class":63},[53,9048,2002],{"class":120},[53,9050,1321],{"class":131},[53,9052,135],{"class":63},[53,9054,9055,9057,9060,9062,9065,9067,9070,9072,9075],{"class":55,"line":87},[53,9056,9031],{"class":63},[53,9058,9059],{"class":124},"draw",[53,9061,128],{"class":63},[53,9063,9064],{"class":278},"\"mpl\"",[53,9066,170],{"class":63},[53,9068,9069],{"class":2050},"scale",[53,9071,121],{"class":120},[53,9073,9074],{"class":131},"0.5",[53,9076,135],{"class":63},[379,9078],{"alt":9079,"src":9080},"Output of the previous code cell","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-01.avif",[7555,9082,9084],{"id":9083},"time-evolution","Time evolution",[12,9086,9087,9088,9188,9189,28],{},"We can realize the time-evolution operator generated by a given Hamiltonian: ",[53,9089,9091,9120],{"className":9090},[2418],[53,9092,9094],{"className":9093},[2422],[2424,9095,9096],{"xmlns":2426},[2428,9097,9098,9118],{},[2431,9099,9100,9102,9104],{},[2437,9101,3174],{},[2584,9103,121],{},[2434,9105,9106,9108],{},[2437,9107,3181],{},[2431,9109,9110,9112,9114,9116],{},[2584,9111,2586],{},[2437,9113,3188],{},[2437,9115,3033],{},[2437,9117,3193],{},[2444,9119,3196],{"encoding":2446},[53,9121,9123,9141],{"className":9122,"ariaHidden":2452},[2451],[53,9124,9126,9129,9132,9135,9138],{"className":9125},[2456],[53,9127],{"className":9128,"style":2461},[2460],[53,9130,3174],{"className":9131,"style":3209},[2465,2506],[53,9133],{"className":9134,"style":2814},[2609],[53,9136,121],{"className":9137},[2818],[53,9139],{"className":9140,"style":2814},[2609],[53,9142,9144,9147],{"className":9143},[2456],[53,9145],{"className":9146,"style":3225},[2460],[53,9148,9150,9153],{"className":9149},[2465],[53,9151,3181],{"className":9152},[2465,2506],[53,9154,9156],{"className":9155},[2474],[53,9157,9159],{"className":9158},[2478],[53,9160,9162],{"className":9161},[2482],[53,9163,9165],{"className":9164,"style":3225},[2486],[53,9166,9167,9170],{"style":2490},[53,9168],{"className":9169,"style":2495},[2494],[53,9171,9173],{"className":9172},[2499,2500,2501,2502],[53,9174,9176,9179,9182,9185],{"className":9175},[2465,2502],[53,9177,2586],{"className":9178},[2465,2502],[53,9180,3188],{"className":9181},[2465,2506,2502],[53,9183,3033],{"className":9184,"style":3048},[2465,2506,2502],[53,9186,3193],{"className":9187},[2465,2506,2502]," via the ",[19,9190,9192],{"href":9191},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fqiskit.synthesis.LieTrotter","Lie-Trotter approximation",[43,9194,9196],{"className":45,"code":9195,"language":47,"meta":48,"style":48},"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",[50,9197,9198,9215,9219,9224,9236,9267,9271,9275,9287,9299],{"__ignoreMap":48},[53,9199,9200,9203,9205,9208,9210,9213],{"class":55,"line":56},[53,9201,9202],{"class":63},"t ",[53,9204,121],{"class":120},[53,9206,9207],{"class":124}," Parameter",[53,9209,128],{"class":63},[53,9211,9212],{"class":278},"\"t\"",[53,9214,135],{"class":63},[53,9216,9217],{"class":55,"line":73},[53,9218,104],{"emptyLinePlaceholder":103},[53,9220,9221],{"class":55,"line":87},[53,9222,9223],{"class":110},"## Create the time-evo op circuit\n",[53,9225,9226,9229,9231,9234],{"class":55,"line":100},[53,9227,9228],{"class":63},"evol_gate ",[53,9230,121],{"class":120},[53,9232,9233],{"class":124}," PauliEvolutionGate",[53,9235,1271],{"class":63},[53,9237,9238,9241,9244,9246,9249,9252,9254,9257,9259,9262,9264],{"class":55,"line":107},[53,9239,9240],{"class":63}," H_op, ",[53,9242,9243],{"class":2050},"time",[53,9245,121],{"class":120},[53,9247,9248],{"class":63},"t, ",[53,9250,9251],{"class":2050},"synthesis",[53,9253,121],{"class":120},[53,9255,9256],{"class":124},"LieTrotter",[53,9258,128],{"class":63},[53,9260,9261],{"class":2050},"reps",[53,9263,121],{"class":120},[53,9265,9266],{"class":63},"num_trotter_steps)\n",[53,9268,9269],{"class":55,"line":114},[53,9270,135],{"class":63},[53,9272,9273],{"class":55,"line":138},[53,9274,104],{"emptyLinePlaceholder":103},[53,9276,9277,9280,9282,9285],{"class":55,"line":158},[53,9278,9279],{"class":63},"qr ",[53,9281,121],{"class":120},[53,9283,9284],{"class":124}," QuantumRegister",[53,9286,8678],{"class":63},[53,9288,9289,9292,9294,9296],{"class":55,"line":182},[53,9290,9291],{"class":63},"qc_evol ",[53,9293,121],{"class":120},[53,9295,125],{"class":124},[53,9297,9298],{"class":63},"(qr)\n",[53,9300,9301,9304,9306,9309,9312,9314],{"class":55,"line":209},[53,9302,9303],{"class":63},"qc_evol.",[53,9305,1834],{"class":124},[53,9307,9308],{"class":63},"(evol_gate, ",[53,9310,9311],{"class":2050},"qargs",[53,9313,121],{"class":120},[53,9315,9316],{"class":63},"qr)\n",[43,9318,9321],{"className":9319,"code":9320,"language":6934,"meta":48},[8426],"\u003Cqiskit.circuit.instructionset.InstructionSet at 0x11eef9be0>\n",[50,9322,9320],{"__ignoreMap":48},[7555,9324,9326],{"id":9325},"hadamard-test","Hadamard 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|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 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are controlled operations that prepare ",[53,10861,10863,10885],{"className":10862},[2418],[53,10864,10866],{"className":10865},[2422],[2424,10867,10868],{"xmlns":2426},[2428,10869,10870,10882],{},[2431,10871,10872,10874,10880],{},[2437,10873,2644],{"mathvariant":2643},[3063,10875,10876,10878],{},[2437,10877,3688],{},[2437,10879,3188],{},[2584,10881,2651],{"stretchy":2650},[2444,10883,10884],{"encoding":2446},"|\\psi_i\\rangle",[53,10886,10888],{"className":10887,"ariaHidden":2452},[2451],[53,10889,10891,10894,10897,10937],{"className":10890},[2456],[53,10892],{"className":10893,"style":2664},[2460],[53,10895,2644],{"className":10896},[2465],[53,10898,10900,10903],{"className":10899},[2465],[53,10901,3688],{"className":10902,"style":2671},[2465,2506],[53,10904,10906],{"className":10905},[2474],[53,10907,10909,10929],{"className":10908},[2478,3095],[53,10910,10912,10926],{"className":10911},[2482],[53,10913,10915],{"className":10914,"style":6697},[2486],[53,10916,10917,10920],{"style":4467},[53,10918],{"className":10919,"style":2495},[2494],[53,10921,10923],{"className":10922},[2499,2500,2501,2502],[53,10924,3188],{"className":10925},[2465,2506,2502],[53,10927,3120],{"className":10928},[3119],[53,10930,10932],{"className":10931},[2482],[53,10933,10935],{"className":10934,"style":3127},[2486],[53,10936],{},[53,10938,2651],{"className":10939},[2675],[53,10941,10943,10965],{"className":10942},[2418],[53,10944,10946],{"className":10945},[2422],[2424,10947,10948],{"xmlns":2426},[2428,10949,10950,10962],{},[2431,10951,10952,10954,10960],{},[2437,10953,2644],{"mathvariant":2643},[3063,10955,10956,10958],{},[2437,10957,3688],{},[2437,10959,2267],{},[2584,10961,2651],{"stretchy":2650},[2444,10963,10964],{"encoding":2446},"|\\psi_j\\rangle",[53,10966,10968],{"className":10967,"ariaHidden":2452},[2451],[53,10969,10971,10974,10977,11017],{"className":10970},[2456],[53,10972],{"className":10973,"style":6846},[2460],[53,10975,2644],{"className":10976},[2465],[53,10978,10980,10983],{"className":10979},[2465],[53,10981,3688],{"className":10982,"style":2671},[2465,2506],[53,10984,10986],{"className":10985},[2474],[53,10987,10989,11009],{"className":10988},[2478,3095],[53,10990,10992,11006],{"className":10991},[2482],[53,10993,10995],{"className":10994,"style":6697},[2486],[53,10996,10997,11000],{"style":4467},[53,10998],{"className":10999,"style":2495},[2494],[53,11001,11003],{"className":11002},[2499,2500,2501,2502],[53,11004,2267],{"className":11005,"style":6718},[2465,2506,2502],[53,11007,3120],{"className":11008},[3119],[53,11010,11012],{"className":11011},[2482],[53,11013,11015],{"className":11014,"style":6728},[2486],[53,11016],{},[53,11018,2651],{"className":11019},[2675]," vectors of the unitary Krylov space, with ",[53,11022,11024,11108],{"className":11023},[2418],[53,11025,11027],{"className":11026},[2422],[2424,11028,11029],{"xmlns":2426},[2428,11030,11031,11105],{},[2431,11032,11033,11035,11041,11043,11045,11063,11065,11067,11069,11071,11089,11095,11097,11099],{},[2437,11034,2644],{"mathvariant":2643},[3063,11036,11037,11039],{},[2437,11038,3688],{},[2437,11040,3521],{},[2584,11042,2651],{"stretchy":2650},[2584,11044,121],{},[2434,11046,11047,11049],{},[2437,11048,3181],{},[2431,11050,11051,11053,11055,11057,11059,11061],{},[2584,11052,2586],{},[2437,11054,3188],{},[2437,11056,3033],{},[2437,11058,3521],{},[2437,11060,4854],{},[2437,11062,3193],{},[2437,11064,2644],{"mathvariant":2643},[2437,11066,3688],{},[2584,11068,2651],{"stretchy":2650},[2584,11070,121],{},[2434,11072,11073,11075],{},[2437,11074,3181],{},[2431,11076,11077,11079,11081,11083,11085,11087],{},[2584,11078,2586],{},[2437,11080,3188],{},[2437,11082,3033],{},[2437,11084,3521],{},[2437,11086,4854],{},[2437,11088,3193],{},[3063,11090,11091,11093],{},[2437,11092,3174],{},[2437,11094,3688],{},[2437,11096,2644],{"mathvariant":2643},[2588,11098,149],{},[2434,11100,11101,11103],{},[2584,11102,2651],{"stretchy":2650},[2437,11104,7576],{},[2444,11106,11107],{"encoding":2446},"|\\psi_k\\rangle = e^{-i H k dt } \\vert \\psi \\rangle = e^{-i H k dt } U_{\\psi} \\vert 0 \\rangle^N",[53,11109,11111,11173,11245],{"className":11110,"ariaHidden":2452},[2451],[53,11112,11114,11117,11120,11161,11164,11167,11170],{"className":11113},[2456],[53,11115],{"className":11116,"style":2664},[2460],[53,11118,2644],{"className":11119},[2465],[53,11121,11123,11126],{"className":11122},[2465],[53,11124,3688],{"className":11125,"style":2671},[2465,2506],[53,11127,11129],{"className":11128},[2474],[53,11130,11132,11153],{"className":11131},[2478,3095],[53,11133,11135,11150],{"className":11134},[2482],[53,11136,11139],{"className":11137,"style":11138},[2486],"height:0.3361em;",[53,11140,11141,11144],{"style":4467},[53,11142],{"className":11143,"style":2495},[2494],[53,11145,11147],{"className":11146},[2499,2500,2501,2502],[53,11148,3521],{"className":11149,"style":3587},[2465,2506,2502],[53,11151,3120],{"className":11152},[3119],[53,11154,11156],{"className":11155},[2482],[53,11157,11159],{"className":11158,"style":3127},[2486],[53,11160],{},[53,11162,2651],{"className":11163},[2675],[53,11165],{"className":11166,"style":2814},[2609],[53,11168,121],{"className":11169},[2818],[53,11171],{"className":11172,"style":2814},[2609],[53,11174,11176,11180,11227,11230,11233,11236,11239,11242],{"className":11175},[2456],[53,11177],{"className":11178,"style":11179},[2460],"height:1.0991em;vertical-align:-0.25em;",[53,11181,11183,11186],{"className":11182},[2465],[53,11184,3181],{"className":11185},[2465,2506],[53,11187,11189],{"className":11188},[2474],[53,11190,11192],{"className":11191},[2478],[53,11193,11195],{"className":11194},[2482],[53,11196,11198],{"className":11197,"style":3557},[2486],[53,11199,11200,11203],{"style":2490},[53,11201],{"className":11202,"style":2495},[2494],[53,11204,11206],{"className":11205},[2499,2500,2501,2502],[53,11207,11209,11212,11215,11218,11221,11224],{"className":11208},[2465,2502],[53,11210,2586],{"className":11211},[2465,2502],[53,11213,3188],{"className":11214},[2465,2506,2502],[53,11216,3033],{"className":11217,"style":3048},[2465,2506,2502],[53,11219,3521],{"className":11220,"style":3587},[2465,2506,2502],[53,11222,4854],{"className":11223},[2465,2506,2502],[53,11225,3193],{"className":11226},[2465,2506,2502],[53,11228,2644],{"className":11229},[2465],[53,11231,3688],{"className":11232,"style":2671},[2465,2506],[53,11234,2651],{"className":11235},[2675],[53,11237],{"className":11238,"style":2814},[2609],[53,11240,121],{"className":11241},[2818],[53,11243],{"className":11244,"style":2814},[2609],[53,11246,11248,11252,11299,11343,11346],{"className":11247},[2456],[53,11249],{"className":11250,"style":11251},[2460],"height:1.1352em;vertical-align:-0.2861em;",[53,11253,11255,11258],{"className":11254},[2465],[53,11256,3181],{"className":11257},[2465,2506],[53,11259,11261],{"className":11260},[2474],[53,11262,11264],{"className":11263},[2478],[53,11265,11267],{"className":11266},[2482],[53,11268,11270],{"className":11269,"style":3557},[2486],[53,11271,11272,11275],{"style":2490},[53,11273],{"className":11274,"style":2495},[2494],[53,11276,11278],{"className":11277},[2499,2500,2501,2502],[53,11279,11281,11284,11287,11290,11293,11296],{"className":11280},[2465,2502],[53,11282,2586],{"className":11283},[2465,2502],[53,11285,3188],{"className":11286},[2465,2506,2502],[53,11288,3033],{"className":11289,"style":3048},[2465,2506,2502],[53,11291,3521],{"className":11292,"style":3587},[2465,2506,2502],[53,11294,4854],{"className":11295},[2465,2506,2502],[53,11297,3193],{"className":11298},[2465,2506,2502],[53,11300,11302,11305],{"className":11301},[2465],[53,11303,3174],{"className":11304,"style":3209},[2465,2506],[53,11306,11308],{"className":11307},[2474],[53,11309,11311,11335],{"className":11310},[2478,3095],[53,11312,11314,11332],{"className":11313},[2482],[53,11315,11317],{"className":11316,"style":11138},[2486],[53,11318,11320,11323],{"style":11319},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[53,11321],{"className":11322,"style":2495},[2494],[53,11324,11326],{"className":11325},[2499,2500,2501,2502],[53,11327,11329],{"className":11328},[2465,2502],[53,11330,3688],{"className":11331,"style":2671},[2465,2506,2502],[53,11333,3120],{"className":11334},[3119],[53,11336,11338],{"className":11337},[2482],[53,11339,11341],{"className":11340,"style":6728},[2486],[53,11342],{},[53,11344,9630],{"className":11345},[2465],[53,11347,11349,11352],{"className":11348},[2675],[53,11350,2651],{"className":11351},[2675],[53,11353,11355],{"className":11354},[2474],[53,11356,11358],{"className":11357},[2478],[53,11359,11361],{"className":11360},[2482],[53,11362,11364],{"className":11363,"style":3225},[2486],[53,11365,11366,11369],{"style":2490},[53,11367],{"className":11368,"style":2495},[2494],[53,11370,11372],{"className":11371},[2499,2500,2501,2502],[53,11373,7576],{"className":11374,"style":3209},[2465,2506,2502],". To measure ",[53,11377,11379,11392],{"className":11378},[2418],[53,11380,11382],{"className":11381},[2422],[2424,11383,11384],{"xmlns":2426},[2428,11385,11386,11390],{},[2431,11387,11388],{},[2437,11389,7626],{},[2444,11391,7626],{"encoding":2446},[53,11393,11395],{"className":11394,"ariaHidden":2452},[2451],[53,11396,11398,11401],{"className":11397},[2456],[53,11399],{"className":11400,"style":2461},[2460],[53,11402,7626],{"className":11403,"style":7771},[2465,2506],", first apply ",[53,11406,11408,11421],{"className":11407},[2418],[53,11409,11411],{"className":11410},[2422],[2424,11412,11413],{"xmlns":2426},[2428,11414,11415,11419],{},[2431,11416,11417],{},[2437,11418,3033],{},[2444,11420,3033],{"encoding":2446},[53,11422,11424],{"className":11423,"ariaHidden":2452},[2451],[53,11425,11427,11430],{"className":11426},[2456],[53,11428],{"className":11429,"style":2461},[2460],[53,11431,3033],{"className":11432,"style":3048},[2465,2506],[53,11434,11436],{"className":11435},[2679],[53,11437,11439,11545],{"className":11438},[2418],[53,11440,11442],{"className":11441},[2422],[2424,11443,11444],{"xmlns":2426,"display":2688},[2428,11445,11446,11542],{},[9347,11447,11448],{"rowspacing":9349,"columnspacing":9350},[9352,11449,11450],{},[9355,11451,11452],{},[9358,11453,11454],{"scriptlevel":149,"displaystyle":2452},[2431,11455,11456,11458,11460,11466,11468,11470,11472,11474,11476,11482,11484,11486,11488,11490,11496,11498,11500,11502,11508,11510,11512,11514,11516,11518,11524,11526,11528,11530,11532,11538,11540],{},[2584,11457,9382],{},[2609,11459],{"width":9350},[9386,11461,11462,11464],{},[2588,11463,173],{},[2588,11465,132],{},[2437,11467,2644],{"mathvariant":2643},[2588,11469,149],{},[2584,11471,2651],{"stretchy":2650},[2584,11473,128],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[2437,11475,2644],{"mathvariant":2643},[3063,11477,11478,11480],{},[2437,11479,3688],{},[2437,11481,2267],{},[2584,11483,2651],{"stretchy":2650},[2584,11485,2002],{},[2437,11487,3069],{},[2437,11489,2644],{"mathvariant":2643},[3063,11491,11492,11494],{},[2437,11493,3688],{},[2437,11495,3188],{},[2584,11497,2651],{"stretchy":2650},[2584,11499,3544],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[2584,11501,2002],{},[9386,11503,11504,11506],{},[2588,11505,173],{},[2588,11507,132],{},[2437,11509,2644],{"mathvariant":2643},[2588,11511,173],{},[2584,11513,2651],{"stretchy":2650},[2584,11515,128],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[2437,11517,2644],{"mathvariant":2643},[3063,11519,11520,11522],{},[2437,11521,3688],{},[2437,11523,2267],{},[2584,11525,2651],{"stretchy":2650},[2584,11527,2586],{},[2437,11529,3069],{},[2437,11531,2644],{"mathvariant":2643},[3063,11533,11534,11536],{},[2437,11535,3688],{},[2437,11537,3188],{},[2584,11539,2651],{"stretchy":2650},[2584,11541,3544],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[2444,11543,11544],{"encoding":2446},"\\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*}",[53,11546,11548],{"className":11547,"ariaHidden":2452},[2451],[53,11549,11551,11555],{"className":11550},[2456],[53,11552],{"className":11553,"style":11554},[2460],"height:2.0074em;vertical-align:-0.7537em;",[53,11556,11558],{"className":11557},[2465],[53,11559,11561],{"className":11560},[9347],[53,11562,11564],{"className":11563},[9606],[53,11565,11567,11974],{"className":11566},[2478,3095],[53,11568,11570,11971],{"className":11569},[2482],[53,11571,11574],{"className":11572,"style":11573},[2486],"height:1.2537em;",[53,11575,11577,11580],{"style":11576},"top:-3.2537em;",[53,11578],{"className":11579,"style":9623},[2494],[53,11581,11583,11586,11589,11592,11656,11659,11662,11668,11671,11711,11714,11717,11720,11723,11726,11729,11769,11772,11778,11781,11784,11787,11849,11852,11855,11861,11864,11904,11907,11910,11913,11916,11919,11922,11962,11965],{"className":11582},[2465],[53,11584,9382],{"className":11585},[2818],[53,11587],{"className":11588,"style":9669},[2609],[53,11590],{"className":11591,"style":2814},[2609],[53,11593,11595,11598,11653],{"className":11594},[2465],[53,11596],{"className":11597},[2836,9688],[53,11599,11601],{"className":11600},[9386],[53,11602,11604,11644],{"className":11603},[2478,3095],[53,11605,11607,11641],{"className":11606},[2482],[53,11608,11610,11622,11630],{"className":11609,"style":9701},[2486],[53,11611,11613,11616],{"style":11612},"top:-2.314em;",[53,11614],{"className":11615,"style":4205},[2494],[53,11617,11619],{"className":11618},[2465],[53,11620,132],{"className":11621},[2465],[53,11623,11624,11627],{"style":9773},[53,11625],{"className":11626,"style":4205},[2494],[53,11628],{"className":11629,"style":9781},[9780],[53,11631,11632,11635],{"style":9784},[53,11633],{"className":11634,"style":4205},[2494],[53,11636,11638],{"className":11637},[2465],[53,11639,173],{"className":11640},[2465],[53,11642,3120],{"className":11643},[3119],[53,11645,11647],{"className":11646},[2482],[53,11648,11651],{"className":11649,"style":11650},[2486],"height:0.686em;",[53,11652],{},[53,11654],{"className":11655},[2675,9688],[53,11657,9630],{"className":11658},[2465],[53,11660,2651],{"className":11661},[2675],[53,11663,11665],{"className":11664},[2465],[53,11666,128],{"className":11667},[2842,9815],[53,11669,2644],{"className":11670},[2465],[53,11672,11674,11677],{"className":11673},[2465],[53,11675,3688],{"className":11676,"style":2671},[2465,2506],[53,11678,11680],{"className":11679},[2474],[53,11681,11683,11703],{"className":11682},[2478,3095],[53,11684,11686,11700],{"className":11685},[2482],[53,11687,11689],{"className":11688,"style":6697},[2486],[53,11690,11691,11694],{"style":4467},[53,11692],{"className":11693,"style":2495},[2494],[53,11695,11697],{"className":11696},[2499,2500,2501,2502],[53,11698,2267],{"className":11699,"style":6718},[2465,2506,2502],[53,11701,3120],{"className":11702},[3119],[53,11704,11706],{"className":11705},[2482],[53,11707,11709],{"className":11708,"style":6728},[2486],[53,11710],{},[53,11712,2651],{"className":11713},[2675],[53,11715],{"className":11716,"style":2610},[2609],[53,11718,2002],{"className":11719},[2614],[53,11721],{"className":11722,"style":2610},[2609],[53,11724,3069],{"className":11725,"style":3115},[2465,2506],[53,11727,2644],{"className":11728},[2465],[53,11730,11732,11735],{"className":11731},[2465],[53,11733,3688],{"className":11734,"style":2671},[2465,2506],[53,11736,11738],{"className":11737},[2474],[53,11739,11741,11761],{"className":11740},[2478,3095],[53,11742,11744,11758],{"className":11743},[2482],[53,11745,11747],{"className":11746,"style":6697},[2486],[53,11748,11749,11752],{"style":4467},[53,11750],{"className":11751,"style":2495},[2494],[53,11753,11755],{"className":11754},[2499,2500,2501,2502],[53,11756,3188],{"className":11757},[2465,2506,2502],[53,11759,3120],{"className":11760},[3119],[53,11762,11764],{"className":11763},[2482],[53,11765,11767],{"className":11766,"style":3127},[2486],[53,11768],{},[53,11770,2651],{"className":11771},[2675],[53,11773,11775],{"className":11774},[2465],[53,11776,3544],{"className":11777},[2842,9815],[53,11779],{"className":11780,"style":2610},[2609],[53,11782,2002],{"className":11783},[2614],[53,11785],{"className":11786,"style":2610},[2609],[53,11788,11790,11793,11846],{"className":11789},[2465],[53,11791],{"className":11792},[2836,9688],[53,11794,11796],{"className":11795},[9386],[53,11797,11799,11838],{"className":11798},[2478,3095],[53,11800,11802,11835],{"className":11801},[2482],[53,11803,11805,11816,11824],{"className":11804,"style":9701},[2486],[53,11806,11807,11810],{"style":11612},[53,11808],{"className":11809,"style":4205},[2494],[53,11811,11813],{"className":11812},[2465],[53,11814,132],{"className":11815},[2465],[53,11817,11818,11821],{"style":9773},[53,11819],{"className":11820,"style":4205},[2494],[53,11822],{"className":11823,"style":9781},[9780],[53,11825,11826,11829],{"style":9784},[53,11827],{"className":11828,"style":4205},[2494],[53,11830,11832],{"className":11831},[2465],[53,11833,173],{"className":11834},[2465],[53,11836,3120],{"className":11837},[3119],[53,11839,11841],{"className":11840},[2482],[53,11842,11844],{"className":11843,"style":11650},[2486],[53,11845],{},[53,11847],{"className":11848},[2675,9688],[53,11850,9834],{"className":11851},[2465],[53,11853,2651],{"className":11854},[2675],[53,11856,11858],{"className":11857},[2465],[53,11859,128],{"className":11860},[2842,9815],[53,11862,2644],{"className":11863},[2465],[53,11865,11867,11870],{"className":11866},[2465],[53,11868,3688],{"className":11869,"style":2671},[2465,2506],[53,11871,11873],{"className":11872},[2474],[53,11874,11876,11896],{"className":11875},[2478,3095],[53,11877,11879,11893],{"className":11878},[2482],[53,11880,11882],{"className":11881,"style":6697},[2486],[53,11883,11884,11887],{"style":4467},[53,11885],{"className":11886,"style":2495},[2494],[53,11888,11890],{"className":11889},[2499,2500,2501,2502],[53,11891,2267],{"className":11892,"style":6718},[2465,2506,2502],[53,11894,3120],{"className":11895},[3119],[53,11897,11899],{"className":11898},[2482],[53,11900,11902],{"className":11901,"style":6728},[2486],[53,11903],{},[53,11905,2651],{"className":11906},[2675],[53,11908],{"className":11909,"style":2610},[2609],[53,11911,2586],{"className":11912},[2614],[53,11914],{"className":11915,"style":2610},[2609],[53,11917,3069],{"className":11918,"style":3115},[2465,2506],[53,11920,2644],{"className":11921},[2465],[53,11923,11925,11928],{"className":11924},[2465],[53,11926,3688],{"className":11927,"style":2671},[2465,2506],[53,11929,11931],{"className":11930},[2474],[53,11932,11934,11954],{"className":11933},[2478,3095],[53,11935,11937,11951],{"className":11936},[2482],[53,11938,11940],{"className":11939,"style":6697},[2486],[53,11941,11942,11945],{"style":4467},[53,11943],{"className":11944,"style":2495},[2494],[53,11946,11948],{"className":11947},[2499,2500,2501,2502],[53,11949,3188],{"className":11950},[2465,2506,2502],[53,11952,3120],{"className":11953},[3119],[53,11955,11957],{"className":11956},[2482],[53,11958,11960],{"className":11959,"style":3127},[2486],[53,11961],{},[53,11963,2651],{"className":11964},[2675],[53,11966,11968],{"className":11967},[2465],[53,11969,3544],{"className":11970},[2842,9815],[53,11972,3120],{"className":11973},[3119],[53,11975,11977],{"className":11976},[2482],[53,11978,11981],{"className":11979,"style":11980},[2486],"height:0.7537em;",[53,11982],{},[12,11984,11985],{},"... then measure:",[53,11987,11989],{"className":11988},[2679],[53,11990,11992,12174],{"className":11991},[2418],[53,11993,11995],{"className":11994},[2422],[2424,11996,11997],{"xmlns":2426,"display":2688},[2428,11998,11999,12171],{},[9347,12000,12001],{"rowspacing":9349,"columnspacing":9350},[9352,12002,12003],{},[9355,12004,12005],{},[9358,12006,12007],{"scriptlevel":149,"displaystyle":2452},[9347,12008,12012,12120],{"rowspacing":12009,"columnalign":12010,"columnspacing":12011},"0.25em","right left","0em",[9352,12013,12014,12031],{},[9355,12015,12016],{},[9358,12017,12018],{"scriptlevel":149,"displaystyle":2452},[2431,12019,12020,12023,12025,12027,12029],{},[2584,12021,12022],{},"⇒",[2609,12024],{"width":9350},[2584,12026,4320],{"stretchy":2650},[2437,12028,7626],{},[2584,12030,2651],{"stretchy":2650},[9355,12032,12033],{},[9358,12034,12035],{"scriptlevel":149,"displaystyle":2452},[2431,12036,12037,12039,12041,12048,12051,12054,12056,12062,12064,12066,12068,12070,12076,12078,12084,12086,12088,12090,12096,12098,12100,12102,12104,12110,12112,12118],{},[2431,12038],{},[2584,12040,121],{},[9386,12042,12043,12045],{},[2588,12044,173],{},[2588,12046,12047],{},"4",[2584,12049,128],{"fence":2650,"stretchy":2452,"minsize":12050,"maxsize":12050},"3em",[2584,12052,12053],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},"∥",[2437,12055,2644],{"mathvariant":2643},[3063,12057,12058,12060],{},[2437,12059,3688],{},[2437,12061,2267],{},[2584,12063,2651],{"stretchy":2650},[2584,12065,2002],{},[2437,12067,3069],{},[2437,12069,2644],{"mathvariant":2643},[3063,12071,12072,12074],{},[2437,12073,3688],{},[2437,12075,3188],{},[2584,12077,2651],{"stretchy":2650},[2434,12079,12080,12082],{},[2584,12081,12053],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[2588,12083,132],{},[2584,12085,2586],{},[2584,12087,12053],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[2437,12089,2644],{"mathvariant":2643},[3063,12091,12092,12094],{},[2437,12093,3688],{},[2437,12095,2267],{},[2584,12097,2651],{"stretchy":2650},[2584,12099,2586],{},[2437,12101,3069],{},[2437,12103,2644],{"mathvariant":2643},[3063,12105,12106,12108],{},[2437,12107,3688],{},[2437,12109,3188],{},[2584,12111,2651],{"stretchy":2650},[2434,12113,12114,12116],{},[2584,12115,12053],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[2588,12117,132],{},[2584,12119,3544],{"fence":2650,"stretchy":2452,"minsize":12050,"maxsize":12050},[9352,12121,12122,12128],{},[9355,12123,12124],{},[9358,12125,12126],{"scriptlevel":149,"displaystyle":2452},[2431,12127],{},[9355,12129,12130],{},[9358,12131,12132],{"scriptlevel":149,"displaystyle":2452},[2431,12133,12134,12136,12138,12141,12144,12146,12152,12154,12156,12158,12164,12166,12169],{},[2431,12135],{},[2584,12137,121],{},[6062,12139,12140],{},"Re",[2584,12142,12143],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},"[",[2584,12145,4320],{"stretchy":2650},[3063,12147,12148,12150],{},[2437,12149,3688],{},[2437,12151,2267],{},[2437,12153,2644],{"mathvariant":2643},[2437,12155,3069],{},[2437,12157,2644],{"mathvariant":2643},[3063,12159,12160,12162],{},[2437,12161,3688],{},[2437,12163,3188],{},[2584,12165,2651],{"stretchy":2650},[2584,12167,12168],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},"]",[2437,12170,28],{"mathvariant":2643},[2444,12172,12173],{"encoding":2446},"\\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*}",[53,12175,12177],{"className":12176,"ariaHidden":2452},[2451],[53,12178,12180,12184],{"className":12179},[2456],[53,12181],{"className":12182,"style":12183},[2460],"height:5.1em;vertical-align:-2.3em;",[53,12185,12187],{"className":12186},[2465],[53,12188,12190],{"className":12189},[9347],[53,12191,12193],{"className":12192},[9606],[53,12194,12196,12966],{"className":12195},[2478,3095],[53,12197,12199,12963],{"className":12198},[2482],[53,12200,12203],{"className":12201,"style":12202},[2486],"height:2.8em;",[53,12204,12206,12210],{"style":12205},"top:-4.8em;",[53,12207],{"className":12208,"style":12209},[2494],"height:4.8em;",[53,12211,12213],{"className":12212},[2465],[53,12214,12216],{"className":12215},[2465],[53,12217,12219,12280],{"className":12218},[9347],[53,12220,12223],{"className":12221},[12222],"col-align-r",[53,12224,12226,12271],{"className":12225},[2478,3095],[53,12227,12229,12268],{"className":12228},[2482],[53,12230,12232,12259],{"className":12231,"style":12202},[2486],[53,12233,12234,12238],{"style":12205},[53,12235],{"className":12236,"style":12237},[2494],"height:3.75em;",[53,12239,12241,12244,12247,12250,12253,12256],{"className":12240},[2465],[53,12242,12022],{"className":12243},[2818],[53,12245],{"className":12246,"style":9669},[2609],[53,12248],{"className":12249,"style":2814},[2609],[53,12251,4320],{"className":12252},[2836],[53,12254,7626],{"className":12255,"style":7771},[2465,2506],[53,12257,2651],{"className":12258},[2675],[53,12260,12262,12265],{"style":12261},"top:-2.1em;",[53,12263],{"className":12264,"style":12237},[2494],[53,12266],{"className":12267},[2465],[53,12269,3120],{"className":12270},[3119],[53,12272,12274],{"className":12273},[2482],[53,12275,12278],{"className":12276,"style":12277},[2486],"height:2.3em;",[53,12279],{},[53,12281,12284],{"className":12282},[12283],"col-align-l",[53,12285,12287,12955],{"className":12286},[2478,3095],[53,12288,12290,12952],{"className":12289},[2482],[53,12291,12293,12816],{"className":12292,"style":12202},[2486],[53,12294,12295,12298],{"style":12205},[53,12296],{"className":12297,"style":12237},[2494],[53,12299,12301,12304,12307,12310,12313,12375,12382,12428,12431,12471,12474,12477,12480,12483,12486,12489,12529,12532,12597,12600,12603,12606,12643,12646,12686,12689,12692,12695,12698,12701,12704,12744,12747,12810],{"className":12300},[2465],[53,12302],{"className":12303},[2465],[53,12305],{"className":12306,"style":2814},[2609],[53,12308,121],{"className":12309},[2818],[53,12311],{"className":12312,"style":2814},[2609],[53,12314,12316,12319,12372],{"className":12315},[2465],[53,12317],{"className":12318},[2836,9688],[53,12320,12322],{"className":12321},[9386],[53,12323,12325,12364],{"className":12324},[2478,3095],[53,12326,12328,12361],{"className":12327},[2482],[53,12329,12331,12342,12350],{"className":12330,"style":9701},[2486],[53,12332,12333,12336],{"style":11612},[53,12334],{"className":12335,"style":4205},[2494],[53,12337,12339],{"className":12338},[2465],[53,12340,12047],{"className":12341},[2465],[53,12343,12344,12347],{"style":9773},[53,12345],{"className":12346,"style":4205},[2494],[53,12348],{"className":12349,"style":9781},[9780],[53,12351,12352,12355],{"style":9784},[53,12353],{"className":12354,"style":4205},[2494],[53,12356,12358],{"className":12357},[2465],[53,12359,173],{"className":12360},[2465],[53,12362,3120],{"className":12363},[3119],[53,12365,12367],{"className":12366},[2482],[53,12368,12370],{"className":12369,"style":11650},[2486],[53,12371],{},[53,12373],{"className":12374},[2675,9688],[53,12376,12378],{"className":12377},[2465],[53,12379,128],{"className":12380},[2842,12381],"size4",[53,12383,12385],{"className":12384},[2465],[53,12386,12389],{"className":12387},[2842,12388],"mult",[53,12390,12392,12419],{"className":12391},[2478,3095],[53,12393,12395,12416],{"className":12394},[2482],[53,12396,12399],{"className":12397,"style":12398},[2486],"height:1.15em;",[53,12400,12402,12406],{"style":12401},"top:-3.15em;",[53,12403],{"className":12404,"style":12405},[2494],"height:3.8em;",[53,12407,12409],{"style":12408},"width:0.556em;height:1.8em;",[6176,12410,12413],{"xmlns":6178,"width":12411,"height":9397,"viewBox":12412},"0.556em","0 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+ b\\|^2 = \\langle a + b | a + b \\rangle = \\|a\\|^2 + \\|b\\|^2 + 2\\text{Re}\\langle a | b \\rangle",[53,13068,13070,13091,13140,13161,13185,13206,13256,13306],{"className":13069,"ariaHidden":2452},[2451],[53,13071,13073,13076,13079,13082,13085,13088],{"className":13072},[2456],[53,13074],{"className":13075,"style":2664},[2460],[53,13077,2644],{"className":13078},[2465],[53,13080,19],{"className":13081},[2465,2506],[53,13083],{"className":13084,"style":2610},[2609],[53,13086,2002],{"className":13087},[2614],[53,13089],{"className":13090,"style":2610},[2609],[53,13092,13094,13098,13101,13131,13134,13137],{"className":13093},[2456],[53,13095],{"className":13096,"style":13097},[2460],"height:1.0641em;vertical-align:-0.25em;",[53,13099,12997],{"className":13100},[2465,2506],[53,13102,13104,13107],{"className":13103},[2465],[53,13105,12053],{"className":13106},[2465],[53,13108,13110],{"className":13109},[2474],[53,13111,13113],{"className":13112},[2478],[53,13114,13116],{"className":13115},[2482],[53,13117,13120],{"className":13118,"style":13119},[2486],"height:0.8141em;",[53,13121,13122,13125],{"style":2490},[53,13123],{"className":13124,"style":2495},[2494],[53,13126,13128],{"className":13127},[2499,2500,2501,2502],[53,13129,132],{"className":13130},[2465,2502],[53,13132],{"className":13133,"style":2814},[2609],[53,13135,121],{"className":13136},[2818],[53,13138],{"className":13139,"style":2814},[2609],[53,13141,13143,13146,13149,13152,13155,13158],{"className":13142},[2456],[53,13144],{"className":13145,"style":2664},[2460],[53,13147,4320],{"className":13148},[2836],[53,13150,19],{"className":13151},[2465,2506],[53,13153],{"className":13154,"style":2610},[2609],[53,13156,2002],{"className":13157},[2614],[53,13159],{"className":13160,"style":2610},[2609],[53,13162,13164,13167,13170,13173,13176,13179,13182],{"className":13163},[2456],[53,13165],{"className":13166,"style":2664},[2460],[53,13168,12997],{"className":13169},[2465,2506],[53,13171,2644],{"className":13172},[2465],[53,13174,19],{"className":13175},[2465,2506],[53,13177],{"className":13178,"style":2610},[2609],[53,13180,2002],{"className":13181},[2614],[53,13183],{"className":13184,"style":2610},[2609],[53,13186,13188,13191,13194,13197,13200,13203],{"className":13187},[2456],[53,13189],{"className":13190,"style":2664},[2460],[53,13192,12997],{"className":13193},[2465,2506],[53,13195,2651],{"className":13196},[2675],[53,13198],{"className":13199,"style":2814},[2609],[53,13201,121],{"className":13202},[2818],[53,13204],{"className":13205,"style":2814},[2609],[53,13207,13209,13212,13215,13218,13247,13250,13253],{"className":13208},[2456],[53,13210],{"className":13211,"style":13097},[2460],[53,13213,12053],{"className":13214},[2465],[53,13216,19],{"className":13217},[2465,2506],[53,13219,13221,13224],{"className":13220},[2465],[53,13222,12053],{"className":13223},[2465],[53,13225,13227],{"className":13226},[2474],[53,13228,13230],{"className":13229},[2478],[53,13231,13233],{"className":13232},[2482],[53,13234,13236],{"className":13235,"style":13119},[2486],[53,13237,13238,13241],{"style":2490},[53,13239],{"className":13240,"style":2495},[2494],[53,13242,13244],{"className":13243},[2499,2500,2501,2502],[53,13245,132],{"className":13246},[2465,2502],[53,13248],{"className":13249,"style":2610},[2609],[53,13251,2002],{"className":13252},[2614],[53,13254],{"className":13255,"style":2610},[2609],[53,13257,13259,13262,13265,13268,13297,13300,13303],{"className":13258},[2456],[53,13260],{"className":13261,"style":13097},[2460],[53,13263,12053],{"className":13264},[2465],[53,13266,12997],{"className":13267},[2465,2506],[53,13269,13271,13274],{"className":13270},[2465],[53,13272,12053],{"className":13273},[2465],[53,13275,13277],{"className":13276},[2474],[53,13278,13280],{"className":13279},[2478],[53,13281,13283],{"className":13282},[2482],[53,13284,13286],{"className":13285,"style":13119},[2486],[53,13287,13288,13291],{"style":2490},[53,13289],{"className":13290,"style":2495},[2494],[53,13292,13294],{"className":13293},[2499,2500,2501,2502],[53,13295,132],{"className":13296},[2465,2502],[53,13298],{"className":13299,"style":2610},[2609],[53,13301,2002],{"className":13302},[2614],[53,13304],{"className":13305,"style":2610},[2609],[53,13307,13309,13312,13315,13321,13324,13327,13330,13333],{"className":13308},[2456],[53,13310],{"className":13311,"style":2664},[2460],[53,13313,132],{"className":13314},[2465],[53,13316,13318],{"className":13317},[2465,6934],[53,13319,12140],{"className":13320},[2465],[53,13322,4320],{"className":13323},[2836],[53,13325,19],{"className":13326},[2465,2506],[53,13328,2644],{"className":13329},[2465],[53,13331,12997],{"className":13332},[2465,2506],[53,13334,2651],{"className":13335},[2675],". Similarly, measuring ",[53,13338,13340,13353],{"className":13339},[2418],[53,13341,13343],{"className":13342},[2422],[2424,13344,13345],{"xmlns":2426},[2428,13346,13347,13351],{},[2431,13348,13349],{},[2437,13350,7641],{},[2444,13352,7641],{"encoding":2446},[53,13354,13356],{"className":13355,"ariaHidden":2452},[2451],[53,13357,13359,13362],{"className":13358},[2456],[53,13360],{"className":13361,"style":2461},[2460],[53,13363,7641],{"className":13364,"style":2610},[2465,2506]," yields",[53,13367,13369],{"className":13368},[2679],[53,13370,13372,13431],{"className":13371},[2418],[53,13373,13375],{"className":13374},[2422],[2424,13376,13377],{"xmlns":2426,"display":2688},[2428,13378,13379,13428],{},[9347,13380,13381],{"rowspacing":9349,"columnspacing":9350},[9352,13382,13383],{},[9355,13384,13385],{},[9358,13386,13387],{"scriptlevel":149,"displaystyle":2452},[2431,13388,13389,13391,13393,13395,13397,13400,13402,13404,13410,13412,13414,13416,13422,13424,13426],{},[2584,13390,4320],{"stretchy":2650},[2437,13392,7641],{},[2584,13394,2651],{"stretchy":2650},[2584,13396,121],{},[6062,13398,13399],{},"Im",[2584,13401,12143],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[2584,13403,4320],{"stretchy":2650},[3063,13405,13406,13408],{},[2437,13407,3688],{},[2437,13409,2267],{},[2437,13411,2644],{"mathvariant":2643},[2437,13413,3069],{},[2437,13415,2644],{"mathvariant":2643},[3063,13417,13418,13420],{},[2437,13419,3688],{},[2437,13421,3188],{},[2584,13423,2651],{"stretchy":2650},[2584,13425,12168],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[2437,13427,28],{"mathvariant":2643},[2444,13429,13430],{"encoding":2446},"\\begin{equation*}\n \\langle Y\\rangle = \\text{Im}\\Big[\\langle\\psi_j| P|\\psi_i\\rangle\\Big].\n\\end{equation*}",[53,13432,13434],{"className":13433,"ariaHidden":2452},[2451],[53,13435,13437,13441],{"className":13436},[2456],[53,13438],{"className":13439,"style":13440},[2460],"height:1.8em;vertical-align:-0.65em;",[53,13442,13444],{"className":13443},[2465],[53,13445,13447],{"className":13446},[9347],[53,13448,13450],{"className":13449},[9606],[53,13451,13453,13605],{"className":13452},[2478,3095],[53,13454,13456,13602],{"className":13455},[2482],[53,13457,13459],{"className":13458,"style":12398},[2486],[53,13460,13461,13465],{"style":12401},[53,13462],{"className":13463,"style":13464},[2494],"height:3.15em;",[53,13466,13468,13471,13474,13477,13480,13483,13486,13492,13498,13501,13541,13544,13547,13550,13590,13593,13599],{"className":13467},[2465],[53,13469,4320],{"className":13470},[2836],[53,13472,7641],{"className":13473,"style":2610},[2465,2506],[53,13475,2651],{"className":13476},[2675],[53,13478],{"className":13479,"style":2814},[2609],[53,13481,121],{"className":13482},[2818],[53,13484],{"className":13485,"style":2814},[2609],[53,13487,13489],{"className":13488},[2465,6934],[53,13490,13399],{"className":13491},[2465],[53,13493,13495],{"className":13494},[2465],[53,13496,12143],{"className":13497},[2842,9815],[53,13499,4320],{"className":13500},[2836],[53,13502,13504,13507],{"className":13503},[2465],[53,13505,3688],{"className":13506,"style":2671},[2465,2506],[53,13508,13510],{"className":13509},[2474],[53,13511,13513,13533],{"className":13512},[2478,3095],[53,13514,13516,13530],{"className":13515},[2482],[53,13517,13519],{"className":13518,"style":6697},[2486],[53,13520,13521,13524],{"style":4467},[53,13522],{"className":13523,"style":2495},[2494],[53,13525,13527],{"className":13526},[2499,2500,2501,2502],[53,13528,2267],{"className":13529,"style":6718},[2465,2506,2502],[53,13531,3120],{"className":13532},[3119],[53,13534,13536],{"className":13535},[2482],[53,13537,13539],{"className":13538,"style":6728},[2486],[53,13540],{},[53,13542,2644],{"className":13543},[2465],[53,13545,3069],{"className":13546,"style":3115},[2465,2506],[53,13548,2644],{"className":13549},[2465],[53,13551,13553,13556],{"className":13552},[2465],[53,13554,3688],{"className":13555,"style":2671},[2465,2506],[53,13557,13559],{"className":13558},[2474],[53,13560,13562,13582],{"className":13561},[2478,3095],[53,13563,13565,13579],{"className":13564},[2482],[53,13566,13568],{"className":13567,"style":6697},[2486],[53,13569,13570,13573],{"style":4467},[53,13571],{"className":13572,"style":2495},[2494],[53,13574,13576],{"className":13575},[2499,2500,2501,2502],[53,13577,3188],{"className":13578},[2465,2506,2502],[53,13580,3120],{"className":13581},[3119],[53,13583,13585],{"className":13584},[2482],[53,13586,13588],{"className":13587,"style":3127},[2486],[53,13589],{},[53,13591,2651],{"className":13592},[2675],[53,13594,13596],{"className":13595},[2465],[53,13597,12168],{"className":13598},[2842,9815],[53,13600,28],{"className":13601},[2465],[53,13603,3120],{"className":13604},[3119],[53,13606,13608],{"className":13607},[2482],[53,13609,13611],{"className":13610,"style":12425},[2486],[53,13612],{},[43,13614,13616],{"className":45,"code":13615,"language":47,"meta":48,"style":48},"## 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",[50,13617,13618,13622,13632,13657,13661,13665,13670,13681,13705,13709,13723,13727,13732,13743,13755,13776,13789,13805,13817,13832,13854,13858,13863,13888,13892,13897,13913,13924,13937,13962,13974,13978,13984,13989,13993],{"__ignoreMap":48},[53,13619,13620],{"class":55,"line":56},[53,13621,9223],{"class":110},[53,13623,13624,13626,13628,13630],{"class":55,"line":73},[53,13625,9228],{"class":63},[53,13627,121],{"class":120},[53,13629,9233],{"class":124},[53,13631,1271],{"class":63},[53,13633,13634,13636,13638,13640,13643,13645,13647,13649,13651,13653,13655],{"class":55,"line":87},[53,13635,9240],{"class":63},[53,13637,9243],{"class":2050},[53,13639,121],{"class":120},[53,13641,13642],{"class":63},"dt, ",[53,13644,9251],{"class":2050},[53,13646,121],{"class":120},[53,13648,9256],{"class":124},[53,13650,128],{"class":63},[53,13652,9261],{"class":2050},[53,13654,121],{"class":120},[53,13656,9266],{"class":63},[53,13658,13659],{"class":55,"line":100},[53,13660,135],{"class":63},[53,13662,13663],{"class":55,"line":107},[53,13664,104],{"emptyLinePlaceholder":103},[53,13666,13667],{"class":55,"line":114},[53,13668,13669],{"class":110},"## Create the time-evo op dagger circuit\n",[53,13671,13672,13675,13677,13679],{"class":55,"line":138},[53,13673,13674],{"class":63},"evol_gate_d ",[53,13676,121],{"class":120},[53,13678,9233],{"class":124},[53,13680,1271],{"class":63},[53,13682,13683,13685,13687,13689,13691,13693,13695,13697,13699,13701,13703],{"class":55,"line":158},[53,13684,9240],{"class":63},[53,13686,9243],{"class":2050},[53,13688,121],{"class":120},[53,13690,13642],{"class":63},[53,13692,9251],{"class":2050},[53,13694,121],{"class":120},[53,13696,9256],{"class":124},[53,13698,128],{"class":63},[53,13700,9261],{"class":2050},[53,13702,121],{"class":120},[53,13704,9266],{"class":63},[53,13706,13707],{"class":55,"line":182},[53,13708,135],{"class":63},[53,13710,13711,13713,13715,13718,13721],{"class":55,"line":209},[53,13712,13674],{"class":63},[53,13714,121],{"class":120},[53,13716,13717],{"class":63}," evol_gate_d.",[53,13719,13720],{"class":124},"inverse",[53,13722,329],{"class":63},[53,13724,13725],{"class":55,"line":226},[53,13726,104],{"emptyLinePlaceholder":103},[53,13728,13729],{"class":55,"line":231},[53,13730,13731],{"class":110},"# Put pieces together\n",[53,13733,13734,13737,13739,13741],{"class":55,"line":245},[53,13735,13736],{"class":63},"qc_reg ",[53,13738,121],{"class":120},[53,13740,9284],{"class":124},[53,13742,8678],{"class":63},[53,13744,13745,13748,13750,13752],{"class":55,"line":258},[53,13746,13747],{"class":63},"qc_temp ",[53,13749,121],{"class":120},[53,13751,125],{"class":124},[53,13753,13754],{"class":63},"(qc_reg)\n",[53,13756,13757,13760,13763,13766,13769,13771,13774],{"class":55,"line":263},[53,13758,13759],{"class":63},"qc_temp.",[53,13761,13762],{"class":124},"compose",[53,13764,13765],{"class":63},"(qc_state_prep, ",[53,13767,13768],{"class":2050},"inplace",[53,13770,121],{"class":120},[53,13772,13773],{"class":131},"True",[53,13775,135],{"class":63},[53,13777,13778,13780,13782,13784,13786],{"class":55,"line":269},[53,13779,1525],{"class":59},[53,13781,8111],{"class":63},[53,13783,1531],{"class":59},[53,13785,8116],{"class":120},[53,13787,13788],{"class":63},"(num_trotter_steps):\n",[53,13790,13791,13794,13796,13798,13800,13802],{"class":55,"line":284},[53,13792,13793],{"class":63}," qc_temp.",[53,13795,1834],{"class":124},[53,13797,9308],{"class":63},[53,13799,9311],{"class":2050},[53,13801,121],{"class":120},[53,13803,13804],{"class":63},"qc_reg)\n",[53,13806,13807,13809,13811,13813,13815],{"class":55,"line":296},[53,13808,1525],{"class":59},[53,13810,8111],{"class":63},[53,13812,1531],{"class":59},[53,13814,8116],{"class":120},[53,13816,13788],{"class":63},[53,13818,13819,13821,13823,13826,13828,13830],{"class":55,"line":309},[53,13820,13793],{"class":63},[53,13822,1834],{"class":124},[53,13824,13825],{"class":63},"(evol_gate_d, ",[53,13827,9311],{"class":2050},[53,13829,121],{"class":120},[53,13831,13804],{"class":63},[53,13833,13834,13836,13838,13841,13843,13846,13848,13850,13852],{"class":55,"line":323},[53,13835,13759],{"class":63},[53,13837,13762],{"class":124},[53,13839,13840],{"class":63},"(qc_state_prep.",[53,13842,13720],{"class":124},[53,13844,13845],{"class":63},"(), ",[53,13847,13768],{"class":2050},[53,13849,121],{"class":120},[53,13851,13773],{"class":131},[53,13853,135],{"class":63},[53,13855,13856],{"class":55,"line":332},[53,13857,104],{"emptyLinePlaceholder":103},[53,13859,13860],{"class":55,"line":344},[53,13861,13862],{"class":110},"# Create controlled version of the circuit\n",[53,13864,13865,13868,13870,13873,13876,13879,13882,13884,13886],{"class":55,"line":357},[53,13866,13867],{"class":63},"controlled_U ",[53,13869,121],{"class":120},[53,13871,13872],{"class":63}," qc_temp.",[53,13874,13875],{"class":124},"to_gate",[53,13877,13878],{"class":63},"().",[53,13880,13881],{"class":124},"control",[53,13883,128],{"class":63},[53,13885,173],{"class":131},[53,13887,135],{"class":63},[53,13889,13890],{"class":55,"line":1247},[53,13891,104],{"emptyLinePlaceholder":103},[53,13893,13894],{"class":55,"line":1252},[53,13895,13896],{"class":110},"# Create hadamard test circuit for real part\n",[53,13898,13899,13901,13903,13905,13907,13909,13911],{"class":55,"line":1257},[53,13900,9279],{"class":63},[53,13902,121],{"class":120},[53,13904,9284],{"class":124},[53,13906,1999],{"class":63},[53,13908,2002],{"class":120},[53,13910,1321],{"class":131},[53,13912,135],{"class":63},[53,13914,13915,13918,13920,13922],{"class":55,"line":1262},[53,13916,13917],{"class":63},"qc_real ",[53,13919,121],{"class":120},[53,13921,125],{"class":124},[53,13923,9298],{"class":63},[53,13925,13926,13929,13931,13933,13935],{"class":55,"line":1274},[53,13927,13928],{"class":63},"qc_real.",[53,13930,144],{"class":124},[53,13932,128],{"class":63},[53,13934,149],{"class":131},[53,13936,135],{"class":63},[53,13938,13939,13941,13943,13946,13949,13951,13953,13955,13957,13959],{"class":55,"line":1284},[53,13940,13928],{"class":63},[53,13942,1834],{"class":124},[53,13944,13945],{"class":63},"(controlled_U, ",[53,13947,13948],{"class":120},"list",[53,13950,128],{"class":63},[53,13952,2099],{"class":120},[53,13954,1999],{"class":63},[53,13956,2002],{"class":120},[53,13958,1321],{"class":131},[53,13960,13961],{"class":63},")))\n",[53,13963,13964,13966,13968,13970,13972],{"class":55,"line":1292},[53,13965,13928],{"class":63},[53,13967,144],{"class":124},[53,13969,128],{"class":63},[53,13971,149],{"class":131},[53,13973,135],{"class":63},[53,13975,13976],{"class":55,"line":1307},[53,13977,104],{"emptyLinePlaceholder":103},[53,13979,13980,13982],{"class":55,"line":1326},[53,13981,8420],{"class":120},[53,13983,1271],{"class":63},[53,13985,13986],{"class":55,"line":1344},[53,13987,13988],{"class":278}," \"Circuit for calculating the real part of the overlap in S via Hadamard test\"\n",[53,13990,13991],{"class":55,"line":1360},[53,13992,135],{"class":63},[53,13994,13995,13997,13999,14001,14003,14005,14008,14011,14013,14015,14017,14019,14021],{"class":55,"line":1366},[53,13996,13928],{"class":63},[53,13998,9059],{"class":124},[53,14000,128],{"class":63},[53,14002,9064],{"class":278},[53,14004,170],{"class":63},[53,14006,14007],{"class":2050},"fold",[53,14009,14010],{"class":120},"=-",[53,14012,173],{"class":131},[53,14014,170],{"class":63},[53,14016,9069],{"class":2050},[53,14018,121],{"class":120},[53,14020,9074],{"class":131},[53,14022,135],{"class":63},[43,14024,14027],{"className":14025,"code":14026,"language":6934,"meta":48},[8426],"Circuit for calculating the real part of the overlap in S via Hadamard test\n",[50,14028,14026],{"__ignoreMap":48},[379,14030],{"alt":9079,"src":14031},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-02.avif",[12,14033,14034],{},"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)",[43,14036,14038],{"className":45,"code":14037,"language":47,"meta":48,"style":48},"print(\n \"Number of layers of 2Q operations\",\n qc_real.decompose(reps=2).depth(lambda x: x[0].num_qubits == 2),\n)\n",[50,14039,14040,14046,14053,14098],{"__ignoreMap":48},[53,14041,14042,14044],{"class":55,"line":56},[53,14043,8420],{"class":120},[53,14045,1271],{"class":63},[53,14047,14048,14051],{"class":55,"line":73},[53,14049,14050],{"class":278}," \"Number of layers of 2Q operations\"",[53,14052,1304],{"class":63},[53,14054,14055,14058,14061,14063,14065,14067,14069,14072,14075,14077,14080,14083,14086,14088,14091,14093,14095],{"class":55,"line":87},[53,14056,14057],{"class":63}," qc_real.",[53,14059,14060],{"class":124},"decompose",[53,14062,128],{"class":63},[53,14064,9261],{"class":2050},[53,14066,121],{"class":120},[53,14068,132],{"class":131},[53,14070,14071],{"class":63},").",[53,14073,14074],{"class":124},"depth",[53,14076,128],{"class":63},[53,14078,14079],{"class":59},"lambda",[53,14081,14082],{"class":1277}," x",[53,14084,14085],{"class":63},": x[",[53,14087,149],{"class":131},[53,14089,14090],{"class":63},"].num_qubits ",[53,14092,1618],{"class":120},[53,14094,2211],{"class":131},[53,14096,14097],{"class":63},"),\n",[53,14099,14100],{"class":55,"line":100},[53,14101,135],{"class":63},[43,14103,14106],{"className":14104,"code":14105,"language":6934,"meta":48},[8426],"Number of layers of 2Q operations 112753\n",[50,14107,14105],{"__ignoreMap":48},[30,14109,14111],{"id":14110},"step-2-optimize-problem-for-quantum-hardware-execution","Step 2: Optimize problem for quantum hardware execution",[2408,14113,14115],{"id":14114},"efficient-hadamard-test","Efficient Hadamard test",[12,14117,14118],{},"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:",[12,14120,14121],{},[6603,14122],{"alt":14123,"src":14124},"fig3.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-08.avif",[12,14126,14127,14128,14199,14200,14266,14267,14295,14296,14361,14362,14404,14405,14684,14685,14726,14727,14768],{},"Assume we can classically calculate ",[53,14129,14131,14149],{"className":14130},[2418],[53,14132,14134],{"className":14133},[2422],[2424,14135,14136],{"xmlns":2426},[2428,14137,14138,14146],{},[2431,14139,14140],{},[3063,14141,14142,14144],{},[2437,14143,6077],{},[2588,14145,149],{},[2444,14147,14148],{"encoding":2446},"E_0",[53,14150,14152],{"className":14151,"ariaHidden":2452},[2451],[53,14153,14155,14158],{"className":14154},[2456],[53,14156],{"className":14157,"style":3082},[2460],[53,14159,14161,14164],{"className":14160},[2465],[53,14162,6077],{"className":14163,"style":5787},[2465,2506],[53,14165,14167],{"className":14166},[2474],[53,14168,14170,14191],{"className":14169},[2478,3095],[53,14171,14173,14188],{"className":14172},[2482],[53,14174,14177],{"className":14175,"style":14176},[2486],"height:0.3011em;",[53,14178,14179,14182],{"style":5909},[53,14180],{"className":14181,"style":2495},[2494],[53,14183,14185],{"className":14184},[2499,2500,2501,2502],[53,14186,149],{"className":14187},[2465,2502],[53,14189,3120],{"className":14190},[3119],[53,14192,14194],{"className":14193},[2482],[53,14195,14197],{"className":14196,"style":3127},[2486],[53,14198],{},", the eigenvalue of ",[53,14201,14203,14225],{"className":14202},[2418],[53,14204,14206],{"className":14205},[2422],[2424,14207,14208],{"xmlns":2426},[2428,14209,14210,14222],{},[2431,14211,14212,14214,14216],{},[2437,14213,2644],{"mathvariant":2643},[2588,14215,149],{},[2434,14217,14218,14220],{},[2584,14219,2651],{"stretchy":2650},[2437,14221,7576],{},[2444,14223,14224],{"encoding":2446},"|0\\rangle^N",[53,14226,14228],{"className":14227,"ariaHidden":2452},[2451],[53,14229,14231,14234,14237],{"className":14230},[2456],[53,14232],{"className":14233,"style":5145},[2460],[53,14235,9630],{"className":14236},[2465],[53,14238,14240,14243],{"className":14239},[2675],[53,14241,2651],{"className":14242},[2675],[53,14244,14246],{"className":14245},[2474],[53,14247,14249],{"className":14248},[2478],[53,14250,14252],{"className":14251},[2482],[53,14253,14255],{"className":14254,"style":3225},[2486],[53,14256,14257,14260],{"style":2490},[53,14258],{"className":14259,"style":2495},[2494],[53,14261,14263],{"className":14262},[2499,2500,2501,2502],[53,14264,7576],{"className":14265,"style":3209},[2465,2506,2502]," under the Hamiltonian ",[53,14268,14270,14283],{"className":14269},[2418],[53,14271,14273],{"className":14272},[2422],[2424,14274,14275],{"xmlns":2426},[2428,14276,14277,14281],{},[2431,14278,14279],{},[2437,14280,3033],{},[2444,14282,3033],{"encoding":2446},[53,14284,14286],{"className":14285,"ariaHidden":2452},[2451],[53,14287,14289,14292],{"className":14288},[2456],[53,14290],{"className":14291,"style":2461},[2460],[53,14293,3033],{"className":14294,"style":3048},[2465,2506],".\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 ",[53,14297,14299,14320],{"className":14298},[2418],[53,14300,14302],{"className":14301},[2422],[2424,14303,14304],{"xmlns":2426},[2428,14305,14306,14318],{},[2431,14307,14308,14310,14312],{},[2437,14309,2644],{"mathvariant":2643},[2588,14311,149],{},[2434,14313,14314,14316],{},[2584,14315,2651],{"stretchy":2650},[2437,14317,7576],{},[2444,14319,14224],{"encoding":2446},[53,14321,14323],{"className":14322,"ariaHidden":2452},[2451],[53,14324,14326,14329,14332],{"className":14325},[2456],[53,14327],{"className":14328,"style":5145},[2460],[53,14330,9630],{"className":14331},[2465],[53,14333,14335,14338],{"className":14334},[2675],[53,14336,2651],{"className":14337},[2675],[53,14339,14341],{"className":14340},[2474],[53,14342,14344],{"className":14343},[2478],[53,14345,14347],{"className":14346},[2482],[53,14348,14350],{"className":14349,"style":3225},[2486],[53,14351,14352,14355],{"style":2490},[53,14353],{"className":14354,"style":2495},[2494],[53,14356,14358],{"className":14357},[2499,2500,2501,2502],[53,14359,7576],{"className":14360,"style":3209},[2465,2506,2502]," 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 ",[53,14363,14365,14383],{"className":14364},[2418],[53,14366,14368],{"className":14367},[2422],[2424,14369,14370],{"xmlns":2426},[2428,14371,14372,14380],{},[2431,14373,14374,14376,14378],{},[6062,14375,6908],{},[6062,14377,6911],{},[2437,14379,3688],{},[2444,14381,14382],{"encoding":2446},"\\text{Prep} \\; \\psi",[53,14384,14386],{"className":14385,"ariaHidden":2452},[2451],[53,14387,14389,14392,14398,14401],{"className":14388},[2456],[53,14390],{"className":14391,"style":6930},[2460],[53,14393,14395],{"className":14394},[2465,6934],[53,14396,6908],{"className":14397},[2465],[53,14399],{"className":14400,"style":2814},[2609],[53,14402,3688],{"className":14403,"style":2671},[2465,2506],", prepares the desired reference state ",[53,14406,14408,14486],{"className":14407},[2418],[53,14409,14411],{"className":14410},[2422],[2424,14412,14413],{"xmlns":2426},[2428,14414,14415,14483],{},[2431,14416,14417,14433,14435,14437,14439,14441,14449,14451,14469,14475],{},[14418,14419,14420,14422,14431],"mpadded",{},[2437,14421,2644],{"mathvariant":2643},[2431,14423,14424,14426,14429],{},[2437,14425,12],{},[2437,14427,14428],{},"s",[2437,14430,3188],{},[2584,14432,2651],{"stretchy":2650},[2584,14434,121],{},[6062,14436,6908],{},[6062,14438,6911],{},[2437,14440,3688],{},[14418,14442,14443,14445,14447],{},[2437,14444,2644],{"mathvariant":2643},[2588,14446,149],{},[2584,14448,2651],{"stretchy":2650},[2584,14450,121],{},[2434,14452,14453,14455],{},[2437,14454,3181],{},[2431,14456,14457,14459,14461,14463,14465,14467],{},[2584,14458,2586],{},[2437,14460,3188],{},[2437,14462,3033],{},[2588,14464,149],{},[2437,14466,4854],{},[2437,14468,3193],{},[3063,14470,14471,14473],{},[2437,14472,3174],{},[2437,14474,3688],{},[14418,14476,14477,14479,14481],{},[2437,14478,2644],{"mathvariant":2643},[2588,14480,149],{},[2584,14482,2651],{"stretchy":2650},[2444,14484,14485],{"encoding":2446},"\\ket{psi} = \\text{Prep} \\; \\psi \\ket{0} = e^{-i H 0 dt} U_{\\psi} \\ket{0}",[53,14487,14489,14525,14570],{"className":14488,"ariaHidden":2452},[2451],[53,14490,14492,14495,14516,14519,14522],{"className":14491},[2456],[53,14493],{"className":14494,"style":2664},[2460],[53,14496,14498,14501,14513],{"className":14497},[2832],[53,14499,2644],{"className":14500},[2465],[53,14502,14504,14507,14510],{"className":14503},[2465],[53,14505,12],{"className":14506},[2465,2506],[53,14508,14428],{"className":14509},[2465,2506],[53,14511,3188],{"className":14512},[2465,2506],[53,14514,2651],{"className":14515},[2675],[53,14517],{"className":14518,"style":2814},[2609],[53,14520,121],{"className":14521},[2818],[53,14523],{"className":14524,"style":2814},[2609],[53,14526,14528,14531,14537,14540,14543,14546,14561,14564,14567],{"className":14527},[2456],[53,14529],{"className":14530,"style":2664},[2460],[53,14532,14534],{"className":14533},[2465,6934],[53,14535,6908],{"className":14536},[2465],[53,14538],{"className":14539,"style":2814},[2609],[53,14541,3688],{"className":14542,"style":2671},[2465,2506],[53,14544],{"className":14545,"style":2860},[2609],[53,14547,14549,14552,14558],{"className":14548},[2832],[53,14550,2644],{"className":14551},[2465],[53,14553,14555],{"className":14554},[2465],[53,14556,149],{"className":14557},[2465],[53,14559,2651],{"className":14560},[2675],[53,14562],{"className":14563,"style":2814},[2609],[53,14565,121],{"className":14566},[2818],[53,14568],{"className":14569,"style":2814},[2609],[53,14571,14573,14576,14623,14666,14669],{"className":14572},[2456],[53,14574],{"className":14575,"style":11251},[2460],[53,14577,14579,14582],{"className":14578},[2465],[53,14580,3181],{"className":14581},[2465,2506],[53,14583,14585],{"className":14584},[2474],[53,14586,14588],{"className":14587},[2478],[53,14589,14591],{"className":14590},[2482],[53,14592,14594],{"className":14593,"style":3557},[2486],[53,14595,14596,14599],{"style":2490},[53,14597],{"className":14598,"style":2495},[2494],[53,14600,14602],{"className":14601},[2499,2500,2501,2502],[53,14603,14605,14608,14611,14614,14617,14620],{"className":14604},[2465,2502],[53,14606,2586],{"className":14607},[2465,2502],[53,14609,3188],{"className":14610},[2465,2506,2502],[53,14612,3033],{"className":14613,"style":3048},[2465,2506,2502],[53,14615,149],{"className":14616},[2465,2502],[53,14618,4854],{"className":14619},[2465,2506,2502],[53,14621,3193],{"className":14622},[2465,2506,2502],[53,14624,14626,14629],{"className":14625},[2465],[53,14627,3174],{"className":14628,"style":3209},[2465,2506],[53,14630,14632],{"className":14631},[2474],[53,14633,14635,14658],{"className":14634},[2478,3095],[53,14636,14638,14655],{"className":14637},[2482],[53,14639,14641],{"className":14640,"style":11138},[2486],[53,14642,14643,14646],{"style":11319},[53,14644],{"className":14645,"style":2495},[2494],[53,14647,14649],{"className":14648},[2499,2500,2501,2502],[53,14650,14652],{"className":14651},[2465,2502],[53,14653,3688],{"className":14654,"style":2671},[2465,2506,2502],[53,14656,3120],{"className":14657},[3119],[53,14659,14661],{"className":14660},[2482],[53,14662,14664],{"className":14663,"style":6728},[2486],[53,14665],{},[53,14667],{"className":14668,"style":2860},[2609],[53,14670,14672,14675,14681],{"className":14671},[2832],[53,14673,2644],{"className":14674},[2465],[53,14676,14678],{"className":14677},[2465],[53,14679,149],{"className":14680},[2465],[53,14682,2651],{"className":14683},[2675],", for example, to prepare the HF state for chemistry ",[53,14686,14688,14705],{"className":14687},[2418],[53,14689,14691],{"className":14690},[2422],[2424,14692,14693],{"xmlns":2426},[2428,14694,14695,14703],{},[2431,14696,14697,14699,14701],{},[6062,14698,6908],{},[6062,14700,6911],{},[2437,14702,3688],{},[2444,14704,14382],{"encoding":2446},[53,14706,14708],{"className":14707,"ariaHidden":2452},[2451],[53,14709,14711,14714,14720,14723],{"className":14710},[2456],[53,14712],{"className":14713,"style":6930},[2460],[53,14715,14717],{"className":14716},[2465,6934],[53,14718,6908],{"className":14719},[2465],[53,14721],{"className":14722,"style":2814},[2609],[53,14724,3688],{"className":14725,"style":2671},[2465,2506]," would be a product of single-qubit NOTs, so controlled-",[53,14728,14730,14747],{"className":14729},[2418],[53,14731,14733],{"className":14732},[2422],[2424,14734,14735],{"xmlns":2426},[2428,14736,14737,14745],{},[2431,14738,14739,14741,14743],{},[6062,14740,6908],{},[6062,14742,6911],{},[2437,14744,3688],{},[2444,14746,14382],{"encoding":2446},[53,14748,14750],{"className":14749,"ariaHidden":2452},[2451],[53,14751,14753,14756,14762,14765],{"className":14752},[2456],[53,14754],{"className":14755,"style":6930},[2460],[53,14757,14759],{"className":14758},[2465,6934],[53,14760,6908],{"className":14761},[2465],[53,14763],{"className":14764,"style":2814},[2609],[53,14766,3688],{"className":14767,"style":2671},[2465,2506]," is just a product of CNOTs.\nThen the circuit above implements the following state prior to measurement:",[53,14770,14772],{"className":14771},[2679],[53,14773,14775,15395],{"className":14774},[2418],[53,14776,14778],{"className":14777},[2422],[2424,14779,14780],{"xmlns":2426,"display":2688},[2428,14781,14782,15392],{},[9347,14783,14784],{"rowspacing":9349,"columnspacing":9350},[9352,14785,14786,14790,15385,15388],{},[9355,14787],{"className":14788},[14789],"mtr-glue",[9355,14791,14792],{},[9358,14793,14794],{"scriptlevel":149,"displaystyle":2452},[9347,14795,14796,14900,14967,15054,15137,15255],{"rowspacing":12009,"columnalign":12010,"columnspacing":12011},[9352,14797,14798,14836],{},[9355,14799,14800],{},[9358,14801,14802],{"scriptlevel":149,"displaystyle":2452},[2431,14803,14804,14812,14824],{},[14418,14805,14806,14808,14810],{},[2437,14807,2644],{"mathvariant":2643},[2588,14809,149],{},[2584,14811,2651],{"stretchy":2650},[2434,14813,14814,14822],{},[14418,14815,14816,14818,14820],{},[2437,14817,2644],{"mathvariant":2643},[2588,14819,149],{},[2584,14821,2651],{"stretchy":2650},[2437,14823,7576],{},[4166,14825,14826,14830],{},[2584,14827,14829],{"stretchy":2452,"minsize":14828},"3.0em","→",[14418,14831,14834],{"width":14832,"lspace":14833},"+0.6em","0.3em",[2437,14835,3033],{},[9355,14837,14838],{},[9358,14839,14840],{"scriptlevel":149,"displaystyle":2452},[2431,14841,14842,14844,14852],{},[2431,14843],{},[9386,14845,14846,14848],{},[2588,14847,173],{},[9391,14849,14850],{},[2588,14851,132],{},[2431,14853,14854,14856,14864,14876,14878,14886,14898],{},[2584,14855,128],{"fence":2452},[14418,14857,14858,14860,14862],{},[2437,14859,2644],{"mathvariant":2643},[2588,14861,149],{},[2584,14863,2651],{"stretchy":2650},[2434,14865,14866,14874],{},[14418,14867,14868,14870,14872],{},[2437,14869,2644],{"mathvariant":2643},[2588,14871,149],{},[2584,14873,2651],{"stretchy":2650},[2437,14875,7576],{},[2584,14877,2002],{},[14418,14879,14880,14882,14884],{},[2437,14881,2644],{"mathvariant":2643},[2588,14883,173],{},[2584,14885,2651],{"stretchy":2650},[2434,14887,14888,14896],{},[14418,14889,14890,14892,14894],{},[2437,14891,2644],{"mathvariant":2643},[2588,14893,149],{},[2584,14895,2651],{"stretchy":2650},[2437,14897,7576],{},[2584,14899,3544],{"fence":2452},[9352,14901,14902,14915],{},[9355,14903,14904],{},[9358,14905,14906],{"scriptlevel":149,"displaystyle":2452},[4166,14907,14908,14910],{},[2584,14909,14829],{"stretchy":2452,"minsize":14828},[14418,14911,14912],{"width":14832,"lspace":14833},[6062,14913,14914],{},"1-ctrl-init",[9355,14916,14917],{},[9358,14918,14919],{"scriptlevel":149,"displaystyle":2452},[2431,14920,14921,14923,14931],{},[2431,14922],{},[9386,14924,14925,14927],{},[2588,14926,173],{},[9391,14928,14929],{},[2588,14930,132],{},[2431,14932,14933,14935,14937,14939,14941,14943,14945,14951,14953,14955,14957,14959,14961,14963,14965],{},[2584,14934,128],{"fence":2452},[2437,14936,2644],{"mathvariant":2643},[2588,14938,149],{},[2584,14940,2651],{"stretchy":2650},[2437,14942,2644],{"mathvariant":2643},[2588,14944,149],{},[2434,14946,14947,14949],{},[2584,14948,2651],{"stretchy":2650},[2437,14950,7576],{},[2584,14952,2002],{},[2437,14954,2644],{"mathvariant":2643},[2588,14956,173],{},[2584,14958,2651],{"stretchy":2650},[2437,14960,2644],{"mathvariant":2643},[2437,14962,3688],{},[2584,14964,2651],{"stretchy":2650},[2584,14966,3544],{"fence":2452},[9352,14968,14969,14981],{},[9355,14970,14971],{},[9358,14972,14973],{"scriptlevel":149,"displaystyle":2452},[4166,14974,14975,14977],{},[2584,14976,14829],{"stretchy":2452,"minsize":14828},[14418,14978,14979],{"width":14832,"lspace":14833},[2437,14980,3174],{},[9355,14982,14983],{},[9358,14984,14985],{"scriptlevel":149,"displaystyle":2452},[2431,14986,14987,14989,14997],{},[2431,14988],{},[9386,14990,14991,14993],{},[2588,14992,173],{},[9391,14994,14995],{},[2588,14996,132],{},[2431,14998,14999,15001,15012,15020,15032,15034,15042,15044,15052],{},[2584,15000,128],{"fence":2452},[2434,15002,15003,15005],{},[2437,15004,3181],{},[2431,15006,15007,15009],{},[2437,15008,3188],{},[2437,15010,15011],{},"ϕ",[14418,15013,15014,15016,15018],{},[2437,15015,2644],{"mathvariant":2643},[2588,15017,149],{},[2584,15019,2651],{"stretchy":2650},[2434,15021,15022,15030],{},[14418,15023,15024,15026,15028],{},[2437,15025,2644],{"mathvariant":2643},[2588,15027,149],{},[2584,15029,2651],{"stretchy":2650},[2437,15031,7576],{},[2584,15033,2002],{},[14418,15035,15036,15038,15040],{},[2437,15037,2644],{"mathvariant":2643},[2588,15039,173],{},[2584,15041,2651],{"stretchy":2650},[2437,15043,3174],{},[14418,15045,15046,15048,15050],{},[2437,15047,2644],{"mathvariant":2643},[2437,15049,3688],{},[2584,15051,2651],{"stretchy":2650},[2584,15053,3544],{"fence":2452},[9352,15055,15056,15069],{},[9355,15057,15058],{},[9358,15059,15060],{"scriptlevel":149,"displaystyle":2452},[4166,15061,15062,15064],{},[2584,15063,14829],{"stretchy":2452,"minsize":14828},[14418,15065,15066],{"width":14832,"lspace":14833},[6062,15067,15068],{},"0-ctrl-init",[9355,15070,15071],{},[9358,15072,15073],{"scriptlevel":149,"displaystyle":2452},[2431,15074,15075,15077,15085],{},[2431,15076],{},[9386,15078,15079,15081],{},[2588,15080,173],{},[9391,15082,15083],{},[2588,15084,132],{},[2431,15086,15087,15089,15099,15107,15115,15117,15125,15127,15135],{},[2584,15088,128],{"fence":2452},[2434,15090,15091,15093],{},[2437,15092,3181],{},[2431,15094,15095,15097],{},[2437,15096,3188],{},[2437,15098,15011],{},[14418,15100,15101,15103,15105],{},[2437,15102,2644],{"mathvariant":2643},[2588,15104,149],{},[2584,15106,2651],{"stretchy":2650},[14418,15108,15109,15111,15113],{},[2437,15110,2644],{"mathvariant":2643},[2437,15112,3688],{},[2584,15114,2651],{"stretchy":2650},[2584,15116,2002],{},[14418,15118,15119,15121,15123],{},[2437,15120,2644],{"mathvariant":2643},[2588,15122,173],{},[2584,15124,2651],{"stretchy":2650},[2437,15126,3174],{},[14418,15128,15129,15131,15133],{},[2437,15130,2644],{"mathvariant":2643},[2437,15132,3688],{},[2584,15134,2651],{"stretchy":2650},[2584,15136,3544],{"fence":2452},[9352,15138,15139,15145],{},[9355,15140,15141],{},[9358,15142,15143],{"scriptlevel":149,"displaystyle":2452},[2584,15144,121],{"lspace":12011,"rspace":12011},[9355,15146,15147],{},[9358,15148,15149],{"scriptlevel":149,"displaystyle":2452},[2431,15150,15151,15153,15159],{},[2431,15152],{},[9386,15154,15155,15157],{},[2588,15156,173],{},[2588,15158,132],{},[2431,15160,15161,15163,15171,15207,15209,15217,15253],{},[2584,15162,128],{"fence":2452},[14418,15164,15165,15167,15169],{},[2437,15166,2644],{"mathvariant":2643},[2584,15168,2002],{"lspace":12011,"rspace":12011},[2584,15170,2651],{"stretchy":2650},[2431,15172,15173,15175,15185,15193,15195,15197,15205],{},[2584,15174,128],{"fence":2452},[2434,15176,15177,15179],{},[2437,15178,3181],{},[2431,15180,15181,15183],{},[2437,15182,3188],{},[2437,15184,15011],{},[14418,15186,15187,15189,15191],{},[2437,15188,2644],{"mathvariant":2643},[2437,15190,3688],{},[2584,15192,2651],{"stretchy":2650},[2584,15194,2002],{},[2437,15196,3174],{},[14418,15198,15199,15201,15203],{},[2437,15200,2644],{"mathvariant":2643},[2437,15202,3688],{},[2584,15204,2651],{"stretchy":2650},[2584,15206,3544],{"fence":2452},[2584,15208,2002],{},[14418,15210,15211,15213,15215],{},[2437,15212,2644],{"mathvariant":2643},[2584,15214,2586],{"lspace":12011,"rspace":12011},[2584,15216,2651],{"stretchy":2650},[2431,15218,15219,15221,15231,15239,15241,15243,15251],{},[2584,15220,128],{"fence":2452},[2434,15222,15223,15225],{},[2437,15224,3181],{},[2431,15226,15227,15229],{},[2437,15228,3188],{},[2437,15230,15011],{},[14418,15232,15233,15235,15237],{},[2437,15234,2644],{"mathvariant":2643},[2437,15236,3688],{},[2584,15238,2651],{"stretchy":2650},[2584,15240,2586],{},[2437,15242,3174],{},[14418,15244,15245,15247,15249],{},[2437,15246,2644],{"mathvariant":2643},[2437,15248,3688],{},[2584,15250,2651],{"stretchy":2650},[2584,15252,3544],{"fence":2452},[2584,15254,3544],{"fence":2452},[9352,15256,15257,15263],{},[9355,15258,15259],{},[9358,15260,15261],{"scriptlevel":149,"displaystyle":2452},[2584,15262,121],{"lspace":12011,"rspace":12011},[9355,15264,15265],{},[9358,15266,15267],{"scriptlevel":149,"displaystyle":2452},[2431,15268,15269,15271,15277],{},[2431,15270],{},[9386,15272,15273,15275],{},[2588,15274,173],{},[2588,15276,132],{},[2431,15278,15279,15281,15293,15331,15333,15345,15383],{},[2584,15280,128],{"fence":2452},[14418,15282,15283,15285,15291],{},[2437,15284,2644],{"mathvariant":2643},[2431,15286,15287,15289],{},[2584,15288,2002],{},[2437,15290,3188],{},[2584,15292,2651],{"stretchy":2650},[2431,15294,15295,15297,15307,15315,15317,15319,15321,15329],{},[2584,15296,128],{"fence":2452},[2434,15298,15299,15301],{},[2437,15300,3181],{},[2431,15302,15303,15305],{},[2437,15304,3188],{},[2437,15306,15011],{},[14418,15308,15309,15311,15313],{},[2437,15310,2644],{"mathvariant":2643},[2437,15312,3688],{},[2584,15314,2651],{"stretchy":2650},[2584,15316,2586],{},[2437,15318,3188],{},[2437,15320,3174],{},[14418,15322,15323,15325,15327],{},[2437,15324,2644],{"mathvariant":2643},[2437,15326,3688],{},[2584,15328,2651],{"stretchy":2650},[2584,15330,3544],{"fence":2452},[2584,15332,2002],{},[14418,15334,15335,15337,15343],{},[2437,15336,2644],{"mathvariant":2643},[2431,15338,15339,15341],{},[2584,15340,2586],{},[2437,15342,3188],{},[2584,15344,2651],{"stretchy":2650},[2431,15346,15347,15349,15359,15367,15369,15371,15373,15381],{},[2584,15348,128],{"fence":2452},[2434,15350,15351,15353],{},[2437,15352,3181],{},[2431,15354,15355,15357],{},[2437,15356,3188],{},[2437,15358,15011],{},[14418,15360,15361,15363,15365],{},[2437,15362,2644],{"mathvariant":2643},[2437,15364,3688],{},[2584,15366,2651],{"stretchy":2650},[2584,15368,2002],{},[2437,15370,3188],{},[2437,15372,3174],{},[14418,15374,15375,15377,15379],{},[2437,15376,2644],{"mathvariant":2643},[2437,15378,3688],{},[2584,15380,2651],{"stretchy":2650},[2584,15382,3544],{"fence":2452},[2584,15384,3544],{"fence":2452},[9355,15386],{"className":15387},[14789],[9355,15389],{"className":15390},[15391],"mml-eqn-num",[2444,15393,15394],{"encoding":2446},"\\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}",[53,15396,15398,17503],{"className":15397,"ariaHidden":2452},[2451],[53,15399,15401,15405],{"className":15400},[2456],[53,15402],{"className":15403,"style":15404},[2460],"height:14.5206em;vertical-align:-7.0103em;",[53,15406,15408],{"className":15407},[9347],[53,15409,15411],{"className":15410},[9606],[53,15412,15414,17495],{"className":15413},[2478,3095],[53,15415,15417,17492],{"className":15416},[2482],[53,15418,15421],{"className":15419,"style":15420},[2486],"height:7.5103em;",[53,15422,15424,15428],{"style":15423},"top:-9.5103em;",[53,15425],{"className":15426,"style":15427},[2494],"height:9.5103em;",[53,15429,15431],{"className":15430},[2465],[53,15432,15434],{"className":15433},[2465],[53,15435,15437,15800],{"className":15436},[9347],[53,15438,15440],{"className":15439},[12222],[53,15441,15443,15791],{"className":15442},[2478,3095],[53,15444,15446,15788],{"className":15445},[2482],[53,15447,15449,15580,15643,15702,15764,15776],{"className":15448,"style":15420},[2486],[53,15450,15451,15454],{"style":15423},[53,15452],{"className":15453,"style":9623},[2494],[53,15455,15457,15472,15475,15516,15519],{"className":15456},[2465],[53,15458,15460,15463,15469],{"className":15459},[2832],[53,15461,2644],{"className":15462},[2465],[53,15464,15466],{"className":15465},[2465],[53,15467,149],{"className":15468},[2465],[53,15470,2651],{"className":15471},[2675],[53,15473],{"className":15474,"style":2860},[2609],[53,15476,15478,15493],{"className":15477},[2832],[53,15479,15481,15484,15490],{"className":15480},[2832],[53,15482,2644],{"className":15483},[2465],[53,15485,15487],{"className":15486},[2465],[53,15488,149],{"className":15489},[2465],[53,15491,2651],{"className":15492},[2675],[53,15494,15496],{"className":15495},[2474],[53,15497,15499],{"className":15498},[2478],[53,15500,15502],{"className":15501},[2482],[53,15503,15505],{"className":15504,"style":7717},[2486],[53,15506,15507,15510],{"style":7740},[53,15508],{"className":15509,"style":2495},[2494],[53,15511,15513],{"className":15512},[2499,2500,2501,2502],[53,15514,7576],{"className":15515,"style":3209},[2465,2506,2502],[53,15517],{"className":15518,"style":2814},[2609],[53,15520,15523],{"className":15521},[2818,15522],"x-arrow",[53,15524,15526,15571],{"className":15525},[2478,3095],[53,15527,15529,15568],{"className":15528},[2482],[53,15530,15533,15549],{"className":15531,"style":15532},[2486],"height:1.1003em;",[53,15534,15536,15539],{"style":15535},"top:-3.322em;",[53,15537],{"className":15538,"style":2495},[2494],[53,15540,15543],{"className":15541},[2499,2500,2501,2502,15542],"x-arrow-pad",[53,15544,15546],{"className":15545},[2465,2502],[53,15547,3033],{"className":15548,"style":3048},[2465,2506,2502],[53,15550,15553,15556],{"className":15551,"style":15552},[9728],"top:-2.689em;",[53,15554],{"className":15555,"style":2495},[2494],[53,15557,15560],{"className":15558,"style":15559},[9748],"height:0.522em;min-width:1.469em;",[6176,15561,15565],{"xmlns":6178,"width":9752,"height":15562,"viewBox":15563,"preserveAspectRatio":15564},"0.522em","0 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we have used the classical simulable phase shift ",[53,17540,17542,17592],{"className":17541},[2418],[53,17543,17545],{"className":17544},[2422],[2424,17546,17547],{"xmlns":2426},[2428,17548,17549,17589],{},[2431,17550,17551,17553,17565,17567,17577],{},[2437,17552,3174],{},[2434,17554,17555,17563],{},[14418,17556,17557,17559,17561],{},[2437,17558,2644],{"mathvariant":2643},[2588,17560,149],{},[2584,17562,2651],{"stretchy":2650},[2437,17564,7576],{},[2584,17566,121],{},[2434,17568,17569,17571],{},[2437,17570,3181],{},[2431,17572,17573,17575],{},[2437,17574,3188],{},[2437,17576,15011],{},[2434,17578,17579,17587],{},[14418,17580,17581,17583,17585],{},[2437,17582,2644],{"mathvariant":2643},[2588,17584,149],{},[2584,17586,2651],{"stretchy":2650},[2437,17588,7576],{},[2444,17590,17591],{"encoding":2446}," U\\ket{0}^N = e^{i\\phi}\\ket{0}^N",[53,17593,17595,17658],{"className":17594,"ariaHidden":2452},[2451],[53,17596,17598,17602,17605,17608,17649,17652,17655],{"className":17597},[2456],[53,17599],{"className":17600,"style":17601},[2460],"height:1.2312em;vertical-align:-0.25em;",[53,17603,3174],{"className":17604,"style":3209},[2465,2506],[53,17606],{"className":17607,"style":2860},[2609],[53,17609,17611,17626],{"className":17610},[2832],[53,17612,17614,17617,17623],{"className":17613},[2832],[53,17615,2644],{"className":17616},[2465],[53,17618,17620],{"className":17619},[2465],[53,17621,149],{"className":17622},[2465],[53,17624,2651],{"className":17625},[2675],[53,17627,17629],{"className":17628},[2474],[53,17630,17632],{"className":17631},[2478],[53,17633,17635],{"className":17634},[2482],[53,17636,17638],{"className":17637,"style":7717},[2486],[53,17639,17640,17643],{"style":7740},[53,17641],{"className":17642,"style":2495},[2494],[53,17644,17646],{"className":17645},[2499,2500,2501,2502],[53,17647,7576],{"className":17648,"style":3209},[2465,2506,2502],[53,17650],{"className":17651,"style":2814},[2609],[53,17653,121],{"className":17654},[2818],[53,17656],{"className":17657,"style":2814},[2609],[53,17659,17661,17664,17699,17702],{"className":17660},[2456],[53,17662],{"className":17663,"style":17601},[2460],[53,17665,17667,17670],{"className":17666},[2465],[53,17668,3181],{"className":17669},[2465,2506],[53,17671,17673],{"className":17672},[2474],[53,17674,17676],{"className":17675},[2478],[53,17677,17679],{"className":17678},[2482],[53,17680,17682],{"className":17681,"style":3557},[2486],[53,17683,17684,17687],{"style":2490},[53,17685],{"className":17686,"style":2495},[2494],[53,17688,17690],{"className":17689},[2499,2500,2501,2502],[53,17691,17693,17696],{"className":17692},[2465,2502],[53,17694,3188],{"className":17695},[2465,2506,2502],[53,17697,15011],{"className":17698},[2465,2506,2502],[53,17700],{"className":17701,"style":2860},[2609],[53,17703,17705,17720],{"className":17704},[2832],[53,17706,17708,17711,17717],{"className":17707},[2832],[53,17709,2644],{"className":17710},[2465],[53,17712,17714],{"className":17713},[2465],[53,17715,149],{"className":17716},[2465],[53,17718,2651],{"className":17719},[2675],[53,17721,17723],{"className":17722},[2474],[53,17724,17726],{"className":17725},[2478],[53,17727,17729],{"className":17728},[2482],[53,17730,17732],{"className":17731,"style":7717},[2486],[53,17733,17734,17737],{"style":7740},[53,17735],{"className":17736,"style":2495},[2494],[53,17738,17740],{"className":17739},[2499,2500,2501,2502],[53,17741,7576],{"className":17742,"style":3209},[2465,2506,2502]," in the third line. Therefore the expectation values are obtained as",[53,17745,17747],{"className":17746},[2679],[53,17748,17750,18060],{"className":17749},[2418],[53,17751,17753],{"className":17752},[2422],[2424,17754,17755],{"xmlns":2426,"display":2688},[2428,17756,17757,18057],{},[9347,17758,17759],{"rowspacing":9349,"columnspacing":9350},[9352,17760,17761,17764,18051,18054],{},[9355,17762],{"className":17763},[14789],[9355,17765,17766],{},[9358,17767,17768],{"scriptlevel":149,"displaystyle":2452},[9347,17769,17770,17888,17991],{"rowspacing":12009,"columnalign":12010,"columnspacing":12011},[9352,17771,17772,17789],{},[9355,17773,17774],{},[9358,17775,17776],{"scriptlevel":149,"displaystyle":2452},[2431,17777,17778,17780,17782,17785,17787],{},[2584,17779,4320],{"stretchy":2650},[2437,17781,7626],{},[2584,17783,17784],{},"⊗",[2437,17786,3069],{},[2584,17788,2651],{"stretchy":2650},[9355,17790,17791],{},[9358,17792,17793],{"scriptlevel":149,"displaystyle":2452},[2431,17794,17795,17797,17799,17805,17807,17850,17852],{},[2431,17796],{},[2584,17798,121],{},[9386,17800,17801,17803],{},[2588,17802,173],{},[2588,17804,12047],{},[2584,17806,128],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[2431,17808,17809,17811,17823,17831,17833,17841,17848],{},[2584,17810,128],{"fence":2452},[2434,17812,17813,17815],{},[2437,17814,3181],{},[2431,17816,17817,17819,17821],{},[2584,17818,2586],{},[2437,17820,3188],{},[2437,17822,15011],{},[14418,17824,17825,17827,17829],{},[2584,17826,4320],{"stretchy":2650},[2437,17828,3688],{},[2437,17830,2644],{"mathvariant":2643},[2584,17832,2002],{},[14418,17834,17835,17837,17839],{},[2584,17836,4320],{"stretchy":2650},[2437,17838,3688],{},[2437,17840,2644],{"mathvariant":2643},[2434,17842,17843,17845],{},[2437,17844,3174],{},[2584,17846,17847],{},"†",[2584,17849,3544],{"fence":2452},[2437,17851,3069],{},[2431,17853,17854,17856,17866,17874,17876,17878,17886],{},[2584,17855,128],{"fence":2452},[2434,17857,17858,17860],{},[2437,17859,3181],{},[2431,17861,17862,17864],{},[2437,17863,3188],{},[2437,17865,15011],{},[14418,17867,17868,17870,17872],{},[2437,17869,2644],{"mathvariant":2643},[2437,17871,3688],{},[2584,17873,2651],{"stretchy":2650},[2584,17875,2002],{},[2437,17877,3174],{},[14418,17879,17880,17882,17884],{},[2437,17881,2644],{"mathvariant":2643},[2437,17883,3688],{},[2584,17885,2651],{"stretchy":2650},[2584,17887,3544],{"fence":2452},[9352,17889,17890,17896],{},[9355,17891,17892],{},[9358,17893,17894],{"scriptlevel":149,"displaystyle":2452},[2431,17895],{},[9355,17897,17898],{},[9358,17899,17900],{"scriptlevel":149,"displaystyle":2452},[2431,17901,17902,17904,17907,17909,17951,17953,17989],{},[2431,17903],{},[2609,17905],{"width":17906},"2em",[2584,17908,2586],{},[2431,17910,17911,17913,17925,17933,17935,17943,17949],{},[2584,17912,128],{"fence":2452},[2434,17914,17915,17917],{},[2437,17916,3181],{},[2431,17918,17919,17921,17923],{},[2584,17920,2586],{},[2437,17922,3188],{},[2437,17924,15011],{},[14418,17926,17927,17929,17931],{},[2584,17928,4320],{"stretchy":2650},[2437,17930,3688],{},[2437,17932,2644],{"mathvariant":2643},[2584,17934,2586],{},[14418,17936,17937,17939,17941],{},[2584,17938,4320],{"stretchy":2650},[2437,17940,3688],{},[2437,17942,2644],{"mathvariant":2643},[2434,17944,17945,17947],{},[2437,17946,3174],{},[2584,17948,17847],{},[2584,17950,3544],{"fence":2452},[2437,17952,3069],{},[2431,17954,17955,17957,17967,17975,17977,17979,17987],{},[2584,17956,128],{"fence":2452},[2434,17958,17959,17961],{},[2437,17960,3181],{},[2431,17962,17963,17965],{},[2437,17964,3188],{},[2437,17966,15011],{},[14418,17968,17969,17971,17973],{},[2437,17970,2644],{"mathvariant":2643},[2437,17972,3688],{},[2584,17974,2651],{"stretchy":2650},[2584,17976,2586],{},[2437,17978,3174],{},[14418,17980,17981,17983,17985],{},[2437,17982,2644],{"mathvariant":2643},[2437,17984,3688],{},[2584,17986,2651],{"stretchy":2650},[2584,17988,3544],{"fence":2452},[2584,17990,3544],{"fence":2650,"stretchy":2452,"minsize":9397,"maxsize":9397},[9352,17992,17993,17999],{},[9355,17994,17995],{},[9358,17996,17997],{"scriptlevel":149,"displaystyle":2452},[2431,17998],{},[9355,18000,18001],{},[9358,18002,18003],{"scriptlevel":149,"displaystyle":2452},[2431,18004,18005,18007,18009,18011,18049],{},[2431,18006],{},[2584,18008,121],{},[6062,18010,12140],{},[2431,18012,18013,18015,18027,18035,18037,18039,18047],{},[2584,18014,12143],{"fence":2452},[2434,18016,18017,18019],{},[2437,18018,3181],{},[2431,18020,18021,18023,18025],{},[2584,18022,2586],{},[2437,18024,3188],{},[2437,18026,15011],{},[14418,18028,18029,18031,18033],{},[2584,18030,4320],{"stretchy":2650},[2437,18032,3688],{},[2437,18034,2644],{"mathvariant":2643},[2437,18036,3069],{},[2437,18038,3174],{},[14418,18040,18041,18043,18045],{},[2437,18042,2644],{"mathvariant":2643},[2437,18044,3688],{},[2584,18046,2651],{"stretchy":2650},[2584,18048,12168],{"fence":2452},[2584,18050,2714],{"separator":2452},[9355,18052],{"className":18053},[14789],[9355,18055],{"className":18056},[15391],[2444,18058,18059],{"encoding":2446},"\\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}",[53,18061,18063,18932],{"className":18062,"ariaHidden":2452},[2451],[53,18064,18066,18070],{"className":18065},[2456],[53,18067],{"className":18068,"style":18069},[2460],"height:5.6666em;vertical-align:-2.5833em;",[53,18071,18073],{"className":18072},[9347],[53,18074,18076],{"className":18075},[9606],[53,18077,18079,18924],{"className":18078},[2478,3095],[53,18080,18082,18921],{"className":18081},[2482],[53,18083,18086],{"className":18084,"style":18085},[2486],"height:3.0833em;",[53,18087,18089,18093],{"style":18088},"top:-5.0833em;",[53,18090],{"className":18091,"style":18092},[2494],"height:5.0833em;",[53,18094,18096],{"className":18095},[2465],[53,18097,18099],{"className":18098},[2465],[53,18100,18102,18173],{"className":18101},[9347],[53,18103,18105],{"className":18104},[12222],[53,18106,18108,18164],{"className":18107},[2478,3095],[53,18109,18111,18161],{"className":18110},[2482],[53,18112,18114,18143,18152],{"className":18113,"style":18085},[2486],[53,18115,18116,18119],{"style":18088},[53,18117],{"className":18118,"style":9623},[2494],[53,18120,18122,18125,18128,18131,18134,18137,18140],{"className":18121},[2465],[53,18123,4320],{"className":18124},[2836],[53,18126,7626],{"className":18127,"style":7771},[2465,2506],[53,18129],{"className":18130,"style":2610},[2609],[53,18132,17784],{"className":18133},[2614],[53,18135],{"className":18136,"style":2610},[2609],[53,18138,3069],{"className":18139,"style":3115},[2465,2506],[53,18141,2651],{"className":18142},[2675],[53,18144,18146,18149],{"style":18145},"top:-2.9473em;",[53,18147],{"className":18148,"style":9623},[2494],[53,18150],{"className":18151},[2465],[53,18153,18155,18158],{"style":18154},"top:-1.0982em;",[53,18156],{"className":18157,"style":9623},[2494],[53,18159],{"className":18160},[2465],[53,18162,3120],{"className":18163},[3119],[53,18165,18167],{"className":18166},[2482],[53,18168,18171],{"className":18169,"style":18170},[2486],"height:2.5833em;",[53,18172],{},[53,18174,18176],{"className":18175},[12283],[53,18177,18179,18913],{"className":18178},[2478,3095],[53,18180,18182,18910],{"className":18181},[2482],[53,18183,18185,18510,18777],{"className":18184,"style":18085},[2486],[53,18186,18187,18190],{"style":18088},[53,18188],{"className":18189,"style":9623},[2494],[53,18191,18193,18196,18199,18202,18205,18267,18273,18276,18403,18406,18409,18412],{"className":18192},[2465],[53,18194],{"className":18195},[2465],[53,18197],{"className":18198,"style":2814},[2609],[53,18200,121],{"className":18201},[2818],[53,18203],{"className":18204,"style":2814},[2609],[53,18206,18208,18211,18264],{"className":18207},[2465],[53,18209],{"className":18210},[2836,9688],[53,18212,18214],{"className":18213},[9386],[53,18215,18217,18256],{"className":18216},[2478,3095],[53,18218,18220,18253],{"className":18219},[2482],[53,18221,18223,18234,18242],{"className":18222,"style":9701},[2486],[53,18224,18225,18228],{"style":11612},[53,18226],{"className":18227,"style":4205},[2494],[53,18229,18231],{"className":18230},[2465],[53,18232,12047],{"className":18233},[2465],[53,18235,18236,18239],{"style":9773},[53,18237],{"className":18238,"style":4205},[2494],[53,18240],{"className":18241,"style":9781},[9780],[53,18243,18244,18247],{"style":9784},[53,18245],{"className":18246,"style":4205},[2494],[53,18248,18250],{"className":18249},[2465],[53,18251,173],{"className":18252},[2465],[53,18254,3120],{"className":18255},[3119],[53,18257,18259],{"className":18258},[2482],[53,18260,18262],{"className":18261,"style":11650},[2486],[53,18263],{},[53,18265],{"className":18266},[2675,9688],[53,18268,18270],{"className":18269},[2465],[53,18271,128],{"className":18272},[2842,9815],[53,18274],{"className":18275,"style":2860},[2609],[53,18277,18279,18285,18323,18326,18341,18344,18347,18350,18365,18368,18397],{"className":18278},[2832],[53,18280,18282],{"className":18281,"style":2838},[2836,2837],[53,18283,128],{"className":18284},[2842,2843],[53,18286,18288,18291],{"className":18287},[2465],[53,18289,3181],{"className":18290},[2465,2506],[53,18292,18294],{"className":18293},[2474],[53,18295,18297],{"className":18296},[2478],[53,18298,18300],{"className":18299},[2482],[53,18301,18303],{"className":18302,"style":4935},[2486],[53,18304,18305,18308],{"style":2801},[53,18306],{"className":18307,"style":2495},[2494],[53,18309,18311],{"className":18310},[2499,2500,2501,2502],[53,18312,18314,18317,18320],{"className":18313},[2465,2502],[53,18315,2586],{"className":18316},[2465,2502],[53,18318,3188],{"className":18319},[2465,2506,2502],[53,18321,15011],{"className":18322},[2465,2506,2502],[53,18324],{"className":18325,"style":2860},[2609],[53,18327,18329,18332,18338],{"className":18328},[2832],[53,18330,4320],{"className":18331},[2836],[53,18333,18335],{"className":18334},[2465],[53,18336,3688],{"className":18337,"style":2671},[2465,2506],[53,18339,2644],{"className":18340},[2465],[53,18342],{"className"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laystyle":2452},[2431,19228,19229,19231,19233,19235,19273],{},[2431,19230],{},[2584,19232,121],{},[6062,19234,13399],{},[2431,19236,19237,19239,19251,19259,19261,19263,19271],{},[2584,19238,12143],{"fence":2452},[2434,19240,19241,19243],{},[2437,19242,3181],{},[2431,19244,19245,19247,19249],{},[2584,19246,2586],{},[2437,19248,3188],{},[2437,19250,15011],{},[14418,19252,19253,19255,19257],{},[2584,19254,4320],{"stretchy":2650},[2437,19256,3688],{},[2437,19258,2644],{"mathvariant":2643},[2437,19260,3069],{},[2437,19262,3174],{},[14418,19264,19265,19267,19269],{},[2437,19266,2644],{"mathvariant":2643},[2437,19268,3688],{},[2584,19270,2651],{"stretchy":2650},[2584,19272,12168],{"fence":2452},[2437,19274,28],{"mathvariant":2643},[9355,19276],{"className":19277},[14789],[9355,19279],{"className":19280},[15391],[2444,19282,19283],{"encoding":2446},"\\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}",[53,19285,19287,20166],{"className":19286,"ariaHidden":2452},[2451],[53,19288,19290,19293],{"className":19289},[2456],[53,19291],{"className":19292,"style":18069},[2460],[53,19294,19296],{"className":19295},[9347],[53,19297,19299],{"className":19298},[9606],[53,19300,19302,20158],{"className":19301},[2478,3095],[53,19303,19305,20155],{"className":19304},[2482],[53,19306,19308],{"className":19307,"style":18085},[2486],[53,19309,19310,19313],{"style":18088},[53,19311],{"className":19312,"style":18092},[2494],[53,19314,19316],{"className":19315},[2465],[53,19317,19319],{"className":19318},[2465],[53,19320,19322,19390],{"className":19321},[9347],[53,19323,19325],{"className":19324},[12222],[53,19326,19328,19382],{"className":19327},[2478,3095],[53,19329,19331,19379],{"className":19330},[2482],[53,19332,19334,19363,19371],{"className":19333,"style":18085},[2486],[53,19335,19336,19339],{"style":18088},[53,19337],{"className":19338,"style":9623},[2494],[53,19340,19342,19345,19348,19351,19354,19357,19360],{"className":19341},[2465],[53,19343,4320],{"className":19344},[2836],[53,19346,7641],{"className":19347,"style":2610},[2465,2506],[53,19349],{"className":19350,"style":2610},[2609],[53,19352,17784],{"className":19353},[2614],[53,19355],{"className":19356,"style":2610},[2609],[53,19358,3069],{"className":19359,"style":3115},[2465,2506],[53,19361,2651],{"className":19362},[2675],[53,19364,19365,19368],{"style":18145},[53,19366],{"className":19367,"style":9623},[2494],[53,19369],{"className":19370},[2465],[53,19372,19373,19376],{"style":18154},[53,19374],{"className":19375,"style":9623},[2494],[53,19377],{"className":19378},[2465],[53,19380,3120],{"className":19381},[3119],[53,19383,19385],{"className":19384},[2482],[53,19386,19388],{"className":19387,"style":18170},[2486],[53,19389],{},[53,19391,19393],{"className":19392},[12283],[53,19394,19396,20147],{"className":19395},[2478,3095],[53,19397,19399,20144],{"className":19398},[2482],[53,19400,19402,19736,20011],{"className":19401,"style":18085},[2486],[53,19403,19404,19407],{"style":18088},[53,19405],{"className":19406,"style":9623},[2494],[53,19408,19410,19413,19416,19419,19422,19484,19490,19493,19626,19629,19632,19635],{"className":19409},[2465],[53,19411],{"className":19412},[2465],[53,19414],{"className":19415,"style":2814},[2609],[53,19417,121],{"className":19418},[2818],[53,19420],{"className":19421,"style":2814},[2609],[53,19423,19425,19428,19481],{"className":19424},[2465],[53,19426],{"className":19427},[2836,9688],[53,19429,19431],{"className":19430},[9386],[53,19432,19434,19473],{"className":19433},[2478,3095],[53,19435,19437,19470],{"className":19436},[2482],[53,19438,19440,19451,19459],{"className":19439,"style":9701},[2486],[53,19441,19442,19445],{"style":11612},[53,19443],{"className":19444,"style":4205},[2494],[53,19446,19448],{"className":19447},[2465],[53,19449,12047],{"className":19450},[2465],[53,19452,19453,19456],{"style":9773},[53,19454],{"className":19455,"style":4205},[2494],[53,19457],{"className":19458,"style":9781},[9780],[53,19460,19461,19464],{"style":9784},[53,19462],{"className":19463,"style":4205},[2494],[53,19465,19467],{"className":19466},[2465],[53,19468,173],{"className":19469},[2465],[53,19471,3120],{"className":19472},[3119],[53,19474,19476],{"className":19475},[2482],[53,19477,19479],{"className":19478,"style":11650},[2486],[53,19480],{},[53,19482],{"className":19483},[2675,9688],[53,19485,19487],{"className":19486},[2465],[53,19488,128],{"className":19489},[2842,9815],[53,19491],{"className":19492,"style":2860},[2609],[53,19494,19496,19502,19540,19543,19558,19561,19564,19567,19570,19573,19588,19591,19620],{"className":19495},[2832],[53,19497,19499],{"className":19498,"style":2838},[2836,2837],[53,19500,128],{"className":19501},[2842,2843],[53,19503,19505,19508],{"className":19504},[2465],[53,19506,3181],{"className":19507},[2465,2506],[53,19509,19511],{"className":19510},[2474],[53,19512,19514],{"className":19513},[2478],[53,19515,19517],{"className":19516},[2482],[53,19518,19520],{"className":19519,"style":4935},[2486],[53,19521,19522,19525],{"style":2801},[53,19523],{"className":19524,"style":2495},[2494],[53,19526,19528],{"className":19527},[2499,2500,2501,2502],[53,19529,19531,19534,19537],{"className":19530},[2465,2502],[53,19532,2586],{"className":19533},[2465,2502],[53,19535,3188],{"className":19536},[2465,2506,2502],[53,19538,15011],{"className":19539},[2465,2506,2502],[53,19541],{"className":19542,"style":2860},[2609],[53,19544,19546,19549,19555],{"className":19545},[2832],[53,19547,4320],{"className":19548},[2836],[53,19550,19552],{"className":19551},[2465],[53,19553,3688],{"className":19554,"style":2671},[2465,2506],[53,19556,2644],{"className":19557},[2465],[53,19559],{"className":19560,"style":2610},[2609],[53,19562,2002],{"className":19563},[2614],[53,19565],{"className":19566,"style":2610},[2609],[53,19568,3188],{"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,[2842,2843],[53,20139],{"className":20140,"style":2860},[2609],[53,20142,28],{"className":20143},[2465],[53,20145,3120],{"className":20146},[3119],[53,20148,20150],{"className":20149},[2482],[53,20151,20153],{"className":20152,"style":18170},[2486],[53,20154],{},[53,20156,3120],{"className":20157},[3119],[53,20159,20161],{"className":20160},[2482],[53,20162,20164],{"className":20163,"style":18170},[2486],[53,20165],{},[53,20167,20169],{"className":20168},[17506],[53,20170,20172,20189],{"className":20171},[2478,3095],[53,20173,20175,20186],{"className":20174},[2482],[53,20176,20178],{"className":20177,"style":18085},[2486],[53,20179,20180,20183],{"style":18088},[53,20181],{"className":20182,"style":18092},[2494],[53,20184],{"className":20185},[17524],[53,20187,3120],{"className":20188},[3119],[53,20190,20192],{"className":20191},[2482],[53,20193,20195],{"className":20194,"style":18170},[2486],[53,20196],{},[12,20198,20199,20200,20241],{},"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 ",[53,20201,20203,20220],{"className":20202},[2418],[53,20204,20206],{"className":20205},[2422],[2424,20207,20208],{"xmlns":2426},[2428,20209,20210,20218],{},[2431,20211,20212,20214,20216],{},[6062,20213,6908],{},[6062,20215,6911],{},[2437,20217,3688],{},[2444,20219,14382],{"encoding":2446},[53,20221,20223],{"className":20222,"ariaHidden":2452},[2451],[53,20224,20226,20229,20235,20238],{"className":20225},[2456],[53,20227],{"className":20228,"style":6930},[2460],[53,20230,20232],{"className":20231},[2465,6934],[53,20233,6908],{"className":20234},[2465],[53,20236],{"className":20237,"style":2814},[2609],[53,20239,3688],{"className":20240,"style":2671},[2465,2506]," and not controlled time evolutions. Reframing our calculation as above will allow us to greatly reduce the depth of the resulting circuits.",[2408,20243,20245],{"id":20244},"decompose-time-evolution-operator-with-trotter-decomposition","Decompose time-evolution operator with Trotter decomposition",[12,20247,20248,20249,170,20330,170,20411,20492,20493,20522,20523,20647,20648,20705,20706,20739,20740,20789,20790,20793],{},"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 ",[53,20250,20252,20275],{"className":20251},[2418],[53,20253,20255],{"className":20254},[2422],[2424,20256,20257],{"xmlns":2426},[2428,20258,20259,20272],{},[2431,20260,20261],{},[3063,20262,20263,20266],{},[2437,20264,20265],{},"R",[2431,20267,20268,20270],{},[2437,20269,9034],{},[2437,20271,9034],{},[2444,20273,20274],{"encoding":2446},"R_{xx}",[53,20276,20278],{"className":20277,"ariaHidden":2452},[2451],[53,20279,20281,20284],{"className":20280},[2456],[53,20282],{"className":20283,"style":3082},[2460],[53,20285,20287,20291],{"className":20286},[2465],[53,20288,20265],{"className":20289,"style":20290},[2465,2506],"margin-right:0.0077em;",[53,20292,20294],{"className":20293},[2474],[53,20295,20297,20322],{"className":20296},[2478,3095],[53,20298,20300,20319],{"className":20299},[2482],[53,20301,20303],{"className":20302,"style":4402},[2486],[53,20304,20306,20309],{"style":20305},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[53,20307],{"className":20308,"style":2495},[2494],[53,20310,20312],{"className":20311},[2499,2500,2501,2502],[53,20313,20315],{"className":20314},[2465,2502],[53,20316,20318],{"className":20317},[2465,2506,2502],"xx",[53,20320,3120],{"className":20321},[3119],[53,20323,20325],{"className":20324},[2482],[53,20326,20328],{"className":20327,"style":3127},[2486],[53,20329],{},[53,20331,20333,20356],{"className":20332},[2418],[53,20334,20336],{"className":20335},[2422],[2424,20337,20338],{"xmlns":2426},[2428,20339,20340,20353],{},[2431,20341,20342],{},[3063,20343,20344,20346],{},[2437,20345,20265],{},[2431,20347,20348,20351],{},[2437,20349,20350],{},"y",[2437,20352,20350],{},[2444,20354,20355],{"encoding":2446},"R_{yy}",[53,20357,20359],{"className":20358,"ariaHidden":2452},[2451],[53,20360,20362,20365],{"className":20361},[2456],[53,20363],{"className":20364,"style":7862},[2460],[53,20366,20368,20371],{"className":20367},[2465],[53,20369,20265],{"className":20370,"style":20290},[2465,2506],[53,20372,20374],{"className":20373},[2474],[53,20375,20377,20403],{"className":20376},[2478,3095],[53,20378,20380,20400],{"className":20379},[2482],[53,20381,20383],{"className":20382,"style":4402},[2486],[53,20384,20385,20388],{"style":20305},[53,20386],{"className":20387,"style":2495},[2494],[53,20389,20391],{"className":20390},[2499,2500,2501,2502],[53,20392,20394,20397],{"className":20393},[2465,2502],[53,20395,20350],{"className":20396,"style":2671},[2465,2506,2502],[53,20398,20350],{"className":20399,"style":2671},[2465,2506,2502],[53,20401,3120],{"className":20402},[3119],[53,20404,20406],{"className":20405},[2482],[53,20407,20409],{"className":20408,"style":6728},[2486],[53,20410],{},[53,20412,20414,20436],{"className":20413},[2418],[53,20415,20417],{"className":20416},[2422],[2424,20418,20419],{"xmlns":2426},[2428,20420,20421,20433],{},[2431,20422,20423],{},[3063,20424,20425,20427],{},[2437,20426,20265],{},[2431,20428,20429,20431],{},[2437,20430,214],{},[2437,20432,214],{},[2444,20434,20435],{"encoding":2446},"R_{zz}",[53,20437,20439],{"className":20438,"ariaHidden":2452},[2451],[53,20440,20442,20445],{"className":20441},[2456],[53,20443],{"className":20444,"style":3082},[2460],[53,20446,20448,20451],{"className":20447},[2465],[53,20449,20265],{"className":20450,"style":20290},[2465,2506],[53,20452,20454],{"className":20453},[2474],[53,20455,20457,20484],{"className":20456},[2478,3095],[53,20458,20460,20481],{"className":20459},[2482],[53,20461,20463],{"className":20462,"style":4402},[2486],[53,20464,20465,20468],{"style":20305},[53,20466],{"className":20467,"style":2495},[2494],[53,20469,20471],{"className":20470},[2499,2500,2501,2502],[53,20472,20474,20478],{"className":20473},[2465,2502],[53,20475,214],{"className":20476,"style":20477},[2465,2506,2502],"margin-right:0.044em;",[53,20479,214],{"className":20480,"style":20477},[2465,2506,2502],[53,20482,3120],{"className":20483},[3119],[53,20485,20487],{"className":20486},[2482],[53,20488,20490],{"className":20489,"style":3127},[2486],[53,20491],{}," with a parametrized angle ",[53,20494,20496,20509],{"className":20495},[2418],[53,20497,20499],{"className":20498},[2422],[2424,20500,20501],{"xmlns":2426},[2428,20502,20503,20507],{},[2431,20504,20505],{},[2437,20506,3193],{},[2444,20508,3193],{"encoding":2446},[53,20510,20512],{"className":20511,"ariaHidden":2452},[2451],[53,20513,20515,20519],{"className":20514},[2456],[53,20516],{"className":20517,"style":20518},[2460],"height:0.6151em;",[53,20520,3193],{"className":20521},[2465,2506]," which correspond to the approximate implementation of ",[53,20524,20526,20570],{"className":20525},[2418],[53,20527,20529],{"className":20528},[2422],[2424,20530,20531],{"xmlns":2426},[2428,20532,20533,20567],{},[2431,20534,20535],{},[2434,20536,20537,20539],{},[2437,20538,3181],{},[2431,20540,20541,20543,20545,20547,20549,20551,20553,20555,20557,20559,20561,20563,20565],{},[2584,20542,2586],{},[2437,20544,3188],{},[2584,20546,128],{"stretchy":2650},[2437,20548,7626],{},[2437,20550,7626],{},[2584,20552,2002],{},[2437,20554,7641],{},[2437,20556,7641],{},[2584,20558,2002],{},[2437,20560,7659],{},[2437,20562,7659],{},[2584,20564,3544],{"stretchy":2650},[2437,20566,3193],{},[2444,20568,20569],{"encoding":2446},"e^{-i (XX + YY + ZZ) t}",[53,20571,20573],{"className":20572,"ariaHidden":2452},[2451],[53,20574,20576,20579],{"className":20575},[2456],[53,20577],{"className":20578,"style":3603},[2460],[53,20580,20582,20585],{"className":20581},[2465],[53,20583,3181],{"className":20584},[2465,2506],[53,20586,20588],{"className":20587},[2474],[53,20589,20591],{"className":20590},[2478],[53,20592,20594],{"className":20593},[2482],[53,20595,20597],{"className":20596,"style":3603},[2486],[53,20598,20599,20602],{"style":2490},[53,20600],{"className":20601,"style":2495},[2494],[53,20603,20605],{"className":20604},[2499,2500,2501,2502],[53,20606,20608,20611,20614,20617,20620,20623,20626,20629,20632,20635,20638,20641,20644],{"className":20607},[2465,2502],[53,20609,2586],{"className":20610},[2465,2502],[53,20612,3188],{"className":20613},[2465,2506,2502],[53,20615,128],{"className":20616},[2836,2502],[53,20618,7626],{"className":20619,"style":7771},[2465,2506,2502],[53,20621,7626],{"className":20622,"style":7771},[2465,2506,2502],[53,20624,2002],{"className":20625},[2614,2502],[53,20627,7641],{"className":20628,"style":2610},[2465,2506,2502],[53,20630,7641],{"className":20631,"style":2610},[2465,2506,2502],[53,20633,2002],{"className":20634},[2614,2502],[53,20636,7659],{"className":20637,"style":7969},[2465,2506,2502],[53,20639,7659],{"className":20640,"style":7969},[2465,2506,2502],[53,20642,3544],{"className":20643},[2675,2502],[53,20645,3193],{"className":20646},[2465,2506,2502],". 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 ",[53,20649,20651,20672],{"className":20650},[2418],[53,20652,20654],{"className":20653},[2422],[2424,20655,20656],{"xmlns":2426},[2428,20657,20658,20669],{},[2431,20659,20660,20662,20665,20667],{},[2588,20661,132],{},[2584,20663,20664],{},"∗",[2437,20666,4854],{},[2437,20668,3193],{},[2444,20670,20671],{"encoding":2446},"2*dt",[53,20673,20675,20693],{"className":20674,"ariaHidden":2452},[2451],[53,20676,20678,20681,20684,20687,20690],{"className":20677},[2456],[53,20679],{"className":20680,"style":2624},[2460],[53,20682,132],{"className":20683},[2465],[53,20685],{"className":20686,"style":2610},[2609],[53,20688,20664],{"className":20689},[2614],[53,20691],{"className":20692,"style":2610},[2609],[53,20694,20696,20699,20702],{"className":20695},[2456],[53,20697],{"className":20698,"style":5329},[2460],[53,20700,4854],{"className":20701},[2465,2506],[53,20703,3193],{"className":20704},[2465,2506]," to achieve a time-evolution of ",[53,20707,20709,20724],{"className":20708},[2418],[53,20710,20712],{"className":20711},[2422],[2424,20713,20714],{"xmlns":2426},[2428,20715,20716,20722],{},[2431,20717,20718,20720],{},[2437,20719,4854],{},[2437,20721,3193],{},[2444,20723,5358],{"encoding":2446},[53,20725,20727],{"className":20726,"ariaHidden":2452},[2451],[53,20728,20730,20733,20736],{"className":20729},[2456],[53,20731],{"className":20732,"style":5329},[2460],[53,20734,4854],{"className":20735},[2465,2506],[53,20737,3193],{"className":20738},[2465,2506],". 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 ",[53,20741,20743,20765],{"className":20742},[2418],[53,20744,20746],{"className":20745},[2422],[2424,20747,20748],{"xmlns":2426},[2428,20749,20750,20762],{},[2431,20751,20752,20754,20756,20758,20760],{},[2437,20753,5752],{},[2437,20755,3174],{},[2584,20757,128],{"stretchy":2650},[2588,20759,132],{},[2584,20761,3544],{"stretchy":2650},[2444,20763,20764],{"encoding":2446},"SU(2)",[53,20766,20768],{"className":20767,"ariaHidden":2452},[2451],[53,20769,20771,20774,20777,20780,20783,20786],{"className":20770},[2456],[53,20772],{"className":20773,"style":2664},[2460],[53,20775,5752],{"className":20776,"style":5787},[2465,2506],[53,20778,3174],{"className":20779,"style":3209},[2465,2506],[53,20781,128],{"className":20782},[2836],[53,20784,132],{"className":20785},[2465],[53,20787,3544],{"className":20788},[2675]," unitary. This gives a much shallower circuit than what is obtained using the generic ",[50,20791,20792],{},"PauliEvolutionGate()"," functionality.",[43,20795,20797],{"className":45,"code":20796,"language":47,"meta":48,"style":48},"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",[50,20798,20799,20813,20817,20822,20837,20856,20873,20890,20915,20919,20928,20965,20999,21007,21011,21021,21032,21046,21058,21068,21087,21096,21111,21115,21125,21138,21156,21171,21177,21190,21194],{"__ignoreMap":48},[53,20800,20801,20803,20805,20807,20809,20811],{"class":55,"line":56},[53,20802,9202],{"class":63},[53,20804,121],{"class":120},[53,20806,9207],{"class":124},[53,20808,128],{"class":63},[53,20810,9212],{"class":278},[53,20812,135],{"class":63},[53,20814,20815],{"class":55,"line":73},[53,20816,104],{"emptyLinePlaceholder":103},[53,20818,20819],{"class":55,"line":87},[53,20820,20821],{"class":110},"# Create instruction for rotation about XX+YY-ZZ:\n",[53,20823,20824,20827,20829,20831,20833,20835],{"class":55,"line":100},[53,20825,20826],{"class":63},"Rxyz_circ ",[53,20828,121],{"class":120},[53,20830,125],{"class":124},[53,20832,128],{"class":63},[53,20834,132],{"class":131},[53,20836,135],{"class":63},[53,20838,20839,20842,20845,20848,20850,20852,20854],{"class":55,"line":107},[53,20840,20841],{"class":63},"Rxyz_circ.",[53,20843,20844],{"class":124},"rxx",[53,20846,20847],{"class":63},"(t, ",[53,20849,149],{"class":131},[53,20851,170],{"class":63},[53,20853,173],{"class":131},[53,20855,135],{"class":63},[53,20857,20858,20860,20863,20865,20867,20869,20871],{"class":55,"line":114},[53,20859,20841],{"class":63},[53,20861,20862],{"class":124},"ryy",[53,20864,20847],{"class":63},[53,20866,149],{"class":131},[53,20868,170],{"class":63},[53,20870,173],{"class":131},[53,20872,135],{"class":63},[53,20874,20875,20877,20880,20882,20884,20886,20888],{"class":55,"line":138},[53,20876,20841],{"class":63},[53,20878,20879],{"class":124},"rzz",[53,20881,20847],{"class":63},[53,20883,149],{"class":131},[53,20885,170],{"class":63},[53,20887,173],{"class":131},[53,20889,135],{"class":63},[53,20891,20892,20895,20897,20900,20903,20905,20908,20910,20913],{"class":55,"line":158},[53,20893,20894],{"class":63},"Rxyz_instr ",[53,20896,121],{"class":120},[53,20898,20899],{"class":63}," Rxyz_circ.",[53,20901,20902],{"class":124},"to_instruction",[53,20904,128],{"class":63},[53,20906,20907],{"class":2050},"label",[53,20909,121],{"class":120},[53,20911,20912],{"class":278},"\"RXX+YY+ZZ\"",[53,20914,135],{"class":63},[53,20916,20917],{"class":55,"line":182},[53,20918,104],{"emptyLinePlaceholder":103},[53,20920,20921,20924,20926],{"class":55,"line":209},[53,20922,20923],{"class":63},"interaction_list ",[53,20925,121],{"class":120},[53,20927,2072],{"class":63},[53,20929,20930,20933,20935,20937,20939,20941,20943,20945,20947,20949,20951,20954,20956,20958,20960,20962],{"class":55,"line":226},[53,20931,20932],{"class":63}," [[i, i ",[53,20934,2002],{"class":120},[53,20936,1321],{"class":131},[53,20938,8101],{"class":63},[53,20940,1525],{"class":59},[53,20942,8141],{"class":63},[53,20944,1531],{"class":59},[53,20946,8116],{"class":120},[53,20948,128],{"class":63},[53,20950,149],{"class":131},[53,20952,20953],{"class":63},", n_qubits ",[53,20955,2262],{"class":120},[53,20957,1321],{"class":131},[53,20959,170],{"class":63},[53,20961,132],{"class":131},[53,20963,20964],{"class":63},")],\n",[53,20966,20967,20969,20971,20973,20975,20977,20979,20981,20983,20985,20987,20989,20991,20993,20995,20997],{"class":55,"line":231},[53,20968,20932],{"class":63},[53,20970,2002],{"class":120},[53,20972,1321],{"class":131},[53,20974,8101],{"class":63},[53,20976,1525],{"class":59},[53,20978,8141],{"class":63},[53,20980,1531],{"class":59},[53,20982,8116],{"class":120},[53,20984,128],{"class":63},[53,20986,173],{"class":131},[53,20988,20953],{"class":63},[53,20990,2262],{"class":120},[53,20992,1321],{"class":131},[53,20994,170],{"class":63},[53,20996,132],{"class":131},[53,20998,20964],{"class":63},[53,21000,21001,21004],{"class":55,"line":245},[53,21002,21003],{"class":63},"] ",[53,21005,21006],{"class":110},"# linear chain\n",[53,21008,21009],{"class":55,"line":258},[53,21010,104],{"emptyLinePlaceholder":103},[53,21012,21013,21015,21017,21019],{"class":55,"line":263},[53,21014,9279],{"class":63},[53,21016,121],{"class":120},[53,21018,9284],{"class":124},[53,21020,8678],{"class":63},[53,21022,21023,21026,21028,21030],{"class":55,"line":269},[53,21024,21025],{"class":63},"trotter_step_circ ",[53,21027,121],{"class":120},[53,21029,125],{"class":124},[53,21031,9298],{"class":63},[53,21033,21034,21036,21039,21041,21043],{"class":55,"line":284},[53,21035,1525],{"class":59},[53,21037,21038],{"class":63}," i, color ",[53,21040,1531],{"class":59},[53,21042,1852],{"class":120},[53,21044,21045],{"class":63},"(interaction_list):\n",[53,21047,21048,21050,21053,21055],{"class":55,"line":296},[53,21049,1811],{"class":59},[53,21051,21052],{"class":63}," interaction ",[53,21054,1531],{"class":59},[53,21056,21057],{"class":63}," color:\n",[53,21059,21060,21063,21065],{"class":55,"line":309},[53,21061,21062],{"class":63}," trotter_step_circ.",[53,21064,1834],{"class":124},[53,21066,21067],{"class":63},"(Rxyz_instr, interaction)\n",[53,21069,21070,21072,21074,21076,21078,21081,21083,21085],{"class":55,"line":323},[53,21071,1612],{"class":59},[53,21073,8141],{"class":63},[53,21075,5572],{"class":120},[53,21077,2196],{"class":120},[53,21079,21080],{"class":63},"(interaction_list) ",[53,21082,2262],{"class":120},[53,21084,1321],{"class":131},[53,21086,1623],{"class":63},[53,21088,21089,21091,21094],{"class":55,"line":332},[53,21090,21062],{"class":63},[53,21092,21093],{"class":124},"barrier",[53,21095,329],{"class":63},[53,21097,21098,21101,21103,21106,21109],{"class":55,"line":344},[53,21099,21100],{"class":63},"reverse_trotter_step_circ ",[53,21102,121],{"class":120},[53,21104,21105],{"class":63}," trotter_step_circ.",[53,21107,21108],{"class":124},"reverse_ops",[53,21110,329],{"class":63},[53,21112,21113],{"class":55,"line":357},[53,21114,104],{"emptyLinePlaceholder":103},[53,21116,21117,21119,21121,21123],{"class":55,"line":1247},[53,21118,9291],{"class":63},[53,21120,121],{"class":120},[53,21122,125],{"class":124},[53,21124,9298],{"class":63},[53,21126,21127,21129,21132,21134,21136],{"class":55,"line":1252},[53,21128,1525],{"class":59},[53,21130,21131],{"class":63}," step ",[53,21133,1531],{"class":59},[53,21135,8116],{"class":120},[53,21137,13788],{"class":63},[53,21139,21140,21142,21144,21146,21148,21151,21154],{"class":55,"line":1257},[53,21141,1612],{"class":59},[53,21143,21131],{"class":63},[53,21145,8664],{"class":120},[53,21147,2211],{"class":131},[53,21149,21150],{"class":120}," ==",[53,21152,21153],{"class":131}," 0",[53,21155,1623],{"class":63},[53,21157,21158,21161,21163,21166,21168],{"class":55,"line":1262},[53,21159,21160],{"class":63}," qc_evol ",[53,21162,121],{"class":120},[53,21164,21165],{"class":63}," qc_evol.",[53,21167,13762],{"class":124},[53,21169,21170],{"class":63},"(trotter_step_circ)\n",[53,21172,21173,21175],{"class":55,"line":1274},[53,21174,1695],{"class":59},[53,21176,1623],{"class":63},[53,21178,21179,21181,21183,21185,21187],{"class":55,"line":1284},[53,21180,21160],{"class":63},[53,21182,121],{"class":120},[53,21184,21165],{"class":63},[53,21186,13762],{"class":124},[53,21188,21189],{"class":63},"(reverse_trotter_step_circ)\n",[53,21191,21192],{"class":55,"line":1292},[53,21193,104],{"emptyLinePlaceholder":103},[53,21195,21196,21198,21200,21202,21204,21206,21208,21210,21212,21214,21216,21218,21220,21222,21224],{"class":55,"line":1307},[53,21197,9303],{"class":63},[53,21199,14060],{"class":124},[53,21201,13878],{"class":63},[53,21203,9059],{"class":124},[53,21205,128],{"class":63},[53,21207,9064],{"class":278},[53,21209,170],{"class":63},[53,21211,14007],{"class":2050},[53,21213,14010],{"class":120},[53,21215,173],{"class":131},[53,21217,170],{"class":63},[53,21219,9069],{"class":2050},[53,21221,121],{"class":120},[53,21223,9074],{"class":131},[53,21225,135],{"class":63},[379,21227],{"alt":9079,"src":21228},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-03.avif",[2408,21230,21232],{"id":21231},"use-an-optimized-circuit-for-state-preparation","Use an optimized circuit for state preparation",[43,21234,21236],{"className":45,"code":21235,"language":47,"meta":48,"style":48},"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",[50,21237,21238,21247,21268,21285,21295],{"__ignoreMap":48},[53,21239,21240,21243,21245],{"class":55,"line":56},[53,21241,21242],{"class":63},"control ",[53,21244,121],{"class":120},[53,21246,2141],{"class":131},[53,21248,21249,21252,21254,21256,21258,21260,21262,21264,21266],{"class":55,"line":73},[53,21250,21251],{"class":63},"excitation ",[53,21253,121],{"class":120},[53,21255,1990],{"class":120},[53,21257,1999],{"class":63},[53,21259,193],{"class":120},[53,21261,2211],{"class":131},[53,21263,203],{"class":63},[53,21265,2002],{"class":120},[53,21267,1979],{"class":131},[53,21269,21270,21273,21275,21277,21279,21281,21283],{"class":55,"line":87},[53,21271,21272],{"class":63},"controlled_state_prep ",[53,21274,121],{"class":120},[53,21276,125],{"class":124},[53,21278,1999],{"class":63},[53,21280,2002],{"class":120},[53,21282,1321],{"class":131},[53,21284,135],{"class":63},[53,21286,21287,21290,21292],{"class":55,"line":100},[53,21288,21289],{"class":63},"controlled_state_prep.",[53,21291,163],{"class":124},[53,21293,21294],{"class":63},"(control, excitation)\n",[53,21296,21297,21299,21301,21303,21305,21307,21309,21311,21313,21315,21317,21319,21321],{"class":55,"line":107},[53,21298,21289],{"class":63},[53,21300,9059],{"class":124},[53,21302,128],{"class":63},[53,21304,9064],{"class":278},[53,21306,170],{"class":63},[53,21308,14007],{"class":2050},[53,21310,14010],{"class":120},[53,21312,173],{"class":131},[53,21314,170],{"class":63},[53,21316,9069],{"class":2050},[53,21318,121],{"class":120},[53,21320,9074],{"class":131},[53,21322,135],{"class":63},[379,21324],{"alt":9079,"src":21325},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-04.avif",[2408,21327,21329,21330,10694,21390,21450],{"id":21328},"template-circuits-for-calculating-matrix-elements-of-stildess-and-htildehh-via-hadamard-test","Template circuits for calculating matrix elements of ",[53,21331,21333,21350],{"className":21332},[2418],[53,21334,21336],{"className":21335},[2422],[2424,21337,21338],{"xmlns":2426},[2428,21339,21340,21348],{},[2431,21341,21342],{},[4166,21343,21344,21346],{"accent":2452},[2437,21345,5752],{},[2584,21347,4172],{},[2444,21349,5757],{"encoding":2446},[53,21351,21353],{"className":21352,"ariaHidden":2452},[2451],[53,21354,21356,21359],{"className":21355},[2456],[53,21357],{"className":21358,"style":4185},[2460],[53,21360,21362],{"className":21361},[2465,4189],[53,21363,21365],{"className":21364},[2478],[53,21366,21368],{"className":21367},[2482],[53,21369,21371,21379],{"className":21370,"style":4185},[2486],[53,21372,21373,21376],{"style":4201},[53,21374],{"className":21375,"style":4205},[2494],[53,21377,5752],{"className":21378,"style":5787},[2465,2506],[53,21380,21381,21384],{"style":4211},[53,21382],{"className":21383,"style":4205},[2494],[53,21385,21387],{"className":21386,"style":5796},[4218],[53,21388,4172],{"className":21389},[2465],[53,21391,21393,21410],{"className":21392},[2418],[53,21394,21396],{"className":21395},[2422],[2424,21397,21398],{"xmlns":2426},[2428,21399,21400,21408],{},[2431,21401,21402],{},[4166,21403,21404,21406],{"accent":2452},[2437,21405,3033],{},[2584,21407,4172],{},[2444,21409,4175],{"encoding":2446},[53,21411,21413],{"className":21412,"ariaHidden":2452},[2451],[53,21414,21416,21419],{"className":21415},[2456],[53,21417],{"className":21418,"style":4185},[2460],[53,21420,21422],{"className":21421},[2465,4189],[53,21423,21425],{"className":21424},[2478],[53,21426,21428],{"className":21427},[2482],[53,21429,21431,21439],{"className":21430,"style":4185},[2486],[53,21432,21433,21436],{"style":4201},[53,21434],{"className":21435,"style":4205},[2494],[53,21437,3033],{"className":21438,"style":3048},[2465,2506],[53,21440,21441,21444],{"style":4211},[53,21442],{"className":21443,"style":4205},[2494],[53,21445,21447],{"className":21446,"style":4219},[4218],[53,21448,4172],{"className":21449},[2465]," via Hadamard test",[12,21452,21453],{},"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.",[43,21455,21457],{"className":45,"code":21456,"language":47,"meta":48,"style":48},"# Parameters for the template circuits\nparameters = []\nfor idx in range(1, krylov_dim):\n parameters.append(2 * dt_circ * (idx))\n",[50,21458,21459,21464,21473,21491],{"__ignoreMap":48},[53,21460,21461],{"class":55,"line":56},[53,21462,21463],{"class":110},"# Parameters for the template circuits\n",[53,21465,21466,21469,21471],{"class":55,"line":73},[53,21467,21468],{"class":63},"parameters ",[53,21470,121],{"class":120},[53,21472,1805],{"class":63},[53,21474,21475,21477,21480,21482,21484,21486,21488],{"class":55,"line":87},[53,21476,1525],{"class":59},[53,21478,21479],{"class":63}," idx ",[53,21481,1531],{"class":59},[53,21483,8116],{"class":120},[53,21485,128],{"class":63},[53,21487,173],{"class":131},[53,21489,21490],{"class":63},", krylov_dim):\n",[53,21492,21493,21496,21498,21500,21502,21504,21507,21509],{"class":55,"line":100},[53,21494,21495],{"class":63}," parameters.",[53,21497,1834],{"class":124},[53,21499,128],{"class":63},[53,21501,132],{"class":131},[53,21503,8124],{"class":120},[53,21505,21506],{"class":63}," dt_circ ",[53,21508,2286],{"class":120},[53,21510,21511],{"class":63}," (idx))\n",[43,21513,21515],{"className":45,"code":21514,"language":47,"meta":48,"style":48},"# 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",[50,21516,21517,21522,21538,21548,21560,21592,21600,21635,21643,21655,21663,21693,21697,21709,21713],{"__ignoreMap":48},[53,21518,21519],{"class":55,"line":56},[53,21520,21521],{"class":110},"# Create modified hadamard test circuit\n",[53,21523,21524,21526,21528,21530,21532,21534,21536],{"class":55,"line":73},[53,21525,9279],{"class":63},[53,21527,121],{"class":120},[53,21529,9284],{"class":124},[53,21531,1999],{"class":63},[53,21533,2002],{"class":120},[53,21535,1321],{"class":131},[53,21537,135],{"class":63},[53,21539,21540,21542,21544,21546],{"class":55,"line":87},[53,21541,117],{"class":63},[53,21543,121],{"class":120},[53,21545,125],{"class":124},[53,21547,9298],{"class":63},[53,21549,21550,21552,21554,21556,21558],{"class":55,"line":100},[53,21551,141],{"class":63},[53,21553,144],{"class":124},[53,21555,128],{"class":63},[53,21557,149],{"class":131},[53,21559,135],{"class":63},[53,21561,21562,21564,21566,21569,21571,21573,21575,21577,21579,21581,21584,21586,21588,21590],{"class":55,"line":107},[53,21563,141],{"class":63},[53,21565,13762],{"class":124},[53,21567,21568],{"class":63},"(controlled_state_prep, ",[53,21570,13948],{"class":120},[53,21572,128],{"class":63},[53,21574,2099],{"class":120},[53,21576,1999],{"class":63},[53,21578,2002],{"class":120},[53,21580,1321],{"class":131},[53,21582,21583],{"class":63},")), ",[53,21585,13768],{"class":2050},[53,21587,121],{"class":120},[53,21589,13773],{"class":131},[53,21591,135],{"class":63},[53,21593,21594,21596,21598],{"class":55,"line":114},[53,21595,141],{"class":63},[53,21597,21093],{"class":124},[53,21599,329],{"class":63},[53,21601,21602,21604,21606,21609,21611,21613,21615,21617,21619,21621,21623,21625,21627,21629,21631,21633],{"class":55,"line":138},[53,21603,141],{"class":63},[53,21605,13762],{"class":124},[53,21607,21608],{"class":63},"(qc_evol, ",[53,21610,13948],{"class":120},[53,21612,128],{"class":63},[53,21614,2099],{"class":120},[53,21616,128],{"class":63},[53,21618,173],{"class":131},[53,21620,20953],{"class":63},[53,21622,2002],{"class":120},[53,21624,1321],{"class":131},[53,21626,21583],{"class":63},[53,21628,13768],{"class":2050},[53,21630,121],{"class":120},[53,21632,13773],{"class":131},[53,21634,135],{"class":63},[53,21636,21637,21639,21641],{"class":55,"line":158},[53,21638,141],{"class":63},[53,21640,21093],{"class":124},[53,21642,329],{"class":63},[53,21644,21645,21647,21649,21651,21653],{"class":55,"line":182},[53,21646,141],{"class":63},[53,21648,9034],{"class":124},[53,21650,128],{"class":63},[53,21652,149],{"class":131},[53,21654,135],{"class":63},[53,21656,21657,21659,21661],{"class":55,"line":209},[53,21658,141],{"class":63},[53,21660,13762],{"class":124},[53,21662,1271],{"class":63},[53,21664,21665,21668,21670,21672,21674,21676,21678,21680,21682,21684,21686,21688,21690],{"class":55,"line":226},[53,21666,21667],{"class":63}," controlled_state_prep.",[53,21669,13720],{"class":124},[53,21671,13845],{"class":63},[53,21673,13948],{"class":120},[53,21675,128],{"class":63},[53,21677,2099],{"class":120},[53,21679,1999],{"class":63},[53,21681,2002],{"class":120},[53,21683,1321],{"class":131},[53,21685,21583],{"class":63},[53,21687,13768],{"class":2050},[53,21689,121],{"class":120},[53,21691,21692],{"class":131},"True\n",[53,21694,21695],{"class":55,"line":231},[53,21696,135],{"class":63},[53,21698,21699,21701,21703,21705,21707],{"class":55,"line":245},[53,21700,141],{"class":63},[53,21702,9034],{"class":124},[53,21704,128],{"class":63},[53,21706,149],{"class":131},[53,21708,135],{"class":63},[53,21710,21711],{"class":55,"line":258},[53,21712,104],{"emptyLinePlaceholder":103},[53,21714,21715,21717,21719,21721,21723,21725,21727,21729,21731,21733,21735],{"class":55,"line":263},[53,21716,141],{"class":63},[53,21718,14060],{"class":124},[53,21720,13878],{"class":63},[53,21722,9059],{"class":124},[53,21724,128],{"class":63},[53,21726,9064],{"class":278},[53,21728,170],{"class":63},[53,21730,14007],{"class":2050},[53,21732,14010],{"class":120},[53,21734,173],{"class":131},[53,21736,135],{"class":63},[379,21738],{"alt":9079,"src":21739},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-05.avif",[43,21741,21743],{"className":45,"code":21742,"language":47,"meta":48,"style":48},"print(\n \"The optimized circuit has 2Q gates depth: \",\n qc.decompose().decompose().depth(lambda x: x[0].num_qubits == 2),\n)\n",[50,21744,21745,21751,21758,21791],{"__ignoreMap":48},[53,21746,21747,21749],{"class":55,"line":56},[53,21748,8420],{"class":120},[53,21750,1271],{"class":63},[53,21752,21753,21756],{"class":55,"line":73},[53,21754,21755],{"class":278}," \"The optimized circuit has 2Q gates depth: \"",[53,21757,1304],{"class":63},[53,21759,21760,21763,21765,21767,21769,21771,21773,21775,21777,21779,21781,21783,21785,21787,21789],{"class":55,"line":87},[53,21761,21762],{"class":63}," qc.",[53,21764,14060],{"class":124},[53,21766,13878],{"class":63},[53,21768,14060],{"class":124},[53,21770,13878],{"class":63},[53,21772,14074],{"class":124},[53,21774,128],{"class":63},[53,21776,14079],{"class":59},[53,21778,14082],{"class":1277},[53,21780,14085],{"class":63},[53,21782,149],{"class":131},[53,21784,14090],{"class":63},[53,21786,1618],{"class":120},[53,21788,2211],{"class":131},[53,21790,14097],{"class":63},[53,21792,21793],{"class":55,"line":100},[53,21794,135],{"class":63},[43,21796,21799],{"className":21797,"code":21798,"language":6934,"meta":48},[8426],"The optimized circuit has 2Q gates depth: 74\n",[50,21800,21798],{"__ignoreMap":48},[12,21802,21803],{},"We have considerably reduced the depth of the Hadamard test with a combination of Trotter approximation and uncontrolled unitaries",[30,21805,21807],{"id":21806},"step-3-execute-using-qiskit-primitives","Step 3: Execute using Qiskit primitives",[12,21809,21810],{},"Instantiate the backend and set runtime parameters",[43,21812,21814],{"className":45,"code":21813,"language":47,"meta":48,"style":48},"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",[50,21815,21816,21828,21862,21868,21882,21890,21911],{"__ignoreMap":48},[53,21817,21818,21821,21823,21826],{"class":55,"line":56},[53,21819,21820],{"class":63},"service ",[53,21822,121],{"class":120},[53,21824,21825],{"class":124}," QiskitRuntimeService",[53,21827,329],{"class":63},[53,21829,21830,21833,21835,21838,21841,21843,21846,21848,21850,21852,21855,21857,21860],{"class":55,"line":73},[53,21831,21832],{"class":63},"backend ",[53,21834,121],{"class":120},[53,21836,21837],{"class":63}," service.",[53,21839,21840],{"class":124},"least_busy",[53,21842,128],{"class":63},[53,21844,21845],{"class":2050},"operational",[53,21847,121],{"class":120},[53,21849,13773],{"class":131},[53,21851,170],{"class":63},[53,21853,21854],{"class":2050},"simulator",[53,21856,121],{"class":120},[53,21858,21859],{"class":131},"False",[53,21861,135],{"class":63},[53,21863,21864,21866],{"class":55,"line":87},[53,21865,1540],{"class":59},[53,21867,1229],{"class":63},[53,21869,21870,21873,21876,21879],{"class":55,"line":100},[53,21871,21872],{"class":278}," \"if_else\"",[53,21874,21875],{"class":59}," not",[53,21877,21878],{"class":59}," in",[53,21880,21881],{"class":63}," backend.target.operation_names\n",[53,21883,21884,21887],{"class":55,"line":107},[53,21885,21886],{"class":63},"): ",[53,21888,21889],{"class":110},"# Needed as \"op_name\" could be \"if_else\"\n",[53,21891,21892,21895,21898,21901,21904,21906,21909],{"class":55,"line":114},[53,21893,21894],{"class":63}," backend.target.",[53,21896,21897],{"class":124},"add_instruction",[53,21899,21900],{"class":63},"(IfElseOp, ",[53,21902,21903],{"class":2050},"name",[53,21905,121],{"class":120},[53,21907,21908],{"class":278},"\"if_else\"",[53,21910,135],{"class":63},[53,21912,21913,21915],{"class":55,"line":138},[53,21914,8420],{"class":120},[53,21916,21917],{"class":63},"(backend.name)\n",[2408,21919,21921],{"id":21920},"transpiling-to-a-qpu","Transpiling to a QPU",[12,21923,21924],{},"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",[43,21926,21928],{"className":45,"code":21927,"language":47,"meta":48,"style":48},"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",[50,21929,21930,21940,21964,21983,21987,22003,22017,22033,22048,22073,22085,22096,22103,22114,22121,22126,22130,22140,22162,22189,22203,22213,22223,22229,22238,22244,22248,22252,22256,22269,22284,22300,22310],{"__ignoreMap":48},[53,21931,21932,21935,21937],{"class":55,"line":56},[53,21933,21934],{"class":63},"target ",[53,21936,121],{"class":120},[53,21938,21939],{"class":63}," backend.target\n",[53,21941,21942,21945,21947,21950,21953,21955,21958,21960,21962],{"class":55,"line":73},[53,21943,21944],{"class":63},"cmap ",[53,21946,121],{"class":120},[53,21948,21949],{"class":63}," target.",[53,21951,21952],{"class":124},"build_coupling_map",[53,21954,128],{"class":63},[53,21956,21957],{"class":2050},"filter_idle_qubits",[53,21959,121],{"class":120},[53,21961,13773],{"class":131},[53,21963,135],{"class":63},[53,21965,21966,21969,21971,21974,21977,21980],{"class":55,"line":87},[53,21967,21968],{"class":63},"cmap_list ",[53,21970,121],{"class":120},[53,21972,21973],{"class":120}," list",[53,21975,21976],{"class":63},"(cmap.",[53,21978,21979],{"class":124},"get_edges",[53,21981,21982],{"class":63},"())\n",[53,21984,21985],{"class":55,"line":100},[53,21986,104],{"emptyLinePlaceholder":103},[53,21988,21989,21992,21994,21997,22000],{"class":55,"line":107},[53,21990,21991],{"class":63},"cust_cmap_list ",[53,21993,121],{"class":120},[53,21995,21996],{"class":63}," copy.",[53,21998,21999],{"class":124},"deepcopy",[53,22001,22002],{"class":63},"(cmap_list)\n",[53,22004,22005,22007,22010,22012,22014],{"class":55,"line":114},[53,22006,1525],{"class":59},[53,22008,22009],{"class":63}," q ",[53,22011,1531],{"class":59},[53,22013,8116],{"class":120},[53,22015,22016],{"class":63},"(target.num_qubits):\n",[53,22018,22019,22022,22024,22027,22030],{"class":55,"line":138},[53,22020,22021],{"class":63}," meas_err ",[53,22023,121],{"class":120},[53,22025,22026],{"class":63}," target[",[53,22028,22029],{"class":278},"\"measure\"",[53,22031,22032],{"class":63},"][(q,)].error\n",[53,22034,22035,22038,22040,22043,22045],{"class":55,"line":158},[53,22036,22037],{"class":63}," t2 ",[53,22039,121],{"class":120},[53,22041,22042],{"class":63}," target.qubit_properties[q].t2 ",[53,22044,2286],{"class":120},[53,22046,22047],{"class":131}," 1e6\n",[53,22049,22050,22052,22055,22057,22060,22063,22066,22068,22071],{"class":55,"line":182},[53,22051,1612],{"class":59},[53,22053,22054],{"class":63}," meas_err ",[53,22056,1546],{"class":120},[53,22058,22059],{"class":131}," 0.02",[53,22061,22062],{"class":59}," or",[53,22064,22065],{"class":63}," t2 ",[53,22067,5572],{"class":120},[53,22069,22070],{"class":131}," 100",[53,22072,1623],{"class":63},[53,22074,22075,22077,22080,22082],{"class":55,"line":209},[53,22076,2162],{"class":59},[53,22078,22079],{"class":63}," q_pair ",[53,22081,1531],{"class":59},[53,22083,22084],{"class":63}," cmap_list:\n",[53,22086,22087,22089,22091,22093],{"class":55,"line":226},[53,22088,2193],{"class":59},[53,22090,22009],{"class":63},[53,22092,1531],{"class":59},[53,22094,22095],{"class":63}," q_pair:\n",[53,22097,22098,22101],{"class":55,"line":231},[53,22099,22100],{"class":59}," try",[53,22102,1623],{"class":63},[53,22104,22105,22108,22111],{"class":55,"line":245},[53,22106,22107],{"class":63}," cust_cmap_list.",[53,22109,22110],{"class":124},"remove",[53,22112,22113],{"class":63},"(q_pair)\n",[53,22115,22116,22119],{"class":55,"line":258},[53,22117,22118],{"class":59}," except",[53,22120,1623],{"class":63},[53,22122,22123],{"class":55,"line":263},[53,22124,22125],{"class":59}," continue\n",[53,22127,22128],{"class":55,"line":269},[53,22129,104],{"emptyLinePlaceholder":103},[53,22131,22132,22134,22136,22138],{"class":55,"line":284},[53,22133,1525],{"class":59},[53,22135,22009],{"class":63},[53,22137,1531],{"class":59},[53,22139,22084],{"class":63},[53,22141,22142,22145,22147,22149,22152,22155,22158,22160],{"class":55,"line":296},[53,22143,22144],{"class":63}," op_name ",[53,22146,121],{"class":120},[53,22148,21973],{"class":120},[53,22150,22151],{"class":63},"(target.",[53,22153,22154],{"class":124},"operation_names_for_qargs",[53,22156,22157],{"class":63},"(q))[",[53,22159,149],{"class":131},[53,22161,1689],{"class":63},[53,22163,22164,22167,22169,22171,22174,22177,22179,22182,22184,22186],{"class":55,"line":309},[53,22165,22166],{"class":63}," twoq_gate_err ",[53,22168,121],{"class":120},[53,22170,22026],{"class":63},[53,22172,22173],{"class":59},"f",[53,22175,22176],{"class":278},"\"",[53,22178,2705],{"class":131},[53,22180,22181],{"class":63},"op_name",[53,22183,2767],{"class":131},[53,22185,22176],{"class":278},[53,22187,22188],{"class":63},"][q].error\n",[53,22190,22191,22193,22196,22198,22201],{"class":55,"line":323},[53,22192,1612],{"class":59},[53,22194,22195],{"class":63}," twoq_gate_err ",[53,22197,1546],{"class":120},[53,22199,22200],{"class":131}," 0.005",[53,22202,1623],{"class":63},[53,22204,22205,22207,22209,22211],{"class":55,"line":332},[53,22206,2162],{"class":59},[53,22208,22079],{"class":63},[53,22210,1531],{"class":59},[53,22212,22084],{"class":63},[53,22214,22215,22217,22219,22221],{"class":55,"line":344},[53,22216,2193],{"class":59},[53,22218,22009],{"class":63},[53,22220,1618],{"class":120},[53,22222,22095],{"class":63},[53,22224,22225,22227],{"class":55,"line":357},[53,22226,22100],{"class":59},[53,22228,1623],{"class":63},[53,22230,22231,22233,22235],{"class":55,"line":1247},[53,22232,22107],{"class":63},[53,22234,22110],{"class":124},[53,22236,22237],{"class":63},"(q)\n",[53,22239,22240,22242],{"class":55,"line":1252},[53,22241,22118],{"class":59},[53,22243,1623],{"class":63},[53,22245,22246],{"class":55,"line":1257},[53,22247,22125],{"class":59},[53,22249,22250],{"class":55,"line":1262},[53,22251,104],{"emptyLinePlaceholder":103},[53,22253,22254],{"class":55,"line":1274},[53,22255,104],{"emptyLinePlaceholder":103},[53,22257,22258,22261,22263,22266],{"class":55,"line":1284},[53,22259,22260],{"class":63},"cust_cmap ",[53,22262,121],{"class":120},[53,22264,22265],{"class":124}," CouplingMap",[53,22267,22268],{"class":63},"(cust_cmap_list)\n",[53,22270,22271,22274,22276,22279,22282],{"class":55,"line":1292},[53,22272,22273],{"class":63},"cust_target ",[53,22275,121],{"class":120},[53,22277,22278],{"class":63}," Target.",[53,22280,22281],{"class":124},"from_configuration",[53,22283,1271],{"class":63},[53,22285,22286,22289,22291,22294,22297],{"class":55,"line":1307},[53,22287,22288],{"class":2050}," basis_gates",[53,22290,121],{"class":120},[53,22292,22293],{"class":63},"backend.",[53,22295,22296],{"class":124},"configuration",[53,22298,22299],{"class":63},"().basis_gates,\n",[53,22301,22302,22305,22307],{"class":55,"line":1326},[53,22303,22304],{"class":2050}," coupling_map",[53,22306,121],{"class":120},[53,22308,22309],{"class":63},"cust_cmap,\n",[53,22311,22312],{"class":55,"line":1344},[53,22313,135],{"class":63},[12,22315,22316],{},"Then transpile the virtual circuit to the best physical layout in this new target",[43,22318,22320],{"className":45,"code":22319,"language":47,"meta":48,"style":48},"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",[50,22321,22322,22334,22346,22357,22367,22376,22380,22384,22399,22403,22436,22452,22458,22465,22472,22477,22482,22505,22518,22523,22528],{"__ignoreMap":48},[53,22323,22324,22327,22329,22331],{"class":55,"line":56},[53,22325,22326],{"class":63},"basis_gates ",[53,22328,121],{"class":120},[53,22330,21973],{"class":120},[53,22332,22333],{"class":63},"(target.operation_names)\n",[53,22335,22336,22339,22341,22344],{"class":55,"line":73},[53,22337,22338],{"class":63},"pm ",[53,22340,121],{"class":120},[53,22342,22343],{"class":124}," generate_preset_pass_manager",[53,22345,1271],{"class":63},[53,22347,22348,22351,22353,22355],{"class":55,"line":87},[53,22349,22350],{"class":2050}," optimization_level",[53,22352,121],{"class":120},[53,22354,8121],{"class":131},[53,22356,1304],{"class":63},[53,22358,22359,22362,22364],{"class":55,"line":100},[53,22360,22361],{"class":2050}," target",[53,22363,121],{"class":120},[53,22365,22366],{"class":63},"cust_target,\n",[53,22368,22369,22371,22373],{"class":55,"line":107},[53,22370,22288],{"class":2050},[53,22372,121],{"class":120},[53,22374,22375],{"class":63},"basis_gates,\n",[53,22377,22378],{"class":55,"line":114},[53,22379,135],{"class":63},[53,22381,22382],{"class":55,"line":138},[53,22383,104],{"emptyLinePlaceholder":103},[53,22385,22386,22389,22391,22394,22397],{"class":55,"line":158},[53,22387,22388],{"class":63},"qc_trans ",[53,22390,121],{"class":120},[53,22392,22393],{"class":63}," pm.",[53,22395,22396],{"class":124},"run",[53,22398,242],{"class":63},[53,22400,22401],{"class":55,"line":182},[53,22402,104],{"emptyLinePlaceholder":103},[53,22404,22405,22407,22409,22412,22415,22417,22419,22421,22423,22425,22427,22429,22431,22433],{"class":55,"line":209},[53,22406,8420],{"class":120},[53,22408,128],{"class":63},[53,22410,22411],{"class":278},"\"depth\"",[53,22413,22414],{"class":63},", qc_trans.",[53,22416,14074],{"class":124},[53,22418,128],{"class":63},[53,22420,14079],{"class":59},[53,22422,14082],{"class":1277},[53,22424,14085],{"class":63},[53,22426,149],{"class":131},[53,22428,14090],{"class":63},[53,22430,1618],{"class":120},[53,22432,2211],{"class":131},[53,22434,22435],{"class":63},"))\n",[53,22437,22438,22440,22442,22445,22447,22450],{"class":55,"line":226},[53,22439,8420],{"class":120},[53,22441,128],{"class":63},[53,22443,22444],{"class":278},"\"num 2q ops\"",[53,22446,22414],{"class":63},[53,22448,22449],{"class":124},"count_ops",[53,22451,21982],{"class":63},[53,22453,22454,22456],{"class":55,"line":231},[53,22455,8420],{"class":120},[53,22457,1271],{"class":63},[53,22459,22460,22463],{"class":55,"line":245},[53,22461,22462],{"class":278}," \"physical qubits\"",[53,22464,1304],{"class":63},[53,22466,22467,22470],{"class":55,"line":258},[53,22468,22469],{"class":120}," sorted",[53,22471,1271],{"class":63},[53,22473,22474],{"class":55,"line":263},[53,22475,22476],{"class":63}," [\n",[53,22478,22479],{"class":55,"line":269},[53,22480,22481],{"class":63}," idx\n",[53,22483,22484,22487,22490,22492,22495,22498,22500,22503],{"class":55,"line":284},[53,22485,22486],{"class":59}," for",[53,22488,22489],{"class":63}," idx, qb ",[53,22491,1531],{"class":59},[53,22493,22494],{"class":63}," qc_trans.layout.initial_layout.",[53,22496,22497],{"class":124},"get_physical_bits",[53,22499,13878],{"class":63},[53,22501,22502],{"class":124},"items",[53,22504,329],{"class":63},[53,22506,22507,22509,22512,22515],{"class":55,"line":296},[53,22508,2193],{"class":59},[53,22510,22511],{"class":63}," qb._register.name ",[53,22513,22514],{"class":120},"!=",[53,22516,22517],{"class":278}," \"ancilla\"\n",[53,22519,22520],{"class":55,"line":309},[53,22521,22522],{"class":63}," ]\n",[53,22524,22525],{"class":55,"line":323},[53,22526,22527],{"class":63}," ),\n",[53,22529,22530],{"class":55,"line":332},[53,22531,135],{"class":63},[43,22533,22536],{"className":22534,"code":22535,"language":6934,"meta":48},[8426],"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",[50,22537,22535],{"__ignoreMap":48},[2408,22539,22541],{"id":22540},"create-pubs-for-execution-with-estimator","Create PUBs for execution with Estimator",[43,22543,22545],{"className":45,"code":22544,"language":47,"meta":48,"style":48},"# 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",[50,22546,22547,22552,22571,22588,22592,22604,22609,22617,22629,22633,22646,22660,22665,22672,22686,22694,22704,22711,22724,22728,22738,22742,22746,22751,22756,22765,22779,22784,22810,22833,22843,22852,22856,22865,22869,22878,22890,22902,22915,22930,22934,22944,22948,22952,22957,22972,22976,22980,22985,22990,22995],{"__ignoreMap":48},[53,22548,22549],{"class":55,"line":56},[53,22550,22551],{"class":110},"# Define observables to measure for S\n",[53,22553,22554,22557,22559,22562,22564,22567,22569],{"class":55,"line":73},[53,22555,22556],{"class":63},"observable_S_real ",[53,22558,121],{"class":120},[53,22560,22561],{"class":278}," \"I\"",[53,22563,8124],{"class":120},[53,22565,22566],{"class":63}," (n_qubits) ",[53,22568,2002],{"class":120},[53,22570,8216],{"class":278},[53,22572,22573,22576,22578,22580,22582,22584,22586],{"class":55,"line":87},[53,22574,22575],{"class":63},"observable_S_imag ",[53,22577,121],{"class":120},[53,22579,22561],{"class":278},[53,22581,8124],{"class":120},[53,22583,22566],{"class":63},[53,22585,2002],{"class":120},[53,22587,8283],{"class":278},[53,22589,22590],{"class":55,"line":100},[53,22591,104],{"emptyLinePlaceholder":103},[53,22593,22594,22597,22599,22602],{"class":55,"line":107},[53,22595,22596],{"class":63},"observable_op_real ",[53,22598,121],{"class":120},[53,22600,22601],{"class":124}," SparsePauliOp",[53,22603,1271],{"class":63},[53,22605,22606],{"class":55,"line":114},[53,22607,22608],{"class":63}," observable_S_real\n",[53,22610,22611,22614],{"class":55,"line":138},[53,22612,22613],{"class":63},") ",[53,22615,22616],{"class":110},"# define a sparse pauli operator for the observable\n",[53,22618,22619,22622,22624,22626],{"class":55,"line":158},[53,22620,22621],{"class":63},"observable_op_imag ",[53,22623,121],{"class":120},[53,22625,22601],{"class":124},[53,22627,22628],{"class":63},"(observable_S_imag)\n",[53,22630,22631],{"class":55,"line":182},[53,22632,104],{"emptyLinePlaceholder":103},[53,22634,22635,22638,22640,22643],{"class":55,"line":209},[53,22636,22637],{"class":63},"layout ",[53,22639,121],{"class":120},[53,22641,22642],{"class":63}," qc_trans.layout ",[53,22644,22645],{"class":110},"# get layout of transpiled circuit\n",[53,22647,22648,22650,22652,22655,22658],{"class":55,"line":226},[53,22649,22596],{"class":63},[53,22651,121],{"class":120},[53,22653,22654],{"class":63}," observable_op_real.",[53,22656,22657],{"class":124},"apply_layout",[53,22659,1271],{"class":63},[53,22661,22662],{"class":55,"line":231},[53,22663,22664],{"class":63}," layout\n",[53,22666,22667,22669],{"class":55,"line":245},[53,22668,22613],{"class":63},[53,22670,22671],{"class":110},"# apply physical layout to the observable\n",[53,22673,22674,22676,22678,22681,22683],{"class":55,"line":258},[53,22675,22621],{"class":63},[53,22677,121],{"class":120},[53,22679,22680],{"class":63}," observable_op_imag.",[53,22682,22657],{"class":124},[53,22684,22685],{"class":63},"(layout)\n",[53,22687,22688,22690,22692],{"class":55,"line":263},[53,22689,22556],{"class":63},[53,22691,121],{"class":120},[53,22693,1229],{"class":63},[53,22695,22696,22699,22702],{"class":55,"line":269},[53,22697,22698],{"class":63}," observable_op_real.paulis.",[53,22700,22701],{"class":124},"to_labels",[53,22703,329],{"class":63},[53,22705,22706,22708],{"class":55,"line":284},[53,22707,22613],{"class":63},[53,22709,22710],{"class":110},"# get the label of the physical observable\n",[53,22712,22713,22715,22717,22720,22722],{"class":55,"line":296},[53,22714,22575],{"class":63},[53,22716,121],{"class":120},[53,22718,22719],{"class":63}," observable_op_imag.paulis.",[53,22721,22701],{"class":124},[53,22723,329],{"class":63},[53,22725,22726],{"class":55,"line":309},[53,22727,104],{"emptyLinePlaceholder":103},[53,22729,22730,22733,22735],{"class":55,"line":323},[53,22731,22732],{"class":63},"observables_S ",[53,22734,121],{"class":120},[53,22736,22737],{"class":63}," [[observable_S_real], [observable_S_imag]]\n",[53,22739,22740],{"class":55,"line":332},[53,22741,104],{"emptyLinePlaceholder":103},[53,22743,22744],{"class":55,"line":344},[53,22745,104],{"emptyLinePlaceholder":103},[53,22747,22748],{"class":55,"line":357},[53,22749,22750],{"class":110},"# Define observables to measure for H\n",[53,22752,22753],{"class":55,"line":1247},[53,22754,22755],{"class":110},"# Hamiltonian terms to measure\n",[53,22757,22758,22761,22763],{"class":55,"line":1252},[53,22759,22760],{"class":63},"observable_list ",[53,22762,121],{"class":120},[53,22764,1805],{"class":63},[53,22766,22767,22769,22772,22774,22776],{"class":55,"line":1257},[53,22768,1525],{"class":59},[53,22770,22771],{"class":63}," pauli, coeff ",[53,22773,1531],{"class":59},[53,22775,1534],{"class":120},[53,22777,22778],{"class":63},"(H_op.paulis, H_op.coeffs):\n",[53,22780,22781],{"class":55,"line":1262},[53,22782,22783],{"class":110}," # print(pauli)\n",[53,22785,22786,22789,22791,22794,22796,22798,22801,22804,22806,22808],{"class":55,"line":1274},[53,22787,22788],{"class":63}," observable_H_real ",[53,22790,121],{"class":120},[53,22792,22793],{"class":63}," pauli[::",[53,22795,2262],{"class":120},[53,22797,173],{"class":131},[53,22799,22800],{"class":63},"].",[53,22802,22803],{"class":124},"to_label",[53,22805,1572],{"class":63},[53,22807,2002],{"class":120},[53,22809,8216],{"class":278},[53,22811,22812,22815,22817,22819,22821,22823,22825,22827,22829,22831],{"class":55,"line":1284},[53,22813,22814],{"class":63}," observable_H_imag ",[53,22816,121],{"class":120},[53,22818,22793],{"class":63},[53,22820,2262],{"class":120},[53,22822,173],{"class":131},[53,22824,22800],{"class":63},[53,22826,22803],{"class":124},[53,22828,1572],{"class":63},[53,22830,2002],{"class":120},[53,22832,8283],{"class":278},[53,22834,22835,22838,22840],{"class":55,"line":1292},[53,22836,22837],{"class":63}," observable_list.",[53,22839,1834],{"class":124},[53,22841,22842],{"class":63},"([observable_H_real])\n",[53,22844,22845,22847,22849],{"class":55,"line":1307},[53,22846,22837],{"class":63},[53,22848,1834],{"class":124},[53,22850,22851],{"class":63},"([observable_H_imag])\n",[53,22853,22854],{"class":55,"line":1326},[53,22855,104],{"emptyLinePlaceholder":103},[53,22857,22858,22860,22862],{"class":55,"line":1344},[53,22859,22637],{"class":63},[53,22861,121],{"class":120},[53,22863,22864],{"class":63}," qc_trans.layout\n",[53,22866,22867],{"class":55,"line":1360},[53,22868,104],{"emptyLinePlaceholder":103},[53,22870,22871,22874,22876],{"class":55,"line":1366},[53,22872,22873],{"class":63},"observable_trans_list ",[53,22875,121],{"class":120},[53,22877,1805],{"class":63},[53,22879,22880,22882,22885,22887],{"class":55,"line":1372},[53,22881,1525],{"class":59},[53,22883,22884],{"class":63}," observable ",[53,22886,1531],{"class":59},[53,22888,22889],{"class":63}," observable_list:\n",[53,22891,22892,22895,22897,22899],{"class":55,"line":1377},[53,22893,22894],{"class":63}," observable_op ",[53,22896,121],{"class":120},[53,22898,22601],{"class":124},[53,22900,22901],{"class":63},"(observable)\n",[53,22903,22904,22906,22908,22911,22913],{"class":55,"line":1383},[53,22905,22894],{"class":63},[53,22907,121],{"class":120},[53,22909,22910],{"class":63}," observable_op.",[53,22912,22657],{"class":124},[53,22914,22685],{"class":63},[53,22916,22917,22920,22922,22925,22927],{"class":55,"line":1389},[53,22918,22919],{"class":63}," observable_trans_list.",[53,22921,1834],{"class":124},[53,22923,22924],{"class":63},"([observable_op.paulis.",[53,22926,22701],{"class":124},[53,22928,22929],{"class":63},"()])\n",[53,22931,22932],{"class":55,"line":1395},[53,22933,104],{"emptyLinePlaceholder":103},[53,22935,22936,22939,22941],{"class":55,"line":1401},[53,22937,22938],{"class":63},"observables_H ",[53,22940,121],{"class":120},[53,22942,22943],{"class":63}," observable_trans_list\n",[53,22945,22946],{"class":55,"line":1407},[53,22947,104],{"emptyLinePlaceholder":103},[53,22949,22950],{"class":55,"line":1413},[53,22951,104],{"emptyLinePlaceholder":103},[53,22953,22954],{"class":55,"line":1419},[53,22955,22956],{"class":110},"# Define a sweep over parameter values\n",[53,22958,22959,22962,22964,22966,22969],{"class":55,"line":1425},[53,22960,22961],{"class":63},"params ",[53,22963,121],{"class":120},[53,22965,1511],{"class":63},[53,22967,22968],{"class":124},"vstack",[53,22970,22971],{"class":63},"(parameters).T\n",[53,22973,22974],{"class":55,"line":1431},[53,22975,104],{"emptyLinePlaceholder":103},[53,22977,22978],{"class":55,"line":1437},[53,22979,104],{"emptyLinePlaceholder":103},[53,22981,22982],{"class":55,"line":1443},[53,22983,22984],{"class":110},"# Estimate the expectation value for all combinations of\n",[53,22986,22987],{"class":55,"line":1448},[53,22988,22989],{"class":110},"# observables and parameter values, where the pub result will have\n",[53,22991,22992],{"class":55,"line":1454},[53,22993,22994],{"class":110},"# shape (# observables, # parameter values).\n",[53,22996,22997,23000,23002,23005,23007],{"class":55,"line":1460},[53,22998,22999],{"class":63},"pub ",[53,23001,121],{"class":120},[53,23003,23004],{"class":63}," (qc_trans, observables_S ",[53,23006,2002],{"class":120},[53,23008,23009],{"class":63}," observables_H, params)\n",[2408,23011,23013],{"id":23012},"run-circuits","Run circuits",[12,23015,23016,23017,23068],{},"Circuits for ",[53,23018,23020,23038],{"className":23019},[2418],[53,23021,23023],{"className":23022},[2422],[2424,23024,23025],{"xmlns":2426},[2428,23026,23027,23035],{},[2431,23028,23029,23031,23033],{},[2437,23030,3193],{},[2584,23032,121],{},[2588,23034,149],{},[2444,23036,23037],{"encoding":2446},"t=0",[53,23039,23041,23059],{"className":23040,"ariaHidden":2452},[2451],[53,23042,23044,23047,23050,23053,23056],{"className":23043},[2456],[53,23045],{"className":23046,"style":20518},[2460],[53,23048,3193],{"className":23049},[2465,2506],[53,23051],{"className":23052,"style":2814},[2609],[53,23054,121],{"className":23055},[2818],[53,23057],{"className":23058,"style":2814},[2609],[53,23060,23062,23065],{"className":23061},[2456],[53,23063],{"className":23064,"style":2624},[2460],[53,23066,149],{"className":23067},[2465]," are classically calculable",[43,23070,23072],{"className":45,"code":23071,"language":47,"meta":48,"style":48},"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",[50,23073,23074,23095,23099,23103,23108,23126,23146,23150,23165,23184,23188,23192,23197,23218,23222,23231,23250,23255,23270,23294,23298,23313,23335,23339,23360,23364,23369,23384,23388,23392],{"__ignoreMap":48},[53,23075,23076,23079,23081,23084,23087,23090,23092],{"class":55,"line":56},[53,23077,23078],{"class":63},"qc_cliff ",[53,23080,121],{"class":120},[53,23082,23083],{"class":63}," qc.",[53,23085,23086],{"class":124},"assign_parameters",[53,23088,23089],{"class":63},"({t: ",[53,23091,149],{"class":131},[53,23093,23094],{"class":63},"})\n",[53,23096,23097],{"class":55,"line":73},[53,23098,104],{"emptyLinePlaceholder":103},[53,23100,23101],{"class":55,"line":87},[53,23102,104],{"emptyLinePlaceholder":103},[53,23104,23105],{"class":55,"line":100},[53,23106,23107],{"class":110},"# Get expectation values from experiment\n",[53,23109,23110,23113,23115,23118,23121,23124],{"class":55,"line":107},[53,23111,23112],{"class":63},"S_expval_real ",[53,23114,121],{"class":120},[53,23116,23117],{"class":124}," StabilizerState",[53,23119,23120],{"class":63},"(qc_cliff).",[53,23122,23123],{"class":124},"expectation_value",[53,23125,1271],{"class":63},[53,23127,23128,23131,23133,23135,23137,23139,23141,23144],{"class":55,"line":114},[53,23129,23130],{"class":124}," Pauli",[53,23132,128],{"class":63},[53,23134,8098],{"class":278},[53,23136,8124],{"class":120},[53,23138,22566],{"class":63},[53,23140,2002],{"class":120},[53,23142,23143],{"class":278}," \"X\"",[53,23145,135],{"class":63},[53,23147,23148],{"class":55,"line":138},[53,23149,135],{"class":63},[53,23151,23152,23155,23157,23159,23161,23163],{"class":55,"line":158},[53,23153,23154],{"class":63},"S_expval_imag ",[53,23156,121],{"class":120},[53,23158,23117],{"class":124},[53,23160,23120],{"class":63},[53,23162,23123],{"class":124},[53,23164,1271],{"class":63},[53,23166,23167,23169,23171,23173,23175,23177,23179,23182],{"class":55,"line":182},[53,23168,23130],{"class":124},[53,23170,128],{"class":63},[53,23172,8098],{"class":278},[53,23174,8124],{"class":120},[53,23176,22566],{"class":63},[53,23178,2002],{"class":120},[53,23180,23181],{"class":278}," \"Y\"",[53,23183,135],{"class":63},[53,23185,23186],{"class":55,"line":209},[53,23187,135],{"class":63},[53,23189,23190],{"class":55,"line":226},[53,23191,104],{"emptyLinePlaceholder":103},[53,23193,23194],{"class":55,"line":231},[53,23195,23196],{"class":110},"# Get expectation values\n",[53,23198,23199,23202,23204,23207,23209,23211,23213,23215],{"class":55,"line":245},[53,23200,23201],{"class":63},"S_expval ",[53,23203,121],{"class":120},[53,23205,23206],{"class":63}," S_expval_real ",[53,23208,2002],{"class":120},[53,23210,1321],{"class":131},[53,23212,2267],{"class":59},[53,23214,8124],{"class":120},[53,23216,23217],{"class":63}," S_expval_imag\n",[53,23219,23220],{"class":55,"line":258},[53,23221,104],{"emptyLinePlaceholder":103},[53,23223,23224,23227,23229],{"class":55,"line":263},[53,23225,23226],{"class":63},"H_expval ",[53,23228,121],{"class":120},[53,23230,2141],{"class":131},[53,23232,23233,23235,23238,23240,23242,23244,23247],{"class":55,"line":269},[53,23234,1525],{"class":59},[53,23236,23237],{"class":63}," obs_idx, (pauli, coeff) ",[53,23239,1531],{"class":59},[53,23241,1852],{"class":120},[53,23243,128],{"class":63},[53,23245,23246],{"class":120},"zip",[53,23248,23249],{"class":63},"(H_op.paulis, H_op.coeffs)):\n",[53,23251,23252],{"class":55,"line":284},[53,23253,23254],{"class":110}," # Get expectation values from experiment\n",[53,23256,23257,23260,23262,23264,23266,23268],{"class":55,"line":296},[53,23258,23259],{"class":63}," expval_real ",[53,23261,121],{"class":120},[53,23263,23117],{"class":124},[53,23265,23120],{"class":63},[53,23267,23123],{"class":124},[53,23269,1271],{"class":63},[53,23271,23272,23275,23278,23280,23282,23284,23286,23288,23290,23292],{"class":55,"line":309},[53,23273,23274],{"class":124}," Pauli",[53,23276,23277],{"class":63},"(pauli[::",[53,23279,2262],{"class":120},[53,23281,173],{"class":131},[53,23283,22800],{"class":63},[53,23285,22803],{"class":124},[53,23287,1572],{"class":63},[53,23289,2002],{"class":120},[53,23291,23143],{"class":278},[53,23293,135],{"class":63},[53,23295,23296],{"class":55,"line":323},[53,23297,1555],{"class":63},[53,23299,23300,23303,23305,23307,23309,23311],{"class":55,"line":332},[53,23301,23302],{"class":63}," expval_imag ",[53,23304,121],{"class":120},[53,23306,23117],{"class":124},[53,23308,23120],{"class":63},[53,23310,23123],{"class":124},[53,23312,1271],{"class":63},[53,23314,23315,23317,23319,23321,23323,23325,23327,23329,23331,23333],{"class":55,"line":344},[53,23316,23274],{"class":124},[53,23318,23277],{"class":63},[53,23320,2262],{"class":120},[53,23322,173],{"class":131},[53,23324,22800],{"class":63},[53,23326,22803],{"class":124},[53,23328,1572],{"class":63},[53,23330,2002],{"class":120},[53,23332,23181],{"class":278},[53,23334,135],{"class":63},[53,23336,23337],{"class":55,"line":357},[53,23338,1555],{"class":63},[53,23340,23341,23344,23346,23349,23351,23353,23355,23357],{"class":55,"line":1247},[53,23342,23343],{"class":63}," expval ",[53,23345,121],{"class":120},[53,23347,23348],{"class":63}," expval_real ",[53,23350,2002],{"class":120},[53,23352,1321],{"class":131},[53,23354,2267],{"class":59},[53,23356,8124],{"class":120},[53,23358,23359],{"class":63}," expval_imag\n",[53,23361,23362],{"class":55,"line":1252},[53,23363,104],{"emptyLinePlaceholder":103},[53,23365,23366],{"class":55,"line":1257},[53,23367,23368],{"class":110}," # Fill-in matrix elements\n",[53,23370,23371,23374,23376,23379,23381],{"class":55,"line":1262},[53,23372,23373],{"class":63}," H_expval ",[53,23375,2180],{"class":120},[53,23377,23378],{"class":63}," coeff ",[53,23380,2286],{"class":120},[53,23382,23383],{"class":63}," expval\n",[53,23385,23386],{"class":55,"line":1274},[53,23387,104],{"emptyLinePlaceholder":103},[53,23389,23390],{"class":55,"line":1284},[53,23391,104],{"emptyLinePlaceholder":103},[53,23393,23394,23396],{"class":55,"line":1292},[53,23395,8420],{"class":120},[53,23397,23398],{"class":63},"(H_expval)\n",[43,23400,23403],{"className":23401,"code":23402,"language":6934,"meta":48},[8426],"(25+0j)\n",[50,23404,23402],{"__ignoreMap":48},[12,23406,23407,23408,10694,23436,23496],{},"Execute circuits for ",[53,23409,23411,23424],{"className":23410},[2418],[53,23412,23414],{"className":23413},[2422],[2424,23415,23416],{"xmlns":2426},[2428,23417,23418,23422],{},[2431,23419,23420],{},[2437,23421,5752],{},[2444,23423,5752],{"encoding":2446},[53,23425,23427],{"className":23426,"ariaHidden":2452},[2451],[53,23428,23430,23433],{"className":23429},[2456],[53,23431],{"className":23432,"style":2461},[2460],[53,23434,5752],{"className":23435,"style":5787},[2465,2506],[53,23437,23439,23456],{"className":23438},[2418],[53,23440,23442],{"className":23441},[2422],[2424,23443,23444],{"xmlns":2426},[2428,23445,23446,23454],{},[2431,23447,23448],{},[4166,23449,23450,23452],{"accent":2452},[2437,23451,3033],{},[2584,23453,4172],{},[2444,23455,4175],{"encoding":2446},[53,23457,23459],{"className":23458,"ariaHidden":2452},[2451],[53,23460,23462,23465],{"className":23461},[2456],[53,23463],{"className":23464,"style":4185},[2460],[53,23466,23468],{"className":23467},[2465,4189],[53,23469,23471],{"className":23470},[2478],[53,23472,23474],{"className":23473},[2482],[53,23475,23477,23485],{"className":23476,"style":4185},[2486],[53,23478,23479,23482],{"style":4201},[53,23480],{"className":23481,"style":4205},[2494],[53,23483,3033],{"className":23484,"style":3048},[2465,2506],[53,23486,23487,23490],{"style":4211},[53,23488],{"className":23489,"style":4205},[2494],[53,23491,23493],{"className":23492,"style":4219},[4218],[53,23494,4172],{"className":23495},[2465]," with Estimator",[43,23498,23500],{"className":45,"code":23499,"language":47,"meta":48,"style":48},"# 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",[50,23501,23502,23507,23517,23526,23536,23559,23586,23590,23594,23604,23619,23630,23638,23646,23654,23659,23670,23684,23691,23703,23711,23717,23723,23727,23733,23745,23762,23766,23771,23784,23792,23799,23811,23815,23819,23847,23851,23855],{"__ignoreMap":48},[53,23503,23504],{"class":55,"line":56},[53,23505,23506],{"class":110},"# Experiment options\n",[53,23508,23509,23512,23514],{"class":55,"line":73},[53,23510,23511],{"class":63},"num_randomizations ",[53,23513,121],{"class":120},[53,23515,23516],{"class":131}," 300\n",[53,23518,23519,23522,23524],{"class":55,"line":87},[53,23520,23521],{"class":63},"num_randomizations_learning ",[53,23523,121],{"class":120},[53,23525,8064],{"class":131},[53,23527,23528,23531,23533],{"class":55,"line":100},[53,23529,23530],{"class":63},"shots_per_randomization ",[53,23532,121],{"class":120},[53,23534,23535],{"class":131}," 100\n",[53,23537,23538,23541,23543,23545,23547,23549,23552,23554,23557],{"class":55,"line":107},[53,23539,23540],{"class":63},"noise_factors ",[53,23542,121],{"class":120},[53,23544,8322],{"class":63},[53,23546,173],{"class":131},[53,23548,170],{"class":63},[53,23550,23551],{"class":131},"1.2",[53,23553,170],{"class":63},[53,23555,23556],{"class":131},"1.4",[53,23558,1689],{"class":63},[53,23560,23561,23564,23566,23568,23570,23572,23574,23576,23579,23581,23584],{"class":55,"line":114},[53,23562,23563],{"class":63},"learning_pair_depths ",[53,23565,121],{"class":120},[53,23567,8322],{"class":63},[53,23569,149],{"class":131},[53,23571,170],{"class":63},[53,23573,12047],{"class":131},[53,23575,170],{"class":63},[53,23577,23578],{"class":131},"24",[53,23580,170],{"class":63},[53,23582,23583],{"class":131},"48",[53,23585,1689],{"class":63},[53,23587,23588],{"class":55,"line":138},[53,23589,104],{"emptyLinePlaceholder":103},[53,23591,23592],{"class":55,"line":158},[53,23593,104],{"emptyLinePlaceholder":103},[53,23595,23596,23599,23601],{"class":55,"line":182},[53,23597,23598],{"class":63},"experimental_opts ",[53,23600,121],{"class":120},[53,23602,23603],{"class":63}," {}\n",[53,23605,23606,23609,23612,23614,23616],{"class":55,"line":209},[53,23607,23608],{"class":63},"experimental_opts[",[53,23610,23611],{"class":278},"\"resilience\"",[53,23613,8101],{"class":63},[53,23615,121],{"class":120},[53,23617,23618],{"class":63}," {\n",[53,23620,23621,23624,23626,23628],{"class":55,"line":226},[53,23622,23623],{"class":278}," \"measure_mitigation\"",[53,23625,1298],{"class":63},[53,23627,13773],{"class":131},[53,23629,1304],{"class":63},[53,23631,23632,23635],{"class":55,"line":231},[53,23633,23634],{"class":278}," \"measure_noise_learning\"",[53,23636,23637],{"class":63},": {\n",[53,23639,23640,23643],{"class":55,"line":245},[53,23641,23642],{"class":278}," \"num_randomizations\"",[53,23644,23645],{"class":63},": num_randomizations_learning,\n",[53,23647,23648,23651],{"class":55,"line":258},[53,23649,23650],{"class":278}," \"shots_per_randomization\"",[53,23652,23653],{"class":63},": shots_per_randomization,\n",[53,23655,23656],{"class":55,"line":263},[53,23657,23658],{"class":63}," },\n",[53,23660,23661,23664,23666,23668],{"class":55,"line":269},[53,23662,23663],{"class":278}," \"zne_mitigation\"",[53,23665,1298],{"class":63},[53,23667,13773],{"class":131},[53,23669,1304],{"class":63},[53,23671,23672,23675,23678,23681],{"class":55,"line":284},[53,23673,23674],{"class":278}," \"zne\"",[53,23676,23677],{"class":63},": {",[53,23679,23680],{"class":278},"\"noise_factors\"",[53,23682,23683],{"class":63},": noise_factors},\n",[53,23685,23686,23689],{"class":55,"line":296},[53,23687,23688],{"class":278}," \"layer_noise_learning\"",[53,23690,23637],{"class":63},[53,23692,23693,23696,23698,23701],{"class":55,"line":309},[53,23694,23695],{"class":278}," \"max_layers_to_learn\"",[53,23697,1298],{"class":63},[53,23699,23700],{"class":131},"10",[53,23702,1304],{"class":63},[53,23704,23705,23708],{"class":55,"line":323},[53,23706,23707],{"class":278}," \"layer_pair_depths\"",[53,23709,23710],{"class":63},": learning_pair_depths,\n",[53,23712,23713,23715],{"class":55,"line":332},[53,23714,23650],{"class":278},[53,23716,23653],{"class":63},[53,23718,23719,23721],{"class":55,"line":344},[53,23720,23642],{"class":278},[53,23722,23645],{"class":63},[53,23724,23725],{"class":55,"line":357},[53,23726,23658],{"class":63},[53,23728,23729,23731],{"class":55,"line":1247},[53,23730,23674],{"class":278},[53,23732,23637],{"class":63},[53,23734,23735,23738,23740,23743],{"class":55,"line":1252},[53,23736,23737],{"class":278}," \"amplifier\"",[53,23739,1298],{"class":63},[53,23741,23742],{"class":278},"\"pea\"",[53,23744,1304],{"class":63},[53,23746,23747,23750,23753,23755,23757,23759],{"class":55,"line":1257},[53,23748,23749],{"class":278}," \"extrapolated_noise_factors\"",[53,23751,23752],{"class":63},": [",[53,23754,149],{"class":131},[53,23756,8101],{"class":63},[53,23758,2002],{"class":120},[53,23760,23761],{"class":63}," noise_factors,\n",[53,23763,23764],{"class":55,"line":1262},[53,23765,23658],{"class":63},[53,23767,23768],{"class":55,"line":1274},[53,23769,23770],{"class":63},"}\n",[53,23772,23773,23775,23778,23780,23782],{"class":55,"line":1284},[53,23774,23608],{"class":63},[53,23776,23777],{"class":278},"\"twirling\"",[53,23779,8101],{"class":63},[53,23781,121],{"class":120},[53,23783,23618],{"class":63},[53,23785,23786,23789],{"class":55,"line":1292},[53,23787,23788],{"class":278}," \"num_randomizations\"",[53,23790,23791],{"class":63},": num_randomizations,\n",[53,23793,23794,23797],{"class":55,"line":1307},[53,23795,23796],{"class":278}," \"shots_per_randomization\"",[53,23798,23653],{"class":63},[53,23800,23801,23804,23806,23809],{"class":55,"line":1326},[53,23802,23803],{"class":278}," \"strategy\"",[53,23805,1298],{"class":63},[53,23807,23808],{"class":278},"\"all\"",[53,23810,1304],{"class":63},[53,23812,23813],{"class":55,"line":1344},[53,23814,23770],{"class":63},[53,23816,23817],{"class":55,"line":1360},[53,23818,104],{"emptyLinePlaceholder":103},[53,23820,23821,23824,23826,23829,23831,23834,23836,23839,23842,23844],{"class":55,"line":1366},[53,23822,23823],{"class":63},"estimator ",[53,23825,121],{"class":120},[53,23827,23828],{"class":124}," Estimator",[53,23830,128],{"class":63},[53,23832,23833],{"class":2050},"mode",[53,23835,121],{"class":120},[53,23837,23838],{"class":63},"backend, ",[53,23840,23841],{"class":2050},"options",[53,23843,121],{"class":120},[53,23845,23846],{"class":63},"experimental_opts)\n",[53,23848,23849],{"class":55,"line":1372},[53,23850,104],{"emptyLinePlaceholder":103},[53,23852,23853],{"class":55,"line":1377},[53,23854,104],{"emptyLinePlaceholder":103},[53,23856,23857,23860,23862,23865,23867],{"class":55,"line":1383},[53,23858,23859],{"class":63},"job ",[53,23861,121],{"class":120},[53,23863,23864],{"class":63}," estimator.",[53,23866,22396],{"class":124},[53,23868,23869],{"class":63},"([pub])\n",[30,23871,23873],{"id":23872},"step-4-post-process-and-return-result-in-desired-classical-format","Step 4: Post-process and return result in desired classical format",[43,23875,23877],{"className":45,"code":23876,"language":47,"meta":48,"style":48},"results = job.result()[0]\n",[50,23878,23879],{"__ignoreMap":48},[53,23880,23881,23884,23886,23889,23892,23895,23897],{"class":55,"line":56},[53,23882,23883],{"class":63},"results ",[53,23885,121],{"class":120},[53,23887,23888],{"class":63}," job.",[53,23890,23891],{"class":124},"result",[53,23893,23894],{"class":63},"()[",[53,23896,149],{"class":131},[53,23898,1689],{"class":63},[2408,23900,23902],{"id":23901},"calculate-effective-hamiltonian-and-overlap-matrices","Calculate Effective Hamiltonian and Overlap matrices",[12,23904,23905,23906,23942],{},"First calculate the phase accumulated by the ",[53,23907,23909,23927],{"className":23908},[2418],[53,23910,23912],{"className":23911},[2422],[2424,23913,23914],{"xmlns":2426},[2428,23915,23916,23924],{},[2431,23917,23918,23920,23922],{},[2437,23919,2644],{"mathvariant":2643},[2588,23921,149],{},[2584,23923,2651],{"stretchy":2650},[2444,23925,23926],{"encoding":2446},"\\vert 0 \\rangle",[53,23928,23930],{"className":23929,"ariaHidden":2452},[2451],[53,23931,23933,23936,23939],{"className":23932},[2456],[53,23934],{"className":23935,"style":2664},[2460],[53,23937,9630],{"className":23938},[2465],[53,23940,2651],{"className":23941},[2675]," state during the uncontrolled time evolution",[43,23944,23946],{"className":45,"code":23945,"language":47,"meta":48,"style":48},"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",[50,23947,23948,23957,24012,24029],{"__ignoreMap":48},[53,23949,23950,23953,23955],{"class":55,"line":56},[53,23951,23952],{"class":63},"prefactors ",[53,23954,121],{"class":120},[53,23956,2072],{"class":63},[53,23958,23959,23962,23965,23967,23969,23971,23973,23975,23977,23980,23982,23985,23987,23989,23991,23993,23995,23998,24000,24003,24005,24007,24009],{"class":55,"line":73},[53,23960,23961],{"class":63}," np.",[53,23963,23964],{"class":124},"exp",[53,23966,128],{"class":63},[53,23968,2262],{"class":120},[53,23970,173],{"class":131},[53,23972,2267],{"class":59},[53,23974,8124],{"class":120},[53,23976,8710],{"class":120},[53,23978,23979],{"class":63},"([c ",[53,23981,1525],{"class":59},[53,23983,23984],{"class":63}," p, c ",[53,23986,1531],{"class":59},[53,23988,1819],{"class":63},[53,23990,1822],{"class":124},[53,23992,1572],{"class":63},[53,23994,1540],{"class":59},[53,23996,23997],{"class":278}," \"Z\"",[53,23999,21878],{"class":59},[53,24001,24002],{"class":63}," p]) ",[53,24004,2286],{"class":120},[53,24006,8141],{"class":63},[53,24008,2286],{"class":120},[53,24010,24011],{"class":63}," dt)\n",[53,24013,24014,24016,24018,24020,24022,24024,24026],{"class":55,"line":87},[53,24015,1811],{"class":59},[53,24017,8141],{"class":63},[53,24019,1531],{"class":59},[53,24021,8116],{"class":120},[53,24023,128],{"class":63},[53,24025,173],{"class":131},[53,24027,24028],{"class":63},", krylov_dim)\n",[53,24030,24031],{"class":55,"line":100},[53,24032,1689],{"class":63},[12,24034,24035,24036],{},"Once we have the results of the circuit executions we can post-process the data to calculate the matrix elements of ",[53,24037,24039,24052],{"className":24038},[2418],[53,24040,24042],{"className":24041},[2422],[2424,24043,24044],{"xmlns":2426},[2428,24045,24046,24050],{},[2431,24047,24048],{},[2437,24049,5752],{},[2444,24051,5752],{"encoding":2446},[53,24053,24055],{"className":24054,"ariaHidden":2452},[2451],[53,24056,24058,24061],{"className":24057},[2456],[53,24059],{"className":24060,"style":2461},[2460],[53,24062,5752],{"className":24063,"style":5787},[2465,2506],[43,24065,24067],{"className":45,"code":24066,"language":47,"meta":48,"style":48},"# 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",[50,24068,24069,24074,24094,24114,24118,24123,24142,24146,24164,24169,24176,24192,24196,24202,24206,24211,24229,24249,24253,24272,24276,24280,24300,24304,24309,24339,24351,24366,24372],{"__ignoreMap":48},[53,24070,24071],{"class":55,"line":56},[53,24072,24073],{"class":110},"# Assemble S, the overlap matrix of dimension D:\n",[53,24075,24076,24079,24081,24083,24085,24088,24090,24092],{"class":55,"line":73},[53,24077,24078],{"class":63},"S_first_row ",[53,24080,121],{"class":120},[53,24082,1511],{"class":63},[53,24084,2044],{"class":124},[53,24086,24087],{"class":63},"(krylov_dim, ",[53,24089,2051],{"class":2050},[53,24091,2054],{"class":120},[53,24093,135],{"class":63},[53,24095,24096,24099,24101,24103,24105,24107,24109,24111],{"class":55,"line":87},[53,24097,24098],{"class":63},"S_first_row[",[53,24100,149],{"class":131},[53,24102,8101],{"class":63},[53,24104,121],{"class":120},[53,24106,1321],{"class":131},[53,24108,8208],{"class":120},[53,24110,21153],{"class":131},[53,24112,24113],{"class":59},"j\n",[53,24115,24116],{"class":55,"line":100},[53,24117,104],{"emptyLinePlaceholder":103},[53,24119,24120],{"class":55,"line":107},[53,24121,24122],{"class":110},"# Add in ancilla-only measurements:\n",[53,24124,24125,24127,24129,24131,24133,24136,24138,24140],{"class":55,"line":114},[53,24126,1525],{"class":59},[53,24128,8141],{"class":63},[53,24130,1531],{"class":59},[53,24132,8116],{"class":120},[53,24134,24135],{"class":63},"(krylov_dim ",[53,24137,2262],{"class":120},[53,24139,1321],{"class":131},[53,24141,1778],{"class":63},[53,24143,24144],{"class":55,"line":138},[53,24145,23254],{"class":110},[53,24147,24148,24150,24152,24155,24157,24159,24161],{"class":55,"line":158},[53,24149,23259],{"class":63},[53,24151,121],{"class":120},[53,24153,24154],{"class":63}," results.data.evs[",[53,24156,149],{"class":131},[53,24158,1650],{"class":63},[53,24160,149],{"class":131},[53,24162,24163],{"class":63},"][\n",[53,24165,24166],{"class":55,"line":182},[53,24167,24168],{"class":63}," i\n",[53,24170,24171,24173],{"class":55,"line":209},[53,24172,2122],{"class":63},[53,24174,24175],{"class":110},"# automatic extrapolated evs if ZNE is used\n",[53,24177,24178,24180,24182,24184,24186,24188,24190],{"class":55,"line":226},[53,24179,23302],{"class":63},[53,24181,121],{"class":120},[53,24183,24154],{"class":63},[53,24185,173],{"class":131},[53,24187,1650],{"class":63},[53,24189,149],{"class":131},[53,24191,24163],{"class":63},[53,24193,24194],{"class":55,"line":231},[53,24195,24168],{"class":63},[53,24197,24198,24200],{"class":55,"line":245},[53,24199,2122],{"class":63},[53,24201,24175],{"class":110},[53,24203,24204],{"class":55,"line":258},[53,24205,104],{"emptyLinePlaceholder":103},[53,24207,24208],{"class":55,"line":263},[53,24209,24210],{"class":110}," # Get expectation values\n",[53,24212,24213,24215,24217,24219,24221,24223,24225,24227],{"class":55,"line":269},[53,24214,23343],{"class":63},[53,24216,121],{"class":120},[53,24218,23348],{"class":63},[53,24220,2002],{"class":120},[53,24222,1321],{"class":131},[53,24224,2267],{"class":59},[53,24226,8124],{"class":120},[53,24228,23359],{"class":63},[53,24230,24231,24234,24236,24238,24240,24242,24245,24247],{"class":55,"line":284},[53,24232,24233],{"class":63}," S_first_row[i ",[53,24235,2002],{"class":120},[53,24237,1321],{"class":131},[53,24239,8101],{"class":63},[53,24241,2180],{"class":120},[53,24243,24244],{"class":63}," prefactors[i] ",[53,24246,2286],{"class":120},[53,24248,23383],{"class":63},[53,24250,24251],{"class":55,"line":296},[53,24252,104],{"emptyLinePlaceholder":103},[53,24254,24255,24258,24260,24263,24266,24269],{"class":55,"line":309},[53,24256,24257],{"class":63},"S_first_row_list ",[53,24259,121],{"class":120},[53,24261,24262],{"class":63}," S_first_row.",[53,24264,24265],{"class":124},"tolist",[53,24267,24268],{"class":63},"() ",[53,24270,24271],{"class":110},"# for saving purposes\n",[53,24273,24274],{"class":55,"line":323},[53,24275,104],{"emptyLinePlaceholder":103},[53,24277,24278],{"class":55,"line":332},[53,24279,104],{"emptyLinePlaceholder":103},[53,24281,24282,24285,24287,24289,24291,24294,24296,24298],{"class":55,"line":344},[53,24283,24284],{"class":63},"S_circ ",[53,24286,121],{"class":120},[53,24288,1511],{"class":63},[53,24290,2044],{"class":124},[53,24292,24293],{"class":63},"((krylov_dim, krylov_dim), ",[53,24295,2051],{"class":2050},[53,24297,2054],{"class":120},[53,24299,135],{"class":63},[53,24301,24302],{"class":55,"line":357},[53,24303,104],{"emptyLinePlaceholder":103},[53,24305,24306],{"class":55,"line":1247},[53,24307,24308],{"class":110},"# Distribute entries from first row across matrix:\n",[53,24310,24311,24313,24316,24318,24320,24323,24325,24327,24330,24333,24335,24337],{"class":55,"line":1252},[53,24312,1525],{"class":59},[53,24314,24315],{"class":63}," i, j ",[53,24317,1531],{"class":59},[53,24319,2091],{"class":63},[53,24321,24322],{"class":124},"product",[53,24324,128],{"class":63},[53,24326,2099],{"class":120},[53,24328,24329],{"class":63},"(krylov_dim), ",[53,24331,24332],{"class":2050},"repeat",[53,24334,121],{"class":120},[53,24336,132],{"class":131},[53,24338,1778],{"class":63},[53,24340,24341,24343,24345,24348],{"class":55,"line":1257},[53,24342,1612],{"class":59},[53,24344,8141],{"class":63},[53,24346,24347],{"class":120},">=",[53,24349,24350],{"class":63}," j:\n",[53,24352,24353,24356,24358,24361,24363],{"class":55,"line":1262},[53,24354,24355],{"class":63}," S_circ[j, i] ",[53,24357,121],{"class":120},[53,24359,24360],{"class":63}," S_first_row[i ",[53,24362,2262],{"class":120},[53,24364,24365],{"class":63}," j]\n",[53,24367,24368,24370],{"class":55,"line":1274},[53,24369,1695],{"class":59},[53,24371,1623],{"class":63},[53,24373,24374,24376,24378,24380,24382,24385,24387],{"class":55,"line":1284},[53,24375,24355],{"class":63},[53,24377,121],{"class":120},[53,24379,1511],{"class":63},[53,24381,1569],{"class":124},[53,24383,24384],{"class":63},"(S_first_row[j ",[53,24386,2262],{"class":120},[53,24388,24389],{"class":63}," i])\n",[43,24391,24393],{"className":45,"code":24392,"language":47,"meta":48,"style":48},"Matrix(S_circ)\n",[50,24394,24395],{"__ignoreMap":48},[53,24396,24397,24400],{"class":55,"line":56},[53,24398,24399],{"class":124},"Matrix",[53,24401,24402],{"class":63},"(S_circ)\n",[12,24404,24405,24406],{},"And the matrix elements of ",[53,24407,24409,24426],{"className":24408},[2418],[53,24410,24412],{"className":24411},[2422],[2424,24413,24414],{"xmlns":2426},[2428,24415,24416,24424],{},[2431,24417,24418],{},[4166,24419,24420,24422],{"accent":2452},[2437,24421,3033],{},[2584,24423,4172],{},[2444,24425,4175],{"encoding":2446},[53,24427,24429],{"className":24428,"ariaHidden":2452},[2451],[53,24430,24432,24435],{"className":24431},[2456],[53,24433],{"className":24434,"style":4185},[2460],[53,24436,24438],{"className":24437},[2465,4189],[53,24439,24441],{"className":24440},[2478],[53,24442,24444],{"className":24443},[2482],[53,24445,24447,24455],{"className":24446,"style":4185},[2486],[53,24448,24449,24452],{"style":4201},[53,24450],{"className":24451,"style":4205},[2494],[53,24453,3033],{"className":24454,"style":3048},[2465,2506],[53,24456,24457,24460],{"style":4211},[53,24458],{"className":24459,"style":4205},[2494],[53,24461,24463],{"className":24462,"style":4219},[4218],[53,24464,4172],{"className":24465},[2465],[43,24467,24469],{"className":45,"code":24468,"language":47,"meta":48,"style":48},"# 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",[50,24470,24471,24475,24494,24508,24512,24528,24533,24551,24556,24580,24585,24592,24622,24626,24632,24636,24641,24660,24683,24687,24701,24705,24724,24728,24732,24758,24768,24782,24788],{"__ignoreMap":48},[53,24472,24473],{"class":55,"line":56},[53,24474,24073],{"class":110},[53,24476,24477,24480,24482,24484,24486,24488,24490,24492],{"class":55,"line":73},[53,24478,24479],{"class":63},"H_first_row ",[53,24481,121],{"class":120},[53,24483,1511],{"class":63},[53,24485,2044],{"class":124},[53,24487,24087],{"class":63},[53,24489,2051],{"class":2050},[53,24491,2054],{"class":120},[53,24493,135],{"class":63},[53,24495,24496,24499,24501,24503,24505],{"class":55,"line":87},[53,24497,24498],{"class":63},"H_first_row[",[53,24500,149],{"class":131},[53,24502,8101],{"class":63},[53,24504,121],{"class":120},[53,24506,24507],{"class":63}," H_expval\n",[53,24509,24510],{"class":55,"line":100},[53,24511,104],{"emptyLinePlaceholder":103},[53,24513,24514,24516,24518,24520,24522,24524,24526],{"class":55,"line":107},[53,24515,1525],{"class":59},[53,24517,23237],{"class":63},[53,24519,1531],{"class":59},[53,24521,1852],{"class":120},[53,24523,128],{"class":63},[53,24525,23246],{"class":120},[53,24527,23249],{"class":63},[53,24529,24530],{"class":55,"line":114},[53,24531,24532],{"class":110}," # Add in ancilla-only measurements:\n",[53,24534,24535,24537,24539,24541,24543,24545,24547,24549],{"class":55,"line":138},[53,24536,1811],{"class":59},[53,24538,8141],{"class":63},[53,24540,1531],{"class":59},[53,24542,8116],{"class":120},[53,24544,24135],{"class":63},[53,24546,2262],{"class":120},[53,24548,1321],{"class":131},[53,24550,1778],{"class":63},[53,24552,24553],{"class":55,"line":158},[53,24554,24555],{"class":110}," # Get expectation values from experiment\n",[53,24557,24558,24561,24563,24565,24567,24569,24571,24573,24576,24578],{"class":55,"line":182},[53,24559,24560],{"class":63}," expval_real ",[53,24562,121],{"class":120},[53,24564,24154],{"class":63},[53,24566,132],{"class":131},[53,24568,8208],{"class":120},[53,24570,2211],{"class":131},[53,24572,8124],{"class":120},[53,24574,24575],{"class":63}," obs_idx][",[53,24577,149],{"class":131},[53,24579,24163],{"class":63},[53,24581,24582],{"class":55,"line":209},[53,24583,24584],{"class":63}," i\n",[53,24586,24587,24590],{"class":55,"line":226},[53,24588,24589],{"class":63}," ] ",[53,24591,24175],{"class":110},[53,24593,24594,24597,24599,24601,24603,24605,24607,24609,24612,24614,24616,24618,24620],{"class":55,"line":231},[53,24595,24596],{"class":63}," expval_imag ",[53,24598,121],{"class":120},[53,24600,24154],{"class":63},[53,24602,132],{"class":131},[53,24604,8208],{"class":120},[53,24606,2211],{"class":131},[53,24608,8124],{"class":120},[53,24610,24611],{"class":63}," obs_idx ",[53,24613,2002],{"class":120},[53,24615,1321],{"class":131},[53,24617,1650],{"class":63},[53,24619,149],{"class":131},[53,24621,24163],{"class":63},[53,24623,24624],{"class":55,"line":245},[53,24625,24584],{"class":63},[53,24627,24628,24630],{"class":55,"line":258},[53,24629,24589],{"class":63},[53,24631,24175],{"class":110},[53,24633,24634],{"class":55,"line":263},[53,24635,104],{"emptyLinePlaceholder":103},[53,24637,24638],{"class":55,"line":269},[53,24639,24640],{"class":110}," # Get expectation values\n",[53,24642,24643,24646,24648,24650,24652,24654,24656,24658],{"class":55,"line":284},[53,24644,24645],{"class":63}," expval ",[53,24647,121],{"class":120},[53,24649,23348],{"class":63},[53,24651,2002],{"class":120},[53,24653,1321],{"class":131},[53,24655,2267],{"class":59},[53,24657,8124],{"class":120},[53,24659,23359],{"class":63},[53,24661,24662,24665,24667,24669,24671,24673,24675,24677,24679,24681],{"class":55,"line":296},[53,24663,24664],{"class":63}," H_first_row[i ",[53,24666,2002],{"class":120},[53,24668,1321],{"class":131},[53,24670,8101],{"class":63},[53,24672,2180],{"class":120},[53,24674,24244],{"class":63},[53,24676,2286],{"class":120},[53,24678,23378],{"class":63},[53,24680,2286],{"class":120},[53,24682,23383],{"class":63},[53,24684,24685],{"class":55,"line":309},[53,24686,104],{"emptyLinePlaceholder":103},[53,24688,24689,24692,24694,24697,24699],{"class":55,"line":323},[53,24690,24691],{"class":63},"H_first_row_list ",[53,24693,121],{"class":120},[53,24695,24696],{"class":63}," H_first_row.",[53,24698,24265],{"class":124},[53,24700,329],{"class":63},[53,24702,24703],{"class":55,"line":332},[53,24704,104],{"emptyLinePlaceholder":103},[53,24706,24707,24710,24712,24714,24716,24718,24720,24722],{"class":55,"line":344},[53,24708,24709],{"class":63},"H_eff_circ ",[53,24711,121],{"class":120},[53,24713,1511],{"class":63},[53,24715,2044],{"class":124},[53,24717,24293],{"class":63},[53,24719,2051],{"class":2050},[53,24721,2054],{"class":120},[53,24723,135],{"class":63},[53,24725,24726],{"class":55,"line":357},[53,24727,104],{"emptyLinePlaceholder":103},[53,24729,24730],{"class":55,"line":1247},[53,24731,24308],{"class":110},[53,24733,24734,24736,24738,24740,24742,24744,24746,24748,24750,24752,24754,24756],{"class":55,"line":1252},[53,24735,1525],{"class":59},[53,24737,24315],{"class":63},[53,24739,1531],{"class":59},[53,24741,2091],{"class":63},[53,24743,24322],{"class":124},[53,24745,128],{"class":63},[53,24747,2099],{"class":120},[53,24749,24329],{"class":63},[53,24751,24332],{"class":2050},[53,24753,121],{"class":120},[53,24755,132],{"class":131},[53,24757,1778],{"class":63},[53,24759,24760,24762,24764,24766],{"class":55,"line":1257},[53,24761,1612],{"class":59},[53,24763,8141],{"class":63},[53,24765,24347],{"class":120},[53,24767,24350],{"class":63},[53,24769,24770,24773,24775,24778,24780],{"class":55,"line":1262},[53,24771,24772],{"class":63}," H_eff_circ[j, i] ",[53,24774,121],{"class":120},[53,24776,24777],{"class":63}," H_first_row[i ",[53,24779,2262],{"class":120},[53,24781,24365],{"class":63},[53,24783,24784,24786],{"class":55,"line":1274},[53,24785,1695],{"class":59},[53,24787,1623],{"class":63},[53,24789,24790,24792,24794,24796,24798,24801,24803],{"class":55,"line":1284},[53,24791,24772],{"class":63},[53,24793,121],{"class":120},[53,24795,1511],{"class":63},[53,24797,1569],{"class":124},[53,24799,24800],{"class":63},"(H_first_row[j ",[53,24802,2262],{"class":120},[53,24804,24389],{"class":63},[43,24806,24808],{"className":45,"code":24807,"language":47,"meta":48,"style":48},"Matrix(H_eff_circ)\n",[50,24809,24810],{"__ignoreMap":48},[53,24811,24812,24814],{"class":55,"line":56},[53,24813,24399],{"class":124},[53,24815,24816],{"class":63},"(H_eff_circ)\n",[12,24818,24819,24820,24880],{},"Finally, we can solve the generalized eigenvalue problem for ",[53,24821,24823,24840],{"className":24822},[2418],[53,24824,24826],{"className":24825},[2422],[2424,24827,24828],{"xmlns":2426},[2428,24829,24830,24838],{},[2431,24831,24832],{},[4166,24833,24834,24836],{"accent":2452},[2437,24835,3033],{},[2584,24837,4172],{},[2444,24839,4175],{"encoding":2446},[53,24841,24843],{"className":24842,"ariaHidden":2452},[2451],[53,24844,24846,24849],{"className":24845},[2456],[53,24847],{"className":24848,"style":4185},[2460],[53,24850,24852],{"className":24851},[2465,4189],[53,24853,24855],{"className":24854},[2478],[53,24856,24858],{"className":24857},[2482],[53,24859,24861,24869],{"className":24860,"style":4185},[2486],[53,24862,24863,24866],{"style":4201},[53,24864],{"className":24865,"style":4205},[2494],[53,24867,3033],{"className":24868,"style":3048},[2465,2506],[53,24870,24871,24874],{"style":4211},[53,24872],{"className":24873,"style":4205},[2494],[53,24875,24877],{"className":24876,"style":4219},[4218],[53,24878,4172],{"className":24879},[2465],":",[12,24882,24883],{},[53,24884,24886,24922],{"className":24885},[2418],[53,24887,24889],{"className":24888},[2422],[2424,24890,24891],{"xmlns":2426},[2428,24892,24893,24919],{},[2431,24894,24895,24901,24907,24909,24911,24913],{},[4166,24896,24897,24899],{"accent":2452},[2437,24898,3033],{},[2584,24900,4172],{},[4166,24902,24903,24905],{"accent":2452},[2437,24904,6069],{},[2584,24906,6072],{},[2584,24908,121],{},[2437,24910,6069],{},[2437,24912,5752],{},[4166,24914,24915,24917],{"accent":2452},[2437,24916,6069],{},[2584,24918,6072],{},[2444,24920,24921],{"encoding":2446},"\\tilde{H} \\vec{c} = c S \\vec{c}",[53,24923,24925,25006],{"className":24924,"ariaHidden":2452},[2451],[53,24926,24928,24931,24962,24997,25000,25003],{"className":24927},[2456],[53,24929],{"className":24930,"style":4185},[2460],[53,24932,24934],{"className":24933},[2465,4189],[53,24935,24937],{"className":24936},[2478],[53,24938,24940],{"className":24939},[2482],[53,24941,24943,24951],{"className":24942,"style":4185},[2486],[53,24944,24945,24948],{"style":4201},[53,24946],{"className":24947,"style":4205},[2494],[53,24949,3033],{"className":24950,"style":3048},[2465,2506],[53,24952,24953,24956],{"style":4211},[53,24954],{"className":24955,"style":4205},[2494],[53,24957,24959],{"className":24958,"style":4219},[4218],[53,24960,4172],{"className":24961},[2465],[53,24963,24965],{"className":24964},[2465,4189],[53,24966,24968],{"className":24967},[2478],[53,24969,24971],{"className":24970},[2482],[53,24972,24974,24982],{"className":24973,"style":6152},[2486],[53,24975,24976,24979],{"style":4201},[53,24977],{"className":24978,"style":4205},[2494],[53,24980,6069],{"className":24981},[2465,2506],[53,24983,24984,24987],{"style":4201},[53,24985],{"className":24986,"style":4205},[2494],[53,24988,24990],{"className":24989,"style":6169},[4218],[53,24991,24993],{"className":24992,"style":6174},[6173],[6176,24994,24995],{"xmlns":6178,"width":6179,"height":6180,"style":6181,"viewBox":6182,"preserveAspectRatio":6183},[6185,24996],{"d":6187},[53,24998],{"className":24999,"style":2814},[2609],[53,25001,121],{"className":25002},[2818],[53,25004],{"className":25005,"style":2814},[2609],[53,25007,25009,25012,25015,25018],{"className":25008},[2456],[53,25010],{"className":25011,"style":6152},[2460],[53,25013,6069],{"className":25014},[2465,2506],[53,25016,5752],{"className":25017,"style":5787},[2465,2506],[53,25019,25021],{"className":25020},[2465,4189],[53,25022,25024],{"className":25023},[2478],[53,25025,25027],{"className":25026},[2482],[53,25028,25030,25038],{"className":25029,"style":6152},[2486],[53,25031,25032,25035],{"style":4201},[53,25033],{"className":25034,"style":4205},[2494],[53,25036,6069],{"className":25037},[2465,2506],[53,25039,25040,25043],{"style":4201},[53,25041],{"className":25042,"style":4205},[2494],[53,25044,25046],{"className":25045,"style":6169},[4218],[53,25047,25049],{"className":25048,"style":6174},[6173],[6176,25050,25051],{"xmlns":6178,"width":6179,"height":6180,"style":6181,"viewBox":6182,"preserveAspectRatio":6183},[6185,25052],{"d":6187},[12,25054,25055,25056],{},"and get an estimate of the ground state energy ",[53,25057,25059,25083],{"className":25058},[2418],[53,25060,25062],{"className":25061},[2422],[2424,25063,25064],{"xmlns":2426},[2428,25065,25066,25080],{},[2431,25067,25068],{},[3063,25069,25070,25072],{},[2437,25071,6069],{},[2431,25073,25074,25076,25078],{},[2437,25075,4312],{},[2437,25077,3188],{},[2437,25079,4315],{},[2444,25081,25082],{"encoding":2446},"c_{min}",[53,25084,25086],{"className":25085,"ariaHidden":2452},[2451],[53,25087,25089,25093],{"className":25088},[2456],[53,25090],{"className":25091,"style":25092},[2460],"height:0.5806em;vertical-align:-0.15em;",[53,25094,25096,25099],{"className":25095},[2465],[53,25097,6069],{"className":25098},[2465,2506],[53,25100,25102],{"className":25101},[2474],[53,25103,25105,25129],{"className":25104},[2478,3095],[53,25106,25108,25126],{"className":25107},[2482],[53,25109,25111],{"className":25110,"style":6697},[2486],[53,25112,25113,25116],{"style":5293},[53,25114],{"className":25115,"style":2495},[2494],[53,25117,25119],{"className":25118},[2499,2500,2501,2502],[53,25120,25122],{"className":25121},[2465,2502],[53,25123,25125],{"className":25124},[2465,2506,2502],"min",[53,25127,3120],{"className":25128},[3119],[53,25130,25132],{"className":25131},[2482],[53,25133,25135],{"className":25134,"style":3127},[2486],[53,25136],{},[43,25138,25140],{"className":45,"code":25139,"language":47,"meta":48,"style":48},"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",[50,25141,25142,25151,25175,25180,25191,25204,25208,25218],{"__ignoreMap":48},[53,25143,25144,25147,25149],{"class":55,"line":56},[53,25145,25146],{"class":63},"gnd_en_circ_est_list ",[53,25148,121],{"class":120},[53,25150,1805],{"class":63},[53,25152,25153,25155,25158,25160,25162,25164,25166,25169,25171,25173],{"class":55,"line":73},[53,25154,1525],{"class":59},[53,25156,25157],{"class":63}," d ",[53,25159,1531],{"class":59},[53,25161,8116],{"class":120},[53,25163,128],{"class":63},[53,25165,173],{"class":131},[53,25167,25168],{"class":63},", krylov_dim ",[53,25170,2002],{"class":120},[53,25172,1321],{"class":131},[53,25174,1778],{"class":63},[53,25176,25177],{"class":55,"line":87},[53,25178,25179],{"class":110}," # Solve generalized eigenvalue problem for different size of the Krylov space\n",[53,25181,25182,25185,25187,25189],{"class":55,"line":100},[53,25183,25184],{"class":63}," gnd_en_circ_est ",[53,25186,121],{"class":120},[53,25188,1268],{"class":124},[53,25190,1271],{"class":63},[53,25192,25193,25196,25199,25201],{"class":55,"line":107},[53,25194,25195],{"class":63}," H_eff_circ[:d, :d], S_circ[:d, :d], ",[53,25197,25198],{"class":2050},"threshold",[53,25200,121],{"class":120},[53,25202,25203],{"class":131},"9e-1\n",[53,25205,25206],{"class":55,"line":114},[53,25207,1555],{"class":63},[53,25209,25210,25213,25215],{"class":55,"line":138},[53,25211,25212],{"class":63}," gnd_en_circ_est_list.",[53,25214,1834],{"class":124},[53,25216,25217],{"class":63},"(gnd_en_circ_est)\n",[53,25219,25220,25222,25224,25227],{"class":55,"line":158},[53,25221,1955],{"class":120},[53,25223,128],{"class":63},[53,25225,25226],{"class":278},"\"The estimated ground state energy is: \"",[53,25228,25229],{"class":63},", gnd_en_circ_est)\n",[43,25231,25234],{"className":25232,"code":25233,"language":6934,"meta":48},[8426],"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",[50,25235,25233],{"__ignoreMap":48},[12,25237,25238],{},"For a single-particle sector, we can efficiently calculate the ground state of this sector of the Hamiltonian classically",[43,25240,25242],{"className":45,"code":25241,"language":47,"meta":48,"style":48},"gs_en = single_particle_gs(H_op, n_qubits)\n",[50,25243,25244],{"__ignoreMap":48},[53,25245,25246,25249,25251,25253],{"class":55,"line":56},[53,25247,25248],{"class":63},"gs_en ",[53,25250,121],{"class":120},[53,25252,1765],{"class":124},[53,25254,25255],{"class":63},"(H_op, n_qubits)\n",[43,25257,25260],{"className":25258,"code":25259,"language":6934,"meta":48},[8426],"n_sys_qubits 30\nn_exc 1 , subspace dimension 31\nsingle particle ground state energy: 21.021912418526906\n",[50,25261,25259],{"__ignoreMap":48},[43,25263,25265],{"className":45,"code":25264,"language":47,"meta":48,"style":48},"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",[50,25266,25267,25277,25294,25299,25311,25323,25335,25339,25347,25363,25373,25384,25395,25406,25410,25447,25456,25470,25484,25493,25498,25502],{"__ignoreMap":48},[53,25268,25269,25272,25275],{"class":55,"line":56},[53,25270,25271],{"class":63},"plt.",[53,25273,25274],{"class":124},"plot",[53,25276,1271],{"class":63},[53,25278,25279,25282,25284,25286,25288,25290,25292],{"class":55,"line":73},[53,25280,25281],{"class":120}," range",[53,25283,128],{"class":63},[53,25285,173],{"class":131},[53,25287,25168],{"class":63},[53,25289,2002],{"class":120},[53,25291,1321],{"class":131},[53,25293,14097],{"class":63},[53,25295,25296],{"class":55,"line":87},[53,25297,25298],{"class":63}," gnd_en_circ_est_list,\n",[53,25300,25301,25304,25306,25309],{"class":55,"line":100},[53,25302,25303],{"class":2050}," color",[53,25305,121],{"class":120},[53,25307,25308],{"class":278},"\"blue\"",[53,25310,1304],{"class":63},[53,25312,25313,25316,25318,25321],{"class":55,"line":107},[53,25314,25315],{"class":2050}," linestyle",[53,25317,121],{"class":120},[53,25319,25320],{"class":278},"\"-.\"",[53,25322,1304],{"class":63},[53,25324,25325,25328,25330,25333],{"class":55,"line":114},[53,25326,25327],{"class":2050}," label",[53,25329,121],{"class":120},[53,25331,25332],{"class":278},"\"KQD estimate\"",[53,25334,1304],{"class":63},[53,25336,25337],{"class":55,"line":138},[53,25338,135],{"class":63},[53,25340,25341,25343,25345],{"class":55,"line":158},[53,25342,25271],{"class":63},[53,25344,25274],{"class":124},[53,25346,1271],{"class":63},[53,25348,25349,25351,25353,25355,25357,25359,25361],{"class":55,"line":182},[53,25350,25281],{"class":120},[53,25352,128],{"class":63},[53,25354,173],{"class":131},[53,25356,25168],{"class":63},[53,25358,2002],{"class":120},[53,25360,1321],{"class":131},[53,25362,14097],{"class":63},[53,25364,25365,25368,25370],{"class":55,"line":209},[53,25366,25367],{"class":63}," [gs_en] ",[53,25369,2286],{"class":120},[53,25371,25372],{"class":63}," krylov_dim,\n",[53,25374,25375,25377,25379,25382],{"class":55,"line":226},[53,25376,25303],{"class":2050},[53,25378,121],{"class":120},[53,25380,25381],{"class":278},"\"red\"",[53,25383,1304],{"class":63},[53,25385,25386,25388,25390,25393],{"class":55,"line":231},[53,25387,25315],{"class":2050},[53,25389,121],{"class":120},[53,25391,25392],{"class":278},"\"-\"",[53,25394,1304],{"class":63},[53,25396,25397,25399,25401,25404],{"class":55,"line":245},[53,25398,25327],{"class":2050},[53,25400,121],{"class":120},[53,25402,25403],{"class":278},"\"exact\"",[53,25405,1304],{"class":63},[53,25407,25408],{"class":55,"line":258},[53,25409,135],{"class":63},[53,25411,25412,25414,25417,25419,25421,25423,25425,25427,25429,25431,25433,25435,25437,25439,25441,25443,25445],{"class":55,"line":263},[53,25413,25271],{"class":63},[53,25415,25416],{"class":124},"xticks",[53,25418,128],{"class":63},[53,25420,2099],{"class":120},[53,25422,128],{"class":63},[53,25424,173],{"class":131},[53,25426,25168],{"class":63},[53,25428,2002],{"class":120},[53,25430,1321],{"class":131},[53,25432,2108],{"class":63},[53,25434,2099],{"class":120},[53,25436,128],{"class":63},[53,25438,173],{"class":131},[53,25440,25168],{"class":63},[53,25442,2002],{"class":120},[53,25444,1321],{"class":131},[53,25446,22435],{"class":63},[53,25448,25449,25451,25454],{"class":55,"line":269},[53,25450,25271],{"class":63},[53,25452,25453],{"class":124},"legend",[53,25455,329],{"class":63},[53,25457,25458,25460,25463,25465,25468],{"class":55,"line":284},[53,25459,25271],{"class":63},[53,25461,25462],{"class":124},"xlabel",[53,25464,128],{"class":63},[53,25466,25467],{"class":278},"\"Krylov space dimension\"",[53,25469,135],{"class":63},[53,25471,25472,25474,25477,25479,25482],{"class":55,"line":296},[53,25473,25271],{"class":63},[53,25475,25476],{"class":124},"ylabel",[53,25478,128],{"class":63},[53,25480,25481],{"class":278},"\"Energy\"",[53,25483,135],{"class":63},[53,25485,25486,25488,25491],{"class":55,"line":309},[53,25487,25271],{"class":63},[53,25489,25490],{"class":124},"title",[53,25492,1271],{"class":63},[53,25494,25495],{"class":55,"line":323},[53,25496,25497],{"class":278}," \"Estimating Ground state energy with Krylov Quantum Diagonalization\"\n",[53,25499,25500],{"class":55,"line":332},[53,25501,135],{"class":63},[53,25503,25504,25506,25509],{"class":55,"line":344},[53,25505,25271],{"class":63},[53,25507,25508],{"class":124},"show",[53,25510,329],{"class":63},[30,25512,25514],{"id":25513},"appendix-krylov-subspace-from-real-time-evolutions","Appendix: Krylov subspace from real time-evolutions",[12,25516,25517],{},"The unitary Krylov space is defined as",[53,25519,25521],{"className":25520},[2679],[53,25522,25524,25632],{"className":25523},[2418],[53,25525,25527],{"className":25526},[2422],[2424,25528,25529],{"xmlns":2426,"display":2688},[2428,25530,25531,25629],{},[2431,25532,25533,25539,25541,25543,25545,25547,25549,25551,25553,25555,25557],{},[3063,25534,25535,25537],{},[2437,25536,2440],{"mathvariant":2439},[2437,25538,3174],{},[2584,25540,128],{"stretchy":2650},[2437,25542,3033],{},[2584,25544,2714],{"separator":2452},[2437,25546,2644],{"mathvariant":2643},[2437,25548,3688],{},[2584,25550,2651],{"stretchy":2650},[2584,25552,3544],{"stretchy":2650},[2584,25554,121],{},[6062,25556,53],{},[2431,25558,25559,25561,25563,25565,25567,25569,25588,25590,25592,25594,25596,25599,25601,25621,25623,25625,25627],{},[2584,25560,2705],{"fence":2452},[2437,25562,2644],{"mathvariant":2643},[2437,25564,3688],{},[2584,25566,2651],{"stretchy":2650},[2584,25568,2714],{"separator":2452},[2434,25570,25571,25573],{},[2437,25572,3181],{},[2431,25574,25575,25577,25579,25581,25584,25586],{},[2584,25576,2586],{},[2437,25578,3188],{},[2437,25580,3033],{},[6062,25582,25583],{}," ",[2437,25585,4854],{},[2437,25587,3193],{},[2437,25589,2644],{"mathvariant":2643},[2437,25591,3688],{},[2584,25593,2651],{"stretchy":2650},[2584,25595,2714],{"separator":2452},[2584,25597,25598],{},"…",[2584,25600,2714],{"separator":2452},[2434,25602,25603,25605],{},[2437,25604,3181],{},[2431,25606,25607,25609,25611,25613,25615,25617,25619],{},[2584,25608,2586],{},[2437,25610,3188],{},[2437,25612,2111],{},[2437,25614,3033],{},[6062,25616,25583],{},[2437,25618,4854],{},[2437,25620,3193],{},[2437,25622,2644],{"mathvariant":2643},[2437,25624,3688],{},[2584,25626,2651],{"stretchy":2650},[2584,25628,2767],{"fence":2452},[2444,25630,25631],{"encoding":2446},"\\mathcal{K}_U(H, |\\psi\\rangle) = \\text{span}\\left\\{ |\\psi\\rangle, e^{-iH\\,dt} |\\psi\\rangle, \\dots, e^{-irH\\,dt} |\\psi\\rangle \\right\\}",[53,25633,25635,25712],{"className":25634,"ariaHidden":2452},[2451],[53,25636,25638,25641,25681,25684,25687,25690,25693,25696,25699,25703,25706,25709],{"className":25637},[2456],[53,25639],{"className":25640,"style":2664},[2460],[53,25642,25644,25647],{"className":25643},[2465],[53,25645,2440],{"className":25646,"style":2470},[2465,2469],[53,25648,25650],{"className":25649},[2474],[53,25651,25653,25673],{"className":25652},[2478,3095],[53,25654,25656,25670],{"className":25655},[2482],[53,25657,25659],{"className":25658,"style":3102},[2486],[53,25660,25661,25664],{"style":3105},[53,25662],{"className":25663,"style":2495},[2494],[53,25665,25667],{"className":25666},[2499,2500,2501,2502],[53,25668,3174],{"className":25669,"style":3209},[2465,2506,2502],[53,25671,3120],{"className":25672},[3119],[53,25674,25676],{"className":25675},[2482],[53,25677,25679],{"className":25678,"style":3127},[2486],[53,25680],{},[53,25682,128],{"className":25683},[2836],[53,25685,3033],{"className":25686,"style":3048},[2465,2506],[53,25688,2714],{"className":25689},[2856],[53,25691],{"className":25692,"style":2860},[2609],[53,25694,2644],{"className":25695},[2465],[53,25697,3688],{"className":25698,"style":2671},[2465,2506],[53,25700,25702],{"className":25701},[2675],"⟩)",[53,25704],{"className":25705,"style":2814},[2609],[53,25707,121],{"className":25708},[2818],[53,25710],{"className":25711,"style":2814},[2609],[53,25713,25715,25719,25725,25728],{"className":25714},[2456],[53,25716],{"className":25717,"style":25718},[2460],"height:1.2491em;vertical-align:-0.35em;",[53,25720,25722],{"className":25721},[2465,6934],[53,25723,53],{"className":25724},[2465],[53,25726],{"className":25727,"style":2860},[2609],[53,25729,25731,25737,25740,25743,25746,25749,25752,25800,25803,25806,25809,25812,25815,25818,25821,25824,25827,25877,25880,25883,25886],{"className":25730},[2832],[53,25732,25734],{"className":25733,"style":2838},[2836,2837],[53,25735,2705],{"className":25736},[2842,2843],[53,25738,2644],{"className":25739},[2465],[53,25741,3688],{"className":25742,"style":2671},[2465,2506],[53,25744,2651],{"className":25745},[2675],[53,25747,2714],{"className":25748},[2856],[53,25750],{"className":25751,"style":2860},[2609],[53,25753,25755,25758],{"className":25754},[2465],[53,25756,3181],{"className":25757},[2465,2506],[53,25759,25761],{"className":25760},[2474],[53,25762,25764],{"className":25763},[2478],[53,25765,25767],{"className":25766},[2482],[53,25768,25770],{"className":25769,"style":4935},[2486],[53,25771,25772,25775],{"style":2801},[53,25773],{"className":25774,"style":2495},[2494],[53,25776,25778],{"className":25777},[2499,2500,2501,2502],[53,25779,25781,25784,25787,25790,25794,25797],{"className":25780},[2465,2502],[53,25782,2586],{"className":25783},[2465,2502],[53,25785,3188],{"className":25786},[2465,2506,2502],[53,25788,3033],{"className":25789,"style":3048},[2465,2506,2502],[53,25791],{"className":25792,"style":25793},[2609,2502],"margin-right:0.1952em;",[53,25795,4854],{"className":25796},[2465,2506,2502],[53,25798,3193],{"className":25799},[2465,2506,2502],[53,25801,2644],{"className":25802},[2465],[53,25804,3688],{"className":25805,"style":2671},[2465,2506],[53,25807,2651],{"className":25808},[2675],[53,25810,2714],{"className":25811},[2856],[53,25813],{"className":25814,"style":2860},[2609],[53,25816,25598],{"className":25817},[2832],[53,25819],{"className":25820,"style":2860},[2609],[53,25822,2714],{"className":25823},[2856],[53,25825],{"className":25826,"style":2860},[2609],[53,25828,25830,25833],{"className":25829},[2465],[53,25831,3181],{"className":25832},[2465,2506],[53,25834,25836],{"className":25835},[2474],[53,25837,25839],{"className":25838},[2478],[53,25840,25842],{"className":25841},[2482],[53,25843,25845],{"className":25844,"style":4935},[2486],[53,25846,25847,25850],{"style":2801},[53,25848],{"className":25849,"style":2495},[2494],[53,25851,25853],{"className":25852},[2499,2500,2501,2502],[53,25854,25856,25859,25862,25865,25868,25871,25874],{"className":25855},[2465,2502],[53,25857,2586],{"className":25858},[2465,2502],[53,25860,3188],{"className":25861},[2465,2506,2502],[53,25863,2111],{"className":25864,"style":2507},[2465,2506,2502],[53,25866,3033],{"className":25867,"style":3048},[2465,2506,2502],[53,25869],{"className":25870,"style":25793},[2609,2502],[53,25872,4854],{"className":25873},[2465,2506,2502],[53,25875,3193],{"className":25876},[2465,2506,2502],[53,25878,2644],{"className":25879},[2465],[53,25881,3688],{"className":25882,"style":2671},[2465,2506],[53,25884,2651],{"className":25885},[2675],[53,25887,25889],{"className":25888,"style":2838},[2675,2837],[53,25890,2767],{"className":25891},[2842,2843],[12,25893,25894,25895,25928,25929,25957,25958,26017],{},"for some timestep ",[53,25896,25898,25913],{"className":25897},[2418],[53,25899,25901],{"className":25900},[2422],[2424,25902,25903],{"xmlns":2426},[2428,25904,25905,25911],{},[2431,25906,25907,25909],{},[2437,25908,4854],{},[2437,25910,3193],{},[2444,25912,5358],{"encoding":2446},[53,25914,25916],{"className":25915,"ariaHidden":2452},[2451],[53,25917,25919,25922,25925],{"className":25918},[2456],[53,25920],{"className":25921,"style":5329},[2460],[53,25923,4854],{"className":25924},[2465,2506],[53,25926,3193],{"className":25927},[2465,2506]," that we will determine later. Temporarily assume ",[53,25930,25932,25945],{"className":25931},[2418],[53,25933,25935],{"className":25934},[2422],[2424,25936,25937],{"xmlns":2426},[2428,25938,25939,25943],{},[2431,25940,25941],{},[2437,25942,2111],{},[2444,25944,2111],{"encoding":2446},[53,25946,25948],{"className":25947,"ariaHidden":2452},[2451],[53,25949,25951,25954],{"className":25950},[2456],[53,25952],{"className":25953,"style":2534},[2460],[53,25955,2111],{"className":25956,"style":2507},[2465,2506]," is even: then define ",[53,25959,25961,25983],{"className":25960},[2418],[53,25962,25964],{"className":25963},[2422],[2424,25965,25966],{"xmlns":2426},[2428,25967,25968,25980],{},[2431,25969,25970,25972,25974,25976,25978],{},[2437,25971,4854],{},[2584,25973,121],{},[2437,25975,2111],{},[2437,25977,193],{"mathvariant":2643},[2588,25979,132],{},[2444,25981,25982],{"encoding":2446},"d=r\u002F2",[53,25984,25986,26004],{"className":25985,"ariaHidden":2452},[2451],[53,25987,25989,25992,25995,25998,26001],{"className":25988},[2456],[53,25990],{"className":25991,"style":5329},[2460],[53,25993,4854],{"className":25994},[2465,2506],[53,25996],{"className":25997,"style":2814},[2609],[53,25999,121],{"className":26000},[2818],[53,26002],{"className":26003,"style":2814},[2609],[53,26005,26007,26010,26013],{"className":26006},[2456],[53,26008],{"className":26009,"style":2664},[2460],[53,26011,2111],{"className":26012,"style":2507},[2465,2506],[53,26014,26016],{"className":26015},[2465],"\u002F2",". Notice that when we project the Hamiltonian into the Krylov space above, it is indistinguishable from the Krylov space",[53,26019,26021],{"className":26020},[2679],[53,26022,26024,26202],{"className":26023},[2418],[53,26025,26027],{"className":26026},[2422],[2424,26028,26029],{"xmlns":2426,"display":2688},[2428,26030,26031,26199],{},[2431,26032,26033,26039,26041,26043,26045,26047,26049,26051,26053,26055,26057,26197],{},[3063,26034,26035,26037],{},[2437,26036,2440],{"mathvariant":2439},[2437,26038,3174],{},[2584,26040,128],{"stretchy":2650},[2437,26042,3033],{},[2584,26044,2714],{"separator":2452},[2437,26046,2644],{"mathvariant":2643},[2437,26048,3688],{},[2584,26050,2651],{"stretchy":2650},[2584,26052,3544],{"stretchy":2650},[2584,26054,121],{},[6062,26056,53],{},[2431,26058,26059,26061,26083,26085,26087,26089,26091,26117,26119,26121,26123,26125,26127,26129,26157,26159,26161,26163,26165,26189,26191,26193,26195],{},[2584,26060,2705],{"fence":2452},[2434,26062,26063,26065],{},[2437,26064,3181],{},[2431,26066,26067,26069,26071,26073,26075,26077,26079,26081],{},[2437,26068,3188],{},[6062,26070,25583],{},[2437,26072,4854],{},[6062,26074,25583],{},[2437,26076,3033],{},[6062,26078,25583],{},[2437,26080,4854],{},[2437,26082,3193],{},[2437,26084,2644],{"mathvariant":2643},[2437,26086,3688],{},[2584,26088,2651],{"stretchy":2650},[2584,26090,2714],{"separator":2452},[2434,26092,26093,26095],{},[2437,26094,3181],{},[2431,26096,26097,26099,26101,26103,26105,26107,26109,26111,26113,26115],{},[2437,26098,3188],{},[2584,26100,128],{"stretchy":2650},[2437,26102,4854],{},[2584,26104,2586],{},[2588,26106,173],{},[2584,26108,3544],{"stretchy":2650},[2437,26110,3033],{},[6062,26112,25583],{},[2437,26114,4854],{},[2437,26116,3193],{},[2437,26118,2644],{"mathvariant":2643},[2437,26120,3688],{},[2584,26122,2651],{"stretchy":2650},[2584,26124,2714],{"separator":2452},[2584,26126,25598],{},[2584,26128,2714],{"separator":2452},[2434,26130,26131,26133],{},[2437,26132,3181],{},[2431,26134,26135,26137,26139,26141,26143,26145,26147,26149,26151,26153,26155],{},[2584,26136,2586],{},[2437,26138,3188],{},[2584,26140,128],{"stretchy":2650},[2437,26142,4854],{},[2584,26144,2586],{},[2588,26146,173],{},[2584,26148,3544],{"stretchy":2650},[2437,26150,3033],{},[6062,26152,25583],{},[2437,26154,4854],{},[2437,26156,3193],{},[2437,26158,2644],{"mathvariant":2643},[2437,26160,3688],{},[2584,26162,2651],{"stretchy":2650},[2584,26164,2714],{"separator":2452},[2434,26166,26167,26169],{},[2437,26168,3181],{},[2431,26170,26171,26173,26175,26177,26179,26181,26183,26185,26187],{},[2584,26172,2586],{},[2437,26174,3188],{},[6062,26176,25583],{},[2437,26178,4854],{},[6062,26180,25583],{},[2437,26182,3033],{},[6062,26184,25583],{},[2437,26186,4854],{},[2437,26188,3193],{},[2437,26190,2644],{"mathvariant":2643},[2437,26192,3688],{},[2584,26194,2651],{"stretchy":2650},[2584,26196,2767],{"fence":2452},[2584,26198,2714],{"separator":2452},[2444,26200,26201],{"encoding":2446},"\\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\\},",[53,26203,26205,26281],{"className":26204,"ariaHidden":2452},[2451],[53,26206,26208,26211,26251,26254,26257,26260,26263,26266,26269,26272,26275,26278],{"className":26207},[2456],[53,26209],{"className":26210,"style":2664},[2460],[53,26212,26214,26217],{"className":26213},[2465],[53,26215,2440],{"className":26216,"style":2470},[2465,2469],[53,26218,26220],{"className":26219},[2474],[53,26221,26223,26243],{"className":26222},[2478,3095],[53,26224,26226,26240],{"className":26225},[2482],[53,26227,26229],{"className":26228,"style":3102},[2486],[53,26230,26231,26234],{"style":3105},[53,26232],{"className":26233,"style":2495},[2494],[53,26235,26237],{"className":26236},[2499,2500,2501,2502],[53,26238,3174],{"className":26239,"style":3209},[2465,2506,2502],[53,26241,3120],{"className":26242},[3119],[53,26244,26246],{"className":26245},[2482],[53,26247,26249],{"className":26248,"style":3127},[2486],[53,26250],{},[53,26252,128],{"className":26253},[2836],[53,26255,3033],{"className":26256,"style":3048},[2465,2506],[53,26258,2714],{"className":26259},[2856],[53,26261],{"className":26262,"style":2860},[2609],[53,26264,2644],{"className":26265},[2465],[53,26267,3688],{"className":26268,"style":2671},[2465,2506],[53,26270,25702],{"className":26271},[2675],[53,26273],{"className":26274,"style":2814},[2609],[53,26276,121],{"className":26277},[2818],[53,26279],{"className":26280,"style":2814},[2609],[53,26282,26284,26287,26293,26296,26608,26611],{"className":26283},[2456],[53,26285],{"className":26286,"style":13440},[2460],[53,26288,26290],{"className":26289},[2465,6934],[53,26291,53],{"className":26292},[2465],[53,26294],{"className":26295,"style":2860},[2609],[53,26297,26299,26305,26358,26361,26364,26367,26370,26373,26433,26436,26439,26442,26445,26448,26451,26454,26457,26460,26522,26525,26528,26531,26534,26537,26593,26596,26599,26602],{"className":26298},[2832],[53,26300,26302],{"className":26301,"style":2838},[2836,2837],[53,26303,2705],{"className":26304},[2842,9815],[53,26306,26308,26311],{"className":26307},[2465],[53,26309,3181],{"className":26310},[2465,2506],[53,26312,26314],{"className":26313},[2474],[53,26315,26317],{"className":26316},[2478],[53,26318,26320],{"className":26319},[2482],[53,26321,26323],{"className":26322,"style":4935},[2486],[53,26324,26325,26328],{"style":2801},[53,26326],{"className":26327,"style":2495},[2494],[53,26329,26331],{"className":26330},[2499,2500,2501,2502],[53,26332,26334,26337,26340,26343,26346,26349,26352,26355],{"className":26333},[2465,2502],[53,26335,3188],{"className":26336},[2465,2506,2502],[53,26338],{"className":26339,"style":25793},[2609,2502],[53,26341,4854],{"className":26342},[2465,2506,2502],[53,26344],{"className":26345,"style":25793},[2609,2502],[53,26347,3033],{"className":26348,"style":3048},[2465,2506,2502],[53,26350],{"className":26351,"style":25793},[2609,2502],[53,26353,4854],{"className":26354},[2465,2506,2502],[53,26356,3193],{"className":26357},[2465,2506,2502],[53,26359,2644],{"className":26360},[2465],[53,26362,3688],{"className":26363,"style":2671},[2465,2506],[53,26365,2651],{"className":26366},[2675],[53,26368,2714],{"className":26369},[2856],[53,26371],{"className":26372,"style":2860},[2609],[53,26374,26376,26379],{"className":26375},[2465],[53,26377,3181],{"className":26378},[2465,2506],[53,26380,26382],{"className":26381},[2474],[53,26383,26385],{"className":26384},[2478],[53,26386,26388],{"className":26387},[2482],[53,26389,26392],{"className":26390,"style":26391},[2486],"height:0.938em;",[53,26393,26394,26397],{"style":2801},[53,26395],{"className":26396,"style":2495},[2494],[53,26398,26400],{"className":26399},[2499,2500,2501,2502],[53,26401,26403,26406,26409,26412,26415,26418,26421,26424,26427,26430],{"className":26402},[2465,2502],[53,26404,3188],{"className":26405},[2465,2506,2502],[53,26407,128],{"className":26408},[2836,2502],[53,26410,4854],{"className":26411},[2465,2506,2502],[53,26413,2586],{"className":26414},[2614,2502],[53,26416,173],{"className":26417},[2465,2502],[53,26419,3544],{"className":26420},[2675,2502],[53,26422,3033],{"className":26423,"style":3048},[2465,2506,2502],[53,26425],{"className":26426,"style":25793},[2609,2502],[53,26428,4854],{"className":26429},[2465,2506,2502],[53,26431,3193],{"className":26432},[2465,2506,2502],[53,26434,2644],{"className":26435},[2465],[53,26437,3688],{"className":26438,"style":2671},[2465,2506],[53,26440,2651],{"className":26441},[2675],[53,26443,2714],{"className":26444},[2856],[53,26446],{"className":26447,"style":2860},[2609],[53,26449,25598],{"className":26450},[2832],[53,26452],{"className":26453,"style":2860},[2609],[53,26455,2714],{"className":26456},[2856],[53,26458],{"className":26459,"style":2860},[2609],[53,26461,26463,26466],{"className":26462},[2465],[53,26464,3181],{"className":26465},[2465,2506],[53,26467,26469],{"className":26468},[2474],[53,26470,26472],{"className":26471},[2478],[53,26473,26475],{"className":26474},[2482],[53,26476,26478],{"className":26477,"style":26391},[2486],[53,26479,26480,26483],{"style":2801},[53,26481],{"className":26482,"style":2495},[2494],[53,26484,26486],{"className":26485},[2499,2500,2501,2502],[53,26487,26489,26492,26495,26498,26501,26504,26507,26510,26513,26516,26519],{"className":26488},[2465,2502],[53,26490,2586],{"className":26491},[2465,2502],[53,26493,3188],{"className":26494},[2465,2506,2502],[53,26496,128],{"className":26497},[2836,2502],[53,26499,4854],{"className":26500},[2465,2506,2502],[53,26502,2586],{"className":26503},[2614,2502],[53,26505,173],{"className":26506},[2465,2502],[53,26508,3544],{"className":26509},[2675,2502],[53,26511,3033],{"className":26512,"style":3048},[2465,2506,2502],[53,26514],{"className":26515,"style":25793},[2609,2502],[53,26517,4854],{"className":26518},[2465,2506,2502],[53,26520,3193],{"className":26521},[2465,2506,2502],[53,26523,2644],{"className":26524},[2465],[53,26526,3688],{"className":26527,"style":2671},[2465,2506],[53,26529,2651],{"className":26530},[2675],[53,26532,2714],{"className":26533},[2856],[53,26535],{"className":26536,"style":2860},[2609],[53,26538,26540,26543],{"className":26539},[2465],[53,26541,3181],{"className":26542},[2465,2506],[53,26544,26546],{"className":26545},[2474],[53,26547,26549],{"className":26548},[2478],[53,26550,26552],{"className":26551},[2482],[53,26553,26555],{"className":26554,"style":4935},[2486],[53,26556,26557,26560],{"style":2801},[53,26558],{"className":26559,"style":2495},[2494],[53,26561,26563],{"className":26562},[2499,2500,2501,2502],[53,26564,26566,26569,26572,26575,26578,26581,26584,26587,26590],{"className":26565},[2465,2502],[53,26567,2586],{"className":26568},[2465,2502],[53,26570,3188],{"className":26571},[2465,2506,2502],[53,26573],{"className":26574,"style":25793},[2609,2502],[53,26576,4854],{"className":26577},[2465,2506,2502],[53,26579],{"className":26580,"style":25793},[2609,2502],[53,26582,3033],{"className":26583,"style":3048},[2465,2506,2502],[53,26585],{"className":26586,"style":25793},[2609,2502],[53,26588,4854],{"className":26589},[2465,2506,2502],[53,26591,3193],{"className":26592},[2465,2506,2502],[53,26594,2644],{"className":26595},[2465],[53,26597,3688],{"className":26598,"style":2671},[2465,2506],[53,26600,2651],{"className":26601},[2675],[53,26603,26605],{"className":26604,"style":2838},[2675,2837],[53,26606,2767],{"className":26607},[2842,9815],[53,26609],{"className":26610,"style":2860},[2609],[53,26612,2714],{"className":26613},[2856],[12,26615,26616,26617,26645],{},"that is, where all the time-evolutions are shifted backward by ",[53,26618,26620,26633],{"className":26619},[2418],[53,26621,26623],{"className":26622},[2422],[2424,26624,26625],{"xmlns":2426},[2428,26626,26627,26631],{},[2431,26628,26629],{},[2437,26630,4854],{},[2444,26632,4854],{"encoding":2446},[53,26634,26636],{"className":26635,"ariaHidden":2452},[2451],[53,26637,26639,26642],{"className":26638},[2456],[53,26640],{"className":26641,"style":5329},[2460],[53,26643,4854],{"className":26644},[2465,2506]," timesteps.\nThe reason it is indistinguishable is because the matrix elements",[53,26647,26649],{"className":26648},[2679],[53,26650,26652,26784],{"className":26651},[2418],[53,26653,26655],{"className":26654},[2422],[2424,26656,26657],{"xmlns":2426,"display":2688},[2428,26658,26659,26781],{},[2431,26660,26661,26677,26679,26681,26683,26685,26707,26709,26733,26735,26737,26739,26741,26743,26745,26747,26749,26775,26777,26779],{},[3063,26662,26663,26669],{},[4166,26664,26665,26667],{"accent":2452},[2437,26666,3033],{},[2584,26668,4172],{},[2431,26670,26671,26673,26675],{},[2437,26672,2267],{},[2584,26674,2714],{"separator":2452},[2437,26676,3521],{},[2584,26678,121],{},[2584,26680,4320],{"stretchy":2650},[2437,26682,3688],{},[2437,26684,2644],{"mathvariant":2643},[2434,26686,26687,26689],{},[2437,26688,3181],{},[2431,26690,26691,26693,26695,26697,26699,26701,26703,26705],{},[2437,26692,3188],{},[6062,26694,25583],{},[2437,26696,2267],{},[6062,26698,25583],{},[2437,26700,3033],{},[6062,26702,25583],{},[2437,26704,4854],{},[2437,26706,3193],{},[2437,26708,3033],{},[2434,26710,26711,26713],{},[2437,26712,3181],{},[2431,26714,26715,26717,26719,26721,26723,26725,26727,26729,26731],{},[2584,26716,2586],{},[2437,26718,3188],{},[6062,26720,25583],{},[2437,26722,3521],{},[6062,26724,25583],{},[2437,26726,3033],{},[6062,26728,25583],{},[2437,26730,4854],{},[2437,26732,3193],{},[2437,26734,2644],{"mathvariant":2643},[2437,26736,3688],{},[2584,26738,2651],{"stretchy":2650},[2584,26740,121],{},[2584,26742,4320],{"stretchy":2650},[2437,26744,3688],{},[2437,26746,2644],{"mathvariant":2643},[2437,26748,3033],{},[2434,26750,26751,26753],{},[2437,26752,3181],{},[2431,26754,26755,26757,26759,26761,26763,26765,26767,26769,26771,26773],{},[2437,26756,3188],{},[2584,26758,128],{"stretchy":2650},[2437,26760,2267],{},[2584,26762,2586],{},[2437,26764,3521],{},[2584,26766,3544],{"stretchy":2650},[2437,26768,3033],{},[6062,26770,25583],{},[2437,26772,4854],{},[2437,26774,3193],{},[2437,26776,2644],{"mathvariant":2643},[2437,26778,3688],{},[2584,26780,2651],{"stretchy":2650},[2444,26782,26783],{"encoding":2446},"\\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",[53,26785,26787,26879,27024],{"className":26786,"ariaHidden":2452},[2451],[53,26788,26790,26793,26870,26873,26876],{"className":26789},[2456],[53,26791],{"className":26792,"style":6650},[2460],[53,26794,26796,26827],{"className":26795},[2465],[53,26797,26799],{"className":26798},[2465,4189],[53,26800,26802],{"className":26801},[2478],[53,26803,26805],{"className":26804},[2482],[53,26806,26808,26816],{"className":26807,"style":4185},[2486],[53,26809,26810,26813],{"style":4201},[53,26811],{"className":26812,"style":4205},[2494],[53,26814,3033],{"className":26815,"style":3048},[2465,2506],[53,26817,26818,26821],{"style":4211},[53,26819],{"className":26820,"style":4205},[2494],[53,26822,26824],{"className":26823,"style":4219},[4218],[53,26825,4172],{"className":26826},[2465],[53,26828,26830],{"className":26829},[2474],[53,26831,26833,26862],{"className":26832},[2478,3095],[53,26834,26836,26859],{"className":26835},[2482],[53,26837,26839],{"className":26838,"style":11138},[2486],[53,26840,26841,26844],{"style":4405},[53,26842],{"className":26843,"style":2495},[2494],[53,26845,26847],{"className":26846},[2499,2500,2501,2502],[53,26848,26850,26853,26856],{"className":26849},[2465,2502],[53,26851,2267],{"className":26852,"style":6718},[2465,2506,2502],[53,26854,2714],{"className":26855},[2856,2502],[53,26857,3521],{"className":26858,"style":3587},[2465,2506,2502],[53,26860,3120],{"className":26861},[3119],[53,26863,26865],{"className":26864},[2482],[53,26866,26868],{"className":26867,"style":6728},[2486],[53,26869],{},[53,26871],{"className":26872,"style":2814},[2609],[53,26874,121],{"className":26875},[2818],[53,26877],{"className":26878,"style":2814},[2609],[53,26880,26882,26885,26888,26891,26894,26947,26950,27006,27009,27012,27015,27018,27021],{"className":26881},[2456],[53,26883],{"className":26884,"style":4907},[2460],[53,26886,4320],{"className":26887},[2836],[53,26889,3688],{"className":26890,"style":2671},[2465,2506],[53,26892,2644],{"className":26893},[2465],[53,26895,26897,26900],{"className":26896},[2465],[53,26898,3181],{"className":26899},[2465,2506],[53,26901,26903],{"className":26902},[2474],[53,26904,26906],{"className":26905},[2478],[53,26907,26909],{"className":26908},[2482],[53,26910,26912],{"className":26911,"style":4935},[2486],[53,26913,26914,26917],{"style":2801},[53,26915],{"className":26916,"style":2495},[2494],[53,26918,26920],{"className":26919},[2499,2500,2501,2502],[53,26921,26923,26926,26929,26932,26935,26938,26941,26944],{"className":26922},[2465,2502],[53,26924,3188],{"className":26925},[2465,2506,2502],[53,26927],{"className":26928,"style":25793},[2609,2502],[53,26930,2267],{"className":26931,"style":6718},[2465,2506,2502],[53,26933],{"className":26934,"style":25793},[2609,2502],[53,26936,3033],{"className":26937,"style":3048},[2465,2506,2502],[53,26939],{"className":26940,"style":25793},[2609,2502],[53,26942,4854],{"className":26943},[2465,2506,2502],[53,26945,3193],{"className":26946},[2465,2506,2502],[53,26948,3033],{"className":26949,"style":3048},[2465,2506],[53,26951,26953,26956],{"className":26952},[2465],[53,26954,3181],{"className":26955},[2465,2506],[53,26957,26959],{"className":26958},[2474],[53,26960,26962],{"className":26961},[2478],[53,26963,26965],{"className":26964},[2482],[53,26966,26968],{"className":26967,"style":4935},[2486],[53,26969,26970,26973],{"style":2801},[53,26971],{"className":26972,"style":2495},[2494],[53,26974,26976],{"className":26975},[2499,2500,2501,2502],[53,26977,26979,26982,26985,26988,26991,26994,26997,27000,27003],{"className":26978},[2465,2502],[53,26980,2586],{"className":26981},[2465,2502],[53,26983,3188],{"className":26984},[2465,2506,2502],[53,26986],{"className":26987,"style":25793},[2609,2502],[53,26989,3521],{"className":26990,"style":3587},[2465,2506,2502],[53,26992],{"className":26993,"style":25793},[2609,2502],[53,26995,3033],{"className":26996,"style":3048},[2465,2506,2502],[53,26998],{"className":26999,"style":25793},[2609,2502],[53,27001,4854],{"className":27002},[2465,2506,2502],[53,27004,3193],{"className":27005},[2465,2506,2502],[53,27007,2644],{"className":27008},[2465],[53,27010,3688],{"className":27011,"style":2671},[2465,2506],[53,27013,2651],{"className":27014},[2675],[53,27016],{"className":27017,"style":2814},[2609],[53,27019,121],{"className":27020},[2818],[53,27022],{"className":27023,"style":2814},[2609],[53,27025,27027,27031,27034,27037,27040,27043,27102,27105,27108],{"className":27026},[2456],[53,27028],{"className":27029,"style":27030},[2460],"height:1.188em;vertical-align:-0.25em;",[53,27032,4320],{"className":27033},[2836],[53,27035,3688],{"className":27036,"style":2671},[2465,2506],[53,27038,2644],{"className":27039},[2465],[53,27041,3033],{"className":27042,"style":3048},[2465,2506],[53,27044,27046,27049],{"className":27045},[2465],[53,27047,3181],{"className":27048},[2465,2506],[53,27050,27052],{"className":27051},[2474],[53,27053,27055],{"className":27054},[2478],[53,27056,27058],{"className":27057},[2482],[53,27059,27061],{"className":27060,"style":26391},[2486],[53,27062,27063,27066],{"style":2801},[53,27064],{"className":27065,"style":2495},[2494],[53,27067,27069],{"className":27068},[2499,2500,2501,2502],[53,27070,27072,27075,27078,27081,27084,27087,27090,27093,27096,27099],{"className":27071},[2465,2502],[53,27073,3188],{"className":27074},[2465,2506,2502],[53,27076,128],{"className":27077},[2836,2502],[53,27079,2267],{"className":27080,"style":6718},[2465,2506,2502],[53,27082,2586],{"className":27083},[2614,2502],[53,27085,3521],{"className":27086,"style":3587},[2465,2506,2502],[53,27088,3544],{"className":27089},[2675,2502],[53,27091,3033],{"className":27092,"style":3048},[2465,2506,2502],[53,27094],{"className":27095,"style":25793},[2609,2502],[53,27097,4854],{"className":27098},[2465,2506,2502],[53,27100,3193],{"className":27101},[2465,2506,2502],[53,27103,2644],{"className":27104},[2465],[53,27106,3688],{"className":27107,"style":2671},[2465,2506],[53,27109,2651],{"className":27110},[2675],[12,27112,27113,27114,27142,27143,28],{},"are invariant under overall shifts of the evolution time, since the time-evolutions commute with the Hamiltonian. For odd ",[53,27115,27117,27130],{"className":27116},[2418],[53,27118,27120],{"className":27119},[2422],[2424,27121,27122],{"xmlns":2426},[2428,27123,27124,27128],{},[2431,27125,27126],{},[2437,27127,2111],{},[2444,27129,2111],{"encoding":2446},[53,27131,27133],{"className":27132,"ariaHidden":2452},[2451],[53,27134,27136,27139],{"className":27135},[2456],[53,27137],{"className":27138,"style":2534},[2460],[53,27140,2111],{"className":27141,"style":2507},[2465,2506],", we can use the analysis for ",[53,27144,27146,27163],{"className":27145},[2418],[53,27147,27149],{"className":27148},[2422],[2424,27150,27151],{"xmlns":2426},[2428,27152,27153,27161],{},[2431,27154,27155,27157,27159],{},[2437,27156,2111],{},[2584,27158,2586],{},[2588,27160,173],{},[2444,27162,2592],{"encoding":2446},[53,27164,27166,27184],{"className":27165,"ariaHidden":2452},[2451],[53,27167,27169,27172,27175,27178,27181],{"className":27168},[2456],[53,27170],{"className":27171,"style":2602},[2460],[53,27173,2111],{"className":27174,"style":2507},[2465,2506],[53,27176],{"className":27177,"style":2610},[2609],[53,27179,2586],{"className":27180},[2614],[53,27182],{"className":27183,"style":2610},[2609],[53,27185,27187,27190],{"className":27186},[2456],[53,27188],{"className":27189,"style":2624},[2460],[53,27191,173],{"className":27192},[2465],[12,27194,27195,27196,24880],{},"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 ",[19,27197,27198],{"href":979},"[3]",[12,27200,27201,27205,27206,27235,27236,27264],{},[27202,27203,27204],"strong",{},"Claim 1:"," there exists a function ",[53,27207,27209,27222],{"className":27208},[2418],[53,27210,27212],{"className":27211},[2422],[2424,27213,27214],{"xmlns":2426},[2428,27215,27216,27220],{},[2431,27217,27218],{},[2437,27219,22173],{},[2444,27221,22173],{"encoding":2446},[53,27223,27225],{"className":27224,"ariaHidden":2452},[2451],[53,27226,27228,27231],{"className":27227},[2456],[53,27229],{"className":27230,"style":6930},[2460],[53,27232,22173],{"className":27233,"style":27234},[2465,2506],"margin-right:0.1076em;"," such that for energies ",[53,27237,27239,27252],{"className":27238},[2418],[53,27240,27242],{"className":27241},[2422],[2424,27243,27244],{"xmlns":2426},[2428,27245,27246,27250],{},[2431,27247,27248],{},[2437,27249,6077],{},[2444,27251,6077],{"encoding":2446},[53,27253,27255],{"className":27254,"ariaHidden":2452},[2451],[53,27256,27258,27261],{"className":27257},[2456],[53,27259],{"className":27260,"style":2461},[2460],[53,27262,6077],{"className":27263,"style":5787},[2465,2506]," in the spectral range of the Hamiltonian (that is, between the ground state energy and the maximum energy)...",[27266,27267,27268,27377,27685],"ol",{},[998,27269,27270],{},[53,27271,27273,27301],{"className":27272},[2418],[53,27274,27276],{"className":27275},[2422],[2424,27277,27278],{"xmlns":2426},[2428,27279,27280,27298],{},[2431,27281,27282,27284,27286,27292,27294,27296],{},[2437,27283,22173],{},[2584,27285,128],{"stretchy":2650},[3063,27287,27288,27290],{},[2437,27289,6077],{},[2588,27291,149],{},[2584,27293,3544],{"stretchy":2650},[2584,27295,121],{},[2588,27297,173],{},[2444,27299,27300],{"encoding":2446},"f(E_0)=1",[53,27302,27304,27368],{"className":27303,"ariaHidden":2452},[2451],[53,27305,27307,27310,27313,27316,27356,27359,27362,27365],{"className":27306},[2456],[53,27308],{"className":27309,"style":2664},[2460],[53,27311,22173],{"className":27312,"style":27234},[2465,2506],[53,27314,128],{"className":27315},[2836],[53,27317,27319,27322],{"className":27318},[2465],[53,27320,6077],{"className":27321,"style":5787},[2465,2506],[53,27323,27325],{"className":27324},[2474],[53,27326,27328,27348],{"className":27327},[2478,3095],[53,27329,27331,27345],{"className":27330},[2482],[53,27332,27334],{"className":27333,"style":14176},[2486],[53,27335,27336,27339],{"style":5909},[53,27337],{"className":27338,"style":2495},[2494],[53,27340,27342],{"className":27341},[2499,2500,2501,2502],[53,27343,149],{"className":27344},[2465,2502],[53,27346,3120],{"className":27347},[3119],[53,27349,27351],{"className":27350},[2482],[53,27352,27354],{"className":27353,"style":3127},[2486],[53,27355],{},[53,27357,3544],{"className":27358},[2675],[53,27360],{"className":27361,"style":2814},[2609],[53,27363,121],{"className":27364},[2818],[53,27366],{"className":27367,"style":2814},[2609],[53,27369,27371,27374],{"className":27370},[2456],[53,27372],{"className":27373,"style":2624},[2460],[53,27375,173],{"className":27376},[2465],[998,27378,27379,27539,27540,27568,27569,27614,27615,27684],{},[53,27380,27382,27432],{"className":27381},[2418],[53,27383,27385],{"className":27384},[2422],[2424,27386,27387],{"xmlns":2426},[2428,27388,27389,27429],{},[2431,27390,27391,27393,27395,27397,27399,27401,27403,27406,27408],{},[2437,27392,2644],{"mathvariant":2643},[2437,27394,22173],{},[2584,27396,128],{"stretchy":2650},[2437,27398,6077],{},[2584,27400,3544],{"stretchy":2650},[2437,27402,2644],{"mathvariant":2643},[2584,27404,27405],{},"≤",[2588,27407,132],{},[2434,27409,27410,27423],{},[2431,27411,27412,27414,27416,27418,27421],{},[2584,27413,128],{"fence":2452},[2588,27415,173],{},[2584,27417,2002],{},[2437,27419,27420],{},"δ",[2584,27422,3544],{"fence":2452},[2431,27424,27425,27427],{},[2584,27426,2586],{},[2437,27428,4854],{},[2444,27430,27431],{"encoding":2446},"|f(E)|\\le2\\left(1 + \\delta\\right)^{-d}",[53,27433,27435,27468],{"className":27434,"ariaHidden":2452},[2451],[53,27436,27438,27441,27444,27447,27450,27453,27456,27459,27462,27465],{"className":27437},[2456],[53,27439],{"className":27440,"style":2664},[2460],[53,27442,2644],{"className":27443},[2465],[53,27445,22173],{"className":27446,"style":27234},[2465,2506],[53,27448,128],{"className":27449},[2836],[53,27451,6077],{"className":27452,"style":5787},[2465,2506],[53,27454,3544],{"className":27455},[2675],[53,27457,2644],{"className":27458},[2465],[53,27460],{"className":27461,"style":2814},[2609],[53,27463,27405],{"className":27464},[2818],[53,27466],{"className":27467,"style":2814},[2609],[53,27469,27471,27475,27478,27481],{"className":27470},[2456],[53,27472],{"className":27473,"style":27474},[2460],"height:1.239em;vertical-align:-0.25em;",[53,27476,132],{"className":27477},[2465],[53,27479],{"className":27480,"style":2860},[2609],[53,27482,27484,27509],{"className":27483},[2832],[53,27485,27487,27490,27493,27496,27499,27502,27506],{"className":27486},[2832],[53,27488,128],{"className":27489,"style":2838},[2836,2837],[53,27491,173],{"className":27492},[2465],[53,27494],{"className":27495,"style":2610},[2609],[53,27497,2002],{"className":27498},[2614],[53,27500],{"className":27501,"style":2610},[2609],[53,27503,27420],{"className":27504,"style":27505},[2465,2506],"margin-right:0.0379em;",[53,27507,3544],{"className":27508,"style":2838},[2675,2837],[53,27510,27512],{"className":27511},[2474],[53,27513,27515],{"className":27514},[2478],[53,27516,27518],{"className":27517},[2482],[53,27519,27522],{"className":27520,"style":27521},[2486],"height:0.989em;",[53,27523,27524,27527],{"style":7740},[53,27525],{"className":27526,"style":2495},[2494],[53,27528,27530],{"className":27529},[2499,2500,2501,2502],[53,27531,27533,27536],{"className":27532},[2465,2502],[53,27534,2586],{"className":27535},[2465,2502],[53,27537,4854],{"className":27538},[2465,2506,2502]," for all values of ",[53,27541,27543,27556],{"className":27542},[2418],[53,27544,27546],{"className":27545},[2422],[2424,27547,27548],{"xmlns":2426},[2428,27549,27550,27554],{},[2431,27551,27552],{},[2437,27553,6077],{},[2444,27555,6077],{"encoding":2446},[53,27557,27559],{"className":27558,"ariaHidden":2452},[2451],[53,27560,27562,27565],{"className":27561},[2456],[53,27563],{"className":27564,"style":2461},[2460],[53,27566,6077],{"className":27567,"style":5787},[2465,2506]," that lie ",[53,27570,27572,27589],{"className":27571},[2418],[53,27573,27575],{"className":27574},[2422],[2424,27576,27577],{"xmlns":2426},[2428,27578,27579,27586],{},[2431,27580,27581,27584],{},[2584,27582,27583],{},"≥",[2437,27585,27420],{},[2444,27587,27588],{"encoding":2446},"\\ge\\delta",[53,27590,27592,27605],{"className":27591,"ariaHidden":2452},[2451],[53,27593,27595,27599,27602],{"className":27594},[2456],[53,27596],{"className":27597,"style":27598},[2460],"height:0.7719em;vertical-align:-0.136em;",[53,27600,27583],{"className":27601},[2818],[53,27603],{"className":27604,"style":2814},[2609],[53,27606,27608,27611],{"className":27607},[2456],[53,27609],{"className":27610,"style":5329},[2460],[53,27612,27420],{"className":27613,"style":27505},[2465,2506]," away from ",[53,27616,27618,27635],{"className":27617},[2418],[53,27619,27621],{"className":27620},[2422],[2424,27622,27623],{"xmlns":2426},[2428,27624,27625,27633],{},[2431,27626,27627],{},[3063,27628,27629,27631],{},[2437,27630,6077],{},[2588,27632,149],{},[2444,27634,14148],{"encoding":2446},[53,27636,27638],{"className":27637,"ariaHidden":2452},[2451],[53,27639,27641,27644],{"className":27640},[2456],[53,27642],{"className":27643,"style":3082},[2460],[53,27645,27647,27650],{"className":27646},[2465],[53,27648,6077],{"className":27649,"style":5787},[2465,2506],[53,27651,27653],{"className":27652},[2474],[53,27654,27656,27676],{"className":27655},[2478,3095],[53,27657,27659,27673],{"className":27658},[2482],[53,27660,27662],{"className":27661,"style":14176},[2486],[53,27663,27664,27667],{"style":5909},[53,27665],{"className":27666,"style":2495},[2494],[53,27668,27670],{"className":27669},[2499,2500,2501,2502],[53,27671,149],{"className":27672},[2465,2502],[53,27674,3120],{"className":27675},[3119],[53,27677,27679],{"className":27678},[2482],[53,27680,27682],{"className":27681,"style":3127},[2486],[53,27683],{},", that is, it is exponentially suppressed",[998,27686,27687,27731,27732,27819,27820],{},[53,27688,27690,27710],{"className":27689},[2418],[53,27691,27693],{"className":27692},[2422],[2424,27694,27695],{"xmlns":2426},[2428,27696,27697,27707],{},[2431,27698,27699,27701,27703,27705],{},[2437,27700,22173],{},[2584,27702,128],{"stretchy":2650},[2437,27704,6077],{},[2584,27706,3544],{"stretchy":2650},[2444,27708,27709],{"encoding":2446},"f(E)",[53,27711,27713],{"className":27712,"ariaHidden":2452},[2451],[53,27714,27716,27719,27722,27725,27728],{"className":27715},[2456],[53,27717],{"className":27718,"style":2664},[2460],[53,27720,22173],{"className":27721,"style":27234},[2465,2506],[53,27723,128],{"className":27724},[2836],[53,27726,6077],{"className":27727,"style":5787},[2465,2506],[53,27729,3544],{"className":27730},[2675]," is a linear combination of ",[53,27733,27735,27765],{"className":27734},[2418],[53,27736,27738],{"className":27737},[2422],[2424,27739,27740],{"xmlns":2426},[2428,27741,27742,27762],{},[2431,27743,27744],{},[2434,27745,27746,27748],{},[2437,27747,3181],{},[2431,27749,27750,27752,27754,27756,27758,27760],{},[2437,27751,3188],{},[2437,27753,2267],{},[2437,27755,6077],{},[6062,27757,25583],{},[2437,27759,4854],{},[2437,27761,3193],{},[2444,27763,27764],{"encoding":2446},"e^{ijE\\,dt}",[53,27766,27768],{"className":27767,"ariaHidden":2452},[2451],[53,27769,27771,27774],{"className":27770},[2456],[53,27772],{"className":27773,"style":3557},[2460],[53,27775,27777,27780],{"className":27776},[2465],[53,27778,3181],{"className":27779},[2465,2506],[53,27781,27783],{"className":27782},[2474],[53,27784,27786],{"className":27785},[2478],[53,27787,27789],{"className":27788},[2482],[53,27790,27792],{"className":27791,"style":3557},[2486],[53,27793,27794,27797],{"style":2490},[53,27795],{"className":27796,"style":2495},[2494],[53,27798,27800],{"className":27799},[2499,2500,2501,2502],[53,27801,27803,27807,27810,27813,27816],{"className":27802},[2465,2502],[53,27804,27806],{"className":27805,"style":6718},[2465,2506,2502],"ij",[53,27808,6077],{"className":27809,"style":5787},[2465,2506,2502],[53,27811],{"className":27812,"style":25793},[2609,2502],[53,27814,4854],{"className":27815},[2465,2506,2502],[53,27817,3193],{"className":27818},[2465,2506,2502]," for ",[53,27821,27823,27873],{"className":27822},[2418],[53,27824,27826],{"className":27825},[2422],[2424,27827,27828],{"xmlns":2426},[2428,27829,27830,27870],{},[2431,27831,27832,27834,27836,27838,27840,27842,27844,27846,27848,27850,27852,27854,27856,27858,27860,27862,27864,27866,27868],{},[2437,27833,2267],{},[2584,27835,121],{},[2584,27837,2586],{},[2437,27839,4854],{},[2584,27841,2714],{"separator":2452},[2584,27843,2586],{},[2437,27845,4854],{},[2584,27847,2002],{},[2588,27849,173],{},[2584,27851,2714],{"separator":2452},[2437,27853,28],{"mathvariant":2643},[2437,27855,28],{"mathvariant":2643},[2437,27857,28],{"mathvariant":2643},[2584,27859,2714],{"separator":2452},[2437,27861,4854],{},[2584,27863,2586],{},[2588,27865,173],{},[2584,27867,2714],{"separator":2452},[2437,27869,4854],{},[2444,27871,27872],{"encoding":2446},"j=-d,-d+1,...,d-1,d",[53,27874,27876,27895,27928,27964],{"className":27875,"ariaHidden":2452},[2451],[53,27877,27879,27883,27886,27889,27892],{"className":27878},[2456],[53,27880],{"className":27881,"style":27882},[2460],"height:0.854em;vertical-align:-0.1944em;",[53,27884,2267],{"className":27885,"style":6718},[2465,2506],[53,27887],{"className":27888,"style":2814},[2609],[53,27890,121],{"className":27891},[2818],[53,27893],{"className":27894,"style":2814},[2609],[53,27896,27898,27901,27904,27907,27910,27913,27916,27919,27922,27925],{"className":27897},[2456],[53,27899],{"className":27900,"style":6930},[2460],[53,27902,2586],{"className":27903},[2465],[53,27905,4854],{"className":27906},[2465,2506],[53,27908,2714],{"className":27909},[2856],[53,27911],{"className":27912,"style":2860},[2609],[53,27914,2586],{"className":27915},[2465],[53,27917,4854],{"className":27918},[2465,2506],[53,27920],{"className":27921,"style":2610},[2609],[53,27923,2002],{"className":27924},[2614],[53,27926],{"className":27927,"style":2610},[2609],[53,27929,27931,27934,27937,27940,27943,27946,27949,27952,27955,27958,27961],{"className":27930},[2456],[53,27932],{"className":27933,"style":6930},[2460],[53,27935,173],{"className":27936},[2465],[53,27938,2714],{"className":27939},[2856],[53,27941],{"className":27942,"style":2860},[2609],[53,27944,2927],{"className":27945},[2465],[53,27947,2714],{"className":27948},[2856],[53,27950],{"className":27951,"style":2860},[2609],[53,27953,4854],{"className":27954},[2465,2506],[53,27956],{"className":27957,"style":2610},[2609],[53,27959,2586],{"className":27960},[2614],[53,27962],{"className":27963,"style":2610},[2609],[53,27965,27967,27970,27973,27976,27979],{"className":27966},[2456],[53,27968],{"className":27969,"style":6930},[2460],[53,27971,173],{"className":27972},[2465],[53,27974,2714],{"className":27975},[2856],[53,27977],{"className":27978,"style":2860},[2609],[53,27980,4854],{"className":27981},[2465,2506],[12,27983,27984,27985,28044,28045,28084],{},"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 ",[53,27986,27988,28014],{"className":27987},[2418],[53,27989,27991],{"className":27990},[2422],[2424,27992,27993],{"xmlns":2426},[2428,27994,27995,28011],{},[2431,27996,27997,27999,28001,28003,28005,28007,28009],{},[2437,27998,22173],{},[2584,28000,128],{"stretchy":2650},[2437,28002,3033],{},[2584,28004,3544],{"stretchy":2650},[2437,28006,2644],{"mathvariant":2643},[2437,28008,3688],{},[2584,28010,2651],{"stretchy":2650},[2444,28012,28013],{"encoding":2446},"f(H)|\\psi\\rangle",[53,28015,28017],{"className":28016,"ariaHidden":2452},[2451],[53,28018,28020,28023,28026,28029,28032,28035,28038,28041],{"className":28019},[2456],[53,28021],{"className":28022,"style":2664},[2460],[53,28024,22173],{"className":28025,"style":27234},[2465,2506],[53,28027,128],{"className":28028},[2836],[53,28030,3033],{"className":28031,"style":3048},[2465,2506],[53,28033,3544],{"className":28034},[2675],[53,28036,2644],{"className":28037},[2465],[53,28039,3688],{"className":28040,"style":2671},[2465,2506],[53,28042,2651],{"className":28043},[2675],". This is our low-energy state. To see why, write ",[53,28046,28048,28066],{"className":28047},[2418],[53,28049,28051],{"className":28050},[2422],[2424,28052,28053],{"xmlns":2426},[2428,28054,28055,28063],{},[2431,28056,28057,28059,28061],{},[2437,28058,2644],{"mathvariant":2643},[2437,28060,3688],{},[2584,28062,2651],{"stretchy":2650},[2444,28064,28065],{"encoding":2446},"|\\psi\\rangle",[53,28067,28069],{"className":28068,"ariaHidden":2452},[2451],[53,28070,28072,28075,28078,28081],{"className":28071},[2456],[53,28073],{"className":28074,"style":2664},[2460],[53,28076,2644],{"className":28077},[2465],[53,28079,3688],{"className":28080,"style":2671},[2465,2506],[53,28082,2651],{"className":28083},[2675]," in the energy eigenbasis:",[53,28086,28088],{"className":28087},[2679],[53,28089,28091,28145],{"className":28090},[2418],[53,28092,28094],{"className":28093},[2422],[2424,28095,28096],{"xmlns":2426,"display":2688},[2428,28097,28098,28142],{},[2431,28099,28100,28102,28104,28106,28108,28123,28130,28132,28138,28140],{},[2437,28101,2644],{"mathvariant":2643},[2437,28103,3688],{},[2584,28105,2651],{"stretchy":2650},[2584,28107,121],{},[28109,28110,28111,28113,28121],"munderover",{},[2584,28112,7422],{},[2431,28114,28115,28117,28119],{},[2437,28116,3521],{},[2584,28118,121],{},[2588,28120,149],{},[2437,28122,7576],{},[3063,28124,28125,28128],{},[2437,28126,28127],{},"γ",[2437,28129,3521],{},[2437,28131,2644],{"mathvariant":2643},[3063,28133,28134,28136],{},[2437,28135,6077],{},[2437,28137,3521],{},[2584,28139,2651],{"stretchy":2650},[2584,28141,2714],{"separator":2452},[2444,28143,28144],{"encoding":2446},"|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_k|E_k\\rangle,",[53,28146,28148,28172],{"className":28147,"ariaHidden":2452},[2451],[53,28149,28151,28154,28157,28160,28163,28166,28169],{"className":28150},[2456],[53,28152],{"className":28153,"style":2664},[2460],[53,28155,2644],{"className":28156},[2465],[53,28158,3688],{"className":28159,"style":2671},[2465,2506],[53,28161,2651],{"className":28162},[2675],[53,28164],{"className":28165,"style":2814},[2609],[53,28167,121],{"className":28168},[2818],[53,28170],{"className":28171,"style":2814},[2609],[53,28173,28175,28179,28254,28257,28299,28302,28342,28345],{"className":28174},[2456],[53,28176],{"className":28177,"style":28178},[2460],"height:3.1304em;vertical-align:-1.3021em;",[53,28180,28183],{"className":28181},[7465,28182],"op-limits",[53,28184,28186,28245],{"className":28185},[2478,3095],[53,28187,28189,28242],{"className":28188},[2482],[53,28190,28193,28215,28227],{"className":28191,"style":28192},[2486],"height:1.8283em;",[53,28194,28196,28200],{"style":28195},"top:-1.8479em;margin-left:0em;",[53,28197],{"className":28198,"style":28199},[2494],"height:3.05em;",[53,28201,28203],{"className":28202},[2499,2500,2501,2502],[53,28204,28206,28209,28212],{"className":28205},[2465,2502],[53,28207,3521],{"className":28208,"style":3587},[2465,2506,2502],[53,28210,121],{"className":28211},[2818,2502],[53,28213,149],{"className":28214},[2465,2502],[53,28216,28218,28221],{"style":28217},"top:-3.05em;",[53,28219],{"className":28220,"style":28199},[2494],[53,28222,28223],{},[53,28224,7422],{"className":28225},[7465,7469,28226],"large-op",[53,28228,28230,28233],{"style":28229},"top:-4.3em;margin-left:0em;",[53,28231],{"className":28232,"style":28199},[2494],[53,28234,28236],{"className":28235},[2499,2500,2501,2502],[53,28237,28239],{"className":28238},[2465,2502],[53,28240,7576],{"className":28241,"style":3209},[2465,2506,2502],[53,28243,3120],{"className":28244},[3119],[53,28246,28248],{"className":28247},[2482],[53,28249,28252],{"className":28250,"style":28251},[2486],"height:1.3021em;",[53,28253],{},[53,28255],{"className":28256,"style":2860},[2609],[53,28258,28260,28264],{"className":28259},[2465],[53,28261,28127],{"className":28262,"style":28263},[2465,2506],"margin-right:0.0556em;",[53,28265,28267],{"className":28266},[2474],[53,28268,28270,28291],{"className":28269},[2478,3095],[53,28271,28273,28288],{"className":28272},[2482],[53,28274,28276],{"className":28275,"style":11138},[2486],[53,28277,28279,28282],{"style":28278},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[53,28280],{"className":28281,"style":2495},[2494],[53,28283,28285],{"className":28284},[2499,2500,2501,2502],[53,28286,3521],{"className":28287,"style":3587},[2465,2506,2502],[53,28289,3120],{"className":28290},[3119],[53,28292,28294],{"className":28293},[2482],[53,28295,28297],{"className":28296,"style":3127},[2486],[53,28298],{},[53,28300,2644],{"className":28301},[2465],[53,28303,28305,28308],{"className":28304},[2465],[53,28306,6077],{"className":28307,"style":5787},[2465,2506],[53,28309,28311],{"className":28310},[2474],[53,28312,28314,28334],{"className":28313},[2478,3095],[53,28315,28317,28331],{"className":28316},[2482],[53,28318,28320],{"className":28319,"style":11138},[2486],[53,28321,28322,28325],{"style":5909},[53,28323],{"className":28324,"style":2495},[2494],[53,28326,28328],{"className":28327},[2499,2500,2501,2502],[53,28329,3521],{"className":28330,"style":3587},[2465,2506,2502],[53,28332,3120],{"className":28333},[3119],[53,28335,28337],{"className":28336},[2482],[53,28338,28340],{"className":28339,"style":3127},[2486],[53,28341],{},[53,28343,2651],{"className":28344},[2675],[53,28346,2714],{"className":28347},[2856],[12,28349,28350,28351,28431,28432,28503,28504,28542,28543,28601],{},"where ",[53,28352,28354,28376],{"className":28353},[2418],[53,28355,28357],{"className":28356},[2422],[2424,28358,28359],{"xmlns":2426},[2428,28360,28361,28373],{},[2431,28362,28363,28365,28371],{},[2437,28364,2644],{"mathvariant":2643},[3063,28366,28367,28369],{},[2437,28368,6077],{},[2437,28370,3521],{},[2584,28372,2651],{"stretchy":2650},[2444,28374,28375],{"encoding":2446},"|E_k\\rangle",[53,28377,28379],{"className":28378,"ariaHidden":2452},[2451],[53,28380,28382,28385,28388,28428],{"className":28381},[2456],[53,28383],{"className":28384,"style":2664},[2460],[53,28386,2644],{"className":28387},[2465],[53,28389,28391,28394],{"className":28390},[2465],[53,28392,6077],{"className":28393,"style":5787},[2465,2506],[53,28395,28397],{"className":28396},[2474],[53,28398,28400,28420],{"className":28399},[2478,3095],[53,28401,28403,28417],{"className":28402},[2482],[53,28404,28406],{"className":28405,"style":11138},[2486],[53,28407,28408,28411],{"style":5909},[53,28409],{"className":28410,"style":2495},[2494],[53,28412,28414],{"className":28413},[2499,2500,2501,2502],[53,28415,3521],{"className":28416,"style":3587},[2465,2506,2502],[53,28418,3120],{"className":28419},[3119],[53,28421,28423],{"className":28422},[2482],[53,28424,28426],{"className":28425,"style":3127},[2486],[53,28427],{},[53,28429,2651],{"className":28430},[2675]," is the kth energy eigenstate and ",[53,28433,28435,28453],{"className":28434},[2418],[53,28436,28438],{"className":28437},[2422],[2424,28439,28440],{"xmlns":2426},[2428,28441,28442,28450],{},[2431,28443,28444],{},[3063,28445,28446,28448],{},[2437,28447,28127],{},[2437,28449,3521],{},[2444,28451,28452],{"encoding":2446},"\\gamma_k",[53,28454,28456],{"className":28455,"ariaHidden":2452},[2451],[53,28457,28459,28463],{"className":28458},[2456],[53,28460],{"className":28461,"style":28462},[2460],"height:0.625em;vertical-align:-0.1944em;",[53,28464,28466,28469],{"className":28465},[2465],[53,28467,28127],{"className":28468,"style":28263},[2465,2506],[53,28470,28472],{"className":28471},[2474],[53,28473,28475,28495],{"className":28474},[2478,3095],[53,28476,28478,28492],{"className":28477},[2482],[53,28479,28481],{"className":28480,"style":11138},[2486],[53,28482,28483,28486],{"style":28278},[53,28484],{"className":28485,"style":2495},[2494],[53,28487,28489],{"className":28488},[2499,2500,2501,2502],[53,28490,3521],{"className":28491,"style":3587},[2465,2506,2502],[53,28493,3120],{"className":28494},[3119],[53,28496,28498],{"className":28497},[2482],[53,28499,28501],{"className":28500,"style":3127},[2486],[53,28502],{}," is its amplitude in the initial state ",[53,28505,28507,28524],{"className":28506},[2418],[53,28508,28510],{"className":28509},[2422],[2424,28511,28512],{"xmlns":2426},[2428,28513,28514,28522],{},[2431,28515,28516,28518,28520],{},[2437,28517,2644],{"mathvariant":2643},[2437,28519,3688],{},[2584,28521,2651],{"stretchy":2650},[2444,28523,28065],{"encoding":2446},[53,28525,28527],{"className":28526,"ariaHidden":2452},[2451],[53,28528,28530,28533,28536,28539],{"className":28529},[2456],[53,28531],{"className":28532,"style":2664},[2460],[53,28534,2644],{"className":28535},[2465],[53,28537,3688],{"className":28538,"style":2671},[2465,2506],[53,28540,2651],{"className":28541},[2675],". Expressed in terms of this, ",[53,28544,28546,28571],{"className":28545},[2418],[53,28547,28549],{"className":28548},[2422],[2424,28550,28551],{"xmlns":2426},[2428,28552,28553,28569],{},[2431,28554,28555,28557,28559,28561,28563,28565,28567],{},[2437,28556,22173],{},[2584,28558,128],{"stretchy":2650},[2437,28560,3033],{},[2584,28562,3544],{"stretchy":2650},[2437,28564,2644],{"mathvariant":2643},[2437,28566,3688],{},[2584,28568,2651],{"stretchy":2650},[2444,28570,28013],{"encoding":2446},[53,28572,28574],{"className":28573,"ariaHidden":2452},[2451],[53,28575,28577,28580,28583,28586,28589,28592,28595,28598],{"className":28576},[2456],[53,28578],{"className":28579,"style":2664},[2460],[53,28581,22173],{"className":28582,"style":27234},[2465,2506],[53,28584,128],{"className":28585},[2836],[53,28587,3033],{"className":28588,"style":3048},[2465,2506],[53,28590,3544],{"className":28591},[2675],[53,28593,2644],{"className":28594},[2465],[53,28596,3688],{"className":28597,"style":2671},[2465,2506],[53,28599,2651],{"className":28600},[2675]," is given by",[53,28603,28605],{"className":28604},[2679],[53,28606,28608,28680],{"className":28607},[2418],[53,28609,28611],{"className":28610},[2422],[2424,28612,28613],{"xmlns":2426,"display":2688},[2428,28614,28615,28677],{},[2431,28616,28617,28619,28621,28623,28625,28627,28629,28631,28633,28647,28653,28655,28657,28663,28665,28667,28673,28675],{},[2437,28618,22173],{},[2584,28620,128],{"stretchy":2650},[2437,28622,3033],{},[2584,28624,3544],{"stretchy":2650},[2437,28626,2644],{"mathvariant":2643},[2437,28628,3688],{},[2584,28630,2651],{"stretchy":2650},[2584,28632,121],{},[28109,28634,28635,28637,28645],{},[2584,28636,7422],{},[2431,28638,28639,28641,28643],{},[2437,28640,3521],{},[2584,28642,121],{},[2588,28644,149],{},[2437,28646,7576],{},[3063,28648,28649,28651],{},[2437,28650,28127],{},[2437,28652,3521],{},[2437,28654,22173],{},[2584,28656,128],{"stretchy":2650},[3063,28658,28659,28661],{},[2437,28660,6077],{},[2437,28662,3521],{},[2584,28664,3544],{"stretchy":2650},[2437,28666,2644],{"mathvariant":2643},[3063,28668,28669,28671],{},[2437,28670,6077],{},[2437,28672,3521],{},[2584,28674,2651],{"stretchy":2650},[2584,28676,2714],{"separator":2452},[2444,28678,28679],{"encoding":2446},"f(H)|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_kf(E_k)|E_k\\rangle,",[53,28681,28683,28719],{"className":28682,"ariaHidden":2452},[2451],[53,28684,28686,28689,28692,28695,28698,28701,28704,28707,28710,28713,28716],{"className":28685},[2456],[53,28687],{"className":28688,"style":2664},[2460],[53,28690,22173],{"className":28691,"style":27234},[2465,2506],[53,28693,128],{"className":28694},[2836],[53,28696,3033],{"className":28697,"style":3048},[2465,2506],[53,28699,3544],{"className":28700},[2675],[53,28702,2644],{"className":28703},[2465],[53,28705,3688],{"className":28706,"style":2671},[2465,2506],[53,28708,2651],{"className":28709},[2675],[53,28711],{"className":28712,"style":2814},[2609],[53,28714,121],{"className":28715},[2818],[53,28717],{"className":28718,"style":2814},[2609],[53,28720,28722,28725,28792,28795,28835,28838,28841,28881,28884,28887,28927,28930],{"className":28721},[2456],[53,28723],{"className":28724,"style":28178},[2460],[53,28726,28728],{"className":28727},[7465,28182],[53,28729,28731,28784],{"className":28730},[2478,3095],[53,28732,28734,28781],{"className":28733},[2482],[53,28735,28737,28757,28767],{"className":28736,"style":28192},[2486],[53,28738,28739,28742],{"style":28195},[53,28740],{"className":28741,"style":28199},[2494],[53,28743,28745],{"className":28744},[2499,2500,2501,2502],[53,28746,28748,28751,28754],{"className":28747},[2465,2502],[53,28749,3521],{"className":28750,"style":3587},[2465,2506,2502],[53,28752,121],{"className":28753},[2818,2502],[53,28755,149],{"className":28756},[2465,2502],[53,28758,28759,28762],{"style":28217},[53,28760],{"className":28761,"style":28199},[2494],[53,28763,28764],{},[53,28765,7422],{"className":28766},[7465,7469,28226],[53,28768,28769,28772],{"style":28229},[53,28770],{"className":28771,"style":28199},[2494],[53,28773,28775],{"className":28774},[2499,2500,2501,2502],[53,28776,28778],{"className":28777},[2465,2502],[53,28779,7576],{"className":28780,"style":3209},[2465,2506,2502],[53,28782,3120],{"className":28783},[3119],[53,28785,28787],{"className":28786},[2482],[53,28788,28790],{"className":28789,"style":28251},[2486],[53,28791],{},[53,28793],{"className":28794,"style":2860},[2609],[53,28796,28798,28801],{"className":28797},[2465],[53,28799,28127],{"className":28800,"style":28263},[2465,2506],[53,28802,28804],{"className":28803},[2474],[53,28805,28807,28827],{"className":28806},[2478,3095],[53,28808,28810,28824],{"className":28809},[2482],[53,28811,28813],{"className":28812,"style":11138},[2486],[53,28814,28815,28818],{"style":28278},[53,28816],{"className":28817,"style":2495},[2494],[53,28819,28821],{"className":28820},[2499,2500,2501,2502],[53,28822,3521],{"className":28823,"style":3587},[2465,2506,2502],[53,28825,3120],{"className":28826},[3119],[53,28828,28830],{"className":28829},[2482],[53,28831,28833],{"className":28832,"style":3127},[2486],[53,28834],{},[53,28836,22173],{"className":28837,"style":27234},[2465,2506],[53,28839,128],{"className":28840},[2836],[53,28842,28844,28847],{"className":28843},[2465],[53,28845,6077],{"className":28846,"style":5787},[2465,2506],[53,28848,28850],{"className":28849},[2474],[53,28851,28853,28873],{"className":28852},[2478,3095],[53,28854,28856,28870],{"className":28855},[2482],[53,28857,28859],{"className":28858,"style":11138},[2486],[53,28860,28861,28864],{"style":5909},[53,28862],{"className":28863,"style":2495},[2494],[53,28865,28867],{"className":28866},[2499,2500,2501,2502],[53,28868,3521],{"className":28869,"style":3587},[2465,2506,2502],[53,28871,3120],{"className":28872},[3119],[53,28874,28876],{"className":28875},[2482],[53,28877,28879],{"className":28878,"style":3127},[2486],[53,28880],{},[53,28882,3544],{"className":28883},[2675],[53,28885,2644],{"className":28886},[2465],[53,28888,28890,28893],{"className":28889},[2465],[53,28891,6077],{"className":28892,"style":5787},[2465,2506],[53,28894,28896],{"className":28895},[2474],[53,28897,28899,28919],{"className":28898},[2478,3095],[53,28900,28902,28916],{"className":28901},[2482],[53,28903,28905],{"className":28904,"style":11138},[2486],[53,28906,28907,28910],{"style":5909},[53,28908],{"className":28909,"style":2495},[2494],[53,28911,28913],{"className":28912},[2499,2500,2501,2502],[53,28914,3521],{"className":28915,"style":3587},[2465,2506,2502],[53,28917,3120],{"className":28918},[3119],[53,28920,28922],{"className":28921},[2482],[53,28923,28925],{"className":28924,"style":3127},[2486],[53,28926],{},[53,28928,2651],{"className":28929},[2675],[53,28931,2714],{"className":28932},[2856],[12,28934,28935,28936,28964,28965,29035,29036,29115],{},"using the fact that we can replace ",[53,28937,28939,28952],{"className":28938},[2418],[53,28940,28942],{"className":28941},[2422],[2424,28943,28944],{"xmlns":2426},[2428,28945,28946,28950],{},[2431,28947,28948],{},[2437,28949,3033],{},[2444,28951,3033],{"encoding":2446},[53,28953,28955],{"className":28954,"ariaHidden":2452},[2451],[53,28956,28958,28961],{"className":28957},[2456],[53,28959],{"className":28960,"style":2461},[2460],[53,28962,3033],{"className":28963,"style":3048},[2465,2506]," by ",[53,28966,28968,28986],{"className":28967},[2418],[53,28969,28971],{"className":28970},[2422],[2424,28972,28973],{"xmlns":2426},[2428,28974,28975,28983],{},[2431,28976,28977],{},[3063,28978,28979,28981],{},[2437,28980,6077],{},[2437,28982,3521],{},[2444,28984,28985],{"encoding":2446},"E_k",[53,28987,28989],{"className":28988,"ariaHidden":2452},[2451],[53,28990,28992,28995],{"className":28991},[2456],[53,28993],{"className":28994,"style":3082},[2460],[53,28996,28998,29001],{"className":28997},[2465],[53,28999,6077],{"className":29000,"style":5787},[2465,2506],[53,29002,29004],{"className":29003},[2474],[53,29005,29007,29027],{"className":29006},[2478,3095],[53,29008,29010,29024],{"className":29009},[2482],[53,29011,29013],{"className":29012,"style":11138},[2486],[53,29014,29015,29018],{"style":5909},[53,29016],{"className":29017,"style":2495},[2494],[53,29019,29021],{"className":29020},[2499,2500,2501,2502],[53,29022,3521],{"className":29023,"style":3587},[2465,2506,2502],[53,29025,3120],{"className":29026},[3119],[53,29028,29030],{"className":29029},[2482],[53,29031,29033],{"className":29032,"style":3127},[2486],[53,29034],{}," when it acts on the eigenstate ",[53,29037,29039,29060],{"className":29038},[2418],[53,29040,29042],{"className":29041},[2422],[2424,29043,29044],{"xmlns":2426},[2428,29045,29046,29058],{},[2431,29047,29048,29050,29056],{},[2437,29049,2644],{"mathvariant":2643},[3063,29051,29052,29054],{},[2437,29053,6077],{},[2437,29055,3521],{},[2584,29057,2651],{"stretchy":2650},[2444,29059,28375],{"encoding":2446},[53,29061,29063],{"className":29062,"ariaHidden":2452},[2451],[53,29064,29066,29069,29072,29112],{"className":29065},[2456],[53,29067],{"className":29068,"style":2664},[2460],[53,29070,2644],{"className":29071},[2465],[53,29073,29075,29078],{"className":29074},[2465],[53,29076,6077],{"className":29077,"style":5787},[2465,2506],[53,29079,29081],{"className":29080},[2474],[53,29082,29084,29104],{"className":29083},[2478,3095],[53,29085,29087,29101],{"className":29086},[2482],[53,29088,29090],{"className":29089,"style":11138},[2486],[53,29091,29092,29095],{"style":5909},[53,29093],{"className":29094,"style":2495},[2494],[53,29096,29098],{"className":29097},[2499,2500,2501,2502],[53,29099,3521],{"className":29100,"style":3587},[2465,2506,2502],[53,29102,3120],{"className":29103},[3119],[53,29105,29107],{"className":29106},[2482],[53,29108,29110],{"className":29109,"style":3127},[2486],[53,29111],{},[53,29113,2651],{"className":29114},[2675],". The energy error of this state is therefore",[53,29117,29119],{"className":29118},[2679],[53,29120,29122,29211],{"className":29121},[2418],[53,29123,29125],{"className":29124},[2422],[2424,29126,29127],{"xmlns":2426,"display":2688},[2428,29128,29129,29208],{},[2431,29130,29131,29134,29136],{},[6062,29132,29133],{},"energy error",[2584,29135,121],{},[9386,29137,29138,29182],{},[2431,29139,29140,29142,29144,29146,29148,29150,29152,29154,29156,29158,29160,29166,29168,29170,29172,29174,29176,29178,29180],{},[2584,29141,4320],{"stretchy":2650},[2437,29143,3688],{},[2437,29145,2644],{"mathvariant":2643},[2437,29147,22173],{},[2584,29149,128],{"stretchy":2650},[2437,29151,3033],{},[2584,29153,3544],{"stretchy":2650},[2584,29155,128],{"stretchy":2650},[2437,29157,3033],{},[2584,29159,2586],{},[3063,29161,29162,29164],{},[2437,29163,6077],{},[2588,29165,149],{},[2584,29167,3544],{"stretchy":2650},[2437,29169,22173],{},[2584,29171,128],{"stretchy":2650},[2437,29173,3033],{},[2584,29175,3544],{"stretchy":2650},[2437,29177,2644],{"mathvariant":2643},[2437,29179,3688],{},[2584,29181,2651],{"stretchy":2650},[2431,29183,29184,29186,29188,29190,29192,29194,29196,29202,29204,29206],{},[2584,29185,4320],{"stretchy":2650},[2437,29187,3688],{},[2437,29189,2644],{"mathvariant":2643},[2437,29191,22173],{},[2584,29193,128],{"stretchy":2650},[2437,29195,3033],{},[2434,29197,29198,29200],{},[2584,29199,3544],{"stretchy":2650},[2588,29201,132],{},[2437,29203,2644],{"mathvariant":2643},[2437,29205,3688],{},[2584,29207,2651],{"stretchy":2650},[2444,29209,29210],{"encoding":2446},"\\text{energy error} = \\frac{\\langle\\psi|f(H)(H-E_0)f(H)|\\psi\\rangle}{\\langle\\psi|f(H)^2|\\psi\\rangle}",[53,29212,29214,29235],{"className":29213,"ariaHidden":2452},[2451],[53,29215,29217,29220,29226,29229,29232],{"className":29216},[2456],[53,29218],{"className":29219,"style":28462},[2460],[53,29221,29223],{"className":29222},[2465,6934],[53,29224,29133],{"className":29225},[2465],[53,29227],{"className":29228,"style":2814},[2609],[53,29230,121],{"className":29231},[2818],[53,29233],{"className":29234,"style":2814},[2609],[53,29236,29238,29242],{"className":29237},[2456],[53,29239],{"className":29240,"style":29241},[2460],"height:2.363em;vertical-align:-0.936em;",[53,29243,29245,29248,29455],{"className":29244},[2465],[53,29246],{"className":29247},[2836,9688],[53,29249,29251],{"className":29250},[9386],[53,29252,29254,29446],{"className":29253},[2478,3095],[53,29255,29257,29443],{"className":29256},[2482],[53,29258,29261,29327,29335],{"className":29259,"style":29260},[2486],"height:1.427em;",[53,29262,29263,29266],{"style":11612},[53,29264],{"className":29265,"style":4205},[2494],[53,29267,29269,29272,29275,29278,29281,29284,29287,29318,29321,29324],{"className":29268},[2465],[53,29270,4320],{"className":29271},[2836],[53,29273,3688],{"className":29274,"style":2671},[2465,2506],[53,29276,2644],{"className":29277},[2465],[53,29279,22173],{"className":29280,"style":27234},[2465,2506],[53,29282,128],{"className":29283},[2836],[53,29285,3033],{"className":29286,"style":3048},[2465,2506],[53,29288,29290,29293],{"className":29289},[2675],[53,29291,3544],{"className":29292},[2675],[53,29294,29296],{"className":29295},[2474],[53,29297,29299],{"className":29298},[2478],[53,29300,29302],{"className":29301},[2482],[53,29303,29306],{"className":29304,"style":29305},[2486],"height:0.7401em;",[53,29307,29309,29312],{"style":29308},"top:-2.989em;margin-right:0.05em;",[53,29310],{"className":29311,"style":2495},[2494],[53,29313,29315],{"className":29314},[2499,2500,2501,2502],[53,29316,132],{"className":29317},[2465,2502],[53,29319,2644],{"className":29320},[2465],[53,29322,3688],{"className":29323,"style":2671},[2465,2506],[53,29325,2651],{"className":29326},[2675],[53,29328,29329,29332],{"style":9773},[53,29330],{"className":29331,"style":4205},[2494],[53,29333],{"className":29334,"style":9781},[9780],[53,29336,29337,29340],{"style":9784},[53,29338],{"className":29339,"style":4205},[2494],[53,29341,29343,29346,29349,29352,29355,29358,29361,29364,29367,29370,29373,29376,29379,29419,29422,29425,29428,29431,29434,29437,29440],{"className":29342},[2465],[53,29344,4320],{"className":29345},[2836],[53,29347,3688],{"className":29348,"style":2671},[2465,2506],[53,29350,2644],{"className":29351},[2465],[53,29353,22173],{"className":29354,"style":27234},[2465,2506],[53,29356,128],{"className":29357},[2836],[53,29359,3033],{"className":29360,"style":3048},[2465,2506],[53,29362,3544],{"className":29363},[2675],[53,29365,128],{"className":29366},[2836],[53,29368,3033],{"className":29369,"style":3048},[2465,2506],[53,29371],{"className":29372,"style":2610},[2609],[53,29374,2586],{"className":29375},[2614],[53,29377],{"className":29378,"style":2610},[2609],[53,29380,29382,29385],{"className":29381},[2465],[53,29383,6077],{"className":29384,"style":5787},[2465,2506],[53,29386,29388],{"className":29387},[2474],[53,29389,29391,29411],{"className":29390},[2478,3095],[53,29392,29394,29408],{"className":29393},[2482],[53,29395,29397],{"className":29396,"style":14176},[2486],[53,29398,29399,29402],{"style":5909},[53,29400],{"className":29401,"style":2495},[2494],[53,29403,29405],{"className":29404},[2499,2500,2501,2502],[53,29406,149],{"className":29407},[2465,2502],[53,29409,3120],{"className":29410},[3119],[53,29412,29414],{"className":29413},[2482],[53,29415,29417],{"className":29416,"style":3127},[2486],[53,29418],{},[53,29420,3544],{"className":29421},[2675],[53,29423,22173],{"className":29424,"style":27234},[2465,2506],[53,29426,128],{"className":29427},[2836],[53,29429,3033],{"className":29430,"style":3048},[2465,2506],[53,29432,3544],{"className":29433},[2675],[53,29435,2644],{"className":29436},[2465],[53,29438,3688],{"className":29439,"style":2671},[2465,2506],[53,29441,2651],{"className":29442},[2675],[53,29444,3120],{"className":29445},[3119],[53,29447,29449],{"className":29448},[2482],[53,29450,29453],{"className":29451,"style":29452},[2486],"height:0.936em;",[53,29454],{},[53,29456],{"className":29457},[2675,9688],[53,29459,29461],{"className":29460},[2679],[53,29462,29464,29592],{"className":29463},[2418],[53,29465,29467],{"className":29466},[2422],[2424,29468,29469],{"xmlns":2426,"display":2688},[2428,29470,29471,29589],{},[2431,29472,29473,29475,29587],{},[2584,29474,121],{},[9386,29476,29477,29541],{},[2431,29478,29479,29493,29495,29501,29507,29509,29511,29517,29523,29525,29531,29533,29539],{},[28109,29480,29481,29483,29491],{},[2584,29482,7422],{},[2431,29484,29485,29487,29489],{},[2437,29486,3521],{},[2584,29488,121],{},[2588,29490,149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\\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}.",[53,29593,29595,29607],{"className":29594,"ariaHidden":2452},[2451],[53,29596,29598,29601,29604],{"className":29597},[2456],[53,29599],{"className":29600,"style":4623},[2460],[53,29602,121],{"className":29603},[2818],[53,29605],{"className":29606,"style":2814},[2609],[53,29608,29610,29614,30195],{"className":29609},[2456],[53,29611],{"className":29612,"style":29613},[2460],"height:2.8419em;vertical-align:-1.1709em;",[53,29615,29617,29620,30192],{"className":29616},[2465],[53,29618],{"className":29619},[2836,9688],[53,29621,29623],{"className":29622},[9386],[53,29624,29626,30183],{"className":29625},[2478,3095],[53,29627,29629,30180],{"className":29628},[2482],[53,29630,29633,29855,29863],{"className":29631,"style":29632},[2486],"height:1.6709em;",[53,29634,29636,29639],{"style":29635},"top:-2.1288em;",[53,29637],{"className":29638,"style":4205},[2494],[53,29640,29642,29705,29708,29711,29751,29780,29783,29786,29826],{"className":29641},[2465],[53,29643,29645,29648],{"className":29644},[7465],[53,29646,7422],{"className":29647,"style":7471},[7465,7469,7470],[53,29649,29651],{"className":29650},[2474],[53,29652,29654,29697],{"className":29653},[2478,3095],[53,29655,29657,29694],{"className":29656},[2482],[53,29658,29660,29680],{"className":29659,"style":7717},[2486],[53,29661,29662,29665],{"style":7487},[53,29663],{"className":29664,"style":2495},[2494],[53,29666,29668],{"className":29667},[2499,2500,2501,2502],[53,29669,29671,29674,29677],{"className":29670},[2465,2502],[53,29672,3521],{"className":29673,"style":3587},[2465,2506,2502],[53,29675,121],{"className":29676},[2818,2502],[53,29678,149],{"className":29679},[2465,2502],[53,29681,29682,29685],{"style":7740},[53,29683],{"className":29684,"style":2495},[2494],[53,29686,29688],{"className":29687},[2499,2500,2501,2502],[53,29689,29691],{"className":29690},[2465,2502],[53,29692,7576],{"className":29693,"style":3209},[2465,2506,2502],[53,29695,3120],{"className":29696},[3119],[53,29698,29700],{"className":29699},[2482],[53,29701,29703],{"className":29702,"style":7506},[2486],[53,29704],{},[53,29706],{"className":29707,"style":2860},[2609],[53,29709,2644],{"className":29710},[2465],[53,29712,29714,29717],{"className":29713},[2465],[53,29715,28127],{"className":29716,"style":28263},[2465,2506],[53,29718,29720],{"className":29719},[2474],[53,29721,29723,29743],{"className":29722},[2478,3095],[53,29724,29726,29740],{"className":29725},[2482],[53,29727,29729],{"className":29728,"style":11138},[2486],[53,29730,29731,29734],{"style":28278},[53,29732],{"className":29733,"style":2495},[2494],[53,29735,29737],{"className":29736},[2499,2500,2501,2502],[53,29738,3521],{"className":29739,"style":3587},[2465,2506,2502],[53,29741,3120],{"className":29742},[3119],[53,29744,29746],{"className":29745},[2482],[53,29747,29749],{"className":29748,"style":3127},[2486],[53,29750],{},[53,29752,29754,29757],{"className":29753},[2465],[53,29755,2644],{"className":29756},[2465],[53,29758,29760],{"className":29759},[2474],[53,29761,29763],{"className":29762},[2478],[53,29764,29766],{"className":29765},[2482],[53,29767,29769],{"className":29768,"style":29305},[2486],[53,29770,29771,29774],{"style":29308},[53,29772],{"className":29773,"style":2495},[2494],[53,29775,29777],{"className":29776},[2499,2500,2501,2502],[53,29778,132],{"className":29779},[2465,2502],[53,29781,22173],{"className":29782,"style":27234},[2465,2506],[53,29784,128],{"className":29785},[2836],[53,29787,29789,29792],{"className":29788},[2465],[53,29790,6077],{"className":29791,"style":5787},[2465,2506],[53,29793,29795],{"className":29794},[2474],[53,29796,29798,29818],{"className":29797},[2478,3095],[53,29799,29801,29815],{"className":29800},[2482],[53,29802,29804],{"className":29803,"style":11138},[2486],[53,29805,29806,29809],{"style":5909},[53,29807],{"className":29808,"style":2495},[2494],[53,29810,29812],{"className":29811},[2499,2500,2501,2502],[53,29813,3521],{"className":29814,"style":3587},[2465,2506,2502],[53,29816,3120],{"className":29817},[3119],[53,29819,29821],{"className":29820},[2482],[53,29822,29824],{"className":29823,"style":3127},[2486],[53,29825],{},[53,29827,29829,29832],{"className":29828},[2675],[53,29830,3544],{"className":29831},[2675],[53,29833,29835],{"className":29834},[2474],[53,29836,29838],{"className":29837},[2478],[53,29839,29841],{"className":29840},[2482],[53,29842,29844],{"className":29843,"style":29305},[2486],[53,29845,29846,29849],{"style":29308},[53,29847],{"className":29848,"style":2495},[2494],[53,29850,29852],{"className":29851},[2499,2500,2501,2502],[53,29853,132],{"className":29854},[2465,2502],[53,29856,29857,29860],{"style":9773},[53,29858],{"className":29859,"style":4205},[2494],[53,29861],{"className":29862,"style":9781},[9780],[53,29864,29866,29869],{"style":29865},"top:-3.6897em;",[53,29867],{"className":29868,"style":4205},[2494],[53,29870,29872,29935,29938,29941,29981,30010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turn this into an upper bound that is easier to understand, we first separate the sum in the numerator into terms with 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and terms with 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for every 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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 ",[53,33884,33886,33912],{"className":33885},[2418],[53,33887,33889],{"className":33888},[2422],[2424,33890,33891],{"xmlns":2426},[2428,33892,33893,33909],{},[2431,33894,33895,33897,33903],{},[2437,33896,2644],{"mathvariant":2643},[3063,33898,33899,33901],{},[2437,33900,28127],{},[2588,33902,149],{},[2434,33904,33905,33907],{},[2437,33906,2644],{"mathvariant":2643},[2588,33908,132],{},[2444,33910,33911],{"encoding":2446},"|\\gamma_0|^2",[53,33913,33915],{"className":33914,"ariaHidden":2452},[2451],[53,33916,33918,33921,33924,33964],{"className":33917},[2456],[53,33919],{"className":33920,"style":13097},[2460],[53,33922,2644],{"className":33923},[2465],[53,33925,33927,33930],{"className":33926},[2465],[53,33928,28127],{"className":33929,"style":28263},[2465,2506],[53,33931,33933],{"className":33932},[2474],[53,33934,33936,33956],{"className":33935},[2478,3095],[53,33937,33939,33953],{"className":33938},[2482],[53,33940,33942],{"className":33941,"style":14176},[2486],[53,33943,33944,33947],{"style":28278},[53,33945],{"className":33946,"style":2495},[2494],[53,33948,33950],{"className":33949},[2499,2500,2501,2502],[53,33951,149],{"className":33952},[2465,2502],[53,33954,3120],{"className":33955},[3119],[53,33957,33959],{"className":33958},[2482],[53,33960,33962],{"className":33961,"style":3127},[2486],[53,33963],{},[53,33965,33967,33970],{"className":33966},[2465],[53,33968,2644],{"className":33969},[2465],[53,33971,33973],{"className":33972},[2474],[53,33974,33976],{"className":33975},[2478],[53,33977,33979],{"className":33978},[2482],[53,33980,33982],{"className":33981,"style":13119},[2486],[53,33983,33984,33987],{"style":2490},[53,33985],{"className":33986,"style":2495},[2494],[53,33988,33990],{"className":33989},[2499,2500,2501,2502],[53,33991,132],{"className":33992},[2465,2502],", since ",[53,33995,33997,34029],{"className":33996},[2418],[53,33998,34000],{"className":33999},[2422],[2424,34001,34002],{"xmlns":2426},[2428,34003,34004,34026],{},[2431,34005,34006,34008,34010,34016,34022,34024],{},[2437,34007,22173],{},[2584,34009,128],{"stretchy":2650},[3063,34011,34012,34014],{},[2437,34013,6077],{},[2588,34015,149],{},[2434,34017,34018,34020],{},[2584,34019,3544],{"stretchy":2650},[2588,34021,132],{},[2584,34023,121],{},[2588,34025,173],{},[2444,34027,34028],{"encoding":2446},"f(E_0)^2=1",[53,34030,34032,34122],{"className":34031,"ariaHidden":2452},[2451],[53,34033,34035,34038,34041,34044,34084,34113,34116,34119],{"className":34034},[2456],[53,34036],{"className":34037,"style":13097},[2460],[53,34039,22173],{"className":34040,"style":27234},[2465,2506],[53,34042,128],{"className":34043},[2836],[53,34045,34047,34050],{"className":34046},[2465],[53,34048,6077],{"className":34049,"style":5787},[2465,2506],[53,34051,34053],{"className":34052},[2474],[53,34054,34056,34076],{"className":34055},[2478,3095],[53,34057,34059,34073],{"className":34058},[2482],[53,34060,34062],{"className":34061,"style":14176},[2486],[53,34063,34064,34067],{"style":5909},[53,34065],{"className":34066,"style":2495},[2494],[53,34068,34070],{"className":34069},[2499,2500,2501,2502],[53,34071,149],{"className":34072},[2465,2502],[53,34074,3120],{"className":34075},[3119],[53,34077,34079],{"className":34078},[2482],[53,34080,34082],{"className":34081,"style":3127},[2486],[53,34083],{},[53,34085,34087,34090],{"className":34086},[2675],[53,34088,3544],{"className":34089},[2675],[53,34091,34093],{"className":34092},[2474],[53,34094,34096],{"className":34095},[2478],[53,34097,34099],{"className":34098},[2482],[53,34100,34102],{"className":34101,"style":13119},[2486],[53,34103,34104,34107],{"style":2490},[53,34105],{"className":34106,"style":2495},[2494],[53,34108,34110],{"className":34109},[2499,2500,2501,2502],[53,34111,132],{"className":34112},[2465,2502],[53,34114],{"className":34115,"style":2814},[2609],[53,34117,121],{"className":34118},[2818],[53,34120],{"className":34121,"style":2814},[2609],[53,34123,34125,34128],{"className":34124},[2456],[53,34126],{"className":34127,"style":2624},[2460],[53,34129,173],{"className":34130},[2465],": adding everything back together, this gives",[53,34133,34135],{"className":34134},[2679],[53,34136,34138,34252],{"className":34137},[2418],[53,34139,34141],{"className":34140},[2422],[2424,34142,34143],{"xmlns":2426,"display":2688},[2428,34144,34145,34249],{},[2431,34146,34147,34149,34151,34153,34155,34175,34199,34201,34207,34213,34215,34217,34223,34229,34231,34237,34239,34245,34247],{},[6062,34148,29133],{},[2584,34150,27405],{},[2437,34152,27420],{},[2584,34154,2002],{},[9386,34156,34157,34159],{},[2588,34158,173],{},[2431,34160,34161,34163,34169],{},[2437,34162,2644],{"mathvariant":2643},[3063,34164,34165,34167],{},[2437,34166,28127],{},[2588,34168,149],{},[2434,34170,34171,34173],{},[2437,34172,2644],{"mathvariant":2643},[2588,34174,132],{},[30536,34176,34177,34179],{},[2584,34178,7422],{},[2431,34180,34181,34187,34189,34195,34197],{},[3063,34182,34183,34185],{},[2437,34184,6077],{},[2437,34186,3521],{},[2584,34188,1546],{},[3063,34190,34191,34193],{},[2437,34192,6077],{},[2588,34194,149],{},[2584,34196,2002],{},[2437,34198,27420],{},[2437,34200,2644],{"mathvariant":2643},[3063,34202,34203,34205],{},[2437,34204,28127],{},[2437,34206,3521],{},[2434,34208,34209,34211],{},[2437,34210,2644],{"mathvariant":2643},[2588,34212,132],{},[2437,34214,22173],{},[2584,34216,128],{"stretchy":2650},[3063,34218,34219,34221],{},[2437,34220,6077],{},[2437,34222,3521],{},[2434,34224,34225,34227],{},[2584,34226,3544],{"stretchy":2650},[2588,34228,132],{},[2584,34230,128],{"stretchy":2650},[3063,34232,34233,34235],{},[2437,34234,6077],{},[2437,34236,3521],{},[2584,34238,2586],{},[3063,34240,34241,34243],{},[2437,34242,6077],{},[2588,34244,149],{},[2584,34246,3544],{"stretchy":2650},[2437,34248,28],{"mathvariant":2643},[2444,34250,34251],{"encoding":2446},"\\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).",[53,34253,34255,34277,34296,34774],{"className":34254,"ariaHidden":2452},[2451],[53,34256,34258,34262,34268,34271,34274],{"className":34257},[2456],[53,34259],{"className":34260,"style":34261},[2460],"height:0.8304em;vertical-align:-0.1944em;",[53,34263,34265],{"className":34264},[2465,6934],[53,34266,29133],{"className":34267},[2465],[53,34269],{"className":34270,"style":2814},[2609],[53,34272,27405],{"className":34273},[2818],[53,34275],{"className":34276,"style":2814},[2609],[53,34278,34280,34284,34287,34290,34293],{"className":34279},[2456],[53,34281],{"className":34282,"style":34283},[2460],"height:0.7778em;vertical-align:-0.0833em;",[53,34285,27420],{"className":34286,"style":27505},[2465,2506],[53,34288],{"className":34289,"style":2610},[2609],[53,34291,2002],{"className":34292},[2614],[53,34294],{"className":34295,"style":2610},[2609],[53,34297,34299,34303,34434,34437,34572,34575,34578,34618,34647,34650,34653,34693,34722,34725,34765,34768,34771],{"className":34298},[2456],[53,34300],{"className":34301,"style":34302},[2460],"height:2.7294em;vertical-align:-1.408em;",[53,34304,34306,34309,34431],{"className":34305},[2465],[53,34307],{"className":34308},[2836,9688],[53,34310,34312],{"className":34311},[9386],[53,34313,34315,34423],{"className":34314},[2478,3095],[53,34316,34318,34420],{"className":34317},[2482],[53,34319,34321,34401,34409],{"className":34320,"style":9701},[2486],[53,34322,34323,34326],{"style":11612},[53,34324],{"className":34325,"style":4205},[2494],[53,34327,34329,34332,34372],{"className":34328},[2465],[53,34330,2644],{"className":34331},[2465],[53,34333,34335,34338],{"className":34334},[2465],[53,34336,28127],{"className":34337,"style":28263},[2465,2506],[53,34339,34341],{"className":34340},[2474],[53,34342,34344,34364],{"className":34343},[2478,3095],[53,34345,34347,34361],{"className":34346},[2482],[53,34348,34350],{"className":34349,"style":14176},[2486],[53,34351,34352,34355],{"style":28278},[53,34353],{"className":34354,"style":2495},[2494],[53,34356,34358],{"className":34357},[2499,2500,2501,2502],[53,34359,149],{"className":34360},[2465,2502],[53,34362,3120],{"className":34363},[3119],[53,34365,34367],{"className":34366},[2482],[53,34368,34370],{"className":34369,"style":3127},[2486],[53,34371],{},[53,34373,34375,34378],{"className":34374},[2465],[53,34376,2644],{"className":34377},[2465],[53,34379,34381],{"className":34380},[2474],[53,34382,34384],{"className":34383},[2478],[53,34385,34387],{"className":34386},[2482],[53,34388,34390],{"className":34389,"style":29305},[2486],[53,34391,34392,34395],{"style":29308},[53,34393],{"className":34394,"style":2495},[2494],[53,34396,34398],{"className":34397},[2499,2500,2501,2502],[53,34399,132],{"className":34400},[2465,2502],[53,34402,34403,34406],{"style":9773},[53,34404],{"className":34405,"style":4205},[2494],[53,34407],{"className":34408,"style":9781},[9780],[53,34410,34411,34414],{"style":9784},[53,34412],{"className":34413,"style":4205},[2494],[53,34415,34417],{"className":34416},[2465],[53,34418,173],{"className":34419},[2465],[53,34421,3120],{"className":34422},[3119],[53,34424,34426],{"className":34425},[2482],[53,34427,34429],{"className":34428,"style":29452},[2486],[53,34430],{},[53,34432],{"className":34433},[2675,9688],[53,34435],{"className":34436,"style":2860},[2609],[53,34438,34440],{"className":34439},[7465,28182],[53,34441,34443,34563],{"className":34442},[2478,3095],[53,34444,34446,34560],{"className":34445},[2482],[53,34447,34450,34550],{"className":34448,"style":34449},[2486],"height:1.05em;",[53,34451,34452,34455],{"style":28195},[53,34453],{"className":34454,"style":28199},[2494],[53,34456,34458],{"className":34457},[2499,2500,2501,2502],[53,34459,34461,34501,34504,34544,34547],{"className":34460},[2465,2502],[53,34462,34464,34467],{"className":34463},[2465,2502],[53,34465,6077],{"className":34466,"style":5787},[2465,2506,2502],[53,34468,34470],{"className":34469},[2474],[53,34471,34473,34493],{"className":34472},[2478,3095],[53,34474,34476,34490],{"className":34475},[2482],[53,34477,34479],{"className":34478,"style":31119},[2486],[53,34480,34481,34484],{"style":31122},[53,34482],{"className":34483,"style":4707},[2494],[53,34485,34487],{"className":34486},[2499,4711,2843,2502],[53,34488,3521],{"className":34489,"style":3587},[2465,2506,2502],[53,34491,3120],{"className":34492},[3119],[53,34494,34496],{"className":34495},[2482],[53,34497,34499],{"className":34498,"style":31141},[2486],[53,34500],{},[53,34502,1546],{"className":34503},[2818,2502],[53,34505,34507,34510],{"className":34506},[2465,2502],[53,34508,6077],{"className":34509,"style":5787},[2465,2506,2502],[53,34511,34513],{"className":34512},[2474],[53,34514,34516,34536],{"className":34515},[2478,3095],[53,34517,34519,34533],{"className":34518},[2482],[53,34520,34522],{"className":34521,"style":31165},[2486],[53,34523,34524,34527],{"style":31168},[53,34525],{"className":345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{"className":34825},[2465],[12,34827,34828,34829,34898,34899,34927,34928,35149,35150,35318,35319,35608],{},"To simplify what is left, notice that for all these ",[53,34830,34832,34849],{"className":34831},[2418],[53,34833,34835],{"className":34834},[2422],[2424,34836,34837],{"xmlns":2426},[2428,34838,34839,34847],{},[2431,34840,34841],{},[3063,34842,34843,34845],{},[2437,34844,6077],{},[2437,34846,3521],{},[2444,34848,28985],{"encoding":2446},[53,34850,34852],{"className":34851,"ariaHidden":2452},[2451],[53,34853,34855,34858],{"className":34854},[2456],[53,34856],{"className":34857,"style":3082},[2460],[53,34859,34861,34864],{"className":34860},[2465],[53,34862,6077],{"className":34863,"style":5787},[2465,2506],[53,34865,34867],{"className":34866},[2474],[53,34868,34870,34890],{"className":34869},[2478,3095],[53,34871,34873,34887],{"className":34872},[2482],[53,34874,34876],{"className":34875,"style":11138},[2486],[53,34877,34878,34881],{"style":5909},[53,34879],{"className":34880,"style":2495},[2494],[53,34882,34884],{"className":34883},[2499,2500,2501,2502],[53,34885,3521],{"className":34886,"style":3587},[2465,2506,2502],[53,34888,3120],{"className":34889},[3119],[53,34891,34893],{"className":34892},[2482],[53,34894,34896],{"className":34895,"style":3127},[2486],[53,34897],{},", by the definition of ",[53,34900,34902,34915],{"className":34901},[2418],[53,34903,34905],{"className":34904},[2422],[2424,34906,34907],{"xmlns":2426},[2428,34908,34909,34913],{},[2431,34910,34911],{},[2437,34912,22173],{},[2444,34914,22173],{"encoding":2446},[53,34916,34918],{"className":34917,"ariaHidden":2452},[2451],[53,34919,34921,34924],{"className":34920},[2456],[53,34922],{"className":34923,"style":6930},[2460],[53,34925,22173],{"className":34926,"style":27234},[2465,2506]," we know that ",[53,34929,34931,34985],{"className":34930},[2418],[53,34932,34934],{"className":34933},[2422],[2424,34935,34936],{"xmlns":2426},[2428,34937,34938,34982],{},[2431,34939,34940,34942,34944,34950,34956,34958,34960],{},[2437,34941,22173],{},[2584,34943,128],{"stretchy":2650},[3063,34945,34946,34948],{},[2437,34947,6077],{},[2437,34949,3521],{},[2434,34951,34952,34954],{},[2584,34953,3544],{"stretchy":2650},[2588,34955,132],{},[2584,34957,27405],{},[2588,34959,12047],{},[2434,34961,34962,34974],{},[2431,34963,34964,34966,34968,34970,34972],{},[2584,34965,128],{"fence":2452},[2588,34967,173],{},[2584,34969,2002],{},[2437,34971,27420],{},[2584,34973,3544],{"fence":2452},[2431,34975,34976,34978,34980],{},[2584,34977,2586],{},[2588,34979,132],{},[2437,34981,4854],{},[2444,34983,34984],{"encoding":2446},"f(E_k)^2 \\le 4\\left(1 + \\delta\\right)^{-2d}",[53,34986,34988,35078],{"className":34987,"ariaHidden":2452},[2451],[53,34989,34991,34994,34997,35000,35040,35069,35072,35075],{"className":34990},[2456],[53,34992],{"className":34993,"style":13097},[2460],[53,34995,22173],{"className":34996,"style":27234},[2465,2506],[53,34998,128],{"className":34999},[2836],[53,35001,35003,35006],{"className":35002},[2465],[53,35004,6077],{"className":35005,"style":5787},[2465,2506],[53,35007,35009],{"className":35008},[2474],[53,35010,35012,35032],{"className":35011},[2478,3095],[53,35013,35015,35029],{"className":35014},[2482],[53,35016,35018],{"className":35017,"style":11138},[2486],[53,35019,35020,35023],{"style":5909},[53,35021],{"className":35022,"style":2495},[2494],[53,35024,35026],{"className":35025},[2499,2500,2501,2502],[53,35027,3521],{"className":35028,"style":3587},[2465,2506,2502],[53,35030,3120],{"className":35031},[3119],[53,35033,35035],{"className":35034},[2482],[53,35036,35038],{"className":35037,"style":3127},[2486],[53,35039],{},[53,35041,35043,35046],{"className":35042},[2675],[53,35044,3544],{"className":35045},[2675],[53,35047,35049],{"className":35048},[2474],[53,35050,35052],{"className":35051},[2478],[53,35053,35055],{"className":35054},[2482],[53,35056,35058],{"className":35057,"style":13119},[2486],[53,35059,35060,35063],{"style":2490},[53,35061],{"className":35062,"style":2495},[2494],[53,35064,35066],{"className":35065},[2499,2500,2501,2502],[53,35067,132],{"className":35068},[2465,2502],[53,35070],{"className":35071,"style":2814},[2609],[53,35073,27405],{"className":35074},[2818],[53,35076],{"className":35077,"style":2814},[2609],[53,35079,35081,35084,35087,35090],{"className":35080},[2456],[53,35082],{"className":35083,"style":27474},[2460],[53,35085,12047],{"className":35086},[2465],[53,35088],{"className":35089,"style":2860},[2609],[53,35091,35093,35117],{"className":35092},[2832],[53,35094,35096,35099,35102,35105,35108,35111,35114],{"className":35095},[2832],[53,35097,128],{"className":35098,"style":2838},[2836,2837],[53,35100,173],{"className":35101},[2465],[53,35103],{"className":35104,"style":2610},[2609],[53,35106,2002],{"className":35107},[2614],[53,35109],{"className":35110,"style":2610},[2609],[53,35112,27420],{"className":35113,"style":27505},[2465,2506],[53,35115,3544],{"className":35116,"style":2838},[2675,2837],[53,35118,35120],{"className":35119},[2474],[53,35121,35123],{"className":35122},[2478],[53,35124,35126],{"className":35125},[2482],[53,35127,35129],{"className":35128,"style":27521},[2486],[53,35130,35131,35134],{"style":7740},[53,35132],{"className":35133,"style":2495},[2494],[53,35135,35137],{"className":35136},[2499,2500,2501,2502],[53,35138,35140,35143,35146],{"className":35139},[2465,2502],[53,35141,2586],{"className":35142},[2465,2502],[53,35144,132],{"className":35145},[2465,2502],[53,35147,4854],{"className":35148},[2465,2506,2502],". Additionally upper bounding ",[53,35151,35153,35189],{"className":35152},[2418],[53,35154,35156],{"className":35155},[2422],[2424,35157,35158],{"xmlns":2426},[2428,35159,35160,35186],{},[2431,35161,35162,35168,35170,35176,35178,35180,35182,35184],{},[3063,35163,35164,35166],{},[2437,35165,6077],{},[2437,35167,3521],{},[2584,35169,2586],{},[3063,35171,35172,35174],{},[2437,35173,6077],{},[2588,35175,149],{},[2584,35177,5572],{},[2588,35179,132],{},[2437,35181,12053],{"mathvariant":2643},[2437,35183,3033],{},[2437,35185,12053],{"mathvariant":2643},[2444,35187,35188],{"encoding":2446},"E_k-E_0\u003C2\\|H\\|",[53,35190,35192,35247,35302],{"className":35191,"ariaHidden":2452},[2451],[53,35193,35195,35198,35238,35241,35244],{"className":35194},[2456],[53,35196],{"className":35197,"style":3082},[2460],[53,35199,35201,35204],{"className":35200},[2465],[53,35202,6077],{"className":35203,"style":5787},[2465,2506],[53,35205,35207],{"className":35206},[2474],[53,35208,35210,35230],{"className":35209},[2478,3095],[53,35211,35213,35227],{"className":35212},[2482],[53,35214,35216],{"className":35215,"style":11138},[2486],[53,35217,35218,35221],{"style":5909},[53,35219],{"className":35220,"style":2495},[2494],[53,35222,35224],{"className":35223},[2499,2500,2501,2502],[53,35225,3521],{"className":35226,"style":3587},[2465,2506,2502],[53,35228,3120],{"className":35229},[3119],[53,35231,35233],{"className":35232},[2482],[53,35234,35236],{"className":35235,"style":3127},[2486],[53,35237],{},[53,35239],{"className":35240,"style":2610},[2609],[53,35242,2586],{"className":35243},[2614],[53,35245],{"className":35246,"style":2610},[2609],[53,35248,35250,35253,35293,35296,35299],{"className":35249},[2456],[53,35251],{"className":35252,"style":3082},[2460],[53,35254,35256,35259],{"className":35255},[2465],[53,35257,6077],{"className":35258,"style":5787},[2465,2506],[53,35260,35262],{"className":35261},[2474],[53,35263,35265,35285],{"className":35264},[2478,3095],[53,35266,35268,35282],{"className":35267},[2482],[53,35269,35271],{"className":35270,"style":14176},[2486],[53,35272,35273,35276],{"style":5909},[53,35274],{"className":35275,"style":2495},[2494],[53,35277,35279],{"className":35278},[2499,2500,2501,2502],[53,35280,149],{"className":35281},[2465,2502],[53,35283,3120],{"className":35284},[3119],[53,35286,35288],{"className":35287},[2482],[53,35289,35291],{"className":35290,"style":3127},[2486],[53,35292],{},[53,35294],{"className":35295,"style":2814},[2609],[53,35297,5572],{"className":35298},[2818],[53,35300],{"className":35301,"style":2814},[2609],[53,35303,35305,35308,35312,35315],{"className":35304},[2456],[53,35306],{"className":35307,"style":2664},[2460],[53,35309,35311],{"className":35310},[2465],"2∥",[53,35313,3033],{"className":35314,"style":3048},[2465,2506],[53,35316,12053],{"className":35317},[2465]," and upper bounding ",[53,35320,35322,35376],{"className":35321},[2418],[53,35323,35325],{"className":35324},[2422],[2424,35326,35327],{"xmlns":2426},[2428,35328,35329,35373],{},[2431,35330,35331,35355,35357,35363,35369,35371],{},[3063,35332,35333,35335],{},[2584,35334,7422],{},[2431,35336,35337,35343,35345,35351,35353],{},[3063,35338,35339,35341],{},[2437,35340,6077],{},[2437,35342,3521],{},[2584,35344,1546],{},[3063,35346,35347,35349],{},[2437,35348,6077],{},[2588,35350,149],{},[2584,35352,2002],{},[2437,35354,27420],{},[2437,35356,2644],{"mathvariant":2643},[3063,35358,35359,35361],{},[2437,35360,28127],{},[2437,35362,3521],{},[2434,35364,35365,35367],{},[2437,35366,2644],{"mathvariant":2643},[2588,35368,132],{},[2584,35370,5572],{},[2588,35372,173],{},[2444,35374,35375],{"encoding":2446},"\\sum_{E_k>E_0+\\delta}|\\gamma_k|^2\u003C1",[53,35377,35379,35599],{"className":35378,"ariaHidden":2452},[2451],[53,35380,35382,35386,35515,35518,35521,35561,35590,35593,35596],{"className":35381},[2456],[53,35383],{"className":35384,"style":35385},[2460],"height:1.2197em;vertical-align:-0.4056em;",[53,35387,35389,35392],{"className":35388},[7465],[53,35390,7422],{"className":35391,"style":7471},[7465,7469,7470],[53,35393,35395],{"className":35394},[2474],[53,35396,35398,35507],{"className":35397},[2478,3095],[53,35399,35401,35504],{"className":35400},[2482],[53,35402,35404],{"className":35403,"style":31089},[2486],[53,35405,35406,35409],{"style":7487},[53,35407],{"className":35408,"style":2495},[2494],[53,35410,35412],{"className":35411},[2499,2500,2501,2502],[53,35413,35415,35455,35458,35498,35501],{"className":35414},[2465,2502],[53,35416,35418,35421],{"className":35417},[2465,2502],[53,35419,6077],{"className":35420,"style":5787},[2465,2506,2502],[53,35422,35424],{"className":35423},[2474],[53,35425,35427,35447],{"className":35426},[2478,3095],[53,35428,35430,35444],{"className":35429},[2482],[53,35431,35433],{"className":35432,"style":31119},[2486],[53,35434,35435,35438],{"style":31122},[53,35436],{"className":35437,"style":4707},[2494],[53,35439,35441],{"className":35440},[2499,4711,2843,2502],[53,35442,3521],{"className":35443,"style":3587},[2465,2506,2502],[53,35445,3120],{"className":35446},[3119],[53,35448,35450],{"className":35449},[2482],[53,35451,35453],{"className":35452,"style":31141},[2486],[53,35454],{},[53,35456,1546],{"className":35457},[2818,2502],[53,35459,35461,35464],{"className":35460},[2465,2502],[53,35462,6077],{"className":35463,"style":5787},[2465,2506,2502],[53,35465,35467],{"className":35466},[2474],[53,35468,35470,35490],{"className":35469},[2478,3095],[53,35471,35473,35487],{"className":35472},[2482],[53,35474,35476],{"className":35475,"style":31165},[2486],[53,35477,35478,35481],{"style":31168},[53,35479],{"className":35480,"style":4707},[2494],[53,35482,35484],{"className":35483},[2499,4711,2843,2502],[53,35485,149],{"className":35486},[2465,2502],[53,35488,3120],{"className":35489},[3119],[53,35491,35493],{"className":35492},[2482],[53,35494,35496],{"className":35495,"style":4724},[2486],[53,35497],{},[53,35499,2002],{"className":35500},[2614,2502],[53,35502,27420],{"className":35503,"style":27505},[2465,2506,2502],[53,35505,3120],{"className":35506},[3119],[53,35508,35510],{"className":35509},[2482],[53,35511,35513],{"className":35512,"style":31204},[2486],[53,35514],{},[53,35516],{"className":35517,"style":2860},[2609],[53,35519,2644],{"className":35520},[2465],[53,35522,35524,35527],{"className":35523},[2465],[53,35525,28127],{"className":35526,"style":28263},[2465,2506],[53,35528,35530],{"className":35529},[2474],[53,35531,35533,35553],{"className":35532},[2478,3095],[53,35534,35536,35550],{"className":35535},[2482],[53,35537,35539],{"className":35538,"style":11138},[2486],[53,35540,35541,35544],{"style":28278},[53,35542],{"className":35543,"style":2495},[2494],[53,35545,35547],{"className":35546},[2499,2500,2501,2502],[53,35548,3521],{"className":35549,"style":3587},[2465,2506,2502],[53,35551,3120],{"className":35552},[3119],[53,35554,35556],{"className":35555},[2482],[53,35557,35559],{"className":35558,"style":3127},[2486],[53,35560],{},[53,35562,35564,35567],{"className":35563},[2465],[53,35565,2644],{"className":35566},[2465],[53,35568,35570],{"className":35569},[2474],[53,35571,35573],{"className":35572},[2478],[53,35574,35576],{"className":35575},[2482],[53,35577,35579],{"className":35578,"style":13119},[2486],[53,35580,35581,35584],{"style":2490},[53,35582],{"className":35583,"style":2495},[2494],[53,35585,35587],{"className":35586},[2499,2500,2501,2502],[53,35588,132],{"className":35589},[2465,2502],[53,35591],{"className":35592,"style":2814},[2609],[53,35594,5572],{"className":35595},[2818],[53,35597],{"className":35598,"style":2814},[2609],[53,35600,35602,35605],{"className":35601},[2456],[53,35603],{"className":35604,"style":2624},[2460],[53,35606,173],{"className":35607},[2465]," gives",[53,35610,35612],{"className":35611},[2679],[53,35613,35615,35686],{"className":35614},[2418],[53,35616,35618],{"className":35617},[2422],[2424,35619,35620],{"xmlns":2426,"display":2688},[2428,35621,35622,35683],{},[2431,35623,35624,35626,35628,35630,35632,35653,35655,35657,35659,35681],{},[6062,35625,29133],{},[2584,35627,27405],{},[2437,35629,27420],{},[2584,35631,2002],{},[9386,35633,35634,35637],{},[2588,35635,35636],{},"8",[2431,35638,35639,35641,35647],{},[2437,35640,2644],{"mathvariant":2643},[3063,35642,35643,35645],{},[2437,35644,28127],{},[2588,35646,149],{},[2434,35648,35649,35651],{},[2437,35650,2644],{"mathvariant":2643},[2588,35652,132],{},[2437,35654,12053],{"mathvariant":2643},[2437,35656,3033],{},[2437,35658,12053],{"mathvariant":2643},[2434,35660,35661,35673],{},[2431,35662,35663,35665,35667,35669,35671],{},[2584,35664,128],{"fence":2452},[2588,35666,173],{},[2584,35668,2002],{},[2437,35670,27420],{},[2584,35672,3544],{"fence":2452},[2431,35674,35675,35677,35679],{},[2584,35676,2586],{},[2588,35678,132],{},[2437,35680,4854],{},[2437,35682,28],{"mathvariant":2643},[2444,35684,35685],{"encoding":2446},"\\text{energy error} \\le \\delta + \\frac{8}{|\\gamma_0|^2}\\|H\\|\\left(1 + \\delta\\right)^{-2d}.",[53,35687,35689,35710,35728],{"className":35688,"ariaHidden":2452},[2451],[53,35690,35692,35695,35701,35704,35707],{"className":35691},[2456],[53,35693],{"className":35694,"style":34261},[2460],[53,35696,35698],{"className":35697},[2465,6934],[53,35699,29133],{"className":35700},[2465],[53,35702],{"className":35703,"style":2814},[2609],[53,35705,27405],{"className":35706},[2818],[53,35708],{"className":35709,"style":2814},[2609],[53,35711,35713,35716,35719,35722,35725],{"className":35712},[2456],[53,35714],{"className":35715,"style":34283},[2460],[53,35717,27420],{"className":35718,"style":27505},[2465,2506],[53,35720],{"className":35721,"style":2610},[2609],[53,35723,2002],{"className":35724},[2614],[53,35726],{"className":35727,"style":2610},[2609],[53,35729,35731,35735,35866,35869,35872,35875,35878,35937,35940],{"className":35730},[2456],[53,35732],{"className":35733,"style":35734},[2460],"height:2.2574em;vertical-align:-0.936em;",[53,35736,35738,35741,35863],{"className":35737},[2465],[53,35739],{"className":35740},[2836,9688],[53,35742,35744],{"className":35743},[9386],[53,35745,35747,35855],{"className":35746},[2478,3095],[53,35748,35750,35852],{"className":35749},[2482],[53,35751,35753,35833,35841],{"className":35752,"style":9701},[2486],[53,35754,35755,35758],{"style":11612},[53,35756],{"className":35757,"style":4205},[2494],[53,35759,35761,35764,35804],{"className":35760},[2465],[53,35762,2644],{"className":35763},[2465],[53,35765,35767,35770],{"className":35766},[2465],[53,35768,28127],{"className":35769,"style":28263},[2465,2506],[53,35771,35773],{"className":35772},[2474],[53,35774,35776,35796],{"className":35775},[2478,3095],[53,35777,35779,35793],{"className":35778},[2482],[53,35780,35782],{"className":35781,"style":14176},[2486],[53,35783,35784,35787],{"style":28278},[53,35785],{"className":35786,"style":2495},[2494],[53,35788,35790],{"className":35789},[2499,2500,2501,2502],[53,35791,149],{"className":35792},[2465,2502],[53,35794,3120],{"className":35795},[3119],[53,35797,35799],{"className":35798},[2482],[53,35800,35802],{"className":35801,"style":3127},[2486],[53,35803],{},[53,35805,35807,35810],{"className":35806},[2465],[53,35808,2644],{"className":35809},[2465],[53,35811,35813],{"className":35812},[2474],[53,35814,35816],{"className":35815},[2478],[53,35817,35819],{"className":35818},[2482],[53,35820,35822],{"className":35821,"style":29305},[2486],[53,35823,35824,35827],{"style":29308},[53,35825],{"className":35826,"style":2495},[2494],[53,35828,35830],{"className":35829},[2499,2500,2501,2502],[53,35831,132],{"className":35832},[2465,2502],[53,35834,35835,35838],{"style":9773},[53,35836],{"className":35837,"style":4205},[2494],[53,35839],{"className":35840,"style":9781},[9780],[53,35842,35843,35846],{"style":9784},[53,35844],{"className":35845,"style":4205},[2494],[53,35847,35849],{"className":35848},[2465],[53,35850,35636],{"className":35851},[2465],[53,35853,3120],{"className":35854},[3119],[53,35856,35858],{"className":35857},[2482],[53,35859,35861],{"className":35860,"style":29452},[2486],[53,35862],{},[53,35864],{"className":35865},[2675,9688],[53,35867,12053],{"className":35868},[2465],[53,35870,3033],{"className":35871,"style":3048},[2465,2506],[53,35873,12053],{"className":35874},[2465],[53,35876],{"className":35877,"style":2860},[2609],[53,35879,35881,35905],{"className":35880},[2832],[53,35882,35884,35887,35890,35893,35896,35899,35902],{"className":35883},[2832],[53,35885,128],{"className":35886,"style":2838},[2836,2837],[53,35888,173],{"className":35889},[2465],[53,35891],{"className":35892,"style":2610},[2609],[53,35894,2002],{"className":35895},[2614],[53,35897],{"className":35898,"style":2610},[2609],[53,35900,27420],{"className":35901,"style":27505},[2465,2506],[53,35903,3544],{"className":35904,"style":2838},[2675,2837],[53,35906,35908],{"className":35907},[2474],[53,35909,35911],{"className":35910},[2478],[53,35912,35914],{"className":35913},[2482],[53,35915,35917],{"className":35916,"style":27521},[2486],[53,35918,35919,35922],{"style":7740},[53,35920],{"className":35921,"style":2495},[2494],[53,35923,35925],{"className":35924},[2499,2500,2501,2502],[53,35926,35928,35931,35934],{"className":35927},[2465,2502],[53,35929,2586],{"className":35930},[2465,2502],[53,35932,132],{"className":35933},[2465,2502],[53,35935,4854],{"className":35936},[2465,2506,2502],[53,35938],{"className":35939,"style":2860},[2609],[53,35941,28],{"className":35942},[2465],[12,35944,35945,35946,35998,35999,36027,36028,36084,36085,36240,36241,36269],{},"This holds for any ",[53,35947,35949,35967],{"className":35948},[2418],[53,35950,35952],{"className":35951},[2422],[2424,35953,35954],{"xmlns":2426},[2428,35955,35956,35964],{},[2431,35957,35958,35960,35962],{},[2437,35959,27420],{},[2584,35961,1546],{},[2588,35963,149],{},[2444,35965,35966],{"encoding":2446},"\\delta>0",[53,35968,35970,35989],{"className":35969,"ariaHidden":2452},[2451],[53,35971,35973,35977,35980,35983,35986],{"className":35972},[2456],[53,35974],{"className":35975,"style":35976},[2460],"height:0.7335em;vertical-align:-0.0391em;",[53,35978,27420],{"className":35979,"style":27505},[2465,2506],[53,35981],{"className":35982,"style":2814},[2609],[53,35984,1546],{"className":35985},[2818],[53,35987],{"className":35988,"style":2814},[2609],[53,35990,35992,35995],{"className":35991},[2456],[53,35993],{"className":35994,"style":2624},[2460],[53,35996,149],{"className":35997},[2465],", so if we set ",[53,36000,36002,36015],{"className":36001},[2418],[53,36003,36005],{"className":36004},[2422],[2424,36006,36007],{"xmlns":2426},[2428,36008,36009,36013],{},[2431,36010,36011],{},[2437,36012,27420],{},[2444,36014,32146],{"encoding":2446},[53,36016,36018],{"className":36017,"ariaHidden":2452},[2451],[53,36019,36021,36024],{"className":36020},[2456],[53,36022],{"className":36023,"style":5329},[2460],[53,36025,27420],{"className":36026,"style":27505},[2465,2506]," equal to our goal error, then the error bound above converges towards that exponentially with the Krylov dimension ",[53,36029,36031,36051],{"className":36030},[2418],[53,36032,36034],{"className":36033},[2422],[2424,36035,36036],{"xmlns":2426},[2428,36037,36038,36048],{},[2431,36039,36040,36042,36044,36046],{},[2588,36041,132],{},[2437,36043,4854],{},[2584,36045,121],{},[2437,36047,2111],{},[2444,36049,36050],{"encoding":2446},"2d=r",[53,36052,36054,36075],{"className":36053,"ariaHidden":2452},[2451],[53,36055,36057,36060,36063,36066,36069,36072],{"className":36056},[2456],[53,36058],{"className":36059,"style":5329},[2460],[53,36061,132],{"className":36062},[2465],[53,36064,4854],{"className":36065},[2465,2506],[53,36067],{"className":36068,"style":2814},[2609],[53,36070,121],{"className":36071},[2818],[53,36073],{"className":36074,"style":2814},[2609],[53,36076,36078,36081],{"className":36077},[2456],[53,36079],{"className":36080,"style":2534},[2460],[53,36082,2111],{"className":36083,"style":2507},[2465,2506],". Also note that if ",[53,36086,36088,36118],{"className":36087},[2418],[53,36089,36091],{"className":36090},[2422],[2424,36092,36093],{"xmlns":2426},[2428,36094,36095,36115],{},[2431,36096,36097,36099,36101,36107,36109],{},[2437,36098,27420],{},[2584,36100,5572],{},[3063,36102,36103,36105],{},[2437,36104,6077],{},[2588,36106,173],{},[2584,36108,2586],{},[3063,36110,36111,36113],{},[2437,36112,6077],{},[2588,36114,149],{},[2444,36116,36117],{"encoding":2446},"\\delta\u003CE_1-E_0",[53,36119,36121,36139,36194],{"className":36120,"ariaHidden":2452},[2451],[53,36122,36124,36127,36130,36133,36136],{"className":36123},[2456],[53,36125],{"className":36126,"style":35976},[2460],[53,36128,27420],{"className":36129,"style":27505},[2465,2506],[53,36131],{"className":36132,"style":2814},[2609],[53,36134,5572],{"className":36135},[2818],[53,36137],{"className":36138,"style":2814},[2609],[53,36140,36142,36145,36185,36188,36191],{"className":36141},[2456],[53,36143],{"className":36144,"style":3082},[2460],[53,36146,36148,36151],{"className":36147},[2465],[53,36149,6077],{"className":36150,"style":5787},[2465,2506],[53,36152,36154],{"className":36153},[2474],[53,36155,36157,36177],{"className":36156},[2478,3095],[53,36158,36160,36174],{"className":36159},[2482],[53,36161,36163],{"className":36162,"style":14176},[2486],[53,36164,36165,36168],{"style":5909},[53,36166],{"className":36167,"style":2495},[2494],[53,36169,36171],{"className":36170},[2499,2500,2501,2502],[53,36172,173],{"className":36173},[2465,2502],[53,36175,3120],{"className":36176},[3119],[53,36178,36180],{"className":36179},[2482],[53,36181,36183],{"className":36182,"style":3127},[2486],[53,36184],{},[53,36186],{"className":36187,"style":2610},[2609],[53,36189,2586],{"className":36190},[2614],[53,36192],{"className":36193,"style":2610},[2609],[53,36195,36197,36200],{"className":36196},[2456],[53,36198],{"className":36199,"style":3082},[2460],[53,36201,36203,36206],{"className":36202},[2465],[53,36204,6077],{"className":36205,"style":5787},[2465,2506],[53,36207,36209],{"className":36208},[2474],[53,36210,36212,36232],{"className":36211},[2478,3095],[53,36213,36215,36229],{"className":36214},[2482],[53,36216,36218],{"className":36217,"style":14176},[2486],[53,36219,36220,36223],{"style":5909},[53,36221],{"className":36222,"style":2495},[2494],[53,36224,36226],{"className":36225},[2499,2500,2501,2502],[53,36227,149],{"className":36228},[2465,2502],[53,36230,3120],{"className":36231},[3119],[53,36233,36235],{"className":36234},[2482],[53,36236,36238],{"className":36237,"style":3127},[2486],[53,36239],{}," then the ",[53,36242,36244,36257],{"className":36243},[2418],[53,36245,36247],{"className":36246},[2422],[2424,36248,36249],{"xmlns":2426},[2428,36250,36251,36255],{},[2431,36252,36253],{},[2437,36254,27420],{},[2444,36256,32146],{"encoding":2446},[53,36258,36260],{"className":36259,"ariaHidden":2452},[2451],[53,36261,36263,36266],{"className":36262},[2456],[53,36264],{"className":36265,"style":5329},[2460],[53,36267,27420],{"className":36268,"style":27505},[2465,2506]," term actually goes away entirely in the above bound.",[12,36271,36272,36273,36331],{},"To complete the argument, we first note that the above is just the energy error of the particular state ",[53,36274,36276,36301],{"className":36275},[2418],[53,36277,36279],{"className":36278},[2422],[2424,36280,36281],{"xmlns":2426},[2428,36282,36283,36299],{},[2431,36284,36285,36287,36289,36291,36293,36295,36297],{},[2437,36286,22173],{},[2584,36288,128],{"stretchy":2650},[2437,36290,3033],{},[2584,36292,3544],{"stretchy":2650},[2437,36294,2644],{"mathvariant":2643},[2437,36296,3688],{},[2584,36298,2651],{"stretchy":2650},[2444,36300,28013],{"encoding":2446},[53,36302,36304],{"className":36303,"ariaHidden":2452},[2451],[53,36305,36307,36310,36313,36316,36319,36322,36325,36328],{"className":36306},[2456],[53,36308],{"className":36309,"style":2664},[2460],[53,36311,22173],{"className":36312,"style":27234},[2465,2506],[53,36314,128],{"className":36315},[2836],[53,36317,3033],{"className":36318,"style":3048},[2465,2506],[53,36320,3544],{"className":36321},[2675],[53,36323,2644],{"className":36324},[2465],[53,36326,3688],{"className":36327,"style":2671},[2465,2506],[53,36329,2651],{"className":36330},[2675],", 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.",[12,36333,36334,36335,10694,36337,36340],{},"A similar analysis as the above can be carried out that additionally accounts for noise and the thresholding procedure discussed in the notebook. See ",[19,36336,8442],{"href":979},[19,36338,36339],{"href":979},"[4]"," for this analysis.",[30,36342,36344],{"id":36343},"appendix-proof-of-claim-1","Appendix: proof of Claim 1",[12,36346,36347,36348,36350,36351,36425,36426,36514,36515,36543],{},"The following is mostly derived from ",[19,36349,27198],{"href":979},", Theorem 3.1: let ",[53,36352,36354,36376],{"className":36353},[2418],[53,36355,36357],{"className":36356},[2422],[2424,36358,36359],{"xmlns":2426},[2428,36360,36361,36373],{},[2431,36362,36363,36365,36367,36369,36371],{},[2588,36364,149],{},[2584,36366,5572],{},[2437,36368,19],{},[2584,36370,5572],{},[2437,36372,12997],{},[2444,36374,36375],{"encoding":2446},"0 \u003C a \u003C b",[53,36377,36379,36398,36416],{"className":36378,"ariaHidden":2452},[2451],[53,36380,36382,36386,36389,36392,36395],{"className":36381},[2456],[53,36383],{"className":36384,"style":36385},[2460],"height:0.6835em;vertical-align:-0.0391em;",[53,36387,149],{"className":36388},[2465],[53,36390],{"className":36391,"style":2814},[2609],[53,36393,5572],{"className":36394},[2818],[53,36396],{"className":36397,"style":2814},[2609],[53,36399,36401,36404,36407,36410,36413],{"className":36400},[2456],[53,36402],{"className":36403,"style":5590},[2460],[53,36405,19],{"className":36406},[2465,2506],[53,36408],{"className":36409,"style":2814},[2609],[53,36411,5572],{"className":36412},[2818],[53,36414],{"className":36415,"style":2814},[2609],[53,36417,36419,36422],{"className":36418},[2456],[53,36420],{"className":36421,"style":5329},[2460],[53,36423,12997],{"className":36424},[2465,2506]," and let ",[53,36427,36429,36450],{"className":36428},[2418],[53,36430,36432],{"className":36431},[2422],[2424,36433,36434],{"xmlns":2426},[2428,36435,36436,36447],{},[2431,36437,36438],{},[3670,36439,36440,36443,36445],{},[2437,36441,36442],{"mathvariant":2643},"Π",[2437,36444,4854],{},[2584,36446,20664],{},[2444,36448,36449],{"encoding":2446},"\\Pi^*_d",[53,36451,36453],{"className":36452,"ariaHidden":2452},[2451],[53,36454,36456,36460],{"className":36455},[2456],[53,36457],{"className":36458,"style":36459},[2460],"height:0.9718em;vertical-align:-0.2831em;",[53,36461,36463,36466],{"className":36462},[2465],[53,36464,36442],{"className":36465},[2465],[53,36467,36469],{"className":36468},[2474],[53,36470,36472,36505],{"className":36471},[2478,3095],[53,36473,36475,36502],{"className":36474},[2482],[53,36476,36479,36491],{"className":36477,"style":36478},[2486],"height:0.6887em;",[53,36480,36482,36485],{"style":36481},"top:-2.4169em;margin-left:0em;margin-right:0.05em;",[53,36483],{"className":36484,"style":2495},[2494],[53,36486,36488],{"className":36487},[2499,2500,2501,2502],[53,36489,4854],{"className":36490},[2465,2506,2502],[53,36492,36493,36496],{"style":2490},[53,36494],{"className":36495,"style":2495},[2494],[53,36497,36499],{"className":36498},[2499,2500,2501,2502],[53,36500,20664],{"className":36501},[2614,2502],[53,36503,3120],{"className":36504},[3119],[53,36506,36508],{"className":36507},[2482],[53,36509,36512],{"className":36510,"style":36511},[2486],"height:0.2831em;",[53,36513],{}," be the space of residual polynomials (polynomials whose value at 0 is 1) of degree at most ",[53,36516,36518,36531],{"className":36517},[2418],[53,36519,36521],{"className":36520},[2422],[2424,36522,36523],{"xmlns":2426},[2428,36524,36525,36529],{},[2431,36526,36527],{},[2437,36528,4854],{},[2444,36530,4854],{"encoding":2446},[53,36532,36534],{"className":36533,"ariaHidden":2452},[2451],[53,36535,36537,36540],{"className":36536},[2456],[53,36538],{"className":36539,"style":5329},[2460],[53,36541,4854],{"className":36542},[2465,2506],". The solution to",[53,36545,36547],{"className":36546},[2679],[53,36548,36550,36644],{"className":36549},[2418],[53,36551,36553],{"className":36552},[2422],[2424,36554,36555],{"xmlns":2426,"display":2688},[2428,36556,36557,36641],{},[2431,36558,36559,36562,36564,36566,36568,36570,36572,36574,36576,36578,36602,36627,36629,36631,36633,36635,36637,36639],{},[2437,36560,36561],{},"β",[2584,36563,128],{"stretchy":2650},[2437,36565,19],{},[2584,36567,2714],{"separator":2452},[2437,36569,12997],{},[2584,36571,2714],{"separator":2452},[2437,36573,4854],{},[2584,36575,3544],{"stretchy":2650},[2584,36577,121],{},[30536,36579,36580,36587],{},[2431,36581,36582,36584],{},[2437,36583,25125],{},[2584,36585,36586],{},"⁡",[2431,36588,36589,36591,36594],{},[2437,36590,12],{},[2584,36592,36593],{},"∈",[3670,36595,36596,36598,36600],{},[2437,36597,36442],{"mathvariant":2643},[2437,36599,4854],{},[2584,36601,20664],{},[30536,36603,36604,36611],{},[2431,36605,36606,36609],{},[2437,36607,36608],{},"max",[2584,36610,36586],{},[2431,36612,36613,36615,36617,36619,36621,36623,36625],{},[2437,36614,9034],{},[2584,36616,36593],{},[2584,36618,12143],{"stretchy":2650},[2437,36620,19],{},[2584,36622,2714],{"separator":2452},[2437,36624,12997],{},[2584,36626,12168],{"stretchy":2650},[2437,36628,2644],{"mathvariant":2643},[2437,36630,12],{},[2584,36632,128],{"stretchy":2650},[2437,36634,9034],{},[2584,36636,3544],{"stretchy":2650},[2437,36638,2644],{"mathvariant":2643},[2609,36640],{"width":9350},[2444,36642,36643],{"encoding":2446},"\\beta(a, b, d) = \\min_{p \\in \\Pi^*_d} \\max_{x \\in [a, b]} |p(x)| \\quad",[53,36645,36647,36693],{"className":36646,"ariaHidden":2452},[2451],[53,36648,36650,36653,36657,36660,36663,36666,36669,36672,36675,36678,36681,36684,36687,36690],{"className":36649},[2456],[53,36651],{"className":36652,"style":2664},[2460],[53,36654,36561],{"className":36655,"style":36656},[2465,2506],"margin-right:0.0528em;",[53,36658,128],{"className":36659},[2836],[53,36661,19],{"className":36662},[2465,2506],[53,36664,2714],{"className":36665},[2856],[53,36667],{"className":36668,"style":2860},[2609],[53,36670,12997],{"className":36671},[2465,2506],[53,36673,2714],{"className":36674},[2856],[53,36676],{"className":36677,"style":2860},[2609],[53,36679,4854],{"className":36680},[2465,2506],[53,36682,3544],{"className":36683},[2675],[53,36685],{"className":36686,"style":2814},[2609],[53,36688,121],{"className":36689},[2818],[53,36691],{"className":36692,"style":2814},[2609],[53,36694,36696,36700,36808,36811,36878,36881,36884,36887,36890,36893,36896,36899],{"className":36695},[2456],[53,36697],{"className":36698,"style":36699},[2460],"height:1.7374em;vertical-align:-0.9874em;",[53,36701,36703],{"className":36702},[7465,28182],[53,36704,36706,36799],{"className":36705},[2478,3095],[53,36707,36709,36796],{"className":36708},[2482],[53,36710,36713,36786],{"className":36711,"style":36712},[2486],"height:0.6679em;",[53,36714,36716,36719],{"style":36715},"top:-2.3557em;margin-left:0em;",[53,36717],{"className":36718,"style":4205},[2494],[53,36720,36722],{"className":36721},[2499,2500,2501,2502],[53,36723,36725,36728,36731],{"className":36724},[2465,2502],[53,36726,12],{"className":36727},[2465,2506,2502],[53,36729,36593],{"className":36730},[2818,2502],[53,36732,36734,36737],{"className":36733},[2465,2502],[53,36735,36442],{"className":36736},[2465,2502],[53,36738,36740],{"className":36739},[2474],[53,36741,36743,36777],{"className":36742},[2478,3095],[53,36744,36746,36774],{"className":36745},[2482],[53,36747,36750,36762],{"className":36748,"style":36749},[2486],"height:0.6771em;",[53,36751,36753,36756],{"style":36752},"top:-2.1528em;margin-left:0em;margin-right:0.0714em;",[53,36754],{"className":36755,"style":4707},[2494],[53,36757,36759],{"className":36758},[2499,4711,2843,2502],[53,36760,4854],{"className":36761},[2465,2506,2502],[53,36763,36765,36768],{"style":36764},"top:-2.8448em;margin-right:0.0714em;",[53,36766],{"className":36767,"style":4707},[2494],[53,36769,36771],{"className":36770},[2499,4711,2843,2502],[53,36772,20664],{"className":36773},[2614,2502],[53,36775,3120],{"className":36776},[3119],[53,36778,36780],{"className":36779},[2482],[53,36781,36784],{"className":36782,"style":36783},[2486],"height:0.3472em;",[53,36785],{},[53,36787,36788,36791],{"style":4201},[53,36789],{"className":36790,"style":4205},[2494],[53,36792,36793],{},[53,36794,25125],{"className":36795},[7465],[53,36797,3120],{"className":36798},[3119],[53,36800,36802],{"className":36801},[2482],[53,36803,36806],{"className":36804,"style":36805},[2486],"height:0.9874em;",[53,36807],{},[53,36809],{"className":36810,"style":2860},[2609],[53,36812,36814],{"className":36813},[7465,28182],[53,36815,36817,36869],{"className":36816},[2478,3095],[53,36818,36820,36866],{"className":36819},[2482],[53,36821,36823,36856],{"className":36822,"style":2534},[2486],[53,36824,36826,36829],{"style":36825},"top:-2.309em;margin-left:0em;",[53,36827],{"className":36828,"style":4205},[2494],[53,36830,36832],{"className":36831},[2499,2500,2501,2502],[53,36833,36835,36838,36841,36844,36847,36850,36853],{"className":36834},[2465,2502],[53,36836,9034],{"className":36837},[2465,2506,2502],[53,36839,36593],{"className":36840},[2818,2502],[53,36842,12143],{"className":36843},[2836,2502],[53,36845,19],{"className":36846},[2465,2506,2502],[53,36848,2714],{"className":36849},[2856,2502],[53,36851,12997],{"className":36852},[2465,2506,2502],[53,36854,12168],{"className":36855},[2675,2502],[53,36857,36858,36861],{"style":4201},[53,36859],{"className":36860,"style":4205},[2494],[53,36862,36863],{},[53,36864,36608],{"className":36865},[7465],[53,36867,3120],{"className":36868},[3119],[53,36870,36872],{"className":36871},[2482],[53,36873,36876],{"className":36874,"style":36875},[2486],"height:0.966em;",[53,36877],{},[53,36879],{"className":36880,"style":2860},[2609],[53,36882,2644],{"className":36883},[2465],[53,36885,12],{"className":36886},[2465,2506],[53,36888,128],{"className":36889},[2836],[53,36891,9034],{"className":36892},[2465,2506],[53,36894,3544],{"className":36895},[2675],[53,36897,2644],{"className":36898},[2465],[53,36900],{"className":36901,"style":9669},[2609],[12,36903,36904],{},"is",[53,36906,36908],{"className":36907},[2679],[53,36909,36911,37014],{"className":36910},[2418],[53,36912,36914],{"className":36913},[2422],[2424,36915,36916],{"xmlns":2426,"display":2688},[2428,36917,36918,37011],{},[2431,36919,36920,36926,36928,36930,36932,36934,37007,37009],{},[2434,36921,36922,36924],{},[2437,36923,12],{},[2584,36925,20664],{},[2584,36927,128],{"stretchy":2650},[2437,36929,9034],{},[2584,36931,3544],{"stretchy":2650},[2584,36933,121],{},[9386,36935,36936,36975],{},[2431,36937,36938,36945],{},[3063,36939,36940,36943],{},[2437,36941,36942],{},"T",[2437,36944,4854],{},[2431,36946,36947,36949,36973],{},[2584,36948,128],{"fence":2452},[9386,36950,36951,36965],{},[2431,36952,36953,36955,36957,36959,36961,36963],{},[2437,36954,12997],{},[2584,36956,2002],{},[2437,36958,19],{},[2584,36960,2586],{},[2588,36962,132],{},[2437,36964,9034],{},[2431,36966,36967,36969,36971],{},[2437,36968,12997],{},[2584,36970,2586],{},[2437,36972,19],{},[2584,36974,3544],{"fence":2452},[2431,36976,36977,36983],{},[3063,36978,36979,36981],{},[2437,36980,36942],{},[2437,36982,4854],{},[2431,36984,36985,36987,37005],{},[2584,36986,128],{"fence":2452},[9386,36988,36989,36997],{},[2431,36990,36991,36993,36995],{},[2437,36992,12997],{},[2584,36994,2002],{},[2437,36996,19],{},[2431,36998,36999,37001,37003],{},[2437,37000,12997],{},[2584,37002,2586],{},[2437,37004,19],{},[2584,37006,3544],{"fence":2452},[2584,37008,2714],{"separator":2452},[2609,37010],{"width":9350},[2444,37012,37013],{"encoding":2446},"p^*(x) = \\frac{T_d\\left(\\frac{b + a - 2x}{b - a}\\right)}{T_d\\left(\\frac{b + a}{b - a}\\right)}, \\quad",[53,37015,37017,37071],{"className":37016,"ariaHidden":2452},[2451],[53,37018,37020,37023,37053,37056,37059,37062,37065,37068],{"className":37019},[2456],[53,37021],{"className":37022,"style":2664},[2460],[53,37024,37026,37029],{"className":37025},[2465],[53,37027,12],{"className":37028},[2465,2506],[53,37030,37032],{"className":37031},[2474],[53,37033,37035],{"className":37034},[2478],[53,37036,37038],{"className":37037},[2482],[53,37039,37042],{"className":37040,"style":37041},[2486],"height:0.7387em;",[53,37043,37044,37047],{"style":2801},[53,37045],{"className":37046,"style":2495},[2494],[53,37048,37050],{"className":37049},[2499,2500,2501,2502],[53,37051,20664],{"className":37052},[2614,2502],[53,37054,128],{"className":37055},[2836],[53,37057,9034],{"className":37058},[2465,2506],[53,37060,3544],{"className":37061},[2675],[53,37063],{"className":37064,"style":2814},[2609],[53,37066,121],{"className":37067},[2818],[53,37069],{"className":37070,"style":2814},[2609],[53,37072,37074,37078,37427,37430],{"className":37073},[2456],[53,37075],{"className":37076,"style":37077},[2460],"height:2.8469em;vertical-align:-1.1734em;",[53,37079,37081,37084,37424],{"className":37080},[2465],[53,37082],{"className":37083},[2836,9688],[53,37085,37087],{"className":37086},[9386],[53,37088,37090,37415],{"className":37089},[2478,3095],[53,37091,37093,37412],{"className":37092},[2482],[53,37094,37097,37248,37256],{"className":37095,"style":37096},[2486],"height:1.6734em;",[53,37098,37100,37103],{"style":37099},"top:-2.2299em;",[53,37101],{"className":37102,"style":4205},[2494],[53,37104,37106,37146,37149],{"className":37105},[2465],[53,37107,37109,37112],{"className":37108},[2465],[53,37110,36942],{"className":37111,"style":3115},[2465,2506],[53,37113,37115],{"className":37114},[2474],[53,37116,37118,37138],{"className":37117},[2478,3095],[53,37119,37121,37135],{"className":37120},[2482],[53,37122,37124],{"className":37123,"style":11138},[2486],[53,37125,37126,37129],{"style":7532},[53,37127],{"className":37128,"style":2495},[2494],[53,37130,37132],{"className":37131},[2499,2500,2501,2502],[53,37133,4854],{"className":37134},[2465,2506,2502],[53,37136,3120],{"className":37137},[3119],[53,37139,37141],{"className":37140},[2482],[53,37142,37144],{"className":37143,"style":3127},[2486],[53,37145],{},[53,37147],{"className":37148,"style":2860},[2609],[53,37150,37152,37158,37242],{"className":37151},[2832],[53,37153,37155],{"className":37154,"style":2838},[2836,2837],[53,37156,128],{"className":37157},[2842,2843],[53,37159,37161,37164,37239],{"className":37160},[2465],[53,37162],{"className":37163},[2836,9688],[53,37165,37167],{"className":37166},[9386],[53,37168,37170,37230],{"className":37169},[2478,3095],[53,37171,37173,37227],{"className":37172},[2482],[53,37174,37177,37198,37206],{"className":37175,"style":37176},[2486],"height:0.8801em;",[53,37178,37180,37183],{"style":37179},"top:-2.655em;",[53,37181],{"className":37182,"style":4205},[2494],[53,37184,37186],{"className":37185},[2499,2500,2501,2502],[53,37187,37189,37192,37195],{"className":37188},[2465,2502],[53,37190,12997],{"className":37191},[2465,2506,2502],[53,37193,2586],{"className":37194},[2614,2502],[53,37196,19],{"className":37197},[2465,2506,2502],[53,37199,37200,37203],{"style":9773},[53,37201],{"className":37202,"style":4205},[2494],[53,37204],{"className":37205,"style":9781},[9780],[53,37207,37209,37212],{"style":37208},"top:-3.394em;",[53,37210],{"className":37211,"style":4205},[2494],[53,37213,37215],{"className":37214},[2499,2500,2501,2502],[53,37216,37218,37221,37224],{"className":37217},[2465,2502],[53,37219,12997],{"className":37220},[2465,2506,2502],[53,37222,2002],{"className":37223},[2614,2502],[53,37225,19],{"className":37226},[2465,2506,2502],[53,37228,3120],{"className":37229},[3119],[53,37231,37233],{"className":37232},[2482],[53,37234,37237],{"className":37235,"style":37236},[2486],"height:0.4033em;",[53,37238],{},[53,37240],{"className":37241},[2675,9688],[53,37243,37245],{"className":37244,"style":2838},[2675,2837],[53,37246,3544],{"className":37247},[2842,2843],[53,37249,37250,37253],{"style":9773},[53,37251],{"className":37252,"style":4205},[2494],[53,37254],{"className":37255,"style":9781},[9780],[53,37257,37259,37262],{"style":37258},"top:-3.7933em;",[53,37260],{"className":37261,"style":4205},[2494],[53,37263,37265,37305,37308],{"className":37264},[2465],[53,37266,37268,37271],{"className":37267},[2465],[53,37269,36942],{"className":37270,"style":3115},[2465,2506],[53,37272,37274],{"className":37273},[2474],[53,37275,37277,37297],{"className":37276},[2478,3095],[53,37278,37280,37294],{"className":37279},[2482],[53,37281,37283],{"className":37282,"style":11138},[2486],[53,37284,37285,37288],{"style":7532},[53,37286],{"className":37287,"style":2495},[2494],[53,37289,37291],{"className":37290},[2499,2500,2501,2502],[53,37292,4854],{"className":37293},[2465,2506,2502],[53,37295,3120],{"className":37296},[3119],[53,37298,37300],{"className":37299},[2482],[53,37301,37303],{"className":37302,"style":3127},[2486],[53,37304],{},[53,37306],{"className":37307,"style":2860},[2609],[53,37309,37311,37317,37406],{"className":37310},[2832],[53,37312,37314],{"className":37313,"style":2838},[2836,2837],[53,37315,128],{"className":37316},[2842,2843],[53,37318,37320,37323,37403],{"className":37319},[2465],[53,37321],{"className":37322},[2836,9688],[53,37324,37326],{"className":37325},[9386],[53,37327,37329,37395],{"className":37328},[2478,3095],[53,37330,37332,37392],{"className":37331},[2482],[53,37333,37335,37355,37363],{"className":37334,"style":37176},[2486],[53,37336,37337,37340],{"style":37179},[53,37338],{"className":37339,"style":4205},[2494],[53,37341,37343],{"className":37342},[2499,2500,2501,2502],[53,37344,37346,37349,37352],{"className":37345},[2465,2502],[53,37347,12997],{"className":37348},[2465,2506,2502],[53,37350,2586],{"className":37351},[2614,2502],[53,37353,19],{"className":37354},[2465,2506,2502],[53,37356,37357,37360],{"style":9773},[53,37358],{"className":37359,"style":4205},[2494],[53,37361],{"className":37362,"style":9781},[9780],[53,37364,37365,37368],{"style":37208},[53,37366],{"className":37367,"style":4205},[2494],[53,37369,37371],{"className":37370},[2499,2500,2501,2502],[53,37372,37374,37377,37380,37383,37386,37389],{"className":37373},[2465,2502],[53,37375,12997],{"className":37376},[2465,2506,2502],[53,37378,2002],{"className":37379},[2614,2502],[53,37381,19],{"className":37382},[2465,2506,2502],[53,37384,2586],{"className":37385},[2614,2502],[53,37387,132],{"className":37388},[2465,2502],[53,37390,9034],{"className":37391},[2465,2506,2502],[53,37393,3120],{"className":37394},[3119],[53,37396,37398],{"className":37397},[2482],[53,37399,37401],{"className":37400,"style":37236},[2486],[53,37402],{},[53,37404],{"className":37405},[2675,9688],[53,37407,37409],{"className":37408,"style":2838},[2675,2837],[53,37410,3544],{"className":37411},[2842,2843],[53,37413,3120],{"className":37414},[3119],[53,37416,37418],{"className":37417},[2482],[53,37419,37422],{"className":37420,"style":37421},[2486],"height:1.1734em;",[53,37423],{},[53,37425],{"className":37426},[2675,9688],[53,37428,2714],{"className":37429},[2856],[53,37431],{"className":37432,"style":9669},[2609],[12,37434,37435],{},"and the corresponding minimal value is",[53,37437,37439],{"className":37438},[2679],[53,37440,37442,37510],{"className":37441},[2418],[53,37443,37445],{"className":37444},[2422],[2424,37446,37447],{"xmlns":2426,"display":2688},[2428,37448,37449,37507],{},[2431,37450,37451,37453,37455,37457,37459,37461,37463,37465,37467,37469,37481,37505],{},[2437,37452,36561],{},[2584,37454,128],{"stretchy":2650},[2437,37456,19],{},[2584,37458,2714],{"separator":2452},[2437,37460,12997],{},[2584,37462,2714],{"separator":2452},[2437,37464,4854],{},[2584,37466,3544],{"stretchy":2650},[2584,37468,121],{},[3670,37470,37471,37473,37475],{},[2437,37472,36942],{},[2437,37474,4854],{},[2431,37476,37477,37479],{},[2584,37478,2586],{},[2588,37480,173],{},[2431,37482,37483,37485,37503],{},[2584,37484,128],{"fence":2452},[9386,37486,37487,37495],{},[2431,37488,37489,37491,37493],{},[2437,37490,12997],{},[2584,37492,2002],{},[2437,37494,19],{},[2431,37496,37497,37499,37501],{},[2437,37498,12997],{},[2584,37500,2586],{},[2437,37502,19],{},[2584,37504,3544],{"fence":2452},[2437,37506,28],{"mathvariant":2643},[2444,37508,37509],{"encoding":2446},"\\beta(a, b, d) = T_d^{-1}\\left(\\frac{b + a}{b - a}\\right).",[53,37511,37513,37558],{"className":37512,"ariaHidden":2452},[2451],[53,37514,37516,37519,37522,37525,37528,37531,37534,37537,37540,37543,37546,37549,37552,37555],{"className":37515},[2456],[53,37517],{"className":37518,"style":2664},[2460],[53,37520,36561],{"className":37521,"style":36656},[2465,2506],[53,37523,128],{"className":37524},[2836],[53,37526,19],{"className":37527},[2465,2506],[53,37529,2714],{"className":37530},[2856],[53,37532],{"className":37533,"style":2860},[2609],[53,37535,12997],{"className":37536},[2465,2506],[53,37538,2714],{"className":37539},[2856],[53,37541],{"className":37542,"style":2860},[2609],[53,37544,4854],{"className":37545},[2465,2506],[53,37547,3544],{"className":37548},[2675],[53,37550],{"className":37551,"style":2814},[2609],[53,37553,121],{"className":37554},[2818],[53,37556],{"className":37557,"style":2814},[2609],[53,37559,37561,37565,37624,37627,37730,37733],{"className":37560},[2456],[53,37562],{"className":37563,"style":37564},[2460],"height:2.4em;vertical-align:-0.95em;",[53,37566,37568,37571],{"className":37567},[2465],[53,37569,36942],{"className":37570,"style":3115},[2465,2506],[53,37572,37574],{"className":37573},[2474],[53,37575,37577,37615],{"className":37576},[2478,3095],[53,37578,37580,37612],{"className":37579},[2482],[53,37581,37583,37595],{"className":37582,"style":2897},[2486],[53,37584,37586,37589],{"style":37585},"top:-2.4086em;margin-left:-0.1389em;margin-right:0.05em;",[53,37587],{"className":37588,"style":2495},[2494],[53,37590,37592],{"className":37591},[2499,2500,2501,2502],[53,37593,4854],{"className":37594},[2465,2506,2502],[53,37596,37597,37600],{"style":2801},[53,37598],{"className":37599,"style":2495},[2494],[53,37601,37603],{"className":37602},[2499,2500,2501,2502],[53,37604,37606,37609],{"className":37605},[2465,2502],[53,37607,2586],{"className":37608},[2465,2502],[53,37610,173],{"className":37611},[2465,2502],[53,37613,3120],{"className":37614},[3119],[53,37616,37618],{"className":37617},[2482],[53,37619,37622],{"className":37620,"style":37621},[2486],"height:0.2914em;",[53,37623],{},[53,37625],{"className":37626,"style":2860},[2609],[53,37628,37630,37636,37724],{"className":37629},[2832],[53,37631,37633],{"className":37632,"style":2838},[2836,2837],[53,37634,128],{"className":37635},[2842,2501],[53,37637,37639,37642,37721],{"className":37638},[2465],[53,37640],{"className":37641},[2836,9688],[53,37643,37645],{"className":37644},[9386],[53,37646,37648,37712],{"className":37647},[2478,3095],[53,37649,37651,37709],{"className":37650},[2482],[53,37652,37655,37678,37686],{"className":37653,"style":37654},[2486],"height:1.3714em;",[53,37656,37657,37660],{"style":11612},[53,37658],{"className":37659,"style":4205},[2494],[53,37661,37663,37666,37669,37672,37675],{"className":37662},[2465],[53,37664,12997],{"className":37665},[2465,2506],[53,37667],{"className":37668,"style":2610},[2609],[53,37670,2586],{"className":37671},[2614],[53,37673],{"className":37674,"style":2610},[2609],[53,37676,19],{"className":37677},[2465,2506],[53,37679,37680,37683],{"style":9773},[53,37681],{"className":37682,"style":4205},[2494],[53,37684],{"className":37685,"style":9781},[9780],[53,37687,37688,37691],{"style":9784},[53,37689],{"className":37690,"style":4205},[2494],[53,37692,37694,37697,37700,37703,37706],{"className":37693},[2465],[53,37695,12997],{"className":37696},[2465,2506],[53,37698],{"className":37699,"style":2610},[2609],[53,37701,2002],{"className":37702},[2614],[53,37704],{"className":37705,"style":2610},[2609],[53,37707,19],{"className":37708},[2465,2506],[53,37710,3120],{"className":37711},[3119],[53,37713,37715],{"className":37714},[2482],[53,37716,37719],{"className":37717,"style":37718},[2486],"height:0.7693em;",[53,37720],{},[53,37722],{"className":37723},[2675,9688],[53,37725,37727],{"className":37726,"style":2838},[2675,2837],[53,37728,3544],{"className":37729},[2842,2501],[53,37731],{"className":37732,"style":2860},[2609],[53,37734,28],{"className":37735},[2465],[12,37737,37738,37739,37791],{},"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 ",[53,37740,37742,37764],{"className":37741},[2418],[53,37743,37745],{"className":37744},[2422],[2424,37746,37747],{"xmlns":2426},[2428,37748,37749,37761],{},[2431,37750,37751,37753,37755,37757,37759],{},[2584,37752,12143],{"stretchy":2650},[2588,37754,149],{},[2584,37756,2714],{"separator":2452},[2588,37758,173],{},[2584,37760,12168],{"stretchy":2650},[2444,37762,37763],{"encoding":2446},"[0,1]",[53,37765,37767],{"className":37766,"ariaHidden":2452},[2451],[53,37768,37770,37773,37776,37779,37782,37785,37788],{"className":37769},[2456],[53,37771],{"className":37772,"style":2664},[2460],[53,37774,12143],{"className":37775},[2836],[53,37777,149],{"className":37778},[2465],[53,37780,2714],{"className":37781},[2856],[53,37783],{"className":37784,"style":2860},[2609],[53,37786,173],{"className":37787},[2465],[53,37789,12168],{"className":37790},[2675],": define",[53,37793,37795],{"className":37794},[2679],[53,37796,37798,37861],{"className":37797},[2418],[53,37799,37801],{"className":37800},[2422],[2424,37802,37803],{"xmlns":2426,"display":2688},[2428,37804,37805,37858],{},[2431,37806,37807,37810,37812,37814,37816,37818,37856],{},[2437,37808,37809],{},"g",[2584,37811,128],{"stretchy":2650},[2437,37813,6077],{},[2584,37815,3544],{"stretchy":2650},[2584,37817,121],{},[9386,37819,37820,37854],{},[2431,37821,37822,37824,37826,37829,37831,37834,37836,37838,37840,37846,37848,37850,37852],{},[2588,37823,173],{},[2584,37825,2586],{},[2437,37827,37828],{},"cos",[2584,37830,36586],{},[2584,37832,128],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},"1.2em",[2584,37835,128],{"stretchy":2650},[2437,37837,6077],{},[2584,37839,2586],{},[3063,37841,37842,37844],{},[2437,37843,6077],{},[2588,37845,149],{},[2584,37847,3544],{"stretchy":2650},[2437,37849,4854],{},[2437,37851,3193],{},[2584,37853,3544],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2588,37855,132],{},[2584,37857,2714],{"separator":2452},[2444,37859,37860],{"encoding":2446},"g(E) = \\frac{1-\\cos\\big((E-E_0)dt\\big)}{2},",[53,37862,37864,37891],{"className":37863,"ariaHidden":2452},[2451],[53,37865,37867,37870,37873,37876,37879,37882,37885,37888],{"className":37866},[2456],[53,37868],{"className":37869,"style":2664},[2460],[53,37871,37809],{"className":37872,"style":2671},[2465,2506],[53,37874,128],{"className":37875},[2836],[53,37877,6077],{"className":37878,"style":5787},[2465,2506],[53,37880,3544],{"className":37881},[2675],[53,37883],{"className":37884,"style":2814},[2609],[53,37886,121],{"className":37887},[2818],[53,37889],{"className":37890,"style":2814},[2609],[53,37892,37894,37898,38053],{"className":37893},[2456],[53,37895],{"className":37896,"style":37897},[2460],"height:2.276em;vertical-align:-0.686em;",[53,37899,37901,37904,38050],{"className":37900},[2465],[53,37902],{"className":37903},[2836,9688],[53,37905,37907],{"className":37906},[9386],[53,37908,37910,38042],{"className":37909},[2478,3095],[53,37911,37913,38039],{"className":37912},[2482],[53,37914,37917,37928,37936],{"className":37915,"style":37916},[2486],"height:1.59em;",[53,37918,37919,37922],{"style":11612},[53,37920],{"className":37921,"style":4205},[2494],[53,37923,37925],{"className":37924},[2465],[53,37926,132],{"className":37927},[2465],[53,37929,37930,37933],{"style":9773},[53,37931],{"className":37932,"style":4205},[2494],[53,37934],{"className":37935,"style":9781},[9780],[53,37937,37939,37942],{"style":37938},"top:-3.74em;",[53,37940],{"className":37941,"style":4205},[2494],[53,37943,37945,37948,37951,37954,37957,37960,37963,37969,37972,37975,37978,37981,37984,38024,38027,38030,38033],{"className":37944},[2465],[53,37946,173],{"className":37947},[2465],[53,37949],{"className":37950,"style":2610},[2609],[53,37952,2586],{"className":37953},[2614],[53,37955],{"className":37956,"style":2610},[2609],[53,37958,37828],{"className":37959},[7465],[53,37961],{"className":37962,"style":2860},[2609],[53,37964,37966],{"className":37965},[2465],[53,37967,128],{"className":37968},[2842,2843],[53,37970,128],{"className":37971},[2836],[53,37973,6077],{"className":37974,"style":5787},[2465,2506],[53,37976],{"className":37977,"style":2610},[2609],[53,37979,2586],{"className":37980},[2614],[53,37982],{"className":37983,"style":2610},[2609],[53,37985,37987,37990],{"className":37986},[2465],[53,37988,6077],{"className":37989,"style":5787},[2465,2506],[53,37991,37993],{"className":37992},[2474],[53,37994,37996,38016],{"className":37995},[2478,3095],[53,37997,37999,38013],{"className":37998},[2482],[53,38000,38002],{"className":38001,"style":14176},[2486],[53,38003,38004,38007],{"style":5909},[53,38005],{"className":38006,"style":2495},[2494],[53,38008,38010],{"className":38009},[2499,2500,2501,2502],[53,38011,149],{"className":38012},[2465,2502],[53,38014,3120],{"className":38015},[3119],[53,38017,38019],{"className":38018},[2482],[53,38020,38022],{"className":38021,"style":3127},[2486],[53,38023],{},[53,38025,3544],{"className":38026},[2675],[53,38028,4854],{"className":38029},[2465,2506],[53,38031,3193],{"className":38032},[2465,2506],[53,38034,38036],{"className":38035},[2465],[53,38037,3544],{"className":38038},[2842,2843],[53,38040,3120],{"className":38041},[3119],[53,38043,38045],{"className":38044},[2482],[53,38046,38048],{"className":38047,"style":11650},[2486],[53,38049],{},[53,38051],{"className":38052},[2675,9688],[53,38054,2714],{"className":38055},[2856],[12,38057,28350,38058,38091,38092,38298,38299,10694,38406,38450,38451,38479,38480,28],{},[53,38059,38061,38076],{"className":38060},[2418],[53,38062,38064],{"className":38063},[2422],[2424,38065,38066],{"xmlns":2426},[2428,38067,38068,38074],{},[2431,38069,38070,38072],{},[2437,38071,4854],{},[2437,38073,3193],{},[2444,38075,5358],{"encoding":2446},[53,38077,38079],{"className":38078,"ariaHidden":2452},[2451],[53,38080,38082,38085,38088],{"className":38081},[2456],[53,38083],{"className":38084,"style":5329},[2460],[53,38086,4854],{"className":38087},[2465,2506],[53,38089,3193],{"className":38090},[2465,2506]," is a timestep such that ",[53,38093,38095,38139],{"className":38094},[2418],[53,38096,38098],{"className":38097},[2422],[2424,38099,38100],{"xmlns":2426},[2428,38101,38102,38136],{},[2431,38103,38104,38106,38108,38110,38116,38118,38120,38122,38128,38130,38132,38134],{},[2584,38105,2586],{},[2437,38107,8458],{},[2584,38109,5572],{},[3063,38111,38112,38114],{},[2437,38113,6077],{},[2588,38115,149],{},[2437,38117,4854],{},[2437,38119,3193],{},[2584,38121,5572],{},[3063,38123,38124,38126],{},[2437,38125,6077],{},[6062,38127,36608],{},[2437,38129,4854],{},[2437,38131,3193],{},[2584,38133,5572],{},[2437,38135,8458],{},[2444,38137,38138],{"encoding":2446},"-\\pi \u003C E_0dt \u003C E_\\text{max}dt \u003C \\pi",[53,38140,38142,38163,38225,38289],{"className":38141,"ariaHidden":2452},[2451],[53,38143,38145,38148,38151,38154,38157,38160],{"className":38144},[2456],[53,38146],{"className":38147,"style":2602},[2460],[53,38149,2586],{"className":38150},[2465],[53,38152,8458],{"className":38153,"style":2671},[2465,2506],[53,38155],{"className":38156,"style":2814},[2609],[53,38158,5572],{"className":38159},[2818],[53,38161],{"className":38162,"style":2814},[2609],[53,38164,38166,38170,38210,38213,38216,38219,38222],{"className":38165},[2456],[53,38167],{"className":38168,"style":38169},[2460],"height:0.8444em;vertical-align:-0.15em;",[53,38171,38173,38176],{"className":38172},[2465],[53,38174,6077],{"className":38175,"style":5787},[2465,2506],[53,38177,38179],{"className":38178},[2474],[53,38180,38182,38202],{"className":38181},[2478,3095],[53,38183,38185,38199],{"className":38184},[2482],[53,38186,38188],{"className":38187,"style":14176},[2486],[53,38189,38190,38193],{"style":5909},[53,38191],{"className":38192,"style":2495},[2494],[53,38194,38196],{"className":38195},[2499,2500,2501,2502],[53,38197,149],{"className":38198},[2465,2502],[53,38200,3120],{"className":38201},[3119],[53,38203,38205],{"className":38204},[2482],[53,38206,38208],{"className":38207,"style":3127},[2486],[53,38209],{},[53,38211,4854],{"className":38212},[2465,2506],[53,38214,3193],{"className":38215},[2465,2506],[53,38217],{"className":38218,"style":2814},[2609],[53,38220,5572],{"className":38221},[2818],[53,38223],{"className":38224,"style":2814},[2609],[53,38226,38228,38231,38274,38277,38280,38283,38286],{"className":38227},[2456],[53,38229],{"className":38230,"style":38169},[2460],[53,38232,38234,38237],{"className":38233},[2465],[53,38235,6077],{"className":38236,"style":5787},[2465,2506],[53,38238,38240],{"className":38239},[2474],[53,38241,38243,38266],{"className":38242},[2478,3095],[53,38244,38246,38263],{"className":38245},[2482],[53,38247,38249],{"className":38248,"style":4402},[2486],[53,38250,38251,38254],{"style":5909},[53,38252],{"className":38253,"style":2495},[2494],[53,38255,38257],{"className":38256},[2499,2500,2501,2502],[53,38258,38260],{"className":38259},[2465,6934,2502],[53,38261,36608],{"className":38262},[2465,2502],[53,38264,3120],{"className":38265},[3119],[53,38267,38269],{"className":38268},[2482],[53,38270,38272],{"className":38271,"style":3127},[2486],[53,38273],{},[53,38275,4854],{"className":38276},[2465,2506],[53,38278,3193],{"className":38279},[2465,2506],[53,38281],{"className":38282,"style":2814},[2609],[53,38284,5572],{"className":38285},[2818],[53,38287],{"className":38288,"style":2814},[2609],[53,38290,38292,38295],{"className":38291},[2456],[53,38293],{"className":38294,"style":2534},[2460],[53,38296,8458],{"className":38297,"style":2671},[2465,2506],".\nNotice that ",[53,38300,38302,38330],{"className":38301},[2418],[53,38303,38305],{"className":38304},[2422],[2424,38306,38307],{"xmlns":2426},[2428,38308,38309,38327],{},[2431,38310,38311,38313,38315,38321,38323,38325],{},[2437,38312,37809],{},[2584,38314,128],{"stretchy":2650},[3063,38316,38317,38319],{},[2437,38318,6077],{},[2588,38320,149],{},[2584,38322,3544],{"stretchy":2650},[2584,38324,121],{},[2588,38326,149],{},[2444,38328,38329],{"encoding":2446},"g(E_0)=0",[53,38331,38333,38397],{"className":38332,"ariaHidden":2452},[2451],[53,38334,38336,38339,38342,38345,38385,38388,38391,38394],{"className":38335},[2456],[53,38337],{"className":38338,"style":2664},[2460],[53,38340,37809],{"className":38341,"style":2671},[2465,2506],[53,38343,128],{"className":38344},[2836],[53,38346,38348,38351],{"className":38347},[2465],[53,38349,6077],{"className":38350,"style":5787},[2465,2506],[53,38352,38354],{"className":38353},[2474],[53,38355,38357,38377],{"className":38356},[2478,3095],[53,38358,38360,38374],{"className":38359},[2482],[53,38361,38363],{"className":38362,"style":14176},[2486],[53,38364,38365,38368],{"style":5909},[53,38366],{"className":38367,"style":2495},[2494],[53,38369,38371],{"className":38370},[2499,2500,2501,2502],[53,38372,149],{"className":38373},[2465,2502],[53,38375,3120],{"className":38376},[3119],[53,38378,38380],{"className":38379},[2482],[53,38381,38383],{"className":38382,"style":3127},[2486],[53,38384],{},[53,38386,3544],{"className":38387},[2675],[53,38389],{"className":38390,"style":2814},[2609],[53,38392,121],{"className":38393},[2818],[53,38395],{"className":38396,"style":2814},[2609],[53,38398,38400,38403],{"className":38399},[2456],[53,38401],{"className":38402,"style":2624},[2460],[53,38404,149],{"className":38405},[2465],[53,38407,38409,38429],{"className":38408},[2418],[53,38410,38412],{"className":38411},[2422],[2424,38413,38414],{"xmlns":2426},[2428,38415,38416,38426],{},[2431,38417,38418,38420,38422,38424],{},[2437,38419,37809],{},[2584,38421,128],{"stretchy":2650},[2437,38423,6077],{},[2584,38425,3544],{"stretchy":2650},[2444,38427,38428],{"encoding":2446},"g(E)",[53,38430,38432],{"className":38431,"ariaHidden":2452},[2451],[53,38433,38435,38438,38441,38444,38447],{"className":38434},[2456],[53,38436],{"className":38437,"style":2664},[2460],[53,38439,37809],{"className":38440,"style":2671},[2465,2506],[53,38442,128],{"className":38443},[2836],[53,38445,6077],{"className":38446,"style":5787},[2465,2506],[53,38448,3544],{"className":38449},[2675]," grows as ",[53,38452,38454,38467],{"className":38453},[2418],[53,38455,38457],{"className":38456},[2422],[2424,38458,38459],{"xmlns":2426},[2428,38460,38461,38465],{},[2431,38462,38463],{},[2437,38464,6077],{},[2444,38466,6077],{"encoding":2446},[53,38468,38470],{"className":38469,"ariaHidden":2452},[2451],[53,38471,38473,38476],{"className":38472},[2456],[53,38474],{"className":38475,"style":2461},[2460],[53,38477,6077],{"className":38478,"style":5787},[2465,2506]," moves away from ",[53,38481,38483,38500],{"className":38482},[2418],[53,38484,38486],{"className":38485},[2422],[2424,38487,38488],{"xmlns":2426},[2428,38489,38490,38498],{},[2431,38491,38492],{},[3063,38493,38494,38496],{},[2437,38495,6077],{},[2588,38497,149],{},[2444,38499,14148],{"encoding":2446},[53,38501,38503],{"className":38502,"ariaHidden":2452},[2451],[53,38504,38506,38509],{"className":38505},[2456],[53,38507],{"className":38508,"style":3082},[2460],[53,38510,38512,38515],{"className":38511},[2465],[53,38513,6077],{"className":38514,"style":5787},[2465,2506],[53,38516,38518],{"className":38517},[2474],[53,38519,38521,38541],{"className":38520},[2478,3095],[53,38522,38524,38538],{"className":38523},[2482],[53,38525,38527],{"className":38526,"style":14176},[2486],[53,38528,38529,38532],{"style":5909},[53,38530],{"className":38531,"style":2495},[2494],[53,38533,38535],{"className":38534},[2499,2500,2501,2502],[53,38536,149],{"className":38537},[2465,2502],[53,38539,3120],{"className":38540},[3119],[53,38542,38544],{"className":38543},[2482],[53,38545,38547],{"className":38546,"style":3127},[2486],[53,38548],{},[12,38550,38551,38552,38626,38627,170,38756,38807],{},"Now using the polynomial ",[53,38553,38555,38579],{"className":38554},[2418],[53,38556,38558],{"className":38557},[2422],[2424,38559,38560],{"xmlns":2426},[2428,38561,38562,38576],{},[2431,38563,38564,38570,38572,38574],{},[2434,38565,38566,38568],{},[2437,38567,12],{},[2584,38569,20664],{},[2584,38571,128],{"stretchy":2650},[2437,38573,9034],{},[2584,38575,3544],{"stretchy":2650},[2444,38577,38578],{"encoding":2446},"p^*(x)",[53,38580,38582],{"className":38581,"ariaHidden":2452},[2451],[53,38583,38585,38588,38617,38620,38623],{"className":38584},[2456],[53,38586],{"className":38587,"style":2664},[2460],[53,38589,38591,38594],{"className":38590},[2465],[53,38592,12],{"className":38593},[2465,2506],[53,38595,38597],{"className":38596},[2474],[53,38598,38600],{"className":38599},[2478],[53,38601,38603],{"className":38602},[2482],[53,38604,38606],{"className":38605,"style":36478},[2486],[53,38607,38608,38611],{"style":2490},[53,38609],{"className":38610,"style":2495},[2494],[53,38612,38614],{"className":38613},[2499,2500,2501,2502],[53,38615,20664],{"className":38616},[2614,2502],[53,38618,128],{"className":38619},[2836],[53,38621,9034],{"className":38622},[2465,2506],[53,38624,3544],{"className":38625},[2675]," with the parameters a, b, d set to ",[53,38628,38630,38662],{"className":38629},[2418],[53,38631,38633],{"className":38632},[2422],[2424,38634,38635],{"xmlns":2426},[2428,38636,38637,38659],{},[2431,38638,38639,38641,38643,38645,38647,38653,38655,38657],{},[2437,38640,19],{},[2584,38642,121],{},[2437,38644,37809],{},[2584,38646,128],{"stretchy":2650},[3063,38648,38649,38651],{},[2437,38650,6077],{},[2588,38652,149],{},[2584,38654,2002],{},[2437,38656,27420],{},[2584,38658,3544],{"stretchy":2650},[2444,38660,38661],{"encoding":2446},"a = g(E_0 + \\delta)",[53,38663,38665,38683,38744],{"className":38664,"ariaHidden":2452},[2451],[53,38666,38668,38671,38674,38677,38680],{"className":38667},[2456],[53,38669],{"className":38670,"style":2534},[2460],[53,38672,19],{"className":38673},[2465,2506],[53,38675],{"className":38676,"style":2814},[2609],[53,38678,121],{"className":38679},[2818],[53,38681],{"className":38682,"style":2814},[2609],[53,38684,38686,38689,38692,38695,38735,38738,38741],{"className":38685},[2456],[53,38687],{"className":38688,"style":2664},[2460],[53,38690,37809],{"className":38691,"style":2671},[2465,2506],[53,38693,128],{"className":38694},[2836],[53,38696,38698,38701],{"className":38697},[2465],[53,38699,6077],{"className":38700,"style":5787},[2465,2506],[53,38702,38704],{"className":38703},[2474],[53,38705,38707,38727],{"className":38706},[2478,3095],[53,38708,38710,38724],{"className":38709},[2482],[53,38711,38713],{"className":38712,"style":14176},[2486],[53,38714,38715,38718],{"style":5909},[53,38716],{"className":38717,"style":2495},[2494],[53,38719,38721],{"className":38720},[2499,2500,2501,2502],[53,38722,149],{"className":38723},[2465,2502],[53,38725,3120],{"className":38726},[3119],[53,38728,38730],{"className":38729},[2482],[53,38731,38733],{"className":38732,"style":3127},[2486],[53,38734],{},[53,38736],{"className":38737,"style":2610},[2609],[53,38739,2002],{"className":38740},[2614],[53,38742],{"className":38743,"style":2610},[2609],[53,38745,38747,38750,38753],{"className":38746},[2456],[53,38748],{"className":38749,"style":2664},[2460],[53,38751,27420],{"className":38752,"style":27505},[2465,2506],[53,38754,3544],{"className":38755},[2675],[53,38757,38759,38777],{"className":38758},[2418],[53,38760,38762],{"className":38761},[2422],[2424,38763,38764],{"xmlns":2426},[2428,38765,38766,38774],{},[2431,38767,38768,38770,38772],{},[2437,38769,12997],{},[2584,38771,121],{},[2588,38773,173],{},[2444,38775,38776],{"encoding":2446},"b = 1",[53,38778,38780,38798],{"className":38779,"ariaHidden":2452},[2451],[53,38781,38783,38786,38789,38792,38795],{"className":38782},[2456],[53,38784],{"className":38785,"style":5329},[2460],[53,38787,12997],{"className":38788},[2465,2506],[53,38790],{"className":38791,"style":2814},[2609],[53,38793,121],{"className":38794},[2818],[53,38796],{"className":38797,"style":2814},[2609],[53,38799,38801,38804],{"className":38800},[2456],[53,38802],{"className":38803,"style":2624},[2460],[53,38805,173],{"className":38806},[2465],", and d = int(r\u002F2), we define the function:",[53,38809,38811],{"className":38810},[2679],[53,38812,38814,39016],{"className":38813},[2418],[53,38815,38817],{"className":38816},[2422],[2424,38818,38819],{"xmlns":2426,"display":2688},[2428,38820,38821,39013],{},[2431,38822,38823,38825,38827,38829,38831,38833,38839,38853,38855],{},[2437,38824,22173],{},[2584,38826,128],{"stretchy":2650},[2437,38828,6077],{},[2584,38830,3544],{"stretchy":2650},[2584,38832,121],{},[2434,38834,38835,38837],{},[2437,38836,12],{},[2584,38838,20664],{},[2431,38840,38841,38843,38845,38847,38849,38851],{},[2584,38842,128],{"fence":2452},[2437,38844,37809],{},[2584,38846,128],{"stretchy":2650},[2437,38848,6077],{},[2584,38850,3544],{"stretchy":2650},[2584,38852,3544],{"fence":2452},[2584,38854,121],{},[9386,38856,38857,38947],{},[2431,38858,38859,38865],{},[3063,38860,38861,38863],{},[2437,38862,36942],{},[2437,38864,4854],{},[2431,38866,38867,38869,38871,38873,38875,38945],{},[2584,38868,128],{"fence":2452},[2588,38870,173],{},[2584,38872,2002],{},[2588,38874,132],{},[9386,38876,38877,38923],{},[2431,38878,38879,38881,38883,38885,38887,38889,38891,38897,38899,38901,38903,38905,38907,38909,38911,38913,38915,38917,38919,38921],{},[2437,38880,37828],{},[2584,38882,36586],{},[2584,38884,128],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2584,38886,128],{"stretchy":2650},[2437,38888,6077],{},[2584,38890,2586],{},[3063,38892,38893,38895],{},[2437,38894,6077],{},[2588,38896,149],{},[2584,38898,3544],{"stretchy":2650},[2437,38900,4854],{},[2437,38902,3193],{},[2584,38904,3544],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2584,38906,2586],{},[2437,38908,37828],{},[2584,38910,36586],{},[2584,38912,128],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2437,38914,27420],{},[6062,38916,25583],{},[2437,38918,4854],{},[2437,38920,3193],{},[2584,38922,3544],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2431,38924,38925,38927,38929,38931,38933,38935,38937,38939,38941,38943],{},[2588,38926,173],{},[2584,38928,2002],{},[2437,38930,37828],{},[2584,38932,36586],{},[2584,38934,128],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2437,38936,27420],{},[6062,38938,25583],{},[2437,38940,4854],{},[2437,38942,3193],{},[2584,38944,3544],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2584,38946,3544],{"fence":2452},[2431,38948,38949,38955],{},[3063,38950,38951,38953],{},[2437,38952,36942],{},[2437,38954,4854],{},[2431,38956,38957,38959,38961,38963,38965,39011],{},[2584,38958,128],{"fence":2452},[2588,38960,173],{},[2584,38962,2002],{},[2588,38964,132],{},[9386,38966,38967,38989],{},[2431,38968,38969,38971,38973,38975,38977,38979,38981,38983,38985,38987],{},[2588,38970,173],{},[2584,38972,2586],{},[2437,38974,37828],{},[2584,38976,36586],{},[2584,38978,128],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2437,38980,27420],{},[6062,38982,25583],{},[2437,38984,4854],{},[2437,38986,3193],{},[2584,38988,3544],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2431,38990,38991,38993,38995,38997,38999,39001,39003,39005,39007,39009],{},[2588,38992,173],{},[2584,38994,2002],{},[2437,38996,37828],{},[2584,38998,36586],{},[2584,39000,128],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2437,39002,27420],{},[6062,39004,25583],{},[2437,39006,4854],{},[2437,39008,3193],{},[2584,39010,3544],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2584,39012,3544],{"fence":2452},[2444,39014,39015],{"encoding":2446},"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 + 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We can see by inserting 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that ",[53,40063,40065,40084],{"className":40064},[2418],[53,40066,40068],{"className":40067},[2422],[2424,40069,40070],{"xmlns":2426},[2428,40071,40072,40082],{},[2431,40073,40074,40076,40078,40080],{},[2437,40075,22173],{},[2584,40077,128],{"stretchy":2650},[2437,40079,6077],{},[2584,40081,3544],{"stretchy":2650},[2444,40083,27709],{"encoding":2446},[53,40085,40087],{"className":40086,"ariaHidden":2452},[2451],[53,40088,40090,40093,40096,40099,40102],{"className":40089},[2456],[53,40091],{"className":40092,"style":2664},[2460],[53,40094,22173],{"className":40095,"style":27234},[2465,2506],[53,40097,128],{"className":40098},[2836],[53,40100,6077],{"className":40101,"style":5787},[2465,2506],[53,40103,3544],{"className":40104},[2675]," is a trigonometric polynomial of degree ",[53,40107,40109,40122],{"className":40108},[2418],[53,40110,40112],{"className":40111},[2422],[2424,40113,40114],{"xmlns":2426},[2428,40115,40116,40120],{},[2431,40117,40118],{},[2437,40119,4854],{},[2444,40121,4854],{"encoding":2446},[53,40123,40125],{"className":40124,"ariaHidden":2452},[2451],[53,40126,40128,40131],{"className":40127},[2456],[53,40129],{"className":40130,"style":5329},[2460],[53,40132,4854],{"className":40133},[2465,2506],", that is, a linear combination of ",[53,40136,40138,40167],{"className":40137},[2418],[53,40139,40141],{"className":40140},[2422],[2424,40142,40143],{"xmlns":2426},[2428,40144,40145,40165],{},[2431,40146,40147],{},[2434,40148,40149,40151],{},[2437,40150,3181],{},[2431,40152,40153,40155,40157,40159,40161,40163],{},[2437,40154,3188],{},[2437,40156,2267],{},[2437,40158,6077],{},[6062,40160,25583],{},[2437,40162,4854],{},[2437,40164,3193],{},[2444,40166,27764],{"encoding":2446},[53,40168,40170],{"className":40169,"ariaHidden":2452},[2451],[53,40171,40173,40176],{"className":40172},[2456],[53,40174],{"className":40175,"style":3557},[2460],[53,40177,40179,40182],{"className":40178},[2465],[53,40180,3181],{"className":40181},[2465,2506],[53,40183,40185],{"className":40184},[2474],[53,40186,40188],{"className":40187},[2478],[53,40189,40191],{"className":40190},[2482],[53,40192,40194],{"className":40193,"style":3557},[2486],[53,40195,40196,40199],{"style":2490},[53,40197],{"className":40198,"style":2495},[2494],[53,40200,40202],{"className":40201},[2499,2500,2501,2502],[53,40203,40205,40208,40211,40214,40217],{"className":40204},[2465,2502],[53,40206,27806],{"className":40207,"style":6718},[2465,2506,2502],[53,40209,6077],{"className":40210,"style":5787},[2465,2506,2502],[53,40212],{"className":40213,"style":25793},[2609,2502],[53,40215,4854],{"className":40216},[2465,2506,2502],[53,40218,3193],{"className":40219},[2465,2506,2502],[53,40221,40223,40272],{"className":40222},[2418],[53,40224,40226],{"className":40225},[2422],[2424,40227,40228],{"xmlns":2426},[2428,40229,40230,40270],{},[2431,40231,40232,40234,40236,40238,40240,40242,40244,40246,40248,40250,40252,40254,40256,40258,40260,40262,40264,40266,40268],{},[2437,40233,2267],{},[2584,40235,121],{},[2584,40237,2586],{},[2437,40239,4854],{},[2584,40241,2714],{"separator":2452},[2584,40243,2586],{},[2437,40245,4854],{},[2584,40247,2002],{},[2588,40249,173],{},[2584,40251,2714],{"separator":2452},[2437,40253,28],{"mathvariant":2643},[2437,40255,28],{"mathvariant":2643},[2437,40257,28],{"mathvariant":2643},[2584,40259,2714],{"separator":2452},[2437,40261,4854],{},[2584,40263,2586],{},[2588,40265,173],{},[2584,40267,2714],{"separator":2452},[2437,40269,4854],{},[2444,40271,27872],{"encoding":2446},[53,40273,40275,40293,40326,40362],{"className":40274,"ariaHidden":2452},[2451],[53,40276,40278,40281,40284,40287,40290],{"className":40277},[2456],[53,40279],{"className":40280,"style":27882},[2460],[53,40282,2267],{"className":40283,"style":6718},[2465,2506],[53,40285],{"className":40286,"style":2814},[2609],[53,40288,121],{"className":40289},[2818],[53,40291],{"className":40292,"style":2814},[2609],[53,40294,40296,40299,40302,40305,40308,40311,40314,40317,40320,40323],{"className":40295},[2456],[53,40297],{"className":40298,"style":6930},[2460],[53,40300,2586],{"className":40301},[2465],[53,40303,4854],{"className":40304},[2465,2506],[53,40306,2714],{"className":40307},[2856],[53,40309],{"className":40310,"style":2860},[2609],[53,40312,2586],{"className":40313},[2465],[53,40315,4854],{"className":40316},[2465,2506],[53,40318],{"className":40319,"style":2610},[2609],[53,40321,2002],{"className":40322},[2614],[53,40324],{"className":40325,"style":2610},[2609],[53,40327,40329,40332,40335,40338,40341,40344,40347,40350,40353,40356,40359],{"className":40328},[2456],[53,40330],{"className":40331,"style":6930},[2460],[53,40333,173],{"className":40334},[2465],[53,40336,2714],{"className":40337},[2856],[53,40339],{"className":40340,"style":2860},[2609],[53,40342,2927],{"className":40343},[2465],[53,40345,2714],{"className":40346},[2856],[53,40348],{"className":40349,"style":2860},[2609],[53,40351,4854],{"className":40352},[2465,2506],[53,40354],{"className":40355,"style":2610},[2609],[53,40357,2586],{"className":40358},[2614],[53,40360],{"className":40361,"style":2610},[2609],[53,40363,40365,40368,40371,40374,40377],{"className":40364},[2456],[53,40366],{"className":40367,"style":6930},[2460],[53,40369,173],{"className":40370},[2465],[53,40372,2714],{"className":40373},[2856],[53,40375],{"className":40376,"style":2860},[2609],[53,40378,4854],{"className":40379},[2465,2506],". Furthermore, from the definition of ",[53,40382,40384,40407],{"className":40383},[2418],[53,40385,40387],{"className":40386},[2422],[2424,40388,40389],{"xmlns":2426},[2428,40390,40391,40405],{},[2431,40392,40393,40399,40401,40403],{},[2434,40394,40395,40397],{},[2437,40396,12],{},[2584,40398,20664],{},[2584,40400,128],{"stretchy":2650},[2437,40402,9034],{},[2584,40404,3544],{"stretchy":2650},[2444,40406,38578],{"encoding":2446},[53,40408,40410],{"className":40409,"ariaHidden":2452},[2451],[53,40411,40413,40416,40445,40448,40451],{"className":40412},[2456],[53,40414],{"className":40415,"style":2664},[2460],[53,40417,40419,40422],{"className":40418},[2465],[53,40420,12],{"className":40421},[2465,2506],[53,40423,40425],{"className":40424},[2474],[53,40426,40428],{"className":40427},[2478],[53,40429,40431],{"className":40430},[2482],[53,40432,40434],{"className":40433,"style":36478},[2486],[53,40435,40436,40439],{"style":2490},[53,40437],{"className":40438,"style":2495},[2494],[53,40440,40442],{"className":40441},[2499,2500,2501,2502],[53,40443,20664],{"className":40444},[2614,2502],[53,40446,128],{"className":40447},[2836],[53,40449,9034],{"className":40450},[2465,2506],[53,40452,3544],{"className":40453},[2675]," above we have that ",[53,40456,40458,40496],{"className":40457},[2418],[53,40459,40461],{"className":40460},[2422],[2424,40462,40463],{"xmlns":2426},[2428,40464,40465,40493],{},[2431,40466,40467,40469,40471,40477,40479,40481,40483,40485,40487,40489,40491],{},[2437,40468,22173],{},[2584,40470,128],{"stretchy":2650},[3063,40472,40473,40475],{},[2437,40474,6077],{},[2588,40476,149],{},[2584,40478,3544],{"stretchy":2650},[2584,40480,121],{},[2437,40482,12],{},[2584,40484,128],{"stretchy":2650},[2588,40486,149],{},[2584,40488,3544],{"stretchy":2650},[2584,40490,121],{},[2588,40492,173],{},[2444,40494,40495],{"encoding":2446},"f(E_0)=p(0)=1",[53,40497,40499,40563,40590],{"className":40498,"ariaHidden":2452},[2451],[53,40500,40502,40505,40508,40511,40551,40554,40557,40560],{"className":40501},[2456],[53,40503],{"className":40504,"style":2664},[2460],[53,40506,22173],{"className":40507,"style":27234},[2465,2506],[53,40509,128],{"className":40510},[2836],[53,40512,40514,40517],{"className":40513},[2465],[53,40515,6077],{"className":40516,"style":5787},[2465,2506],[53,40518,40520],{"className":40519},[2474],[53,40521,40523,40543],{"className":40522},[2478,3095],[53,40524,40526,40540],{"className":40525},[2482],[53,40527,40529],{"className":40528,"style":14176},[2486],[53,40530,40531,40534],{"style":5909},[53,40532],{"className":40533,"style":2495},[2494],[53,40535,40537],{"className":40536},[2499,2500,2501,2502],[53,40538,149],{"className":40539},[2465,2502],[53,40541,3120],{"className":40542},[3119],[53,40544,40546],{"className":40545},[2482],[53,40547,40549],{"className":40548,"style":3127},[2486],[53,40550],{},[53,40552,3544],{"className":40553},[2675],[53,40555],{"className":40556,"style":2814},[2609],[53,40558,121],{"className":40559},[2818],[53,40561],{"className":40562,"style":2814},[2609],[53,40564,40566,40569,40572,40575,40578,40581,40584,40587],{"className":40565},[2456],[53,40567],{"className":40568,"style":2664},[2460],[53,40570,12],{"className":40571},[2465,2506],[53,40573,128],{"className":40574},[2836],[53,40576,149],{"className":40577},[2465],[53,40579,3544],{"className":40580},[2675],[53,40582],{"className":40583,"style":2814},[2609],[53,40585,121],{"className":40586},[2818],[53,40588],{"className":40589,"style":2814},[2609],[53,40591,40593,40596],{"className":40592},[2456],[53,40594],{"className":40595,"style":2624},[2460],[53,40597,173],{"className":40598},[2465]," and for any ",[53,40601,40603,40616],{"className":40602},[2418],[53,40604,40606],{"className":40605},[2422],[2424,40607,40608],{"xmlns":2426},[2428,40609,40610,40614],{},[2431,40611,40612],{},[2437,40613,6077],{},[2444,40615,6077],{"encoding":2446},[53,40617,40619],{"className":40618,"ariaHidden":2452},[2451],[53,40620,40622,40625],{"className":40621},[2456],[53,40623],{"className":40624,"style":2461},[2460],[53,40626,6077],{"className":40627,"style":5787},[2465,2506]," in the spectral range such that ",[53,40630,40632,40662],{"className":40631},[2418],[53,40633,40635],{"className":40634},[2422],[2424,40636,40637],{"xmlns":2426},[2428,40638,40639,40659],{},[2431,40640,40641,40643,40645,40647,40653,40655,40657],{},[2437,40642,2644],{"mathvariant":2643},[2437,40644,6077],{},[2584,40646,2586],{},[3063,40648,40649,40651],{},[2437,40650,6077],{},[2588,40652,149],{},[2437,40654,2644],{"mathvariant":2643},[2584,40656,1546],{},[2437,40658,27420],{},[2444,40660,40661],{"encoding":2446},"\\vert E-E_0 \\vert > \\delta",[53,40663,40665,40686,40744],{"className":40664,"ariaHidden":2452},[2451],[53,40666,40668,40671,40674,40677,40680,40683],{"className":40667},[2456],[53,40669],{"className":40670,"style":2664},[2460],[53,40672,2644],{"className":40673},[2465],[53,40675,6077],{"className":40676,"style":5787},[2465,2506],[53,40678],{"className":40679,"style":2610},[2609],[53,40681,2586],{"className":40682},[2614],[53,40684],{"className":40685,"style":2610},[2609],[53,40687,40689,40692,40732,40735,40738,40741],{"className":40688},[2456],[53,40690],{"className":40691,"style":2664},[2460],[53,40693,40695,40698],{"className":40694},[2465],[53,40696,6077],{"className":40697,"style":5787},[2465,2506],[53,40699,40701],{"className":40700},[2474],[53,40702,40704,40724],{"className":40703},[2478,3095],[53,40705,40707,40721],{"className":40706},[2482],[53,40708,40710],{"className":40709,"style":14176},[2486],[53,40711,40712,40715],{"style":5909},[53,40713],{"className":40714,"style":2495},[2494],[53,40716,40718],{"className":40717},[2499,2500,2501,2502],[53,40719,149],{"className":40720},[2465,2502],[53,40722,3120],{"className":40723},[3119],[53,40725,40727],{"className":40726},[2482],[53,40728,40730],{"className":40729,"style":3127},[2486],[53,40731],{},[53,40733,2644],{"className":40734},[2465],[53,40736],{"className":40737,"style":2814},[2609],[53,40739,1546],{"className":40740},[2818],[53,40742],{"className":40743,"style":2814},[2609],[53,40745,40747,40750],{"className":40746},[2456],[53,40748],{"className":40749,"style":5329},[2460],[53,40751,27420],{"className":40752,"style":27505},[2465,2506]," we have",[53,40755,40757],{"className":40756},[2679],[53,40758,40760,40874],{"className":40759},[2418],[53,40761,40763],{"className":40762},[2422],[2424,40764,40765],{"xmlns":2426,"display":2688},[2428,40766,40767,40871],{},[2431,40768,40769,40771,40773,40775,40777,40779,40781,40783,40785,40787,40789,40791,40793,40795,40797,40799,40801,40813],{},[2437,40770,2644],{"mathvariant":2643},[2437,40772,22173],{},[2584,40774,128],{"stretchy":2650},[2437,40776,6077],{},[2584,40778,3544],{"stretchy":2650},[2437,40780,2644],{"mathvariant":2643},[2584,40782,27405],{},[2437,40784,36561],{},[2584,40786,128],{"stretchy":2650},[2437,40788,19],{},[2584,40790,2714],{"separator":2452},[2437,40792,12997],{},[2584,40794,2714],{"separator":2452},[2437,40796,4854],{},[2584,40798,3544],{"stretchy":2650},[2584,40800,121],{},[3670,40802,40803,40805,40807],{},[2437,40804,36942],{},[2437,40806,4854],{},[2431,40808,40809,40811],{},[2584,40810,2586],{},[2588,40812,173],{},[2431,40814,40815,40817,40819,40821,40823,40869],{},[2584,40816,128],{"fence":2452},[2588,40818,173],{},[2584,40820,2002],{},[2588,40822,132],{},[9386,40824,40825,40847],{},[2431,40826,40827,40829,40831,40833,40835,40837,40839,40841,40843,40845],{},[2588,40828,173],{},[2584,40830,2586],{},[2437,40832,37828],{},[2584,40834,36586],{},[2584,40836,128],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2437,40838,27420],{},[6062,40840,25583],{},[2437,40842,4854],{},[2437,40844,3193],{},[2584,40846,3544],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2431,40848,40849,40851,40853,40855,40857,40859,40861,40863,40865,40867],{},[2588,40850,173],{},[2584,40852,2002],{},[2437,40854,37828],{},[2584,40856,36586],{},[2584,40858,128],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2437,40860,27420],{},[6062,40862,25583],{},[2437,40864,4854],{},[2437,40866,3193],{},[2584,40868,3544],{"fence":2650,"stretchy":2452,"minsize":37833,"maxsize":37833},[2584,40870,3544],{"fence":2452},[2444,40872,40873],{"encoding":2446},"|f(E)| \\le \\beta(a, b, d) = T_d^{-1}\\left(1 + 2\\frac{1-\\cos\\big(\\delta\\,dt\\big)}{1 + 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2\\left(1 + \\delta\\right)^{-d} = 2\\left(1 + \\delta\\right)^{-\\lfloor k\u002F2\\rfloor}.",[53,41273,41275,41287,41364],{"className":41274,"ariaHidden":2452},[2451],[53,41276,41278,41281,41284],{"className":41277},[2456],[53,41279],{"className":41280,"style":27598},[2460],[53,41282,27405],{"className":41283},[2818],[53,41285],{"className":41286,"style":2814},[2609],[53,41288,41290,41293,41296,41299,41355,41358,41361],{"className":41289},[2456],[53,41291],{"className":41292,"style":27474},[2460],[53,41294,132],{"className":41295},[2465],[53,41297],{"className":41298,"style":2860},[2609],[53,41300,41302,41326],{"className":41301},[2832],[53,41303,41305,41308,41311,41314,41317,41320,41323],{"className":41304},[2832],[53,41306,128],{"className":41307,"style":2838},[2836,2837],[53,41309,173],{"className":41310},[2465],[53,41312],{"className":41313,"style":2610},[2609],[53,41315,2002],{"className":41316},[2614],[53,41318],{"className":41319,"style":2610},[2609],[53,41321,27420],{"className":41322,"style":27505},[2465,2506],[53,41324,3544],{"className":41325,"style":2838},[2675,2837],[53,41327,41329],{"className":41328},[2474],[53,41330,41332],{"className":41331},[2478],[53,41333,41335],{"className":41334},[2482],[53,41336,41338],{"className":41337,"style":27521},[2486],[53,41339,41340,41343],{"style":7740},[53,41341],{"className":41342,"style":2495},[2494],[53,41344,41346],{"className":41345},[2499,2500,2501,2502],[53,41347,41349,41352],{"className":41348},[2465,2502],[53,41350,2586],{"className":41351},[2465,2502],[53,41353,4854],{"className":41354},[2465,2506,2502],[53,41356],{"className":41357,"style":2814},[2609],[53,41359,121],{"className":41360},[2818],[53,41362],{"className":41363,"style":2814},[2609],[53,41365,41367,41371,41374,41377,41443,41446],{"className":41366},[2456],[53,41368],{"className":41369,"style":41370},[2460],"height:1.2779em;vertical-align:-0.25em;",[53,41372,132],{"className":41373},[2465],[53,41375],{"className":41376,"style":2860},[2609],[53,41378,41380,41404],{"className":41379},[2832],[53,41381,41383,41386,41389,41392,41395,41398,41401],{"className":41382},[2832],[53,41384,128],{"className":41385,"style":2838},[2836,2837],[53,41387,173],{"className":41388},[2465],[53,41390],{"className":41391,"style":2610},[2609],[53,41393,2002],{"className":41394},[2614],[53,41396],{"className":41397,"style":2610},[2609],[53,41399,27420],{"className":41400,"style":27505},[2465,2506],[53,41402,3544],{"className":41403,"style":2838},[2675,2837],[53,41405,41407],{"className":41406},[2474],[53,41408,41410],{"className":41409},[2478],[53,41411,41413],{"className":41412},[2482],[53,41414,41417],{"className":41415,"style":41416},[2486],"height:1.0279em;",[53,41418,41419,41422],{"style":7740},[53,41420],{"className":41421,"style":2495},[2494],[53,41423,41425],{"className":41424},[2499,2500,2501,2502],[53,41426,41428,41431,41434,41437,41440],{"className":41427},[2465,2502],[53,41429,2586],{"className":41430},[2465,2502],[53,41432,41257],{"className":41433},[2836,2502],[53,41435,3521],{"className":41436,"style":3587},[2465,2506,2502],[53,41438,26016],{"className":41439},[2465,2502],[53,41441,41266],{"className":41442},[2675,2502],[53,41444],{"className":41445,"style":2860},[2609],[53,41447,28],{"className":41448},[2465],[30,41450,41452],{"id":41451},"references","References",[12,41454,41455,41457,41458],{},[53,41456,173],{}," N. Yoshioka, M. Amico, W. Kirby et al. \"Diagonalization of large many-body Hamiltonians on a quantum processor\". ",[19,41459,41461],{"href":41460},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.14431","arXiv:2407.14431",[12,41463,41464,41466],{},[53,41465,132],{}," 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).",[12,41468,41469,41471],{},[53,41470,8121],{}," Å. Björck. \"Numerical methods in matrix computations\". Texts in Applied Mathematics. Springer International Publishing. (2014).",[12,41473,41474,41476],{},[53,41475,12047],{}," William Kirby. \"Analysis of quantum Krylov algorithms with errors\". Quantum 8, 1457 (2024).",[30,41478,41480],{"id":41479},"tutorial-survey","Tutorial survey",[12,41482,41483],{},"Please take this short survey to provide feedback on this tutorial. Your insights will help us improve our content offerings and user experience.",[12,41485,41486],{},[19,41487,41489],{"href":41488},"https:\u002F\u002Fyour.feedback.ibm.com\u002Fjfe\u002Fform\u002FSV_82nennpKIjjD8rQ","Link to survey",[30,41491,488],{"id":487},[12,41493,41494,41495,28],{},"Get ",[19,41496,930],{"href":929},[501,41498,41499],{},"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":48,"searchDepth":73,"depth":73,"links":41501},[41502,41503,41504,41505,41509,41516,41521,41524,41525,41526,41527,41528],{"id":954,"depth":73,"text":955},{"id":989,"depth":73,"text":990},{"id":1014,"depth":73,"text":1015},{"id":2405,"depth":73,"text":2406,"children":41506},[41507,41508],{"id":2410,"depth":87,"text":2411},{"id":3979,"depth":87,"text":3980},{"id":14110,"depth":73,"text":14111,"children":41510},[41511,41512,41513,41514],{"id":14114,"depth":87,"text":14115},{"id":20244,"depth":87,"text":20245},{"id":21231,"depth":87,"text":21232},{"id":21328,"depth":87,"text":41515},"Template circuits for calculating matrix elements of S~\\tilde{S}S~ and H~\\tilde{H}H~ via Hadamard test",{"id":21806,"depth":73,"text":21807,"children":41517},[41518,41519,41520],{"id":21920,"depth":87,"text":21921},{"id":22540,"depth":87,"text":22541},{"id":23012,"depth":87,"text":23013},{"id":23872,"depth":73,"text":23873,"children":41522},[41523],{"id":23901,"depth":87,"text":23902},{"id":25513,"depth":73,"text":25514},{"id":36343,"depth":73,"text":36344},{"id":41451,"depth":73,"text":41452},{"id":41479,"depth":73,"text":41480},{"id":487,"depth":73,"text":488},[517,518,519,41530],"Krylov diagonalization",[41532],{"username":918,"name":41533,"role":41534,"bio":41535,"links":41536},"Mirko Amico","Researcher, Extropic · formerly IBM Research","A researcher at Extropic, previously at IBM Research, and a Qiskit Pulse expert.",[41537],{"label":41538,"href":929},"Original tutorial ↗",{"username":918,"name":41533,"role":41540},"Researcher, Extropic","Implement the Krylov Quantum Diagonalization Algorithm (KQD) within the context of Qiskit patterns.","Deep dive · 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original notebook"," and shared under CC-BY-4.0. The text, code, and figures below are Onri's, as published.",[12,41569,41570],{},"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",[53,41572,41574],{"className":41573},[2679],[53,41575,41577,41608],{"className":41576},[2418],[53,41578,41580],{"className":41579},[2422],[2424,41581,41582],{"xmlns":2426,"display":2688},[2428,41583,41584,41605],{},[2431,41585,41586,41592,41594,41596,41598,41601,41603],{},[3063,41587,41588,41590],{},[2437,41589,3069],{},[2437,41591,3181],{},[2584,41593,128],{"stretchy":2650},[2437,41595,22173],{},[2584,41597,2714],{"separator":2452},[2437,41599,41600],{},"τ",[2584,41602,3544],{"stretchy":2650},[2584,41604,2714],{"separator":2452},[2444,41606,41607],{"encoding":2446},"P_e(f,\\tau),",[53,41609,41611],{"className":41610,"ariaHidden":2452},[2451],[53,41612,41614,41617,41657,41660,41663,41666,41669,41673,41676],{"className":41613},[2456],[53,41615],{"className":41616,"style":2664},[2460],[53,41618,41620,41623],{"className":41619},[2465],[53,41621,3069],{"className":41622,"style":3115},[2465,2506],[53,41624,41626],{"className":41625},[2474],[53,41627,41629,41649],{"className":41628},[2478,3095],[53,41630,41632,41646],{"className":41631},[2482],[53,41633,41635],{"className":41634,"style":4402},[2486],[53,41636,41637,41640],{"style":7532},[53,41638],{"className":41639,"style":2495},[2494],[53,41641,41643],{"className":41642},[2499,2500,2501,2502],[53,41644,3181],{"className":41645},[2465,2506,2502],[53,41647,3120],{"className":41648},[3119],[53,41650,41652],{"className":41651},[2482],[53,41653,41655],{"className":41654,"style":3127},[2486],[53,41656],{},[53,41658,128],{"className":41659},[2836],[53,41661,22173],{"className":41662,"style":27234},[2465,2506],[53,41664,2714],{"className":41665},[2856],[53,41667],{"className":41668,"style":2860},[2609],[53,41670,41600],{"className":41671,"style":41672},[2465,2506],"margin-right:0.1132em;",[53,41674,3544],{"className":41675},[2675],[53,41677,2714],{"className":41678},[2856],[12,41680,28350,41681,41709,41710,41739],{},[53,41682,41684,41697],{"className":41683},[2418],[53,41685,41687],{"className":41686},[2422],[2424,41688,41689],{"xmlns":2426},[2428,41690,41691,41695],{},[2431,41692,41693],{},[2437,41694,22173],{},[2444,41696,22173],{"encoding":2446},[53,41698,41700],{"className":41699,"ariaHidden":2452},[2451],[53,41701,41703,41706],{"className":41702},[2456],[53,41704],{"className":41705,"style":6930},[2460],[53,41707,22173],{"className":41708,"style":27234},[2465,2506]," denotes the applied drive frequency and ",[53,41711,41713,41727],{"className":41712},[2418],[53,41714,41716],{"className":41715},[2422],[2424,41717,41718],{"xmlns":2426},[2428,41719,41720,41724],{},[2431,41721,41722],{},[2437,41723,41600],{},[2444,41725,41726],{"encoding":2446},"\\tau",[53,41728,41730],{"className":41729,"ariaHidden":2452},[2451],[53,41731,41733,41736],{"className":41732},[2456],[53,41734],{"className":41735,"style":2534},[2460],[53,41737,41600],{"className":41738,"style":41672},[2465,2506]," 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.",[30,41741,41743],{"id":41742},"nomenclature-and-definitions","Nomenclature and Definitions",[12,41745,41746],{},"The following table defines the physical and mathematical quantities that recur throughout the visualization workflow.",[41748,41749,41750,41763],"table",{},[41751,41752,41753],"thead",{},[41754,41755,41756,41760],"tr",{},[41757,41758,41759],"th",{},"Symbol or Term",[41757,41761,41762],{},"Meaning",[41764,41765,41766,41813,41859,41905,42032,42109,42187,42222,42443,42540,42577,42623,42774,42851,43076,43111,43204,43212],"tbody",{},[41754,41767,41768,41810],{},[41769,41770,41771],"td",{},[53,41772,41774,41792],{"className":41773},[2418],[53,41775,41777],{"className":41776},[2422],[2424,41778,41779],{"xmlns":2426},[2428,41780,41781,41789],{},[2431,41782,41783,41785,41787],{},[2584,41784,2644],{"stretchy":2650},[2437,41786,37809],{},[2584,41788,2651],{"stretchy":2650},[2444,41790,41791],{"encoding":2446},"\\lvert g\\rangle",[53,41793,41795],{"className":41794,"ariaHidden":2452},[2451],[53,41796,41798,41801,41804,41807],{"className":41797},[2456],[53,41799],{"className":41800,"style":2664},[2460],[53,41802,2644],{"className":41803},[2836],[53,41805,37809],{"className":41806,"style":2671},[2465,2506],[53,41808,2651],{"className":41809},[2675],[41769,41811,41812],{},"Ground state of the qubit",[41754,41814,41815,41856],{},[41769,41816,41817],{},[53,41818,41820,41838],{"className":41819},[2418],[53,41821,41823],{"className":41822},[2422],[2424,41824,41825],{"xmlns":2426},[2428,41826,41827,41835],{},[2431,41828,41829,41831,41833],{},[2584,41830,2644],{"stretchy":2650},[2437,41832,3181],{},[2584,41834,2651],{"stretchy":2650},[2444,41836,41837],{"encoding":2446},"\\lvert e\\rangle",[53,41839,41841],{"className":41840,"ariaHidden":2452},[2451],[53,41842,41844,41847,41850,41853],{"className":41843},[2456],[53,41845],{"className":41846,"style":2664},[2460],[53,41848,2644],{"className":41849},[2836],[53,41851,3181],{"className":41852},[2465,2506],[53,41854,2651],{"className":41855},[2675],[41769,41857,41858],{},"Excited state of the qubit",[41754,41860,41861,41902],{},[41769,41862,41863],{},[53,41864,41866,41884],{"className":41865},[2418],[53,41867,41869],{"className":41868},[2422],[2424,41870,41871],{"xmlns":2426},[2428,41872,41873,41881],{},[2431,41874,41875,41877,41879],{},[2584,41876,2644],{"stretchy":2650},[2437,41878,3688],{},[2584,41880,2651],{"stretchy":2650},[2444,41882,41883],{"encoding":2446},"\\lvert \\psi\\rangle",[53,41885,41887],{"className":41886,"ariaHidden":2452},[2451],[53,41888,41890,41893,41896,41899],{"className":41889},[2456],[53,41891],{"className":41892,"style":2664},[2460],[53,41894,2644],{"className":41895},[2836],[53,41897,3688],{"className":41898,"style":2671},[2465,2506],[53,41900,2651],{"className":41901},[2675],[41769,41903,41904],{},"Quantum state vector in a two-dimensional Hilbert space",[41754,41906,41907,41953],{},[41769,41908,41909],{},[53,41910,41912,41931],{"className":41911},[2418],[53,41913,41915],{"className":41914},[2422],[2424,41916,41917],{"xmlns":2426},[2428,41918,41919,41928],{},[2431,41920,41921,41924,41926],{},[2437,41922,41923],{},"α",[2584,41925,2714],{"separator":2452},[2437,41927,36561],{},[2444,41929,41930],{"encoding":2446},"\\alpha,\\beta",[53,41932,41934],{"className":41933,"ariaHidden":2452},[2451],[53,41935,41937,41940,41944,41947,41950],{"className":41936},[2456],[53,41938],{"className":41939,"style":6930},[2460],[53,41941,41923],{"className":41942,"style":41943},[2465,2506],"margin-right:0.0037em;",[53,41945,2714],{"className":41946},[2856],[53,41948],{"className":41949,"style":2860},[2609],[53,41951,36561],{"className":41952,"style":36656},[2465,2506],[41769,41954,41955,41956,10694,41994],{},"Complex probability amplitudes for ",[53,41957,41959,41976],{"className":41958},[2418],[53,41960,41962],{"className":41961},[2422],[2424,41963,41964],{"xmlns":2426},[2428,41965,41966,41974],{},[2431,41967,41968,41970,41972],{},[2584,41969,2644],{"stretchy":2650},[2437,41971,37809],{},[2584,41973,2651],{"stretchy":2650},[2444,41975,41791],{"encoding":2446},[53,41977,41979],{"className":41978,"ariaHidden":2452},[2451],[53,41980,41982,41985,41988,41991],{"className":41981},[2456],[53,41983],{"className":41984,"style":2664},[2460],[53,41986,2644],{"className":41987},[2836],[53,41989,37809],{"className":41990,"style":2671},[2465,2506],[53,41992,2651],{"className":41993},[2675],[53,41995,41997,42014],{"className":41996},[2418],[53,41998,42000],{"className":41999},[2422],[2424,42001,42002],{"xmlns":2426},[2428,42003,42004,42012],{},[2431,42005,42006,42008,42010],{},[2584,42007,2644],{"stretchy":2650},[2437,42009,3181],{},[2584,42011,2651],{"stretchy":2650},[2444,42013,41837],{"encoding":2446},[53,42015,42017],{"className":42016,"ariaHidden":2452},[2451],[53,42018,42020,42023,42026,42029],{"className":42019},[2456],[53,42021],{"className":42022,"style":2664},[2460],[53,42024,2644],{"className":42025},[2836],[53,42027,3181],{"className":42028},[2465,2506],[53,42030,2651],{"className":42031},[2675],[41754,42033,42034,42106],{},[41769,42035,42036],{},[53,42037,42039,42057],{"className":42038},[2418],[53,42040,42042],{"className":42041},[2422],[2424,42043,42044],{"xmlns":2426},[2428,42045,42046,42054],{},[2431,42047,42048],{},[3063,42049,42050,42052],{},[2437,42051,3069],{},[2437,42053,3181],{},[2444,42055,42056],{"encoding":2446},"P_e",[53,42058,42060],{"className":42059,"ariaHidden":2452},[2451],[53,42061,42063,42066],{"className":42062},[2456],[53,42064],{"className":42065,"style":3082},[2460],[53,42067,42069,42072],{"className":42068},[2465],[53,42070,3069],{"className":42071,"style":3115},[2465,2506],[53,42073,42075],{"className":42074},[2474],[53,42076,42078,42098],{"className":42077},[2478,3095],[53,42079,42081,42095],{"className":42080},[2482],[53,42082,42084],{"className":42083,"style":4402},[2486],[53,42085,42086,42089],{"style":7532},[53,42087],{"className":42088,"style":2495},[2494],[53,42090,42092],{"className":42091},[2499,2500,2501,2502],[53,42093,3181],{"className":42094},[2465,2506,2502],[53,42096,3120],{"className":42097},[3119],[53,42099,42101],{"className":42100},[2482],[53,42102,42104],{"className":42103,"style":3127},[2486],[53,42105],{},[41769,42107,42108],{},"Probability of measuring the qubit in the excited state",[41754,42110,42111,42184],{},[41769,42112,42113],{},[53,42114,42116,42134],{"className":42115},[2418],[53,42117,42119],{"className":42118},[2422],[2424,42120,42121],{"xmlns":2426},[2428,42122,42123,42131],{},[2431,42124,42125],{},[3063,42126,42127,42129],{},[2437,42128,22173],{},[2588,42130,149],{},[2444,42132,42133],{"encoding":2446},"f_0",[53,42135,42137],{"className":42136,"ariaHidden":2452},[2451],[53,42138,42140,42143],{"className":42139},[2456],[53,42141],{"className":42142,"style":6930},[2460],[53,42144,42146,42149],{"className":42145},[2465],[53,42147,22173],{"className":42148,"style":27234},[2465,2506],[53,42150,42152],{"className":42151},[2474],[53,42153,42155,42176],{"className":42154},[2478,3095],[53,42156,42158,42173],{"className":42157},[2482],[53,42159,42161],{"className":42160,"style":14176},[2486],[53,42162,42164,42167],{"style":42163},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[53,42165],{"className":42166,"style":2495},[2494],[53,42168,42170],{"className":42169},[2499,2500,2501,2502],[53,42171,149],{"className":42172},[2465,2502],[53,42174,3120],{"className":42175},[3119],[53,42177,42179],{"className":42178},[2482],[53,42180,42182],{"className":42181,"style":3127},[2486],[53,42183],{},[41769,42185,42186],{},"Natural transition frequency of the qubit",[41754,42188,42189,42219],{},[41769,42190,42191],{},[53,42192,42194,42207],{"className":42193},[2418],[53,42195,42197],{"className":42196},[2422],[2424,42198,42199],{"xmlns":2426},[2428,42200,42201,42205],{},[2431,42202,42203],{},[2437,42204,22173],{},[2444,42206,22173],{"encoding":2446},[53,42208,42210],{"className":42209,"ariaHidden":2452},[2451],[53,42211,42213,42216],{"className":42212},[2456],[53,42214],{"className":42215,"style":6930},[2460],[53,42217,22173],{"className":42218,"style":27234},[2465,2506],[41769,42220,42221],{},"Applied drive frequency",[41754,42223,42224,42297],{},[41769,42225,42226],{},[53,42227,42229,42248],{"className":42228},[2418],[53,42230,42232],{"className":42231},[2422],[2424,42233,42234],{"xmlns":2426},[2428,42235,42236,42245],{},[2431,42237,42238],{},[3063,42239,42240,42243],{},[2437,42241,42242],{},"ω",[2588,42244,149],{},[2444,42246,42247],{"encoding":2446},"\\omega_0",[53,42249,42251],{"className":42250,"ariaHidden":2452},[2451],[53,42252,42254,42257],{"className":42253},[2456],[53,42255],{"className":42256,"style":25092},[2460],[53,42258,42260,42263],{"className":42259},[2465],[53,42261,42242],{"className":42262,"style":2671},[2465,2506],[53,42264,42266],{"className":42265},[2474],[53,42267,42269,42289],{"className":42268},[2478,3095],[53,42270,42272,42286],{"className":42271},[2482],[53,42273,42275],{"className":42274,"style":14176},[2486],[53,42276,42277,42280],{"style":4467},[53,42278],{"className":42279,"style":2495},[2494],[53,42281,42283],{"className":42282},[2499,2500,2501,2502],[53,42284,149],{"className":42285},[2465,2502],[53,42287,3120],{"className":42288},[3119],[53,42290,42292],{"className":42291},[2482],[53,42293,42295],{"className":42294,"style":3127},[2486],[53,42296],{},[41769,42298,42299,42300],{},"Angular transition frequency, ",[53,42301,42303,42333],{"className":42302},[2418],[53,42304,42306],{"className":42305},[2422],[2424,42307,42308],{"xmlns":2426},[2428,42309,42310,42330],{},[2431,42311,42312,42318,42320,42322,42324],{},[3063,42313,42314,42316],{},[2437,42315,42242],{},[2588,42317,149],{},[2584,42319,121],{},[2588,42321,132],{},[2437,42323,8458],{},[3063,42325,42326,42328],{},[2437,42327,22173],{},[2588,42329,149],{},[2444,42331,42332],{"encoding":2446},"\\omega_0=2\\pi f_0",[53,42334,42336,42391],{"className":42335,"ariaHidden":2452},[2451],[53,42337,42339,42342,42382,42385,42388],{"className":42338},[2456],[53,42340],{"className":42341,"style":25092},[2460],[53,42343,42345,42348],{"className":42344},[2465],[53,42346,42242],{"className":42347,"style":2671},[2465,2506],[53,42349,42351],{"className":42350},[2474],[53,42352,42354,42374],{"className":42353},[2478,3095],[53,42355,42357,42371],{"className":42356},[2482],[53,42358,42360],{"className":42359,"style":14176},[2486],[53,42361,42362,42365],{"style":4467},[53,42363],{"className":42364,"style":2495},[2494],[53,42366,42368],{"className":42367},[2499,2500,2501,2502],[53,42369,149],{"className":42370},[2465,2502],[53,42372,3120],{"className":42373},[3119],[53,42375,42377],{"className":42376},[2482],[53,42378,42380],{"className":42379,"style":3127},[2486],[53,42381],{},[53,42383],{"className":42384,"style":2814},[2609],[53,42386,121],{"className":42387},[2818],[53,42389],{"className":42390,"style":2814},[2609],[53,42392,42394,42397,42400,42403],{"className":42393},[2456],[53,42395],{"className":42396,"style":6930},[2460],[53,42398,132],{"className":42399},[2465],[53,42401,8458],{"className":42402,"style":2671},[2465,2506],[53,42404,42406,42409],{"className":42405},[2465],[53,42407,22173],{"className":42408,"style":27234},[2465,2506],[53,42410,42412],{"className":42411},[2474],[53,42413,42415,42435],{"className":42414},[2478,3095],[53,42416,42418,42432],{"className":42417},[2482],[53,42419,42421],{"className":42420,"style":14176},[2486],[53,42422,42423,42426],{"style":42163},[53,42424],{"className":42425,"style":2495},[2494],[53,42427,42429],{"className":42428},[2499,2500,2501,2502],[53,42430,149],{"className":42431},[2465,2502],[53,42433,3120],{"className":42434},[3119],[53,42436,42438],{"className":42437},[2482],[53,42439,42441],{"className":42440,"style":3127},[2486],[53,42442],{},[41754,42444,42445,42476],{},[41769,42446,42447],{},[53,42448,42450,42464],{"className":42449},[2418],[53,42451,42453],{"className":42452},[2422],[2424,42454,42455],{"xmlns":2426},[2428,42456,42457,42461],{},[2431,42458,42459],{},[2437,42460,42242],{},[2444,42462,42463],{"encoding":2446},"\\omega",[53,42465,42467],{"className":42466,"ariaHidden":2452},[2451],[53,42468,42470,42473],{"className":42469},[2456],[53,42471],{"className":42472,"style":2534},[2460],[53,42474,42242],{"className":42475,"style":2671},[2465,2506],[41769,42477,42478,42479],{},"Angular drive frequency, ",[53,42480,42482,42504],{"className":42481},[2418],[53,42483,42485],{"className":42484},[2422],[2424,42486,42487],{"xmlns":2426},[2428,42488,42489,42501],{},[2431,42490,42491,42493,42495,42497,42499],{},[2437,42492,42242],{},[2584,42494,121],{},[2588,42496,132],{},[2437,42498,8458],{},[2437,42500,22173],{},[2444,42502,42503],{"encoding":2446},"\\omega=2\\pi f",[53,42505,42507,42525],{"className":42506,"ariaHidden":2452},[2451],[53,42508,42510,42513,42516,42519,42522],{"className":42509},[2456],[53,42511],{"className":42512,"style":2534},[2460],[53,42514,42242],{"className":42515,"style":2671},[2465,2506],[53,42517],{"className":42518,"style":2814},[2609],[53,42520,121],{"className":42521},[2818],[53,42523],{"className":42524,"style":2814},[2609],[53,42526,42528,42531,42534,42537],{"className":42527},[2456],[53,42529],{"className":42530,"style":6930},[2460],[53,42532,132],{"className":42533},[2465],[53,42535,8458],{"className":42536,"style":2671},[2465,2506],[53,42538,22173],{"className":42539,"style":27234},[2465,2506],[41754,42541,42542,42574],{},[41769,42543,42544],{},[53,42545,42547,42562],{"className":42546},[2418],[53,42548,42550],{"className":42549},[2422],[2424,42551,42552],{"xmlns":2426},[2428,42553,42554,42559],{},[2431,42555,42556],{},[2437,42557,42558],{"mathvariant":2643},"Ω",[2444,42560,42561],{"encoding":2446},"\\Omega",[53,42563,42565],{"className":42564,"ariaHidden":2452},[2451],[53,42566,42568,42571],{"className":42567},[2456],[53,42569],{"className":42570,"style":2461},[2460],[53,42572,42558],{"className":42573},[2465],[41769,42575,42576],{},"On-resonance Rabi angular frequency",[41754,42578,42579,42620],{},[41769,42580,42581],{},[53,42582,42584,42604],{"className":42583},[2418],[53,42585,42587],{"className":42586},[2422],[2424,42588,42589],{"xmlns":2426},[2428,42590,42591,42601],{},[2431,42592,42593,42595,42597,42599],{},[2437,42594,42558],{"mathvariant":2643},[2437,42596,193],{"mathvariant":2643},[2588,42598,132],{},[2437,42600,8458],{},[2444,42602,42603],{"encoding":2446},"\\Omega\u002F2\\pi",[53,42605,42607],{"className":42606,"ariaHidden":2452},[2451],[53,42608,42610,42613,42617],{"className":42609},[2456],[53,42611],{"className":42612,"style":2664},[2460],[53,42614,42616],{"className":42615},[2465],"Ω\u002F2",[53,42618,8458],{"className":42619,"style":2671},[2465,2506],[41769,42621,42622],{},"On-resonance Rabi frequency in ordinary frequency units",[41754,42624,42625,42657],{},[41769,42626,42627],{},[53,42628,42630,42645],{"className":42629},[2418],[53,42631,42633],{"className":42632},[2422],[2424,42634,42635],{"xmlns":2426},[2428,42636,42637,42642],{},[2431,42638,42639],{},[2437,42640,42641],{"mathvariant":2643},"Δ",[2444,42643,42644],{"encoding":2446},"\\Delta",[53,42646,42648],{"className":42647,"ariaHidden":2452},[2451],[53,42649,42651,42654],{"className":42650},[2456],[53,42652],{"className":42653,"style":2461},[2460],[53,42655,42641],{"className":42656},[2465],[41769,42658,42659,42660],{},"Drive detuning, ",[53,42661,42663,42689],{"className":42662},[2418],[53,42664,42666],{"className":42665},[2422],[2424,42667,42668],{"xmlns":2426},[2428,42669,42670,42686],{},[2431,42671,42672,42674,42676,42678,42680],{},[2437,42673,42641],{"mathvariant":2643},[2584,42675,121],{},[2437,42677,42242],{},[2584,42679,2586],{},[3063,42681,42682,42684],{},[2437,42683,42242],{},[2588,42685,149],{},[2444,42687,42688],{"encoding":2446},"\\Delta=\\omega-\\omega_0",[53,42690,42692,42710,42728],{"className":42691,"ariaHidden":2452},[2451],[53,42693,42695,42698,42701,42704,42707],{"className":42694},[2456],[53,42696],{"className":42697,"style":2461},[2460],[53,42699,42641],{"className":42700},[2465],[53,42702],{"className":42703,"style":2814},[2609],[53,42705,121],{"className":42706},[2818],[53,42708],{"className":42709,"style":2814},[2609],[53,42711,42713,42716,42719,42722,42725],{"className":42712},[2456],[53,42714],{"className":42715,"style":2602},[2460],[53,42717,42242],{"className":42718,"style":2671},[2465,2506],[53,42720],{"className":42721,"style":2610},[2609],[53,42723,2586],{"className":42724},[2614],[53,42726],{"className":42727,"style":2610},[2609],[53,42729,42731,42734],{"className":42730},[2456],[53,42732],{"className":42733,"style":25092},[2460],[53,42735,42737,42740],{"className":42736},[2465],[53,42738,42242],{"className":42739,"style":2671},[2465,2506],[53,42741,42743],{"className":42742},[2474],[53,42744,42746,42766],{"className":42745},[2478,3095],[53,42747,42749,42763],{"className":42748},[2482],[53,42750,42752],{"className":42751,"style":14176},[2486],[53,42753,42754,42757],{"style":4467},[53,42755],{"className":42756,"style":2495},[2494],[53,42758,42760],{"className":42759},[2499,2500,2501,2502],[53,42761,149],{"className":42762},[2465,2502],[53,42764,3120],{"className":42765},[3119],[53,42767,42769],{"className":42768},[2482],[53,42770,42772],{"className":42771,"style":3127},[2486],[53,42773],{},[41754,42775,42776,42848],{},[41769,42777,42778],{},[53,42779,42781,42799],{"className":42780},[2418],[53,42782,42784],{"className":42783},[2422],[2424,42785,42786],{"xmlns":2426},[2428,42787,42788,42796],{},[2431,42789,42790],{},[3063,42791,42792,42794],{},[2437,42793,42558],{"mathvariant":2643},[2437,42795,20265],{},[2444,42797,42798],{"encoding":2446},"\\Omega_R",[53,42800,42802],{"className":42801,"ariaHidden":2452},[2451],[53,42803,42805,42808],{"className":42804},[2456],[53,42806],{"className":42807,"style":3082},[2460],[53,42809,42811,42814],{"className":42810},[2465],[53,42812,42558],{"className":42813},[2465],[53,42815,42817],{"className":42816},[2474],[53,42818,42820,42840],{"className":42819},[2478,3095],[53,42821,42823,42837],{"className":42822},[2482],[53,42824,42826],{"className":42825,"style":3102},[2486],[53,42827,42828,42831],{"style":5293},[53,42829],{"className":42830,"style":2495},[2494],[53,42832,42834],{"className":42833},[2499,2500,2501,2502],[53,42835,20265],{"className":42836,"style":20290},[2465,2506,2502],[53,42838,3120],{"className":42839},[3119],[53,42841,42843],{"className":42842},[2482],[53,42844,42846],{"className":42845,"style":3127},[2486],[53,42847],{},[41769,42849,42850],{},"Generalized Rabi angular frequency",[41754,42852,42853,42928],{},[41769,42854,42855],{},[53,42856,42858,42877],{"className":42857},[2418],[53,42859,42861],{"className":42860},[2422],[2424,42862,42863],{"xmlns":2426},[2428,42864,42865,42874],{},[2431,42866,42867],{},[3063,42868,42869,42872],{},[2437,42870,42871],{},"ν",[2437,42873,20265],{},[2444,42875,42876],{"encoding":2446},"\\nu_R",[53,42878,42880],{"className":42879,"ariaHidden":2452},[2451],[53,42881,42883,42886],{"className":42882},[2456],[53,42884],{"className":42885,"style":25092},[2460],[53,42887,42889,42893],{"className":42888},[2465],[53,42890,42871],{"className":42891,"style":42892},[2465,2506],"margin-right:0.0637em;",[53,42894,42896],{"className":42895},[2474],[53,42897,42899,42920],{"className":42898},[2478,3095],[53,42900,42902,42917],{"className":42901},[2482],[53,42903,42905],{"className":42904,"style":3102},[2486],[53,42906,42908,42911],{"style":42907},"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;",[53,42909],{"className":42910,"style":2495},[2494],[53,42912,42914],{"className":42913},[2499,2500,2501,2502],[53,42915,20265],{"className":42916,"style":20290},[2465,2506,2502],[53,42918,3120],{"className":42919},[3119],[53,42921,42923],{"className":42922},[2482],[53,42924,42926],{"className":42925,"style":3127},[2486],[53,42927],{},[41769,42929,42930,42931],{},"Generalized Rabi frequency, ",[53,42932,42934,42966],{"className":42933},[2418],[53,42935,42937],{"className":42936},[2422],[2424,42938,42939],{"xmlns":2426},[2428,42940,42941,42963],{},[2431,42942,42943,42949,42951,42957,42959,42961],{},[3063,42944,42945,42947],{},[2437,42946,42871],{},[2437,42948,20265],{},[2584,42950,121],{},[3063,42952,42953,42955],{},[2437,42954,42558],{"mathvariant":2643},[2437,42956,20265],{},[2437,42958,193],{"mathvariant":2643},[2588,42960,132],{},[2437,42962,8458],{},[2444,42964,42965],{"encoding":2446},"\\nu_R=\\Omega_R\u002F2\\pi",[53,42967,42969,43024],{"className":42968,"ariaHidden":2452},[2451],[53,42970,42972,42975,43015,43018,43021],{"className":42971},[2456],[53,42973],{"className":42974,"style":25092},[2460],[53,42976,42978,42981],{"className":42977},[2465],[53,42979,42871],{"className":42980,"style":42892},[2465,2506],[53,42982,42984],{"className":42983},[2474],[53,42985,42987,43007],{"className":42986},[2478,3095],[53,42988,42990,43004],{"className":42989},[2482],[53,42991,42993],{"className":42992,"style":3102},[2486],[53,42994,42995,42998],{"style":42907},[53,42996],{"className":42997,"style":2495},[2494],[53,42999,43001],{"className":43000},[2499,2500,2501,2502],[53,43002,20265],{"className":43003,"style":20290},[2465,2506,2502],[53,43005,3120],{"className":43006},[3119],[53,43008,43010],{"className":43009},[2482],[53,43011,43013],{"className":43012,"style":3127},[2486],[53,43014],{},[53,43016],{"className":43017,"style":2814},[2609],[53,43019,121],{"className":43020},[2818],[53,43022],{"className":43023,"style":2814},[2609],[53,43025,43027,43030,43070,43073],{"className":43026},[2456],[53,43028],{"className":43029,"style":2664},[2460],[53,43031,43033,43036],{"className":43032},[2465],[53,43034,42558],{"className":43035},[2465],[53,43037,43039],{"className":43038},[2474],[53,43040,43042,43062],{"className":43041},[2478,3095],[53,43043,43045,43059],{"className":43044},[2482],[53,43046,43048],{"className":43047,"style":3102},[2486],[53,43049,43050,43053],{"style":5293},[53,43051],{"className":43052,"style":2495},[2494],[53,43054,43056],{"className":43055},[2499,2500,2501,2502],[53,43057,20265],{"className":43058,"style":20290},[2465,2506,2502],[53,43060,3120],{"className":43061},[3119],[53,43063,43065],{"className":43064},[2482],[53,43066,43068],{"className":43067,"style":3127},[2486],[53,43069],{},[53,43071,26016],{"className":43072},[2465],[53,43074,8458],{"className":43075,"style":2671},[2465,2506],[41754,43077,43078,43108],{},[41769,43079,43080],{},[53,43081,43083,43096],{"className":43082},[2418],[53,43084,43086],{"className":43085},[2422],[2424,43087,43088],{"xmlns":2426},[2428,43089,43090,43094],{},[2431,43091,43092],{},[2437,43093,41600],{},[2444,43095,41726],{"encoding":2446},[53,43097,43099],{"className":43098,"ariaHidden":2452},[2451],[53,43100,43102,43105],{"className":43101},[2456],[53,43103],{"className":43104,"style":2534},[2460],[53,43106,41600],{"className":43107,"style":41672},[2465,2506],[41769,43109,43110],{},"Pulse duration",[41754,43112,43113,43201],{},[41769,43114,43115],{},[53,43116,43118,43138],{"className":43117},[2418],[53,43119,43121],{"className":43120},[2422],[2424,43122,43123],{"xmlns":2426},[2428,43124,43125,43135],{},[2431,43126,43127],{},[3670,43128,43129,43131,43133],{},[2437,43130,36942],{},[2588,43132,132],{},[2584,43134,20664],{},[2444,43136,43137],{"encoding":2446},"T_2^\\ast",[53,43139,43141],{"className":43140,"ariaHidden":2452},[2451],[53,43142,43144,43148],{"className":43143},[2456],[53,43145],{"className":43146,"style":43147},[2460],"height:0.9368em;vertical-align:-0.2481em;",[53,43149,43151,43154],{"className":43150},[2465],[53,43152,36942],{"className":43153,"style":3115},[2465,2506],[53,43155,43157],{"className":43156},[2474],[53,43158,43160,43192],{"className":43159},[2478,3095],[53,43161,43163,43189],{"className":43162},[2482],[53,43164,43166,43178],{"className":43165,"style":36478},[2486],[53,43167,43169,43172],{"style":43168},"top:-2.4519em;margin-left:-0.1389em;margin-right:0.05em;",[53,43170],{"className":43171,"style":2495},[2494],[53,43173,43175],{"className":43174},[2499,2500,2501,2502],[53,43176,132],{"className":43177},[2465,2502],[53,43179,43180,43183],{"style":2490},[53,43181],{"className":43182,"style":2495},[2494],[53,43184,43186],{"className":43185},[2499,2500,2501,2502],[53,43187,20664],{"className":43188},[2614,2502],[53,43190,3120],{"className":43191},[3119],[53,43193,43195],{"className":43194},[2482],[53,43196,43199],{"className":43197,"style":43198},[2486],"height:0.2481em;",[53,43200],{},[41769,43202,43203],{},"Phenomenological dephasing time used as an optional damping parameter",[41754,43205,43206,43209],{},[41769,43207,43208],{},"RWA",[41769,43210,43211],{},"Rotating-wave approximation",[41754,43213,43214,43217],{},[41769,43215,43216],{},"FFT",[41769,43218,43219],{},"Fast Fourier transform",[30,43221,43223],{"id":43222},"physical-model","Physical Model",[12,43225,43226,43227,43265,43266,43304],{},"A qubit can be modeled as a two-level quantum system with a ground state ",[53,43228,43230,43247],{"className":43229},[2418],[53,43231,43233],{"className":43232},[2422],[2424,43234,43235],{"xmlns":2426},[2428,43236,43237,43245],{},[2431,43238,43239,43241,43243],{},[2584,43240,2644],{"stretchy":2650},[2437,43242,37809],{},[2584,43244,2651],{"stretchy":2650},[2444,43246,41791],{"encoding":2446},[53,43248,43250],{"className":43249,"ariaHidden":2452},[2451],[53,43251,43253,43256,43259,43262],{"className":43252},[2456],[53,43254],{"className":43255,"style":2664},[2460],[53,43257,2644],{"className":43258},[2836],[53,43260,37809],{"className":43261,"style":2671},[2465,2506],[53,43263,2651],{"className":43264},[2675]," and an excited state ",[53,43267,43269,43286],{"className":43268},[2418],[53,43270,43272],{"className":43271},[2422],[2424,43273,43274],{"xmlns":2426},[2428,43275,43276,43284],{},[2431,43277,43278,43280,43282],{},[2584,43279,2644],{"stretchy":2650},[2437,43281,3181],{},[2584,43283,2651],{"stretchy":2650},[2444,43285,41837],{"encoding":2446},[53,43287,43289],{"className":43288,"ariaHidden":2452},[2451],[53,43290,43292,43295,43298,43301],{"className":43291},[2456],[53,43293],{"className":43294,"style":2664},[2460],[53,43296,2644],{"className":43297},[2836],[53,43299,3181],{"className":43300},[2465,2506],[53,43302,2651],{"className":43303},[2675],". In the absence of a microwave drive, the energy splitting is",[53,43306,43308],{"className":43307},[2679],[53,43309,43311,43350],{"className":43310},[2418],[53,43312,43314],{"className":43313},[2422],[2424,43315,43316],{"xmlns":2426,"display":2688},[2428,43317,43318,43347],{},[2431,43319,43320,43326,43328,43334,43336,43339,43345],{},[3063,43321,43322,43324],{},[2437,43323,6077],{},[2437,43325,3181],{},[2584,43327,2586],{},[3063,43329,43330,43332],{},[2437,43331,6077],{},[2437,43333,37809],{},[2584,43335,121],{},[2437,43337,43338],{"mathvariant":2643},"ℏ",[3063,43340,43341,43343],{},[2437,43342,42242],{},[2588,43344,149],{},[2584,43346,2714],{"separator":2452},[2444,43348,43349],{"encoding":2446},"E_e-E_g=\\hbar\\omega_0,",[53,43351,43353,43408,43463],{"className":43352,"ariaHidden":2452},[2451],[53,43354,43356,43359,43399,43402,43405],{"className":43355},[2456],[53,43357],{"className":43358,"style":3082},[2460],[53,43360,43362,43365],{"className":43361},[2465],[53,43363,6077],{"className":43364,"style":5787},[2465,2506],[53,43366,43368],{"className":43367},[2474],[53,43369,43371,43391],{"className":43370},[2478,3095],[53,43372,43374,43388],{"className":43373},[2482],[53,43375,43377],{"className":43376,"style":4402},[2486],[53,43378,43379,43382],{"style":5909},[53,43380],{"className":43381,"style":2495},[2494],[53,43383,43385],{"className":43384},[2499,2500,2501,2502],[53,43386,3181],{"className":43387},[2465,2506,2502],[53,43389,3120],{"className":43390},[3119],[53,43392,43394],{"className":43393},[2482],[53,43395,43397],{"className":43396,"style":3127},[2486],[53,43398],{},[53,43400],{"className":43401,"style":2610},[2609],[53,43403,2586],{"className":43404},[2614],[53,43406],{"className":43407,"style":2610},[2609],[53,43409,43411,43414,43454,43457,43460],{"className":43410},[2456],[53,43412],{"className":43413,"style":7862},[2460],[53,43415,43417,43420],{"className":43416},[2465],[53,43418,6077],{"className":43419,"style":5787},[2465,2506],[53,43421,43423],{"className":43422},[2474],[53,43424,43426,43446],{"className":43425},[2478,3095],[53,43427,43429,43443],{"className":43428},[2482],[53,43430,43432],{"className":43431,"style":4402},[2486],[53,43433,43434,43437],{"style":5909},[53,43435],{"className":43436,"style":2495},[2494],[53,43438,43440],{"className":43439},[2499,2500,2501,2502],[53,43441,37809],{"className":43442,"style":2671},[2465,2506,2502],[53,43444,3120],{"className":43445},[3119],[53,43447,43449],{"className":43448},[2482],[53,43450,43452],{"className":43451,"style":6728},[2486],[53,43453],{},[53,43455],{"className":43456,"style":2814},[2609],[53,43458,121],{"className":43459},[2818],[53,43461],{"className":43462,"style":2814},[2609],[53,43464,43466,43470,43473,43513],{"className":43465},[2456],[53,43467],{"className":43468,"style":43469},[2460],"height:0.8833em;vertical-align:-0.1944em;",[53,43471,43338],{"className":43472},[2465],[53,43474,43476,43479],{"className":43475},[2465],[53,43477,42242],{"className":43478,"style":2671},[2465,2506],[53,43480,43482],{"className":43481},[2474],[53,43483,43485,43505],{"className":43484},[2478,3095],[53,43486,43488,43502],{"className":43487},[2482],[53,43489,43491],{"className":43490,"style":14176},[2486],[53,43492,43493,43496],{"style":4467},[53,43494],{"className":43495,"style":2495},[2494],[53,43497,43499],{"className":43498},[2499,2500,2501,2502],[53,43500,149],{"className":43501},[2465,2502],[53,43503,3120],{"className":43504},[3119],[53,43506,43508],{"className":43507},[2482],[53,43509,43511],{"className":43510,"style":3127},[2486],[53,43512],{},[53,43514,2714],{"className":43515},[2856],[12,43517,43518],{},"where the angular transition frequency is",[53,43520,43522],{"className":43521},[2679],[53,43523,43525,43557],{"className":43524},[2418],[53,43526,43528],{"className":43527},[2422],[2424,43529,43530],{"xmlns":2426,"display":2688},[2428,43531,43532,43554],{},[2431,43533,43534,43540,43542,43544,43546,43552],{},[3063,43535,43536,43538],{},[2437,43537,42242],{},[2588,43539,149],{},[2584,43541,121],{},[2588,43543,132],{},[2437,43545,8458],{},[3063,43547,43548,43550],{},[2437,43549,22173],{},[2588,43551,149],{},[2437,43553,28],{"mathvariant":2643},[2444,43555,43556],{"encoding":2446},"\\omega_0=2\\pi f_0.",[53,43558,43560,43615],{"className":43559,"ariaHidden":2452},[2451],[53,43561,43563,43566,43606,43609,43612],{"className":43562},[2456],[53,43564],{"className":43565,"style":25092},[2460],[53,43567,43569,43572],{"className":43568},[2465],[53,43570,42242],{"className":43571,"style":2671},[2465,2506],[53,43573,43575],{"className":43574},[2474],[53,43576,43578,43598],{"className":43577},[2478,3095],[53,43579,43581,43595],{"className":43580},[2482],[53,43582,43584],{"className":43583,"style":14176},[2486],[53,43585,43586,43589],{"style":4467},[53,43587],{"className":43588,"style":2495},[2494],[53,43590,43592],{"className":43591},[2499,2500,2501,2502],[53,43593,149],{"className":43594},[2465,2502],[53,43596,3120],{"className":43597},[3119],[53,43599,43601],{"className":43600},[2482],[53,43602,43604],{"className":43603,"style":3127},[2486],[53,43605],{},[53,43607],{"className":43608,"style":2814},[2609],[53,43610,121],{"className":43611},[2818],[53,43613],{"className":43614,"style":2814},[2609],[53,43616,43618,43621,43624,43627,43667],{"className":43617},[2456],[53,43619],{"className":43620,"style":6930},[2460],[53,43622,132],{"className":43623},[2465],[53,43625,8458],{"className":43626,"style":2671},[2465,2506],[53,43628,43630,43633],{"className":43629},[2465],[53,43631,22173],{"className":43632,"style":27234},[2465,2506],[53,43634,43636],{"className":43635},[2474],[53,43637,43639,43659],{"className":43638},[2478,3095],[53,43640,43642,43656],{"className":43641},[2482],[53,43643,43645],{"className":43644,"style":14176},[2486],[53,43646,43647,43650],{"style":42163},[53,43648],{"className":43649,"style":2495},[2494],[53,43651,43653],{"className":43652},[2499,2500,2501,2502],[53,43654,149],{"className":43655},[2465,2502],[53,43657,3120],{"className":43658},[3119],[53,43660,43662],{"className":43661},[2482],[53,43663,43665],{"className":43664,"style":3127},[2486],[53,43666],{},[53,43668,28],{"className":43669},[2465],[12,43671,43672],{},"The default configuration uses a transition frequency of",[53,43674,43676],{"className":43675},[2679],[53,43677,43679,43715],{"className":43678},[2418],[53,43680,43682],{"className":43681},[2422],[2424,43683,43684],{"xmlns":2426,"display":2688},[2428,43685,43686,43712],{},[2431,43687,43688,43694,43696,43699,43701,43710],{},[3063,43689,43690,43692],{},[2437,43691,22173],{},[2588,43693,149],{},[2584,43695,121],{},[2588,43697,43698],{},"5.000",[6062,43700,6064],{},[2431,43702,43703,43706,43708],{},[2437,43704,43705],{"mathvariant":2643},"G",[2437,43707,3033],{"mathvariant":2643},[2437,43709,214],{"mathvariant":2643},[2437,43711,28],{"mathvariant":2643},[2444,43713,43714],{"encoding":2446},"f_0=5.000~\\mathrm{GHz}.",[53,43716,43718,43773],{"className":43717,"ariaHidden":2452},[2451],[53,43719,43721,43724,43764,43767,43770],{"className":43720},[2456],[53,43722],{"className":43723,"style":6930},[2460],[53,43725,43727,43730],{"className":43726},[2465],[53,43728,22173],{"className":43729,"style":27234},[2465,2506],[53,43731,43733],{"className":43732},[2474],[53,43734,43736,43756],{"className":43735},[2478,3095],[53,43737,43739,43753],{"className":43738},[2482],[53,43740,43742],{"className":43741,"style":14176},[2486],[53,43743,43744,43747],{"style":42163},[53,43745],{"className":43746,"style":2495},[2494],[53,43748,43750],{"className":43749},[2499,2500,2501,2502],[53,43751,149],{"className":43752},[2465,2502],[53,43754,3120],{"className":43755},[3119],[53,43757,43759],{"className":43758},[2482],[53,43760,43762],{"className":43761,"style":3127},[2486],[53,43763],{},[53,43765],{"className":43766,"style":2814},[2609],[53,43768,121],{"className":43769},[2818],[53,43771],{"className":43772,"style":2814},[2609],[53,43774,43776,43779,43782,43786,43794],{"className":43775},[2456],[53,43777],{"className":43778,"style":2461},[2460],[53,43780,43698],{"className":43781},[2465],[53,43783,6064],{"className":43784},[2609,43785],"nobreak",[53,43787,43789],{"className":43788},[2465],[53,43790,43793],{"className":43791},[2465,43792],"mathrm","GHz",[53,43795,28],{"className":43796},[2465],[12,43798,43799,43800,43828,43829,10694,43867,43905],{},"Applying a microwave drive with frequency ",[53,43801,43803,43816],{"className":43802},[2418],[53,43804,43806],{"className":43805},[2422],[2424,43807,43808],{"xmlns":2426},[2428,43809,43810,43814],{},[2431,43811,43812],{},[2437,43813,22173],{},[2444,43815,22173],{"encoding":2446},[53,43817,43819],{"className":43818,"ariaHidden":2452},[2451],[53,43820,43822,43825],{"className":43821},[2456],[53,43823],{"className":43824,"style":6930},[2460],[53,43826,22173],{"className":43827,"style":27234},[2465,2506]," induces coherent rotations between ",[53,43830,43832,43849],{"className":43831},[2418],[53,43833,43835],{"className":43834},[2422],[2424,43836,43837],{"xmlns":2426},[2428,43838,43839,43847],{},[2431,43840,43841,43843,43845],{},[2584,43842,2644],{"stretchy":2650},[2437,43844,37809],{},[2584,43846,2651],{"stretchy":2650},[2444,43848,41791],{"encoding":2446},[53,43850,43852],{"className":43851,"ariaHidden":2452},[2451],[53,43853,43855,43858,43861,43864],{"className":43854},[2456],[53,43856],{"className":43857,"style":2664},[2460],[53,43859,2644],{"className":43860},[2836],[53,43862,37809],{"className":43863,"style":2671},[2465,2506],[53,43865,2651],{"className":43866},[2675],[53,43868,43870,43887],{"className":43869},[2418],[53,43871,43873],{"className":43872},[2422],[2424,43874,43875],{"xmlns":2426},[2428,43876,43877,43885],{},[2431,43878,43879,43881,43883],{},[2584,43880,2644],{"stretchy":2650},[2437,43882,3181],{},[2584,43884,2651],{"stretchy":2650},[2444,43886,41837],{"encoding":2446},[53,43888,43890],{"className":43889,"ariaHidden":2452},[2451],[53,43891,43893,43896,43899,43902],{"className":43892},[2456],[53,43894],{"className":43895,"style":2664},[2460],[53,43897,2644],{"className":43898},[2836],[53,43900,3181],{"className":43901},[2465,2506],[53,43903,2651],{"className":43904},[2675],". Exact resonance occurs when",[53,43907,43909],{"className":43908},[2679],[53,43910,43912,43936],{"className":43911},[2418],[53,43913,43915],{"className":43914},[2422],[2424,43916,43917],{"xmlns":2426,"display":2688},[2428,43918,43919,43933],{},[2431,43920,43921,43923,43925,43931],{},[2437,43922,22173],{},[2584,43924,121],{},[3063,43926,43927,43929],{},[2437,43928,22173],{},[2588,43930,149],{},[2437,43932,28],{"mathvariant":2643},[2444,43934,43935],{"encoding":2446},"f=f_0.",[53,43937,43939,43957],{"className":43938,"ariaHidden":2452},[2451],[53,43940,43942,43945,43948,43951,43954],{"className":43941},[2456],[53,43943],{"className":43944,"style":6930},[2460],[53,43946,22173],{"className":43947,"style":27234},[2465,2506],[53,43949],{"className":43950,"style":2814},[2609],[53,43952,121],{"className":43953},[2818],[53,43955],{"className":43956,"style":2814},[2609],[53,43958,43960,43963,44003],{"className":43959},[2456],[53,43961],{"className":43962,"style":6930},[2460],[53,43964,43966,43969],{"className":43965},[2465],[53,43967,22173],{"className":43968,"style":27234},[2465,2506],[53,43970,43972],{"className":43971},[2474],[53,43973,43975,43995],{"className":43974},[2478,3095],[53,43976,43978,43992],{"className":43977},[2482],[53,43979,43981],{"className":43980,"style":14176},[2486],[53,43982,43983,43986],{"style":42163},[53,43984],{"className":43985,"style":2495},[2494],[53,43987,43989],{"className":43988},[2499,2500,2501,2502],[53,43990,149],{"className":43991},[2465,2502],[53,43993,3120],{"className":43994},[3119],[53,43996,43998],{"className":43997},[2482],[53,43999,44001],{"className":44000,"style":3127},[2486],[53,44002],{},[53,44004,28],{"className":44005},[2465],[12,44007,44008],{},"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.",[30,44010,44012],{"id":44011},"mathematical-foundation","Mathematical Foundation",[12,44014,44015],{},"The quantum state is a normalized vector in a two-dimensional Hilbert space,",[53,44017,44019],{"className":44018},[2679],[53,44020,44022,44062],{"className":44021},[2418],[53,44023,44025],{"className":44024},[2422],[2424,44026,44027],{"xmlns":2426,"display":2688},[2428,44028,44029,44059],{},[2431,44030,44031,44033,44035,44037,44039,44041,44043,44045,44047,44049,44051,44053,44055,44057],{},[2584,44032,2644],{"stretchy":2650},[2437,44034,3688],{},[2584,44036,2651],{"stretchy":2650},[2584,44038,121],{},[2437,44040,41923],{},[2584,44042,2644],{"stretchy":2650},[2437,44044,37809],{},[2584,44046,2651],{"stretchy":2650},[2584,44048,2002],{},[2437,44050,36561],{},[2584,44052,2644],{"stretchy":2650},[2437,44054,3181],{},[2584,44056,2651],{"stretchy":2650},[2584,44058,2714],{"separator":2452},[2444,44060,44061],{"encoding":2446},"\\lvert \\psi\\rangle=\\alpha\\lvert g\\rangle+\\beta\\lvert e\\rangle,",[53,44063,44065,44089,44116],{"className":44064,"ariaHidden":2452},[2451],[53,44066,44068,44071,44074,44077,44080,44083,44086],{"className":44067},[2456],[53,44069],{"className":44070,"style":2664},[2460],[53,44072,2644],{"className":44073},[2836],[53,44075,3688],{"className":44076,"style":2671},[2465,2506],[53,44078,2651],{"className":44079},[2675],[53,44081],{"className":44082,"style":2814},[2609],[53,44084,121],{"className":44085},[2818],[53,44087],{"className":44088,"style":2814},[2609],[53,44090,44092,44095,44098,44101,44104,44107,44110,44113],{"className":44091},[2456],[53,44093],{"className":44094,"style":2664},[2460],[53,44096,41923],{"className":44097,"style":41943},[2465,2506],[53,44099,2644],{"className":44100},[2836],[53,44102,37809],{"className":44103,"style":2671},[2465,2506],[53,44105,2651],{"className":44106},[2675],[53,44108],{"className":44109,"style":2610},[2609],[53,44111,2002],{"className":44112},[2614],[53,44114],{"className":44115,"style":2610},[2609],[53,44117,44119,44122,44125,44128,44131,44134],{"className":44118},[2456],[53,44120],{"className":44121,"style":2664},[2460],[53,44123,36561],{"className":44124,"style":36656},[2465,2506],[53,44126,2644],{"className":44127},[2836],[53,44129,3181],{"className":44130},[2465,2506],[53,44132,2651],{"className":44133},[2675],[53,44135,2714],{"className":44136},[2856],[12,44138,28350,44139,10694,44168,44197],{},[53,44140,44142,44156],{"className":44141},[2418],[53,44143,44145],{"className":44144},[2422],[2424,44146,44147],{"xmlns":2426},[2428,44148,44149,44153],{},[2431,44150,44151],{},[2437,44152,41923],{},[2444,44154,44155],{"encoding":2446},"\\alpha",[53,44157,44159],{"className":44158,"ariaHidden":2452},[2451],[53,44160,44162,44165],{"className":44161},[2456],[53,44163],{"className":44164,"style":2534},[2460],[53,44166,41923],{"className":44167,"style":41943},[2465,2506],[53,44169,44171,44185],{"className":44170},[2418],[53,44172,44174],{"className":44173},[2422],[2424,44175,44176],{"xmlns":2426},[2428,44177,44178,44182],{},[2431,44179,44180],{},[2437,44181,36561],{},[2444,44183,44184],{"encoding":2446},"\\beta",[53,44186,44188],{"className":44187,"ariaHidden":2452},[2451],[53,44189,44191,44194],{"className":44190},[2456],[53,44192],{"className":44193,"style":6930},[2460],[53,44195,36561],{"className":44196,"style":36656},[2465,2506]," are complex probability amplitudes. Normalization requires",[53,44199,44201],{"className":44200},[2679],[53,44202,44204,44243],{"className":44203},[2418],[53,44205,44207],{"className":44206},[2422],[2424,44208,44209],{"xmlns":2426,"display":2688},[2428,44210,44211,44240],{},[2431,44212,44213,44215,44217,44223,44225,44227,44229,44235,44237],{},[2437,44214,2644],{"mathvariant":2643},[2437,44216,41923],{},[2434,44218,44219,44221],{},[2437,44220,2644],{"mathvariant":2643},[2588,44222,132],{},[2584,44224,2002],{},[2437,44226,2644],{"mathvariant":2643},[2437,44228,36561],{},[2434,44230,44231,44233],{},[2437,44232,2644],{"mathvariant":2643},[2588,44234,132],{},[2584,44236,121],{},[2588,44238,44239],{},"1.",[2444,44241,44242],{"encoding":2446},"|\\alpha|^2+|\\beta|^2=1.",[53,44244,44246,44297,44347],{"className":44245,"ariaHidden":2452},[2451],[53,44247,44249,44253,44256,44259,44288,44291,44294],{"className":44248},[2456],[53,44250],{"className":44251,"style":44252},[2460],"height:1.1141em;vertical-align:-0.25em;",[53,44254,2644],{"className":44255},[2465],[53,44257,41923],{"className":44258,"style":41943},[2465,2506],[53,44260,44262,44265],{"className":44261},[2465],[53,44263,2644],{"className":44264},[2465],[53,44266,44268],{"className":44267},[2474],[53,44269,44271],{"className":44270},[2478],[53,44272,44274],{"className":44273},[2482],[53,44275,44277],{"className":44276,"style":2897},[2486],[53,44278,44279,44282],{"style":2801},[53,44280],{"className":44281,"style":2495},[2494],[53,44283,44285],{"className":44284},[2499,2500,2501,2502],[53,44286,132],{"className":44287},[2465,2502],[53,44289],{"className":44290,"style":2610},[2609],[53,44292,2002],{"className":44293},[2614],[53,44295],{"className":44296,"style":2610},[2609],[53,44298,44300,44303,44306,44309,44338,44341,44344],{"className":44299},[2456],[53,44301],{"className":44302,"style":44252},[2460],[53,44304,2644],{"className":44305},[2465],[53,44307,36561],{"className":44308,"style":36656},[2465,2506],[53,44310,44312,44315],{"className":44311},[2465],[53,44313,2644],{"className":44314},[2465],[53,44316,44318],{"className":44317},[2474],[53,44319,44321],{"className":44320},[2478],[53,44322,44324],{"className":44323},[2482],[53,44325,44327],{"className":44326,"style":2897},[2486],[53,44328,44329,44332],{"style":2801},[53,44330],{"className":44331,"style":2495},[2494],[53,44333,44335],{"className":44334},[2499,2500,2501,2502],[53,44336,132],{"className":44337},[2465,2502],[53,44339],{"className":44340,"style":2814},[2609],[53,44342,121],{"className":44343},[2818],[53,44345],{"className":44346,"style":2814},[2609],[53,44348,44350,44353],{"className":44349},[2456],[53,44351],{"className":44352,"style":2624},[2460],[53,44354,44239],{"className":44355},[2465],[12,44357,44358],{},"The Born rule gives the probability of measuring the excited state,",[53,44360,44362],{"className":44361},[2679],[53,44363,44365,44417],{"className":44364},[2418],[53,44366,44368],{"className":44367},[2422],[2424,44369,44370],{"xmlns":2426,"display":2688},[2428,44371,44372,44414],{},[2431,44373,44374,44380,44382,44384,44386,44388,44390,44392,44394,44400,44402,44404,44406,44412],{},[3063,44375,44376,44378],{},[2437,44377,3069],{},[2437,44379,3181],{},[2584,44381,121],{},[2437,44383,2644],{"mathvariant":2643},[2584,44385,4320],{"stretchy":2650},[2437,44387,3181],{},[2584,44389,2644],{"stretchy":2650},[2437,44391,3688],{},[2584,44393,2651],{"stretchy":2650},[2434,44395,44396,44398],{},[2437,44397,2644],{"mathvariant":2643},[2588,44399,132],{},[2584,44401,121],{},[2437,44403,2644],{"mathvariant":2643},[2437,44405,36561],{},[2434,44407,44408,44410],{},[2437,44409,2644],{"mathvariant":2643},[2588,44411,132],{},[2437,44413,28],{"mathvariant":2643},[2444,44415,44416],{"encoding":2446},"P_e=|\\langle e\\lvert\\psi\\rangle|^2=|\\beta|^2.",[53,44418,44420,44475,44537],{"className":44419,"ariaHidden":2452},[2451],[53,44421,44423,44426,44466,44469,44472],{"className":44422},[2456],[53,44424],{"className":44425,"style":3082},[2460],[53,44427,44429,44432],{"className":44428},[2465],[53,44430,3069],{"className":44431,"style":3115},[2465,2506],[53,44433,44435],{"className":44434},[2474],[53,44436,44438,44458],{"className":44437},[2478,3095],[53,44439,44441,44455],{"className":44440},[2482],[53,44442,44444],{"className":44443,"style":4402},[2486],[53,44445,44446,44449],{"style":7532},[53,44447],{"className":44448,"style":2495},[2494],[53,44450,44452],{"className":44451},[2499,2500,2501,2502],[53,44453,3181],{"className":44454},[2465,2506,2502],[53,44456,3120],{"className":44457},[3119],[53,44459,44461],{"className":44460},[2482],[53,44462,44464],{"className":44463,"style":3127},[2486],[53,44465],{},[53,44467],{"className":44468,"style":2814},[2609],[53,44470,121],{"className":44471},[2818],[53,44473],{"className":44474,"style":2814},[2609],[53,44476,44478,44481,44484,44487,44490,44493,44496,44499,44528,44531,44534],{"className":44477},[2456],[53,44479],{"className":44480,"style":44252},[2460],[53,44482,2644],{"className":44483},[2465],[53,44485,4320],{"className":44486},[2836],[53,44488,3181],{"className":44489},[2465,2506],[53,44491,2644],{"className":44492},[2836],[53,44494,3688],{"className":44495,"style":2671},[2465,2506],[53,44497,2651],{"className":44498},[2675],[53,44500,44502,44505],{"className":44501},[2465],[53,44503,2644],{"className":44504},[2465],[53,44506,44508],{"className":44507},[2474],[53,44509,44511],{"className":44510},[2478],[53,44512,44514],{"className":44513},[2482],[53,44515,44517],{"className":44516,"style":2897},[2486],[53,44518,44519,44522],{"style":2801},[53,44520],{"className":44521,"style":2495},[2494],[53,44523,44525],{"className":44524},[2499,2500,2501,2502],[53,44526,132],{"className":44527},[2465,2502],[53,44529],{"className":44530,"style":2814},[2609],[53,44532,121],{"className":44533},[2818],[53,44535],{"className":44536,"style":2814},[2609],[53,44538,44540,44543,44546,44549,44578],{"className":44539},[2456],[53,44541],{"className":44542,"style":44252},[2460],[53,44544,2644],{"className":44545},[2465],[53,44547,36561],{"className":44548,"style":36656},[2465,2506],[53,44550,44552,44555],{"className":44551},[2465],[53,44553,2644],{"className":44554},[2465],[53,44556,44558],{"className":44557},[2474],[53,44559,44561],{"className":44560},[2478],[53,44562,44564],{"className":44563},[2482],[53,44565,44567],{"className":44566,"style":2897},[2486],[53,44568,44569,44572],{"style":2801},[53,44570],{"className":44571,"style":2495},[2494],[53,44573,44575],{"className":44574},[2499,2500,2501,2502],[53,44576,132],{"className":44577},[2465,2502],[53,44579,28],{"className":44580},[2465],[12,44582,44583],{},"Pauli matrices then provide a natural operator basis for the two-level system,",[53,44585,44587],{"className":44586},[2679],[53,44588,44590,44754],{"className":44589},[2418],[53,44591,44593],{"className":44592},[2422],[2424,44594,44595],{"xmlns":2426,"display":2688},[2428,44596,44597,44751],{},[2431,44598,44599,44606,44608,44645,44647,44649,44655,44657,44697,44699,44701,44707,44709,44749],{},[3063,44600,44601,44604],{},[2437,44602,44603],{},"σ",[2437,44605,9034],{},[2584,44607,121],{},[2431,44609,44610,44612,44643],{},[2584,44611,12143],{"fence":2452},[9347,44613,44615,44629],{"rowspacing":9349,"columnalign":44614,"columnspacing":9350},"center center",[9352,44616,44617,44623],{},[9355,44618,44619],{},[9358,44620,44621],{"scriptlevel":149,"displaystyle":2650},[2588,44622,149],{},[9355,44624,44625],{},[9358,44626,44627],{"scriptlevel":149,"displaystyle":2650},[2588,44628,173],{},[9352,44630,44631,44637],{},[9355,44632,44633],{},[9358,44634,44635],{"scriptlevel":149,"displaystyle":2650},[2588,44636,173],{},[9355,44638,44639],{},[9358,44640,44641],{"scriptlevel":149,"displaystyle":2650},[2588,44642,149],{},[2584,44644,12168],{"fence":2452},[2584,44646,2714],{"separator":2452},[2609,44648],{"width":17906},[3063,44650,44651,44653],{},[2437,44652,44603],{},[2437,44654,20350],{},[2584,44656,121],{},[2431,44658,44659,44661,44695],{},[2584,44660,12143],{"fence":2452},[9347,44662,44663,44681],{"rowspacing":9349,"columnalign":44614,"columnspacing":9350},[9352,44664,44665,44671],{},[9355,44666,44667],{},[9358,44668,44669],{"scriptlevel":149,"displaystyle":2650},[2588,44670,149],{},[9355,44672,44673],{},[9358,44674,44675],{"scriptlevel":149,"displaystyle":2650},[2431,44676,44677,44679],{},[2584,44678,2586],{},[2437,44680,3188],{},[9352,44682,44683,44689],{},[9355,44684,44685],{},[9358,44686,44687],{"scriptlevel":149,"displaystyle":2650},[2437,44688,3188],{},[9355,44690,44691],{},[9358,44692,44693],{"scriptlevel":149,"displaystyle":2650},[2588,44694,149],{},[2584,44696,12168],{"fence":2452},[2584,44698,2714],{"separator":2452},[2609,44700],{"width":17906},[3063,44702,44703,44705],{},[2437,44704,44603],{},[2437,44706,214],{},[2584,44708,121],{},[2431,44710,44711,44713,44747],{},[2584,44712,12143],{"fence":2452},[9347,44714,44715,44729],{"rowspacing":9349,"columnalign":44614,"columnspacing":9350},[9352,44716,44717,44723],{},[9355,44718,44719],{},[9358,44720,44721],{"scriptlevel":149,"displaystyle":2650},[2588,44722,173],{},[9355,44724,44725],{},[9358,44726,44727],{"scriptlevel":149,"displaystyle":2650},[2588,44728,149],{},[9352,44730,44731,44737],{},[9355,44732,44733],{},[9358,44734,44735],{"scriptlevel":149,"displaystyle":2650},[2588,44736,149],{},[9355,44738,44739],{},[9358,44740,44741],{"scriptlevel":149,"displaystyle":2650},[2431,44742,44743,44745],{},[2584,44744,2586],{},[2588,44746,173],{},[2584,44748,12168],{"fence":2452},[2437,44750,28],{"mathvariant":2643},[2444,44752,44753],{"encoding":2446},"\\sigma_x=\\begin{bmatrix}0 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-1\\end{bmatrix}.",[53,44755,44757,44812,45002,45189],{"className":44756,"ariaHidden":2452},[2451],[53,44758,44760,44763,44803,44806,44809],{"className":44759},[2456],[53,44761],{"className":44762,"style":25092},[2460],[53,44764,44766,44769],{"className":44765},[2465],[53,44767,44603],{"className":44768,"style":2671},[2465,2506],[53,44770,44772],{"className":44771},[2474],[53,44773,44775,44795],{"className":44774},[2478,3095],[53,44776,44778,44792],{"className":44777},[2482],[53,44779,44781],{"className":44780,"style":4402},[2486],[53,44782,44783,44786],{"style":4467},[53,44784],{"className":44785,"style":2495},[2494],[53,44787,44789],{"className":44788},[2499,2500,2501,2502],[53,44790,9034],{"className":44791},[2465,2506,2502],[53,44793,3120],{"className":44794},[3119],[53,44796,44798],{"className":44797},[2482],[53,44799,44801],{"className":44800,"style":3127},[2486],[53,44802],{},[53,44804],{"className":44805,"style":2814},[2609],[53,44807,121],{"className":44808},[2818],[53,44810],{"className":44811,"style":2814},[2609],[53,44813,44815,44818,44941,44944,44947,44950,44953,44993,44996,44999],{"className":44814},[2456],[53,44816],{"className":44817,"style":37564},[2460],[53,44819,44821,44827,44935],{"className":44820},[2832],[53,44822,44824],{"className":44823,"style":2838},[2836,2837],[53,44825,12143],{"className":44826},[2842,2501],[53,44828,44830],{"className":44829},[2465],[53,44831,44833,44882,44887,44890],{"className":44832},[9347],[53,44834,44836],{"className":44835},[9606],[53,44837,44839,44873],{"className":44838},[2478,3095],[53,44840,44842,44870],{"className":44841},[2482],[53,44843,44846,44858],{"className":44844,"style":44845},[2486],"height:1.45em;",[53,44847,44849,44852],{"style":44848},"top:-3.61em;",[53,44850],{"className":44851,"style":4205},[2494],[53,44853,44855],{"className":44854},[2465],[53,44856,149],{"className":44857},[2465],[53,44859,44861,44864],{"style":44860},"top:-2.41em;",[53,44862],{"className":44863,"style":4205},[2494],[53,44865,44867],{"className":44866},[2465],[53,44868,173],{"className":44869},[2465],[53,44871,3120],{"className":44872},[3119],[53,44874,44876],{"className":44875},[2482],[53,44877,44880],{"className":44878,"style":44879},[2486],"height:0.95em;",[53,44881],{},[53,44883],{"className":44884,"style":44886},[44885],"arraycolsep","width:0.5em;",[53,44888],{"className":44889,"style":44886},[44885],[53,44891,44893],{"className":44892},[9606],[53,44894,44896,44927],{"className":44895},[2478,3095],[53,44897,44899,44924],{"className":44898},[2482],[53,44900,44902,44913],{"className":44901,"style":44845},[2486],[53,44903,44904,44907],{"style":44848},[53,44905],{"className":44906,"style":4205},[2494],[53,44908,44910],{"className":44909},[2465],[53,44911,173],{"className":44912},[2465],[53,44914,44915,44918],{"style":44860},[53,44916],{"className":44917,"style":4205},[2494],[53,44919,44921],{"className":44920},[2465],[53,44922,149],{"className":44923},[2465],[53,44925,3120],{"className":44926},[3119],[53,44928,44930],{"className":44929},[2482],[53,44931,44933],{"className":44932,"style":44879},[2486],[53,44934],{},[53,44936,44938],{"className":44937,"style":2838},[2675,2837],[53,44939,12168],{"className":44940},[2842,2501],[53,44942],{"className":44943,"style":2860},[2609],[53,44945,2714],{"className":44946},[2856],[53,44948],{"className":44949,"style":18524},[2609],[53,44951],{"className":44952,"style":2860},[2609],[53,44954,44956,44959],{"className":44955},[2465],[53,44957,44603],{"className":44958,"style":2671},[2465,2506],[53,44960,44962],{"className":44961},[2474],[53,44963,44965,44985],{"className":44964},[2478,3095],[53,44966,44968,44982],{"className":44967},[2482],[53,44969,44971],{"className":44970,"style":4402},[2486],[53,44972,44973,44976],{"style":4467},[53,44974],{"className":44975,"style":2495},[2494],[53,44977,44979],{"className":44978},[2499,2500,2501,2502],[53,44980,20350],{"className":44981,"style":2671},[2465,2506,2502],[53,44983,3120],{"className":44984},[3119],[53,44986,44988],{"className":44987},[2482],[53,44989,44991],{"className":44990,"style":6728},[2486],[53,44992],{},[53,44994],{"className":44995,"style":2814},[2609],[53,44997,121],{"className":44998},[2818],[53,45000],{"className":45001,"style":2814},[2609],[53,45003,45005,45008,45128,45131,45134,45137,45140,45180,45183,45186],{"className":45004},[2456],[53,45006],{"className":45007,"style":37564},[2460],[53,45009,45011,45017,45122],{"className":45010},[2832],[53,45012,45014],{"className":45013,"style":2838},[2836,2837],[53,45015,12143],{"className":45016},[2842,2501],[53,45018,45020],{"className":45019},[2465],[53,45021,45023,45068,45071,45074],{"className":45022},[9347],[53,45024,45026],{"className":45025},[9606],[53,45027,45029,45060],{"className":45028},[2478,3095],[53,45030,45032,45057],{"className":45031},[2482],[53,45033,45035,45046],{"className":45034,"style":44845},[2486],[53,45036,45037,45040],{"style":44848},[53,45038],{"className":45039,"style":4205},[2494],[53,45041,45043],{"cl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Applied detuning supplies the longitudinal control component.",[30,45359,45361],{"id":45360},"rotating-frame-hamiltonian","Rotating-Frame Hamiltonian",[12,45363,45364],{},"In the laboratory frame, a minimal semiclassical Hamiltonian for the driven qubit is",[53,45366,45368],{"className":45367},[2679],[53,45369,45371,45439],{"className":45370},[2418],[53,45372,45374],{"className":45373},[2422],[2424,45375,45376],{"xmlns":2426,"display":2688},[2428,45377,45378,45436],{},[2431,45379,45380,45382,45384,45386,45388,45390,45404,45410,45412,45414,45416,45418,45420,45422,45424,45426,45428,45434],{},[2437,45381,3033],{},[2584,45383,128],{"stretchy":2650},[2437,45385,3193],{},[2584,45387,3544],{"stretchy":2650},[2584,45389,121],{},[9386,45391,45392,45402],{},[2431,45393,45394,45396],{},[2437,45395,43338],{"mathvariant":2643},[3063,45397,45398,45400],{},[2437,45399,42242],{},[2588,45401,149],{},[2588,45403,132],{},[3063,45405,45406,45408],{},[2437,45407,44603],{},[2437,45409,214],{},[2584,45411,2002],{},[2437,45413,43338],{"mathvariant":2643},[2437,45415,42558],{"mathvariant":2643},[2437,45417,37828],{},[2584,45419,36586],{},[2584,45421,128],{"stretchy":2650},[2437,45423,42242],{},[2437,45425,3193],{},[2584,45427,3544],{"stretchy":2650},[3063,45429,45430,45432],{},[2437,45431,44603],{},[2437,45433,9034],{},[2584,45435,2714],{"separator":2452},[2444,45437,45438],{"encoding":2446},"H(t)=\\frac{\\hbar\\omega_0}{2}\\sigma_z+\\hbar\\Omega\\cos(\\omega t)\\sigma_x,",[53,45440,45442,45469,45628],{"className":45441,"ariaHidden":2452},[2451],[53,45443,45445,45448,45451,45454,45457,45460,45463,45466],{"className":45444},[2456],[53,45446],{"className":45447,"style":2664},[2460],[53,45449,3033],{"className":45450,"style":3048},[2465,2506],[53,45452,128],{"className":45453},[2836],[53,45455,3193],{"className":45456},[2465,2506],[53,45458,3544],{"className":45459},[2675],[53,45461],{"className":45462,"style":2814},[2609],[53,45464,121],{"className":45465},[2818],[53,45467],{"className":45468,"style":2814},[2609],[53,45470,45472,45476,45579,45619,45622,45625],{"className":45471},[2456],[53,45473],{"className":45474,"style":45475},[2460],"height:2.0519em;vertical-align:-0.686em;",[53,45477,45479,45482,45576],{"className":45478},[2465],[53,45480],{"className":45481},[2836,9688],[53,45483,45485],{"className":45484},[9386],[53,45486,45488,45568],{"className":45487},[2478,3095],[53,45489,45491,45565],{"className":45490},[2482],[53,45492,45495,45506,45514],{"className":45493,"style":45494},[2486],"height:1.3659em;",[53,45496,45497,45500],{"style":11612},[53,45498],{"className":45499,"style":4205},[2494],[53,45501,45503],{"className":45502},[2465],[53,45504,132],{"className":45505},[2465],[53,45507,45508,45511],{"style":9773},[53,45509],{"className":45510,"style":4205},[2494],[53,45512],{"className":45513,"style":9781},[9780],[53,45515,45516,45519],{"style":9784},[53,45517],{"className":45518,"style":4205},[2494],[53,45520,45522,45525],{"className":45521},[2465],[53,45523,43338],{"className":45524},[2465],[53,45526,45528,45531],{"className":45527},[2465],[53,45529,42242],{"className":45530,"style":2671},[2465,2506],[53,45532,45534],{"className":45533},[2474],[53,45535,45537,45557],{"className":45536},[2478,3095],[53,45538,45540,45554],{"className":45539},[2482],[53,45541,45543],{"className":45542,"style":14176},[2486],[53,45544,45545,45548],{"style":4467},[53,45546],{"className":45547,"style":2495},[2494],[53,45549,45551],{"className":45550},[2499,2500,2501,2502],[53,45552,149],{"className":45553},[2465,2502],[53,45555,3120],{"className":45556},[3119],[53,45558,45560],{"className":45559},[2482],[53,45561,45563],{"className":45562,"style":3127},[2486],[53,45564],{},[53,45566,3120],{"className":45567},[3119],[53,45569,45571],{"className":45570},[2482],[53,45572,45574],{"className":45573,"style":11650},[2486],[53,45575],{},[53,45577],{"className":45578},[2675,9688],[53,45580,45582,45585],{"className":45581},[2465],[53,45583,44603],{"className":45584,"style":2671},[2465,2506],[53,45586,45588],{"className":45587},[2474],[53,45589,45591,45611],{"className":45590},[2478,3095],[53,45592,45594,45608],{"className":45593},[2482],[53,45595,45597],{"className":45596,"style":4402},[2486],[53,45598,45599,45602],{"style":4467},[53,45600],{"className":45601,"style":2495},[2494],[53,45603,45605],{"className":45604},[2499,2500,2501,2502],[53,45606,214],{"className":45607,"style":20477},[2465,2506,2502],[53,45609,3120],{"className":45610},[3119],[53,45612,45614],{"className":45613},[2482],[53,45615,45617],{"className":45616,"style":3127},[2486],[53,45618],{},[53,45620],{"className":45621,"style":2610},[2609],[53,45623,2002],{"className":45624},[2614],[53,45626],{"className":45627,"style":2610},[2609],[53,45629,45631,45634,45638,45641,45644,45647,45650,45653,45656,45696],{"className":45630},[2456],[53,45632],{"className":45633,"style":2664},[2460],[53,45635,45637],{"className":45636},[2465],"ℏΩ",[53,45639],{"className":45640,"style":2860},[2609],[53,45642,37828],{"className":45643},[7465],[53,45645,128],{"className":45646},[2836],[53,45648,42242],{"className":45649,"style":2671},[2465,2506],[53,45651,3193],{"className":45652},[2465,2506],[53,45654,3544],{"className":45655},[2675],[53,45657,45659,45662],{"className":45658},[2465],[53,45660,44603],{"className":45661,"style":2671},[2465,2506],[53,45663,45665],{"className":45664},[2474],[53,45666,45668,45688],{"className":45667},[2478,3095],[53,45669,45671,45685],{"className":45670},[2482],[53,45672,45674],{"className":45673,"style":4402},[2486],[53,45675,45676,45679],{"style":4467},[53,45677],{"className":45678,"style":2495},[2494],[53,45680,45682],{"className":45681},[2499,2500,2501,2502],[53,45683,9034],{"className":45684},[2465,2506,2502],[53,45686,3120],{"className":45687},[3119],[53,45689,45691],{"className":45690},[2482],[53,45692,45694],{"className":45693,"style":3127},[2486],[53,45695],{},[53,45697,2714],{"className":45698},[2856],[12,45700,28350,45701,45761,45762,45790,45791,45828],{},[53,45702,45704,45725],{"className":45703},[2418],[53,45705,45707],{"className":45706},[2422],[2424,45708,45709],{"xmlns":2426},[2428,45710,45711,45723],{},[2431,45712,45713,45715,45717,45719,45721],{},[2437,45714,42242],{},[2584,45716,121],{},[2588,45718,132],{},[2437,45720,8458],{},[2437,45722,22173],{},[2444,45724,42503],{"encoding":2446},[53,45726,45728,45746],{"className":45727,"ariaHidden":2452},[2451],[53,45729,45731,45734,45737,45740,45743],{"className":45730},[2456],[53,45732],{"className":45733,"style":2534},[2460],[53,45735,42242],{"className":45736,"style":2671},[2465,2506],[53,45738],{"className":45739,"style":2814},[2609],[53,45741,121],{"className":45742},[2818],[53,45744],{"className":45745,"style":2814},[2609],[53,45747,45749,45752,45755,45758],{"className":45748},[2456],[53,45750],{"className":45751,"style":6930},[2460],[53,45753,132],{"className":45754},[2465],[53,45756,8458],{"className":45757,"style":2671},[2465,2506],[53,45759,22173],{"className":45760,"style":27234},[2465,2506]," is the drive angular frequency and ",[53,45763,45765,45778],{"className":45764},[2418],[53,45766,45768],{"className":45767},[2422],[2424,45769,45770],{"xmlns":2426},[2428,45771,45772,45776],{},[2431,45773,45774],{},[2437,45775,42558],{"mathvariant":2643},[2444,45777,42561],{"encoding":2446},[53,45779,45781],{"className":45780,"ariaHidden":2452},[2451],[53,45782,45784,45787],{"className":45783},[2456],[53,45785],{"className":45786,"style":2461},[2460],[53,45788,42558],{"className":45789},[2465]," is the on-resonance Rabi angular frequency. The computational model specifies the drive strength using ",[53,45792,45794,45813],{"className":45793},[2418],[53,45795,45797],{"className":45796},[2422],[2424,45798,45799],{"xmlns":2426},[2428,45800,45801,45811],{},[2431,45802,45803,45805,45807,45809],{},[2437,45804,42558],{"mathvariant":2643},[2437,45806,193],{"mathvariant":2643},[2588,45808,132],{},[2437,45810,8458],{},[2444,45812,42603],{"encoding":2446},[53,45814,45816],{"className":45815,"ariaHidden":2452},[2451],[53,45817,45819,45822,45825],{"className":45818},[2456],[53,45820],{"className":45821,"style":2664},[2460],[53,45823,42616],{"className":45824},[2465],[53,45826,8458],{"className":45827,"style":2671},[2465,2506]," in MHz. Transforming into a frame rotating at the drive frequency and applying the rotating-wave approximation gives the effective time-independent Hamiltonian",[53,45830,45832],{"className":45831},[2679],[53,45833,45835,45894],{"className":45834},[2418],[53,45836,45838],{"className":45837},[2422],[2424,45839,45840],{"xmlns":2426,"display":2688},[2428,45841,45842,45891],{},[2431,45843,45844,45857,45859,45865,45889],{},[3063,45845,45846,45848],{},[2437,45847,3033],{},[2431,45849,45850,45852,45855],{},[2437,45851,20265],{"mathvariant":2643},[2437,45853,45854],{"mathvariant":2643},"W",[2437,45856,2553],{"mathvariant":2643},[2584,45858,121],{},[9386,45860,45861,45863],{},[2437,45862,43338],{"mathvariant":2643},[2588,45864,132],{},[2431,45866,45867,45869,45871,45877,45879,45881,45887],{},[2584,45868,128],{"fence":2452},[2437,45870,42641],{"mathvariant":2643},[3063,45872,45873,45875],{},[2437,45874,44603],{},[2437,45876,214],{},[2584,45878,2002],{},[2437,45880,42558],{"mathvariant":2643},[3063,45882,45883,45885],{},[2437,45884,44603],{},[2437,45886,9034],{},[2584,45888,3544],{"fence":2452},[2584,45890,2714],{"separator":2452},[2444,45892,45893],{"encoding":2446},"H_{\\mathrm{RWA}}=\\frac{\\hbar}{2}\\left(\\Delta\\sigma_z+\\Omega\\sigma_x\\right),",[53,45895,45897,45958],{"className":45896,"ariaHidden":2452},[2451],[53,45898,45900,45903,45949,45952,45955],{"className":45899},[2456],[53,45901],{"className":45902,"style":3082},[2460],[53,45904,45906,45909],{"className":45905},[2465],[53,45907,3033],{"className":45908,"style":3048},[2465,2506],[53,45910,45912],{"className":45911},[2474],[53,45913,45915,45941],{"className":45914},[2478,3095],[53,45916,45918,45938],{"className":45917},[2482],[53,45919,45921],{"className":45920,"style":3102},[2486],[53,45922,45923,45926],{"style":4405},[53,45924],{"className":45925,"style":2495},[2494],[53,45927,45929],{"className":45928},[2499,2500,2501,2502],[53,45930,45932],{"className":45931},[2465,2502],[53,45933,45935],{"className":45934},[2465,2502],[53,45936,43208],{"className":45937},[2465,43792,2502],[53,45939,3120],{"className":45940},[3119],[53,45942,45944],{"className":45943},[2482],[53,45945,45947],{"className":45946,"style":3127},[2486],[53,45948],{},[53,45950],{"className":45951,"style":2814},[2609],[53,45953,121],{"className":45954},[2818],[53,45956],{"className":45957,"style":2814},[2609],[53,45959,45961,45964,46026,46029,46133,46136],{"className":45960},[2456],[53,45962],{"className":45963,"style":45475},[2460],[53,45965,45967,45970,46023],{"className":45966},[2465],[53,45968],{"className":45969},[2836,9688],[53,45971,45973],{"className":45972},[9386],[53,45974,45976,46015],{"className":45975},[2478,3095],[53,45977,45979,46012],{"className":45978},[2482],[53,45980,45982,45993,46001],{"className":45981,"style":45494},[2486],[53,45983,45984,45987],{"style":11612},[53,45985],{"className":45986,"style":4205},[2494],[53,45988,45990],{"className":45989},[2465],[53,45991,132],{"className":45992},[2465],[53,45994,45995,45998],{"style":9773},[53,45996],{"className":45997,"style":4205},[2494],[53,45999],{"className":46000,"style":9781},[9780],[53,46002,46003,46006],{"style":9784},[53,46004],{"className":46005,"style":4205},[2494],[53,46007,46009],{"className":46008},[2465],[53,46010,43338],{"className":46011},[2465],[53,46013,3120],{"className":46014},[3119],[53,46016,46018],{"className":46017},[2482],[53,46019,46021],{"className":46020,"style":11650},[2486],[53,46022],{},[53,46024],{"className":46025},[2675,9688],[53,46027],{"className":46028,"style":2860},[2609],[53,46030,46032,46035,46038,46078,46081,46084,46087,46090,46130],{"className":46031},[2832],[53,46033,128],{"className":46034,"style":2838},[2836,2837],[53,46036,42641],{"className":46037},[2465],[53,46039,46041,46044],{"className":46040},[2465],[53,46042,44603],{"className":46043,"style":2671},[2465,2506],[53,46045,46047],{"className":46046},[2474],[53,46048,46050,46070],{"className":46049},[2478,3095],[53,46051,46053,46067],{"className":46052},[2482],[53,46054,46056],{"className":46055,"style":4402},[2486],[53,46057,46058,46061],{"style":4467},[53,46059],{"className":46060,"style":2495},[2494],[53,46062,46064],{"className":46063},[2499,2500,2501,2502],[53,46065,214],{"className":46066,"style":20477},[2465,2506,2502],[53,46068,3120],{"className":46069},[3119],[53,46071,46073],{"className":46072},[2482],[53,46074,46076],{"className":46075,"style":3127},[2486],[53,46077],{},[53,46079],{"className":46080,"style":2610},[2609],[53,46082,2002],{"className":46083},[2614],[53,46085],{"className":46086,"style":2610},[2609],[53,46088,42558],{"className":46089},[2465],[53,46091,46093,46096],{"className":46092},[2465],[53,46094,44603],{"className":46095,"style":2671},[2465,2506],[53,46097,46099],{"className":46098},[2474],[53,46100,46102,46122],{"className":46101},[2478,3095],[53,46103,46105,46119],{"className":46104},[2482],[53,46106,46108],{"className":46107,"style":4402},[2486],[53,46109,46110,46113],{"style":4467},[53,46111],{"className":46112,"style":2495},[2494],[53,46114,46116],{"className":46115},[2499,2500,2501,2502],[53,46117,9034],{"className":46118},[2465,2506,2502],[53,46120,3120],{"className":46121},[3119],[53,46123,46125],{"className":46124},[2482],[53,46126,46128],{"className":46127,"style":3127},[2486],[53,46129],{},[53,46131,3544],{"className":46132,"style":2838},[2675,2837],[53,46134],{"className":46135,"style":2860},[2609],[53,46137,2714],{"className":46138},[2856],[12,46140,46141],{},"with detuning",[53,46143,46145],{"className":46144},[2679],[53,46146,46148,46196],{"className":46147},[2418],[53,46149,46151],{"className":46150},[2422],[2424,46152,46153],{"xmlns":2426,"display":2688},[2428,46154,46155,46193],{},[2431,46156,46157,46159,46161,46163,46165,46171,46173,46175,46177,46179,46181,46183,46189,46191],{},[2437,46158,42641],{"mathvariant":2643},[2584,46160,121],{},[2437,46162,42242],{},[2584,46164,2586],{},[3063,46166,46167,46169],{},[2437,46168,42242],{},[2588,46170,149],{},[2584,46172,121],{},[2588,46174,132],{},[2437,46176,8458],{},[2584,46178,128],{"stretchy":2650},[2437,46180,22173],{},[2584,46182,2586],{},[3063,46184,46185,46187],{},[2437,46186,22173],{},[2588,46188,149],{},[2584,46190,3544],{"stretchy":2650},[2437,46192,28],{"mathvariant":2643},[2444,46194,46195],{"encoding":2446},"\\Delta=\\omega-\\omega_0=2\\pi(f-f_0).",[53,46197,46199,46217,46235,46290,46317],{"className":46198,"ariaHidden":2452},[2451],[53,46200,46202,46205,46208,46211,46214],{"className":46201},[2456],[53,46203],{"className":46204,"style":2461},[2460],[53,46206,42641],{"className":46207},[2465],[53,46209],{"className":46210,"style":2814},[2609],[53,46212,121],{"className":46213},[2818],[53,46215],{"className":46216,"style":2814},[2609],[53,46218,46220,46223,46226,46229,46232],{"className":46219},[2456],[53,46221],{"className":46222,"style":2602},[2460],[53,46224,42242],{"className":46225,"style":2671},[2465,2506],[53,46227],{"className":46228,"style":2610},[2609],[53,46230,2586],{"className":46231},[2614],[53,46233],{"className":46234,"style":2610},[2609],[53,46236,46238,46241,46281,46284,46287],{"className":46237},[2456],[53,46239],{"className":46240,"style":25092},[2460],[53,46242,46244,46247],{"className":46243},[2465],[53,46245,42242],{"className":46246,"style":2671},[2465,2506],[53,46248,46250],{"className":46249},[2474],[53,46251,46253,46273],{"className":46252},[2478,3095],[53,46254,46256,46270],{"className":46255},[2482],[53,46257,46259],{"className":46258,"style":14176},[2486],[53,46260,46261,46264],{"style":4467},[53,46262],{"className":46263,"style":2495},[2494],[53,46265,46267],{"className":46266},[2499,2500,2501,2502],[53,46268,149],{"className":46269},[2465,2502],[53,46271,3120],{"className":46272},[3119],[53,46274,46276],{"className":46275},[2482],[53,46277,46279],{"className":46278,"style":3127},[2486],[53,46280],{},[53,46282],{"className":46283,"style":2814},[2609],[53,46285,121],{"className":46286},[2818],[53,46288],{"className":46289,"style":2814},[2609],[53,46291,46293,46296,46299,46302,46305,46308,46311,46314],{"className":46292},[2456],[53,46294],{"className":46295,"style":2664},[2460],[53,46297,132],{"className":46298},[2465],[53,46300,8458],{"className":46301,"style":2671},[2465,2506],[53,46303,128],{"className":46304},[2836],[53,46306,22173],{"className":46307,"style":27234},[2465,2506],[53,46309],{"className":46310,"style":2610},[2609],[53,46312,2586],{"className":46313},[2614],[53,46315],{"className":46316,"style":2610},[2609],[53,46318,46320,46323,46363,46366],{"className":46319},[2456],[53,46321],{"className":46322,"style":2664},[2460],[53,46324,46326,46329],{"className":46325},[2465],[53,46327,22173],{"className":46328,"style":27234},[2465,2506],[53,46330,46332],{"className":46331},[2474],[53,46333,46335,46355],{"className":46334},[2478,3095],[53,46336,46338,46352],{"className":46337},[2482],[53,46339,46341],{"className":46340,"style":14176},[2486],[53,46342,46343,46346],{"style":42163},[53,46344],{"className":46345,"style":2495},[2494],[53,46347,46349],{"className":46348},[2499,2500,2501,2502],[53,46350,149],{"className":46351},[2465,2502],[53,46353,3120],{"className":46354},[3119],[53,46356,46358],{"className":46357},[2482],[53,46359,46361],{"className":46360,"style":3127},[2486],[53,46362],{},[53,46364,3544],{"className":46365},[2675],[53,46367,28],{"className":46368},[2465],[12,46370,46371,46372,10694,46400,46428,46429,46457,46458,46486],{},"This Hamiltonian describes precession around an effective control axis with components proportional to ",[53,46373,46375,46388],{"className":46374},[2418],[53,46376,46378],{"className":46377},[2422],[2424,46379,46380],{"xmlns":2426},[2428,46381,46382,46386],{},[2431,46383,46384],{},[2437,46385,42558],{"mathvariant":2643},[2444,46387,42561],{"encoding":2446},[53,46389,46391],{"className":46390,"ariaHidden":2452},[2451],[53,46392,46394,46397],{"className":46393},[2456],[53,46395],{"className":46396,"style":2461},[2460],[53,46398,42558],{"className":46399},[2465],[53,46401,46403,46416],{"className":46402},[2418],[53,46404,46406],{"className":46405},[2422],[2424,46407,46408],{"xmlns":2426},[2428,46409,46410,46414],{},[2431,46411,46412],{},[2437,46413,42641],{"mathvariant":2643},[2444,46415,42644],{"encoding":2446},[53,46417,46419],{"className":46418,"ariaHidden":2452},[2451],[53,46420,46422,46425],{"className":46421},[2456],[53,46423],{"className":46424,"style":2461},[2460],[53,46426,42641],{"className":46427},[2465],". The transverse component ",[53,46430,46432,46445],{"className":46431},[2418],[53,46433,46435],{"className":46434},[2422],[2424,46436,46437],{"xmlns":2426},[2428,46438,46439,46443],{},[2431,46440,46441],{},[2437,46442,42558],{"mathvariant":2643},[2444,46444,42561],{"encoding":2446},[53,46446,46448],{"className":46447,"ariaHidden":2452},[2451],[53,46449,46451,46454],{"className":46450},[2456],[53,46452],{"className":46453,"style":2461},[2460],[53,46455,42558],{"className":46456},[2465]," drives population transfer. A longitudinal component ",[53,46459,46461,46474],{"className":46460},[2418],[53,46462,46464],{"className":46463},[2422],[2424,46465,46466],{"xmlns":2426},[2428,46467,46468,46472],{},[2431,46469,46470],{},[2437,46471,42641],{"mathvariant":2643},[2444,46473,42644],{"encoding":2446},[53,46475,46477],{"className":46476,"ariaHidden":2452},[2451],[53,46478,46480,46483],{"className":46479},[2456],[53,46481],{"className":46482,"style":2461},[2460],[53,46484,42641],{"className":46485},[2465]," tilts the rotation axis away from the resonant direction.",[30,46488,46490],{"id":46489},"excited-state-probability","Excited-State Probability",[12,46492,46493,46494,46532,46533,46561],{},"The simulation assumes that the qubit begins in the ground state ",[53,46495,46497,46514],{"className":46496},[2418],[53,46498,46500],{"className":46499},[2422],[2424,46501,46502],{"xmlns":2426},[2428,46503,46504,46512],{},[2431,46505,46506,46508,46510],{},[2584,46507,2644],{"stretchy":2650},[2437,46509,37809],{},[2584,46511,2651],{"stretchy":2650},[2444,46513,41791],{"encoding":2446},[53,46515,46517],{"className":46516,"ariaHidden":2452},[2451],[53,46518,46520,46523,46526,46529],{"className":46519},[2456],[53,46521],{"className":46522,"style":2664},[2460],[53,46524,2644],{"className":46525},[2836],[53,46527,37809],{"className":46528,"style":2671},[2465,2506],[53,46530,2651],{"className":46531},[2675],". Evolving under the RWA Hamiltonian, the probability of occupying the excited state after pulse duration ",[53,46534,46536,46549],{"className":46535},[2418],[53,46537,46539],{"className":46538},[2422],[2424,46540,46541],{"xmlns":2426},[2428,46542,46543,46547],{},[2431,46544,46545],{},[2437,46546,41600],{},[2444,46548,41726],{"encoding":2446},[53,46550,46552],{"className":46551,"ariaHidden":2452},[2451],[53,46553,46555,46558],{"className":46554},[2456],[53,46556],{"className":46557,"style":2534},[2460],[53,46559,41600],{"className":46560,"style":41672},[2465,2506]," is",[53,46563,46565],{"className":46564},[2679],[53,46566,46568,46647],{"className":46567},[2418],[53,46569,46571],{"className":46570},[2422],[2424,46572,46573],{"xmlns":2426,"display":2688},[2428,46574,46575,46644],{},[2431,46576,46577,46583,46585,46587,46589,46591,46593,46595,46611,46622,46642],{},[3063,46578,46579,46581],{},[2437,46580,3069],{},[2437,46582,3181],{},[2584,46584,128],{"stretchy":2650},[2437,46586,22173],{},[2584,46588,2714],{"separator":2452},[2437,46590,41600],{},[2584,46592,3544],{"stretchy":2650},[2584,46594,121],{},[9386,46596,46597,46603],{},[2434,46598,46599,46601],{},[2437,46600,42558],{"mathvariant":2643},[2588,46602,132],{},[3670,46604,46605,46607,46609],{},[2437,46606,42558],{"mathvariant":2643},[2437,46608,20265],{},[2588,46610,132],{},[2434,46612,46613,46620],{},[2431,46614,46615,46618],{},[2437,46616,46617],{},"sin",[2584,46619,36586],{},[2588,46621,132],{},[2431,46623,46624,46626,46640],{},[2584,46625,128],{"fence":2452},[9386,46627,46628,46638],{},[2431,46629,46630,46636],{},[3063,46631,46632,46634],{},[2437,46633,42558],{"mathvariant":2643},[2437,46635,20265],{},[2437,46637,41600],{},[2588,46639,132],{},[2584,46641,3544],{"fence":2452},[2584,46643,2714],{"separator":2452},[2444,46645,46646],{"encoding":2446},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right),",[53,46648,46650,46723],{"className":46649,"ariaHidden":2452},[2451],[53,46651,46653,46656,46696,46699,46702,46705,46708,46711,46714,46717,46720],{"className":46652},[2456],[53,46654],{"className":46655,"style":2664},[2460],[53,46657,46659,46662],{"className":46658},[2465],[53,46660,3069],{"className":46661,"style":3115},[2465,2506],[53,46663,46665],{"className":46664},[2474],[53,46666,46668,46688],{"className":46667},[2478,3095],[53,46669,46671,46685],{"className":46670},[2482],[53,46672,46674],{"className":46673,"style":4402},[2486],[53,46675,46676,46679],{"style":7532},[53,46677],{"className":46678,"style":2495},[2494],[53,46680,46682],{"className":46681},[2499,2500,2501,2502],[53,46683,3181],{"className":46684},[2465,2506,2502],[53,46686,3120],{"className":46687},[3119],[53,46689,46691],{"className":46690},[2482],[53,46692,46694],{"className":46693,"style":3127},[2486],[53,46695],{},[53,46697,128],{"className":46698},[2836],[53,46700,22173],{"className":46701,"style":27234},[2465,2506],[53,46703,2714],{"className":46704},[2856],[53,46706],{"className":46707,"style":2860},[2609],[53,46709,41600],{"className":46710,"style":41672},[2465,2506],[53,46712,3544],{"className":46713},[2675],[53,46715],{"className":46716,"style":2814},[2609],[53,46718,121],{"className":46719},[2818],[53,46721],{"className":46722,"style":2814},[2609],[53,46724,46726,46730,46872,46875,46906,46909,47027,47030],{"className":46725},[2456],[53,46727],{"className":46728,"style":46729},[2460],"height:2.4706em;vertical-align:-0.9795em;",[53,46731,46733,46736,46869],{"className":46732},[2465],[53,46734],{"className":46735},[2836,9688],[53,46737,46739],{"className":46738},[9386],[53,46740,46742,46860],{"className":46741},[2478,3095],[53,46743,46745,46857],{"className":46744},[2482],[53,46746,46749,46812,46820],{"className":46747,"style":46748},[2486],"height:1.4911em;",[53,46750,46751,46754],{"style":11612},[53,46752],{"className":46753,"style":4205},[2494],[53,46755,46757],{"className":46756},[2465],[53,46758,46760,46763],{"className":46759},[2465],[53,46761,42558],{"className":46762},[2465],[53,46764,46766],{"className":46765},[2474],[53,46767,46769,46803],{"className":46768},[2478,3095],[53,46770,46772,46800],{"className":46771},[2482],[53,46773,46776,46788],{"className":46774,"style":46775},[2486],"height:0.7959em;",[53,46777,46779,46782],{"style":46778},"top:-2.4065em;margin-left:0em;margin-right:0.05em;",[53,46780],{"className":46781,"style":2495},[2494],[53,46783,46785],{"className":46784},[2499,2500,2501,2502],[53,46786,20265],{"className":46787,"style":20290},[2465,2506,2502],[53,46789,46791,46794],{"style":46790},"top:-3.0448em;margin-right:0.05em;",[53,46792],{"className":46793,"style":2495},[2494],[53,46795,46797],{"className":46796},[2499,2500,2501,2502],[53,46798,132],{"className":46799},[2465,2502],[53,46801,3120],{"className":46802},[3119],[53,46804,46806],{"className":46805},[2482],[53,46807,46810],{"className":46808,"style":46809},[2486],"height:0.2935em;",[53,46811],{},[53,46813,46814,46817],{"style":9773},[53,46815],{"className":46816,"style":4205},[2494],[53,46818],{"className":46819,"style":9781},[9780],[53,46821,46822,46825],{"style":9784},[53,46823],{"className":46824,"style":4205},[2494],[53,46826,46828],{"className":46827},[2465],[53,46829,46831,46834],{"className":46830},[2465],[53,46832,42558],{"className":46833},[2465],[53,46835,46837],{"className":46836},[2474],[53,46838,46840],{"className":46839},[2478],[53,46841,46843],{"className":46842},[2482],[53,46844,46846],{"className":46845,"style":13119},[2486],[53,46847,46848,46851],{"style":2490},[53,46849],{"className":46850,"style":2495},[2494],[53,46852,46854],{"className":46853},[2499,2500,2501,2502],[53,46855,132],{"className":46856},[2465,2502],[53,46858,3120],{"className":46859},[3119],[53,46861,46863],{"className":46862},[2482],[53,46864,46867],{"className":46865,"style":46866},[2486],"height:0.9795em;",[53,46868],{},[53,46870],{"className":46871},[2675,9688],[53,46873],{"className":46874,"style":2860},[2609],[53,46876,46878,46881],{"className":46877},[7465],[53,46879,46617],{"className":46880},[7465],[53,46882,46884],{"className":46883},[2474],[53,46885,46887],{"className":46886},[2478],[53,46888,46890],{"className":46889},[2482],[53,46891,46894],{"className":46892,"style":46893},[2486],"height:0.8719em;",[53,46895,46897,46900],{"style":46896},"top:-3.1208em;margin-right:0.05em;",[53,46898],{"className":46899,"style":2495},[2494],[53,46901,46903],{"className":46902},[2499,2500,2501,2502],[53,46904,132],{"className":46905},[2465,2502],[53,46907],{"className":46908,"style":2860},[2609],[53,46910,46912,46918,47021],{"className":46911},[2832],[53,46913,46915],{"className":46914,"style":2838},[2836,2837],[53,46916,128],{"className":46917},[2842,2501],[53,46919,46921,46924,47018],{"className":46920},[2465],[53,46922],{"className":46923},[2836,9688],[53,46925,46927],{"className":46926},[9386],[53,46928,46930,47010],{"className":46929},[2478,3095],[53,46931,46933,47007],{"className":46932},[2482],[53,46934,46937,46948,46956],{"className":46935,"style":46936},[2486],"height:1.3603em;",[53,46938,46939,46942],{"style":11612},[53,46940],{"className":46941,"style":4205},[2494],[53,46943,46945],{"className":46944},[2465],[53,46946,132],{"className":46947},[2465],[53,46949,46950,46953],{"style":9773},[53,46951],{"className":46952,"style":4205},[2494],[53,46954],{"className":46955,"style":9781},[9780],[53,46957,46958,46961],{"style":9784},[53,46959],{"className":46960,"style":4205},[2494],[53,46962,46964,47004],{"className":46963},[2465],[53,46965,46967,46970],{"className":46966},[2465],[53,46968,42558],{"className":46969},[2465],[53,46971,46973],{"className":46972},[2474],[53,46974,46976,46996],{"className":46975},[2478,3095],[53,46977,46979,46993],{"className":46978},[2482],[53,46980,46982],{"className":46981,"style":3102},[2486],[53,46983,46984,46987],{"style":5293},[53,46985],{"className":46986,"style":2495},[2494],[53,46988,46990],{"className":46989},[2499,2500,2501,2502],[53,46991,20265],{"className":46992,"style":20290},[2465,2506,2502],[53,46994,3120],{"className":46995},[3119],[53,46997,46999],{"className":46998},[2482],[53,47000,47002],{"className":47001,"style":3127},[2486],[53,47003],{},[53,47005,41600],{"className":47006,"style":41672},[2465,2506],[53,47008,3120],{"className":47009},[3119],[53,47011,47013],{"className":47012},[2482],[53,47014,47016],{"className":47015,"style":11650},[2486],[53,47017],{},[53,47019],{"className":47020},[2675,9688],[53,47022,47024],{"className":47023,"style":2838},[2675,2837],[53,47025,3544],{"className":47026},[2842,2501],[53,47028],{"className":47029,"style":2860},[2609],[53,47031,2714],{"className":47032},[2856],[12,47034,47035],{},"where",[53,47037,47039],{"className":47038},[2679],[53,47040,47042,47080],{"className":47041},[2418],[53,47043,47045],{"className":47044},[2422],[2424,47046,47047],{"xmlns":2426,"display":2688},[2428,47048,47049,47077],{},[2431,47050,47051,47057,47059],{},[3063,47052,47053,47055],{},[2437,47054,42558],{"mathvariant":2643},[2437,47056,20265],{},[2584,47058,121],{},[9391,47060,47061],{},[2431,47062,47063,47069,47071],{},[2434,47064,47065,47067],{},[2437,47066,42558],{"mathvariant":2643},[2588,47068,132],{},[2584,47070,2002],{},[2434,47072,47073,47075],{},[2437,47074,42641],{"mathvariant":2643},[2588,47076,132],{},[2444,47078,47079],{"encoding":2446},"\\Omega_R=\\sqrt{\\Omega^2+\\Delta^2}",[53,47081,47083,47138],{"className":47082,"ariaHidden":2452},[2451],[53,47084,47086,47089,47129,47132,47135],{"className":47085},[2456],[53,47087],{"className":47088,"style":3082},[2460],[53,47090,47092,47095],{"className":47091},[2465],[53,47093,42558],{"className":47094},[2465],[53,47096,47098],{"className":47097},[2474],[53,47099,47101,47121],{"className":47100},[2478,3095],[53,47102,47104,47118],{"className":47103},[2482],[53,47105,47107],{"className":47106,"style":3102},[2486],[53,47108,47109,47112],{"style":5293},[53,47110],{"className":47111,"style":2495},[2494],[53,47113,47115],{"className":47114},[2499,2500,2501,2502],[53,47116,20265],{"className":47117,"style":20290},[2465,2506,2502],[53,47119,3120],{"className":47120},[3119],[53,47122,47124],{"className":47123},[2482],[53,47125,47127],{"className":47126,"style":3127},[2486],[53,47128],{},[53,47130],{"className":47131,"style":2814},[2609],[53,47133,121],{"className":47134},[2818],[53,47136],{"className":47137,"style":2814},[2609],[53,47139,47141,47145],{"className":47140},[2456],[53,47142],{"className":47143,"style":47144},[2460],"height:1.24em;vertical-align:-0.1777em;",[53,47146,47148],{"className":47147},[2465,9714],[53,47149,47151,47257],{"className":47150},[2478,3095],[53,47152,47154,47254],{"className":47153},[2482],[53,47155,47158,47237],{"className":47156,"style":47157},[2486],"height:1.0623em;",[53,47159,47162,47166],{"className":47160,"style":47161},[9728],"top:-3.2em;",[53,47163],{"className":47164,"style":47165},[2494],"height:3.2em;",[53,47167,47170,47199,47202,47205,47208],{"className":47168,"style":47169},[2465],"padding-left:1em;",[53,47171,47173,47176],{"className":47172},[2465],[53,47174,42558],{"className":47175},[2465],[53,47177,47179],{"className":47178},[2474],[53,47180,47182],{"className":47181},[2478],[53,47183,47185],{"className":47184},[2482],[53,47186,47188],{"className":47187,"style":29305},[2486],[53,47189,47190,47193],{"style":29308},[53,47191],{"className":47192,"style":2495},[2494],[53,47194,47196],{"className":47195},[2499,2500,2501,2502],[53,47197,132],{"className":47198},[2465,2502],[53,47200],{"className":47201,"style":2610},[2609],[53,47203,2002],{"className":47204},[2614],[53,47206],{"className":47207,"style":2610},[2609],[53,47209,47211,47214],{"className":47210},[2465],[53,47212,42641],{"className":47213},[2465],[53,47215,47217],{"className":47216},[2474],[53,47218,47220],{"className":47219},[2478],[53,47221,47223],{"className":47222},[2482],[53,47224,47226],{"className":47225,"style":29305},[2486],[53,47227,47228,47231],{"style":29308},[53,47229],{"className":47230,"style":2495},[2494],[53,47232,47234],{"className":47233},[2499,2500,2501,2502],[53,47235,132],{"className":47236},[2465,2502],[53,47238,47240,47243],{"style":47239},"top:-3.0223em;",[53,47241],{"className":47242,"style":47165},[2494],[53,47244,47247],{"className":47245,"style":47246},[9748],"min-width:1.02em;height:1.28em;",[6176,47248,47251],{"xmlns":6178,"width":9752,"height":47249,"viewBox":47250,"preserveAspectRatio":9755},"1.28em","0 0 400000 1296",[6185,47252],{"d":47253},"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",[53,47255,3120],{"className":47256},[3119],[53,47258,47260],{"className":47259},[2482],[53,47261,47264],{"className":47262,"style":47263},[2486],"height:0.1777em;",[53,47265],{},[12,47267,47268],{},"is the generalized Rabi angular frequency. The amplitude coefficient",[53,47270,47272],{"className":47271},[2679],[53,47273,47275,47303],{"className":47274},[2418],[53,47276,47278],{"className":47277},[2422],[2424,47279,47280],{"xmlns":2426,"display":2688},[2428,47281,47282,47300],{},[2431,47283,47284],{},[9386,47285,47286,47292],{},[2434,47287,47288,47290],{},[2437,47289,42558],{"mathvariant":2643},[2588,47291,132],{},[3670,47293,47294,47296,47298],{},[2437,47295,42558],{"mathvariant":2643},[2437,47297,20265],{},[2588,47299,132],{},[2444,47301,47302],{"encoding":2446},"\\frac{\\Omega^2}{\\Omega_R^2}",[53,47304,47306],{"className":47305,"ariaHidden":2452},[2451],[53,47307,47309,47312],{"className":47308},[2456],[53,47310],{"className":47311,"style":46729},[2460],[53,47313,47315,47318,47445],{"className":47314},[2465],[53,47316],{"className":47317},[2836,9688],[53,47319,47321],{"className":47320},[9386],[53,47322,47324,47437],{"className":47323},[2478,3095],[53,47325,47327,47434],{"className":47326},[2482],[53,47328,47330,47389,47397],{"className":47329,"style":46748},[2486],[53,47331,47332,47335],{"style":11612},[53,47333],{"className":47334,"style":4205},[2494],[53,47336,47338],{"className":47337},[2465],[53,47339,47341,47344],{"className":47340},[2465],[53,47342,42558],{"className":47343},[2465],[53,47345,47347],{"className":47346},[2474],[53,47348,47350,47381],{"className":47349},[2478,3095],[53,47351,47353,47378],{"className":47352},[2482],[53,47354,47356,47367],{"className":47355,"style":46775},[2486],[53,47357,47358,47361],{"style":46778},[53,47359],{"className":47360,"style":2495},[2494],[53,47362,47364],{"className":47363},[2499,2500,2501,2502],[53,47365,20265],{"className":47366,"style":20290},[2465,2506,2502],[53,47368,47369,47372],{"style":46790},[53,47370],{"className":47371,"style":2495},[2494],[53,47373,47375],{"className":47374},[2499,2500,2501,2502],[53,47376,132],{"className":47377},[2465,2502],[53,47379,3120],{"className":47380},[3119],[53,47382,47384],{"className":47383},[2482],[53,47385,47387],{"className":47386,"style":46809},[2486],[53,47388],{},[53,47390,47391,47394],{"style":9773},[53,47392],{"className":47393,"style":4205},[2494],[53,47395],{"className":47396,"style":9781},[9780],[53,47398,47399,47402],{"style":9784},[53,47400],{"className":47401,"style":4205},[2494],[53,47403,47405],{"className":47404},[2465],[53,47406,47408,47411],{"className":47407},[2465],[53,47409,42558],{"className":47410},[2465],[53,47412,47414],{"className":47413},[2474],[53,47415,47417],{"className":47416},[2478],[53,47418,47420],{"className":47419},[2482],[53,47421,47423],{"className":47422,"style":13119},[2486],[53,47424,47425,47428],{"style":2490},[53,47426],{"className":47427,"style":2495},[2494],[53,47429,47431],{"className":47430},[2499,2500,2501,2502],[53,47432,132],{"className":47433},[2465,2502],[53,47435,3120],{"className":47436},[3119],[53,47438,47440],{"className":47439},[2482],[53,47441,47443],{"className":47442,"style":46866},[2486],[53,47444],{},[53,47446],{"className":47447},[2675,9688],[12,47449,47450,47451,47502],{},"sets the maximum possible population transfer. At exact resonance, ",[53,47452,47454,47472],{"className":47453},[2418],[53,47455,47457],{"className":47456},[2422],[2424,47458,47459],{"xmlns":2426},[2428,47460,47461,47469],{},[2431,47462,47463,47465,47467],{},[2437,47464,42641],{"mathvariant":2643},[2584,47466,121],{},[2588,47468,149],{},[2444,47470,47471],{"encoding":2446},"\\Delta=0",[53,47473,47475,47493],{"className":47474,"ariaHidden":2452},[2451],[53,47476,47478,47481,47484,47487,47490],{"className":47477},[2456],[53,47479],{"className":47480,"style":2461},[2460],[53,47482,42641],{"className":47483},[2465],[53,47485],{"className":47486,"style":2814},[2609],[53,47488,121],{"className":47489},[2818],[53,47491],{"className":47492,"style":2814},[2609],[53,47494,47496,47499],{"className":47495},[2456],[53,47497],{"className":47498,"style":2624},[2460],[53,47500,149],{"className":47501},[2465],", so",[53,47504,47506],{"className":47505},[2679],[53,47507,47509,47541],{"className":47508},[2418],[53,47510,47512],{"className":47511},[2422],[2424,47513,47514],{"xmlns":2426,"display":2688},[2428,47515,47516,47538],{},[2431,47517,47518,47534,47536],{},[9386,47519,47520,47526],{},[2434,47521,47522,47524],{},[2437,47523,42558],{"mathvariant":2643},[2588,47525,132],{},[3670,47527,47528,47530,47532],{},[2437,47529,42558],{"mathvariant":2643},[2437,47531,20265],{},[2588,47533,132],{},[2584,47535,121],{},[2588,47537,44239],{},[2444,47539,47540],{"encoding":2446},"\\frac{\\Omega^2}{\\Omega_R^2}=1.",[53,47542,47544,47695],{"className":47543,"ariaHidden":2452},[2451],[53,47545,47547,47550,47686,47689,47692],{"className":47546},[2456],[53,47548],{"className":47549,"style":46729},[2460],[53,47551,47553,47556,47683],{"className":47552},[2465],[53,47554],{"className":47555},[2836,9688],[53,47557,47559],{"className":47558},[9386],[53,47560,47562,47675],{"className":47561},[2478,3095],[53,47563,47565,47672],{"className":47564},[2482],[53,47566,47568,47627,47635],{"className":47567,"style":46748},[2486],[53,47569,47570,47573],{"style":11612},[53,47571],{"className":47572,"style":4205},[2494],[53,47574,47576],{"className":47575},[2465],[53,47577,47579,47582],{"className":47578},[2465],[53,47580,42558],{"className":47581},[2465],[53,47583,47585],{"className":47584},[2474],[53,47586,47588,47619],{"className":47587},[2478,3095],[53,47589,47591,47616],{"className":47590},[2482],[53,47592,47594,47605],{"className":47593,"style":46775},[2486],[53,47595,47596,47599],{"style":46778},[53,47597],{"className":47598,"style":2495},[2494],[53,47600,47602],{"className":47601},[2499,2500,2501,2502],[53,47603,20265],{"className":47604,"style":20290},[2465,2506,2502],[53,47606,47607,47610],{"style":46790},[53,47608],{"className":47609,"style":2495},[2494],[53,47611,47613],{"className":47612},[2499,2500,2501,2502],[53,47614,132],{"className":47615},[2465,2502],[53,47617,3120],{"className":47618},[3119],[53,47620,47622],{"className":47621},[2482],[53,47623,47625],{"className":47624,"style":46809},[2486],[53,47626],{},[53,47628,47629,47632],{"style":9773},[53,47630],{"className":47631,"style":4205},[2494],[53,47633],{"className":47634,"style":9781},[9780],[53,47636,47637,47640],{"style":9784},[53,47638],{"className":47639,"style":4205},[2494],[53,47641,47643],{"className":47642},[2465],[53,47644,47646,47649],{"className":47645},[2465],[53,47647,42558],{"className":47648},[2465],[53,47650,47652],{"className":47651},[2474],[53,47653,47655],{"className":47654},[2478],[53,47656,47658],{"className":47657},[2482],[53,47659,47661],{"className":47660,"style":13119},[2486],[53,47662,47663,47666],{"style":2490},[53,47664],{"className":47665,"style":2495},[2494],[53,47667,47669],{"className":47668},[2499,2500,2501,2502],[53,47670,132],{"className":47671},[2465,2502],[53,47673,3120],{"className":47674},[3119],[53,47676,47678],{"className":47677},[2482],[53,47679,47681],{"className":47680,"style":46866},[2486],[53,47682],{},[53,47684],{"className":47685},[2675,9688],[53,47687],{"className":47688,"style":2814},[2609],[53,47690,121],{"className":47691},[2818],[53,47693],{"className":47694,"style":2814},[2609],[53,47696,47698,47701],{"className":47697},[2456],[53,47699],{"className":47700,"style":2624},[2460],[53,47702,44239],{"className":47703},[2465],[12,47705,47706],{},"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.",[12,47708,47709],{},"At resonance, the excited-state probability simplifies to",[53,47711,47713],{"className":47712},[2679],[53,47714,47716,47770],{"className":47715},[2418],[53,47717,47719],{"className":47718},[2422],[2424,47720,47721],{"xmlns":2426,"display":2688},[2428,47722,47723,47767],{},[2431,47724,47725,47731,47733,47735,47737,47739,47749,47765],{},[3063,47726,47727,47729],{},[2437,47728,3069],{},[2437,47730,3181],{},[2584,47732,128],{"stretchy":2650},[2437,47734,41600],{},[2584,47736,3544],{"stretchy":2650},[2584,47738,121],{},[2434,47740,47741,47747],{},[2431,47742,47743,47745],{},[2437,47744,46617],{},[2584,47746,36586],{},[2588,47748,132],{},[2431,47750,47751,47753,47763],{},[2584,47752,128],{"fence":2452},[9386,47754,47755,47761],{},[2431,47756,47757,47759],{},[2437,47758,42558],{"mathvariant":2643},[2437,47760,41600],{},[2588,47762,132],{},[2584,47764,3544],{"fence":2452},[2437,47766,28],{"mathvariant":2643},[2444,47768,47769],{"encoding":2446},"P_e(\\tau)=\\sin^2\\left(\\frac{\\Omega\\tau}{2}\\right).",[53,47771,47773,47837],{"className":47772,"ariaHidden":2452},[2451],[53,47774,47776,47779,47819,47822,47825,47828,47831,47834],{"className":47775},[2456],[53,47777],{"className":47778,"style":2664},[2460],[53,47780,47782,47785],{"className":47781},[2465],[53,47783,3069],{"className":47784,"style":3115},[2465,2506],[53,47786,47788],{"className":47787},[2474],[53,47789,47791,47811],{"className":47790},[2478,3095],[53,47792,47794,47808],{"className":47793},[2482],[53,47795,47797],{"className":47796,"style":4402},[2486],[53,47798,47799,47802],{"style":7532},[53,47800],{"className":47801,"style":2495},[2494],[53,47803,47805],{"className":47804},[2499,2500,2501,2502],[53,47806,3181],{"className":47807},[2465,2506,2502],[53,47809,3120],{"className":47810},[3119],[53,47812,47814],{"className":47813},[2482],[53,47815,47817],{"className":47816,"style":3127},[2486],[53,47818],{},[53,47820,128],{"className":47821},[2836],[53,47823,41600],{"className":47824,"style":41672},[2465,2506],[53,47826,3544],{"className":47827},[2675],[53,47829],{"className":47830,"style":2814},[2609],[53,47832,121],{"className":47833},[2818],[53,47835],{"className":47836,"style":2814},[2609],[53,47838,47840,47843,47872,47875,47955,47958],{"className":47839},[2456],[53,47841],{"className":47842,"style":37564},[2460],[53,47844,47846,47849],{"className":47845},[7465],[53,47847,46617],{"className":47848},[7465],[53,47850,47852],{"className":47851},[2474],[53,47853,47855],{"className":47854},[2478],[53,47856,47858],{"className":47857},[2482],[53,47859,47861],{"className":47860,"style":46893},[2486],[53,47862,47863,47866],{"style":46896},[53,47864],{"className":47865,"style":2495},[2494],[53,47867,47869],{"className":47868},[2499,2500,2501,2502],[53,47870,132],{"className":47871},[2465,2502],[53,47873],{"className":47874,"style":2860},[2609],[53,47876,47878,47884,47949],{"className":47877},[2832],[53,47879,47881],{"className":47880,"style":2838},[2836,2837],[53,47882,128],{"className":47883},[2842,2501],[53,47885,47887,47890,47946],{"className":47886},[2465],[53,47888],{"className":47889},[2836,9688],[53,47891,47893],{"className":47892},[9386],[53,47894,47896,47938],{"className":47895},[2478,3095],[53,47897,47899,47935],{"className":47898},[2482],[53,47900,47902,47913,47921],{"className":47901,"style":46936},[2486],[53,47903,47904,47907],{"style":11612},[53,47905],{"className":47906,"style":4205},[2494],[53,47908,47910],{"className":47909},[2465],[53,47911,132],{"className":47912},[2465],[53,47914,47915,47918],{"style":9773},[53,47916],{"className":47917,"style":4205},[2494],[53,47919],{"className":47920,"style":9781},[9780],[53,47922,47923,47926],{"style":9784},[53,47924],{"className":47925,"style":4205},[2494],[53,47927,47929,47932],{"className":47928},[2465],[53,47930,42558],{"className":47931},[2465],[53,47933,41600],{"className":47934,"style":41672},[2465,2506],[53,47936,3120],{"className":47937},[3119],[53,47939,47941],{"className":47940},[2482],[53,47942,47944],{"className":47943,"style":11650},[2486],[53,47945],{},[53,47947],{"className":47948},[2675,9688],[53,47950,47952],{"className":47951,"style":2838},[2675,2837],[53,47953,3544],{"className":47954},[2842,2501],[53,47956],{"className":47957,"style":2860},[2609],[53,47959,28],{"className":47960},[2465],[12,47962,47963,47964,47993],{},"The pulse duration required for ideal population inversion defines the standard ",[53,47965,47967,47981],{"className":47966},[2418],[53,47968,47970],{"className":47969},[2422],[2424,47971,47972],{"xmlns":2426},[2428,47973,47974,47978],{},[2431,47975,47976],{},[2437,47977,8458],{},[2444,47979,47980],{"encoding":2446},"\\pi",[53,47982,47984],{"className":47983,"ariaHidden":2452},[2451],[53,47985,47987,47990],{"className":47986},[2456],[53,47988],{"className":47989,"style":2534},[2460],[53,47991,8458],{"className":47992,"style":2671},[2465,2506],"-pulse time,",[53,47995,47997],{"className":47996},[2679],[53,47998,48000,48050],{"className":47999},[2418],[53,48001,48003],{"className":48002},[2422],[2424,48004,48005],{"xmlns":2426,"display":2688},[2428,48006,48007,48047],{},[2431,48008,48009,48015,48017,48023,48025,48045],{},[3063,48010,48011,48013],{},[2437,48012,41600],{},[2437,48014,8458],{},[2584,48016,121],{},[9386,48018,48019,48021],{},[2437,48020,8458],{},[2437,48022,42558],{"mathvariant":2643},[2584,48024,121],{},[9386,48026,48027,48029],{},[2588,48028,173],{},[2431,48030,48031,48033,48035,48037,48039,48041,48043],{},[2588,48032,132],{},[2584,48034,128],{"stretchy":2650},[2437,48036,42558],{"mathvariant":2643},[2437,48038,193],{"mathvariant":2643},[2588,48040,132],{},[2437,48042,8458],{},[2584,48044,3544],{"stretchy":2650},[2437,48046,28],{"mathvariant":2643},[2444,48048,48049],{"encoding":2446},"\\tau_\\pi=\\frac{\\pi}{\\Omega}=\\frac{1}{2(\\Omega\u002F2\\pi)}.",[53,48051,48053,48109,48188],{"className":48052,"ariaHidden":2452},[2451],[53,48054,48056,48059,48100,48103,48106],{"className":48055},[2456],[53,48057],{"className":48058,"style":25092},[2460],[53,48060,48062,48065],{"className":48061},[2465],[53,48063,41600],{"className":48064,"style":41672},[2465,2506],[53,48066,48068],{"className":48067},[2474],[53,48069,48071,48092],{"className":48070},[2478,3095],[53,48072,48074,48089],{"className":48073},[2482],[53,48075,48077],{"className":48076,"style":4402},[2486],[53,48078,48080,48083],{"style":48079},"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;",[53,48081],{"className":48082,"style":2495},[2494],[53,48084,48086],{"className":48085},[2499,2500,2501,2502],[53,48087,8458],{"className":48088,"style":2671},[2465,2506,2502],[53,48090,3120],{"className":48091},[3119],[53,48093,48095],{"className":48094},[2482],[53,48096,48098],{"className":48097,"style":3127},[2486],[53,48099],{},[53,48101],{"className":48102,"style":2814},[2609],[53,48104,121],{"className":48105},[2818],[53,48107],{"className":48108,"style":2814},[2609],[53,48110,48112,48116,48179,48182,48185],{"className":48111},[2456],[53,48113],{"className":48114,"style":48115},[2460],"height:1.7936em;vertical-align:-0.686em;",[53,48117,48119,48122,48176],{"className":48118},[2465],[53,48120],{"className":48121},[2836,9688],[53,48123,48125],{"className":48124},[9386],[53,48126,48128,48168],{"className":48127},[2478,3095],[53,48129,48131,48165],{"className":48130},[2482],[53,48132,48135,48146,48154],{"className":48133,"style":48134},[2486],"height:1.1076em;",[53,48136,48137,48140],{"style":11612},[53,48138],{"className":48139,"style":4205},[2494],[53,48141,48143],{"className":48142},[2465],[53,48144,42558],{"className":48145},[2465],[53,48147,48148,48151],{"style":9773},[53,48149],{"className":48150,"style":4205},[2494],[53,48152],{"className":48153,"style":9781},[9780],[53,48155,48156,48159],{"style":9784},[53,48157],{"className":48158,"style":4205},[2494],[53,48160,48162],{"className":48161},[2465],[53,48163,8458],{"className":48164,"style":2671},[2465,2506],[53,48166,3120],{"className":48167},[3119],[53,48169,48171],{"className":48170},[2482],[53,48172,48174],{"className":48173,"style":11650},[2486],[53,48175],{},[53,48177],{"className":48178},[2675,9688],[53,48180],{"className":48181,"style":2814},[2609],[53,48183,121],{"className":48184},[2818],[53,48186],{"className":48187,"style":2814},[2609],[53,48189,48191,48194,48268],{"className":48190},[2456],[53,48192],{"className":48193,"style":35734},[2460],[53,48195,48197,48200,48265],{"className":48196},[2465],[53,48198],{"className":48199},[2836,9688],[53,48201,48203],{"className":48202},[9386],[53,48204,48206,48257],{"className":48205},[2478,3095],[53,48207,48209,48254],{"className":48208},[2482],[53,48210,48212,48235,48243],{"className":48211,"style":9701},[2486],[53,48213,48214,48217],{"style":11612},[53,48215],{"className":48216,"style":4205},[2494],[53,48218,48220,48223,48226,48229,48232],{"className":48219},[2465],[53,48221,132],{"className":48222},[2465],[53,48224,128],{"className":48225},[2836],[53,48227,42616],{"className":48228},[2465],[53,48230,8458],{"className":48231,"style":2671},[2465,2506],[53,48233,3544],{"className":48234},[2675],[53,48236,48237,48240],{"style":9773},[53,48238],{"className":48239,"style":4205},[2494],[53,48241],{"className":48242,"style":9781},[9780],[53,48244,48245,48248],{"style":9784},[53,48246],{"className":48247,"style":4205},[2494],[53,48249,48251],{"className":48250},[2465],[53,48252,173],{"className":48253},[2465],[53,48255,3120],{"className":48256},[3119],[53,48258,48260],{"className":48259},[2482],[53,48261,48263],{"className":48262,"style":29452},[2486],[53,48264],{},[53,48266],{"className":48267},[2675,9688],[53,48269,28],{"className":48270},[2465],[12,48272,48273],{},"Given the default drive 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ideal ",[53,48431,48433,48446],{"className":48432},[2418],[53,48434,48436],{"className":48435},[2422],[2424,48437,48438],{"xmlns":2426},[2428,48439,48440,48444],{},[2431,48441,48442],{},[2437,48443,8458],{},[2444,48445,47980],{"encoding":2446},[53,48447,48449],{"className":48448,"ariaHidden":2452},[2451],[53,48450,48452,48455],{"className":48451},[2456],[53,48453],{"className":48454,"style":2534},[2460],[53,48456,8458],{"className":48457,"style":2671},[2465,2506],"-pulse time 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full period of the resonant excited-state probability oscillation is",[53,48582,48584],{"className":48583},[2679],[53,48585,48587,48635],{"className":48586},[2418],[53,48588,48590],{"className":48589},[2422],[2424,48591,48592],{"xmlns":2426,"display":2688},[2428,48593,48594,48632],{},[2431,48595,48596,48602,48604,48614,48616,48630],{},[3063,48597,48598,48600],{},[2437,48599,36942],{},[2437,48601,20265],{},[2584,48603,121],{},[9386,48605,48606,48612],{},[2431,48607,48608,48610],{},[2588,48609,132],{},[2437,48611,8458],{},[2437,48613,42558],{"mathvariant":2643},[2584,48615,121],{},[9386,48617,48618,48620],{},[2588,48619,173],{},[2431,48621,48622,48624,48626,48628],{},[2437,48623,42558],{"mathvariant":2643},[2437,48625,193],{"mathvariant":2643},[2588,48627,132],{},[2437,48629,8458],{},[2437,48631,28],{"mathvariant":2643},[2444,48633,48634],{"encoding":2446},"T_R=\\frac{2\\pi}{\\Omega}=\\frac{1}{\\Omega\u002F2\\pi}.",[53,48636,48638,48693,48774],{"className":48637,"ariaHidden":2452},[2451],[53,48639,48641,48644,48684,48687,48690],{"className":48640},[2456],[53,48642],{"className":48643,"style":3082},[2460],[53,48645,48647,48650],{"className":48646},[2465],[53,48648,36942],{"className":48649,"style":3115},[2465,2506],[53,48651,48653],{"className":48652},[2474],[53,48654,48656,48676],{"className":48655},[2478,3095],[53,48657,48659,48673],{"className":48658},[2482],[53,48660,48662],{"className":48661,"style":3102},[2486],[53,48663,48664,48667],{"style":7532},[53,48665],{"className":48666,"style":2495},[2494],[53,48668,48670],{"className":48669},[2499,2500,2501,2502],[53,48671,20265],{"className":48672,"style":20290},[2465,2506,2502],[53,48674,3120],{"className":48675},[3119],[53,48677,48679],{"className":48678},[2482],[53,48680,48682],{"className":48681,"style":3127},[2486],[53,48683],{},[53,48685],{"className":48686,"style":2814},[2609],[53,48688,121],{"className":48689},[2818],[53,48691],{"className":48692,"style":2814},[2609],[53,48694,48696,48700,48765,48768,48771],{"className":48695},[2456],[53,48697],{"className":48698,"style":48699},[2460],"height:2.0074em;vertical-align:-0.686em;",[53,48701,48703,48706,48762],{"className":48702},[2465],[53,48704],{"className":48705},[2836,9688],[53,48707,48709],{"className":48708},[9386],[53,48710,48712,48754],{"className":48711},[2478,3095],[53,48713,48715,48751],{"className":48714},[2482],[53,48716,48718,48729,48737],{"className":48717,"style":9701},[2486],[53,48719,48720,48723],{"style":11612},[53,48721],{"className":48722,"style":4205},[2494],[53,48724,48726],{"className":48725},[2465],[53,48727,42558],{"className":48728},[2465],[53,48730,48731,48734],{"style":9773},[53,48732],{"className":48733,"style":4205},[2494],[53,48735],{"className":48736,"style":9781},[9780],[53,48738,48739,48742],{"style":9784},[53,48740],{"className":48741,"style":4205},[2494],[53,48743,48745,48748],{"className":48744},[2465],[53,48746,132],{"className":48747},[2465],[53,48749,8458],{"className":48750,"style":2671},[2465,2506],[53,48752,3120],{"className":48753},[3119],[53,48755,48757],{"className":48756},[2482],[53,48758,48760],{"className":48759,"style":11650},[2486],[53,48761],{},[53,48763],{"className":48764},[2675,9688],[53,48766],{"className":48767,"style":2814},[2609],[53,48769,121],{"className":48770},[2818],[53,48772],{"className":48773,"style":2814},[2609],[53,48775,48777,48780,48845],{"className":48776},[2456],[53,48778],{"className":48779,"style":35734},[2460],[53,48781,48783,48786,48842],{"className":48782},[2465],[53,48784],{"className":48785},[2836,9688],[53,48787,48789],{"className":48788},[9386],[53,48790,48792,48834],{"className":48791},[2478,3095],[53,48793,48795,48831],{"className":48794},[2482],[53,48796,48798,48812,48820],{"className":48797,"style":9701},[2486],[53,48799,48800,48803],{"style":11612},[53,48801],{"className":48802,"style":4205},[2494],[53,48804,48806,48809],{"className":48805},[2465],[53,48807,42616],{"className":48808},[2465],[53,48810,8458],{"className":48811,"style":2671},[2465,2506],[53,48813,48814,48817],{"style":9773},[53,48815],{"className":48816,"style":4205},[2494],[53,48818],{"className":48819,"style":9781},[9780],[53,48821,48822,48825],{"style":9784},[53,48823],{"className":48824,"style":4205},[2494],[53,48826,48828],{"className":48827},[2465],[53,48829,173],{"className":48830},[2465],[53,48832,3120],{"className":48833},[3119],[53,48835,48837],{"className":48836},[2482],[53,48838,48840],{"className":48839,"style":29452},[2486],[53,48841],{},[53,48843],{"className":48844},[2675,9688],[53,48846,28],{"className":48847},[2465],[12,48849,48850],{},"For the same default drive strength, the full temporal period is",[53,48852,48854],{"className":48853},[2679],[53,48855,48857,48890],{"className":48856},[2418],[53,48858,48860],{"className":48859},[2422],[2424,48861,48862],{"xmlns":2426,"display":2688},[2428,48863,48864,48887],{},[2431,48865,48866,48872,48874,48877,48879,48885],{},[3063,48867,48868,48870],{},[2437,48869,36942],{},[2437,48871,20265],{},[2584,48873,121],{},[2588,48875,48876],{},"50",[6062,48878,6064],{},[2431,48880,48881,48883],{},[2437,48882,4315],{"mathvariant":2643},[2437,48884,14428],{"mathvariant":2643},[2437,48886,28],{"mathvariant":2643},[2444,48888,48889],{"encoding":2446},"T_R=50~\\mathrm{ns}.",[53,48891,48893,48948],{"className":48892,"ariaHidden":2452},[2451],[53,48894,48896,48899,48939,48942,48945],{"className":48895},[2456],[53,48897],{"className":48898,"style":3082},[2460],[53,48900,48902,48905],{"className":48901},[2465],[53,48903,36942],{"className":48904,"style":3115},[2465,2506],[53,48906,48908],{"className":48907},[2474],[53,48909,48911,48931],{"className":48910},[2478,3095],[53,48912,48914,48928],{"className":48913},[2482],[53,48915,48917],{"className":48916,"style":3102},[2486],[53,48918,48919,48922],{"style":7532},[53,48920],{"className":48921,"style":2495},[2494],[53,48923,48925],{"className":48924},[2499,2500,2501,2502],[53,48926,20265],{"className":48927,"style":20290},[2465,2506,2502],[53,48929,3120],{"className":48930},[3119],[53,48932,48934],{"className":48933},[2482],[53,48935,48937],{"className":48936,"style":3127},[2486],[53,48938],{},[53,48940],{"className":48941,"style":2814},[2609],[53,48943,121],{"className":48944},[2818],[53,48946],{"className":48947,"style":2814},[2609],[53,48949,48951,48954,48957,48960,48966],{"className":48950},[2456],[53,48952],{"className":48953,"style":2624},[2460],[53,48955,48876],{"className":48956},[2465],[53,48958,6064],{"className":48959},[2609,43785],[53,48961,48963],{"className":48962},[2465],[53,48964,48574],{"className":48965},[2465,43792],[53,48967,28],{"className":48968},[2465],[30,48970,48972],{"id":48971},"phenomenological-dephasing","Phenomenological Dephasing",[12,48974,48975],{},"The computational model includes an optional phenomenological damping factor,",[53,48977,48979],{"className":48978},[2679],[53,48980,48982,49054],{"className":48981},[2418],[53,48983,48985],{"className":48984},[2422],[2424,48986,48987],{"xmlns":2426,"display":2688},[2428,48988,48989,49051],{},[2431,48990,48991,48997,48999,49001,49003,49005,49007,49009,49015,49017,49019,49021,49023,49025,49027,49029,49049],{},[3063,48992,48993,48995],{},[2437,48994,3069],{},[2437,48996,3181],{},[2584,48998,128],{"stretchy":2650},[2437,49000,22173],{},[2584,49002,2714],{"separator":2452},[2437,49004,41600],{},[2584,49006,3544],{"stretchy":2650},[2584,49008,14829],{},[3063,49010,49011,49013],{},[2437,49012,3069],{},[2437,49014,3181],{},[2584,49016,128],{"stretchy":2650},[2437,49018,22173],{},[2584,49020,2714],{"separator":2452},[2437,49022,41600],{},[2584,49024,3544],{"stretchy":2650},[2437,49026,23964],{},[2584,49028,36586],{},[2431,49030,49031,49033,49035,49047],{},[2584,49032,128],{"fence":2452},[2584,49034,2586],{},[9386,49036,49037,49039],{},[2437,49038,41600],{},[3670,49040,49041,49043,49045],{},[2437,49042,36942],{},[2588,49044,132],{},[2584,49046,20664],{},[2584,49048,3544],{"fence":2452},[2437,49050,28],{"mathvariant":2643},[2444,49052,49053],{"encoding":2446},"P_e(f,\\tau)\\rightarrow P_e(f,\\tau)\\exp\\left(-\\frac{\\tau}{T_2^\\ast}\\right).",[53,49055,49057,49130],{"className":49056,"ariaHidden":2452},[2451],[53,49058,49060,49063,49103,49106,49109,49112,49115,49118,49121,49124,49127],{"className":49059},[2456],[53,49061],{"className":49062,"style":2664},[2460],[53,49064,49066,49069],{"className":49065},[2465],[53,49067,3069],{"className":49068,"style":3115},[2465,2506],[53,49070,49072],{"className":49071},[2474],[53,49073,49075,49095],{"className":49074},[2478,3095],[53,49076,49078,49092],{"className":49077},[2482],[53,49079,49081],{"className":49080,"style":4402},[2486],[53,49082,49083,49086],{"style":7532},[53,49084],{"className":49085,"style":2495},[2494],[53,49087,49089],{"className":49088},[2499,2500,2501,2502],[53,49090,3181],{"className":49091},[2465,2506,2502],[53,49093,3120],{"className":49094},[3119],[53,49096,49098],{"className":49097},[2482],[53,49099,49101],{"className":49100,"style":3127},[2486],[53,49102],{},[53,49104,128],{"className":49105},[2836],[53,49107,22173],{"className":49108,"style":27234},[2465,2506],[53,49110,2714],{"className":49111},[2856],[53,49113],{"className":49114,"style":2860},[2609],[53,49116,41600],{"className":49117,"style":41672},[2465,2506],[53,49119,3544],{"className":49120},[2675],[53,49122],{"className":49123,"style":2814},[2609],[53,49125,14829],{"className":49126},[2818],[53,49128],{"className":49129,"style":2814},[2609],[53,49131,49133,49137,49177,49180,49183,49186,49189,49192,49195,49198,49201,49204,49336,49339],{"className":49132},[2456],[53,49134],{"className":49135,"style":49136},[2460],"height:2.4023em;vertical-align:-0.9523em;",[53,49138,49140,49143],{"className":49139},[2465],[53,49141,3069],{"className":49142,"style":3115},[2465,2506],[53,49144,49146],{"className":49145},[2474],[53,49147,49149,49169],{"className":49148},[2478,3095],[53,49150,49152,49166],{"className":49151},[2482],[53,49153,49155],{"className":49154,"style":4402},[2486],[53,49156,49157,49160],{"style":7532},[53,49158],{"className":49159,"style":2495},[2494],[53,49161,49163],{"className":49162},[2499,2500,2501,2502],[53,49164,3181],{"className":49165},[2465,2506,2502],[53,49167,3120],{"className":49168},[3119],[53,49170,49172],{"className":49171},[2482],[53,49173,49175],{"className":49174,"style":3127},[2486],[53,49176],{},[53,49178,128],{"className":49179},[2836],[53,49181,22173],{"className":49182,"style":27234},[2465,2506],[53,49184,2714],{"className":49185},[2856],[53,49187],{"className":49188,"style":2860},[2609],[53,49190,41600],{"className":49191,"style":41672},[2465,2506],[53,49193,3544],{"className":49194},[2675],[53,49196],{"className":49197,"style":2860},[2609],[53,49199,23964],{"className":49200},[7465],[53,49202],{"className":49203,"style":2860},[2609],[53,49205,49207,49213,49216,49330],{"className":49206},[2832],[53,49208,49210],{"className":49209,"style":2838},[2836,2837],[53,49211,128],{"className":49212},[2842,2501],[53,49214,2586],{"className":49215},[2465],[53,49217,49219,49222,49327],{"className":49218},[2465],[53,49220],{"className":49221},[2836,9688],[53,49223,49225],{"className":49224},[9386],[53,49226,49228,49318],{"className":49227},[2478,3095],[53,49229,49231,49315],{"className":49230},[2482],[53,49232,49234,49296,49304],{"className":49233,"style":48134},[2486],[53,49235,49236,49239],{"style":11612},[53,49237],{"className":49238,"style":4205},[2494],[53,49240,49242],{"className":49241},[2465],[53,49243,49245,49248],{"className":49244},[2465],[53,49246,36942],{"className":49247,"style":3115},[2465,2506],[53,49249,49251],{"className":49250},[2474],[53,49252,49254,49287],{"className":49253},[2478,3095],[53,49255,49257,49284],{"className":49256},[2482],[53,49258,49261,49273],{"className":49259,"style":49260},[2486],"height:0.6705em;",[53,49262,49264,49267],{"style":49263},"top:-2.4337em;margin-left:-0.1389em;margin-right:0.05em;",[53,49265],{"className":49266,"style":2495},[2494],[53,49268,49270],{"className":49269},[2499,2500,2501,2502],[53,49271,132],{"className":49272},[2465,2502],[53,49274,49275,49278],{"style":46790},[53,49276],{"className":49277,"style":2495},[2494],[53,49279,49281],{"className":49280},[2499,2500,2501,2502],[53,49282,20664],{"className":49283},[2614,2502],[53,49285,3120],{"className":49286},[3119],[53,49288,49290],{"className":49289},[2482],[53,49291,49294],{"className":49292,"style":49293},[2486],"height:0.2663em;",[53,49295],{},[53,49297,49298,49301],{"style":9773},[53,49299],{"className":49300,"style":4205},[2494],[53,49302],{"className":49303,"style":9781},[9780],[53,49305,49306,49309],{"style":9784},[53,49307],{"className":49308,"style":4205},[2494],[53,49310,49312],{"className":49311},[2465],[53,49313,41600],{"className":49314,"style":41672},[2465,2506],[53,49316,3120],{"className":49317},[3119],[53,49319,49321],{"className":49320},[2482],[53,49322,49325],{"className":49323,"style":49324},[2486],"height:0.9523em;",[53,49326],{},[53,49328],{"className":49329},[2675,9688],[53,49331,49333],{"className":49332,"style":283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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",[53,49346,49348],{"className":49347},[2679],[53,49349,49351,49386],{"className":49350},[2418],[53,49352,49354],{"className":49353},[2422],[2424,49355,49356],{"xmlns":2426,"display":2688},[2428,49357,49358,49383],{},[2431,49359,49360,49368,49370,49373,49375,49381],{},[3670,49361,49362,49364,49366],{},[2437,49363,36942],{},[2588,49365,132],{},[2584,49367,20664],{},[2584,49369,121],{},[2588,49371,49372],{},"500",[6062,49374,6064],{},[2431,49376,49377,49379],{},[2437,49378,4315],{"mathvariant":2643},[2437,49380,14428],{"mathvariant":2643},[2437,49382,28],{"mathvariant":2643},[2444,49384,49385],{"encoding":2446},"T_2^\\ast=500~\\mathrm{ns}.",[53,49387,49389,49457],{"className":49388,"ariaHidden":2452},[2451],[53,49390,49392,49396,49448,49451,49454],{"className":49391},[2456],[53,49393],{"className":49394,"style":49395},[2460],"height:0.9857em;vertical-align:-0.247em;",[53,49397,49399,49402],{"className":49398},[2465],[53,49400,36942],{"className":49401,"style":3115},[2465,2506],[53,49403,49405],{"className":49404},[2474],[53,49406,49408,49440],{"className":49407},[2478,3095],[53,49409,49411,49437],{"className":49410},[2482],[53,49412,49414,49426],{"className":49413,"style":37041},[2486],[53,49415,49417,49420],{"style":49416},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[53,49418],{"className":49419,"style":2495},[2494],[53,49421,49423],{"className":49422},[2499,2500,2501,2502],[53,49424,132],{"className":49425},[2465,2502],[53,49427,49428,49431],{"style":2801},[53,49429],{"className":49430,"style":2495},[2494],[53,49432,49434],{"className":49433},[2499,2500,2501,2502],[53,49435,20664],{"className":49436},[2614,2502],[53,49438,3120],{"className":49439},[3119],[53,49441,49443],{"className":49442},[2482],[53,49444,49446],{"className":49445,"style":3808},[2486],[53,49447],{},[53,49449],{"className":49450,"style":2814},[2609],[53,49452,121],{"className":49453},[2818],[53,49455],{"className":49456,"style":2814},[2609],[53,49458,49460,49463,49466,49469,49475],{"className":49459},[2456],[53,49461],{"className":49462,"style":2624},[2460],[53,49464,49372],{"className":49465},[2465],[53,49467,6064],{"className":49468},[2609,43785],[53,49470,49472],{"className":49471},[2465],[53,49473,48574],{"className":49474},[2465,43792],[53,49476,28],{"className":49477},[2465],[12,49479,49480,49481,49484],{},"Setting ",[50,49482,49483],{},"T2STAR_NS = None"," disables the phenomenological damping factor.",[30,49486,49488],{"id":49487},"detuning-and-generalized-rabi-frequency","Detuning and Generalized Rabi Frequency",[12,49490,49491],{},"Detuning quantifies the spectral mismatch between the applied drive and the natural qubit transition,",[53,49493,49495],{"className":49494},[2679],[53,49496,49498,49534],{"className":49497},[2418],[53,49499,49501],{"className":49500},[2422],[2424,49502,49503],{"xmlns":2426,"display":2688},[2428,49504,49505,49531],{},[2431,49506,49507,49509,49511,49513,49515,49517,49519,49521,49527,49529],{},[2437,49508,42641],{"mathvariant":2643},[2584,49510,121],{},[2588,49512,132],{},[2437,49514,8458],{},[2584,49516,128],{"stretchy":2650},[2437,49518,22173],{},[2584,49520,2586],{},[3063,49522,49523,49525],{},[2437,49524,22173],{},[2588,49526,149],{},[2584,49528,3544],{"stretchy":2650},[2437,49530,28],{"mathvariant":2643},[2444,49532,49533],{"encoding":2446},"\\Delta=2\\pi(f-f_0).",[53,49535,49537,49555,49582],{"className":49536,"ariaHidden":2452},[2451],[53,49538,49540,49543,49546,49549,49552],{"className":49539},[2456],[53,49541],{"className":49542,"style":2461},[2460],[53,49544,42641],{"className":49545},[2465],[53,49547],{"className":49548,"style":2814},[2609],[53,49550,121],{"className":49551},[2818],[53,49553],{"className":49554,"style":2814},[2609],[53,49556,49558,49561,49564,49567,49570,49573,49576,49579],{"className":49557},[2456],[53,49559],{"className":49560,"style":2664},[2460],[53,49562,132],{"className":49563},[2465],[53,49565,8458],{"className":49566,"style":2671},[2465,2506],[53,49568,128],{"className":49569},[2836],[53,49571,22173],{"className":49572,"style":27234},[2465,2506],[53,49574],{"className":49575,"style":2610},[2609],[53,49577,2586],{"className":49578},[2614],[53,49580],{"className":49581,"style":2610},[2609],[53,49583,49585,49588,49628,49631],{"className":49584},[2456],[53,49586],{"className":49587,"style":2664},[2460],[53,49589,49591,49594],{"className":49590},[2465],[53,49592,22173],{"className":49593,"style":27234},[2465,2506],[53,49595,49597],{"className":49596},[2474],[53,49598,49600,49620],{"className":49599},[2478,3095],[53,49601,49603,49617],{"className":49602},[2482],[53,49604,49606],{"className":49605,"style":14176},[2486],[53,49607,49608,49611],{"style":42163},[53,49609],{"className":49610,"style":2495},[2494],[53,49612,49614],{"className":49613},[2499,2500,2501,2502],[53,49615,149],{"className":49616},[2465,2502],[53,49618,3120],{"className":49619},[3119],[53,49621,49623],{"className":49622},[2482],[53,49624,49626],{"className":49625,"style":3127},[2486],[53,49627],{},[53,49629,3544],{"className":49630},[2675],[53,49632,28],{"className":49633},[2465],[12,49635,49636],{},"Expressed 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At resonance, the generalized frequency equals the applied drive strength,",[53,50183,50185],{"className":50184},[2679],[53,50186,50188,50230],{"className":50187},[2418],[53,50189,50191],{"className":50190},[2422],[2424,50192,50193],{"xmlns":2426,"display":2688},[2428,50194,50195,50227],{},[2431,50196,50197,50203,50205,50211,50213,50215,50225],{},[3063,50198,50199,50201],{},[2437,50200,42871],{},[2437,50202,20265],{},[2584,50204,128],{"stretchy":2650},[3063,50206,50207,50209],{},[2437,50208,22173],{},[2588,50210,149],{},[2584,50212,3544],{"stretchy":2650},[2584,50214,121],{},[9386,50216,50217,50219],{},[2437,50218,42558],{"mathvariant":2643},[2431,50220,50221,50223],{},[2588,50222,132],{},[2437,50224,8458],{},[2437,50226,28],{"mathvariant":2643},[2444,50228,50229],{"encoding":2446},"\\nu_R(f_0)=\\frac{\\Omega}{2\\pi}.",[53,50231,50233,50334],{"className":50232,"ariaHidden":2452},[2451],[53,50234,50236,50239,50279,50282,50322,50325,50328,50331],{"className":50235},[2456],[53,50237],{"className":50238,"style":2664},[2460],[53,50240,50242,50245],{"className":50241},[2465],[53,50243,42871],{"className":50244,"style":42892},[2465,2506],[53,50246,50248],{"className":50247},[2474],[53,50249,50251,50271],{"className":50250},[2478,3095],[53,50252,50254,50268],{"className":50253},[2482],[53,50255,50257],{"className":50256,"style":3102},[2486],[53,50258,50259,50262],{"style":42907},[53,50260],{"className":50261,"style":2495},[2494],[53,50263,50265],{"className":50264},[2499,2500,2501,2502],[53,50266,20265],{"className":50267,"style":20290},[2465,2506,2502],[53,50269,3120],{"className":50270},[3119],[53,50272,50274],{"className":50273},[2482],[53,50275,50277],{"className":50276,"style":3127},[2486],[53,50278],{},[53,50280,128],{"className":50281},[2836],[53,50283,50285,50288],{"className":50284},[2465],[53,50286,22173],{"className":50287,"style":27234},[2465,2506],[53,50289,50291],{"className":50290},[2474],[53,50292,50294,50314],{"className":50293},[2478,3095],[53,50295,50297,50311],{"className":50296},[2482],[53,50298,50300],{"className":50299,"style":14176},[2486],[53,50301,50302,50305],{"style":42163},[53,50303],{"className":50304,"style":2495},[2494],[53,50306,50308],{"className":50307},[2499,2500,2501,2502],[53,50309,149],{"className":50310},[2465,2502],[53,50312,3120],{"className":50313},[3119],[53,50315,50317],{"className":50316},[2482],[53,50318,50320],{"className":50319,"style":3127},[2486],[53,50321],{},[53,50323,3544],{"className":50324},[2675],[53,50326],{"className":50327,"style":2814},[2609],[53,50329,121],{"className":50330},[2818],[53,50332],{"className":50333,"style":2814},[2609],[53,50335,50337,50340,50405],{"className":50336},[2456],[53,50338],{"className":50339,"style":48329},[2460],[53,50341,50343,50346,50402],{"className":50342},[2465],[53,50344],{"className":50345},[2836,9688],[53,50347,50349],{"className":50348},[9386],[53,50350,50352,50394],{"className":50351},[2478,3095],[53,50353,50355,50391],{"className":50354},[2482],[53,50356,50358,50372,50380],{"className":50357,"style":46936},[2486],[53,50359,50360,50363],{"style":11612},[53,50361],{"className":50362,"style":4205},[2494],[53,50364,50366,50369],{"className":50365},[2465],[53,50367,132],{"className":50368},[2465],[53,50370,8458],{"className":50371,"style":2671},[2465,2506],[53,50373,50374,50377],{"style":9773},[53,50375],{"className":50376,"style":4205},[2494],[53,50378],{"className":50379,"style":9781},[9780],[53,50381,50382,50385],{"style":9784},[53,50383],{"className":50384,"style":4205},[2494],[53,50386,50388],{"className":50387},[2465],[53,50389,42558],{"className":50390},[2465],[53,50392,3120],{"className":50393},[3119],[53,50395,50397],{"className":50396},[2482],[53,50398,50400],{"className":50399,"style":11650},[2486],[53,50401],{},[53,50403],{"className":50404},[2675,9688],[53,50406,28],{"className":50407},[2465],[12,50409,50410,50411,50480,50481,50583],{},"As the system moves away from resonance, ",[53,50412,50414,50431],{"className":50413},[2418],[53,50415,50417],{"className":50416},[2422],[2424,50418,50419],{"xmlns":2426},[2428,50420,50421,50429],{},[2431,50422,50423],{},[3063,50424,50425,50427],{},[2437,50426,42871],{},[2437,50428,20265],{},[2444,50430,42876],{"encoding":2446},[53,50432,50434],{"className":50433,"ariaHidden":2452},[2451],[53,50435,50437,50440],{"className":50436},[2456],[53,50438],{"className":50439,"style":25092},[2460],[53,50441,50443,50446],{"className":50442},[2465],[53,50444,42871],{"className":50445,"style":42892},[2465,2506],[53,50447,50449],{"className":50448},[2474],[53,50450,50452,50472],{"className":50451},[2478,3095],[53,50453,50455,50469],{"className":50454},[2482],[53,50456,50458],{"className":50457,"style":3102},[2486],[53,50459,50460,50463],{"style":42907},[53,50461],{"className":50462,"style":2495},[2494],[53,50464,50466],{"className":50465},[2499,2500,2501,2502],[53,50467,20265],{"className":50468,"style":20290},[2465,2506,2502],[53,50470,3120],{"className":50471},[3119],[53,50473,50475],{"className":50474},[2482],[53,50476,50478],{"className":50477,"style":3127},[2486],[53,50479],{}," increases symmetrically with ",[53,50482,50484,50510],{"className":50483},[2418],[53,50485,50487],{"className":50486},[2422],[2424,50488,50489],{"xmlns":2426},[2428,50490,50491,50507],{},[2431,50492,50493,50495,50497,50499,50505],{},[2584,50494,2644],{"stretchy":2650},[2437,50496,22173],{},[2584,50498,2586],{},[3063,50500,50501,50503],{},[2437,50502,22173],{},[2588,50504,149],{},[2584,50506,2644],{"stretchy":2650},[2444,50508,50509],{"encoding":2446},"\\lvert f-f_0\\rvert",[53,50511,50513,50534],{"className":50512,"ariaHidden":2452},[2451],[53,50514,50516,50519,50522,50525,50528,50531],{"className":50515},[2456],[53,50517],{"className":50518,"style":2664},[2460],[53,50520,2644],{"className":50521},[2836],[53,50523,22173],{"className":50524,"style":27234},[2465,2506],[53,50526],{"className":50527,"style":2610},[2609],[53,50529,2586],{"className":50530},[2614],[53,50532],{"className":50533,"style":2610},[2609],[53,50535,50537,50540,50580],{"className":50536},[2456],[53,50538],{"className":50539,"style":2664},[2460],[53,50541,50543,50546],{"className":50542},[2465],[53,50544,22173],{"className":50545,"style":27234},[2465,2506],[53,50547,50549],{"className":50548},[2474],[53,50550,50552,50572],{"className":50551},[2478,3095],[53,50553,50555,50569],{"className":50554},[2482],[53,50556,50558],{"className":50557,"style":14176},[2486],[53,50559,50560,50563],{"style":42163},[53,50561],{"className":50562,"style":2495},[2494],[53,50564,50566],{"className":50565},[2499,2500,2501,2502],[53,50567,149],{"className":50568},[2465,2502],[53,50570,3120],{"className":50571},[3119],[53,50573,50575],{"className":50574},[2482],[53,50576,50578],{"className":50577,"style":3127},[2486],[53,50579],{},[53,50581,2644],{"className":50582},[2675],". 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.",[30,50585,50587],{"id":50586},"quantum-information-interpretation","Quantum Information Interpretation",[12,50589,50590,50591,50661],{},"In quantum information language, a calibrated resonant Rabi pulse implements a single-qubit rotation. At exact resonance, the effective Hamiltonian is proportional to ",[53,50592,50594,50612],{"className":50593},[2418],[53,50595,50597],{"className":50596},[2422],[2424,50598,50599],{"xmlns":2426},[2428,50600,50601,50609],{},[2431,50602,50603],{},[3063,50604,50605,50607],{},[2437,50606,44603],{},[2437,50608,9034],{},[2444,50610,50611],{"encoding":2446},"\\sigma_x",[53,50613,50615],{"className":50614,"ariaHidden":2452},[2451],[53,50616,50618,50621],{"className":50617},[2456],[53,50619],{"className":50620,"style":25092},[2460],[53,50622,50624,50627],{"className":50623},[2465],[53,50625,44603],{"className":50626,"style":2671},[2465,2506],[53,50628,50630],{"className":50629},[2474],[53,50631,50633,50653],{"className":50632},[2478,3095],[53,50634,50636,50650],{"className":50635},[2482],[53,50637,50639],{"className":50638,"style":4402},[2486],[53,50640,50641,50644],{"style":4467},[53,50642],{"className":50643,"style":2495},[2494],[53,50645,50647],{"className":50646},[2499,2500,2501,2502],[53,50648,9034],{"className":50649},[2465,2506,2502],[53,50651,3120],{"className":50652},[3119],[53,50654,50656],{"className":50655},[2482],[53,50657,50659],{"className":50658,"style":3127},[2486],[53,50660],{},", so the microwave pulse implements",[53,50663,50665],{"className":50664},[2679],[53,50666,50668,50725],{"className":50667},[2418],[53,50669,50671],{"className":50670},[2422],[2424,50672,50673],{"xmlns":2426,"display":2688},[2428,50674,50675,50722],{},[2431,50676,50677,50683,50685,50688,50690,50692,50694,50696,50720],{},[3063,50678,50679,50681],{},[2437,50680,20265],{},[2437,50682,9034],{},[2584,50684,128],{"stretchy":2650},[2437,50686,50687],{},"θ",[2584,50689,3544],{"stretchy":2650},[2584,50691,121],{},[2437,50693,23964],{},[2584,50695,36586],{},[2431,50697,50698,50700,50702,50718],{},[2584,50699,128],{"fence":2452},[2584,50701,2586],{},[9386,50703,50704,50716],{},[2431,50705,50706,50708,50710],{},[2437,50707,3188],{},[2437,50709,50687],{},[3063,50711,50712,50714],{},[2437,50713,44603],{},[2437,50715,9034],{},[2588,50717,132],{},[2584,50719,3544],{"fence":2452},[2584,50721,2714],{"separator":2452},[2444,50723,50724],{"encoding":2446},"R_x(\\theta)=\\exp\\left(-\\frac{i\\theta\\sigma_x}{2}\\right),",[53,50726,50728,50792],{"className":50727,"ariaHidden":2452},[2451],[53,50729,50731,50734,50774,50777,50780,50783,50786,50789],{"className":50730},[2456],[53,50732],{"className":50733,"style":2664},[2460],[53,50735,50737,50740],{"className":50736},[2465],[53,50738,20265],{"className":50739,"style":20290},[2465,2506],[53,50741,50743],{"className":50742},[2474],[53,50744,50746,50766],{"className":50745},[2478,3095],[53,50747,50749,50763],{"className":50748},[2482],[53,50750,50752],{"className":50751,"style":4402},[2486],[53,50753,50754,50757],{"style":20305},[53,50755],{"className":50756,"style":2495},[2494],[53,50758,50760],{"className":50759},[2499,2500,2501,2502],[53,50761,9034],{"className":50762},[2465,2506,2502],[53,50764,3120],{"className":50765},[3119],[53,50767,50769],{"className":50768},[2482],[53,50770,50772],{"className":50771,"style":3127},[2486],[53,50773],{},[53,50775,128],{"className":50776},[2836],[53,50778,50687],{"className":50779,"style":2507},[2465,2506],[53,50781,3544],{"className":50782},[2675],[53,50784],{"className":50785,"style":2814},[2609],[53,50787,121],{"className":50788},[2818],[53,50790],{"className":50791,"style":2814},[2609],[53,50793,50795,50798,50801,50804,50927,50930],{"className":50794},[2456],[53,50796],{"className":50797,"style":37564},[2460],[53,50799,23964],{"className":50800},[7465],[53,50802],{"className":50803,"style":2860},[2609],[53,50805,50807,50813,50816,50921],{"className":50806},[2832],[53,50808,50810],{"className":50809,"style":2838},[2836,2837],[53,50811,128],{"className":50812},[2842,2501],[53,50814,2586],{"className":50815},[2465],[53,50817,50819,50822,50918],{"className":50818},[2465],[53,50820],{"className":50821},[2836,9688],[53,50823,50825],{"className":50824},[9386],[53,50826,50828,50910],{"className":50827},[2478,3095],[53,50829,50831,50907],{"className":50830},[2482],[53,50832,50834,50845,50853],{"className":50833,"style":37654},[2486],[53,50835,50836,50839],{"style":11612},[53,50837],{"className":50838,"style":4205},[2494],[53,50840,50842],{"className":50841},[2465],[53,50843,132],{"className":50844},[2465],[53,50846,50847,50850],{"style":9773},[53,50848],{"className":50849,"style":4205},[2494],[53,50851],{"className":50852,"style":9781},[9780],[53,50854,50855,50858],{"style":9784},[53,50856],{"className":50857,"style":4205},[2494],[53,50859,50861,50864,50867],{"className":50860},[2465],[53,50862,3188],{"className":50863},[2465,2506],[53,50865,50687],{"className":50866,"style":2507},[2465,2506],[53,50868,50870,50873],{"className":50869},[2465],[53,50871,44603],{"className":50872,"style":2671},[2465,2506],[53,50874,50876],{"className":50875},[2474],[53,50877,50879,50899],{"className":50878},[2478,3095],[53,50880,50882,50896],{"className":50881},[2482],[53,50883,50885],{"className":50884,"style":4402},[2486],[53,50886,50887,50890],{"style":4467},[53,50888],{"className":50889,"style":2495},[2494],[53,50891,50893],{"className":50892},[2499,2500,2501,2502],[53,50894,9034],{"className":50895},[2465,2506,2502],[53,50897,3120],{"className":50898},[3119],[53,50900,50902],{"className":50901},[2482],[53,50903,50905],{"className":50904,"style":3127},[2486],[53,50906],{},[53,50908,3120],{"className":50909},[3119],[53,50911,50913],{"className":50912},[2482],[53,50914,50916],{"className":50915,"style":11650},[2486],[53,50917],{},[53,50919],{"className":50920},[2675,9688],[53,50922,50924],{"className":50923,"style":2838},[2675,2837],[53,50925,3544],{"className":50926},[2842,2501],[53,50928],{"className":50929,"style":2860},[2609],[53,50931,2714],{"className":50932},[2856],[12,50934,47035],{},[53,50936,50938],{"className":50937},[2679],[53,50939,50941,50963],{"className":50940},[2418],[53,50942,50944],{"className":50943},[2422],[2424,50945,50946],{"xmlns":2426,"display":2688},[2428,50947,50948,50960],{},[2431,50949,50950,50952,50954,50956,50958],{},[2437,50951,50687],{},[2584,50953,121],{},[2437,50955,42558],{"mathvariant":2643},[2437,50957,41600],{},[2437,50959,28],{"mathvariant":2643},[2444,50961,50962],{"encoding":2446},"\\theta=\\Omega\\tau.",[53,50964,50966,50984],{"className":50965,"ariaHidden":2452},[2451],[53,50967,50969,50972,50975,50978,50981],{"className":50968},[2456],[53,50970],{"className":50971,"style":5329},[2460],[53,50973,50687],{"className":50974,"style":2507},[2465,2506],[53,50976],{"className":50977,"style":2814},[2609],[53,50979,121],{"className":50980},[2818],[53,50982],{"className":50983,"style":2814},[2609],[53,50985,50987,50990,50993,50996],{"className":50986},[2456],[53,50988],{"className":50989,"style":2461},[2460],[53,50991,42558],{"className":50992},[2465],[53,50994,41600],{"className":50995,"style":41672},[2465,2506],[53,50997,28],{"className":50998},[2465],[12,51000,51001,51002,51030],{},"A calibrated resonant ",[53,51003,51005,51018],{"className":51004},[2418],[53,51006,51008],{"className":51007},[2422],[2424,51009,51010],{"xmlns":2426},[2428,51011,51012,51016],{},[2431,51013,51014],{},[2437,51015,8458],{},[2444,51017,47980],{"encoding":2446},[53,51019,51021],{"className":51020,"ariaHidden":2452},[2451],[53,51022,51024,51027],{"className":51023},[2456],[53,51025],{"className":51026,"style":2534},[2460],[53,51028,8458],{"className":51029,"style":2671},[2465,2506],"-pulse enacts",[53,51032,51034],{"className":51033},[2679],[53,51035,51037,51075],{"className":51036},[2418],[53,51038,51040],{"className":51039},[2422],[2424,51041,51042],{"xmlns":2426,"display":2688},[2428,51043,51044,51072],{},[2431,51045,51046,51052,51054,51056,51058,51060,51062,51064,51070],{},[3063,51047,51048,51050],{},[2437,51049,20265],{},[2437,51051,9034],{},[2584,51053,128],{"stretchy":2650},[2437,51055,8458],{},[2584,51057,3544],{"stretchy":2650},[2584,51059,121],{},[2584,51061,2586],{},[2437,51063,3188],{},[3063,51065,51066,51068],{},[2437,51067,44603],{},[2437,51069,9034],{},[2584,51071,2714],{"separator":2452},[2444,51073,51074],{"encoding":2446},"R_x(\\pi)=-i\\sigma_x,",[53,51076,51078,51142],{"className":51077,"ariaHidden":2452},[2451],[53,51079,51081,51084,51124,51127,51130,51133,51136,51139],{"className":51080},[2456],[53,51082],{"className":51083,"style":2664},[2460],[53,51085,51087,51090],{"className":51086},[2465],[53,51088,20265],{"className":51089,"style":20290},[2465,2506],[53,51091,51093],{"className":51092},[2474],[53,51094,51096,51116],{"className":51095},[2478,3095],[53,51097,51099,51113],{"className":51098},[2482],[53,51100,51102],{"className":51101,"style":4402},[2486],[53,51103,51104,51107],{"style":20305},[53,51105],{"className":51106,"style":2495},[2494],[53,51108,51110],{"className":51109},[2499,2500,2501,2502],[53,51111,9034],{"className":51112},[2465,2506,2502],[53,51114,3120],{"className":51115},[3119],[53,51117,51119],{"className":51118},[2482],[53,51120,51122],{"className":51121,"style":3127},[2486],[53,51123],{},[53,51125,128],{"className":51126},[2836],[53,51128,8458],{"className":51129,"style":2671},[2465,2506],[53,51131,3544],{"className":51132},[2675],[53,51134],{"className":51135,"style":2814},[2609],[53,51137,121],{"className":51138},[2818],[53,51140],{"className":51141,"style":2814},[2609],[53,51143,51145,51148,51151,51154,51194],{"className":51144},[2456],[53,51146],{"className":51147,"style":27882},[2460],[53,51149,2586],{"className":51150},[2465],[53,51152,3188],{"className":51153},[2465,2506],[53,51155,51157,51160],{"className":51156},[2465],[53,51158,44603],{"className":51159,"style":2671},[2465,2506],[53,51161,51163],{"className":51162},[2474],[53,51164,51166,51186],{"className":51165},[2478,3095],[53,51167,51169,51183],{"className":51168},[2482],[53,51170,51172],{"className":51171,"style":4402},[2486],[53,51173,51174,51177],{"style":4467},[53,51175],{"className":51176,"style":2495},[2494],[53,51178,51180],{"className":51179},[2499,2500,2501,2502],[53,51181,9034],{"className":51182},[2465,2506,2502],[53,51184,3120],{"className":51185},[3119],[53,51187,51189],{"className":51188},[2482],[53,51190,51192],{"className":51191,"style":3127},[2486],[53,51193],{},[53,51195,2714],{"className":51196},[2856],[12,51198,51199,51200,51228,51229,51265],{},"which corresponds to a quantum ",[53,51201,51203,51216],{"className":51202},[2418],[53,51204,51206],{"className":51205},[2422],[2424,51207,51208],{"xmlns":2426},[2428,51209,51210,51214],{},[2431,51211,51212],{},[2437,51213,7626],{},[2444,51215,7626],{"encoding":2446},[53,51217,51219],{"className":51218,"ariaHidden":2452},[2451],[53,51220,51222,51225],{"className":51221},[2456],[53,51223],{"className":51224,"style":2461},[2460],[53,51226,7626],{"className":51227,"style":7771},[2465,2506]," gate up to a global phase. A resonant ",[53,51230,51232,51250],{"className":51231},[2418],[53,51233,51235],{"className":51234},[2422],[2424,51236,51237],{"xmlns":2426},[2428,51238,51239,51247],{},[2431,51240,51241,51243,51245],{},[2437,51242,8458],{},[2437,51244,193],{"mathvariant":2643},[2588,51246,132],{},[2444,51248,51249],{"encoding":2446},"\\pi\u002F2",[53,51251,51253],{"className":51252,"ariaHidden":2452},[2451],[53,51254,51256,51259,51262],{"className":51255},[2456],[53,51257],{"className":51258,"style":2664},[2460],[53,51260,8458],{"className":51261,"style":2671},[2465,2506],[53,51263,26016],{"className":51264},[2465],"-pulse prepares a balanced coherent superposition,",[53,51267,51269],{"className":51268},[2679],[53,51270,51272,51318],{"className":51271},[2418],[53,51273,51275],{"className":51274},[2422],[2424,51276,51277],{"xmlns":2426,"display":2688},[2428,51278,51279,51315],{},[2431,51280,51281,51283,51285,51287,51289,51313],{},[2584,51282,2644],{"stretchy":2650},[2437,51284,37809],{},[2584,51286,2651],{"stretchy":2650},[2584,51288,14829],{},[9386,51290,51291,51309],{},[2431,51292,51293,51295,51297,51299,51301,51303,51305,51307],{},[2584,51294,2644],{"stretchy":2650},[2437,51296,37809],{},[2584,51298,2651],{"stretchy":2650},[2584,51300,2586],{},[2437,51302,3188],{},[2584,51304,2644],{"stretchy":2650},[2437,51306,3181],{},[2584,51308,2651],{"stretchy":2650},[9391,51310,51311],{},[2588,51312,132],{},[2584,51314,2714],{"separator":2452},[2444,51316,51317],{"encoding":2446},"\\lvert g\\rangle\\rightarrow\\frac{\\lvert g\\rangle-i\\lvert e\\rangle}{\\sqrt{2}},",[53,51319,51321,51345],{"className":51320,"ariaHidden":2452},[2451],[53,51322,51324,51327,51330,51333,51336,51339,51342],{"className":51323},[2456],[53,51325],{"className":51326,"style":2664},[2460],[53,51328,2644],{"className":51329},[2836],[53,51331,37809],{"className":51332,"style":2671},[2465,2506],[53,51334,2651],{"className":51335},[2675],[53,51337],{"className":51338,"style":2814},[2609],[53,51340,14829],{"className":51341},[2818],[53,51343],{"className":51344,"style":2814},[2609],[53,51346,51348,51352,51485],{"className":51347},[2456],[53,51349],{"className":51350,"style":51351},[2460],"height:2.357em;vertical-align:-0.93em;",[53,51353,51355,51358,51482],{"className":51354},[2465],[53,51356],{"className":51357},[2836,9688],[53,51359,51361],{"className":51360},[9386],[53,51362,51364,51474],{"className":51363},[2478,3095],[53,51365,51367,51471],{"className":51366},[2482],[53,51368,51370,51425,51433],{"className":51369,"style":29260},[2486],[53,51371,51372,51375],{"style":9704},[53,51373],{"className":51374,"style":4205},[2494],[53,51376,51378],{"className":51377},[2465],[53,51379,51381],{"className":51380},[2465,9714],[53,51382,51384,51417],{"className":51383},[2478,3095],[53,51385,51387,51414],{"className":51386},[2482],[53,51388,51390,51402],{"className":51389,"style":9724},[2486],[53,51391,51393,51396],{"className":51392,"style":4201},[9728],[53,51394],{"className":51395,"style":4205},[2494],[53,51397,51399],{"className":51398,"style":9735},[2465],[53,51400,132],{"className":51401},[2465],[53,51403,51404,51407],{"style":9741},[53,51405],{"className":51406,"style":4205},[2494],[53,51408,51410],{"className":51409,"style":9749},[9748],[6176,51411,51412],{"xmlns":6178,"width":9752,"height":9753,"viewBox":9754,"preserveAspectRatio":9755},[6185,51413],{"d":9758},[53,51415,3120],{"className":51416},[3119],[53,51418,51420],{"className":51419},[2482],[53,51421,51423],{"className":51422,"style":9768},[2486],[53,51424],{},[53,51426,51427,51430],{"style":9773},[53,51428],{"className":51429,"style":4205},[2494],[53,51431],{"className":51432,"style":9781},[9780],[53,51434,51435,51438],{"style":9784},[53,51436],{"className":51437,"style":4205},[2494],[53,51439,51441,51444,51447,51450,51453,51456,51459,51462,51465,51468],{"className":51440},[2465],[53,51442,2644],{"className":51443},[2836],[53,51445,37809],{"className":51446,"style":2671},[2465,2506],[53,51448,2651],{"className":51449},[2675],[53,51451],{"className":51452,"style":2610},[2609],[53,51454,2586],{"className":51455},[2614],[53,51457],{"className":51458,"style":2610},[2609],[53,51460,3188],{"className":51461},[2465,2506],[53,51463,2644],{"className":51464},[2836],[53,51466,3181],{"className":51467},[2465,2506],[53,51469,2651],{"className":51470},[2675],[53,51472,3120],{"className":51473},[3119],[53,51475,51477],{"className":51476},[2482],[53,51478,51480],{"className":51479,"style":9803},[2486],[53,51481],{},[53,51483],{"className":51484},[2675,9688],[53,51486,2714],{"className":51487},[2856],[12,51489,51490],{},"with a phase convention set by the microwave drive phase.",[12,51492,51493],{},"Detuning changes the rotation axis so it is no longer purely transverse. The unitary can be written as",[53,51495,51497],{"className":51496},[2679],[53,51498,51500,51570],{"className":51499},[2418],[53,51501,51503],{"className":51502},[2422],[2424,51504,51505],{"xmlns":2426,"display":2688},[2428,51506,51507,51567],{},[2431,51508,51509,51520,51522,51524,51526,51528,51530,51532,51565],{},[3063,51510,51511,51513],{},[2437,51512,20265],{},[4166,51514,51515,51517],{"accent":2452},[2437,51516,4315],{},[2584,51518,51519],{},"^",[2584,51521,128],{"stretchy":2650},[2437,51523,50687],{},[2584,51525,3544],{"stretchy":2650},[2584,51527,121],{},[2437,51529,23964],{},[2584,51531,36586],{},[2431,51533,51534,51536,51538,51548,51554,51557,51563],{},[2584,51535,128],{"fence":2452},[2584,51537,2586],{},[9386,51539,51540,51546],{},[2431,51541,51542,51544],{},[2437,51543,3188],{},[2437,51545,50687],{},[2588,51547,132],{},[4166,51549,51550,51552],{"accent":2452},[2437,51551,4315],{},[2584,51553,51519],{},[2584,51555,51556],{},"⋅",[4166,51558,51559,51561],{"accent":2452},[2437,51560,44603],{},[2584,51562,6072],{},[2584,51564,3544],{"fence":2452},[2584,51566,2714],{"separator":2452},[2444,51568,51569],{"encoding":2446},"R_{\\hat{n}}(\\theta)=\\exp\\left(-\\frac{i\\theta}{2}\\hat{n}\\cdot\\vec{\\sigma}\\right),",[53,51571,51573,51670],{"className":51572,"ariaHidden":2452},[2451],[53,51574,51576,51579,51652,51655,51658,51661,51664,51667],{"className":51575},[2456],[53,51577],{"className":51578,"style":2664},[2460],[53,51580,51582,51585],{"className":51581},[2465],[53,51583,20265],{"className":51584,"style":20290},[2465,2506],[53,51586,51588],{"className":51587},[2474],[53,51589,51591,51644],{"className":51590},[2478,3095],[53,51592,51594,51641],{"className":51593},[2482],[53,51595,51597],{"className":51596,"style":11138},[2486],[53,51598,51599,51602],{"style":20305},[53,51600],{"className":51601,"style":2495},[2494],[53,51603,51605],{"className":51604},[2499,2500,2501,2502],[53,51606,51608],{"className":51607},[2465,2502],[53,51609,51611],{"className":51610},[2465,4189,2502],[53,51612,51614],{"className":51613},[2478],[53,51615,51617],{"className":51616},[2482],[53,51618,51620,51629],{"className":51619,"style":5329},[2486],[53,51621,51623,51626],{"style":51622},"top:-2.7em;",[53,51624],{"className":51625,"style":2495},[2494],[53,51627,4315],{"className":51628},[2465,2506,2502],[53,51630,51631,51634],{"style":51622},[53,51632],{"className":51633,"style":2495},[2494],[53,51635,51638],{"className":51636,"style":51637},[4218],"left:-0.25em;",[53,51639,51519],{"className":51640},[2465,2502],[53,51642,3120],{"className":51643},[3119],[53,51645,51647],{"className":51646},[2482],[53,51648,51650],{"className":51649,"style":3127},[2486],[53,51651],{},[53,51653,128],{"className":51654},[2836],[53,51656,50687],{"className":51657,"style":2507},[2465,2506],[53,51659,3544],{"className":51660},[2675],[53,51662],{"className":51663,"style":2814},[2609],[53,51665,121],{"className":51666},[2818],[53,51668],{"className":51669,"style":2814},[2609],[53,51671,51673,51676,51679,51682,51841,51844],{"className":51672},[2456],[53,51674],{"className":51675,"style":37564},[2460],[53,51677,23964],{"className":51678},[7465],[53,51680],{"className":51681,"style":2860},[2609],[53,51683,51685,51691,51694,51759,51790,51793,51796,51799,51835],{"className":51684},[2832],[53,51686,51688],{"className":51687,"style":2838},[2836,2837],[53,51689,128],{"className":51690},[2842,2501],[53,51692,2586],{"className":51693},[2465],[53,51695,51697,51700,51756],{"className":51696},[2465],[53,51698],{"className":51699},[2836,9688],[53,51701,51703],{"className":51702},[9386],[53,51704,51706,51748],{"className":51705},[2478,3095],[53,51707,51709,51745],{"className":51708},[2482],[53,51710,51712,51723,51731],{"className":51711,"style":37654},[2486],[53,51713,51714,51717],{"style":11612},[53,51715],{"className":51716,"style":4205},[2494],[53,51718,51720],{"className":51719},[2465],[53,51721,132],{"className":51722},[2465],[53,51724,51725,51728],{"style":9773},[53,51726],{"className":51727,"style":4205},[2494],[53,51729],{"className":51730,"style":9781},[9780],[53,51732,51733,51736],{"style":9784},[53,51734],{"className":51735,"style":4205},[2494],[53,51737,51739,51742],{"className":51738},[2465],[53,51740,3188],{"className":51741},[2465,2506],[53,51743,50687],{"className":51744,"style":2507},[2465,2506],[53,51746,3120],{"className":51747},[3119],[53,51749,51751],{"className":51750},[2482],[53,51752,51754],{"className":51753,"style":11650},[2486],[53,51755],{},[53,51757],{"className":51758},[2675,9688],[53,51760,51762],{"className":51761},[2465,4189],[53,51763,51765],{"className":51764},[2478],[53,51766,51768],{"className":51767},[2482],[53,51769,51771,51779],{"className":51770,"style":5329},[2486],[53,51772,51773,51776],{"style":4201},[53,51774],{"className":51775,"style":4205},[2494],[53,51777,4315],{"className":51778},[2465,2506],[53,51780,51781,51784],{"style":4201},[53,51782],{"className":51783,"style":4205},[2494],[53,51785,51787],{"className":51786,"style":51637},[4218],[53,51788,51519],{"className":51789},[2465],[53,51791],{"className":51792,"style":2610},[2609],[53,51794,51556],{"className":51795},[2614],[53,51797],{"className":51798,"style":2610},[2609],[53,51800,51802],{"className":51801},[2465,4189],[53,51803,51805],{"className":51804},[2478],[53,51806,51808],{"className":51807},[2482],[53,51809,51811,51819],{"className":51810,"style":6152},[2486],[53,51812,51813,51816],{"style":4201},[53,51814],{"className":51815,"style":4205},[2494],[53,51817,44603],{"className":51818,"style":2671},[2465,2506],[53,51820,51821,51824],{"style":4201},[53,51822],{"className":51823,"style":4205},[2494],[53,51825,51828],{"className":51826,"style":51827},[4218],"left:-0.2355em;",[53,51829,51831],{"className":51830,"style":6174},[6173],[6176,51832,51833],{"xmlns":6178,"width":6179,"height":6180,"style":6181,"viewBox":6182,"preserveAspectRatio":6183},[6185,51834],{"d":6187},[53,51836,51838],{"className":51837,"style":2838},[2675,2837],[53,51839,3544],{"className":51840},[2842,2501],[53,51842],{"className":51843,"style":2860},[2609],[53,51845,2714],{"className":51846},[2856],[12,51848,47035],{},[53,51850,51852],{"className":51851},[2679],[53,51853,51855,51917],{"className":51854},[2418],[53,51856,51858],{"className":51857},[2422],[2424,51859,51860],{"xmlns":2426,"display":2688},[2428,51861,51862,51914],{},[2431,51863,51864,51870,51872,51882,51884,51886,51888,51890,51892,51894,51896,51898,51900,51902,51904,51910,51912],{},[4166,51865,51866,51868],{"accent":2452},[2437,51867,4315],{},[2584,51869,51519],{},[2584,51871,121],{},[9386,51873,51874,51876],{},[2588,51875,173],{},[3063,51877,51878,51880],{},[2437,51879,42558],{"mathvariant":2643},[2437,51881,20265],{},[2584,51883,128],{"stretchy":2650},[2437,51885,42558],{"mathvariant":2643},[2584,51887,2714],{"separator":2452},[2588,51889,149],{},[2584,51891,2714],{"separator":2452},[2437,51893,42641],{"mathvariant":2643},[2584,51895,3544],{"stretchy":2650},[2584,51897,2714],{"separator":2452},[2609,51899],{"width":17906},[2437,51901,50687],{},[2584,51903,121],{},[3063,51905,51906,51908],{},[2437,51907,42558],{"mathvariant":2643},[2437,51909,20265],{},[2437,51911,41600],{},[2437,51913,28],{"mathvariant":2643},[2444,51915,51916],{"encoding":2446},"\\hat{n}=\\frac{1}{\\Omega_R}(\\Omega,0,\\Delta), \\qquad \\theta=\\Omega_R\\tau.",[53,51918,51920,51966,52121],{"className":51919,"ariaHidden":2452},[2451],[53,51921,51923,51926,51957,51960,51963],{"className":51922},[2456],[53,51924],{"className":51925,"style":5329},[2460],[53,51927,51929],{"className":51928},[2465,4189],[53,51930,51932],{"className":51931},[2478],[53,51933,51935],{"className":51934},[2482],[53,51936,51938,51946],{"className":51937,"style":5329},[2486],[53,51939,51940,51943],{"style":4201},[53,51941],{"className":51942,"style":4205},[2494],[53,51944,4315],{"className":51945},[2465,2506],[53,51947,51948,51951],{"style":4201},[53,51949],{"className":51950,"style":4205},[2494],[53,51952,51954],{"className":51953,"style":51637},[4218],[53,51955,51519],{"className":51956},[2465],[53,51958],{"className":51959,"style":2814},[2609],[53,51961,121],{"className":51962},[2818],[53,51964],{"className":51965,"style":2814},[2609],[53,51967,51969,51973,52073,52076,52079,52082,52085,52088,52091,52094,52097,52100,52103,52106,52109,52112,52115,52118],{"className":51968},[2456],[53,51970],{"className":51971,"style":51972},[2460],"height:2.1574em;vertical-align:-0.836em;",[53,51974,51976,51979,52070],{"className":51975},[2465],[53,51977],{"className":51978},[2836,9688],[53,51980,51982],{"className":51981},[9386],[53,51983,51985,52061],{"className":51984},[2478,3095],[53,51986,51988,52058],{"className":51987},[2482],[53,51989,51991,52039,52047],{"className":51990,"style":9701},[2486],[53,51992,51993,51996],{"style":11612},[53,51994],{"className":51995,"style":4205},[2494],[53,51997,51999],{"className":51998},[2465],[53,52000,52002,52005],{"className":52001},[2465],[53,52003,42558],{"className":52004},[2465],[53,52006,52008],{"className":52007},[2474],[53,52009,52011,52031],{"className":52010},[2478,3095],[53,52012,52014,52028],{"className":52013},[2482],[53,52015,52017],{"className":52016,"style":3102},[2486],[53,52018,52019,52022],{"style":5293},[53,52020],{"className":52021,"style":2495},[2494],[53,52023,52025],{"className":52024},[2499,2500,2501,2502],[53,52026,20265],{"className":52027,"style":20290},[2465,2506,2502],[53,52029,3120],{"className":52030},[3119],[53,52032,52034],{"className":52033},[2482],[53,52035,52037],{"className":52036,"style":3127},[2486],[53,52038],{},[53,52040,52041,52044],{"style":9773},[53,52042],{"className":52043,"style":4205},[2494],[53,52045],{"className":52046,"style":9781},[9780],[53,52048,52049,52052],{"style":9784},[53,52050],{"className":52051,"style":4205},[2494],[53,52053,52055],{"className":52054},[2465],[53,52056,173],{"className":52057},[2465],[53,52059,3120],{"className":52060},[3119],[53,52062,52064],{"className":52063},[2482],[53,52065,52068],{"className":52066,"style":52067},[2486],"height:0.836em;",[53,52069],{},[53,52071],{"className":52072},[2675,9688],[53,52074,128],{"className":52075},[2836],[53,52077,42558],{"className":52078},[2465],[53,52080,2714],{"className":52081},[2856],[53,52083],{"className":52084,"style":2860},[2609],[53,52086,149],{"className":52087},[2465],[53,52089,2714],{"className":52090},[2856],[53,52092],{"className":52093,"style":2860},[2609],[53,52095,42641],{"className":52096},[2465],[53,52098,3544],{"className":52099},[2675],[53,52101,2714],{"className":52102},[2856],[53,52104],{"className":52105,"style":18524},[2609],[53,52107],{"className":52108,"style":2860},[2609],[53,52110,50687],{"className":52111,"style":2507},[2465,2506],[53,52113],{"className":52114,"style":2814},[2609],[53,52116,121],{"className":52117},[2818],[53,52119],{"className":52120,"style":2814},[2609],[53,52122,52124,52127,52167,52170],{"className":52123},[2456],[53,52125],{"className":52126,"style":3082},[2460],[53,52128,52130,52133],{"className":52129},[2465],[53,52131,42558],{"className":52132},[2465],[53,52134,52136],{"className":52135},[2474],[53,52137,52139,52159],{"className":52138},[2478,3095],[53,52140,52142,52156],{"className":52141},[2482],[53,52143,52145],{"className":52144,"style":3102},[2486],[53,52146,52147,52150],{"style":5293},[53,52148],{"className":52149,"style":2495},[2494],[53,52151,52153],{"className":52152},[2499,2500,2501,2502],[53,52154,20265],{"className":52155,"style":20290},[2465,2506,2502],[53,52157,3120],{"className":52158},[3119],[53,52160,52162],{"className":52161},[2482],[53,52163,52165],{"className":52164,"style":3127},[2486],[53,52166],{},[53,52168,41600],{"className":52169,"style":41672},[2465,2506],[53,52171,28],{"className":52172},[2465],[12,52174,52175],{},"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.",[30,52177,52179],{"id":52178},"bloch-sphere-interpretation","Bloch-Sphere Interpretation",[12,52181,52182],{},"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.",[12,52184,52185,52186,52224,52225,52253,52254,52289],{},"A resonant microwave drive rotates the Bloch vector around a transverse axis. Starting from ",[53,52187,52189,52206],{"className":52188},[2418],[53,52190,52192],{"className":52191},[2422],[2424,52193,52194],{"xmlns":2426},[2428,52195,52196,52204],{},[2431,52197,52198,52200,52202],{},[2584,52199,2644],{"stretchy":2650},[2437,52201,37809],{},[2584,52203,2651],{"stretchy":2650},[2444,52205,41791],{"encoding":2446},[53,52207,52209],{"className":52208,"ariaHidden":2452},[2451],[53,52210,52212,52215,52218,52221],{"className":52211},[2456],[53,52213],{"className":52214,"style":2664},[2460],[53,52216,2644],{"className":52217},[2836],[53,52219,37809],{"className":52220,"style":2671},[2465,2506],[53,52222,2651],{"className":52223},[2675],", a properly timed ",[53,52226,52228,52241],{"className":52227},[2418],[53,52229,52231],{"className":52230},[2422],[2424,52232,52233],{"xmlns":2426},[2428,52234,52235,52239],{},[2431,52236,52237],{},[2437,52238,8458],{},[2444,52240,47980],{"encoding":2446},[53,52242,52244],{"className":52243,"ariaHidden":2452},[2451],[53,52245,52247,52250],{"className":52246},[2456],[53,52248],{"className":52249,"style":2534},[2460],[53,52251,8458],{"className":52252,"style":2671},[2465,2506],"-pulse moves the state to the excited-state pole. A ",[53,52255,52257,52274],{"className":52256},[2418],[53,52258,52260],{"className":52259},[2422],[2424,52261,52262],{"xmlns":2426},[2428,52263,52264,52272],{},[2431,52265,52266,52268,52270],{},[2437,52267,8458],{},[2437,52269,193],{"mathvariant":2643},[2588,52271,132],{},[2444,52273,51249],{"encoding":2446},[53,52275,52277],{"className":52276,"ariaHidden":2452},[2451],[53,52278,52280,52283,52286],{"className":52279},[2456],[53,52281],{"className":52282,"style":2664},[2460],[53,52284,8458],{"className":52285,"style":2671},[2465,2506],[53,52287,26016],{"className":52288},[2465],"-pulse moves the state to the equator, where the ground-state and excited-state measurement probabilities are equal.",[12,52291,52292,52293,52321],{},"Detuning tilts the rotation axis toward the ",[53,52294,52296,52309],{"className":52295},[2418],[53,52297,52299],{"className":52298},[2422],[2424,52300,52301],{"xmlns":2426},[2428,52302,52303,52307],{},[2431,52304,52305],{},[2437,52306,214],{},[2444,52308,214],{"encoding":2446},[53,52310,52312],{"className":52311,"ariaHidden":2452},[2451],[53,52313,52315,52318],{"className":52314},[2456],[53,52316],{"className":52317,"style":2534},[2460],[53,52319,214],{"className":52320,"style":20477},[2465,2506],"-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",[53,52323,52325],{"className":52324},[2679],[53,52326,52328,52358],{"className":52327},[2418],[53,52329,52331],{"className":52330},[2422],[2424,52332,52333],{"xmlns":2426,"display":2688},[2428,52334,52335,52355],{},[2431,52336,52337,52353],{},[9386,52338,52339,52345],{},[2434,52340,52341,52343],{},[2437,52342,42558],{"mathvariant":2643},[2588,52344,132],{},[3670,52346,52347,52349,52351],{},[2437,52348,42558],{"mathvariant":2643},[2437,52350,20265],{},[2588,52352,132],{},[2437,52354,28],{"mathvariant":2643},[2444,52356,52357],{"encoding":2446},"\\frac{\\Omega^2}{\\Omega_R^2}.",[53,52359,52361],{"className":52360,"ariaHidden":2452},[2451],[53,52362,52364,52367,52503],{"className":52363},[2456],[53,52365],{"className":52366,"style":46729},[2460],[53,52368,52370,52373,52500],{"className":52369},[2465],[53,52371],{"className":52372},[2836,9688],[53,52374,52376],{"className":52375},[9386],[53,52377,52379,52492],{"className":52378},[2478,3095],[53,52380,52382,52489],{"className":52381},[2482],[53,52383,52385,52444,52452],{"className":52384,"style":46748},[2486],[53,52386,52387,52390],{"style":11612},[53,52388],{"className":52389,"style":4205},[2494],[53,52391,52393],{"className":52392},[2465],[53,52394,52396,52399],{"className":52395},[2465],[53,52397,42558],{"className":52398},[2465],[53,52400,52402],{"className":52401},[2474],[53,52403,52405,52436],{"className":52404},[2478,3095],[53,52406,52408,52433],{"className":52407},[2482],[53,52409,52411,52422],{"className":52410,"style":46775},[2486],[53,52412,52413,52416],{"style":46778},[53,52414],{"className":52415,"style":2495},[2494],[53,52417,52419],{"className":52418},[2499,2500,2501,2502],[53,52420,20265],{"className":52421,"style":20290},[2465,2506,2502],[53,52423,52424,52427],{"style":46790},[53,52425],{"className":52426,"style":2495},[2494],[53,52428,52430],{"className":52429},[2499,2500,2501,2502],[53,52431,132],{"className":52432},[2465,2502],[53,52434,3120],{"className":52435},[3119],[53,52437,52439],{"className":52438},[2482],[53,52440,52442],{"className":52441,"style":46809},[2486],[53,52443],{},[53,52445,52446,52449],{"style":9773},[53,52447],{"className":52448,"style":4205},[2494],[53,52450],{"className":52451,"style":9781},[9780],[53,52453,52454,52457],{"style":9784},[53,52455],{"className":52456,"style":4205},[2494],[53,52458,52460],{"className":52459},[2465],[53,52461,52463,52466],{"className":52462},[2465],[53,52464,42558],{"className":52465},[2465],[53,52467,52469],{"className":52468},[2474],[53,52470,52472],{"className":52471},[2478],[53,52473,52475],{"className":52474},[2482],[53,52476,52478],{"className":52477,"style":13119},[2486],[53,52479,52480,52483],{"style":2490},[53,52481],{"className":52482,"style":2495},[2494],[53,52484,52486],{"className":52485},[2499,2500,2501,2502],[53,52487,132],{"className":52488},[2465,2502],[53,52490,3120],{"className":52491},[3119],[53,52493,52495],{"className":52494},[2482],[53,52496,52498],{"className":52497,"style":46866},[2486],[53,52499],{},[53,52501],{"className":52502},[2675,9688],[53,52504,28],{"className":52505},[2465],[12,52507,52508,52509,52548],{},"As ",[53,52510,52512,52530],{"className":52511},[2418],[53,52513,52515],{"className":52514},[2422],[2424,52516,52517],{"xmlns":2426},[2428,52518,52519,52527],{},[2431,52520,52521,52523,52525],{},[2584,52522,2644],{"stretchy":2650},[2437,52524,42641],{"mathvariant":2643},[2584,52526,2644],{"stretchy":2650},[2444,52528,52529],{"encoding":2446},"\\lvert\\Delta\\rvert",[53,52531,52533],{"className":52532,"ariaHidden":2452},[2451],[53,52534,52536,52539,52542,52545],{"className":52535},[2456],[53,52537],{"className":52538,"style":2664},[2460],[53,52540,2644],{"className":52541},[2836],[53,52543,42641],{"className":52544},[2465],[53,52546,2644],{"className":52547},[2675]," grows, the rotation axis becomes more longitudinal, the oscillation frequency increases, and the maximum excited-state population decreases.",[30,52550,52552],{"id":52551},"fourier-analysis-dynamics","Fourier Analysis Dynamics",[12,52554,52555,52556,52584,52585,52683],{},"The spectral analysis computes a real Fast Fourier transform along the pulse-duration axis. For each sampled drive frequency ",[53,52557,52559,52572],{"className":52558},[2418],[53,52560,52562],{"className":52561},[2422],[2424,52563,52564],{"xmlns":2426},[2428,52565,52566,52570],{},[2431,52567,52568],{},[2437,52569,22173],{},[2444,52571,22173],{"encoding":2446},[53,52573,52575],{"className":52574,"ariaHidden":2452},[2451],[53,52576,52578,52581],{"className":52577},[2456],[53,52579],{"className":52580,"style":6930},[2460],[53,52582,22173],{"className":52583,"style":27234},[2465,2506],", the trace ",[53,52586,52588,52616],{"className":52587},[2418],[53,52589,52591],{"className":52590},[2422],[2424,52592,52593],{"xmlns":2426},[2428,52594,52595,52613],{},[2431,52596,52597,52603,52605,52607,52609,52611],{},[3063,52598,52599,52601],{},[2437,52600,3069],{},[2437,52602,3181],{},[2584,52604,128],{"stretchy":2650},[2437,52606,41600],{},[2584,52608,2714],{"separator":2452},[2437,52610,22173],{},[2584,52612,3544],{"stretchy":2650},[2444,52614,52615],{"encoding":2446},"P_e(\\tau,f)",[53,52617,52619],{"className":52618,"ariaHidden":2452},[2451],[53,52620,52622,52625,52665,52668,52671,52674,52677,52680],{"className":52621},[2456],[53,52623],{"className":52624,"style":2664},[2460],[53,52626,52628,52631],{"className":52627},[2465],[53,52629,3069],{"className":52630,"style":3115},[2465,2506],[53,52632,52634],{"className":52633},[2474],[53,52635,52637,52657],{"className":52636},[2478,3095],[53,52638,52640,52654],{"className":52639},[2482],[53,52641,52643],{"className":52642,"style":4402},[2486],[53,52644,52645,52648],{"style":7532},[53,52646],{"className":52647,"style":2495},[2494],[53,52649,52651],{"className":52650},[2499,2500,2501,2502],[53,52652,3181],{"className":52653},[2465,2506,2502],[53,52655,3120],{"className":52656},[3119],[53,52658,52660],{"className":52659},[2482],[53,52661,52663],{"className":52662,"style":3127},[2486],[53,52664],{},[53,52666,128],{"className":52667},[2836],[53,52669,41600],{"className":52670,"style":41672},[2465,2506],[53,52672,2714],{"className":52673},[2856],[53,52675],{"className":52676,"style":2860},[2609],[53,52678,22173],{"className":52679,"style":27234},[2465,2506],[53,52681,3544],{"className":52682},[2675]," is treated as a time-domain signal. The spectral amplitude is",[53,52685,52687],{"className":52686},[2679],[53,52688,52690,52753],{"className":52689},[2418],[53,52691,52693],{"className":52692},[2422],[2424,52694,52695],{"xmlns":2426,"display":2688},[2428,52696,52697,52750],{},[2431,52698,52699,52701,52703,52705,52707,52709,52711,52713,52748],{},[2437,52700,5752],{},[2584,52702,128],{"stretchy":2650},[2437,52704,42871],{},[2584,52706,2714],{"separator":2452},[2437,52708,22173],{},[2584,52710,3544],{"stretchy":2650},[2584,52712,121],{},[2431,52714,52715,52717,52724,52746],{},[2584,52716,2644],{"fence":2452},[3063,52718,52719,52722],{},[2437,52720,52721],{"mathvariant":2439},"F",[2437,52723,41600],{},[2431,52725,52726,52728,52734,52736,52738,52740,52742,52744],{},[2584,52727,2705],{"fence":2452},[3063,52729,52730,52732],{},[2437,52731,3069],{},[2437,52733,3181],{},[2584,52735,128],{"stretchy":2650},[2437,52737,41600],{},[2584,52739,2714],{"separator":2452},[2437,52741,22173],{},[2584,52743,3544],{"stretchy":2650},[2584,52745,2767],{"fence":2452},[2584,52747,2644],{"fence":2452},[2437,52749,28],{"mathvariant":2643},[2444,52751,52752],{"encoding":2446},"S(\\nu,f)=\\left|\\mathcal{F}_{\\tau}\\left\\{P_e(\\tau,f)\\right\\}\\right|.",[53,52754,52756,52792],{"className":52755,"ariaHidden":2452},[2451],[53,52757,52759,52762,52765,52768,52771,52774,52777,52780,52783,52786,52789],{"className":52758},[2456],[53,52760],{"className":52761,"style":2664},[2460],[53,52763,5752],{"className":52764,"style":5787},[2465,2506],[53,52766,128],{"className":52767},[2836],[53,52769,42871],{"className":52770,"style":42892},[2465,2506],[53,52772,2714],{"className":52773},[2856],[53,52775],{"className":52776,"style":2860},[2609],[53,52778,22173],{"className":52779,"style":27234},[2465,2506],[53,52781,3544],{"className":52782},[2675],[53,52784],{"className":52785,"style":2814},[2609],[53,52787,121],{"className":52788},[2818],[53,52790],{"className":52791,"style":2814},[2609],[53,52793,52795,52798,52922,52925],{"className":52794},[2456],[53,52796],{"className":52797,"style":2664},[2460],[53,52799,52801,52804,52849,52852,52919],{"className":52800},[2832],[53,52802,2644],{"className":52803,"style":2838},[2836,2837],[53,52805,52807,52811],{"className":52806},[2465],[53,52808,52721],{"className":52809,"style":52810},[2465,2469],"margin-right:0.0993em;",[53,52812,52814],{"className":52813},[2474],[53,52815,52817,52841],{"className":52816},[2478,3095],[53,52818,52820,52838],{"className":52819},[2482],[53,52821,52823],{"className":52822,"style":4402},[2486],[53,52824,52826,52829],{"style":52825},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[53,52827],{"className":52828,"style":2495},[2494],[53,52830,52832],{"className":52831},[2499,2500,2501,2502],[53,52833,52835],{"className":52834},[2465,2502],[53,52836,41600],{"className":52837,"style":41672},[2465,2506,2502],[53,52839,3120],{"className":52840},[3119],[53,52842,52844],{"className":52843},[2482],[53,52845,52847],{"className":52846,"style":3127},[2486],[53,52848],{},[53,52850],{"className":52851,"style":2860},[2609],[53,52853,52855,52858,52898,52901,52904,52907,52910,52913,52916],{"className":52854},[2832],[53,52856,2705],{"className":52857,"style":2838},[2836,2837],[53,52859,52861,52864],{"className":52860},[2465],[53,52862,3069],{"className":52863,"style":3115},[2465,2506],[53,52865,52867],{"className":52866},[2474],[53,52868,52870,52890],{"className":52869},[2478,3095],[53,52871,52873,52887],{"className":52872},[2482],[53,52874,52876],{"className":52875,"style":4402},[2486],[53,52877,52878,52881],{"style":7532},[53,52879],{"className":52880,"style":2495},[2494],[53,52882,52884],{"className":52883},[2499,2500,2501,2502],[53,52885,3181],{"className":52886},[2465,2506,2502],[53,52888,3120],{"className":52889},[3119],[53,52891,52893],{"className":52892},[2482],[53,52894,52896],{"className":52895,"style":3127},[2486],[53,52897],{},[53,52899,128],{"className":52900},[2836],[53,52902,41600],{"className":52903,"style":41672},[2465,2506],[53,52905,2714],{"className":52906},[2856],[53,52908],{"className":52909,"style":2860},[2609],[53,52911,22173],{"className":52912,"style":27234},[2465,2506],[53,52914,3544],{"className":52915},[2675],[53,52917,2767],{"className":52918,"style":2838},[2675,2837],[53,52920,2644],{"className":52921,"style":2838},[2675,2837],[53,52923],{"className":52924,"style":2860},[2609],[53,52926,28],{"className":52927},[2465],[12,52929,52930],{},"This probability expression contains a squared sinusoid. Using the identity",[53,52932,52934],{"className":52933},[2679],[53,52935,52937,53013],{"className":52936},[2418],[53,52938,52940],{"className":52939},[2422],[2424,52941,52942],{"xmlns":2426,"display":2688},[2428,52943,52944,53010],{},[2431,52945,52946,52956,52976,52978,52984],{},[2434,52947,52948,52954],{},[2431,52949,52950,52952],{},[2437,52951,46617],{},[2584,52953,36586],{},[2588,52955,132],{},[2431,52957,52958,52960,52974],{},[2584,52959,128],{"fence":2452},[9386,52961,52962,52972],{},[2431,52963,52964,52970],{},[3063,52965,52966,52968],{},[2437,52967,42558],{"mathvariant":2643},[2437,52969,20265],{},[2437,52971,41600],{},[2588,52973,132],{},[2584,52975,3544],{"fence":2452},[2584,52977,121],{},[9386,52979,52980,52982],{},[2588,52981,173],{},[2588,52983,132],{},[2431,52985,52986,52988,52990,52992,52994,52996,52998,53004,53006,53008],{},[2584,52987,12143],{"fence":2452},[2588,52989,173],{},[2584,52991,2586],{},[2437,52993,37828],{},[2584,52995,36586],{},[2584,52997,128],{"stretchy":2650},[3063,52999,53000,53002],{},[2437,53001,42558],{"mathvariant":2643},[2437,53003,20265],{},[2437,53005,41600],{},[2584,53007,3544],{"stretchy":2650},[2584,53009,12168],{"fence":2452},[2444,53011,53012],{"encoding":2446},"\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right)=\\frac{1}{2}\\left[1-\\cos(\\Omega_R\\tau)\\right]",[53,53014,53016,53180],{"className":53015,"ariaHidden":2452},[2451],[53,53017,53019,53022,53051,53054,53171,53174,53177],{"className":53018},[2456],[53,53020],{"className":53021,"style":37564},[2460],[53,53023,53025,53028],{"className":53024},[7465],[53,53026,46617],{"className":53027},[7465],[53,53029,53031],{"className":53030},[2474],[53,53032,53034],{"className":53033},[2478],[53,53035,53037],{"className":53036},[2482],[53,53038,53040],{"className":53039,"style":46893},[2486],[53,53041,53042,53045],{"style":46896},[53,53043],{"className":53044,"style":2495},[2494],[53,53046,53048],{"className":53047},[2499,2500,2501,2502],[53,53049,132],{"className":53050},[2465,2502],[53,53052],{"className":53053,"style":2860},[2609],[53,53055,53057,53063,53165],{"className":53056},[2832],[53,53058,53060],{"className":53059,"style":2838},[2836,2837],[53,53061,128],{"className":53062},[2842,2501],[53,53064,53066,53069,53162],{"className":53065},[2465],[53,53067],{"className":53068},[2836,9688],[53,53070,53072],{"className":53071},[9386],[53,53073,53075,53154],{"className":53074},[2478,3095],[53,53076,53078,53151],{"className":53077},[2482],[53,53079,53081,53092,53100],{"className":53080,"style":46936},[2486],[53,53082,53083,53086],{"style":11612},[53,53084],{"className":53085,"style":4205},[2494],[53,53087,53089],{"className":53088},[2465],[53,53090,132],{"className":53091},[2465],[53,53093,53094,53097],{"style":9773},[53,53095],{"className":53096,"style":4205},[2494],[53,53098],{"className":53099,"style":9781},[9780],[53,53101,53102,53105],{"style":9784},[53,53103],{"className":53104,"style":4205},[2494],[53,53106,53108,53148],{"className":53107},[2465],[53,53109,53111,53114],{"className":53110},[2465],[53,53112,42558],{"className":53113},[2465],[53,53115,53117],{"className":53116},[2474],[53,53118,53120,53140],{"className":53119},[2478,3095],[53,53121,53123,53137],{"className":53122},[2482],[53,53124,53126],{"className":53125,"style":3102},[2486],[53,53127,53128,53131],{"style":5293},[53,53129],{"className":53130,"style":2495},[2494],[53,53132,53134],{"className":53133},[2499,2500,2501,2502],[53,53135,20265],{"className":53136,"style":20290},[2465,2506,2502],[53,53138,3120],{"className":53139},[3119],[53,53141,53143],{"className":53142},[2482],[53,53144,53146],{"className":53145,"style":3127},[2486],[53,53147],{},[53,53149,41600],{"className":53150,"style":41672},[2465,2506],[53,53152,3120],{"className":53153},[3119],[53,53155,53157],{"className":53156},[2482],[53,53158,53160],{"className":53159,"style":11650},[2486],[53,53161],{},[53,53163],{"className":53164},[2675,9688],[53,53166,53168],{"className":53167,"style":2838},[2675,2837],[53,53169,3544],{"className":53170},[2842,2501],[53,53172],{"className":53173,"style":2814},[2609],[53,53175,121],{"className":53176},[2818],[53,53178],{"className":53179,"style":2814},[2609],[53,53181,53183,53186,53248,53251],{"className":53182},[2456],[53,53184],{"className":53185,"style":48699},[2460],[53,53187,53189,53192,53245],{"className":53188},[2465],[53,53190],{"className":53191},[2836,9688],[53,53193,53195],{"className":53194},[9386],[53,53196,53198,53237],{"className":53197},[2478,3095],[53,53199,53201,53234],{"className":53200},[2482],[53,53202,53204,53215,53223],{"className":53203,"style":9701},[2486],[53,53205,53206,53209],{"style":11612},[53,53207],{"className":53208,"style":4205},[2494],[53,53210,53212],{"className":53211},[2465],[53,53213,132],{"className":53214},[2465],[53,53216,53217,53220],{"style":9773},[53,53218],{"className":53219,"style":4205},[2494],[53,53221],{"className":53222,"style":9781},[9780],[53,53224,53225,53228],{"style":9784},[53,53226],{"className":53227,"style"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that the dominant oscillatory spectral component appears at the generalized frequency",[53,53328,53330],{"className":53329},[2679],[53,53331,53333,53369],{"className":53332},[2418],[53,53334,53336],{"className":53335},[2422],[2424,53337,53338],{"xmlns":2426,"display":2688},[2428,53339,53340,53366],{},[2431,53341,53342,53348,53350,53364],{},[3063,53343,53344,53346],{},[2437,53345,42871],{},[2437,53347,20265],{},[2584,53349,121],{},[9386,53351,53352,53358],{},[3063,53353,53354,53356],{},[2437,53355,42558],{"mathvariant":2643},[2437,53357,20265],{},[2431,53359,53360,53362],{},[2588,53361,132],{},[2437,53363,8458],{},[2437,53365,28],{"mathvariant":2643},[2444,53367,53368],{"encoding":2446},"\\nu_R=\\frac{\\Omega_R}{2\\pi}.",[53,53370,53372,53427],{"className":53371,"ariaHidden":2452},[2451],[53,53373,53375,53378,53418,53421,53424],{"className":53374},[2456],[53,53376],{"className":53377,"style":25092},[2460],[53,53379,53381,53384],{"className":53380},[2465],[53,53382,42871],{"className":53383,"style":42892},[2465,2506],[53,53385,53387],{"className":53386},[2474],[53,53388,53390,53410],{"className":53389},[2478,3095],[53,53391,53393,53407],{"className":53392},[2482],[53,53394,53396],{"className":53395,"style":3102},[2486],[53,53397,53398,53401],{"style":42907},[53,53399],{"className":53400,"style":2495},[2494],[53,53402,53404],{"className":53403},[2499,2500,2501,2502],[53,53405,20265],{"className":53406,"style":20290},[2465,2506,2502],[53,53408,3120],{"className":53409},[3119],[53,53411,53413],{"className":53412},[2482],[53,53414,53416],{"className":53415,"style":3127},[2486],[53,53417],{},[53,53419],{"className":53420,"style":2814},[2609],[53,53422,121],{"className":53423},[2818],[53,53425],{"className":53426,"style":2814},[2609],[53,53428,53430,53433,53535],{"className":53429},[2456],[53,53431],{"className":53432,"style":48329},[2460],[53,53434,53436,53439,53532],{"className":53435},[2465],[53,53437],{"className":53438},[2836,9688],[53,53440,53442],{"className":53441},[9386],[53,53443,53445,53524],{"className":53444},[2478,3095],[53,53446,53448,53521],{"className":53447},[2482],[53,53449,53451,53465,53473],{"className":53450,"style":46936},[2486],[53,53452,53453,53456],{"style":11612},[53,53454],{"className":53455,"style":4205},[2494],[53,53457,53459,53462],{"className":53458},[2465],[53,53460,132],{"className":53461},[2465],[53,53463,8458],{"className":53464,"style":2671},[2465,2506],[53,53466,53467,53470],{"style":9773},[53,53468],{"className":53469,"style":4205},[2494],[53,53471],{"className":53472,"style":9781},[9780],[53,53474,53475,53478],{"style":9784},[53,53476],{"className":53477,"style":4205},[2494],[53,53479,53481],{"className":53480},[2465],[53,53482,53484,53487],{"className":53483},[2465],[53,53485,42558],{"className":53486},[2465],[53,53488,53490],{"className":53489},[2474],[53,53491,53493,53513],{"className":53492},[2478,3095],[53,53494,53496,53510],{"className":53495},[2482],[53,53497,53499],{"className":53498,"style":3102},[2486],[53,53500,53501,53504],{"style":5293},[53,53502],{"className":53503,"style":2495},[2494],[53,53505,53507],{"className":53506},[2499,2500,2501,2502],[53,53508,20265],{"className":53509,"style":20290},[2465,2506,2502],[53,53511,3120],{"className":53512},[3119],[53,53514,53516],{"className":53515},[2482],[53,53517,53519],{"className":53518,"style":3127},[2486],[53,53520],{},[53,53522,3120],{"className":53523},[3119],[53,53525,53527],{"className":53526},[2482],[53,53528,53530],{"className":53529,"style":11650},[2486],[53,53531],{},[53,53533],{"className":53534},[2675,9688],[53,53536,28],{"className":53537},[2465],[12,53539,53540],{},"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.",[30,53542,53544],{"id":53543},"computational-methodology","Computational Methodology",[12,53546,53547,53548,53576,53577,53605,53606,10694,53634,53662,53663,53797,53798,54017],{},"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 ",[53,53549,53551,53564],{"className":53550},[2418],[53,53552,53554],{"className":53553},[2422],[2424,53555,53556],{"xmlns":2426},[2428,53557,53558,53562],{},[2431,53559,53560],{},[2437,53561,22173],{},[2444,53563,22173],{"encoding":2446},[53,53565,53567],{"className":53566,"ariaHidden":2452},[2451],[53,53568,53570,53573],{"className":53569},[2456],[53,53571],{"className":53572,"style":6930},[2460],[53,53574,22173],{"className":53575,"style":27234},[2465,2506]," in GHz and a pulse-duration grid ",[53,53578,53580,53593],{"className":53579},[2418],[53,53581,53583],{"className":53582},[2422],[2424,53584,53585],{"xmlns":2426},[2428,53586,53587,53591],{},[2431,53588,53589],{},[2437,53590,41600],{},[2444,53592,41726],{"encoding":2446},[53,53594,53596],{"className":53595,"ariaHidden":2452},[2451],[53,53597,53599,53602],{"className":53598},[2456],[53,53600],{"className":53601,"style":2534},[2460],[53,53603,41600],{"className":53604,"style":41672},[2465,2506]," in ns, then converts both to SI units. From a dense two-dimensional mesh over ",[53,53607,53609,53622],{"className":53608},[2418],[53,53610,53612],{"className":53611},[2422],[2424,53613,53614],{"xmlns":2426},[2428,53615,53616,53620],{},[2431,53617,53618],{},[2437,53619,22173],{},[2444,53621,22173],{"encoding":2446},[53,53623,53625],{"className":53624,"ariaHidden":2452},[2451],[53,53626,53628,53631],{"className":53627},[2456],[53,53629],{"className":53630,"style":6930},[2460],[53,53632,22173],{"className":53633,"style":27234},[2465,2506],[53,53635,53637,53650],{"className":53636},[2418],[53,53638,53640],{"className":53639},[2422],[2424,53641,53642],{"xmlns":2426},[2428,53643,53644,53648],{},[2431,53645,53646],{},[2437,53647,41600],{},[2444,53649,41726],{"encoding":2446},[53,53651,53653],{"className":53652,"ariaHidden":2452},[2451],[53,53654,53656,53659],{"className":53655},[2456],[53,53657],{"className":53658,"style":2534},[2460],[53,53660,41600],{"className":53661,"style":41672},[2465,2506],", the workflow computes the detuning ",[53,53664,53666,53700],{"className":53665},[2418],[53,53667,53669],{"className":53668},[2422],[2424,53670,53671],{"xmlns":2426},[2428,53672,53673,53697],{},[2431,53674,53675,53677,53679,53681,53683,53685,53687,53689,53695],{},[2437,53676,42641],{"mathvariant":2643},[2584,53678,121],{},[2588,53680,132],{},[2437,53682,8458],{},[2584,53684,128],{"stretchy":2650},[2437,53686,22173],{},[2584,53688,2586],{},[3063,53690,53691,53693],{},[2437,53692,22173],{},[2588,53694,149],{},[2584,53696,3544],{"stretchy":2650},[2444,53698,53699],{"encoding":2446},"\\Delta=2\\pi(f-f_0)",[53,53701,53703,53721,53748],{"className":53702,"ariaHidden":2452},[2451],[53,53704,53706,53709,53712,53715,53718],{"className":53705},[2456],[53,53707],{"className":53708,"style":2461},[2460],[53,53710,42641],{"className":53711},[2465],[53,53713],{"className":53714,"style":2814},[2609],[53,53716,121],{"className":53717},[2818],[53,53719],{"className":53720,"style":2814},[2609],[53,53722,53724,53727,53730,53733,53736,53739,53742,53745],{"className":53723},[2456],[53,53725],{"className":53726,"style":2664},[2460],[53,53728,132],{"className":53729},[2465],[53,53731,8458],{"className":53732,"style":2671},[2465,2506],[53,53734,128],{"className":53735},[2836],[53,53737,22173],{"className":53738,"style":27234},[2465,2506],[53,53740],{"className":53741,"style":2610},[2609],[53,53743,2586],{"className":53744},[2614],[53,53746],{"className":53747,"style":2610},[2609],[53,53749,53751,53754,53794],{"className":53750},[2456],[53,53752],{"className":53753,"style":2664},[2460],[53,53755,53757,53760],{"className":53756},[2465],[53,53758,22173],{"className":53759,"style":27234},[2465,2506],[53,53761,53763],{"className":53762},[2474],[53,53764,53766,53786],{"className":53765},[2478,3095],[53,53767,53769,53783],{"className":53768},[2482],[53,53770,53772],{"className":53771,"style":14176},[2486],[53,53773,53774,53777],{"style":42163},[53,53775],{"className":53776,"style":2495},[2494],[53,53778,53780],{"className":53779},[2499,2500,2501,2502],[53,53781,149],{"className":53782},[2465,2502],[53,53784,3120],{"className":53785},[3119],[53,53787,53789],{"className":53788},[2482],[53,53790,53792],{"className":53791,"style":3127},[2486],[53,53793],{},[53,53795,3544],{"className":53796},[2675],", evaluates the generalized Rabi angular frequency ",[53,53799,53801,53838],{"className":53800},[2418],[53,53802,53804],{"className":53803},[2422],[2424,53805,53806],{"xmlns":2426},[2428,53807,53808,53836],{},[2431,53809,53810,53816,53818],{},[3063,53811,53812,53814],{},[2437,53813,42558],{"mathvariant":2643},[2437,53815,20265],{},[2584,53817,121],{},[9391,53819,53820],{},[2431,53821,53822,53828,53830],{},[2434,53823,53824,53826],{},[2437,53825,42558],{"mathvariant":2643},[2588,53827,132],{},[2584,53829,2002],{},[2434,53831,53832,53834],{},[2437,53833,42641],{"mathvariant":2643},[2588,53835,132],{},[2444,53837,47079],{"encoding":2446},[53,53839,53841,53896],{"className":53840,"ariaHidden":2452},[2451],[53,53842,53844,53847,53887,53890,53893],{"className":53843},[2456],[53,53845],{"className":53846,"style":3082},[2460],[53,53848,53850,53853],{"className":53849},[2465],[53,53851,42558],{"className":53852},[2465],[53,53854,53856],{"className":53855},[2474],[53,53857,53859,53879],{"className":53858},[2478,3095],[53,53860,53862,53876],{"className":53861},[2482],[53,53863,53865],{"className":53864,"style":3102},[2486],[53,53866,53867,53870],{"style":5293},[53,53868],{"className":53869,"style":2495},[2494],[53,53871,53873],{"className":53872},[2499,2500,2501,2502],[53,53874,20265],{"className":53875,"style":20290},[2465,2506,2502],[53,53877,3120],{"className":53878},[3119],[53,53880,53882],{"className":53881},[2482],[53,53883,53885],{"className":53884,"style":3127},[2486],[53,53886],{},[53,53888],{"className":53889,"style":2814},[2609],[53,53891,121],{"className":53892},[2818],[53,53894],{"className":53895,"style":2814},[2609],[53,53897,53899,53903],{"className":53898},[2456],[53,53900],{"className":53901,"style":53902},[2460],"height:1.04em;vertical-align:-0.1266em;",[53,53904,53906],{"className":53905},[2465,9714],[53,53907,53909,54008],{"className":53908},[2478,3095],[53,53910,53912,54005],{"className":53911},[2482],[53,53913,53916,53992],{"className":53914,"style":53915},[2486],"height:0.9134em;",[53,53917,53919,53922],{"className":53918,"style":4201},[9728],[53,53920],{"className":53921,"style":4205},[2494],[53,53923,53925,53954,53957,53960,53963],{"className":53924,"style":9735},[2465],[53,53926,53928,53931],{"className":53927},[2465],[53,53929,42558],{"className":53930},[2465],[53,53932,53934],{"className":53933},[2474],[53,53935,53937],{"className":53936},[2478],[53,53938,53940],{"className":53939},[2482],[53,53941,53943],{"className":53942,"style":29305},[2486],[53,53944,53945,53948],{"style":29308},[53,53946],{"className":53947,"style":2495},[2494],[53,53949,53951],{"className":53950},[2499,2500,2501,2502],[53,53952,132],{"className":53953},[2465,2502],[53,53955],{"className":53956,"style":2610},[2609],[53,53958,2002],{"className":53959},[2614],[53,53961],{"className":53962,"style":2610},[2609],[53,53964,53966,53969],{"className":53965},[2465],[53,53967,42641],{"className":53968},[2465],[53,53970,53972],{"className":53971},[2474],[53,53973,53975],{"className":53974},[2478],[53,53976,53978],{"className":53977},[2482],[53,53979,53981],{"className":53980,"style":29305},[2486],[53,53982,53983,53986],{"style":29308},[53,53984],{"className":53985,"style":2495},[2494],[53,53987,53989],{"className":53988},[2499,2500,2501,2502],[53,53990,132],{"className":53991},[2465,2502],[53,53993,53995,53998],{"style":53994},"top:-2.8734em;",[53,53996],{"className":53997,"style":4205},[2494],[53,53999,54001],{"className":54000,"style":9749},[9748],[6176,54002,54003],{"xmlns":6178,"width":9752,"height":9753,"viewBox":9754,"preserveAspectRatio":9755},[6185,54004],{"d":9758},[53,54006,3120],{"className":54007},[3119],[53,54009,54011],{"className":54010},[2482],[53,54012,54015],{"className":54013,"style":54014},[2486],"height:0.1266em;",[53,54016],{},", and calculates the analytic excited-state probability",[53,54019,54021],{"className":54020},[2679],[53,54022,54024,54102],{"className":54023},[2418],[53,54025,54027],{"className":54026},[2422],[2424,54028,54029],{"xmlns":2426,"display":2688},[2428,54030,54031,54099],{},[2431,54032,54033,54039,54041,54043,54045,54047,54049,54051,54067,54077,54097],{},[3063,54034,54035,54037],{},[2437,54036,3069],{},[2437,54038,3181],{},[2584,54040,128],{"stretchy":2650},[2437,54042,22173],{},[2584,54044,2714],{"separator":2452},[2437,54046,41600],{},[2584,54048,3544],{"stretchy":2650},[2584,54050,121],{},[9386,54052,54053,54059],{},[2434,54054,54055,54057],{},[2437,54056,42558],{"mathvariant":2643},[2588,54058,132],{},[3670,54060,54061,54063,54065],{},[2437,54062,42558],{"mathvariant":2643},[2437,54064,20265],{},[2588,54066,132],{},[2434,54068,54069,54075],{},[2431,54070,54071,54073],{},[2437,54072,46617],{},[2584,54074,36586],{},[2588,54076,132],{},[2431,54078,54079,54081,54095],{},[2584,54080,128],{"fence":2452},[9386,54082,54083,54093],{},[2431,54084,54085,54091],{},[3063,54086,54087,54089],{},[2437,54088,42558],{"mathvariant":2643},[2437,54090,20265],{},[2437,54092,41600],{},[2588,54094,132],{},[2584,54096,3544],{"fence":2452},[2437,54098,28],{"mathvariant":2643},[2444,54100,54101],{"encoding":2446},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right).",[53,54103,54105,54178],{"className":54104,"ariaHidden":2452},[2451],[53,54106,54108,54111,54151,54154,54157,54160,54163,54166,54169,54172,54175],{"className":54107},[2456],[53,54109],{"className":54110,"style":2664},[2460],[53,54112,54114,54117],{"className":54113},[2465],[53,54115,3069],{"className":54116,"style":3115},[2465,2506],[53,54118,54120],{"className":54119},[2474],[53,54121,54123,54143],{"className":54122},[2478,3095],[53,54124,54126,54140],{"className":54125},[2482],[53,54127,54129],{"className":54128,"style":4402},[2486],[53,54130,54131,54134],{"style":7532},[53,54132],{"className":54133,"style":2495},[2494],[53,54135,54137],{"className":54136},[2499,2500,2501,2502],[53,54138,3181],{"className":54139},[2465,2506,2502],[53,54141,3120],{"className":54142},[3119],[53,54144,54146],{"className":54145},[2482],[53,54147,54149],{"className":54148,"style":3127},[2486],[53,54150],{},[53,54152,128],{"className":54153},[2836],[53,54155,22173],{"className":54156,"style":27234},[2465,2506],[53,54158,2714],{"className":54159},[2856],[53,54161],{"className":54162,"style":2860},[2609],[53,54164,41600],{"className":54165,"style":41672},[2465,2506],[53,54167,3544],{"className":54168},[2675],[53,54170],{"className":54171,"style":2814},[2609],[53,54173,121],{"className":54174},[2818],[53,54176],{"className":54177,"style":2814},[2609],[53,54179,54181,54184,54320,54323,54352,54355,54472,54475],{"className":54180},[2456],[53,54182],{"className":54183,"style":46729},[2460],[53,54185,54187,54190,54317],{"className":54186},[2465],[53,54188],{"className":54189},[2836,9688],[53,54191,54193],{"className":54192},[9386],[53,54194,54196,54309],{"className":54195},[2478,3095],[53,54197,54199,54306],{"className":54198},[2482],[53,54200,54202,54261,54269],{"className":54201,"style":46748},[2486],[53,54203,54204,54207],{"style":11612},[53,54205],{"className":54206,"style":4205},[2494],[53,54208,54210],{"className":54209},[2465],[53,54211,54213,54216],{"className":54212},[2465],[53,54214,42558],{"className":54215},[2465],[53,54217,54219],{"className":54218},[2474],[53,54220,54222,54253],{"className":54221},[2478,3095],[53,54223,54225,54250],{"className":54224},[2482],[53,54226,54228,54239],{"className":54227,"style":46775},[2486],[53,54229,54230,54233],{"style":46778},[53,54231],{"className":54232,"style":2495},[2494],[53,54234,54236],{"className":54235},[2499,2500,2501,2502],[53,54237,20265],{"className":54238,"style":20290},[2465,2506,2502],[53,54240,54241,54244],{"style":46790},[53,54242],{"className":54243,"style":2495},[2494],[53,54245,54247],{"className":54246},[2499,2500,2501,2502],[53,54248,132],{"className":54249},[2465,2502],[53,54251,3120],{"className":54252},[3119],[53,54254,54256],{"className":54255},[2482],[53,54257,54259],{"className":54258,"style":46809},[2486],[53,54260],{},[53,54262,54263,54266],{"style":9773},[53,54264],{"className":54265,"style":4205},[2494],[53,54267],{"className":54268,"style":9781},[9780],[53,54270,54271,54274],{"style":9784},[53,54272],{"className":54273,"style":4205},[2494],[53,54275,54277],{"className":54276},[2465],[53,54278,54280,54283],{"className":54279},[2465],[53,54281,42558],{"className":54282},[2465],[53,54284,54286],{"className":54285},[2474],[53,54287,54289],{"className":54288},[2478],[53,54290,54292],{"className":54291},[2482],[53,54293,54295],{"className":54294,"style":13119},[2486],[53,54296,54297,54300],{"style":2490},[53,54298],{"className":54299,"style":2495},[2494],[53,54301,54303],{"className":54302},[2499,2500,2501,2502],[53,54304,132],{"className":54305},[2465,2502],[53,54307,3120],{"className":54308},[3119],[53,54310,54312],{"className":54311},[2482],[53,54313,54315],{"className":54314,"style":46866},[2486],[53,54316],{},[53,54318],{"className":54319},[2675,9688],[53,54321],{"className":54322,"style":2860},[2609],[53,54324,54326,54329],{"className":54325},[7465],[53,54327,46617],{"className":54328},[7465],[53,54330,54332],{"className":54331},[2474],[53,54333,54335],{"className":54334},[2478],[53,54336,54338],{"className":54337},[2482],[53,54339,54341],{"className":54340,"style":46893},[2486],[53,54342,54343,54346],{"style":46896},[53,54344],{"className":54345,"style":2495},[2494],[53,54347,54349],{"className":54348},[2499,2500,2501,2502],[53,54350,132],{"className":54351},[2465,2502],[53,54353],{"className":54354,"style":2860},[2609],[53,54356,54358,54364,54466],{"className":54357},[2832],[53,54359,54361],{"className":54360,"style":2838},[2836,2837],[53,54362,128],{"className":54363},[2842,2501],[53,54365,54367,54370,54463],{"className":54366},[2465],[53,54368],{"className":54369},[2836,9688],[53,54371,54373],{"className":54372},[9386],[53,54374,54376,54455],{"className":54375},[2478,3095],[53,54377,54379,54452],{"className":54378},[2482],[53,54380,54382,54393,54401],{"className":54381,"style":46936},[2486],[53,54383,54384,54387],{"style":11612},[53,54385],{"className":54386,"style":4205},[2494],[53,54388,54390],{"className":54389},[2465],[53,54391,132],{"className":54392},[2465],[53,54394,54395,54398],{"style":9773},[53,54396],{"className":54397,"style":4205},[2494],[53,54399],{"className":54400,"style":9781},[9780],[53,54402,54403,54406],{"style":9784},[53,54404],{"className":54405,"style":4205},[2494],[53,54407,54409,54449],{"className":54408},[2465],[53,54410,54412,54415],{"className":54411},[2465],[53,54413,42558],{"className":54414},[2465],[53,54416,54418],{"className":54417},[2474],[53,54419,54421,54441],{"className":54420},[2478,3095],[53,54422,54424,54438],{"className":54423},[2482],[53,54425,54427],{"className":54426,"style":3102},[2486],[53,54428,54429,54432],{"style":5293},[53,54430],{"className":54431,"style":2495},[2494],[53,54433,54435],{"className":54434},[2499,2500,2501,2502],[53,54436,20265],{"className":54437,"style":20290},[2465,2506,2502],[53,54439,3120],{"className":54440},[3119],[53,54442,54444],{"className":54443},[2482],[53,54445,54447],{"className":54446,"style":3127},[2486],[53,54448],{},[53,54450,41600],{"className":54451,"style":41672},[2465,2506],[53,54453,3120],{"className":54454},[3119],[53,54456,54458],{"className":54457},[2482],[53,54459,54461],{"className":54460,"style":11650},[2486],[53,54462],{},[53,54464],{"className":54465},[2675,9688],[53,54467,54469],{"className":54468,"style":2838},[2675,2837],[53,54470,3544],{"className":54471},[2842,2501],[53,54473],{"className":54474,"style":2860},[2609],[53,54476,28],{"className":54477},[2465],[12,54479,54480,54481,54596],{},"The routine optionally applies the damping envelope ",[53,54482,54484,54518],{"className":54483},[2418],[53,54485,54487],{"className":54486},[2422],[2424,54488,54489],{"xmlns":2426},[2428,54490,54491,54515],{},[2431,54492,54493,54495,54497,54499,54501,54503,54505,54513],{},[2437,54494,23964],{},[2584,54496,36586],{},[2584,54498,128],{"stretchy":2650},[2584,54500,2586],{},[2437,54502,41600],{},[2437,54504,193],{"mathvariant":2643},[3670,54506,54507,54509,54511],{},[2437,54508,36942],{},[2588,54510,132],{},[2584,54512,20664],{},[2584,54514,3544],{"stretchy":2650},[2444,54516,54517],{"encoding":2446},"\\exp(-\\tau\u002FT_2^\\ast)",[53,54519,54521],{"className":54520,"ariaHidden":2452},[2451],[53,54522,54524,54527,54530,54533,54536,54539,54542,54593],{"className":54523},[2456],[53,54525],{"className":54526,"style":2664},[2460],[53,54528,23964],{"className":54529},[7465],[53,54531,128],{"className":54532},[2836],[53,54534,2586],{"className":54535},[2465],[53,54537,41600],{"className":54538,"style":41672},[2465,2506],[53,54540,193],{"className":54541},[2465],[53,54543,54545,54548],{"className":54544},[2465],[53,54546,36942],{"className":54547,"style":3115},[2465,2506],[53,54549,54551],{"className":54550},[2474],[53,54552,54554,54585],{"className":54553},[2478,3095],[53,54555,54557,54582],{"className":54556},[2482],[53,54558,54560,54571],{"className":54559,"style":36478},[2486],[53,54561,54562,54565],{"style":43168},[53,54563],{"className":54564,"style":2495},[2494],[53,54566,54568],{"className":54567},[2499,2500,2501,2502],[53,54569,132],{"className":54570},[2465,2502],[53,54572,54573,54576],{"style":2490},[53,54574],{"className":54575,"style":2495},[2494],[53,54577,54579],{"className":54578},[2499,2500,2501,2502],[53,54580,20664],{"className":54581},[2614,2502],[53,54583,3120],{"className":54584},[3119],[53,54586,54588],{"className":54587},[2482],[53,54589,54591],{"className":54590,"style":43198},[2486],[53,54592],{},[53,54594,3544],{"className":54595},[2675]," and clips the result to the physical interval",[53,54598,54600],{"className":54599},[2679],[53,54601,54603,54629],{"className":54602},[2418],[53,54604,54606],{"className":54605},[2422],[2424,54607,54608],{"xmlns":2426,"display":2688},[2428,54609,54610,54626],{},[2431,54611,54612,54614,54616,54622,54624],{},[2588,54613,149],{},[2584,54615,27405],{},[3063,54617,54618,54620],{},[2437,54619,3069],{},[2437,54621,3181],{},[2584,54623,27405],{},[2588,54625,44239],{},[2444,54627,54628],{"encoding":2446},"0\\le P_e\\le 1.",[53,54630,54632,54651,54706],{"className":54631,"ariaHidden":2452},[2451],[53,54633,54635,54639,54642,54645,54648],{"className":54634},[2456],[53,54636],{"className":54637,"style":54638},[2460],"height:0.7804em;vertical-align:-0.136em;",[53,54640,149],{"className":54641},[2465],[53,54643],{"className":54644,"style":2814},[2609],[53,54646,27405],{"className":54647},[2818],[53,54649],{"className":54650,"style":2814},[2609],[53,54652,54654,54657,54697,54700,54703],{"className":54653},[2456],[53,54655],{"className":54656,"style":3082},[2460],[53,54658,54660,54663],{"className":54659},[2465],[53,54661,3069],{"className":54662,"style":3115},[2465,2506],[53,54664,54666],{"className":54665},[2474],[53,54667,54669,54689],{"className":54668},[2478,3095],[53,54670,54672,54686],{"className":54671},[2482],[53,54673,54675],{"className":54674,"style":4402},[2486],[53,54676,54677,54680],{"style":7532},[53,54678],{"className":54679,"style":2495},[2494],[53,54681,54683],{"className":54682},[2499,2500,2501,2502],[53,54684,3181],{"className":54685},[2465,2506,2502],[53,54687,3120],{"className":54688},[3119],[53,54690,54692],{"className":54691},[2482],[53,54693,54695],{"className":54694,"style":3127},[2486],[53,54696],{},[53,54698],{"className":54699,"style":2814},[2609],[53,54701,27405],{"className":54702},[2818],[53,54704],{"className":54705,"style":2814},[2609],[53,54707,54709,54712],{"className":54708},[2456],[53,54710],{"className":54711,"style":2624},[2460],[53,54713,44239],{"className":54714},[2465],[12,54716,54717],{},"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.",[30,54719,54721],{"id":54720},"graphical-interpretation","Graphical Interpretation",[12,54723,54724,54725,54823],{},"The ",[53,54726,54728,54756],{"className":54727},[2418],[53,54729,54731],{"className":54730},[2422],[2424,54732,54733],{"xmlns":2426},[2428,54734,54735,54753],{},[2431,54736,54737,54743,54745,54747,54749,54751],{},[3063,54738,54739,54741],{},[2437,54740,3069],{},[2437,54742,3181],{},[2584,54744,128],{"stretchy":2650},[2437,54746,22173],{},[2584,54748,2714],{"separator":2452},[2437,54750,41600],{},[2584,54752,3544],{"stretchy":2650},[2444,54754,54755],{"encoding":2446},"P_e(f,\\tau)",[53,54757,54759],{"className":54758,"ariaHidden":2452},[2451],[53,54760,54762,54765,54805,54808,54811,54814,54817,54820],{"className":54761},[2456],[53,54763],{"className":54764,"style":2664},[2460],[53,54766,54768,54771],{"className":54767},[2465],[53,54769,3069],{"className":54770,"style":3115},[2465,2506],[53,54772,54774],{"className":54773},[2474],[53,54775,54777,54797],{"className":54776},[2478,3095],[53,54778,54780,54794],{"className":54779},[2482],[53,54781,54783],{"className":54782,"style":4402},[2486],[53,54784,54785,54788],{"style":7532},[53,54786],{"className":54787,"style":2495},[2494],[53,54789,54791],{"className":54790},[2499,2500,2501,2502],[53,54792,3181],{"className":54793},[2465,2506,2502],[53,54795,3120],{"className":54796},[3119],[53,54798,54800],{"className":54799},[2482],[53,54801,54803],{"className":54802,"style":3127},[2486],[53,54804],{},[53,54806,128],{"className":54807},[2836],[53,54809,22173],{"className":54810,"style":27234},[2465,2506],[53,54812,2714],{"className":54813},[2856],[53,54815],{"className":54816,"style":2860},[2609],[53,54818,41600],{"className":54819,"style":41672},[2465,2506],[53,54821,3544],{"className":54822},[2675]," 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.",[12,54825,54826],{},"The resonant interaction region appears near",[53,54828,54830],{"className":54829},[2679],[53,54831,54833,54856],{"className":54832},[2418],[53,54834,54836],{"className":54835},[2422],[2424,54837,54838],{"xmlns":2426,"display":2688},[2428,54839,54840,54854],{},[2431,54841,54842,54844,54846,54852],{},[2437,54843,22173],{},[2584,54845,121],{},[3063,54847,54848,54850],{},[2437,54849,22173],{},[2588,54851,149],{},[2437,54853,28],{"mathvariant":2643},[2444,54855,43935],{"encoding":2446},[53,54857,54859,54877],{"className":54858,"ariaHidden":2452},[2451],[53,54860,54862,54865,54868,54871,54874],{"className":54861},[2456],[53,54863],{"className":54864,"style":6930},[2460],[53,54866,22173],{"className":54867,"style":27234},[2465,2506],[53,54869],{"className":54870,"style":2814},[2609],[53,54872,121],{"className":54873},[2818],[53,54875],{"className":54876,"style":2814},[2609],[53,54878,54880,54883,54923],{"className":54879},[2456],[53,54881],{"className":54882,"style":6930},[2460],[53,54884,54886,54889],{"className":54885},[2465],[53,54887,22173],{"className":54888,"style":27234},[2465,2506],[53,54890,54892],{"className":54891},[2474],[53,54893,54895,54915],{"className":54894},[2478,3095],[53,54896,54898,54912],{"className":54897},[2482],[53,54899,54901],{"className":54900,"style":14176},[2486],[53,54902,54903,54906],{"style":42163},[53,54904],{"className":54905,"style":2495},[2494],[53,54907,54909],{"className":54908},[2499,2500,2501,2502],[53,54910,149],{"className":54911},[2465,2502],[53,54913,3120],{"className":54914},[3119],[53,54916,54918],{"className":54917},[2482],[53,54919,54921],{"className":54920,"style":3127},[2486],[53,54922],{},[53,54924,28],{"className":54925},[2465],[12,54927,54928,54929,55008],{},"At resonance, contrast is maximal and the pulse duration alone sets the rotation angle. With the default drive strength ",[53,54930,54932,54966],{"className":54931},[2418],[53,54933,54935],{"className":54934},[2422],[2424,54936,54937],{"xmlns":2426},[2428,54938,54939,54963],{},[2431,54940,54941,54943,54945,54947,54949,54951,54953,54955],{},[2437,54942,42558],{"mathvariant":2643},[2437,54944,193],{"mathvariant":2643},[2588,54946,132],{},[2437,54948,8458],{},[2584,54950,121],{},[2588,54952,48303],{},[6062,54954,6064],{},[2431,54956,54957,54959,54961],{},[2437,54958,48310],{"mathvariant":2643},[2437,54960,3033],{"mathvariant":2643},[2437,54962,214],{"mathvariant":2643},[2444,54964,54965],{"encoding":2446},"\\Omega\u002F2\\pi=20~\\mathrm{MHz}",[53,54967,54969,54990],{"className":54968,"ariaHidden":2452},[2451],[53,54970,54972,54975,54978,54981,54984,54987],{"className":54971},[2456],[53,54973],{"className":54974,"style":2664},[2460],[53,54976,42616],{"className":54977},[2465],[53,54979,8458],{"className":54980,"style":2671},[2465,2506],[53,54982],{"className":54983,"style":2814},[2609],[53,54985,121],{"className":54986},[2818],[53,54988],{"className":54989,"style":2814},[2609],[53,54991,54993,54996,54999,55002],{"className":54992},[2456],[53,54994],{"className":54995,"style":2461},[2460],[53,54997,48303],{"className":54998},[2465],[53,55000,6064],{"className":55001},[2609,43785],[53,55003,55005],{"className":55004},[2465],[53,55006,48423],{"className":55007},[2465,43792],", the ideal resonant maxima appear near",[53,55010,55012],{"className":55011},[2679],[53,55013,55015,55088],{"className":55014},[2418],[53,55016,55018],{"className":55017},[2422],[2424,55019,55020],{"xmlns":2426,"display":2688},[2428,55021,55022,55085],{},[2431,55023,55024,55026,55028,55030,55032,55038,55040,55042,55045,55047,55053,55055,55057,55060,55062,55068,55070,55072,55075,55077,55083],{},[2437,55025,41600],{},[2584,55027,121],{},[2588,55029,48484],{},[6062,55031,6064],{},[2431,55033,55034,55036],{},[2437,55035,4315],{"mathvariant":2643},[2437,55037,14428],{"mathvariant":2643},[2584,55039,2714],{"separator":2452},[6062,55041,6064],{},[2588,55043,55044],{},"75",[6062,55046,6064],{},[2431,55048,55049,55051],{},[2437,55050,4315],{"mathvariant":2643},[2437,55052,14428],{"mathvariant":2643},[2584,55054,2714],{"separator":2452},[6062,55056,6064],{},[2588,55058,55059],{},"125",[6062,55061,6064],{},[2431,55063,55064,55066],{},[2437,55065,4315],{"mathvariant":2643},[2437,55067,14428],{"mathvariant":2643},[2584,55069,2714],{"separator":2452},[6062,55071,6064],{},[2588,55073,55074],{},"175",[6062,55076,6064],{},[2431,55078,55079,55081],{},[2437,55080,4315],{"mathvariant":2643},[2437,55082,14428],{"mathvariant":2643},[2584,55084,2714],{"separator":2452},[2444,55086,55087],{"encoding":2446},"\\tau=25~\\mathrm{ns},\\ 75~\\mathrm{ns},\\ 125~\\mathrm{ns},\\ 175~\\mathrm{ns},",[53,55089,55091,55109],{"className":55090,"ariaHidden":2452},[2451],[53,55092,55094,55097,55100,55103,55106],{"className":55093},[2456],[53,55095],{"className":55096,"style":2534},[2460],[53,55098,41600],{"className":55099,"style":41672},[2465,2506],[53,55101],{"className":55102,"style":2814},[2609],[53,55104,121],{"className":55105},[2818],[53,55107],{"className":55108,"style":2814},[2609],[53,55110,55112,55116,55119,55122,55128,55131,55134,55137,55140,55143,55149,55152,55155,55158,55161,55164,55170,55173,55176,55179,55182,55185,55191],{"className":55111},[2456],[53,55113],{"className":55114,"style":55115},[2460],"height:0.8389em;vertical-align:-0.1944em;",[53,55117,48484],{"className":55118},[2465],[53,55120,6064],{"className":55121},[2609,43785],[53,55123,55125],{"className":55124},[2465],[53,55126,48574],{"className":55127},[2465,43792],[53,55129,2714],{"className":55130},[2856],[53,55132,6064],{"className":55133},[2609],[53,55135],{"className":55136,"style":2860},[2609],[53,55138,55044],{"className":55139},[2465],[53,55141,6064],{"className":55142},[2609,43785],[53,55144,55146],{"className":55145},[2465],[53,55147,48574],{"className":55148},[2465,43792],[53,55150,2714],{"className":55151},[2856],[53,55153,6064],{"className":55154},[2609],[53,55156],{"className":55157,"style":2860},[2609],[53,55159,55059],{"className":55160},[2465],[53,55162,6064],{"className":55163},[2609,43785],[53,55165,55167],{"className":55166},[2465],[53,55168,48574],{"className":55169},[2465,43792],[53,55171,2714],{"className":55172},[2856],[53,55174,6064],{"className":55175},[2609],[53,55177],{"className":55178,"style":2860},[2609],[53,55180,55074],{"className":55181},[2465],[53,55183,6064],{"className":55184},[2609,43785],[53,55186,55188],{"className":55187},[2465],[53,55189,48574],{"className":55190},[2465,43792],[53,55192,2714],{"className":55193},[2856],[12,55195,55196],{},"before the optional phenomenological damping factor is applied.",[12,55198,55199,55200,55269,55270,55396],{},"Off resonance, the fringes show lower contrast and faster temporal oscillations. This behavior follows from the generalized Rabi frequency ",[53,55201,55203,55220],{"className":55202},[2418],[53,55204,55206],{"className":55205},[2422],[2424,55207,55208],{"xmlns":2426},[2428,55209,55210,55218],{},[2431,55211,55212],{},[3063,55213,55214,55216],{},[2437,55215,42558],{"mathvariant":2643},[2437,55217,20265],{},[2444,55219,42798],{"encoding":2446},[53,55221,55223],{"className":55222,"ariaHidden":2452},[2451],[53,55224,55226,55229],{"className":55225},[2456],[53,55227],{"className":55228,"style":3082},[2460],[53,55230,55232,55235],{"className":55231},[2465],[53,55233,42558],{"className":55234},[2465],[53,55236,55238],{"className":55237},[2474],[53,55239,55241,55261],{"className":55240},[2478,3095],[53,55242,55244,55258],{"className":55243},[2482],[53,55245,55247],{"className":55246,"style":3102},[2486],[53,55248,55249,55252],{"style":5293},[53,55250],{"className":55251,"style":2495},[2494],[53,55253,55255],{"className":55254},[2499,2500,2501,2502],[53,55256,20265],{"className":55257,"style":20290},[2465,2506,2502],[53,55259,3120],{"className":55260},[3119],[53,55262,55264],{"className":55263},[2482],[53,55265,55267],{"className":55266,"style":3127},[2486],[53,55268],{},", which increases with detuning, and from the amplitude coefficient ",[53,55271,55273,55301],{"className":55272},[2418],[53,55274,55276],{"className":55275},[2422],[2424,55277,55278],{"xmlns":2426},[2428,55279,55280,55298],{},[2431,55281,55282,55288,55290],{},[2434,55283,55284,55286],{},[2437,55285,42558],{"mathvariant":2643},[2588,55287,132],{},[2437,55289,193],{"mathvariant":2643},[3670,55291,55292,55294,55296],{},[2437,55293,42558],{"mathvariant":2643},[2437,55295,20265],{},[2588,55297,132],{},[2444,55299,55300],{"encoding":2446},"\\Omega^2\u002F\\Omega_R^2",[53,55302,55304],{"className":55303,"ariaHidden":2452},[2451],[53,55305,55307,55311,55340,55343],{"className":55306},[2456],[53,55308],{"className":55309,"style":55310},[2460],"height:1.0894em;vertical-align:-0.2753em;",[53,55312,55314,55317],{"className":55313},[2465],[53,55315,42558],{"className":55316},[2465],[53,55318,55320],{"className":55319},[2474],[53,55321,55323],{"className":55322},[2478],[53,55324,55326],{"className":55325},[2482],[53,55327,55329],{"className":55328,"style":13119},[2486],[53,55330,55331,55334],{"style":2490},[53,55332],{"className":55333,"style":2495},[2494],[53,55335,55337],{"className":55336},[2499,2500,2501,2502],[53,55338,132],{"className":55339},[2465,2502],[53,55341,193],{"className":55342},[2465],[53,55344,55346,55349],{"className":55345},[2465],[53,55347,42558],{"className":55348},[2465],[53,55350,55352],{"className":55351},[2474],[53,55353,55355,55387],{"className":55354},[2478,3095],[53,55356,55358,55384],{"className":55357},[2482],[53,55359,55361,55373],{"className":55360,"style":13119},[2486],[53,55362,55364,55367],{"style":55363},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[53,55365],{"className":55366,"style":2495},[2494],[53,55368,55370],{"className":55369},[2499,2500,2501,2502],[53,55371,20265],{"className":55372,"style":20290},[2465,2506,2502],[53,55374,55375,55378],{"style":2490},[53,55376],{"className":55377,"style":2495},[2494],[53,55379,55381],{"className":55380},[2499,2500,2501,2502],[53,55382,132],{"className":55383},[2465,2502],[53,55385,3120],{"className":55386},[3119],[53,55388,55390],{"className":55389},[2482],[53,55391,55394],{"className":55392,"style":55393},[2486],"height:0.2753em;",[53,55395],{},", which decreases with detuning.",[12,55398,55399],{},"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.",[12,55401,55402,55403,55500],{},"The FFT heatmap characterizes the oscillation-frequency content of ",[53,55404,55406,55433],{"className":55405},[2418],[53,55407,55409],{"className":55408},[2422],[2424,55410,55411],{"xmlns":2426},[2428,55412,55413,55431],{},[2431,55414,55415,55421,55423,55425,55427,55429],{},[3063,55416,55417,55419],{},[2437,55418,3069],{},[2437,55420,3181],{},[2584,55422,128],{"stretchy":2650},[2437,55424,41600],{},[2584,55426,2714],{"separator":2452},[2437,55428,22173],{},[2584,55430,3544],{"stretchy":2650},[2444,55432,52615],{"encoding":2446},[53,55434,55436],{"className":55435,"ariaHidden":2452},[2451],[53,55437,55439,55442,55482,55485,55488,55491,55494,55497],{"className":55438},[2456],[53,55440],{"className":55441,"style":2664},[2460],[53,55443,55445,55448],{"className":55444},[2465],[53,55446,3069],{"className":55447,"style":3115},[2465,2506],[53,55449,55451],{"className":55450},[2474],[53,55452,55454,55474],{"className":55453},[2478,3095],[53,55455,55457,55471],{"className":55456},[2482],[53,55458,55460],{"className":55459,"style":4402},[2486],[53,55461,55462,55465],{"style":7532},[53,55463],{"className":55464,"style":2495},[2494],[53,55466,55468],{"className":55467},[2499,2500,2501,2502],[53,55469,3181],{"className":55470},[2465,2506,2502],[53,55472,3120],{"className":55473},[3119],[53,55475,55477],{"className":55476},[2482],[53,55478,55480],{"className":55479,"style":3127},[2486],[53,55481],{},[53,55483,128],{"className":55484},[2836],[53,55486,41600],{"className":55487,"style":41672},[2465,2506],[53,55489,2714],{"className":55490},[2856],[53,55492],{"className":55493,"style":2860},[2609],[53,55495,22173],{"className":55496,"style":27234},[2465,2506],[53,55498,3544],{"className":55499},[2675],". An overlaid theoretical curve follows",[53,55502,55504],{"className":55503},[2679],[53,55505,55507,55579],{"className":55506},[2418],[53,55508,55510],{"className":55509},[2422],[2424,55511,55512],{"xmlns":2426,"display":2688},[2428,55513,55514,55576],{},[2431,55515,55516,55522,55524,55526,55528,55530,55574],{},[3063,55517,55518,55520],{},[2437,55519,42871],{},[2437,55521,20265],{},[2584,55523,128],{"stretchy":2650},[2437,55525,22173],{},[2584,55527,3544],{"stretchy":2650},[2584,55529,121],{},[9391,55531,55532],{},[2431,55533,55534,55554,55556,55558,55560,55562,55568],{},[2434,55535,55536,55552],{},[2431,55537,55538,55540,55550],{},[2584,55539,128],{"fence":2452},[9386,55541,55542,55544],{},[2437,55543,42558],{"mathvariant":2643},[2431,55545,55546,55548],{},[2588,55547,132],{},[2437,55549,8458],{},[2584,55551,3544],{"fence":2452},[2588,55553,132],{},[2584,55555,2002],{},[2584,55557,128],{"stretchy":2650},[2437,55559,22173],{},[2584,55561,2586],{},[3063,55563,55564,55566],{},[2437,55565,22173],{},[2588,55567,149],{},[2434,55569,55570,55572],{},[2584,55571,3544],{"stretchy":2650},[2588,55573,132],{},[2437,55575,28],{"mathvariant":2643},[2444,55577,55578],{"encoding":2446},"\\nu_R(f)=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[53,55580,55582,55646],{"className":55581,"ariaHidden":2452},[2451],[53,55583,55585,55588,55628,55631,55634,55637,55640,55643],{"className":55584},[2456],[53,55586],{"className":55587,"style":2664},[2460],[53,55589,55591,55594],{"className":55590},[2465],[53,55592,42871],{"className":55593,"style":42892},[2465,2506],[53,55595,55597],{"className":55596},[2474],[53,55598,55600,55620],{"className":55599},[2478,3095],[53,55601,55603,55617],{"className":55602},[2482],[53,55604,55606],{"className":55605,"style":3102},[2486],[53,55607,55608,55611],{"style":42907},[53,55609],{"className":55610,"style":2495},[2494],[53,55612,55614],{"className":55613},[2499,2500,2501,2502],[53,55615,20265],{"className":55616,"style":20290},[2465,2506,2502],[53,55618,3120],{"className":55619},[3119],[53,55621,55623],{"className":55622},[2482],[53,55624,55626],{"className":55625,"style":3127},[2486],[53,55627],{},[53,55629,128],{"className":55630},[2836],[53,55632,22173],{"className":55633,"style":27234},[2465,2506],[53,55635,3544],{"className":55636},[2675],[53,55638],{"className":55639,"style":2814},[2609],[53,55641,121],{"className":55642},[2818],[53,55644],{"className":55645,"style":2814},[2609],[53,55647,55649,55652,55895],{"className":55648},[2456],[53,55650],{"className":55651,"style":49921},[2460],[53,55653,55655],{"className":55654},[2465,9714],[53,55656,55658,55887],{"className":55657},[2478,3095],[53,55659,55661,55884],{"className":55660},[2482],[53,55662,55664,55872],{"className":55663,"style":49934},[2486],[53,55665,55667,55670],{"className":55666,"style":49938},[9728],[53,55668],{"className":55669,"style":49942},[2494],[53,55671,55673,55779,55782,55785,55788,55791,55794,55797,55800,55803,55843],{"className":55672,"style":47169},[2465],[53,55674,55676,55756],{"className":55675},[2832],[53,55677,55679,55685,55750],{"className":55678},[2832],[53,55680,55682],{"className":55681,"style":2838},[2836,2837],[53,55683,128],{"className":55684},[2842,2501],[53,55686,55688,55691,55747],{"className":55687},[2465],[53,55689],{"className":55690},[2836,9688],[53,55692,55694],{"className":55693},[9386],[53,55695,55697,55739],{"className":55696},[2478,3095],[53,55698,55700,55736],{"className":55699},[2482],[53,55701,55703,55717,55725],{"className":55702,"style":46936},[2486],[53,55704,55705,55708],{"style":11612},[53,55706],{"className":55707,"style":4205},[2494],[53,55709,55711,55714],{"className":55710},[2465],[53,55712,132],{"className":55713},[2465],[53,55715,8458],{"className":55716,"style":2671},[2465,2506],[53,55718,55719,55722],{"style":9773},[53,55720],{"className":55721,"style":4205},[2494],[53,55723],{"className":55724,"style":9781},[9780],[53,55726,55727,55730],{"style":9784},[53,55728],{"className":55729,"style":4205},[2494],[53,55731,55733],{"className":55732},[2465],[53,55734,42558],{"className":55735},[2465],[53,55737,3120],{"className":55738},[3119],[53,55740,55742],{"className":55741},[2482],[53,55743,55745],{"className":55744,"style":11650},[2486],[53,55746],{},[53,55748],{"className":55749},[2675,9688],[53,55751,55753],{"className":55752,"style":2838},[2675,2837],[53,55754,3544],{"className":55755},[2842,2501],[53,55757,55759],{"className":55758},[2474],[53,55760,55762],{"className":55761},[2478],[53,55763,55765],{"className":55764},[2482],[53,55766,55768],{"className":55767,"style":50041},[2486],[53,55769,55770,55773],{"style":50044},[53,55771],{"className":55772,"style":2495},[2494],[53,55774,55776],{"className":55775},[2499,2500,2501,2502],[53,55777,132],{"className":55778},[2465,2502],[53,55780],{"className":55781,"style":2610},[2609],[53,55783,2002],{"className":55784},[2614],[53,55786],{"className":55787,"style":2610},[2609],[53,55789,128],{"className":55790},[2836],[53,55792,22173],{"className":55793,"style":27234},[2465,2506],[53,55795],{"className":55796,"style":2610},[2609],[53,55798,2586],{"className":55799},[2614],[53,55801],{"className":55802,"style":2610},[2609],[53,55804,55806,55809],{"className":55805},[2465],[53,55807,22173],{"className":55808,"style":27234},[2465,2506],[53,55810,55812],{"className":55811},[2474],[53,55813,55815,55835],{"className":55814},[2478,3095],[53,55816,55818,55832],{"className":55817},[2482],[53,55819,55821],{"className":55820,"style":14176},[2486],[53,55822,55823,55826],{"style":42163},[53,55824],{"className":55825,"style":2495},[2494],[53,55827,55829],{"className":55828},[2499,2500,2501,2502],[53,55830,149],{"className":55831},[2465,2502],[53,55833,3120],{"className":55834},[3119],[53,55836,55838],{"className":55837},[2482],[53,55839,55841],{"className":55840,"style":3127},[2486],[53,55842],{},[53,55844,55846,55849],{"className":55845},[2675],[53,55847,3544],{"className":55848},[2675],[53,55850,55852],{"className":55851},[2474],[53,55853,55855],{"className":55854},[2478],[53,55856,55858],{"className":55857},[2482],[53,55859,55861],{"className":55860,"style":29305},[2486],[53,55862,55863,55866],{"style":29308},[53,55864],{"className":55865,"style":2495},[2494],[53,55867,55869],{"className":55868},[2499,2500,2501,2502],[53,55870,132],{"className":55871},[2465,2502],[53,55873,55874,55877],{"style":50149},[53,55875],{"className":55876,"style":49942},[2494],[53,55878,55880],{"className":55879,"style":50156},[9748],[6176,55881,55882],{"xmlns":6178,"width":9752,"height":50159,"viewBox":50160,"preserveAspectRatio":9755},[6185,55883],{"d":50163},[53,55885,3120],{"className":55886},[3119],[53,55888,55890],{"className":55889},[2482],[53,55891,55893],{"className":55892,"style":50173},[2486],[53,55894],{},[53,55896,28],{"className":55897},[2465],[30,55899,55901],{"id":55900},"primary-modeling-assumptions","Primary Modeling Assumptions",[12,55903,55904],{},"This notebook is an idealized mathematical physics visualization designed for conceptual clarity and calibration intuition.",[41748,55906,55907,55917],{},[41751,55908,55909],{},[41754,55910,55911,55914],{},[41757,55912,55913],{},"Assumption",[41757,55915,55916],{},"Implication",[41764,55918,55919,56003,56011,56018,56026,56063,56153,56310,56358,56366,56374,56382],{},[41754,55920,55921,55924],{},[41769,55922,55923],{},"Two-level truncation",[41769,55925,55926,55927,10694,55965,28],{},"The simulated model includes only ",[53,55928,55930,55947],{"className":55929},[2418],[53,55931,55933],{"className":55932},[2422],[2424,55934,55935],{"xmlns":2426},[2428,55936,55937,55945],{},[2431,55938,55939,55941,55943],{},[2584,55940,2644],{"stretchy":2650},[2437,55942,37809],{},[2584,55944,2651],{"stretchy":2650},[2444,55946,41791],{"encoding":2446},[53,55948,55950],{"className":55949,"ariaHidden":2452},[2451],[53,55951,55953,55956,55959,55962],{"className":55952},[2456],[53,55954],{"className":55955,"style":2664},[2460],[53,55957,2644],{"className":55958},[2836],[53,55960,37809],{"className":55961,"style":2671},[2465,2506],[53,55963,2651],{"className":55964},[2675],[53,55966,55968,55985],{"className":55967},[2418],[53,55969,55971],{"className":55970},[2422],[2424,55972,55973],{"xmlns":2426},[2428,55974,55975,55983],{},[2431,55976,55977,55979,55981],{},[2584,55978,2644],{"stretchy":2650},[2437,55980,3181],{},[2584,55982,2651],{"stretchy":2650},[2444,55984,41837],{"encoding":2446},[53,55986,55988],{"className":55987,"ariaHidden":2452},[2451],[53,55989,55991,55994,55997,56000],{"className":55990},[2456],[53,55992],{"className":55993,"style":2664},[2460],[53,55995,2644],{"className":55996},[2836],[53,55998,3181],{"className":55999},[2465,2506],[53,56001,2651],{"className":56002},[2675],[41754,56004,56005,56008],{},[41769,56006,56007],{},"Classical drive",[41769,56009,56010],{},"The microwave field is prescribed externally instead of quantized as a field mode.",[41754,56012,56013,56015],{},[41769,56014,43211],{},[41769,56016,56017],{},"Fast counter-rotating terms are omitted.",[41754,56019,56020,56023],{},[41769,56021,56022],{},"Square pulse",[41769,56024,56025],{},"The drive is assumed to turn on and off instantaneously.",[41754,56027,56028,56031],{},[41769,56029,56030],{},"Uniform drive amplitude",[41769,56032,56033,56034,56062],{},"The parameter ",[53,56035,56037,56050],{"className":56036},[2418],[53,56038,56040],{"className":56039},[2422],[2424,56041,56042],{"xmlns":2426},[2428,56043,56044,56048],{},[2431,56045,56046],{},[2437,56047,42558],{"mathvariant":2643},[2444,56049,42561],{"encoding":2446},[53,56051,56053],{"className":56052,"ariaHidden":2452},[2451],[53,56054,56056,56059],{"className":56055},[2456],[53,56057],{"className":56058,"style":2461},[2460],[53,56060,42558],{"className":56061},[2465]," remains constant across the frequency sweep.",[41754,56064,56065,56068],{},[41769,56066,56067],{},"Optional damping envelope",[41769,56069,56033,56070,56152],{},[53,56071,56073,56092],{"className":56072},[2418],[53,56074,56076],{"className":56075},[2422],[2424,56077,56078],{"xmlns":2426},[2428,56079,56080,56090],{},[2431,56081,56082],{},[3670,56083,56084,56086,56088],{},[2437,56085,36942],{},[2588,56087,132],{},[2584,56089,20664],{},[2444,56091,43137],{"encoding":2446},[53,56093,56095],{"className":56094,"ariaHidden":2452},[2451],[53,56096,56098,56101],{"className":56097},[2456],[53,56099],{"className":56100,"style":43147},[2460],[53,56102,56104,56107],{"className":56103},[2465],[53,56105,36942],{"className":56106,"style":3115},[2465,2506],[53,56108,56110],{"className":56109},[2474],[53,56111,56113,56144],{"className":56112},[2478,3095],[53,56114,56116,56141],{"className":56115},[2482],[53,56117,56119,56130],{"className":56118,"style":36478},[2486],[53,56120,56121,56124],{"style":43168},[53,56122],{"className":56123,"style":2495},[2494],[53,56125,56127],{"className":56126},[2499,2500,2501,2502],[53,56128,132],{"className":56129},[2465,2502],[53,56131,56132,56135],{"style":2490},[53,56133],{"className":56134,"style":2495},[2494],[53,56136,56138],{"className":56137},[2499,2500,2501,2502],[53,56139,20664],{"className":56140},[2614,2502],[53,56142,3120],{"className":56143},[3119],[53,56145,56147],{"className":56146},[2482],[53,56148,56150],{"className":56149,"style":43198},[2486],[53,56151],{}," enters as a simple exponential contrast envelope.",[41754,56154,56155,56229],{},[41769,56156,56157,56158,56228],{},"No explicit ",[53,56159,56161,56179],{"className":56160},[2418],[53,56162,56164],{"className":56163},[2422],[2424,56165,56166],{"xmlns":2426},[2428,56167,56168,56176],{},[2431,56169,56170],{},[3063,56171,56172,56174],{},[2437,56173,36942],{},[2588,56175,173],{},[2444,56177,56178],{"encoding":2446},"T_1",[53,56180,56182],{"className":56181,"ariaHidden":2452},[2451],[53,56183,56185,56188],{"className":56184},[2456],[53,56186],{"className":56187,"style":3082},[2460],[53,56189,56191,56194],{"className":56190},[2465],[53,56192,36942],{"className":56193,"style":3115},[2465,2506],[53,56195,56197],{"className":56196},[2474],[53,56198,56200,56220],{"className":56199},[2478,3095],[53,56201,56203,56217],{"className":56202},[2482],[53,56204,56206],{"className":56205,"style":14176},[2486],[53,56207,56208,56211],{"style":7532},[53,56209],{"className":56210,"style":2495},[2494],[53,56212,56214],{"className":56213},[2499,2500,2501,2502],[53,56215,173],{"className":56216},[2465,2502],[53,56218,3120],{"className":56219},[3119],[53,56221,56223],{"className":56222},[2482],[53,56224,56226],{"className":56225,"style":3127},[2486],[53,56227],{}," relaxation",[41769,56230,56231,56232,56270,56271,56309],{},"Energy decay from ",[53,56233,56235,56252],{"className":56234},[2418],[53,56236,56238],{"className":56237},[2422],[2424,56239,56240],{"xmlns":2426},[2428,56241,56242,56250],{},[2431,56243,56244,56246,56248],{},[2584,56245,2644],{"stretchy":2650},[2437,56247,3181],{},[2584,56249,2651],{"stretchy":2650},[2444,56251,41837],{"encoding":2446},[53,56253,56255],{"className":56254,"ariaHidden":2452},[2451],[53,56256,56258,56261,56264,56267],{"className":56257},[2456],[53,56259],{"className":56260,"style":2664},[2460],[53,56262,2644],{"className":56263},[2836],[53,56265,3181],{"className":56266},[2465,2506],[53,56268,2651],{"className":56269},[2675]," to ",[53,56272,56274,56291],{"className":56273},[2418],[53,56275,56277],{"className":56276},[2422],[2424,56278,56279],{"xmlns":2426},[2428,56280,56281,56289],{},[2431,56282,56283,56285,56287],{},[2584,56284,2644],{"stretchy":2650},[2437,56286,37809],{},[2584,56288,2651],{"stretchy":2650},[2444,56290,41791],{"encoding":2446},[53,56292,56294],{"className":56293,"ariaHidden":2452},[2451],[53,56295,56297,56300,56303,56306],{"className":56296},[2456],[53,56298],{"className":56299,"style":2664},[2460],[53,56301,2644],{"className":56302},[2836],[53,56304,37809],{"className":56305,"style":2671},[2465,2506],[53,56307,2651],{"className":56308},[2675]," is omitted.",[41754,56311,56312,56315],{},[41769,56313,56314],{},"No higher-state leakage",[41769,56316,56317,56318,56357],{},"Higher levels, including a transmon ",[53,56319,56321,56339],{"className":56320},[2418],[53,56322,56324],{"className":56323},[2422],[2424,56325,56326],{"xmlns":2426},[2428,56327,56328,56336],{},[2431,56329,56330,56332,56334],{},[2584,56331,2644],{"stretchy":2650},[2437,56333,22173],{},[2584,56335,2651],{"stretchy":2650},[2444,56337,56338],{"encoding":2446},"\\lvert f\\rangle",[53,56340,56342],{"className":56341,"ariaHidden":2452},[2451],[53,56343,56345,56348,56351,56354],{"className":56344},[2456],[53,56346],{"className":56347,"style":2664},[2460],[53,56349,2644],{"className":56350},[2836],[53,56352,22173],{"className":56353,"style":27234},[2465,2506],[53,56355,2651],{"className":56356},[2675]," state, are excluded.",[41754,56359,56360,56363],{},[41769,56361,56362],{},"No AC Stark shift",[41769,56364,56365],{},"Drive-induced transition-frequency shifts are excluded.",[41754,56367,56368,56371],{},[41769,56369,56370],{},"No Bloch-Siegert shift",[41769,56372,56373],{},"Counter-rotating corrections are excluded.",[41754,56375,56376,56379],{},[41769,56377,56378],{},"No experimental readout model",[41769,56380,56381],{},"Assignment errors and finite signal-to-noise effects are omitted.",[41754,56383,56384,56387],{},[41769,56385,56386],{},"No pulse-envelope shaping",[41769,56388,56389],{},"Gaussian, DRAG, cosine, and hardware-specific envelopes are omitted.",[12,56391,56392],{},"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.",[30,56394,56396],{"id":56395},"notebook-control-knobs","Notebook Control Knobs",[12,56398,56399],{},"The first code section centralizes the main physical and numerical parameters for direct adjustment.",[41748,56401,56402,56415],{},[41751,56403,56404],{},[41754,56405,56406,56409,56413],{},[41757,56407,56408],{},"Parameter",[41757,56410,56412],{"align":56411},"right","Default",[41757,56414,41762],{},[41764,56416,56417,56501,56554,56569,56584,56599,56614,56629,56643],{},[41754,56418,56419,56424,56428],{},[41769,56420,56421],{},[50,56422,56423],{},"F0_GHZ",[41769,56425,56426],{"align":56411},[50,56427,43698],{},[41769,56429,56430,56431,56500],{},"Qubit transition frequency ",[53,56432,56434,56451],{"className":56433},[2418],[53,56435,56437],{"className":56436},[2422],[2424,56438,56439],{"xmlns":2426},[2428,56440,56441,56449],{},[2431,56442,56443],{},[3063,56444,56445,56447],{},[2437,56446,22173],{},[2588,56448,149],{},[2444,56450,42133],{"encoding":2446},[53,56452,56454],{"className":56453,"ariaHidden":2452},[2451],[53,56455,56457,56460],{"className":56456},[2456],[53,56458],{"className":56459,"style":6930},[2460],[53,56461,56463,56466],{"className":56462},[2465],[53,56464,22173],{"className":56465,"style":27234},[2465,2506],[53,56467,56469],{"className":56468},[2474],[53,56470,56472,56492],{"className":56471},[2478,3095],[53,56473,56475,56489],{"className":56474},[2482],[53,56476,56478],{"className":56477,"style":14176},[2486],[53,56479,56480,56483],{"style":42163},[53,56481],{"className":56482,"style":2495},[2494],[53,56484,56486],{"className":56485},[2499,2500,2501,2502],[53,56487,149],{"className":56488},[2465,2502],[53,56490,3120],{"className":56491},[3119],[53,56493,56495],{"className":56494},[2482],[53,56496,56498],{"className":56497,"style":3127},[2486],[53,56499],{}," in GHz",[41754,56502,56503,56508,56513],{},[41769,56504,56505],{},[50,56506,56507],{},"OMEGA_ONRESONANCE_MHZ",[41769,56509,56510],{"align":56411},[50,56511,56512],{},"20.0",[41769,56514,56515,56516,56553],{},"On-resonance Rabi rate ",[53,56517,56519,56538],{"className":56518},[2418],[53,56520,56522],{"className":56521},[2422],[2424,56523,56524],{"xmlns":2426},[2428,56525,56526,56536],{},[2431,56527,56528,56530,56532,56534],{},[2437,56529,42558],{"mathvariant":2643},[2437,56531,193],{"mathvariant":2643},[2588,56533,132],{},[2437,56535,8458],{},[2444,56537,42603],{"encoding":2446},[53,56539,56541],{"className":56540,"ariaHidden":2452},[2451],[53,56542,56544,56547,56550],{"className":56543},[2456],[53,56545],{"className":56546,"style":2664},[2460],[53,56548,42616],{"className":56549},[2465],[53,56551,8458],{"className":56552,"style":2671},[2465,2506]," in MHz",[41754,56555,56556,56561,56566],{},[41769,56557,56558],{},[50,56559,56560],{},"FREQ_MIN_GHZ",[41769,56562,56563],{"align":56411},[50,56564,56565],{},"4.90",[41769,56567,56568],{},"Minimum swept drive frequency in GHz",[41754,56570,56571,56576,56581],{},[41769,56572,56573],{},[50,56574,56575],{},"FREQ_MAX_GHZ",[41769,56577,56578],{"align":56411},[50,56579,56580],{},"5.10",[41769,56582,56583],{},"Maximum swept drive frequency in GHz",[41754,56585,56586,56591,56596],{},[41769,56587,56588],{},[50,56589,56590],{},"N_FREQ",[41769,56592,56593],{"align":56411},[50,56594,56595],{},"401",[41769,56597,56598],{},"Number of drive-frequency samples",[41754,56600,56601,56606,56611],{},[41769,56602,56603],{},[50,56604,56605],{},"DUR_MIN_NS",[41769,56607,56608],{"align":56411},[50,56609,56610],{},"0.0",[41769,56612,56613],{},"Minimum pulse duration in ns",[41754,56615,56616,56621,56626],{},[41769,56617,56618],{},[50,56619,56620],{},"DUR_MAX_NS",[41769,56622,56623],{"align":56411},[50,56624,56625],{},"200.0",[41769,56627,56628],{},"Maximum pulse duration in ns",[41754,56630,56631,56636,56640],{},[41769,56632,56633],{},[50,56634,56635],{},"N_DUR",[41769,56637,56638],{"align":56411},[50,56639,56595],{},[41769,56641,56642],{},"Number of pulse-duration samples",[41754,56644,56645,56650,56655],{},[41769,56646,56647],{},[50,56648,56649],{},"T2STAR_NS",[41769,56651,56652],{"align":56411},[50,56653,56654],{},"500.0",[41769,56656,56657,56658,56661],{},"Optional damping time in ns, with ",[50,56659,56660],{},"None"," disabling damping",[12,56663,56664],{},"Additional cross-section plotting cells expose localized slice controls.",[41748,56666,56667,56677],{},[41751,56668,56669],{},[41754,56670,56671,56673,56675],{},[41757,56672,56408],{},[41757,56674,56412],{"align":56411},[41757,56676,41762],{},[41764,56678,56679,56778],{},[41754,56680,56681,56686,56690],{},[41769,56682,56683],{},[50,56684,56685],{},"F_CROSS_GHZ",[41769,56687,56688],{"align":56411},[50,56689,43698],{},[41769,56691,56692,56693],{},"Drive-frequency slice used to plot ",[53,56694,56696,56720],{"className":56695},[2418],[53,56697,56699],{"className":56698},[2422],[2424,56700,56701],{"xmlns":2426},[2428,56702,56703,56717],{},[2431,56704,56705,56711,56713,56715],{},[3063,56706,56707,56709],{},[2437,56708,3069],{},[2437,56710,3181],{},[2584,56712,128],{"stretchy":2650},[2437,56714,41600],{},[2584,56716,3544],{"stretchy":2650},[2444,56718,56719],{"encoding":2446},"P_e(\\tau)",[53,56721,56723],{"className":56722,"ariaHidden":2452},[2451],[53,56724,56726,56729,56769,56772,56775],{"className":56725},[2456],[53,56727],{"className":56728,"style":2664},[2460],[53,56730,56732,56735],{"className":56731},[2465],[53,56733,3069],{"className":56734,"style":3115},[2465,2506],[53,56736,56738],{"className":56737},[2474],[53,56739,56741,56761],{"className":56740},[2478,3095],[53,56742,56744,56758],{"className":56743},[2482],[53,56745,56747],{"className":56746,"style":4402},[2486],[53,56748,56749,56752],{"style":7532},[53,56750],{"className":56751,"style":2495},[2494],[53,56753,56755],{"className":56754},[2499,2500,2501,2502],[53,56756,3181],{"className":56757},[2465,2506,2502],[53,56759,3120],{"className":56760},[3119],[53,56762,56764],{"className":56763},[2482],[53,56765,56767],{"className":56766,"style":3127},[2486],[53,56768],{},[53,56770,128],{"className":56771},[2836],[53,56773,41600],{"className":56774,"style":41672},[2465,2506],[53,56776,3544],{"className":56777},[2675],[41754,56779,56780,56785,56790],{},[41769,56781,56782],{},[50,56783,56784],{},"T_CROSS_NS",[41769,56786,56787],{"align":56411},[50,56788,56789],{},"100.0",[41769,56791,56792,56793],{},"Pulse-duration slice used to plot ",[53,56794,56796,56820],{"className":56795},[2418],[53,56797,56799],{"className":56798},[2422],[2424,56800,56801],{"xmlns":2426},[2428,56802,56803,56817],{},[2431,56804,56805,56811,56813,56815],{},[3063,56806,56807,56809],{},[2437,56808,3069],{},[2437,56810,3181],{},[2584,56812,128],{"stretchy":2650},[2437,56814,22173],{},[2584,56816,3544],{"stretchy":2650},[2444,56818,56819],{"encoding":2446},"P_e(f)",[53,56821,56823],{"className":56822,"ariaHidden":2452},[2451],[53,56824,56826,56829,56869,56872,56875],{"className":56825},[2456],[53,56827],{"className":56828,"style":2664},[2460],[53,56830,56832,56835],{"className":56831},[2465],[53,56833,3069],{"className":56834,"style":3115},[2465,2506],[53,56836,56838],{"className":56837},[2474],[53,56839,56841,56861],{"className":56840},[2478,3095],[53,56842,56844,56858],{"className":56843},[2482],[53,56845,56847],{"className":56846,"style":4402},[2486],[53,56848,56849,56852],{"style":7532},[53,56850],{"className":56851,"style":2495},[2494],[53,56853,56855],{"className":56854},[2499,2500,2501,2502],[53,56856,3181],{"className":56857},[2465,2506,2502],[53,56859,3120],{"className":56860},[3119],[53,56862,56864],{"className":56863},[2482],[53,56865,56867],{"className":56866,"style":3127},[2486],[53,56868],{},[53,56870,128],{"className":56871},[2836],[53,56873,22173],{"className":56874,"style":27234},[2465,2506],[53,56876,3544],{"className":56877},[2675],[12,56879,56880],{},"Separate FFT processing controls support spectral analysis.",[41748,56882,56883,56893],{},[41751,56884,56885],{},[41754,56886,56887,56889,56891],{},[41757,56888,56408],{},[41757,56890,56412],{"align":56411},[41757,56892,41762],{},[41764,56894,56895,56909,56924,56938,56953,56968],{},[41754,56896,56897,56902,56906],{},[41769,56898,56899],{},[50,56900,56901],{},"FFT_PAD",[41769,56903,56904],{"align":56411},[50,56905,35636],{},[41769,56907,56908],{},"Zero-padding factor used in the Fourier transform",[41754,56910,56911,56916,56921],{},[41769,56912,56913],{},[50,56914,56915],{},"WINDOW",[41769,56917,56918],{"align":56411},[50,56919,56920],{},"\"hann\"",[41769,56922,56923],{},"Window function applied along the pulse-duration axis",[41754,56925,56926,56931,56935],{},[41769,56927,56928],{},[50,56929,56930],{},"DETREND_MEAN",[41769,56932,56933],{"align":56411},[50,56934,13773],{},[41769,56936,56937],{},"Flag that subtracts the mean before the FFT",[41754,56939,56940,56945,56950],{},[41769,56941,56942],{},[50,56943,56944],{},"AMP_MODE",[41769,56946,56947],{"align":56411},[50,56948,56949],{},"\"magnitude\"",[41769,56951,56952],{},"Display mode for magnitude, power, or dB-scaled amplitude",[41754,56954,56955,56960,56965],{},[41769,56956,56957],{},[50,56958,56959],{},"NORM_MODE",[41769,56961,56962],{"align":56411},[50,56963,56964],{},"\"global\"",[41769,56966,56967],{},"Normalization mode applied globally or per drive-frequency column",[41754,56969,56970,56975,56980],{},[41769,56971,56972],{},[50,56973,56974],{},"FREQ_UNITS",[41769,56976,56977],{"align":56411},[50,56978,56979],{},"\"MHz\"",[41769,56981,56982],{},"Frequency units used on the Fourier-frequency axis",[12,56984,56985],{},"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.",[43,56987,56989],{"className":45,"code":56988,"language":47,"meta":48,"style":48},"# @title Install dependencies\nimport sys\n%pip -q install pyvista pythreejs trame\n",[50,56990,56991,56996,57003],{"__ignoreMap":48},[53,56992,56993],{"class":55,"line":56},[53,56994,56995],{"class":110},"# @title Install dependencies\n",[53,56997,56998,57000],{"class":55,"line":73},[53,56999,60],{"class":59},[53,57001,57002],{"class":63}," sys\n",[53,57004,57005,57007,57010,57012],{"class":55,"line":87},[53,57006,8664],{"class":120},[53,57008,57009],{"class":63},"pip ",[53,57011,2262],{"class":120},[53,57013,57014],{"class":63},"q install pyvista pythreejs trame\n",[43,57016,57018],{"className":45,"code":57017,"language":47,"meta":48,"style":48},"# @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",[50,57019,57020,57025,57032,57043,57053,57064,57076,57080,57087,57103,57113,57121,57130,57134,57149,57162,57166,57171,57187,57203,57219,57235,57251,57267,57283,57298,57317,57321],{"__ignoreMap":48},[53,57021,57022],{"class":55,"line":56},[53,57023,57024],{"class":110},"# @title Controls & imports\n",[53,57026,57027,57029],{"class":55,"line":73},[53,57028,60],{"class":59},[53,57030,57031],{"class":63}," math\n",[53,57033,57034,57036,57038,57040],{"class":55,"line":87},[53,57035,76],{"class":59},[53,57037,1061],{"class":63},[53,57039,60],{"class":59},[53,57041,57042],{"class":63}," Optional, Tuple\n",[53,57044,57045,57047,57049,57051],{"class":55,"line":100},[53,57046,60],{"class":59},[53,57048,64],{"class":63},[53,57050,67],{"class":59},[53,57052,70],{"class":63},[53,57054,57055,57057,57060,57062],{"class":55,"line":107},[53,57056,60],{"class":59},[53,57058,57059],{"class":63}," matplotlib.pyplot ",[53,57061,67],{"class":59},[53,57063,1054],{"class":63},[53,57065,57066,57068,57071,57073],{"class":55,"line":114},[53,57067,60],{"class":59},[53,57069,57070],{"class":63}," pandas ",[53,57072,67],{"class":59},[53,57074,57075],{"class":63}," pd\n",[53,57077,57078],{"class":55,"line":138},[53,57079,104],{"emptyLinePlaceholder":103},[53,57081,57082,57085],{"class":55,"line":158},[53,57083,57084],{"class":59},"try",[53,57086,1623],{"class":63},[53,57088,57089,57092,57095,57097,57100],{"class":55,"line":182},[53,57090,57091],{"class":59}," import",[53,57093,57094],{"class":63}," pyvista ",[53,57096,67],{"class":59},[53,57098,57099],{"class":63}," pv ",[53,57101,57102],{"class":110},"# optional; only for 3D interactive\n",[53,57104,57105,57108,57110],{"class":55,"line":209},[53,57106,57107],{"class":131}," PV_AVAILABLE",[53,57109,1318],{"class":120},[53,57111,57112],{"class":131}," True\n",[53,57114,57115,57118],{"class":55,"line":226},[53,57116,57117],{"class":59},"except",[53,57119,57120],{"class":63}," Exception:\n",[53,57122,57123,57125,57127],{"class":55,"line":231},[53,57124,57107],{"class":131},[53,57126,1318],{"class":120},[53,57128,57129],{"class":131}," False\n",[53,57131,57132],{"class":55,"line":245},[53,57133,104],{"emptyLinePlaceholder":103},[53,57135,57136,57139,57142,57144,57146],{"class":55,"line":258},[53,57137,57138],{"class":63},"plt.rcParams[",[53,57140,57141],{"class":278},"'figure.dpi'",[53,57143,8101],{"class":63},[53,57145,121],{"class":120},[53,57147,57148],{"class":131}," 250\n",[53,57150,57151,57153,57156,57158,57160],{"class":55,"line":263},[53,57152,57138],{"class":63},[53,57154,57155],{"class":278},"'axes.grid'",[53,57157,8101],{"class":63},[53,57159,121],{"class":120},[53,57161,57129],{"class":131},[53,57163,57164],{"class":55,"line":269},[53,57165,104],{"emptyLinePlaceholder":103},[53,57167,57168],{"class":55,"line":284},[53,57169,57170],{"class":110},"# ---------- Control knobs ----------\n",[53,57172,57173,57175,57177,57179,57181,57184],{"class":55,"line":296},[53,57174,56423],{"class":131},[53,57176,1298],{"class":63},[53,57178,1301],{"class":120},[53,57180,1318],{"class":120},[53,57182,57183],{"class":131}," 5.000",[53,57185,57186],{"class":110}," # Qubit transition freq f0 [GHz]\n",[53,57188,57189,57191,57193,57195,57197,57200],{"class":55,"line":309},[53,57190,56507],{"class":131},[53,57192,1298],{"class":63},[53,57194,1301],{"class":120},[53,57196,1318],{"class":120},[53,57198,57199],{"class":131}," 20.0",[53,57201,57202],{"class":110}," # Ω\u002F2π on-resonance [MHz]\n",[53,57204,57205,57207,57209,57211,57213,57216],{"class":55,"line":323},[53,57206,56560],{"class":131},[53,57208,1298],{"class":63},[53,57210,1301],{"class":120},[53,57212,1318],{"class":120},[53,57214,57215],{"class":131}," 4.90",[53,57217,57218],{"class":110}," # Sweep min f [GHz]\n",[53,57220,57221,57223,57225,57227,57229,57232],{"class":55,"line":332},[53,57222,56575],{"class":131},[53,57224,1298],{"class":63},[53,57226,1301],{"class":120},[53,57228,1318],{"class":120},[53,57230,57231],{"class":131}," 5.10",[53,57233,57234],{"class":110}," # Sweep max f [GHz]\n",[53,57236,57237,57239,57241,57243,57245,57248],{"class":55,"line":344},[53,57238,56590],{"class":131},[53,57240,1298],{"class":63},[53,57242,1315],{"class":120},[53,57244,1318],{"class":120},[53,57246,57247],{"class":131}," 401",[53,57249,57250],{"class":110}," # # of frequency points\n",[53,57252,57253,57255,57257,57259,57261,57264],{"class":55,"line":357},[53,57254,56605],{"class":131},[53,57256,1298],{"class":63},[53,57258,1301],{"class":120},[53,57260,1318],{"class":120},[53,57262,57263],{"class":131}," 0.0",[53,57265,57266],{"class":110}," # Sweep min duration τ [ns]\n",[53,57268,57269,57271,57273,57275,57277,57280],{"class":55,"line":1247},[53,57270,56620],{"class":131},[53,57272,1298],{"class":63},[53,57274,1301],{"class":120},[53,57276,1318],{"class":120},[53,57278,57279],{"class":131}," 200.0",[53,57281,57282],{"class":110}," # Sweep max duration τ [ns]\n",[53,57284,57285,57287,57289,57291,57293,57295],{"class":55,"line":1252},[53,57286,56635],{"class":131},[53,57288,1298],{"class":63},[53,57290,1315],{"class":120},[53,57292,1318],{"class":120},[53,57294,57247],{"class":131},[53,57296,57297],{"class":110}," # # of duration points\n",[53,57299,57300,57302,57305,57307,57309,57311,57314],{"class":55,"line":1257},[53,57301,56649],{"class":131},[53,57303,57304],{"class":63},": Optional[",[53,57306,1301],{"class":120},[53,57308,8101],{"class":63},[53,57310,121],{"class":120},[53,57312,57313],{"class":131}," 500.0",[53,57315,57316],{"class":110}," # None to disable exp(-τ\u002FT2*)\n",[53,57318,57319],{"class":55,"line":1262},[53,57320,104],{"emptyLinePlaceholder":103},[53,57322,57323,57325,57327,57330],{"class":55,"line":1274},[53,57324,8420],{"class":120},[53,57326,128],{"class":63},[53,57328,57329],{"class":278},"'Controls loaded.'",[53,57331,135],{"class":63},[43,57333,57336],{"className":57334,"code":57335,"language":6934,"meta":48},[8426],"Controls loaded.\n",[50,57337,57335],{"__ignoreMap":48},[43,57339,57341],{"className":45,"code":57340,"language":47,"meta":48,"style":48},"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",[50,57342,57343,57352,57359,57366,57377,57388,57406,57411,57416,57420,57424,57429,57434,57439,57444,57449,57453,57457,57462,57466,57481,57496,57510,57525,57548,57552,57580,57602,57631,57635,57666,57686,57706,57729,57733,57757,57787,57801,57805],{"__ignoreMap":48},[53,57344,57345,57347,57350],{"class":55,"line":56},[53,57346,1265],{"class":59},[53,57348,57349],{"class":124}," rabi_probability",[53,57351,1271],{"class":63},[53,57353,57354,57357],{"class":55,"line":73},[53,57355,57356],{"class":1277}," f_ghz",[53,57358,1281],{"class":63},[53,57360,57361,57364],{"class":55,"line":87},[53,57362,57363],{"class":1277}," tau_ns",[53,57365,1281],{"class":63},[53,57367,57368,57371,57373,57375],{"class":55,"line":100},[53,57369,57370],{"class":1277}," f0_ghz",[53,57372,1298],{"class":63},[53,57374,1301],{"class":120},[53,57376,1304],{"class":63},[53,57378,57379,57382,57384,57386],{"class":55,"line":107},[53,57380,57381],{"class":1277}," omega_onres_mhz",[53,57383,1298],{"class":63},[53,57385,1301],{"class":120},[53,57387,1304],{"class":63},[53,57389,57390,57393,57395,57397,57399,57401,57404],{"class":55,"line":114},[53,57391,57392],{"class":1277}," t2star_ns",[53,57394,57304],{"class":63},[53,57396,1301],{"class":120},[53,57398,8101],{"class":63},[53,57400,121],{"class":120},[53,57402,57403],{"class":131}," None",[53,57405,1304],{"class":63},[53,57407,57408],{"class":55,"line":138},[53,57409,57410],{"class":63},") -> np.ndarray:\n",[53,57412,57413],{"class":55,"line":158},[53,57414,57415],{"class":278}," \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n",[53,57417,57418],{"class":55,"line":182},[53,57419,104],{"emptyLinePlaceholder":103},[53,57421,57422],{"class":55,"line":209},[53,57423,1380],{"class":278},[53,57425,57426],{"class":55,"line":226},[53,57427,57428],{"class":278}," f_ghz: 1D array of drive frequencies [GHz].\n",[53,57430,57431],{"class":55,"line":231},[53,57432,57433],{"class":278}," tau_ns: 1D array of pulse durations [ns].\n",[53,57435,57436],{"class":55,"line":245},[53,57437,57438],{"class":278}," f0_ghz: Qubit frequency on resonance [GHz].\n",[53,57440,57441],{"class":55,"line":258},[53,57442,57443],{"class":278}," omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n",[53,57445,57446],{"class":55,"line":263},[53,57447,57448],{"class":278}," t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n",[53,57450,57451],{"class":55,"line":269},[53,57452,104],{"emptyLinePlaceholder":103},[53,57454,57455],{"class":55,"line":284},[53,57456,1451],{"class":278},[53,57458,57459],{"class":55,"line":296},[53,57460,57461],{"class":278}," 2D array P_e with shape (len(tau_ns), len(f_ghz)).\n",[53,57463,57464],{"class":55,"line":309},[53,57465,1363],{"class":278},[53,57467,57468,57471,57473,57476,57478],{"class":55,"line":323},[53,57469,57470],{"class":63}," two_pi ",[53,57472,121],{"class":120},[53,57474,57475],{"class":131}," 2.0",[53,57477,8124],{"class":120},[53,57479,57480],{"class":63}," math.pi\n",[53,57482,57483,57486,57488,57491,57493],{"class":55,"line":332},[53,57484,57485],{"class":63}," f ",[53,57487,121],{"class":120},[53,57489,57490],{"class":63}," f_ghz ",[53,57492,2286],{"class":120},[53,57494,57495],{"class":131}," 1e9\n",[53,57497,57498,57501,57503,57506,57508],{"class":55,"line":344},[53,57499,57500],{"class":63}," f0 ",[53,57502,121],{"class":120},[53,57504,57505],{"class":63}," f0_ghz ",[53,57507,2286],{"class":120},[53,57509,57495],{"class":131},[53,57511,57512,57515,57517,57520,57522],{"class":55,"line":357},[53,57513,57514],{"class":63}," tau ",[53,57516,121],{"class":120},[53,57518,57519],{"class":63}," tau_ns ",[53,57521,2286],{"class":120},[53,57523,57524],{"class":131}," 1e-9\n",[53,57526,57527,57530,57532,57535,57537,57540,57542,57545],{"class":55,"line":1247},[53,57528,57529],{"class":63}," omega ",[53,57531,121],{"class":120},[53,57533,57534],{"class":63}," two_pi ",[53,57536,2286],{"class":120},[53,57538,57539],{"class":63}," omega_onres_mhz ",[53,57541,2286],{"class":120},[53,57543,57544],{"class":131}," 1e6",[53,57546,57547],{"class":110}," # rad\u002Fs\n",[53,57549,57550],{"class":55,"line":1252},[53,57551,104],{"emptyLinePlaceholder":103},[53,57553,57554,57557,57560,57562,57564,57567,57570,57573,57575,57578],{"class":55,"line":1257},[53,57555,57556],{"class":63}," F, ",[53,57558,57559],{"class":131},"TAU",[53,57561,1318],{"class":120},[53,57563,1511],{"class":63},[53,57565,57566],{"class":124},"meshgrid",[53,57568,57569],{"class":63},"(f, tau, ",[53,57571,57572],{"class":2050},"indexing",[53,57574,121],{"class":120},[53,57576,57577],{"class":278},"'xy'",[53,57579,135],{"class":63},[53,57581,57582,57585,57587,57589,57591,57594,57596,57599],{"class":55,"line":1262},[53,57583,57584],{"class":63}," Delta ",[53,57586,121],{"class":120},[53,57588,57534],{"class":63},[53,57590,2286],{"class":120},[53,57592,57593],{"class":63}," (F ",[53,57595,2262],{"class":120},[53,57597,57598],{"class":63}," f0) ",[53,57600,57601],{"class":110},"# rad\u002Fs\n",[53,57603,57604,57607,57609,57611,57613,57616,57618,57620,57622,57625,57627,57629],{"class":55,"line":1274},[53,57605,57606],{"class":63}," Omega_R ",[53,57608,121],{"class":120},[53,57610,1511],{"class":63},[53,57612,9714],{"class":124},[53,57614,57615],{"class":63},"(omega",[53,57617,2272],{"class":120},[53,57619,132],{"class":131},[53,57621,8208],{"class":120},[53,57623,57624],{"class":63}," Delta",[53,57626,2272],{"class":120},[53,57628,132],{"class":131},[53,57630,135],{"class":63},[53,57632,57633],{"class":55,"line":1284},[53,57634,104],{"emptyLinePlaceholder":103},[53,57636,57637,57640,57642,57645,57647,57650,57652,57655,57657,57660,57662,57664],{"class":55,"line":1292},[53,57638,57639],{"class":59}," with",[53,57641,1511],{"class":63},[53,57643,57644],{"class":124},"errstate",[53,57646,128],{"class":63},[53,57648,57649],{"class":2050},"divide",[53,57651,121],{"class":120},[53,57653,57654],{"class":278},"'ignore'",[53,57656,170],{"class":63},[53,57658,57659],{"class":2050},"invalid",[53,57661,121],{"class":120},[53,57663,57654],{"class":278},[53,57665,1778],{"class":63},[53,57667,57668,57671,57673,57676,57678,57681,57683],{"class":55,"line":1307},[53,57669,57670],{"class":63}," amp ",[53,57672,121],{"class":120},[53,57674,57675],{"class":63}," (omega ",[53,57677,193],{"class":120},[53,57679,57680],{"class":63}," Omega_R) ",[53,57682,2272],{"class":120},[53,57684,57685],{"class":131}," 2\n",[53,57687,57688,57691,57693,57696,57698,57701,57703],{"class":55,"line":1326},[53,57689,57690],{"class":63}," phase ",[53,57692,121],{"class":120},[53,57694,57695],{"class":131}," 0.5",[53,57697,8124],{"class":120},[53,57699,57700],{"class":63}," Omega_R ",[53,57702,2286],{"class":120},[53,57704,57705],{"class":131}," TAU\n",[53,57707,57708,57711,57713,57716,57718,57720,57722,57725,57727],{"class":55,"line":1344},[53,57709,57710],{"class":63}," Pe ",[53,57712,121],{"class":120},[53,57714,57715],{"class":63}," amp ",[53,57717,2286],{"class":120},[53,57719,1511],{"class":63},[53,57721,46617],{"class":124},[53,57723,57724],{"class":63},"(phase) ",[53,57726,2272],{"class":120},[53,57728,57685],{"class":131},[53,57730,57731],{"class":55,"line":1360},[53,57732,104],{"emptyLinePlaceholder":103},[53,57734,57735,57737,57740,57742,57744,57746,57749,57751,57753,57755],{"class":55,"line":1366},[53,57736,1612],{"class":59},[53,57738,57739],{"class":63}," t2star_ns ",[53,57741,36904],{"class":59},[53,57743,21875],{"class":59},[53,57745,57403],{"class":131},[53,57747,57748],{"class":59}," and",[53,57750,57739],{"class":63},[53,57752,1546],{"class":120},[53,57754,57263],{"class":131},[53,57756,1623],{"class":63},[53,57758,57759,57762,57764,57766,57768,57770,57772,57774,57777,57780,57782,57785],{"class":55,"line":1372},[53,57760,57761],{"class":63}," decay ",[53,57763,121],{"class":120},[53,57765,1511],{"class":63},[53,57767,23964],{"class":124},[53,57769,128],{"class":63},[53,57771,2262],{"class":120},[53,57773,57559],{"class":131},[53,57775,57776],{"class":120}," \u002F",[53,57778,57779],{"class":63}," (t2star_ns ",[53,57781,2286],{"class":120},[53,57783,57784],{"class":131}," 1e-9",[53,57786,22435],{"class":63},[53,57788,57789,57791,57793,57796,57798],{"class":55,"line":1377},[53,57790,57710],{"class":63},[53,57792,121],{"class":120},[53,57794,57795],{"class":63}," Pe ",[53,57797,2286],{"class":120},[53,57799,57800],{"class":63}," decay\n",[53,57802,57803],{"class":55,"line":1383},[53,57804,104],{"emptyLinePlaceholder":103},[53,57806,57807,57809,57811,57814,57817,57819,57821,57824],{"class":55,"line":1389},[53,57808,2399],{"class":59},[53,57810,1511],{"class":63},[53,57812,57813],{"class":124},"clip",[53,57815,57816],{"class":63},"(Pe, ",[53,57818,56610],{"class":131},[53,57820,170],{"class":63},[53,57822,57823],{"class":131},"1.0",[53,57825,135],{"class":63},[43,57827,57829],{"className":45,"code":57828,"language":47,"meta":48,"style":48},"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",[50,57830,57831,57857,57882,57906],{"__ignoreMap":48},[53,57832,57833,57836,57838,57840,57843,57845,57847,57849,57851,57853,57855],{"class":55,"line":56},[53,57834,57835],{"class":63},"freq_ghz ",[53,57837,121],{"class":120},[53,57839,1511],{"class":63},[53,57841,57842],{"class":124},"linspace",[53,57844,128],{"class":63},[53,57846,56560],{"class":131},[53,57848,170],{"class":63},[53,57850,56575],{"class":131},[53,57852,170],{"class":63},[53,57854,56590],{"class":131},[53,57856,135],{"class":63},[53,57858,57859,57862,57864,57866,57868,57870,57872,57874,57876,57878,57880],{"class":55,"line":73},[53,57860,57861],{"class":63},"dur_ns ",[53,57863,121],{"class":120},[53,57865,1511],{"class":63},[53,57867,57842],{"class":124},[53,57869,128],{"class":63},[53,57871,56605],{"class":131},[53,57873,170],{"class":63},[53,57875,56620],{"class":131},[53,57877,170],{"class":63},[53,57879,56635],{"class":131},[53,57881,135],{"class":63},[53,57883,57884,57887,57889,57891,57894,57896,57898,57900,57902,57904],{"class":55,"line":87},[53,57885,57886],{"class":63},"Pe ",[53,57888,121],{"class":120},[53,57890,57349],{"class":124},[53,57892,57893],{"class":63},"(freq_ghz, dur_ns, ",[53,57895,56423],{"class":131},[53,57897,170],{"class":63},[53,57899,56507],{"class":131},[53,57901,170],{"class":63},[53,57903,56649],{"class":131},[53,57905,135],{"class":63},[53,57907,57908,57910,57912,57915],{"class":55,"line":100},[53,57909,8420],{"class":120},[53,57911,128],{"class":63},[53,57913,57914],{"class":278},"'Grid shape (τ × f):'",[53,57916,57917],{"class":63},", Pe.shape)\n",[43,57919,57922],{"className":57920,"code":57921,"language":6934,"meta":48},[8426],"Grid shape (τ × f): (401, 401)\n",[50,57923,57921],{"__ignoreMap":48},[43,57925,57927],{"className":45,"code":57926,"language":47,"meta":48,"style":48},"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",[50,57928,57929,57944,57949,57964,57977,57990,58003,58016,58029,58042,58055,58068,58073,58092,58096],{"__ignoreMap":48},[53,57930,57931,57934,57936,57939,57942],{"class":55,"line":56},[53,57932,57933],{"class":63},"controls_table ",[53,57935,121],{"class":120},[53,57937,57938],{"class":63}," pd.",[53,57940,57941],{"class":124},"DataFrame",[53,57943,1271],{"class":63},[53,57945,57946],{"class":55,"line":73},[53,57947,57948],{"class":63}," [\n",[53,57950,57951,57954,57957,57959,57961],{"class":55,"line":87},[53,57952,57953],{"class":63}," [",[53,57955,57956],{"class":278},"'f0 (GHz)'",[53,57958,170],{"class":63},[53,57960,56423],{"class":131},[53,57962,57963],{"class":63},"],\n",[53,57965,57966,57968,57971,57973,57975],{"class":55,"line":100},[53,57967,57953],{"class":63},[53,57969,57970],{"class":278},"'Ω\u002F2π on-res (MHz)'",[53,57972,170],{"class":63},[53,57974,56507],{"class":131},[53,57976,57963],{"class":63},[53,57978,57979,57981,57984,57986,57988],{"class":55,"line":107},[53,57980,57953],{"class":63},[53,57982,57983],{"class":278},"'f sweep min (GHz)'",[53,57985,170],{"class":63},[53,57987,56560],{"class":131},[53,57989,57963],{"class":63},[53,57991,57992,57994,57997,57999,58001],{"class":55,"line":114},[53,57993,57953],{"class":63},[53,57995,57996],{"class":278},"'f sweep max (GHz)'",[53,57998,170],{"class":63},[53,58000,56575],{"class":131},[53,58002,57963],{"class":63},[53,58004,58005,58007,58010,58012,58014],{"class":55,"line":138},[53,58006,57953],{"class":63},[53,58008,58009],{"class":278},"'N_f'",[53,58011,170],{"class":63},[53,58013,56590],{"class":131},[53,58015,57963],{"class":63},[53,58017,58018,58020,58023,58025,58027],{"class":55,"line":158},[53,58019,57953],{"class":63},[53,58021,58022],{"class":278},"'τ sweep min (ns)'",[53,58024,170],{"class":63},[53,58026,56605],{"class":131},[53,58028,57963],{"class":63},[53,58030,58031,58033,58036,58038,58040],{"class":55,"line":182},[53,58032,57953],{"class":63},[53,58034,58035],{"class":278},"'τ sweep max (ns)'",[53,58037,170],{"class":63},[53,58039,56620],{"class":131},[53,58041,57963],{"class":63},[53,58043,58044,58046,58049,58051,58053],{"class":55,"line":209},[53,58045,57953],{"class":63},[53,58047,58048],{"class":278},"'N_τ'",[53,58050,170],{"class":63},[53,58052,56635],{"class":131},[53,58054,57963],{"class":63},[53,58056,58057,58059,58062,58064,58066],{"class":55,"line":226},[53,58058,57953],{"class":63},[53,58060,58061],{"class":278},"'T2* (ns; None=off)'",[53,58063,170],{"class":63},[53,58065,56649],{"class":131},[53,58067,57963],{"class":63},[53,58069,58070],{"class":55,"line":231},[53,58071,58072],{"class":63}," ],\n",[53,58074,58075,58078,58080,58082,58085,58087,58090],{"class":55,"line":245},[53,58076,58077],{"class":2050}," columns",[53,58079,121],{"class":120},[53,58081,12143],{"class":63},[53,58083,58084],{"class":278},"'Parameter'",[53,58086,170],{"class":63},[53,58088,58089],{"class":278},"'Value'",[53,58091,1689],{"class":63},[53,58093,58094],{"class":55,"line":258},[53,58095,135],{"class":63},[53,58097,58098],{"class":55,"line":263},[53,58099,58100],{"class":63},"controls_table\n",[43,58102,58105],{"className":58103,"code":58104,"language":6934,"meta":48},[8426],"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",[50,58106,58104],{"__ignoreMap":48},[43,58108,58110],{"className":45,"code":58109,"language":47,"meta":48,"style":48},"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",[50,58111,58112,58144,58170,58206,58221,58235,58259,58269],{"__ignoreMap":48},[53,58113,58114,58117,58119,58122,58125,58127,58130,58132,58134,58137,58139,58142],{"class":55,"line":56},[53,58115,58116],{"class":63},"fig, ax ",[53,58118,121],{"class":120},[53,58120,58121],{"class":63}," plt.",[53,58123,58124],{"class":124},"subplots",[53,58126,128],{"class":63},[53,58128,58129],{"class":2050},"figsize",[53,58131,121],{"class":120},[53,58133,128],{"class":63},[53,58135,58136],{"class":131},"7",[53,58138,170],{"class":63},[53,58140,58141],{"class":131},"5",[53,58143,22435],{"class":63},[53,58145,58146,58149,58151,58154,58157,58160,58163,58165,58168],{"class":55,"line":73},[53,58147,58148],{"class":63},"c ",[53,58150,121],{"class":120},[53,58152,58153],{"class":63}," ax.",[53,58155,58156],{"class":124},"pcolormesh",[53,58158,58159],{"class":63},"(freq_ghz, dur_ns, Pe, ",[53,58161,58162],{"class":2050},"shading",[53,58164,121],{"class":120},[53,58166,58167],{"class":278},"'auto'",[53,58169,135],{"class":63},[53,58171,58172,58175,58177,58180,58183,58186,58189,58191,58194,58196,58198,58200,58204],{"class":55,"line":87},[53,58173,58174],{"class":63},"cb ",[53,58176,121],{"class":120},[53,58178,58179],{"class":63}," fig.",[53,58181,58182],{"class":124},"colorbar",[53,58184,58185],{"class":63},"(c, ",[53,58187,58188],{"class":2050},"ax",[53,58190,121],{"class":120},[53,58192,58193],{"class":63},"ax, ",[53,58195,20907],{"class":2050},[53,58197,121],{"class":120},[53,58199,2111],{"class":59},[53,58201,58203],{"class":58202},"sVyAn","'$P_e$'",[53,58205,135],{"class":63},[53,58207,58208,58211,58214,58216,58219],{"class":55,"line":100},[53,58209,58210],{"class":63},"ax.",[53,58212,58213],{"class":124},"set_xlabel",[53,58215,128],{"class":63},[53,58217,58218],{"class":278},"'Drive frequency f (GHz)'",[53,58220,135],{"class":63},[53,58222,58223,58225,58228,58230,58233],{"class":55,"line":107},[53,58224,58210],{"class":63},[53,58226,58227],{"class":124},"set_ylabel",[53,58229,128],{"class":63},[53,58231,58232],{"class":278},"'Pulse duration τ (ns)'",[53,58234,135],{"class":63},[53,58236,58237,58239,58242,58244,58247,58250,58252,58254,58257],{"class":55,"line":114},[53,58238,58210],{"class":63},[53,58240,58241],{"class":124},"set_title",[53,58243,128],{"class":63},[53,58245,58246],{"class":278},"'Rabi excited-state probability $P_e(f,",[53,58248,58249],{"class":120},"\\\\",[53,58251,2714],{"class":278},[53,58253,58249],{"class":120},[53,58255,58256],{"class":278},"tau)$'",[53,58258,135],{"class":63},[53,58260,58261,58264,58267],{"class":55,"line":138},[53,58262,58263],{"class":63},"fig.",[53,58265,58266],{"class":124},"tight_layout",[53,58268,329],{"class":63},[53,58270,58271,58273,58275],{"class":55,"line":158},[53,58272,25271],{"class":63},[53,58274,25508],{"class":124},[53,58276,329],{"class":63},[379,58278],{"alt":58279,"src":58280},"Output 1 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.webp",[43,58282,58284],{"className":45,"code":58283,"language":47,"meta":48,"style":48},"# @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",[50,58285,58286,58291,58296,58300,58307,58314,58320,58324,58329,58334,58338,58347,58351,58360,58372,58384,58396,58408,58412,58416,58442,58457,58464,58471,58475,58479,58489,58499,58503,58508,58518,58522,58531,58551,58560,58580,58584,58593,58602,58611,58620,58629,58633,58643,58653,58663,58673,58683,58687,58697,58707,58717,58727,58737,58741,58751,58769,58773,58777,58801,58806,58825,58829,58833,58842,58858,58866,58871,58882,58887,58891,58900,58905,58921,58932,58937,58941,58950,58954,58958,58965,58974,59004,59015,59020,59028,59033,59038,59045,59052,59059,59066,59073,59080,59088,59093,59103,59107,59115,59123,59127,59131,59137,59143,59149,59155,59161,59165,59175,59179,59183,59187,59198,59202,59212,59222,59234,59246,59251,59258,59272,59277,59290,59306,59326,59334,59340,59345,59356,59362,59374,59380,59385,59390,59395,59418,59424,59444,59449,59454,59464,59476,59488,59500,59520,59526,59550,59573,59589,59594,59621,59626,59650,59672,59693,59698,59712,59727,59745,59750,59755,59770,59776,59782,59788,59793,59798,59830,59857,59883,59888,59902,59917,59922,59927,59941,59946,59952,59958,59964,59969,59974,59979,59993,59998,60009,60017,60027,60032,60037,60042,60049,60065,60074,60083,60092,60101,60124,60130,60135,60140,60145,60155,60164,60173,60178,60184,60200,60211,60226,60236,60243,60252,60257,60279,60284,60301,60318,60323,60336,60350,60355,60363,60368,60373,60383,60391,60399,60410,60421,60430,60436,60446,60456,60465,60474,60485,60497,60508,60513,60518,60523,60533,60540,60548,60559,60570,60579,60585,60595,60605,60614,60623,60632,60644,60656,60665,60670,60675,60680,60690,60697,60704,60715,60726,60735,60741,60751,60759,60766,60773,60782,60791,60800,60809,60820,60832,60844,60853,60858,60863,60868,60878,60885,60892,60904,60915,60924,60930,60946,60951,60960,60970,60979,60988,60994,61003,61015,61028,61040,61045,61050,61055,61074,61080,61089,61096,61103,61111,61122,61133,61144,61149,61158,61165,61172,61180,61201,61212,61224,61235,61246,61251,61256,61261,61275,61287,61293,61298,61306,61316,61321,61333,61344,61354,61365,61370,61375,61387,61398,61407,61418,61423,61428,61438,61444,61458,61463,61473,61478,61490,61495,61500,61523,61528,61551,61562,61601,61611,61623,61643,61661,61666,61671,61683,61693,61703,61708,61713,61721,61731,61746,61757,61769,61774,61781,61790,61804,61815,61827,61832,61837,61847,61855,61865,61886,61897,61909,61914,61921,61930,61949,61960,61972,61977,61986,61991,62001,62009,62018,62032,62043,62069,62074,62081,62090,62103,62114,62137,62142,62147,62161,62169,62178,62191,62213,62224,62229,62236,62245,62258,62279,62290,62295,62300,62310,62320,62328,62338,62347,62359,62371,62383,62388,62395,62404,62413,62424,62435,62446,62451,62458,62467,62476,62487,62498,62519,62524,62531,62540,62549,62560,62571,62592,62597,62602,62607,62617,62628,62640,62652,62664,62676,62681,62686,62695,62700,62710,62725,62740,62745,62757,62782,62807,62832,62857,62883,62890,62896,62921,62942,62964,62969,62976,62982,63004,63025,63046,63051,63058,63064,63092,63097,63104,63110,63137,63142,63149,63155,63178],{"__ignoreMap":48},[53,58287,58288],{"class":55,"line":56},[53,58289,58290],{"class":110},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n",[53,58292,58293],{"class":55,"line":73},[53,58294,58295],{"class":278},"\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n",[53,58297,58298],{"class":55,"line":87},[53,58299,104],{"emptyLinePlaceholder":103},[53,58301,58302,58304],{"class":55,"line":100},[53,58303,60],{"class":59},[53,58305,58306],{"class":63}," importlib.util\n",[53,58308,58309,58311],{"class":55,"line":107},[53,58310,60],{"class":59},[53,58312,58313],{"class":63}," subprocess\n",[53,58315,58316,58318],{"class":55,"line":114},[53,58317,60],{"class":59},[53,58319,57002],{"class":63},[53,58321,58322],{"class":55,"line":138},[53,58323,104],{"emptyLinePlaceholder":103},[53,58325,58326],{"class":55,"line":158},[53,58327,58328],{"class":110},"# -------------------------------------------------------------------------\n",[53,58330,58331],{"class":55,"line":182},[53,58332,58333],{"class":110},"# Control knobs\n",[53,58335,58336],{"class":55,"line":209},[53,58337,58328],{"class":110},[53,58339,58340,58343,58345],{"class":55,"line":226},[53,58341,58342],{"class":131},"INSTALL_BLOCH_DEPENDENCIES",[53,58344,1318],{"class":120},[53,58346,57112],{"class":131},[53,58348,58349],{"class":55,"line":231},[53,58350,104],{"emptyLinePlaceholder":103},[53,58352,58353,58356,58358],{"class":55,"line":245},[53,58354,58355],{"class":131},"BLOCH_REQUIRED_PACKAGES",[53,58357,1318],{"class":120},[53,58359,23618],{"class":63},[53,58361,58362,58365,58367,58370],{"class":55,"line":258},[53,58363,58364],{"class":278}," \"numpy\"",[53,58366,1298],{"class":63},[53,58368,58369],{"class":278},"\"numpy\"",[53,58371,1304],{"class":63},[53,58373,58374,58377,58379,58382],{"class":55,"line":263},[53,58375,58376],{"class":278}," \"matplotlib\"",[53,58378,1298],{"class":63},[53,58380,58381],{"class":278},"\"matplotlib\"",[53,58383,1304],{"class":63},[53,58385,58386,58389,58391,58394],{"class":55,"line":269},[53,58387,58388],{"class":278}," \"scipy\"",[53,58390,1298],{"class":63},[53,58392,58393],{"class":278},"\"scipy\"",[53,58395,1304],{"class":63},[53,58397,58398,58401,58403,58406],{"class":55,"line":284},[53,58399,58400],{"class":278}," \"qutip\"",[53,58402,1298],{"class":63},[53,58404,58405],{"class":278},"\"qutip\"",[53,58407,1304],{"class":63},[53,58409,58410],{"class":55,"line":296},[53,58411,23770],{"class":63},[53,58413,58414],{"class":55,"line":309},[53,58415,104],{"emptyLinePlaceholder":103},[53,58417,58418,58421,58423,58426,58428,58431,58433,58436,58438,58440],{"class":55,"line":323},[53,58419,58420],{"class":131},"BLOCH_F0_GHZ",[53,58422,1318],{"class":120},[53,58424,58425],{"class":120}," globals",[53,58427,13878],{"class":63},[53,58429,58430],{"class":124},"get",[53,58432,128],{"class":63},[53,58434,58435],{"class":278},"\"F0_GHZ\"",[53,58437,170],{"class":63},[53,58439,43698],{"class":131},[53,58441,135],{"class":63},[53,58443,58444,58447,58449,58451,58453,58455],{"class":55,"line":332},[53,58445,58446],{"class":131},"BLOCH_OMEGA_ONRESONANCE_MHZ",[53,58448,1318],{"class":120},[53,58450,58425],{"class":120},[53,58452,13878],{"class":63},[53,58454,58430],{"class":124},[53,58456,1271],{"class":63},[53,58458,58459,58462],{"class":55,"line":344},[53,58460,58461],{"class":278}," \"OMEGA_ONRESONANCE_MHZ\"",[53,58463,1304],{"class":63},[53,58465,58466,58469],{"class":55,"line":357},[53,58467,58468],{"class":131}," 20.0",[53,58470,1304],{"class":63},[53,58472,58473],{"class":55,"line":1247},[53,58474,135],{"class":63},[53,58476,58477],{"class":55,"line":1252},[53,58478,104],{"emptyLinePlaceholder":103},[53,58480,58481,58484,58486],{"class":55,"line":1257},[53,58482,58483],{"class":131},"BLOCH_RESONANT_DETUNING_MHZ",[53,58485,1318],{"class":120},[53,58487,58488],{"class":131}," 0.0\n",[53,58490,58491,58494,58496],{"class":55,"line":1262},[53,58492,58493],{"class":131},"BLOCH_DETUNED_DETUNING_MHZ",[53,58495,1318],{"class":120},[53,58497,58498],{"class":131}," 40.0\n",[53,58500,58501],{"class":55,"line":1274},[53,58502,104],{"emptyLinePlaceholder":103},[53,58504,58505],{"class":55,"line":1284},[53,58506,58507],{"class":110},"# Use None to automatically plot one resonant pi pulse.\n",[53,58509,58510,58513,58515],{"class":55,"line":1292},[53,58511,58512],{"class":131},"BLOCH_DURATION_NS",[53,58514,1318],{"class":120},[53,58516,58517],{"class":131}," None\n",[53,58519,58520],{"class":55,"line":1307},[53,58521,104],{"emptyLinePlaceholder":103},[53,58523,58524,58527,58529],{"class":55,"line":1326},[53,58525,58526],{"class":131},"BLOCH_NUM_TIME_POINTS",[53,58528,1318],{"class":120},[53,58530,23516],{"class":131},[53,58532,58533,58536,58538,58541,58544,58546,58549],{"class":55,"line":1344},[53,58534,58535],{"class":131},"BLOCH_FIGSIZE",[53,58537,1318],{"class":120},[53,58539,58540],{"class":63}," (",[53,58542,58543],{"class":131},"8.8",[53,58545,170],{"class":63},[53,58547,58548],{"class":131},"7.8",[53,58550,135],{"class":63},[53,58552,58553,58556,58558],{"class":55,"line":1360},[53,58554,58555],{"class":131},"BLOCH_DPI",[53,58557,1318],{"class":120},[53,58559,57148],{"class":131},[53,58561,58562,58565,58567,58569,58571,58574,58576,58578],{"class":55,"line":1366},[53,58563,58564],{"class":131},"BLOCH_VIEW",[53,58566,1318],{"class":120},[53,58568,8322],{"class":63},[53,58570,2262],{"class":120},[53,58572,58573],{"class":131},"60",[53,58575,170],{"class":63},[53,58577,48484],{"class":131},[53,58579,1689],{"class":63},[53,58581,58582],{"class":55,"line":1372},[53,58583,104],{"emptyLinePlaceholder":103},[53,58585,58586,58589,58591],{"class":55,"line":1377},[53,58587,58588],{"class":131},"BLOCH_SHOW_CONTROL_AXIS_ARROWS",[53,58590,1318],{"class":120},[53,58592,57112],{"class":131},[53,58594,58595,58598,58600],{"class":55,"line":1383},[53,58596,58597],{"class":131},"BLOCH_SHOW_ARROW_TIP_LABELS",[53,58599,1318],{"class":120},[53,58601,57112],{"class":131},[53,58603,58604,58607,58609],{"class":55,"line":1389},[53,58605,58606],{"class":131},"BLOCH_SHOW_ENDPOINT_MARKERS",[53,58608,1318],{"class":120},[53,58610,57112],{"class":131},[53,58612,58613,58616,58618],{"class":55,"line":1395},[53,58614,58615],{"class":131},"BLOCH_SHOW_LEGEND",[53,58617,1318],{"class":120},[53,58619,57112],{"class":131},[53,58621,58622,58625,58627],{"class":55,"line":1401},[53,58623,58624],{"class":131},"BLOCH_PRINT_SUMMARY",[53,58626,1318],{"class":120},[53,58628,57112],{"class":131},[53,58630,58631],{"class":55,"line":1407},[53,58632,104],{"emptyLinePlaceholder":103},[53,58634,58635,58638,58640],{"class":55,"line":1413},[53,58636,58637],{"class":131},"COLOR_RESONANT_TRAJECTORY",[53,58639,1318],{"class":120},[53,58641,58642],{"class":278}," \"tab:blue\"\n",[53,58644,58645,58648,58650],{"class":55,"line":1419},[53,58646,58647],{"class":131},"COLOR_DETUNED_TRAJECTORY",[53,58649,1318],{"class":120},[53,58651,58652],{"class":278}," \"tab:orange\"\n",[53,58654,58655,58658,58660],{"class":55,"line":1425},[53,58656,58657],{"class":131},"COLOR_RESONANT_AXIS",[53,58659,1318],{"class":120},[53,58661,58662],{"class":278}," \"tab:green\"\n",[53,58664,58665,58668,58670],{"class":55,"line":1431},[53,58666,58667],{"class":131},"COLOR_DETUNED_AXIS",[53,58669,1318],{"class":120},[53,58671,58672],{"class":278}," \"tab:red\"\n",[53,58674,58675,58678,58680],{"class":55,"line":1437},[53,58676,58677],{"class":131},"COLOR_INITIAL_STATE",[53,58679,1318],{"class":120},[53,58681,58682],{"class":278}," \"black\"\n",[53,58684,58685],{"class":55,"line":1443},[53,58686,104],{"emptyLinePlaceholder":103},[53,58688,58689,58692,58694],{"class":55,"line":1448},[53,58690,58691],{"class":131},"TRAJECTORY_LINEWIDTH",[53,58693,1318],{"class":120},[53,58695,58696],{"class":131}," 2.4\n",[53,58698,58699,58702,58704],{"class":55,"line":1454},[53,58700,58701],{"class":131},"CONTROL_AXIS_LINEWIDTH",[53,58703,1318],{"class":120},[53,58705,58706],{"class":131}," 2.2\n",[53,58708,58709,58712,58714],{"class":55,"line":1460},[53,58710,58711],{"class":131},"ENDPOINT_MARKER_SIZE",[53,58713,1318],{"class":120},[53,58715,58716],{"class":131}," 45\n",[53,58718,58719,58722,58724],{"class":55,"line":1465},[53,58720,58721],{"class":131},"ARROW_LENGTH_RATIO",[53,58723,1318],{"class":120},[53,58725,58726],{"class":131}," 0.12\n",[53,58728,58729,58732,58734],{"class":55,"line":1470},[53,58730,58731],{"class":131},"ARROW_LABEL_SCALE",[53,58733,1318],{"class":120},[53,58735,58736],{"class":131}," 1.16\n",[53,58738,58739],{"class":55,"line":1475},[53,58740,104],{"emptyLinePlaceholder":103},[53,58742,58743,58746,58748],{"class":55,"line":1492},[53,58744,58745],{"class":131},"LEGEND_LOCATION",[53,58747,1318],{"class":120},[53,58749,58750],{"class":278}," \"upper left\"\n",[53,58752,58753,58756,58758,58760,58763,58765,58767],{"class":55,"line":1503},[53,58754,58755],{"class":131},"LEGEND_BBOX_TO_ANCHOR",[53,58757,1318],{"class":120},[53,58759,58540],{"class":63},[53,58761,58762],{"class":131},"1.02",[53,58764,170],{"class":63},[53,58766,58762],{"class":131},[53,58768,135],{"class":63},[53,58770,58771],{"class":55,"line":1519},[53,58772,104],{"emptyLinePlaceholder":103},[53,58774,58775],{"class":55,"line":1552},[53,58776,104],{"emptyLinePlaceholder":103},[53,58778,58779,58781,58784,58786,58789,58791,58794,58797,58799],{"class":55,"line":1558},[53,58780,1265],{"class":59},[53,58782,58783],{"class":124}," package_is_available",[53,58785,128],{"class":63},[53,58787,58788],{"class":1277},"package_name",[53,58790,1298],{"class":63},[53,58792,58793],{"class":120},"str",[53,58795,58796],{"class":63},") -> ",[53,58798,1334],{"class":120},[53,58800,1623],{"class":63},[53,58802,58803],{"class":55,"line":1586},[53,58804,58805],{"class":278}," \"\"\"Return True when a package can be imported.\"\"\"\n",[53,58807,58808,58810,58813,58816,58819,58821,58823],{"class":55,"line":1609},[53,58809,2399],{"class":59},[53,58811,58812],{"class":63}," importlib.util.",[53,58814,58815],{"class":124},"find_spec",[53,58817,58818],{"class":63},"(package_name) ",[53,58820,36904],{"class":59},[53,58822,21875],{"class":59},[53,58824,58517],{"class":131},[53,58826,58827],{"class":55,"line":1626},[53,58828,104],{"emptyLinePlaceholder":103},[53,58830,58831],{"class":55,"line":1635},[53,58832,104],{"emptyLinePlaceholder":103},[53,58834,58835,58837,58840],{"class":55,"line":1664},[53,58836,1265],{"class":59},[53,58838,58839],{"class":124}," install_missing_packages",[53,58841,1271],{"class":63},[53,58843,58844,58847,58850,58852,58854,58856],{"class":55,"line":1672},[53,58845,58846],{"class":1277}," required_packages",[53,58848,58849],{"class":63},": dict[",[53,58851,58793],{"class":120},[53,58853,170],{"class":63},[53,58855,58793],{"class":120},[53,58857,57963],{"class":63},[53,58859,58860,58862,58864],{"class":55,"line":1692},[53,58861,58796],{"class":63},[53,58863,56660],{"class":131},[53,58865,1623],{"class":63},[53,58867,58868],{"class":55,"line":1700},[53,58869,58870],{"class":278}," \"\"\"Install missing notebook packages using uv pip with a pip fallback.\"\"\"\n",[53,58872,58873,58875,58877,58880],{"class":55,"line":1707},[53,58874,1612],{"class":59},[53,58876,21875],{"class":59},[53,58878,58879],{"class":131}," INSTALL_BLOCH_DEPENDENCIES",[53,58881,1623],{"class":63},[53,58883,58884],{"class":55,"line":1727},[53,58885,58886],{"class":59}," return\n",[53,58888,58889],{"class":55,"line":1734},[53,58890,104],{"emptyLinePlaceholder":103},[53,58892,58893,58896,58898],{"class":55,"line":1750},[53,58894,58895],{"class":63}," missing_package_names ",[53,58897,121],{"class":120},[53,58899,2072],{"class":63},[53,58901,58902],{"class":55,"line":1755},[53,58903,58904],{"class":63}," pip_name\n",[53,58906,58907,58909,58912,58914,58917,58919],{"class":55,"line":1760},[53,58908,2162],{"class":59},[53,58910,58911],{"class":63}," import_name, pip_name ",[53,58913,1531],{"class":59},[53,58915,58916],{"class":63}," 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resonant_trajectory[",[53,62722,62029],{"class":278},[53,62724,1689],{"class":63},[53,62726,62728,62731,62733,62736,62738],{"class":55,"line":62727},469,[53,62729,62730],{"class":63}," detuned_axis ",[53,62732,121],{"class":120},[53,62734,62735],{"class":63}," detuned_trajectory[",[53,62737,62029],{"class":278},[53,62739,1689],{"class":63},[53,62741,62743],{"class":55,"line":62742},470,[53,62744,104],{"emptyLinePlaceholder":103},[53,62746,62748,62750,62752,62755],{"class":55,"line":62747},471,[53,62749,1955],{"class":120},[53,62751,128],{"class":63},[53,62753,62754],{"class":278},"\"Bloch-sphere trajectory summary\"",[53,62756,135],{"class":63},[53,62758,62760,62762,62764,62766,62769,62772,62775,62777,62780],{"class":55,"line":62759},472,[53,62761,1955],{"class":120},[53,62763,128],{"class":63},[53,62765,22173],{"class":59},[53,62767,62768],{"class":278},"\"Natural transition frequency f0: ",[53,62770,62771],{"class":131},"{BLOCH_F0_GHZ",[53,62773,62774],{"class":59},":.6f",[53,62776,2767],{"class":131},[53,62778,62779],{"class":278}," GHz\"",[53,62781,135],{"class":63},[53,62783,62785,62787,62789,62791,62794,62796,62799,62801,62803,62805],{"class":55,"line":62784},473,[53,62786,1955],{"class":120},[53,62788,128],{"class":63},[53,62790,22173],{"class":59},[53,62792,62793],{"class":278},"\"Resonant drive frequency: ",[53,62795,2705],{"class":131},[53,62797,62798],{"class":63},"resonant_drive_ghz",[53,62800,62774],{"class":59},[53,62802,2767],{"class":131},[53,62804,62779],{"class":278},[53,62806,135],{"class":63},[53,62808,62810,62812,62814,62816,62819,62821,62824,62826,62828,62830],{"class":55,"line":62809},474,[53,62811,1955],{"class":120},[53,62813,128],{"class":63},[53,62815,22173],{"class":59},[53,62817,62818],{"class":278},"\"Detuned drive frequency: ",[53,62820,2705],{"class":131},[53,62822,62823],{"class":63},"detuned_drive_ghz",[53,62825,62774],{"class":59},[53,62827,2767],{"class":131},[53,62829,62779],{"class":278},[53,62831,135],{"class":63},[53,62833,62835,62837,62839,62841,62844,62847,62850,62852,62855],{"class":55,"line":62834},475,[53,62836,1955],{"class":120},[53,62838,128],{"class":63},[53,62840,22173],{"class":59},[53,62842,62843],{"class":278},"\"On-resonance Rabi rate: ",[53,62845,62846],{"class":131},"{BLOCH_OMEGA_ONRESONANCE_MHZ",[53,62848,62849],{"class":59},":.3f",[53,62851,2767],{"class":131},[53,62853,62854],{"class":278}," MHz\"",[53,62856,135],{"class":63},[53,62858,62860,62862,62864,62866,62869,62871,62874,62876,62878,62881],{"class":55,"line":62859},476,[53,62861,1955],{"class":120},[53,62863,128],{"class":63},[53,62865,22173],{"class":59},[53,62867,62868],{"class":278},"\"Plotted duration: ",[53,62870,2705],{"class":131},[53,62872,62873],{"class":63},"bloch_duration_ns",[53,62875,62849],{"class":59},[53,62877,2767],{"class":131},[53,62879,62880],{"class":278}," ns\"",[53,62882,135],{"class":63},[53,62884,62886,62888],{"class":55,"line":62885},477,[53,62887,1955],{"class":120},[53,62889,1271],{"class":63},[53,62891,62893],{"class":55,"line":62892},478,[53,62894,62895],{"class":278}," \"Resonant control axis n_res: \"\n",[53,62897,62899,62902,62905,62907,62910,62912,62914,62916,62918],{"class":55,"line":62898},479,[53,62900,62901],{"class":59}," f",[53,62903,62904],{"class":278},"\"(",[53,62906,2705],{"class":131},[53,62908,62909],{"class":63},"resonant_axis[",[53,62911,149],{"class":131},[53,62913,12168],{"class":63},[53,62915,62774],{"class":59},[53,62917,2767],{"class":131},[53,62919,62920],{"class":278},", \"\n",[53,62922,62924,62926,62928,62930,62932,62934,62936,62938,62940],{"class":55,"line":62923},480,[53,62925,62901],{"class":59},[53,62927,22176],{"class":278},[53,62929,2705],{"class":131},[53,62931,62909],{"class":63},[53,62933,173],{"class":131},[53,62935,12168],{"class":63},[53,62937,62774],{"class":59},[53,62939,2767],{"class":131},[53,62941,62920],{"class":278},[53,62943,62945,62947,62949,62951,62953,62955,62957,62959,62961],{"class":55,"line":62944},481,[53,62946,62901],{"class":59},[53,62948,22176],{"class":278},[53,62950,2705],{"class":131},[53,62952,62909],{"class":63},[53,62954,132],{"class":131},[53,62956,12168],{"class":63},[53,62958,62774],{"class":59},[53,62960,2767],{"class":131},[53,62962,62963],{"class":278},")\"\n",[53,62965,62967],{"class":55,"line":62966},482,[53,62968,1555],{"class":63},[53,62970,62972,62974],{"class":55,"line":62971},483,[53,62973,1955],{"class":120},[53,62975,1271],{"class":63},[53,62977,62979],{"class":55,"line":62978},484,[53,62980,62981],{"class":278}," \"Detuned control axis n_det: \"\n",[53,62983,62985,62987,62989,62991,62994,62996,62998,63000,63002],{"class":55,"line":62984},485,[53,62986,62901],{"class":59},[53,62988,62904],{"class":278},[53,62990,2705],{"class":131},[53,62992,62993],{"class":63},"detuned_axis[",[53,62995,149],{"class":131},[53,62997,12168],{"class":63},[53,62999,62774],{"class":59},[53,63001,2767],{"class":131},[53,63003,62920],{"class":278},[53,63005,63007,63009,63011,63013,63015,63017,63019,63021,63023],{"class":55,"line":63006},486,[53,63008,62901],{"class":59},[53,63010,22176],{"class":278},[53,63012,2705],{"class":131},[53,63014,62993],{"class":63},[53,63016,173],{"class":131},[53,63018,12168],{"class":63},[53,63020,62774],{"class":59},[53,63022,2767],{"class":131},[53,63024,62920],{"class":278},[53,63026,63028,63030,63032,63034,63036,63038,63040,63042,63044],{"class":55,"line":63027},487,[53,63029,62901],{"class":59},[53,63031,22176],{"class":278},[53,63033,2705],{"class":131},[53,63035,62993],{"class":63},[53,63037,132],{"class":131},[53,63039,12168],{"class":63},[53,63041,62774],{"class":59},[53,63043,2767],{"class":131},[53,63045,62963],{"class":278},[53,63047,63049],{"class":55,"line":63048},488,[53,63050,1555],{"class":63},[53,63052,63054,63056],{"class":55,"line":63053},489,[53,63055,1955],{"class":120},[53,63057,1271],{"class":63},[53,63059,63061],{"class":55,"line":63060},490,[53,63062,63063],{"class":278}," \"Resonant final excited-state probability: \"\n",[53,63065,63067,63069,63071,63073,63075,63078,63080,63082,63084,63086,63088,63090],{"class":55,"line":63066},491,[53,63068,62901],{"class":59},[53,63070,22176],{"class":278},[53,63072,2705],{"class":131},[53,63074,61740],{"class":63},[53,63076,63077],{"class":278},"'excited_probability'",[53,63079,1650],{"class":63},[53,63081,2262],{"class":120},[53,63083,173],{"class":131},[53,63085,12168],{"class":63},[53,63087,62774],{"class":59},[53,63089,2767],{"class":131},[53,63091,61622],{"class":278},[53,63093,63095],{"class":55,"line":63094},492,[53,63096,1555],{"class":63},[53,63098,63100,63102],{"class":55,"line":63099},493,[53,63101,1955],{"class":120},[53,63103,1271],{"class":63},[53,63105,63107],{"class":55,"line":63106},494,[53,63108,63109],{"class":278}," \"Detuned final excited-state probability: \"\n",[53,63111,63113,63115,63117,63119,63121,63123,63125,63127,63129,63131,63133,63135],{"class":55,"line":63112},495,[53,63114,62901],{"class":59},[53,63116,22176],{"class":278},[53,63118,2705],{"class":131},[53,63120,61799],{"class":63},[53,63122,63077],{"class":278},[53,63124,1650],{"class":63},[53,63126,2262],{"class":120},[53,63128,173],{"class":131},[53,63130,12168],{"class":63},[53,63132,62774],{"class":59},[53,63134,2767],{"class":131},[53,63136,61622],{"class":278},[53,63138,63140],{"class":55,"line":63139},496,[53,63141,1555],{"class":63},[53,63143,63145,63147],{"class":55,"line":63144},497,[53,63146,1955],{"class":120},[53,63148,1271],{"class":63},[53,63150,63152],{"class":55,"line":63151},498,[53,63153,63154],{"class":278}," \"Detuned generalized Rabi frequency: \"\n",[53,63156,63158,63160,63162,63164,63166,63169,63171,63173,63175],{"class":55,"line":63157},499,[53,63159,62901],{"class":59},[53,63161,22176],{"class":278},[53,63163,2705],{"class":131},[53,63165,61799],{"class":63},[53,63167,63168],{"class":278},"'generalized_rabi_mhz'",[53,63170,12168],{"class":63},[53,63172,62774],{"class":59},[53,63174,2767],{"class":131},[53,63176,63177],{"class":278}," MHz\"\n",[53,63179,63181],{"class":55,"line":63180},500,[53,63182,1555],{"class":63},[379,63184],{"alt":63185,"src":63186},"Output 2 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-02.webp",[43,63188,63191],{"className":63189,"code":63190,"language":6934,"meta":48},[8426],"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",[50,63192,63190],{"__ignoreMap":48},[43,63194,63196],{"className":45,"code":63195,"language":47,"meta":48,"style":48},"# --- 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",[50,63197,63198,63203,63215,63225,63235,63239,63244,63260,63277,63298,63302,63307,63312,63336,63362,63394,63411,63426,63430,63435,63458,63484,63512,63529,63543,63547,63552,63574,63578,63583,63621,63650,63654,63668,63673,63697,63717,63728,63748,63752,63756,63769,63799,63814,63827,63831,63836,63875,63891,63895,63903],{"__ignoreMap":48},[53,63199,63200],{"class":55,"line":56},[53,63201,63202],{"class":110},"# --- High-resolution 3D surface with colormap & legend ---\n",[53,63204,63205,63207,63210,63212],{"class":55,"line":73},[53,63206,76],{"class":59},[53,63208,63209],{"class":63}," mpl_toolkits.mplot3d ",[53,63211,60],{"class":59},[53,63213,63214],{"class":63}," Axes3D\n",[53,63216,63217,63219,63221,63223],{"class":55,"line":87},[53,63218,60],{"class":59},[53,63220,64],{"class":63},[53,63222,67],{"class":59},[53,63224,70],{"class":63},[53,63226,63227,63229,63231,63233],{"class":55,"line":100},[53,63228,60],{"class":59},[53,63230,57059],{"class":63},[53,63232,67],{"class":59},[53,63234,1054],{"class":63},[53,63236,63237],{"class":55,"line":107},[53,63238,104],{"emptyLinePlaceholder":103},[53,63240,63241],{"class":55,"line":114},[53,63242,63243],{"class":110},"# Controls for surface quality and appearance\n",[53,63245,63246,63249,63251,63253,63255,63257],{"class":55,"line":138},[53,63247,63248],{"class":131},"UPSAMPLE_FACTOR",[53,63250,1298],{"class":63},[53,63252,1315],{"class":120},[53,63254,1318],{"class":120},[53,63256,2211],{"class":131},[53,63258,63259],{"class":110}," # 2=~4× more faces; try 3 for even smoother\n",[53,63261,63262,63265,63267,63269,63271,63274],{"class":55,"line":158},[53,63263,63264],{"class":131},"SURFACE_CMAP",[53,63266,1298],{"class":63},[53,63268,58793],{"class":120},[53,63270,1318],{"class":120},[53,63272,63273],{"class":278}," 'inferno'",[53,63275,63276],{"class":110}," # e.g., 'viridis', 'plasma', 'magma', 'inferno'\n",[53,63278,63279,63282,63284,63287,63289,63291,63293,63295],{"class":55,"line":182},[53,63280,63281],{"class":131},"Z_MIN",[53,63283,170],{"class":63},[53,63285,63286],{"class":131},"Z_MAX",[53,63288,1318],{"class":120},[53,63290,57263],{"class":131},[53,63292,170],{"class":63},[53,63294,57823],{"class":131},[53,63296,63297],{"class":110}," # clamp\u002Fnormalize expected Pe range\n",[53,63299,63300],{"class":55,"line":209},[53,63301,104],{"emptyLinePlaceholder":103},[53,63303,63304],{"class":55,"line":226},[53,63305,63306],{"class":110},"# --- Bilinear upsampling (SciPy-free) ---\n",[53,63308,63309],{"class":55,"line":231},[53,63310,63311],{"class":110},"# 1) Upsample frequency axis for each fixed τ\n",[53,63313,63314,63317,63319,63321,63323,63326,63328,63331,63333],{"class":55,"line":245},[53,63315,63316],{"class":63},"freq_hi ",[53,63318,121],{"class":120},[53,63320,1511],{"class":63},[53,63322,57842],{"class":124},[53,63324,63325],{"class":63},"(freq_ghz.",[53,63327,25125],{"class":124},[53,63329,63330],{"class":63},"(), freq_ghz.",[53,63332,36608],{"class":124},[53,63334,63335],{"class":63},"(),\n",[53,63337,63338,63341,63343,63345,63347,63350,63352,63354,63356,63358,63360],{"class":55,"line":258},[53,63339,63340],{"class":131}," UPSAMPLE_FACTOR",[53,63342,8124],{"class":120},[53,63344,58540],{"class":63},[53,63346,1658],{"class":120},[53,63348,63349],{"class":63},"(freq_ghz) ",[53,63351,2262],{"class":120},[53,63353,1321],{"class":131},[53,63355,203],{"class":63},[53,63357,2002],{"class":120},[53,63359,1321],{"class":131},[53,63361,135],{"class":63},[53,63363,63364,63367,63369,63371,63374,63377,63379,63382,63384,63387,63389,63392],{"class":55,"line":263},[53,63365,63366],{"class":63},"Pe_tau_f ",[53,63368,121],{"class":120},[53,63370,1511],{"class":63},[53,63372,63373],{"class":124},"empty",[53,63375,63376],{"class":63},"((",[53,63378,1658],{"class":120},[53,63380,63381],{"class":63},"(dur_ns), ",[53,63383,1658],{"class":120},[53,63385,63386],{"class":63},"(freq_hi)), ",[53,63388,2051],{"class":2050},[53,63390,63391],{"class":120},"=float",[53,63393,135],{"class":63},[53,63395,63396,63398,63400,63402,63404,63406,63408],{"class":55,"line":269},[53,63397,1525],{"class":59},[53,63399,8141],{"class":63},[53,63401,1531],{"class":59},[53,63403,8116],{"class":120},[53,63405,128],{"class":63},[53,63407,1658],{"class":120},[53,63409,63410],{"class":63},"(dur_ns)):\n",[53,63412,63413,63416,63418,63420,63423],{"class":55,"line":284},[53,63414,63415],{"class":63}," Pe_tau_f[i, :] ",[53,63417,121],{"class":120},[53,63419,1511],{"class":63},[53,63421,63422],{"class":124},"interp",[53,63424,63425],{"class":63},"(freq_hi, freq_ghz, Pe[i, :])\n",[53,63427,63428],{"class":55,"line":296},[53,63429,104],{"emptyLinePlaceholder":103},[53,63431,63432],{"class":55,"line":309},[53,63433,63434],{"class":110},"# 2) Upsample τ axis for each fixed frequency\n",[53,63436,63437,63440,63442,63444,63446,63449,63451,63454,63456],{"class":55,"line":323},[53,63438,63439],{"class":63},"dur_hi ",[53,63441,121],{"class":120},[53,63443,1511],{"class":63},[53,63445,57842],{"class":124},[53,63447,63448],{"class":63},"(dur_ns.",[53,63450,25125],{"class":124},[53,63452,63453],{"class":63},"(), dur_ns.",[53,63455,36608],{"class":124},[53,63457,63335],{"class":63},[53,63459,63460,63463,63465,63467,63469,63472,63474,63476,63478,63480,63482],{"class":55,"line":332},[53,63461,63462],{"class":131}," UPSAMPLE_FACTOR",[53,63464,8124],{"class":120},[53,63466,58540],{"class":63},[53,63468,1658],{"class":120},[53,63470,63471],{"class":63},"(dur_ns) ",[53,63473,2262],{"class":120},[53,63475,1321],{"class":131},[53,63477,203],{"class":63},[53,63479,2002],{"class":120},[53,63481,1321],{"class":131},[53,63483,135],{"class":63},[53,63485,63486,63489,63491,63493,63495,63497,63499,63502,63504,63506,63508,63510],{"class":55,"line":344},[53,63487,63488],{"class":63},"Pe_hi ",[53,63490,121],{"class":120},[53,63492,1511],{"class":63},[53,63494,63373],{"class":124},[53,63496,63376],{"class":63},[53,63498,1658],{"class":120},[53,63500,63501],{"class":63},"(dur_hi), ",[53,63503,1658],{"class":120},[53,63505,63386],{"class":63},[53,63507,2051],{"class":2050},[53,63509,63391],{"class":120},[53,63511,135],{"class":63},[53,63513,63514,63516,63518,63520,63522,63524,63526],{"class":55,"line":357},[53,63515,1525],{"class":59},[53,63517,8575],{"class":63},[53,63519,1531],{"class":59},[53,63521,8116],{"class":120},[53,63523,128],{"class":63},[53,63525,1658],{"class":120},[53,63527,63528],{"class":63},"(freq_hi)):\n",[53,63530,63531,63534,63536,63538,63540],{"class":55,"line":1247},[53,63532,63533],{"class":63}," Pe_hi[:, j] ",[53,63535,121],{"class":120},[53,63537,1511],{"class":63},[53,63539,63422],{"class":124},[53,63541,63542],{"class":63},"(dur_hi, dur_ns, Pe_tau_f[:, j])\n",[53,63544,63545],{"class":55,"line":1252},[53,63546,104],{"emptyLinePlaceholder":103},[53,63548,63549],{"class":55,"line":1257},[53,63550,63551],{"class":110},"# Mesh for plotting\n",[53,63553,63554,63557,63559,63561,63563,63566,63568,63570,63572],{"class":55,"line":1262},[53,63555,63556],{"class":63},"F_hi, T_hi ",[53,63558,121],{"class":120},[53,63560,1511],{"class":63},[53,63562,57566],{"class":124},[53,63564,63565],{"class":63},"(freq_hi, dur_hi, ",[53,63567,57572],{"class":2050},[53,63569,121],{"class":120},[53,63571,57577],{"class":278},[53,63573,135],{"class":63},[53,63575,63576],{"class":55,"line":1274},[53,63577,104],{"emptyLinePlaceholder":103},[53,63579,63580],{"class":55,"line":1284},[53,63581,63582],{"class":110},"# --- Plot ---\n",[53,63584,63585,63588,63590,63592,63594,63596,63598,63600,63602,63604,63606,63609,63611,63614,63616,63619],{"class":55,"line":1292},[53,63586,63587],{"class":63},"fig ",[53,63589,121],{"class":120},[53,63591,58121],{"class":63},[53,63593,61512],{"class":124},[53,63595,128],{"class":63},[53,63597,58129],{"class":2050},[53,63599,121],{"class":120},[53,63601,128],{"class":63},[53,63603,35636],{"class":131},[53,63605,170],{"class":63},[53,63607,63608],{"class":131},"6",[53,63610,2108],{"class":63},[53,63612,63613],{"class":2050},"dpi",[53,63615,121],{"class":120},[53,63617,63618],{"class":131},"200",[53,63620,135],{"class":63},[53,63622,63623,63626,63628,63630,63633,63635,63638,63640,63643,63645,63648],{"class":55,"line":1307},[53,63624,63625],{"class":63},"ax ",[53,63627,121],{"class":120},[53,63629,58179],{"class":63},[53,63631,63632],{"class":124},"add_subplot",[53,63634,128],{"class":63},[53,63636,63637],{"class":131},"111",[53,63639,170],{"class":63},[53,63641,63642],{"class":2050},"projection",[53,63644,121],{"class":120},[53,63646,63647],{"class":278},"'3d'",[53,63649,135],{"class":63},[53,63651,63652],{"class":55,"line":1326},[53,63653,104],{"emptyLinePlaceholder":103},[53,63655,63656,63659,63661,63663,63666],{"class":55,"line":1344},[53,63657,63658],{"class":63},"surf ",[53,63660,121],{"class":120},[53,63662,58153],{"class":63},[53,63664,63665],{"class":124},"plot_surface",[53,63667,1271],{"class":63},[53,63669,63670],{"class":55,"line":1360},[53,63671,63672],{"class":63}," F_hi, T_hi, Pe_hi,\n",[53,63674,63675,63678,63680,63682,63684,63687,63689,63691,63694],{"class":55,"line":1366},[53,63676,63677],{"class":2050}," rstride",[53,63679,121],{"class":120},[53,63681,173],{"class":131},[53,63683,170],{"class":63},[53,63685,63686],{"class":2050},"cstride",[53,63688,121],{"class":120},[53,63690,173],{"class":131},[53,63692,63693],{"class":63},", ",[53,63695,63696],{"class":110},"# use every row\u002Fcolumn (after upsampling)\n",[53,63698,63699,63702,63704,63706,63708,63711,63713,63715],{"class":55,"line":1372},[53,63700,63701],{"class":2050}," linewidth",[53,63703,121],{"class":120},[53,63705,149],{"class":131},[53,63707,170],{"class":63},[53,63709,63710],{"class":2050},"antialiased",[53,63712,121],{"class":120},[53,63714,13773],{"class":131},[53,63716,1304],{"class":63},[53,63718,63719,63722,63724,63726],{"class":55,"line":1377},[53,63720,63721],{"class":2050}," cmap",[53,63723,121],{"class":120},[53,63725,63264],{"class":131},[53,63727,1304],{"class":63},[53,63729,63730,63733,63735,63737,63739,63742,63744,63746],{"class":55,"line":1383},[53,63731,63732],{"class":2050}," vmin",[53,63734,121],{"class":120},[53,63736,63281],{"class":131},[53,63738,170],{"class":63},[53,63740,63741],{"class":2050},"vmax",[53,63743,121],{"class":120},[53,63745,63286],{"class":131},[53,63747,1304],{"class":63},[53,63749,63750],{"class":55,"line":1389},[53,63751,135],{"class":63},[53,63753,63754],{"class":55,"line":1395},[53,63755,104],{"emptyLinePlaceholder":103},[53,63757,63758,63760,63762,63764,63767],{"class":55,"line":1401},[53,63759,58210],{"class":63},[53,63761,58213],{"class":124},[53,63763,128],{"class":63},[53,63765,63766],{"class":278},"'f (GHz)'",[53,63768,135],{"class":63},[53,63770,63771,63773,63775,63777,63779,63782,63785,63788,63790,63792,63794,63797],{"class":55,"line":1407},[53,63772,58210],{"class":63},[53,63774,58227],{"class":124},[53,63776,128],{"class":63},[53,63778,2111],{"class":59},[53,63780,63781],{"class":58202},"'$",[53,63783,63784],{"class":120},"\\t",[53,63786,63787],{"class":58202},"au$ ",[53,63789,128],{"class":131},[53,63791,48574],{"class":58202},[53,63793,3544],{"class":131},[53,63795,63796],{"class":58202},"'",[53,63798,135],{"class":63},[53,63800,63801,63803,63806,63808,63810,63812],{"class":55,"line":1413},[53,63802,58210],{"class":63},[53,63804,63805],{"class":124},"set_zlabel",[53,63807,128],{"class":63},[53,63809,2111],{"class":59},[53,63811,58203],{"class":58202},[53,63813,135],{"class":63},[53,63815,63816,63818,63820,63822,63825],{"class":55,"line":1419},[53,63817,58210],{"class":63},[53,63819,58241],{"class":124},[53,63821,128],{"class":63},[53,63823,63824],{"class":278},"'Rabi $P_e$ surface'",[53,63826,135],{"class":63},[53,63828,63829],{"class":55,"line":1425},[53,63830,104],{"emptyLinePlaceholder":103},[53,63832,63833],{"class":55,"line":1431},[53,63834,63835],{"class":110},"# Colorbar (legend for the colormap)\n",[53,63837,63838,63840,63842,63844,63846,63849,63851,63853,63855,63858,63860,63863,63865,63868,63870,63873],{"class":55,"line":1437},[53,63839,58174],{"class":63},[53,63841,121],{"class":120},[53,63843,58179],{"class":63},[53,63845,58182],{"class":124},[53,63847,63848],{"class":63},"(surf, ",[53,63850,58188],{"class":2050},[53,63852,121],{"class":120},[53,63854,58193],{"class":63},[53,63856,63857],{"class":2050},"shrink",[53,63859,121],{"class":120},[53,63861,63862],{"class":131},"0.7",[53,63864,170],{"class":63},[53,63866,63867],{"class":2050},"pad",[53,63869,121],{"class":120},[53,63871,63872],{"class":131},"0.08",[53,63874,135],{"class":63},[53,63876,63877,63880,63883,63885,63887,63889],{"class":55,"line":1443},[53,63878,63879],{"class":63},"cb.",[53,63881,63882],{"class":124},"set_label",[53,63884,128],{"class":63},[53,63886,2111],{"class":59},[53,63888,58203],{"class":58202},[53,63890,135],{"class":63},[53,63892,63893],{"class":55,"line":1448},[53,63894,104],{"emptyLinePlaceholder":103},[53,63896,63897,63899,63901],{"class":55,"line":1454},[53,63898,58263],{"class":63},[53,63900,58266],{"class":124},[53,63902,329],{"class":63},[53,63904,63905,63907,63909],{"class":55,"line":1460},[53,63906,25271],{"class":63},[53,63908,25508],{"class":124},[53,63910,329],{"class":63},[379,63912],{"alt":63913,"src":63914},"Output 3 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-03.webp",[43,63916,63918],{"className":45,"code":63917,"language":47,"meta":48,"style":48},"# @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",[50,63919,63920,63925,63930,63945,63961,63965,63975,63985,63989,64025,64030,64055,64077,64126,64140,64164,64168,64173,64194,64204,64208,64235,64254,64280,64294,64299,64345,64362,64370,64378,64382,64387,64408,64418,64422,64448,64465,64477,64491,64537,64553,64561],{"__ignoreMap":48},[53,63921,63922],{"class":55,"line":56},[53,63923,63924],{"class":110},"# @title Cross-sections\n",[53,63926,63927],{"class":55,"line":73},[53,63928,63929],{"class":110},"# Slicing controls\n",[53,63931,63932,63934,63936,63938,63940,63942],{"class":55,"line":87},[53,63933,56685],{"class":131},[53,63935,1298],{"class":63},[53,63937,1301],{"class":120},[53,63939,1318],{"class":120},[53,63941,57183],{"class":131},[53,63943,63944],{"class":110}," # drive frequency slice [GHz]\n",[53,63946,63947,63949,63951,63953,63955,63958],{"class":55,"line":100},[53,63948,56784],{"class":131},[53,63950,1298],{"class":63},[53,63952,1301],{"class":120},[53,63954,1318],{"class":120},[53,63956,63957],{"class":131}," 100.0",[53,63959,63960],{"class":110}," # duration slice [ns]\n",[53,63962,63963],{"class":55,"line":107},[53,63964,104],{"emptyLinePlaceholder":103},[53,63966,63967,63969,63971,63973],{"class":55,"line":114},[53,63968,60],{"class":59},[53,63970,64],{"class":63},[53,63972,67],{"class":59},[53,63974,70],{"class":63},[53,63976,63977,63979,63981,63983],{"class":55,"line":138},[53,63978,60],{"class":59},[53,63980,57059],{"class":63},[53,63982,67],{"class":59},[53,63984,1054],{"class":63},[53,63986,63987],{"class":55,"line":158},[53,63988,104],{"emptyLinePlaceholder":103},[53,63990,63991,63993,63996,63998,64001,64004,64007,64009,64011,64013,64015,64017,64019,64021,64023],{"class":55,"line":182},[53,63992,1265],{"class":59},[53,63994,63995],{"class":124}," _nearest_index",[53,63997,128],{"class":63},[53,63999,64000],{"class":1277},"vec",[53,64002,64003],{"class":63},": np.ndarray, ",[53,64005,64006],{"class":1277},"value",[53,64008,1298],{"class":63},[53,64010,1301],{"class":120},[53,64012,170],{"class":63},[53,64014,21903],{"class":1277},[53,64016,1298],{"class":63},[53,64018,58793],{"class":120},[53,64020,58796],{"class":63},[53,64022,1315],{"class":120},[53,64024,1623],{"class":63},[53,64026,64027],{"class":55,"line":209},[53,64028,64029],{"class":278}," \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n",[53,64031,64032,64035,64037,64039,64042,64044,64047,64049,64051,64053],{"class":55,"line":226},[53,64033,64034],{"class":63}," vmin, vmax ",[53,64036,121],{"class":120},[53,64038,59516],{"class":120},[53,64040,64041],{"class":63},"(vec.",[53,64043,25125],{"class":124},[53,64045,64046],{"class":63},"()), ",[53,64048,1301],{"class":120},[53,64050,64041],{"class":63},[53,64052,36608],{"class":124},[53,64054,21982],{"class":63},[53,64056,64057,64059,64062,64064,64067,64070,64072,64074],{"class":55,"line":231},[53,64058,1612],{"class":59},[53,64060,64061],{"class":63}," value ",[53,64063,5572],{"class":120},[53,64065,64066],{"class":63}," vmin ",[53,64068,64069],{"class":59},"or",[53,64071,64061],{"class":63},[53,64073,1546],{"class":120},[53,64075,64076],{"class":63}," vmax:\n",[53,64078,64079,64082,64084,64086,64089,64091,64093,64095,64097,64099,64101,64103,64106,64108,64111,64113,64115,64117,64119,64121,64124],{"class":55,"line":245},[53,64080,64081],{"class":120}," print",[53,64083,128],{"class":63},[53,64085,22173],{"class":59},[53,64087,64088],{"class":278},"\"[Note] ",[53,64090,2705],{"class":131},[53,64092,21903],{"class":63},[53,64094,2767],{"class":131},[53,64096,121],{"class":278},[53,64098,2705],{"class":131},[53,64100,64006],{"class":63},[53,64102,2767],{"class":131},[53,64104,64105],{"class":278}," is outside sweep; clamping to [",[53,64107,2705],{"class":131},[53,64109,64110],{"class":63},"vmin",[53,64112,2767],{"class":131},[53,64114,170],{"class":278},[53,64116,2705],{"class":131},[53,64118,63741],{"class":63},[53,64120,2767],{"class":131},[53,64122,64123],{"class":278},"].\"",[53,64125,135],{"class":63},[53,64127,64128,64131,64133,64135,64137],{"class":55,"line":258},[53,64129,64130],{"class":63}," value ",[53,64132,121],{"class":120},[53,64134,1511],{"class":63},[53,64136,57813],{"class":124},[53,64138,64139],{"class":63},"(value, vmin, vmax)\n",[53,64141,64142,64144,64146,64148,64151,64153,64156,64159,64161],{"class":55,"line":263},[53,64143,2399],{"class":59},[53,64145,1990],{"class":120},[53,64147,59813],{"class":63},[53,64149,64150],{"class":124},"argmin",[53,64152,59813],{"class":63},[53,64154,64155],{"class":124},"abs",[53,64157,64158],{"class":63},"(vec ",[53,64160,2262],{"class":120},[53,64162,64163],{"class":63}," value)))\n",[53,64165,64166],{"class":55,"line":269},[53,64167,104],{"emptyLinePlaceholder":103},[53,64169,64170],{"class":55,"line":284},[53,64171,64172],{"class":110},"# --- Slice at f = F_CROSS_GHZ ---\n",[53,64174,64175,64178,64180,64182,64185,64187,64189,64192],{"class":55,"line":296},[53,64176,64177],{"class":63},"j ",[53,64179,121],{"class":120},[53,64181,63995],{"class":124},[53,64183,64184],{"class":63},"(freq_ghz, ",[53,64186,56685],{"class":131},[53,64188,170],{"class":63},[53,64190,64191],{"class":278},"\"F_CROSS_GHZ\"",[53,64193,135],{"class":63},[53,64195,64196,64199,64201],{"class":55,"line":309},[53,64197,64198],{"class":63},"Pe_vs_tau ",[53,64200,121],{"class":120},[53,64202,64203],{"class":63}," Pe[:, j]\n",[53,64205,64206],{"class":55,"line":323},[53,64207,104],{"emptyLinePlaceholder":103},[53,64209,64210,64212,64214,64216,64218,64220,64222,64224,64226,64228,64230,64233],{"class":55,"line":332},[53,64211,58116],{"class":63},[53,64213,121],{"class":120},[53,64215,58121],{"class":63},[53,64217,58124],{"class":124},[53,64219,128],{"class":63},[53,64221,58129],{"class":2050},[53,64223,121],{"class":120},[53,64225,128],{"class":63},[53,64227,63608],{"class":131},[53,64229,170],{"class":63},[53,64231,64232],{"class":131},"3.5",[53,64234,22435],{"class":63},[53,64236,64237,64239,64241,64244,64247,64249,64252],{"class":55,"line":344},[53,64238,58210],{"class":63},[53,64240,25274],{"class":124},[53,64242,64243],{"class":63},"(dur_ns, Pe_vs_tau, ",[53,64245,64246],{"class":2050},"color",[53,64248,121],{"class":120},[53,64250,64251],{"class":278},"'blue'",[53,64253,135],{"class":63},[53,64255,64256,64258,64260,64262,64264,64266,64268,64270,64272,64274,64276,64278],{"class":55,"line":357},[53,64257,58210],{"class":63},[53,64259,58213],{"class":124},[53,64261,128],{"class":63},[53,64263,2111],{"class":59},[53,64265,63781],{"class":58202},[53,64267,63784],{"class":120},[53,64269,63787],{"class":58202},[53,64271,128],{"class":131},[53,64273,48574],{"class":58202},[53,64275,3544],{"class":131},[53,64277,63796],{"class":58202},[53,64279,135],{"class":63},[53,64281,64282,64284,64286,64288,64290,64292],{"class":55,"line":1247},[53,64283,58210],{"class":63},[53,64285,58227],{"class":124},[53,64287,128],{"class":63},[53,64289,2111],{"class":59},[53,64291,58203],{"class":58202},[53,64293,135],{"class":63},[53,64295,64296],{"class":55,"line":1252},[53,64297,64298],{"class":110},"# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\n",[53,64300,64301,64303,64305,64307,64309,64312,64314,64316,64319,64321,64324,64326,64329,64331,64333,64336,64338,64340,64343],{"class":55,"line":1257},[53,64302,58210],{"class":63},[53,64304,58241],{"class":124},[53,64306,128],{"class":63},[53,64308,2111],{"class":59},[53,64310,64311],{"class":58202},"'$P_e",[53,64313,128],{"class":131},[53,64315,63784],{"class":120},[53,64317,64318],{"class":58202},"au",[53,64320,3544],{"class":131},[53,64322,64323],{"class":58202},"$ at f = '",[53,64325,8208],{"class":120},[53,64327,64328],{"class":59}," f",[53,64330,63796],{"class":278},[53,64332,2705],{"class":131},[53,64334,64335],{"class":63},"freq_ghz[j]",[53,64337,62774],{"class":59},[53,64339,2767],{"class":131},[53,64341,64342],{"class":278}," GHz'",[53,64344,135],{"class":63},[53,64346,64347,64349,64352,64354,64356,64358,64360],{"class":55,"line":1262},[53,64348,58210],{"class":63},[53,64350,64351],{"class":124},"set_ylim",[53,64353,128],{"class":63},[53,64355,56610],{"class":131},[53,64357,170],{"class":63},[53,64359,57823],{"class":131},[53,64361,135],{"class":63},[53,64363,64364,64366,64368],{"class":55,"line":1274},[53,64365,58263],{"class":63},[53,64367,58266],{"class":124},[53,64369,329],{"class":63},[53,64371,64372,64374,64376],{"class":55,"line":1284},[53,64373,25271],{"class":63},[53,64375,25508],{"class":124},[53,64377,329],{"class":63},[53,64379,64380],{"class":55,"line":1292},[53,64381,104],{"emptyLinePlaceholder":103},[53,64383,64384],{"class":55,"line":1307},[53,64385,64386],{"class":110},"# --- Slice at τ = T_CROSS_NS ---\n",[53,64388,64389,64392,64394,64396,64399,64401,64403,64406],{"class":55,"line":1326},[53,64390,64391],{"class":63},"i ",[53,64393,121],{"class":120},[53,64395,63995],{"class":124},[53,64397,64398],{"class":63},"(dur_ns, ",[53,64400,56784],{"class":131},[53,64402,170],{"class":63},[53,64404,64405],{"class":278},"\"T_CROSS_NS\"",[53,64407,135],{"class":63},[53,64409,64410,64413,64415],{"class":55,"line":1344},[53,64411,64412],{"class":63},"Pe_vs_f ",[53,64414,121],{"class":120},[53,64416,64417],{"class":63}," Pe[i, :]\n",[53,64419,64420],{"class":55,"line":1360},[53,64421,104],{"emptyLinePlaceholder":103},[53,64423,64424,64426,64428,64430,64432,64434,64436,64438,64440,64442,64444,64446],{"class":55,"line":1366},[53,64425,58116],{"class":63},[53,64427,121],{"class":120},[53,64429,58121],{"class":63},[53,64431,58124],{"class":124},[53,64433,128],{"class":63},[53,64435,58129],{"class":2050},[53,64437,121],{"class":120},[53,64439,128],{"class":63},[53,64441,63608],{"class":131},[53,64443,170],{"class":63},[53,64445,64232],{"class":131},[53,64447,22435],{"class":63},[53,64449,64450,64452,64454,64457,64459,64461,64463],{"class":55,"line":1372},[53,64451,58210],{"class":63},[53,64453,25274],{"class":124},[53,64455,64456],{"class":63},"(freq_ghz, Pe_vs_f, ",[53,64458,64246],{"class":2050},[53,64460,121],{"class":120},[53,64462,64251],{"class":278},[53,64464,135],{"class":63},[53,64466,64467,64469,64471,64473,64475],{"class":55,"line":1377},[53,64468,58210],{"class":63},[53,64470,58213],{"class":124},[53,64472,128],{"class":63},[53,64474,63766],{"class":278},[53,64476,135],{"class":63},[53,64478,64479,64481,64483,64485,64487,64489],{"class":55,"line":1383},[53,64480,58210],{"class":63},[53,64482,58227],{"class":124},[53,64484,128],{"class":63},[53,64486,2111],{"class":59},[53,64488,58203],{"class":58202},[53,64490,135],{"class":63},[53,64492,64493,64495,64497,64499,64501,64503,64505,64507,64509,64512,64514,64517,64519,64521,64523,64525,64528,64530,64532,64535],{"class":55,"line":1389},[53,64494,58210],{"class":63},[53,64496,58241],{"class":124},[53,64498,128],{"class":63},[53,64500,2111],{"class":59},[53,64502,64311],{"class":58202},[53,64504,128],{"class":131},[53,64506,22173],{"class":58202},[53,64508,3544],{"class":131},[53,64510,64511],{"class":58202},"$ at $",[53,64513,63784],{"class":120},[53,64515,64516],{"class":58202},"au$ = '",[53,64518,8208],{"class":120},[53,64520,64328],{"class":59},[53,64522,63796],{"class":278},[53,64524,2705],{"class":131},[53,64526,64527],{"class":63},"dur_ns[i]",[53,64529,62849],{"class":59},[53,64531,2767],{"class":131},[53,64533,64534],{"class":278}," ns'",[53,64536,135],{"class":63},[53,64538,64539,64541,64543,64545,64547,64549,64551],{"class":55,"line":1395},[53,64540,58210],{"class":63},[53,64542,64351],{"class":124},[53,64544,128],{"class":63},[53,64546,56610],{"class":131},[53,64548,170],{"class":63},[53,64550,57823],{"class":131},[53,64552,135],{"class":63},[53,64554,64555,64557,64559],{"class":55,"line":1401},[53,64556,58263],{"class":63},[53,64558,58266],{"class":124},[53,64560,329],{"class":63},[53,64562,64563,64565,64567],{"class":55,"line":1407},[53,64564,25271],{"class":63},[53,64566,25508],{"class":124},[53,64568,329],{"class":63},[379,64570],{"alt":64571,"src":64572},"Output 4 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-04.webp",[379,64574],{"alt":64575,"src":64576},"Output 5 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-05.webp",[43,64578,64580],{"className":45,"code":64579,"language":47,"meta":48,"style":48},"# @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",[50,64581,64582,64587,64597,64607,64611,64616,64632,64648,64664,64680,64696,64712,64716,64721,64747,64765,64778,64799,64803,64812,64819,64826,64841,64856,64871,64886,64901,64916,64921,64926,64930,64934,64939,64944,64949,64954,64959,64964,64969,64974,64978,64982,64987,64992,64996,65013,65030,65050,65054,65059,65092,65102,65145,65157,65161,65166,65175,65190,65197,65234,65248,65252,65257,65292,65317,65321,65326,65344,65362,65368,65381,65385,65390,65408,65448,65462,65468,65490,65494,65511,65542,65546,65551,65570,65584,65590,65603,65607,65614,65618,65623,65634,65639,65649,65659,65669,65679,65689,65699,65703],{"__ignoreMap":48},[53,64583,64584],{"class":55,"line":56},[53,64585,64586],{"class":110},"# @title FFT of Rabi oscillations along tau (controls + computation)\n",[53,64588,64589,64591,64593,64595],{"class":55,"line":73},[53,64590,60],{"class":59},[53,64592,64],{"class":63},[53,64594,67],{"class":59},[53,64596,70],{"class":63},[53,64598,64599,64601,64603,64605],{"class":55,"line":87},[53,64600,60],{"class":59},[53,64602,57059],{"class":63},[53,64604,67],{"class":59},[53,64606,1054],{"class":63},[53,64608,64609],{"class":55,"line":100},[53,64610,104],{"emptyLinePlaceholder":103},[53,64612,64613],{"class":55,"line":107},[53,64614,64615],{"class":110},"# -------- Control knobs (FFT & display) --------\n",[53,64617,64618,64620,64622,64624,64626,64629],{"class":55,"line":114},[53,64619,56901],{"class":131},[53,64621,1298],{"class":63},[53,64623,1315],{"class":120},[53,64625,1318],{"class":120},[53,64627,64628],{"class":131}," 8",[53,64630,64631],{"class":110}," # zero-padding factor (2, 4, 8, ...)\n",[53,64633,64634,64636,64638,64640,64642,64645],{"class":55,"line":138},[53,64635,56915],{"class":131},[53,64637,1298],{"class":63},[53,64639,58793],{"class":120},[53,64641,1318],{"class":120},[53,64643,64644],{"class":278}," \"hann\"",[53,64646,64647],{"class":110}," # \"hann\", \"boxcar\"\n",[53,64649,64650,64652,64654,64656,64658,64661],{"class":55,"line":158},[53,64651,56930],{"class":131},[53,64653,1298],{"class":63},[53,64655,1334],{"class":120},[53,64657,1318],{"class":120},[53,64659,64660],{"class":131}," True",[53,64662,64663],{"class":110}," # subtract mean along τ before FFT\n",[53,64665,64666,64668,64670,64672,64674,64677],{"class":55,"line":182},[53,64667,56944],{"class":131},[53,64669,1298],{"class":63},[53,64671,58793],{"class":120},[53,64673,1318],{"class":120},[53,64675,64676],{"class":278}," \"magnitude\"",[53,64678,64679],{"class":110}," # \"magnitude\" | \"power\" | \"db\"\n",[53,64681,64682,64684,64686,64688,64690,64693],{"class":55,"line":209},[53,64683,56959],{"class":131},[53,64685,1298],{"class":63},[53,64687,58793],{"class":120},[53,64689,1318],{"class":120},[53,64691,64692],{"class":278}," \"global\"",[53,64694,64695],{"class":110}," # \"global\" | \"per_f\" (column-wise)\n",[53,64697,64698,64700,64702,64704,64706,64709],{"class":55,"line":226},[53,64699,56974],{"class":131},[53,64701,1298],{"class":63},[53,64703,58793],{"class":120},[53,64705,1318],{"class":120},[53,64707,64708],{"class":278}," \"MHz\"",[53,64710,64711],{"class":110}," # \"MHz\" or \"GHz\" for Fourier axis\n",[53,64713,64714],{"class":55,"line":231},[53,64715,104],{"emptyLinePlaceholder":103},[53,64717,64718],{"class":55,"line":245},[53,64719,64720],{"class":110},"# -------- Helpers --------\n",[53,64722,64723,64725,64728,64730,64732,64734,64736,64738,64741,64743,64745],{"class":55,"line":258},[53,64724,1265],{"class":59},[53,64726,64727],{"class":124}," _make_window",[53,64729,128],{"class":63},[53,64731,4315],{"class":1277},[53,64733,1298],{"class":63},[53,64735,1315],{"class":120},[53,64737,170],{"class":63},[53,64739,64740],{"class":1277},"kind",[53,64742,1298],{"class":63},[53,64744,58793],{"class":120},[53,64746,57410],{"class":63},[53,64748,64749,64751,64754,64757,64759,64761,64763],{"class":55,"line":263},[53,64750,1612],{"class":59},[53,64752,64753],{"class":63}," kind.",[53,64755,64756],{"class":124},"lower",[53,64758,1572],{"class":63},[53,64760,1618],{"class":120},[53,64762,64644],{"class":278},[53,64764,1623],{"class":63},[53,64766,64767,64770,64772,64775],{"class":55,"line":269},[53,64768,64769],{"class":59}," return",[53,64771,1511],{"class":63},[53,64773,64774],{"class":124},"hanning",[53,64776,64777],{"class":63},"(n)\n",[53,64779,64780,64782,64784,64787,64790,64792,64794,64796],{"class":55,"line":284},[53,64781,2399],{"class":59},[53,64783,1511],{"class":63},[53,64785,64786],{"class":124},"ones",[53,64788,64789],{"class":63},"(n, ",[53,64791,2051],{"class":2050},[53,64793,63391],{"class":120},[53,64795,22613],{"class":63},[53,64797,64798],{"class":110},"# boxcar\n",[53,64800,64801],{"class":55,"line":296},[53,64802,104],{"emptyLinePlaceholder":103},[53,64804,64805,64807,64810],{"class":55,"line":309},[53,64806,1265],{"class":59},[53,64808,64809],{"class":124}," rfft_spectrogram",[53,64811,1271],{"class":63},[53,64813,64814,64817],{"class":55,"line":323},[53,64815,64816],{"class":1277}," Pe",[53,64818,1281],{"class":63},[53,64820,64821,64824],{"class":55,"line":332},[53,64822,64823],{"class":1277}," dur_ns",[53,64825,1281],{"class":63},[53,64827,64828,64831,64833,64835,64837,64839],{"class":55,"line":344},[53,64829,64830],{"class":1277}," pad",[53,64832,1298],{"class":63},[53,64834,1315],{"class":120},[53,64836,1318],{"class":120},[53,64838,64628],{"class":131},[53,64840,1304],{"class":63},[53,64842,64843,64846,64848,64850,64852,64854],{"class":55,"line":357},[53,64844,64845],{"class":1277}," window",[53,64847,1298],{"class":63},[53,64849,58793],{"class":120},[53,64851,1318],{"class":120},[53,64853,64644],{"class":278},[53,64855,1304],{"class":63},[53,64857,64858,64861,64863,64865,64867,64869],{"class":55,"line":1247},[53,64859,64860],{"class":1277}," detrend_mean",[53,64862,1298],{"class":63},[53,64864,1334],{"class":120},[53,64866,1318],{"class":120},[53,64868,64660],{"class":131},[53,64870,1304],{"class":63},[53,64872,64873,64876,64878,64880,64882,64884],{"class":55,"line":1252},[53,64874,64875],{"class":1277}," amp_mode",[53,64877,1298],{"class":63},[53,64879,58793],{"class":120},[53,64881,1318],{"class":120},[53,64883,64676],{"class":278},[53,64885,1304],{"class":63},[53,64887,64888,64891,64893,64895,64897,64899],{"class":55,"line":1257},[53,64889,64890],{"class":1277}," norm_mode",[53,64892,1298],{"class":63},[53,64894,58793],{"class":120},[53,64896,1318],{"class":120},[53,64898,64692],{"class":278},[53,64900,1304],{"class":63},[53,64902,64903,64906,64908,64910,64912,64914],{"class":55,"line":1262},[53,64904,64905],{"class":1277}," freq_units",[53,64907,1298],{"class":63},[53,64909,58793],{"class":120},[53,64911,1318],{"class":120},[53,64913,64708],{"class":278},[53,64915,1304],{"class":63},[53,64917,64918],{"class":55,"line":1274},[53,64919,64920],{"class":63},") -> tuple[np.ndarray, np.ndarray]:\n",[53,64922,64923],{"class":55,"line":1284},[53,64924,64925],{"class":278}," \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n",[53,64927,64928],{"class":55,"line":1292},[53,64929,104],{"emptyLinePlaceholder":103},[53,64931,64932],{"class":55,"line":1307},[53,64933,1380],{"class":278},[53,64935,64936],{"class":55,"line":1326},[53,64937,64938],{"class":278}," Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n",[53,64940,64941],{"class":55,"line":1344},[53,64942,64943],{"class":278}," dur_ns: τ grid [ns], length N_tau, evenly spaced.\n",[53,64945,64946],{"class":55,"line":1360},[53,64947,64948],{"class":278}," pad: zero-padding factor (≥1).\n",[53,64950,64951],{"class":55,"line":1366},[53,64952,64953],{"class":278}," window: \"hann\" or \"boxcar\".\n",[53,64955,64956],{"class":55,"line":1372},[53,64957,64958],{"class":278}," detrend_mean: subtract column-wise mean before FFT.\n",[53,64960,64961],{"class":55,"line":1377},[53,64962,64963],{"class":278}," amp_mode: \"magnitude\", \"power\", or \"db\".\n",[53,64965,64966],{"class":55,"line":1383},[53,64967,64968],{"class":278}," norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n",[53,64970,64971],{"class":55,"line":1389},[53,64972,64973],{"class":278}," freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n",[53,64975,64976],{"class":55,"line":1395},[53,64977,104],{"emptyLinePlaceholder":103},[53,64979,64980],{"class":55,"line":1401},[53,64981,1451],{"class":278},[53,64983,64984],{"class":55,"line":1407},[53,64985,64986],{"class":278}," f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n",[53,64988,64989],{"class":55,"line":1413},[53,64990,64991],{"class":278}," S: 2D nonnegative spectrogram (N_w, N_f).\n",[53,64993,64994],{"class":55,"line":1419},[53,64995,1363],{"class":278},[53,64997,64998,65001,65003,65005,65007,65009,65011],{"class":55,"line":1425},[53,64999,65000],{"class":63}," Pe ",[53,65002,121],{"class":120},[53,65004,1511],{"class":63},[53,65006,59816],{"class":124},[53,65008,57816],{"class":63},[53,65010,1301],{"class":120},[53,65012,135],{"class":63},[53,65014,65015,65018,65020,65022,65024,65026,65028],{"class":55,"line":1431},[53,65016,65017],{"class":63}," dur_ns ",[53,65019,121],{"class":120},[53,65021,1511],{"class":63},[53,65023,59816],{"class":124},[53,65025,64398],{"class":63},[53,65027,1301],{"class":120},[53,65029,135],{"class":63},[53,65031,65032,65035,65038,65040,65042,65044,65047],{"class":55,"line":1437},[53,65033,65034],{"class":59}," assert",[53,65036,65037],{"class":63}," Pe.shape[",[53,65039,149],{"class":131},[53,65041,8101],{"class":63},[53,65043,1618],{"class":120},[53,65045,65046],{"class":63}," dur_ns.size, ",[53,65048,65049],{"class":278},"\"Pe must be shaped (N_tau, N_f).\"\n",[53,65051,65052],{"class":55,"line":1443},[53,65053,104],{"emptyLinePlaceholder":103},[53,65055,65056],{"class":55,"line":1448},[53,65057,65058],{"class":110}," # Uniform sampling assumption (true for np.linspace)\n",[53,65060,65061,65064,65066,65068,65071,65073,65075,65077,65080,65082,65085,65087,65089],{"class":55,"line":1454},[53,65062,65063],{"class":63}," dt ",[53,65065,121],{"class":120},[53,65067,59516],{"class":120},[53,65069,65070],{"class":63},"(dur_ns[",[53,65072,173],{"class":131},[53,65074,8101],{"class":63},[53,65076,2262],{"class":120},[53,65078,65079],{"class":63}," dur_ns[",[53,65081,149],{"class":131},[53,65083,65084],{"class":63},"]) ",[53,65086,2286],{"class":120},[53,65088,57784],{"class":131},[53,65090,65091],{"class":110}," # [s]\n",[53,65093,65094,65097,65099],{"class":55,"line":1460},[53,65095,65096],{"class":63}," n_tau ",[53,65098,121],{"class":120},[53,65100,65101],{"class":63}," dur_ns.size\n",[53,65103,65104,65107,65109,65111,65113,65115,65118,65120,65122,65125,65127,65130,65133,65135,65138,65140,65142],{"class":55,"line":1465},[53,65105,65106],{"class":63}," nfft ",[53,65108,121],{"class":120},[53,65110,1990],{"class":120},[53,65112,128],{"class":63},[53,65114,132],{"class":131},[53,65116,65117],{"class":120}," **",[53,65119,1990],{"class":120},[53,65121,59813],{"class":63},[53,65123,65124],{"class":124},"ceil",[53,65126,59813],{"class":63},[53,65128,65129],{"class":124},"log2",[53,65131,65132],{"class":63},"(n_tau))) ",[53,65134,2286],{"class":120},[53,65136,65137],{"class":120}," max",[53,65139,128],{"class":63},[53,65141,173],{"class":131},[53,65143,65144],{"class":63},", pad))\n",[53,65146,65147,65150,65152,65154],{"class":55,"line":1470},[53,65148,65149],{"class":63}," w ",[53,65151,121],{"class":120},[53,65153,64727],{"class":124},[53,65155,65156],{"class":63},"(n_tau, window)\n",[53,65158,65159],{"class":55,"line":1475},[53,65160,104],{"emptyLinePlaceholder":103},[53,65162,65163],{"class":55,"line":1492},[53,65164,65165],{"class":110}," # Prepare output\n",[53,65167,65168,65171,65173],{"class":55,"line":1503},[53,65169,65170],{"class":63}," S_list ",[53,65172,121],{"class":120},[53,65174,1805],{"class":63},[53,65176,65177,65180,65182,65185,65188],{"class":55,"line":1519},[53,65178,65179],{"class":63}," X ",[53,65181,121],{"class":120},[53,65183,65184],{"class":63}," Pe.",[53,65186,65187],{"class":124},"copy",[53,65189,329],{"class":63},[53,65191,65192,65194],{"class":55,"line":1552},[53,65193,1612],{"class":59},[53,65195,65196],{"class":63}," detrend_mean:\n",[53,65198,65199,65202,65204,65207,65209,65212,65215,65217,65219,65221,65223,65225,65228,65230,65232],{"class":55,"line":1558},[53,65200,65201],{"class":63}," X ",[53,65203,121],{"class":120},[53,65205,65206],{"class":63}," X ",[53,65208,2262],{"class":120},[53,65210,65211],{"class":63}," X.",[53,65213,65214],{"class":124},"mean",[53,65216,128],{"class":63},[53,65218,61067],{"class":2050},[53,65220,121],{"class":120},[53,65222,149],{"class":131},[53,65224,170],{"class":63},[53,65226,65227],{"class":2050},"keepdims",[53,65229,121],{"class":120},[53,65231,13773],{"class":131},[53,65233,135],{"class":63},[53,65235,65236,65238,65241,65244,65246],{"class":55,"line":1586},[53,65237,65179],{"class":63},[53,65239,65240],{"class":120},"*=",[53,65242,65243],{"class":63}," w[:, ",[53,65245,56660],{"class":131},[53,65247,1689],{"class":63},[53,65249,65250],{"class":55,"line":1609},[53,65251,104],{"emptyLinePlaceholder":103},[53,65253,65254],{"class":55,"line":1626},[53,65255,65256],{"class":110}," # Real FFT along τ (axis=0)\n",[53,65258,65259,65262,65264,65267,65270,65273,65275,65277,65280,65282,65284,65286,65289],{"class":55,"line":1635},[53,65260,65261],{"class":63}," Xf ",[53,65263,121],{"class":120},[53,65265,65266],{"class":63}," np.fft.",[53,65268,65269],{"class":124},"rfft",[53,65271,65272],{"class":63},"(X, ",[53,65274,4315],{"class":2050},[53,65276,121],{"class":120},[53,65278,65279],{"class":63},"nfft, ",[53,65281,61067],{"class":2050},[53,65283,121],{"class":120},[53,65285,149],{"class":131},[53,65287,65288],{"class":63},") ",[53,65290,65291],{"class":110},"# shape (N_w, N_f)\n",[53,65293,65294,65297,65299,65301,65304,65307,65309,65311,65314],{"class":55,"line":1664},[53,65295,65296],{"class":63}," f_fft_hz ",[53,65298,121],{"class":120},[53,65300,65266],{"class":63},[53,65302,65303],{"class":124},"rfftfreq",[53,65305,65306],{"class":63},"(nfft, ",[53,65308,4854],{"class":2050},[53,65310,121],{"class":120},[53,65312,65313],{"class":63},"dt) ",[53,65315,65316],{"class":110},"# [Hz]\n",[53,65318,65319],{"class":55,"line":1672},[53,65320,104],{"emptyLinePlaceholder":103},[53,65322,65323],{"class":55,"line":1692},[53,65324,65325],{"class":110}," # Amplitude selection\n",[53,65327,65328,65330,65333,65335,65337,65339,65342],{"class":55,"line":1700},[53,65329,1612],{"class":59},[53,65331,65332],{"class":63}," amp_mode.",[53,65334,64756],{"class":124},[53,65336,1572],{"class":63},[53,65338,1618],{"class":120},[53,65340,65341],{"class":278}," \"power\"",[53,65343,1623],{"class":63},[53,65345,65346,65349,65351,65353,65355,65358,65360],{"class":55,"line":1707},[53,65347,65348],{"class":63}," S ",[53,65350,121],{"class":120},[53,65352,1511],{"class":63},[53,65354,64155],{"class":124},[53,65356,65357],{"class":63},"(Xf) ",[53,65359,2272],{"class":120},[53,65361,57685],{"class":131},[53,65363,65364,65366],{"class":55,"line":1727},[53,65365,1695],{"class":59},[53,65367,1623],{"class":63},[53,65369,65370,65372,65374,65376,65378],{"class":55,"line":1734},[53,65371,65348],{"class":63},[53,65373,121],{"class":120},[53,65375,1511],{"class":63},[53,65377,64155],{"class":124},[53,65379,65380],{"class":63},"(Xf)\n",[53,65382,65383],{"class":55,"line":1750},[53,65384,104],{"emptyLinePlaceholder":103},[53,65386,65387],{"class":55,"line":1755},[53,65388,65389],{"class":110}," # Normalization\n",[53,65391,65392,65394,65397,65399,65401,65403,65406],{"class":55,"line":1760},[53,65393,1612],{"class":59},[53,65395,65396],{"class":63}," norm_mode.",[53,65398,64756],{"class":124},[53,65400,1572],{"class":63},[53,65402,1618],{"class":120},[53,65404,65405],{"class":278}," \"per_f\"",[53,65407,1623],{"class":63},[53,65409,65410,65413,65415,65417,65420,65423,65425,65427,65429,65431,65433,65435,65437,65439,65441,65443,65446],{"class":55,"line":1781},[53,65411,65412],{"class":63}," col_max ",[53,65414,121],{"class":120},[53,65416,1511],{"class":63},[53,65418,65419],{"class":124},"maximum",[53,65421,65422],{"class":63},"(S.",[53,65424,36608],{"class":124},[53,65426,128],{"class":63},[53,65428,61067],{"class":2050},[53,65430,121],{"class":120},[53,65432,149],{"class":131},[53,65434,170],{"class":63},[53,65436,65227],{"class":2050},[53,65438,121],{"class":120},[53,65440,13773],{"class":131},[53,65442,2108],{"class":63},[53,65444,65445],{"class":131},"1e-15",[53,65447,135],{"class":63},[53,65449,65450,65452,65454,65457,65459],{"class":55,"line":1786},[53,65451,65348],{"class":63},[53,65453,121],{"class":120},[53,65455,65456],{"class":63}," S ",[53,65458,193],{"class":120},[53,65460,65461],{"class":63}," col_max\n",[53,65463,65464,65466],{"class":55,"line":1792},[53,65465,1695],{"class":59},[53,65467,1623],{"class":63},[53,65469,65470,65472,65474,65476,65478,65480,65482,65484,65486,65488],{"class":55,"line":1797},[53,65471,65348],{"class":63},[53,65473,121],{"class":120},[53,65475,65456],{"class":63},[53,65477,193],{"class":120},[53,65479,65137],{"class":120},[53,65481,65422],{"class":63},[53,65483,36608],{"class":124},[53,65485,13845],{"class":63},[53,65487,65445],{"class":131},[53,65489,135],{"class":63},[53,65491,65492],{"class":55,"line":1808},[53,65493,104],{"emptyLinePlaceholder":103},[53,65495,65496,65498,65500,65502,65504,65506,65509],{"class":55,"line":1828},[53,65497,1612],{"class":59},[53,65499,65332],{"class":63},[53,65501,64756],{"class":124},[53,65503,1572],{"class":63},[53,65505,1618],{"class":120},[53,65507,65508],{"class":278}," \"db\"",[53,65510,1623],{"class":63},[53,65512,65513,65515,65517,65519,65521,65523,65526,65528,65530,65533,65536,65539],{"class":55,"line":1868},[53,65514,65348],{"class":63},[53,65516,121],{"class":120},[53,65518,57199],{"class":131},[53,65520,8124],{"class":120},[53,65522,1511],{"class":63},[53,65524,65525],{"class":124},"log10",[53,65527,59813],{"class":63},[53,65529,65419],{"class":124},[53,65531,65532],{"class":63},"(S, ",[53,65534,65535],{"class":131},"1e-12",[53,65537,65538],{"class":63},")) ",[53,65540,65541],{"class":110},"# dB, clipped floor\n",[53,65543,65544],{"class":55,"line":1873},[53,65545,104],{"emptyLinePlaceholder":103},[53,65547,65548],{"class":55,"line":1883},[53,65549,65550],{"class":110}," # Units\n",[53,65552,65553,65555,65558,65561,65563,65565,65568],{"class":55,"line":1898},[53,65554,1612],{"class":59},[53,65556,65557],{"class":63}," freq_units.",[53,65559,65560],{"class":124},"upper",[53,65562,1572],{"class":63},[53,65564,1618],{"class":120},[53,65566,65567],{"class":278}," \"GHz\"",[53,65569,1623],{"class":63},[53,65571,65572,65575,65577,65580,65582],{"class":55,"line":1931},[53,65573,65574],{"class":63}," f_fft ",[53,65576,121],{"class":120},[53,65578,65579],{"class":63}," f_fft_hz ",[53,65581,2286],{"class":120},[53,65583,57524],{"class":131},[53,65585,65586,65588],{"class":55,"line":1936},[53,65587,1695],{"class":59},[53,65589,1623],{"class":63},[53,65591,65592,65594,65596,65598,65600],{"class":55,"line":1947},[53,65593,65574],{"class":63},[53,65595,121],{"class":120},[53,65597,65579],{"class":63},[53,65599,2286],{"class":120},[53,65601,65602],{"class":131}," 1e-6\n",[53,65604,65605],{"class":55,"line":1952},[53,65606,104],{"emptyLinePlaceholder":103},[53,65608,65609,65611],{"class":55,"line":1966},[53,65610,2399],{"class":59},[53,65612,65613],{"class":63}," f_fft, S\n",[53,65615,65616],{"class":55,"line":1971},[53,65617,104],{"emptyLinePlaceholder":103},[53,65619,65620],{"class":55,"line":1982},[53,65621,65622],{"class":110},"# Compute spectrogram with current grid\n",[53,65624,65625,65628,65630,65632],{"class":55,"line":2010},[53,65626,65627],{"class":63},"f_fft_axis, S_fft ",[53,65629,121],{"class":120},[53,65631,64809],{"class":124},[53,65633,1271],{"class":63},[53,65635,65636],{"class":55,"line":2029},[53,65637,65638],{"class":63}," Pe, dur_ns,\n",[53,65640,65641,65643,65645,65647],{"class":55,"line":2034},[53,65642,64830],{"class":2050},[53,65644,121],{"class":120},[53,65646,56901],{"class":131},[53,65648,1304],{"class":63},[53,65650,65651,65653,65655,65657],{"class":55,"line":2059},[53,65652,64845],{"class":2050},[53,65654,121],{"class":120},[53,65656,56915],{"class":131},[53,65658,1304],{"class":63},[53,65660,65661,65663,65665,65667],{"class":55,"line":2064},[53,65662,64860],{"class":2050},[53,65664,121],{"class":120},[53,65666,56930],{"class":131},[53,65668,1304],{"class":63},[53,65670,65671,65673,65675,65677],{"class":55,"line":2075},[53,65672,64875],{"class":2050},[53,65674,121],{"class":120},[53,65676,56944],{"class":131},[53,65678,1304],{"class":63},[53,65680,65681,65683,65685,65687],{"class":55,"line":2119},[53,65682,64890],{"class":2050},[53,65684,121],{"class":120},[53,65686,56959],{"class":131},[53,65688,1304],{"class":63},[53,65690,65691,65693,65695,65697],{"class":55,"line":2128},[53,65692,64905],{"class":2050},[53,65694,121],{"class":120},[53,65696,56974],{"class":131},[53,65698,1304],{"class":63},[53,65700,65701],{"class":55,"line":2133},[53,65702,135],{"class":63},[53,65704,65705,65707,65709,65712,65715,65718,65721,65723,65725,65728,65730,65732,65734,65736,65738],{"class":55,"line":2144},[53,65706,8420],{"class":120},[53,65708,128],{"class":63},[53,65710,65711],{"class":278},"\"FFT grid:\"",[53,65713,65714],{"class":63},", S_fft.shape, ",[53,65716,65717],{"class":278},"\"| Fourier freq range:\"",[53,65719,65720],{"class":63},", f_fft_axis[",[53,65722,149],{"class":131},[53,65724,1655],{"class":63},[53,65726,65727],{"class":278},"\"→\"",[53,65729,65720],{"class":63},[53,65731,2262],{"class":120},[53,65733,173],{"class":131},[53,65735,1655],{"class":63},[53,65737,56974],{"class":131},[53,65739,135],{"class":63},[43,65741,65744],{"className":65742,"code":65743,"language":6934,"meta":48},[8426],"FFT grid: (2049, 401) | Fourier freq range: 0.0 → 999.9999999999999 MHz\n",[50,65745,65743],{"__ignoreMap":48},[43,65747,65749],{"className":45,"code":65748,"language":47,"meta":48,"style":48},"# @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",[50,65750,65751,65756,65762,65775,65789,65817,65837,65869,65892,65911,65925,65931],{"__ignoreMap":48},[53,65752,65753],{"class":55,"line":56},[53,65754,65755],{"class":110},"# @title Theoretical Rabi ridge\n",[53,65757,65758,65760],{"class":55,"line":73},[53,65759,60],{"class":59},[53,65761,57031],{"class":63},[53,65763,65764,65767,65769,65771,65773],{"class":55,"line":87},[53,65765,65766],{"class":63},"two_pi ",[53,65768,121],{"class":120},[53,65770,57475],{"class":131},[53,65772,8124],{"class":120},[53,65774,57480],{"class":63},[53,65776,65777,65780,65782,65785,65787],{"class":55,"line":100},[53,65778,65779],{"class":63},"f0 ",[53,65781,121],{"class":120},[53,65783,65784],{"class":131}," F0_GHZ",[53,65786,8124],{"class":120},[53,65788,57495],{"class":131},[53,65790,65791,65794,65796,65798,65800,65803,65805,65808,65811,65814],{"class":55,"line":107},[53,65792,65793],{"class":63},"delta_hz ",[53,65795,121],{"class":120},[53,65797,57534],{"class":63},[53,65799,2286],{"class":120},[53,65801,65802],{"class":63}," (freq_ghz",[53,65804,2286],{"class":120},[53,65806,65807],{"class":131},"1e9",[53,65809,65810],{"class":120}," -",[53,65812,65813],{"class":63}," f0) ",[53,65815,65816],{"class":110},"# rad\u002Fs detuning\n",[53,65818,65819,65822,65824,65826,65828,65831,65833,65835],{"class":55,"line":114},[53,65820,65821],{"class":63},"omega_onres ",[53,65823,121],{"class":120},[53,65825,57534],{"class":63},[53,65827,2286],{"class":120},[53,65829,65830],{"class":131}," OMEGA_ONRESONANCE_MHZ",[53,65832,8124],{"class":120},[53,65834,57544],{"class":131},[53,65836,57547],{"class":110},[53,65838,65839,65842,65844,65846,65848,65851,65853,65855,65857,65860,65862,65864,65867],{"class":55,"line":138},[53,65840,65841],{"class":63},"Omega_R ",[53,65843,121],{"class":120},[53,65845,1511],{"class":63},[53,65847,9714],{"class":124},[53,65849,65850],{"class":63},"(omega_onres",[53,65852,2272],{"class":120},[53,65854,132],{"class":131},[53,65856,8208],{"class":120},[53,65858,65859],{"class":63}," delta_hz",[53,65861,2272],{"class":120},[53,65863,132],{"class":131},[53,65865,65866],{"class":63},") ",[53,65868,57601],{"class":110},[53,65870,65871,65874,65876,65878,65880,65882,65884,65886,65889],{"class":55,"line":158},[53,65872,65873],{"class":63},"ridge_hz ",[53,65875,121],{"class":120},[53,65877,57700],{"class":63},[53,65879,193],{"class":120},[53,65881,58540],{"class":63},[53,65883,59438],{"class":131},[53,65885,8124],{"class":120},[53,65887,65888],{"class":63}," math.pi) ",[53,65890,65891],{"class":110},"# Hz\n",[53,65893,65894,65896,65899,65901,65903,65905,65907,65909],{"class":55,"line":182},[53,65895,1540],{"class":59},[53,65897,65898],{"class":131}," FREQ_UNITS",[53,65900,28],{"class":63},[53,65902,65560],{"class":124},[53,65904,1572],{"class":63},[53,65906,1618],{"class":120},[53,65908,65567],{"class":278},[53,65910,1623],{"class":63},[53,65912,65913,65916,65918,65921,65923],{"class":55,"line":209},[53,65914,65915],{"class":63}," ridge_axis ",[53,65917,121],{"class":120},[53,65919,65920],{"class":63}," ridge_hz ",[53,65922,2286],{"class":120},[53,65924,57524],{"class":131},[53,65926,65927,65929],{"class":55,"line":226},[53,65928,61303],{"class":59},[53,65930,1623],{"class":63},[53,65932,65933,65935,65937,65939,65941,65944],{"class":55,"line":231},[53,65934,65915],{"class":63},[53,65936,121],{"class":120},[53,65938,65920],{"class":63},[53,65940,2286],{"class":120},[53,65942,65943],{"class":131}," 1e-6",[53,65945,65946],{"class":110}," # MHz\n",[43,65948,65950],{"className":45,"code":65949,"language":47,"meta":48,"style":48},"# @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",[50,65951,65952,65957,65985,66018,66038,66081,66085,66131,66144,66165,66203,66221,66229],{"__ignoreMap":48},[53,65953,65954],{"class":55,"line":56},[53,65955,65956],{"class":110},"# @title 2D heatmap\n",[53,65958,65959,65961,65963,65965,65967,65969,65971,65973,65975,65978,65980,65983],{"class":55,"line":73},[53,65960,58116],{"class":63},[53,65962,121],{"class":120},[53,65964,58121],{"class":63},[53,65966,58124],{"class":124},[53,65968,128],{"class":63},[53,65970,58129],{"class":2050},[53,65972,121],{"class":120},[53,65974,128],{"class":63},[53,65976,65977],{"class":131},"7.5",[53,65979,170],{"class":63},[53,65981,65982],{"class":131},"5.5",[53,65984,22435],{"class":63},[53,65986,65987,65990,65992,65994,65996,65999,66001,66003,66006,66008,66011,66013,66016],{"class":55,"line":87},[53,65988,65989],{"class":63},"im ",[53,65991,121],{"class":120},[53,65993,58153],{"class":63},[53,65995,58156],{"class":124},[53,65997,65998],{"class":63},"(freq_ghz, f_fft_axis, S_fft, ",[53,66000,58162],{"class":2050},[53,66002,121],{"class":120},[53,66004,66005],{"class":278},"\"auto\"",[53,66007,170],{"class":63},[53,66009,66010],{"class":2050},"cmap",[53,66012,121],{"class":120},[53,66014,66015],{"class":278},"\"inferno\"",[53,66017,135],{"class":63},[53,66019,66020,66022,66024,66026,66028,66031,66033,66035],{"class":55,"line":100},[53,66021,58174],{"class":63},[53,66023,121],{"class":120},[53,66025,58179],{"class":63},[53,66027,58182],{"class":124},[53,66029,66030],{"class":63},"(im, ",[53,66032,58188],{"class":2050},[53,66034,121],{"class":120},[53,66036,66037],{"class":63},"ax)\n",[53,66039,66040,66042,66044,66046,66049,66051,66053,66056,66059,66062,66064,66066,66069,66071,66074,66076,66079],{"class":55,"line":107},[53,66041,63879],{"class":63},[53,66043,63882],{"class":124},[53,66045,128],{"class":63},[53,66047,66048],{"class":278},"\"FFT amplitude\"",[53,66050,8208],{"class":120},[53,66052,58540],{"class":63},[53,66054,66055],{"class":278},"\" (dB)\"",[53,66057,66058],{"class":59}," if",[53,66060,66061],{"class":131}," AMP_MODE",[53,66063,28],{"class":63},[53,66065,64756],{"class":124},[53,66067,66068],{"class":63},"()",[53,66070,1618],{"class":120},[53,66072,66073],{"class":278},"\"db\"",[53,66075,8375],{"class":59},[53,66077,66078],{"class":278}," \" (arb.)\"",[53,66080,22435],{"class":63},[53,66082,66083],{"class":55,"line":114},[53,66084,104],{"emptyLinePlaceholder":103},[53,66086,66087,66089,66091,66094,66097,66099,66102,66104,66107,66109,66111,66113,66115,66117,66120,66123,66126,66129],{"class":55,"line":138},[53,66088,58210],{"class":63},[53,66090,25274],{"class":124},[53,66092,66093],{"class":63},"(freq_ghz, ridge_axis, ",[53,66095,66096],{"class":278},"\"--\"",[53,66098,170],{"class":63},[53,66100,66101],{"class":2050},"lw",[53,66103,121],{"class":120},[53,66105,66106],{"class":131},"1.5",[53,66108,170],{"class":63},[53,66110,20907],{"class":2050},[53,66112,121],{"class":120},[53,66114,2111],{"class":59},[53,66116,61576],{"class":58202},[53,66118,66119],{"class":120},"\\O",[53,66121,66122],{"class":58202},"mega_R\u002F2",[53,66124,66125],{"class":120},"\\p",[53,66127,66128],{"class":58202},"i$\"",[53,66130,135],{"class":63},[53,66132,66133,66135,66137,66139,66142],{"class":55,"line":158},[53,66134,58210],{"class":63},[53,66136,58213],{"class":124},[53,66138,128],{"class":63},[53,66140,66141],{"class":278},"\"Drive frequency f (GHz)\"",[53,66143,135],{"class":63},[53,66145,66146,66148,66150,66152,66154,66157,66160,66163],{"class":55,"line":182},[53,66147,58210],{"class":63},[53,66149,58227],{"class":124},[53,66151,128],{"class":63},[53,66153,22173],{"class":59},[53,66155,66156],{"class":278},"\"Fourier frequency (",[53,66158,66159],{"class":131},"{FREQ_UNITS}",[53,66161,66162],{"class":278},")\"",[53,66164,135],{"class":63},[53,66166,66167,66169,66171,66173,66176,66178,66180,66183,66186,66188,66191,66193,66196,66198,66201],{"class":55,"line":209},[53,66168,58210],{"class":63},[53,66170,58241],{"class":124},[53,66172,128],{"class":63},[53,66174,66175],{"class":278},"\"Fourier transform along pulse length: $|",[53,66177,58249],{"class":120},[53,66179,2469],{"class":278},[53,66181,66182],{"class":131},"{F}",[53,66184,66185],{"class":278},"_",[53,66187,58249],{"class":120},[53,66189,66190],{"class":278},"tau",[53,66192,58249],{"class":120},[53,66194,66195],{"class":278},"{P_e",[53,66197,58249],{"class":120},[53,66199,66200],{"class":278},"}|$\"",[53,66202,135],{"class":63},[53,66204,66205,66207,66209,66211,66214,66216,66219],{"class":55,"line":226},[53,66206,58210],{"class":63},[53,66208,25453],{"class":124},[53,66210,128],{"class":63},[53,66212,66213],{"class":2050},"loc",[53,66215,121],{"class":120},[53,66217,66218],{"class":278},"\"upper right\"",[53,66220,135],{"class":63},[53,66222,66223,66225,66227],{"class":55,"line":231},[53,66224,58263],{"class":63},[53,66226,58266],{"class":124},[53,66228,329],{"class":63},[53,66230,66231,66233,66235],{"class":55,"line":245},[53,66232,25271],{"class":63},[53,66234,25508],{"class":124},[53,66236,329],{"class":63},[379,66238],{"alt":66239,"src":66240},"Output 6 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-06.webp",[43,66242,66244],{"className":45,"code":66243,"language":47,"meta":48,"style":48},"# --- 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",[50,66245,66246,66251,66261,66271,66281,66285,66290,66307,66324,66341,66358,66362,66367,66384,66400,66404,66409,66422,66426,66431,66469,66490,66494,66514,66527,66541,66545,66550,66555,66590,66619,66623,66628,66662,66672,66682,66686,66690,66716,66720,66725,66759,66784,66796,66806,66824,66842,66852,66868,66872,66876,66889,66908,66944,66957,66996,67000,67005,67032,67036,67044],{"__ignoreMap":48},[53,66247,66248],{"class":55,"line":56},[53,66249,66250],{"class":110},"# --- Adaptive high-res 3D surface ---\n",[53,66252,66253,66255,66257,66259],{"class":55,"line":73},[53,66254,76],{"class":59},[53,66256,63209],{"class":63},[53,66258,60],{"class":59},[53,66260,63214],{"class":63},[53,66262,66263,66265,66267,66269],{"class":55,"line":87},[53,66264,60],{"class":59},[53,66266,64],{"class":63},[53,66268,67],{"class":59},[53,66270,70],{"class":63},[53,66272,66273,66275,66277,66279],{"class":55,"line":100},[53,66274,60],{"class":59},[53,66276,57059],{"class":63},[53,66278,67],{"class":59},[53,66280,1054],{"class":63},[53,66282,66283],{"class":55,"line":107},[53,66284,104],{"emptyLinePlaceholder":103},[53,66286,66287],{"class":55,"line":114},[53,66288,66289],{"class":110},"# ------ Control knobs ------\n",[53,66291,66292,66295,66297,66299,66301,66304],{"class":55,"line":138},[53,66293,66294],{"class":131},"UPSAMPLE_F_FREQ",[53,66296,1298],{"class":63},[53,66298,1315],{"class":120},[53,66300,1318],{"class":120},[53,66302,66303],{"class":131}," 3",[53,66305,66306],{"class":110}," # try 2–3; upsample the drive-frequency axis only\n",[53,66308,66309,66312,66314,66316,66318,66321],{"class":55,"line":158},[53,66310,66311],{"class":131},"MAX_FACES",[53,66313,1298],{"class":63},[53,66315,1315],{"class":120},[53,66317,1318],{"class":120},[53,66319,66320],{"class":131}," 300_000",[53,66322,66323],{"class":110}," # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\n",[53,66325,66326,66329,66331,66333,66335,66338],{"class":55,"line":182},[53,66327,66328],{"class":131},"MIN_ROWS",[53,66330,1298],{"class":63},[53,66332,1315],{"class":120},[53,66334,1318],{"class":120},[53,66336,66337],{"class":131}," 128",[53,66339,66340],{"class":110}," # keep at least this many Fourier-frequency rows\n",[53,66342,66343,66346,66348,66350,66352,66355],{"class":55,"line":209},[53,66344,66345],{"class":131},"MAX_ROWS",[53,66347,1298],{"class":63},[53,66349,1315],{"class":120},[53,66351,1318],{"class":120},[53,66353,66354],{"class":131}," 1024",[53,66356,66357],{"class":110}," # hard cap for rows to keep plotting fast\n",[53,66359,66360],{"class":55,"line":226},[53,66361,104],{"emptyLinePlaceholder":103},[53,66363,66364],{"class":55,"line":231},[53,66365,66366],{"class":110},"# Preserve global color limits so the legend stays consistent\n",[53,66368,66369,66372,66374,66376,66378,66381],{"class":55,"line":245},[53,66370,66371],{"class":63},"vmin ",[53,66373,121],{"class":120},[53,66375,59516],{"class":120},[53,66377,59813],{"class":63},[53,66379,66380],{"class":124},"nanmin",[53,66382,66383],{"class":63},"(S_fft))\n",[53,66385,66386,66389,66391,66393,66395,66398],{"class":55,"line":258},[53,66387,66388],{"class":63},"vmax ",[53,66390,121],{"class":120},[53,66392,59516],{"class":120},[53,66394,59813],{"class":63},[53,66396,66397],{"class":124},"nanmax",[53,66399,66383],{"class":63},[53,66401,66402],{"class":55,"line":263},[53,66403,104],{"emptyLinePlaceholder":103},[53,66405,66406],{"class":55,"line":269},[53,66407,66408],{"class":110},"# Shapes\n",[53,66410,66411,66414,66416,66419],{"class":55,"line":284},[53,66412,66413],{"class":63},"n_rows, n_cols ",[53,66415,121],{"class":120},[53,66417,66418],{"class":63}," S_fft.shape ",[53,66420,66421],{"class":110},"# rows=Fourier freq (ν), cols=drive freq (f)\n",[53,66423,66424],{"class":55,"line":296},[53,66425,104],{"emptyLinePlaceholder":103},[53,66427,66428],{"class":55,"line":309},[53,66429,66430],{"class":110},"# 1) Upsample along the typically smaller axis (drive frequency) only\n",[53,66432,66433,66436,66438,66440,66442,66444,66446,66449,66451,66453,66455,66457,66459,66461,66464,66466],{"class":55,"line":323},[53,66434,66435],{"class":63},"n_cols_up ",[53,66437,121],{"class":120},[53,66439,2371],{"class":120},[53,66441,128],{"class":63},[53,66443,66294],{"class":131},[53,66445,8124],{"class":120},[53,66447,66448],{"class":63}," (n_cols ",[53,66450,2262],{"class":120},[53,66452,1321],{"class":131},[53,66454,203],{"class":63},[53,66456,2002],{"class":120},[53,66458,1321],{"class":131},[53,66460,170],{"class":63},[53,66462,66463],{"class":131},"1200",[53,66465,22613],{"class":63},[53,66467,66468],{"class":110},"# sensible upper bound\n",[53,66470,66471,66473,66475,66477,66479,66481,66483,66485,66487],{"class":55,"line":332},[53,66472,63316],{"class":63},[53,66474,121],{"class":120},[53,66476,1511],{"class":63},[53,66478,57842],{"class":124},[53,66480,63325],{"class":63},[53,66482,25125],{"class":124},[53,66484,63330],{"class":63},[53,66486,36608],{"class":124},[53,66488,66489],{"class":63},"(), n_cols_up)\n",[53,66491,66492],{"class":55,"line":344},[53,66493,104],{"emptyLinePlaceholder":103},[53,66495,66496,66499,66501,66503,66505,66508,66510,66512],{"class":55,"line":357},[53,66497,66498],{"class":63},"S_up_f ",[53,66500,121],{"class":120},[53,66502,1511],{"class":63},[53,66504,63373],{"class":124},[53,66506,66507],{"class":63},"((n_rows, n_cols_up), ",[53,66509,2051],{"class":2050},[53,66511,63391],{"class":120},[53,66513,135],{"class":63},[53,66515,66516,66518,66520,66522,66524],{"class":55,"line":1247},[53,66517,1525],{"class":59},[53,66519,8141],{"class":63},[53,66521,1531],{"class":59},[53,66523,8116],{"class":120},[53,66525,66526],{"class":63},"(n_rows):\n",[53,66528,66529,66532,66534,66536,66538],{"class":55,"line":1252},[53,66530,66531],{"class":63}," S_up_f[i, :] ",[53,66533,121],{"class":120},[53,66535,1511],{"class":63},[53,66537,63422],{"class":124},[53,66539,66540],{"class":63},"(freq_hi, freq_ghz, S_fft[i, :])\n",[53,66542,66543],{"class":55,"line":1257},[53,66544,104],{"emptyLinePlaceholder":103},[53,66546,66547],{"class":55,"line":1262},[53,66548,66549],{"class":110},"# 2) Determine how many Fourier rows we can afford for MAX_FACES\n",[53,66551,66552],{"class":55,"line":1274},[53,66553,66554],{"class":110},"# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\n",[53,66556,66557,66560,66562,66564,66566,66568,66570,66572,66575,66577,66579,66581,66583,66586,66588],{"class":55,"line":1284},[53,66558,66559],{"class":63},"n_rows_cap ",[53,66561,121],{"class":120},[53,66563,1990],{"class":120},[53,66565,128],{"class":63},[53,66567,66311],{"class":131},[53,66569,57776],{"class":120},[53,66571,65137],{"class":120},[53,66573,66574],{"class":63},"(n_cols_up ",[53,66576,2262],{"class":120},[53,66578,1321],{"class":131},[53,66580,170],{"class":63},[53,66582,173],{"class":131},[53,66584,66585],{"class":63},")) ",[53,66587,2002],{"class":120},[53,66589,1979],{"class":131},[53,66591,66592,66595,66597,66599,66601,66603,66606,66608,66610,66612,66614,66616],{"class":55,"line":1292},[53,66593,66594],{"class":63},"n_rows_plot ",[53,66596,121],{"class":120},[53,66598,1990],{"class":120},[53,66600,59813],{"class":63},[53,66602,57813],{"class":124},[53,66604,66605],{"class":63},"(n_rows_cap, ",[53,66607,66328],{"class":131},[53,66609,170],{"class":63},[53,66611,25125],{"class":120},[53,66613,128],{"class":63},[53,66615,66345],{"class":131},[53,66617,66618],{"class":63},", n_rows)))\n",[53,66620,66621],{"class":55,"line":1307},[53,66622,104],{"emptyLinePlaceholder":103},[53,66624,66625],{"class":55,"line":1326},[53,66626,66627],{"class":110},"# Down-sample ν-axis evenly to n_rows_plot\n",[53,66629,66630,66633,66635,66637,66639,66641,66643,66646,66648,66650,66653,66656,66658,66660],{"class":55,"line":1344},[53,66631,66632],{"class":63},"row_idx ",[53,66634,121],{"class":120},[53,66636,1511],{"class":63},[53,66638,57842],{"class":124},[53,66640,128],{"class":63},[53,66642,149],{"class":131},[53,66644,66645],{"class":63},", n_rows ",[53,66647,2262],{"class":120},[53,66649,1321],{"class":131},[53,66651,66652],{"class":63},", n_rows_plot).",[53,66654,66655],{"class":124},"astype",[53,66657,128],{"class":63},[53,66659,1315],{"class":120},[53,66661,135],{"class":63},[53,66663,66664,66667,66669],{"class":55,"line":1360},[53,66665,66666],{"class":63},"f_fft_plot ",[53,66668,121],{"class":120},[53,66670,66671],{"class":63}," f_fft_axis[row_idx]\n",[53,66673,66674,66677,66679],{"class":55,"line":1366},[53,66675,66676],{"class":63},"S_final ",[53,66678,121],{"class":120},[53,66680,66681],{"class":63}," S_up_f[row_idx, :]\n",[53,66683,66684],{"class":55,"line":1372},[53,66685,104],{"emptyLinePlaceholder":103},[53,66687,66688],{"class":55,"line":1377},[53,66689,63551],{"class":110},[53,66691,66692,66695,66698,66700,66702,66704,66707,66709,66711,66714],{"class":55,"line":1383},[53,66693,66694],{"class":63},"F2_hi, ",[53,66696,66697],{"class":131},"FFOUR_hi",[53,66699,1318],{"class":120},[53,66701,1511],{"class":63},[53,66703,57566],{"class":124},[53,66705,66706],{"class":63},"(freq_hi, f_fft_plot, ",[53,66708,57572],{"class":2050},[53,66710,121],{"class":120},[53,66712,66713],{"class":278},"\"xy\"",[53,66715,135],{"class":63},[53,66717,66718],{"class":55,"line":1389},[53,66719,104],{"emptyLinePlaceholder":103},[53,66721,66722],{"class":55,"line":1395},[53,66723,66724],{"class":110},"# 3) Plot (keeps 'inferno')\n",[53,66726,66727,66729,66731,66733,66735,66737,66739,66741,66743,66745,66747,66749,66751,66753,66755,66757],{"class":55,"line":1401},[53,66728,63587],{"class":63},[53,66730,121],{"class":120},[53,66732,58121],{"class":63},[53,66734,61512],{"class":124},[53,66736,128],{"class":63},[53,66738,58129],{"class":2050},[53,66740,121],{"class":120},[53,66742,128],{"class":63},[53,66744,35636],{"class":131},[53,66746,170],{"class":63},[53,66748,63608],{"class":131},[53,66750,2108],{"class":63},[53,66752,63613],{"class":2050},[53,66754,121],{"class":120},[53,66756,63618],{"class":131},[53,66758,135],{"class":63},[53,66760,66761,66763,66765,66767,66769,66771,66773,66775,66777,66779,66782],{"class":55,"line":1407},[53,66762,63625],{"class":63},[53,66764,121],{"class":120},[53,66766,58179],{"class":63},[53,66768,63632],{"class":124},[53,66770,128],{"class":63},[53,66772,63637],{"class":131},[53,66774,170],{"class":63},[53,66776,63642],{"class":2050},[53,66778,121],{"class":120},[53,66780,66781],{"class":278},"\"3d\"",[53,66783,135],{"class":63},[53,66785,66786,66788,66790,66792,66794],{"class":55,"line":1413},[53,66787,63658],{"class":63},[53,66789,121],{"class":120},[53,66791,58153],{"class":63},[53,66793,63665],{"class":124},[53,66795,1271],{"class":63},[53,66797,66798,66801,66803],{"class":55,"line":1419},[53,66799,66800],{"class":63}," F2_hi, ",[53,66802,66697],{"class":131},[53,66804,66805],{"class":63},", S_final,\n",[53,66807,66808,66810,66812,66814,66816,66818,66820,66822],{"class":55,"line":1425},[53,66809,63677],{"class":2050},[53,66811,121],{"class":120},[53,66813,173],{"class":131},[53,66815,170],{"class":63},[53,66817,63686],{"class":2050},[53,66819,121],{"class":120},[53,66821,173],{"class":131},[53,66823,1304],{"class":63},[53,66825,66826,66828,66830,66832,66834,66836,66838,66840],{"class":55,"line":1431},[53,66827,63701],{"class":2050},[53,66829,121],{"class":120},[53,66831,149],{"class":131},[53,66833,170],{"class":63},[53,66835,63710],{"class":2050},[53,66837,121],{"class":120},[53,66839,13773],{"class":131},[53,66841,1304],{"class":63},[53,66843,66844,66846,66848,66850],{"class":55,"line":1437},[53,66845,63721],{"class":2050},[53,66847,121],{"class":120},[53,66849,66015],{"class":278},[53,66851,1304],{"class":63},[53,66853,66854,66856,66858,66861,66863,66865],{"class":55,"line":1443},[53,66855,63732],{"class":2050},[53,66857,121],{"class":120},[53,66859,66860],{"class":63},"vmin, ",[53,66862,63741],{"class":2050},[53,66864,121],{"class":120},[53,66866,66867],{"class":63},"vmax\n",[53,66869,66870],{"class":55,"line":1448},[53,66871,135],{"class":63},[53,66873,66874],{"class":55,"line":1454},[53,66875,104],{"emptyLinePlaceholder":103},[53,66877,66878,66880,66882,66884,66887],{"class":55,"line":1460},[53,66879,58210],{"class":63},[53,66881,58213],{"class":124},[53,66883,128],{"class":63},[53,66885,66886],{"class":278},"\"f (GHz)\"",[53,66888,135],{"class":63},[53,66890,66891,66893,66895,66897,66899,66902,66904,66906],{"class":55,"line":1465},[53,66892,58210],{"class":63},[53,66894,58227],{"class":124},[53,66896,128],{"class":63},[53,66898,22173],{"class":59},[53,66900,66901],{"class":278},"\"Fourier freq (",[53,66903,66159],{"class":131},[53,66905,66162],{"class":278},[53,66907,135],{"class":63},[53,66909,66910,66912,66914,66916,66918,66920,66922,66924,66926,66928,66930,66932,66934,66936,66938,66940,66942],{"class":55,"line":1470},[53,66911,58210],{"class":63},[53,66913,63805],{"class":124},[53,66915,128],{"class":63},[53,66917,66048],{"class":278},[53,66919,8208],{"class":120},[53,66921,58540],{"class":63},[53,66923,66055],{"class":278},[53,66925,66058],{"class":59},[53,66927,66061],{"class":131},[53,66929,28],{"class":63},[53,66931,64756],{"class":124},[53,66933,1572],{"class":63},[53,66935,1618],{"class":120},[53,66937,65508],{"class":278},[53,66939,8375],{"class":59},[53,66941,66078],{"class":278},[53,66943,22435],{"class":63},[53,66945,66946,66948,66950,66952,66955],{"class":55,"line":1475},[53,66947,58210],{"class":63},[53,66949,58241],{"class":124},[53,66951,128],{"class":63},[53,66953,66954],{"class":278},"\"Rabi FFT surface (adaptive resolution)\"",[53,66956,135],{"class":63},[53,66958,66959,66961,66963,66965,66967,66969,66971,66973,66975,66977,66979,66981,66983,66985,66987,66989,66991,66994],{"class":55,"line":1492},[53,66960,58263],{"class":63},[53,66962,58182],{"class":124},[53,66964,63848],{"class":63},[53,66966,58188],{"class":2050},[53,66968,121],{"class":120},[53,66970,58193],{"class":63},[53,66972,63857],{"class":2050},[53,66974,121],{"class":120},[53,66976,63862],{"class":131},[53,66978,170],{"class":63},[53,66980,63867],{"class":2050},[53,66982,121],{"class":120},[53,66984,63872],{"class":131},[53,66986,170],{"class":63},[53,66988,20907],{"class":2050},[53,66990,121],{"class":120},[53,66992,66993],{"class":278},"\"Amplitude\"",[53,66995,135],{"class":63},[53,66997,66998],{"class":55,"line":1503},[53,66999,104],{"emptyLinePlaceholder":103},[53,67001,67002],{"class":55,"line":1519},[53,67003,67004],{"class":110},"# A clearer default view\n",[53,67006,67007,67009,67012,67014,67017,67019,67021,67023,67026,67028,67030],{"class":55,"line":1552},[53,67008,58210],{"class":63},[53,67010,67011],{"class":124},"view_init",[53,67013,128],{"class":63},[53,67015,67016],{"class":2050},"elev",[53,67018,121],{"class":120},[53,67020,48484],{"class":131},[53,67022,170],{"class":63},[53,67024,67025],{"class":2050},"azim",[53,67027,14010],{"class":120},[53,67029,58573],{"class":131},[53,67031,135],{"class":63},[53,67033,67034],{"class":55,"line":1558},[53,67035,104],{"emptyLinePlaceholder":103},[53,67037,67038,67040,67042],{"class":55,"line":1586},[53,67039,58263],{"class":63},[53,67041,58266],{"class":124},[53,67043,329],{"class":63},[53,67045,67046,67048,67050],{"class":55,"line":1609},[53,67047,25271],{"class":63},[53,67049,25508],{"class":124},[53,67051,329],{"class":63},[379,67053],{"alt":67054,"src":67055},"Output 7 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-07.webp",[43,67057,67059],{"className":45,"code":67058,"language":47,"meta":48,"style":48},"# @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",[50,67060,67061,67066,67089,67121,67125,67130,67154,67182,67199,67244,67262,67298,67325,67333,67346,67350,67355,67381,67407,67424,67436,67472,67501],{"__ignoreMap":48},[53,67062,67063],{"class":55,"line":56},[53,67064,67065],{"class":110},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\n",[53,67067,67068,67071,67073,67075,67077,67079,67081,67083,67086],{"class":55,"line":73},[53,67069,67070],{"class":131},"F_STAR_GHZ",[53,67072,1298],{"class":63},[53,67074,1301],{"class":120},[53,67076,1318],{"class":120},[53,67078,59516],{"class":120},[53,67080,128],{"class":63},[53,67082,56423],{"class":131},[53,67084,67085],{"class":63},") ",[53,67087,67088],{"class":110},"# slice near resonance\n",[53,67090,67091,67094,67096,67098,67101,67103,67105,67107,67110,67112,67115,67118],{"class":55,"line":87},[53,67092,67093],{"class":131},"NU_STAR",[53,67095,1318],{"class":120},[53,67097,59516],{"class":120},[53,67099,67100],{"class":63},"(ridge_axis[np.",[53,67102,64150],{"class":124},[53,67104,59813],{"class":63},[53,67106,64155],{"class":124},[53,67108,67109],{"class":63},"(freq_ghz ",[53,67111,2262],{"class":120},[53,67113,67114],{"class":131}," F_STAR_GHZ",[53,67116,67117],{"class":63},"))]) ",[53,67119,67120],{"class":110},"# near Ω_R\u002F2π\n",[53,67122,67123],{"class":55,"line":100},[53,67124,104],{"emptyLinePlaceholder":103},[53,67126,67127],{"class":55,"line":107},[53,67128,67129],{"class":110},"# Slice at f = F_STAR_GHZ\n",[53,67131,67132,67134,67136,67138,67140,67142,67144,67146,67148,67150,67152],{"class":55,"line":114},[53,67133,64177],{"class":63},[53,67135,121],{"class":120},[53,67137,1990],{"class":120},[53,67139,59813],{"class":63},[53,67141,64150],{"class":124},[53,67143,59813],{"class":63},[53,67145,64155],{"class":124},[53,67147,67109],{"class":63},[53,67149,2262],{"class":120},[53,67151,67114],{"class":131},[53,67153,13961],{"class":63},[53,67155,67156,67158,67160,67162,67164,67166,67168,67170,67172,67175,67177,67180],{"class":55,"line":138},[53,67157,58116],{"class":63},[53,67159,121],{"class":120},[53,67161,58121],{"class":63},[53,67163,58124],{"class":124},[53,67165,128],{"class":63},[53,67167,58129],{"class":2050},[53,67169,121],{"class":120},[53,67171,128],{"class":63},[53,67173,67174],{"class":131},"6.0",[53,67176,170],{"class":63},[53,67178,67179],{"class":131},"3.8",[53,67181,22435],{"class":63},[53,67183,67184,67186,67188,67191,67193,67195,67197],{"class":55,"line":158},[53,67185,58210],{"class":63},[53,67187,25274],{"class":124},[53,67189,67190],{"class":63},"(f_fft_axis, S_fft[:, j], ",[53,67192,64246],{"class":2050},[53,67194,121],{"class":120},[53,67196,64251],{"class":278},[53,67198,135],{"class":63},[53,67200,67201,67203,67206,67209,67212,67214,67216,67218,67220,67222,67224,67226,67228,67230,67232,67234,67236,67238,67240,67242],{"class":55,"line":182},[53,67202,58210],{"class":63},[53,67204,67205],{"class":124},"axvline",[53,67207,67208],{"class":63},"(ridge_axis[j], ",[53,67210,67211],{"class":2050},"ls",[53,67213,121],{"class":120},[53,67215,66096],{"class":278},[53,67217,170],{"class":63},[53,67219,66101],{"class":2050},[53,67221,121],{"class":120},[53,67223,57823],{"class":131},[53,67225,170],{"class":63},[53,67227,20907],{"class":2050},[53,67229,121],{"class":120},[53,67231,2111],{"class":59},[53,67233,61576],{"class":58202},[53,67235,66119],{"class":120},[53,67237,66122],{"class":58202},[53,67239,66125],{"class":120},[53,67241,66128],{"class":58202},[53,67243,135],{"class":63},[53,67245,67246,67248,67250,67252,67254,67256,67258,67260],{"class":55,"line":209},[53,67247,58210],{"class":63},[53,67249,58213],{"class":124},[53,67251,128],{"class":63},[53,67253,22173],{"class":59},[53,67255,66156],{"class":278},[53,67257,66159],{"class":131},[53,67259,66162],{"class":278},[53,67261,135],{"class":63},[53,67263,67264,67266,67268,67270,67272,67274,67276,67278,67280,67282,67284,67286,67288,67290,67292,67294,67296],{"class":55,"line":226},[53,67265,58210],{"class":63},[53,67267,58227],{"class":124},[53,67269,128],{"class":63},[53,67271,66048],{"class":278},[53,67273,8208],{"class":120},[53,67275,58540],{"class":63},[53,67277,66055],{"class":278},[53,67279,66058],{"class":59},[53,67281,66061],{"class":131},[53,67283,28],{"class":63},[53,67285,64756],{"class":124},[53,67287,66068],{"class":63},[53,67289,1618],{"class":120},[53,67291,66073],{"class":278},[53,67293,8375],{"class":59},[53,67295,66078],{"class":278},[53,67297,22435],{"class":63},[53,67299,67300,67302,67304,67306,67309,67312,67314,67316,67318,67320,67323],{"class":55,"line":231},[53,67301,58210],{"class":63},[53,67303,58241],{"class":124},[53,67305,128],{"class":63},[53,67307,67308],{"class":59},"fr",[53,67310,67311],{"class":278},"\"Slice at $f=",[53,67313,2705],{"class":131},[53,67315,64335],{"class":63},[53,67317,62774],{"class":59},[53,67319,2767],{"class":131},[53,67321,67322],{"class":278},"$ GHz\"",[53,67324,135],{"class":63},[53,67326,67327,67329,67331],{"class":55,"line":245},[53,67328,58210],{"class":63},[53,67330,25453],{"class":124},[53,67332,329],{"class":63},[53,67334,67335,67337,67339,67342,67344],{"class":55,"line":258},[53,67336,58263],{"class":63},[53,67338,58266],{"class":124},[53,67340,67341],{"class":63},"(); plt.",[53,67343,25508],{"class":124},[53,67345,329],{"class":63},[53,67347,67348],{"class":55,"line":263},[53,67349,104],{"emptyLinePlaceholder":103},[53,67351,67352],{"class":55,"line":269},[53,67353,67354],{"class":110},"# Slice at ν = NU_STAR\n",[53,67356,67357,67359,67361,67363,67365,67367,67369,67371,67374,67376,67379],{"class":55,"line":284},[53,67358,64391],{"class":63},[53,67360,121],{"class":120},[53,67362,1990],{"class":120},[53,67364,59813],{"class":63},[53,67366,64150],{"class":124},[53,67368,59813],{"class":63},[53,67370,64155],{"class":124},[53,67372,67373],{"class":63},"(f_fft_axis ",[53,67375,2262],{"class":120},[53,67377,67378],{"class":131}," NU_STAR",[53,67380,13961],{"class":63},[53,67382,67383,67385,67387,67389,67391,67393,67395,67397,67399,67401,67403,67405],{"class":55,"line":296},[53,67384,58116],{"class":63},[53,67386,121],{"class":120},[53,67388,58121],{"class":63},[53,67390,58124],{"class":124},[53,67392,128],{"class":63},[53,67394,58129],{"class":2050},[53,67396,121],{"class":120},[53,67398,128],{"class":63},[53,67400,67174],{"class":131},[53,67402,170],{"class":63},[53,67404,67179],{"class":131},[53,67406,22435],{"class":63},[53,67408,67409,67411,67413,67416,67418,67420,67422],{"class":55,"line":309},[53,67410,58210],{"class":63},[53,67412,25274],{"class":124},[53,67414,67415],{"class":63},"(freq_ghz, S_fft[i, :], ",[53,67417,64246],{"class":2050},[53,67419,121],{"class":120},[53,67421,64251],{"class":278},[53,67423,135],{"class":63},[53,67425,67426,67428,67430,67432,67434],{"class":55,"line":323},[53,67427,58210],{"class":63},[53,67429,58213],{"class":124},[53,67431,128],{"class":63},[53,67433,66141],{"class":278},[53,67435,135],{"class":63},[53,67437,67438,67440,67442,67444,67446,67448,67450,67452,67454,67456,67458,67460,67462,67464,67466,67468,67470],{"class":55,"line":332},[53,67439,58210],{"class":63},[53,67441,58227],{"class":124},[53,67443,128],{"class":63},[53,67445,66048],{"class":278},[53,67447,8208],{"class":120},[53,67449,58540],{"class":63},[53,67451,66055],{"class":278},[53,67453,66058],{"class":59},[53,67455,66061],{"class":131},[53,67457,28],{"class":63},[53,67459,64756],{"class":124},[53,67461,66068],{"class":63},[53,67463,1618],{"class":120},[53,67465,66073],{"class":278},[53,67467,8375],{"class":59},[53,67469,66078],{"class":278},[53,67471,22435],{"class":63},[53,67473,67474,67476,67478,67480,67482,67485,67487,67490,67492,67494,67497,67499],{"class":55,"line":344},[53,67475,58210],{"class":63},[53,67477,58241],{"class":124},[53,67479,128],{"class":63},[53,67481,67308],{"class":59},[53,67483,67484],{"class":278},"\"Slice at Fourier freq ≈ ",[53,67486,2705],{"class":131},[53,67488,67489],{"class":63},"f_fft_axis[i]",[53,67491,62849],{"class":59},[53,67493,2767],{"class":131},[53,67495,67496],{"class":131}," {FREQ_UNITS}",[53,67498,22176],{"class":278},[53,67500,135],{"class":63},[53,67502,67503,67505,67507,67509,67511],{"class":55,"line":357},[53,67504,58263],{"class":63},[53,67506,58266],{"class":124},[53,67508,67341],{"class":63},[53,67510,25508],{"class":124},[53,67512,329],{"class":63},[379,67514],{"alt":67515,"src":67516},"Output 8 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-08.webp",[379,67518],{"alt":67519,"src":67520},"Output 9 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-09.webp",[30,67522,488],{"id":487},[12,67524,41494,67525,495,67527,28],{},[19,67526,41566],{"href":41565},[19,67528,67530],{"href":67529},"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",[501,67532,67533],{},"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":48,"searchDepth":73,"depth":73,"links":67535},[67536,67537,67538,67539,67540,67541,67542,67543,67544,67545,67546,67547,67548,67549,67550],{"id":41742,"depth":73,"text":41743},{"id":43222,"depth":73,"text":43223},{"id":44011,"depth":73,"text":44012},{"id":45360,"depth":73,"text":45361},{"id":46489,"depth":73,"text":46490},{"id":48971,"depth":73,"text":48972},{"id":49487,"depth":73,"text":49488},{"id":50586,"depth":73,"text":50587},{"id":52178,"depth":73,"text":52179},{"id":52551,"depth":73,"text":52552},{"id":53543,"depth":73,"text":53544},{"id":54720,"depth":73,"text":54721},{"id":55900,"depth":73,"text":55901},{"id":56395,"depth":73,"text":56396},{"id":487,"depth":73,"text":488},[517,518,519,67552],"Rabi oscillations",[67554],{"username":7,"name":523,"role":524,"bio":525,"links":67555},[67556,67557],{"label":528,"href":529},{"label":531,"href":41565},{"username":7,"name":523,"role":533},"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":41555,"description":67559},"blog\u002Fexpert-notes\u002Frabi-oscillation-visualization",[67568,571],"hardware","cLyZBkTx0Kr_9zY5k_7c4mZTRRqgWxllLNGfFbxk58w",{"id":67571,"title":67572,"authors":67573,"body":67574,"breadcrumb":72161,"builders":72163,"byline":72168,"challenge":534,"courseAuthor":534,"courseLead":534,"dek":534,"description":72169,"draft":537,"extension":538,"eyebrow":67560,"finish":534,"fork":534,"hero":534,"heroAlt":534,"heroCta":534,"heroImage":534,"kind":540,"lessonCount":534,"meta":72170,"navigation":103,"newsItems":534,"next":534,"ogImage":72171,"order":534,"outcomes":534,"path":72172,"publishDate":544,"readingTime":72173,"related":72174,"relatedProjects":534,"seo":72175,"stem":72176,"tags":72177,"track":534,"trackName":534,"__hash__":72178},"blog\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications.md","A Sampling-Budget & Bit-String Analogy for Qubit Classifications",[7],{"type":9,"value":67575,"toc":72157},[67576,67583,71998,72004,72007,72010,72016,72022,72025,72027,72145,72147,72154],[12,67577,67578],{},[924,67579,926,67580,41567],{},[19,67581,41566],{"href":67582},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[43,67584,67586],{"className":45,"code":67585,"language":47,"meta":48,"style":48},"# 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",[50,67587,67588,67593,67598,67603,67608,67613,67618,67623,67628,67633,67638,67642,67649,67655,67667,67678,67682,67692,67702,67712,67724,67736,67740,67753,67757,67762,67766,67813,67817,67822,67827,67831,67844,67858,67872,67897,67933,67999,68032,68036,68040,68087,68091,68096,68101,68105,68129,68150,68163,68169,68199,68221,68225,68229,68234,68238,68254,68265,68273,68283,68290,68294,68298,68310,68317,68324,68331,68336,68340,68346,68353,68360,68365,68369,68375,68382,68389,68394,68398,68404,68411,68418,68423,68427,68433,68440,68447,68452,68456,68460,68464,68469,68473,68485,68497,68509,68535,68542,68554,68564,68568,68575,68585,68589,68593,68597,68622,68626,68630,68635,68639,68671,68675,68680,68684,68717,68721,68725,68756,68760,68765,68769,68780,68797,68814,68847,68859,68863,68872,68880,68888,68896,68904,68912,68920,68928,68942,68955,68968,68972,68984,69021,69028,69032,69036,69075,69079,69084,69088,69105,69109,69113,69118,69122,69136,69145,69149,69160,69174,69196,69206,69231,69235,69249,69253,69258,69280,69300,69322,69336,69354,69371,69375,69379,69384,69388,69393,69409,69446,69477,69481,69491,69509,69513,69518,69543,69567,69571,69582,69586,69591,69595,69600,69615,69637,69656,69676,69680,69685,69690,69695,69722,69748,69772,69777,69781,69810,69864,69901,69914,69937,69945,69955,69983,69991,69995,70000,70022,70032,70052,70076,70080,70102,70127,70184,70188,70227,70240,70261,70275,70292,70300,70310,70335,70343,70347,70358,70362,70367,70371,70388,70401,70421,70441,70464,70486,70505,70522,70527,70531,70535,70540,70544,70557,70562,70566,70597,70601,70682,70686,70690,70701,70706,70711,70716,70721,70725,70733,70740,70751,70758,70765,70769,70780,70785,70790,70795,70800,70804,70818,70823,70828,70833,70838,70843,70847,70851,70863,70867,70871,70882,70887,70892,70897,70902,70906,70911,70916,70920,70940,70952,70962,70972,70992,71052,71083,71087,71102,71118,71169,71173,71188,71199,71210,71222,71227,71242,71257,71262,71266,71271,71289,71294,71298,71303,71311,71323,71328,71352,71356,71371,71398,71402,71407,71418,71445,71449,71454,71467,71475,71486,71494,71498,71516,71538,71550,71554,71559,71564,71580,71588,71602,71610,71636,71640,71657,71678,71689,71701,71712,71724,71728,71735,71742,71746,71750,71765,71779,71790,71800,71809,71813,71817,71827,71854,71865,71869,71874,71897,71908,71923,71945,71967],{"__ignoreMap":48},[53,67589,67590],{"class":55,"line":56},[53,67591,67592],{"class":110},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n",[53,67594,67595],{"class":55,"line":73},[53,67596,67597],{"class":110},"# generates tables and figures, so you can iterate further.\n",[53,67599,67600],{"class":55,"line":87},[53,67601,67602],{"class":110},"#\n",[53,67604,67605],{"class":55,"line":100},[53,67606,67607],{"class":110},"# Contents created in the current directory:\n",[53,67609,67610],{"class":55,"line":107},[53,67611,67612],{"class":110},"# - Sampling_Budget_Qubit_Bitstrings.ipynb (Colab notebook)\n",[53,67614,67615],{"class":55,"line":114},[53,67616,67617],{"class":110},"# - summary.csv (summary table)\n",[53,67619,67620],{"class":55,"line":138},[53,67621,67622],{"class":110},"# - bitstrings_\u003CMODEL>.csv (raw bit-strings per model)\n",[53,67624,67625],{"class":55,"line":158},[53,67626,67627],{"class":110},"# - fig_ones_fraction.png (bar chart with Wilson CI)\n",[53,67629,67630],{"class":55,"line":182},[53,67631,67632],{"class":110},"# - fig_required_shots.png (required shots for ±ε)\n",[53,67634,67635],{"class":55,"line":209},[53,67636,67637],{"class":110},"# - simulation_config.json (run parameters)\n",[53,67639,67640],{"class":55,"line":226},[53,67641,104],{"emptyLinePlaceholder":103},[53,67643,67644,67646],{"class":55,"line":231},[53,67645,60],{"class":59},[53,67647,67648],{"class":63}," json\n",[53,67650,67651,67653],{"class":55,"line":245},[53,67652,60],{"class":59},[53,67654,57031],{"class":63},[53,67656,67657,67659,67662,67664],{"class":55,"line":258},[53,67658,76],{"class":59},[53,67660,67661],{"class":63}," dataclasses ",[53,67663,60],{"class":59},[53,67665,67666],{"class":63}," dataclass\n",[53,67668,67669,67671,67673,67675],{"class":55,"line":263},[53,67670,76],{"class":59},[53,67672,1061],{"class":63},[53,67674,60],{"class":59},[53,67676,67677],{"class":63}," List, Dict, Tuple\n",[53,67679,67680],{"class":55,"line":269},[53,67681,104],{"emptyLinePlaceholder":103},[53,67683,67684,67686,67688,67690],{"class":55,"line":284},[53,67685,60],{"class":59},[53,67687,64],{"class":63},[53,67689,67],{"class":59},[53,67691,70],{"class":63},[53,67693,67694,67696,67698,67700],{"class":55,"line":296},[53,67695,60],{"class":59},[53,67697,57070],{"class":63},[53,67699,67],{"class":59},[53,67701,57075],{"class":63},[53,67703,67704,67706,67708,67710],{"class":55,"line":309},[53,67705,60],{"class":59},[53,67707,57059],{"class":63},[53,67709,67],{"class":59},[53,67711,1054],{"class":63},[53,67713,67714,67716,67719,67721],{"class":55,"line":323},[53,67715,76],{"class":59},[53,67717,67718],{"class":63}," IPython.display ",[53,67720,60],{"class":59},[53,67722,67723],{"class":63}," display, Markdown\n",[53,67725,67726,67728,67731,67733],{"class":55,"line":332},[53,67727,60],{"class":59},[53,67729,67730],{"class":63}," nbformat ",[53,67732,67],{"class":59},[53,67734,67735],{"class":63}," nbf\n",[53,67737,67738],{"class":55,"line":344},[53,67739,104],{"emptyLinePlaceholder":103},[53,67741,67742,67744,67746,67748,67750],{"class":55,"line":357},[53,67743,57138],{"class":63},[53,67745,57141],{"class":278},[53,67747,8101],{"class":63},[53,67749,121],{"class":120},[53,67751,67752],{"class":131}," 200\n",[53,67754,67755],{"class":55,"line":1247},[53,67756,104],{"emptyLinePlaceholder":103},[53,67758,67759],{"class":55,"line":1252},[53,67760,67761],{"class":110},"# ---------- Helper math ----------\n",[53,67763,67764],{"class":55,"line":1257},[53,67765,104],{"emptyLinePlaceholder":103},[53,67767,67768,67770,67773,67775,67777,67779,67781,67783,67785,67787,67789,67791,67793,67795,67797,67799,67802,67805,67807,67809,67811],{"class":55,"line":1262},[53,67769,1265],{"class":59},[53,67771,67772],{"class":124}," wilson_ci",[53,67774,128],{"class":63},[53,67776,3521],{"class":1277},[53,67778,1298],{"class":63},[53,67780,1315],{"class":120},[53,67782,170],{"class":63},[53,67784,4315],{"class":1277},[53,67786,1298],{"class":63},[53,67788,1315],{"class":120},[53,67790,170],{"class":63},[53,67792,214],{"class":1277},[53,67794,1298],{"class":63},[53,67796,1301],{"class":120},[53,67798,1318],{"class":120},[53,67800,67801],{"class":131}," 1.96",[53,67803,67804],{"class":63},") -> Tuple[",[53,67806,1301],{"class":120},[53,67808,170],{"class":63},[53,67810,1301],{"class":120},[53,67812,59519],{"class":63},[53,67814,67815],{"class":55,"line":1274},[53,67816,1363],{"class":278},[53,67818,67819],{"class":55,"line":1284},[53,67820,67821],{"class":278}," Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n",[53,67823,67824],{"class":55,"line":1292},[53,67825,67826],{"class":278}," Returns (low, high). Works for edge cases k=0 or k=n.\n",[53,67828,67829],{"class":55,"line":1307},[53,67830,1363],{"class":278},[53,67832,67833,67835,67838,67840,67842],{"class":55,"line":1326},[53,67834,1612],{"class":59},[53,67836,67837],{"class":63}," n ",[53,67839,1618],{"class":120},[53,67841,21153],{"class":131},[53,67843,1623],{"class":63},[53,67845,67846,67848,67850,67852,67854,67856],{"class":55,"line":1344},[53,67847,64769],{"class":59},[53,67849,58540],{"class":63},[53,67851,56610],{"class":131},[53,67853,170],{"class":63},[53,67855,57823],{"class":131},[53,67857,135],{"class":63},[53,67859,67860,67863,67865,67867,67869],{"class":55,"line":1360},[53,67861,67862],{"class":63}," p_hat ",[53,67864,121],{"class":120},[53,67866,1615],{"class":63},[53,67868,193],{"class":120},[53,67870,67871],{"class":63}," n\n",[53,67873,67874,67877,67879,67882,67884,67887,67889,67891,67893,67895],{"class":55,"line":1366},[53,67875,67876],{"class":63}," denom ",[53,67878,121],{"class":120},[53,67880,67881],{"class":131}," 1.0",[53,67883,8208],{"class":120},[53,67885,67886],{"class":63}," (z ",[53,67888,2272],{"class":120},[53,67890,2211],{"class":131},[53,67892,203],{"class":63},[53,67894,193],{"class":120},[53,67896,67871],{"class":63},[53,67898,67899,67902,67904,67907,67909,67911,67913,67915,67917,67919,67921,67923,67925,67928,67930],{"class":55,"line":1372},[53,67900,67901],{"class":63}," center ",[53,67903,121],{"class":120},[53,67905,67906],{"class":63}," (p_hat ",[53,67908,2002],{"class":120},[53,67910,67886],{"class":63},[53,67912,2272],{"class":120},[53,67914,2211],{"class":131},[53,67916,203],{"class":63},[53,67918,193],{"class":120},[53,67920,58540],{"class":63},[53,67922,132],{"class":131},[53,67924,8124],{"class":120},[53,67926,67927],{"class":63}," n)) ",[53,67929,193],{"class":120},[53,67931,67932],{"class":63}," denom\n",[53,67934,67935,67938,67940,67942,67944,67947,67949,67952,67954,67956,67958,67960,67963,67965,67968,67970,67972,67974,67976,67978,67980,67982,67984,67986,67988,67990,67992,67995,67997],{"class":55,"line":1377},[53,67936,67937],{"class":63}," span ",[53,67939,121],{"class":120},[53,67941,67886],{"class":63},[53,67943,2286],{"class":120},[53,67945,67946],{"class":63}," math.",[53,67948,9714],{"class":124},[53,67950,67951],{"class":63},"((p_hat ",[53,67953,2286],{"class":120},[53,67955,58540],{"class":63},[53,67957,173],{"class":131},[53,67959,65810],{"class":120},[53,67961,67962],{"class":63}," p_hat) ",[53,67964,193],{"class":120},[53,67966,67967],{"class":63}," n) ",[53,67969,2002],{"class":120},[53,67971,67886],{"class":63},[53,67973,2272],{"class":120},[53,67975,2211],{"class":131},[53,67977,203],{"class":63},[53,67979,193],{"class":120},[53,67981,58540],{"class":63},[53,67983,12047],{"class":131},[53,67985,8124],{"class":120},[53,67987,67837],{"class":63},[53,67989,2272],{"class":120},[53,67991,2211],{"class":131},[53,67993,67994],{"class":63},"))) ",[53,67996,193],{"class":120},[53,67998,67932],{"class":63},[53,68000,68001,68003,68005,68007,68009,68011,68014,68016,68019,68021,68023,68025,68027,68029],{"class":55,"line":1383},[53,68002,2399],{"class":59},[53,68004,58540],{"class":63},[53,68006,36608],{"class":120},[53,68008,128],{"class":63},[53,68010,56610],{"class":131},[53,68012,68013],{"class":63},", center ",[53,68015,2262],{"class":120},[53,68017,68018],{"class":63}," span), ",[53,68020,25125],{"class":120},[53,68022,128],{"class":63},[53,68024,57823],{"class":131},[53,68026,68013],{"class":63},[53,68028,2002],{"class":120},[53,68030,68031],{"class":63}," span))\n",[53,68033,68034],{"class":55,"line":1389},[53,68035,104],{"emptyLinePlaceholder":103},[53,68037,68038],{"class":55,"line":1395},[53,68039,104],{"emptyLinePlaceholder":103},[53,68041,68042,68044,68047,68049,68051,68053,68055,68057,68060,68062,68064,68066,68069,68071,68073,68075,68077,68079,68081,68083,68085],{"class":55,"line":1401},[53,68043,1265],{"class":59},[53,68045,68046],{"class":124}," shots_for_epsilon",[53,68048,128],{"class":63},[53,68050,12],{"class":1277},[53,68052,1298],{"class":63},[53,68054,1301],{"class":120},[53,68056,170],{"class":63},[53,68058,68059],{"class":1277},"epsilon",[53,68061,1298],{"class":63},[53,68063,1301],{"class":120},[53,68065,1318],{"class":120},[53,68067,68068],{"class":131}," 0.01",[53,68070,170],{"class":63},[53,68072,214],{"class":1277},[53,68074,1298],{"class":63},[53,68076,1301],{"class":120},[53,68078,1318],{"class":120},[53,68080,67801],{"class":131},[53,68082,58796],{"class":63},[53,68084,1315],{"class":120},[53,68086,1623],{"class":63},[53,68088,68089],{"class":55,"line":1407},[53,68090,1363],{"class":278},[53,68092,68093],{"class":55,"line":1413},[53,68094,68095],{"class":278}," Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n",[53,68097,68098],{"class":55,"line":1419},[53,68099,68100],{"class":278}," at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n",[53,68102,68103],{"class":55,"line":1425},[53,68104,1363],{"class":278},[53,68106,68107,68110,68112,68114,68116,68118,68121,68123,68125,68127],{"class":55,"line":1431},[53,68108,68109],{"class":63}," p ",[53,68111,121],{"class":120},[53,68113,2371],{"class":120},[53,68115,128],{"class":63},[53,68117,36608],{"class":120},[53,68119,68120],{"class":63},"(p, ",[53,68122,56610],{"class":131},[53,68124,2108],{"class":63},[53,68126,57823],{"class":131},[53,68128,135],{"class":63},[53,68130,68131,68134,68136,68139,68141,68143,68145,68147],{"class":55,"line":1437},[53,68132,68133],{"class":63}," var ",[53,68135,121],{"class":120},[53,68137,68138],{"class":63}," p ",[53,68140,2286],{"class":120},[53,68142,58540],{"class":63},[53,68144,57823],{"class":131},[53,68146,65810],{"class":120},[53,68148,68149],{"class":63}," p)\n",[53,68151,68152,68154,68157,68159,68161],{"class":55,"line":1443},[53,68153,1612],{"class":59},[53,68155,68156],{"class":63}," var ",[53,68158,1618],{"class":120},[53,68160,57263],{"class":131},[53,68162,1623],{"class":63},[53,68164,68165,68167],{"class":55,"line":1448},[53,68166,64769],{"class":59},[53,68168,1979],{"class":131},[53,68170,68171,68174,68176,68178,68180,68182,68184,68186,68188,68190,68193,68195,68197],{"class":55,"line":1454},[53,68172,68173],{"class":63}," n ",[53,68175,121],{"class":120},[53,68177,67886],{"class":63},[53,68179,2272],{"class":120},[53,68181,2211],{"class":131},[53,68183,203],{"class":63},[53,68185,2286],{"class":120},[53,68187,68156],{"class":63},[53,68189,193],{"class":120},[53,68191,68192],{"class":63}," (epsilon ",[53,68194,2272],{"class":120},[53,68196,2211],{"class":131},[53,68198,135],{"class":63},[53,68200,68201,68203,68205,68207,68209,68211,68213,68216,68218],{"class":55,"line":1460},[53,68202,2399],{"class":59},[53,68204,65137],{"class":120},[53,68206,128],{"class":63},[53,68208,173],{"class":131},[53,68210,170],{"class":63},[53,68212,1315],{"class":120},[53,68214,68215],{"class":63},"(math.",[53,68217,65124],{"class":124},[53,68219,68220],{"class":63},"(n)))\n",[53,68222,68223],{"class":55,"line":1465},[53,68224,104],{"emptyLinePlaceholder":103},[53,68226,68227],{"class":55,"line":1470},[53,68228,104],{"emptyLinePlaceholder":103},[53,68230,68231],{"class":55,"line":1475},[53,68232,68233],{"class":110},"# ---------- Domain model ----------\n",[53,68235,68236],{"class":55,"line":1492},[53,68237,104],{"emptyLinePlaceholder":103},[53,68239,68240,68243,68245,68248,68250,68252],{"class":55,"line":1503},[53,68241,68242],{"class":124},"@dataclass",[53,68244,128],{"class":63},[53,68246,68247],{"class":2050},"frozen",[53,68249,121],{"class":120},[53,68251,13773],{"class":131},[53,68253,135],{"class":63},[53,68255,68256,68259,68263],{"class":55,"line":1519},[53,68257,68258],{"class":59},"class",[53,68260,68262],{"class":68261},"sU0A5"," QubitModel",[53,68264,1623],{"class":63},[53,68266,68267,68270],{"class":55,"line":1552},[53,68268,68269],{"class":63}," name: ",[53,68271,68272],{"class":120},"str\n",[53,68274,68275,68278,68280],{"class":55,"line":1558},[53,68276,68277],{"class":63}," satisfaction_prob: ",[53,68279,1301],{"class":120},[53,68281,68282],{"class":110}," # Probability a single shot meets the analytical criterion (bit = 1)\n",[53,68284,68285,68288],{"class":55,"line":1586},[53,68286,68287],{"class":63}," description: ",[53,68289,68272],{"class":120},[53,68291,68292],{"class":55,"line":1609},[53,68293,104],{"emptyLinePlaceholder":103},[53,68295,68296],{"class":55,"line":1626},[53,68297,104],{"emptyLinePlaceholder":103},[53,68299,68300,68303,68306,68308],{"class":55,"line":1635},[53,68301,68302],{"class":131},"DEFAULT_MODELS",[53,68304,68305],{"class":63},": List[QubitModel] ",[53,68307,121],{"class":120},[53,68309,2072],{"class":63},[53,68311,68312,68315],{"class":55,"line":1664},[53,68313,68314],{"class":124}," QubitModel",[53,68316,1271],{"class":63},[53,68318,68319,68322],{"class":55,"line":1672},[53,68320,68321],{"class":278}," \"Toy working qubit\"",[53,68323,1304],{"class":63},[53,68325,68326,68329],{"class":55,"line":1692},[53,68327,68328],{"class":131}," 0.50",[53,68330,1304],{"class":63},[53,68332,68333],{"class":55,"line":1700},[53,68334,68335],{"class":278}," \"Simple demonstrator; about half shots meet the criterion.\"\n",[53,68337,68338],{"class":55,"line":1707},[53,68339,22527],{"class":63},[53,68341,68342,68344],{"class":55,"line":1727},[53,68343,68314],{"class":124},[53,68345,1271],{"class":63},[53,68347,68348,68351],{"class":55,"line":1734},[53,68349,68350],{"class":278}," \"NISQ qubit\"",[53,68352,1304],{"class":63},[53,68354,68355,68358],{"class":55,"line":1750},[53,68356,68357],{"class":131}," 0.60",[53,68359,1304],{"class":63},[53,68361,68362],{"class":55,"line":1755},[53,68363,68364],{"class":278}," \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n",[53,68366,68367],{"class":55,"line":1760},[53,68368,22527],{"class":63},[53,68370,68371,68373],{"class":55,"line":1781},[53,68372,68314],{"class":124},[53,68374,1271],{"class":63},[53,68376,68377,68380],{"class":55,"line":1786},[53,68378,68379],{"class":278}," \"Fault-tolerant (logical)\"",[53,68381,1304],{"class":63},[53,68383,68384,68387],{"class":55,"line":1792},[53,68385,68386],{"class":131}," 0.90",[53,68388,1304],{"class":63},[53,68390,68391],{"class":55,"line":1797},[53,68392,68393],{"class":278}," \"Error-corrected logical qubit; high single-shot success probability.\"\n",[53,68395,68396],{"class":55,"line":1808},[53,68397,22527],{"class":63},[53,68399,68400,68402],{"class":55,"line":1828},[53,68401,68314],{"class":124},[53,68403,1271],{"class":63},[53,68405,68406,68409],{"class":55,"line":1868},[53,68407,68408],{"class":278}," \"Topologically protected\"",[53,68410,1304],{"class":63},[53,68412,68413,68416],{"class":55,"line":1873},[53,68414,68415],{"class":131}," 0.98",[53,68417,1304],{"class":63},[53,68419,68420],{"class":55,"line":1883},[53,68421,68422],{"class":278}," \"Intrinsic protection; very high single-shot success probability.\"\n",[53,68424,68425],{"class":55,"line":1898},[53,68426,22527],{"class":63},[53,68428,68429,68431],{"class":55,"line":1931},[53,68430,68314],{"class":124},[53,68432,1271],{"class":63},[53,68434,68435,68438],{"class":55,"line":1936},[53,68436,68437],{"class":278}," \"Ideal qubit\"",[53,68439,1304],{"class":63},[53,68441,68442,68445],{"class":55,"line":1947},[53,68443,68444],{"class":131}," 1.00",[53,68446,1304],{"class":63},[53,68448,68449],{"class":55,"line":1952},[53,68450,68451],{"class":278}," \"Theoretical perfect qubit; always successful in a single shot.\"\n",[53,68453,68454],{"class":55,"line":1966},[53,68455,22527],{"class":63},[53,68457,68458],{"class":55,"line":1971},[53,68459,1689],{"class":63},[53,68461,68462],{"class":55,"line":1982},[53,68463,104],{"emptyLinePlaceholder":103},[53,68465,68466],{"class":55,"line":2010},[53,68467,68468],{"class":110},"# ---------- Simulation configuration ----------\n",[53,68470,68471],{"class":55,"line":2029},[53,68472,104],{"emptyLinePlaceholder":103},[53,68474,68475,68478,68481,68483],{"class":55,"line":2034},[53,68476,68477],{"class":131},"CONFIG",[53,68479,68480],{"class":63},": Dict ",[53,68482,121],{"class":120},[53,68484,23618],{"class":63},[53,68486,68487,68490,68492,68495],{"class":55,"line":2059},[53,68488,68489],{"class":278}," \"random_seed\"",[53,68491,1298],{"class":63},[53,68493,68494],{"class":131},"123456",[53,68496,1304],{"class":63},[53,68498,68499,68502,68504,68507],{"class":55,"line":2064},[53,68500,68501],{"class":278}," \"shots_per_model\"",[53,68503,1298],{"class":63},[53,68505,68506],{"class":131},"10000",[53,68508,1304],{"class":63},[53,68510,68511,68514,68516,68519,68521,68524,68526,68529,68532],{"class":55,"line":2075},[53,68512,68513],{"class":278}," \"epsilons\"",[53,68515,23752],{"class":63},[53,68517,68518],{"class":131},"0.02",[53,68520,170],{"class":63},[53,68522,68523],{"class":131},"0.01",[53,68525,170],{"class":63},[53,68527,68528],{"class":131},"0.005",[53,68530,68531],{"class":63},"], ",[53,68533,68534],{"class":110},"# ±2%, ±1%, ±0.5% targets\n",[53,68536,68537,68540],{"class":55,"line":2119},[53,68538,68539],{"class":278}," \"cost_model\"",[53,68541,23637],{"class":63},[53,68543,68544,68547,68549,68552],{"class":55,"line":2128},[53,68545,68546],{"class":278}," \"default_cost_per_shot_usd\"",[53,68548,1298],{"class":63},[53,68550,68551],{"class":131},"0.00005",[53,68553,1304],{"class":63},[53,68555,68556,68559,68561],{"class":55,"line":2133},[53,68557,68558],{"class":278}," \"fixed_overhead_usd\"",[53,68560,1298],{"class":63},[53,68562,68563],{"class":131},"0.10\n",[53,68565,68566],{"class":55,"line":2144},[53,68567,23658],{"class":63},[53,68569,68570,68573],{"class":55,"line":2159},[53,68571,68572],{"class":278}," \"timing\"",[53,68574,23637],{"class":63},[53,68576,68577,68580,68582],{"class":55,"line":2174},[53,68578,68579],{"class":278}," \"time_per_shot_seconds\"",[53,68581,1298],{"class":63},[53,68583,68584],{"class":131},"2e-6\n",[53,68586,68587],{"class":55,"line":2185},[53,68588,59379],{"class":63},[53,68590,68591],{"class":55,"line":2190},[53,68592,23770],{"class":63},[53,68594,68595],{"class":55,"line":2216},[53,68596,104],{"emptyLinePlaceholder":103},[53,68598,68599,68602,68604,68607,68610,68612,68614,68616,68619],{"class":55,"line":2232},[53,68600,68601],{"class":63},"rng ",[53,68603,121],{"class":120},[53,68605,68606],{"class":63}," np.random.",[53,68608,68609],{"class":124},"default_rng",[53,68611,128],{"class":63},[53,68613,68477],{"class":131},[53,68615,12143],{"class":63},[53,68617,68618],{"class":278},"\"random_seed\"",[53,68620,68621],{"class":63},"])\n",[53,68623,68624],{"class":55,"line":2251},[53,68625,104],{"emptyLinePlaceholder":103},[53,68627,68628],{"class":55,"line":2291},[53,68629,104],{"emptyLinePlaceholder":103},[53,68631,68632],{"class":55,"line":2314},[53,68633,68634],{"class":110},"# ---------- Core simulation ----------\n",[53,68636,68637],{"class":55,"line":2320},[53,68638,104],{"emptyLinePlaceholder":103},[53,68640,68641,68643,68646,68648,68650,68652,68654,68656,68659,68661,68663,68665,68668],{"class":55,"line":2328},[53,68642,1265],{"class":59},[53,68644,68645],{"class":124}," simulate_bitstring",[53,68647,128],{"class":63},[53,68649,12],{"class":1277},[53,68651,1298],{"class":63},[53,68653,1301],{"class":120},[53,68655,170],{"class":63},[53,68657,68658],{"class":1277},"shots",[53,68660,1298],{"class":63},[53,68662,1315],{"class":120},[53,68664,170],{"class":63},[53,68666,68667],{"class":1277},"rng_",[53,68669,68670],{"class":63},": np.random.Generator) -> np.ndarray:\n",[53,68672,68673],{"class":55,"line":2337},[53,68674,1363],{"class":278},[53,68676,68677],{"class":55,"line":2342},[53,68678,68679],{"class":278}," Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n",[53,68681,68682],{"class":55,"line":2358},[53,68683,1363],{"class":278},[53,68685,68686,68688,68691,68694,68696,68698,68701,68704,68706,68709,68711,68713,68715],{"class":55,"line":2363},[53,68687,2399],{"class":59},[53,68689,68690],{"class":63}," rng_.",[53,68692,68693],{"class":124},"binomial",[53,68695,128],{"class":63},[53,68697,173],{"class":131},[53,68699,68700],{"class":63},", p, ",[53,68702,68703],{"class":2050},"size",[53,68705,121],{"class":120},[53,68707,68708],{"class":63},"shots).",[53,68710,66655],{"class":124},[53,68712,128],{"class":63},[53,68714,1315],{"class":120},[53,68716,135],{"class":63},[53,68718,68719],{"class":55,"line":2383},[53,68720,104],{"emptyLinePlaceholder":103},[53,68722,68723],{"class":55,"line":2396},[53,68724,104],{"emptyLinePlaceholder":103},[53,68726,68727,68729,68732,68734,68737,68740,68743,68745,68748,68751,68753],{"class":55,"line":59248},[53,68728,1265],{"class":59},[53,68730,68731],{"class":124}," summarize_model",[53,68733,128],{"class":63},[53,68735,68736],{"class":1277},"model",[53,68738,68739],{"class":63},": QubitModel, ",[53,68741,68742],{"class":1277},"bits",[53,68744,64003],{"class":63},[53,68746,68747],{"class":1277},"epsilons",[53,68749,68750],{"class":63},": List[",[53,68752,1301],{"class":120},[53,68754,68755],{"class":63},"]) -> Dict:\n",[53,68757,68758],{"class":55,"line":59253},[53,68759,1363],{"class":278},[53,68761,68762],{"class":55,"line":59260},[53,68763,68764],{"class":278}," Calculate summary statistics for a given model and its simulated bitstring.\n",[53,68766,68767],{"class":55,"line":59274},[53,68768,1363],{"class":278},[53,68770,68771,68773,68775,68777],{"class":55,"line":59279},[53,68772,68173],{"class":63},[53,68774,121],{"class":120},[53,68776,1990],{"class":120},[53,68778,68779],{"class":63},"(bits.size)\n",[53,68781,68782,68785,68787,68789,68792,68795],{"class":55,"line":59292},[53,68783,68784],{"class":63}," ones ",[53,68786,121],{"class":120},[53,68788,1990],{"class":120},[53,68790,68791],{"class":63},"(bits.",[53,68793,68794],{"class":124},"sum",[53,68796,21982],{"class":63},[53,68798,68799,68802,68804,68806,68809,68811],{"class":55,"line":59308},[53,68800,68801],{"class":63}," zeros ",[53,68803,121],{"class":120},[53,68805,1990],{"class":120},[53,68807,68808],{"class":63},"(n ",[53,68810,2262],{"class":120},[53,68812,68813],{"class":63}," ones)\n",[53,68815,68816,68819,68821,68824,68826,68828,68830,68832,68834,68836,68838,68840,68842,68845],{"class":55,"line":59328},[53,68817,68818],{"class":63}," frac ",[53,68820,121],{"class":120},[53,68822,68823],{"class":63}," ones ",[53,68825,193],{"class":120},[53,68827,67837],{"class":63},[53,68829,1540],{"class":59},[53,68831,67837],{"class":63},[53,68833,1546],{"class":120},[53,68835,21153],{"class":131},[53,68837,8375],{"class":59},[53,68839,59516],{"class":120},[53,68841,128],{"class":63},[53,68843,68844],{"class":278},"\"nan\"",[53,68846,135],{"class":63},[53,68848,68849,68852,68854,68856],{"class":55,"line":59336},[53,68850,68851],{"class":63}," lo, hi ",[53,68853,121],{"class":120},[53,68855,67772],{"class":124},[53,68857,68858],{"class":63},"(ones, n)\n",[53,68860,68861],{"class":55,"line":59342},[53,68862,104],{"emptyLinePlaceholder":103},[53,68864,68865,68868,68870],{"class":55,"line":59347},[53,68866,68867],{"class":63}," row ",[53,68869,121],{"class":120},[53,68871,23618],{"class":63},[53,68873,68874,68877],{"class":55,"line":59358},[53,68875,68876],{"class":278}," \"Model\"",[53,68878,68879],{"class":63},": model.name,\n",[53,68881,68882,68885],{"class":55,"line":59364},[53,68883,68884],{"class":278}," \"Description\"",[53,68886,68887],{"class":63},": model.description,\n",[53,68889,68890,68893],{"class":55,"line":59376},[53,68891,68892],{"class":278}," \"Configured p (target)\"",[53,68894,68895],{"class":63},": model.satisfaction_prob,\n",[53,68897,68898,68901],{"class":55,"line":59382},[53,68899,68900],{"class":278}," \"Shots\"",[53,68902,68903],{"class":63},": n,\n",[53,68905,68906,68909],{"class":55,"line":59387},[53,68907,68908],{"class":278}," \"Ones (successes)\"",[53,68910,68911],{"class":63},": ones,\n",[53,68913,68914,68917],{"class":55,"line":59392},[53,68915,68916],{"class":278}," \"Zeros (failures)\"",[53,68918,68919],{"class":63},": zeros,\n",[53,68921,68922,68925],{"class":55,"line":59397},[53,68923,68924],{"class":278}," \"Ones fraction (observed)\"",[53,68926,68927],{"class":63},": frac,\n",[53,68929,68930,68933,68936,68939],{"class":55,"line":59420},[53,68931,68932],{"class":278}," \"Wilson CI 95",[53,68934,68935],{"class":131},"% lo",[53,68937,68938],{"class":278},"w\"",[53,68940,68941],{"class":63},": lo,\n",[53,68943,68944,68946,68949,68952],{"class":55,"line":59426},[53,68945,68932],{"class":278},[53,68947,68948],{"class":131},"% hi",[53,68950,68951],{"class":278},"gh\"",[53,68953,68954],{"class":63},": hi,\n",[53,68956,68957,68960,68963,68965],{"class":55,"line":59446},[53,68958,68959],{"class":278}," \"CI width (95%)\"",[53,68961,68962],{"class":63},": hi ",[53,68964,2262],{"class":120},[53,68966,68967],{"class":63}," lo,\n",[53,68969,68970],{"class":55,"line":59451},[53,68971,59379],{"class":63},[53,68973,68974,68976,68979,68981],{"class":55,"line":59456},[53,68975,1811],{"class":59},[53,68977,68978],{"class":63}," eps ",[53,68980,1531],{"class":59},[53,68982,68983],{"class":63}," epsilons:\n",[53,68985,68986,68989,68991,68994,68996,68998,69001,69003,69005,69007,69009,69012,69014,69016,69018],{"class":55,"line":59466},[53,68987,68988],{"class":63}," row[",[53,68990,22173],{"class":59},[53,68992,68993],{"class":278},"\"Required shots for ±",[53,68995,2705],{"class":131},[53,68997,1315],{"class":120},[53,68999,69000],{"class":63},"(eps",[53,69002,2286],{"class":120},[53,69004,5577],{"class":131},[53,69006,3544],{"class":63},[53,69008,2767],{"class":131},[53,69010,69011],{"class":278},"%\"",[53,69013,8101],{"class":63},[53,69015,121],{"class":120},[53,69017,68046],{"class":124},[53,69019,69020],{"class":63},"(model.satisfaction_prob, eps)\n",[53,69022,69023,69025],{"class":55,"line":59478},[53,69024,2399],{"class":59},[53,69026,69027],{"class":63}," row\n",[53,69029,69030],{"class":55,"line":59490},[53,69031,104],{"emptyLinePlaceholder":103},[53,69033,69034],{"class":55,"line":59502},[53,69035,104],{"emptyLinePlaceholder":103},[53,69037,69038,69040,69043,69045,69047,69049,69051,69053,69056,69058,69060,69062,69065,69067,69069,69071,69073],{"class":55,"line":59522},[53,69039,1265],{"class":59},[53,69041,69042],{"class":124}," apply_costs",[53,69044,128],{"class":63},[53,69046,68658],{"class":1277},[53,69048,1298],{"class":63},[53,69050,1315],{"class":120},[53,69052,170],{"class":63},[53,69054,69055],{"class":1277},"cost_per_shot",[53,69057,1298],{"class":63},[53,69059,1301],{"class":120},[53,69061,170],{"class":63},[53,69063,69064],{"class":1277},"fixed_overhead",[53,69066,1298],{"class":63},[53,69068,1301],{"class":120},[53,69070,58796],{"class":63},[53,69072,1301],{"class":120},[53,69074,1623],{"class":63},[53,69076,69077],{"class":55,"line":59528},[53,69078,1363],{"class":278},[53,69080,69081],{"class":55,"line":59552},[53,69082,69083],{"class":278}," Calculate illustrative cost based on shots and cost model.\n",[53,69085,69086],{"class":55,"line":59575},[53,69087,1363],{"class":278},[53,69089,69090,69092,69095,69097,69100,69102],{"class":55,"line":59591},[53,69091,2399],{"class":59},[53,69093,69094],{"class":63}," fixed_overhead ",[53,69096,2002],{"class":120},[53,69098,69099],{"class":63}," shots ",[53,69101,2286],{"class":120},[53,69103,69104],{"class":63}," cost_per_shot\n",[53,69106,69107],{"class":55,"line":59596},[53,69108,104],{"emptyLinePlaceholder":103},[53,69110,69111],{"class":55,"line":59623},[53,69112,104],{"emptyLinePlaceholder":103},[53,69114,69115],{"class":55,"line":59628},[53,69116,69117],{"class":110},"# ---------- Run the simulation ----------\n",[53,69119,69120],{"class":55,"line":59652},[53,69121,104],{"emptyLinePlaceholder":103},[53,69123,69124,69127,69129,69132,69134],{"class":55,"line":59674},[53,69125,69126],{"class":63},"all_bitstrings: Dict[",[53,69128,58793],{"class":120},[53,69130,69131],{"class":63},", np.ndarray] ",[53,69133,121],{"class":120},[53,69135,23603],{"class":63},[53,69137,69138,69141,69143],{"class":55,"line":59695},[53,69139,69140],{"class":63},"summary_rows: List[Dict] ",[53,69142,121],{"class":120},[53,69144,1805],{"class":63},[53,69146,69147],{"class":55,"line":59700},[53,69148,104],{"emptyLinePlaceholder":103},[53,69150,69151,69153,69155,69158],{"class":55,"line":59714},[53,69152,8420],{"class":120},[53,69154,128],{"class":63},[53,69156,69157],{"class":278},"\"Running simulation...\"",[53,69159,135],{"class":63},[53,69161,69162,69164,69167,69169,69172],{"class":55,"line":59729},[53,69163,1525],{"class":59},[53,69165,69166],{"class":63}," m ",[53,69168,1531],{"class":59},[53,69170,69171],{"class":131}," DEFAULT_MODELS",[53,69173,1623],{"class":63},[53,69175,69176,69179,69181,69183,69186,69188,69190,69193],{"class":55,"line":59747},[53,69177,69178],{"class":63}," bits ",[53,69180,121],{"class":120},[53,69182,68645],{"class":124},[53,69184,69185],{"class":63},"(m.satisfaction_prob, ",[53,69187,68477],{"class":131},[53,69189,12143],{"class":63},[53,69191,69192],{"class":278},"\"shots_per_model\"",[53,69194,69195],{"class":63},"], rng)\n",[53,69197,69198,69201,69203],{"class":55,"line":59752},[53,69199,69200],{"class":63}," all_bitstrings[m.name] ",[53,69202,121],{"class":120},[53,69204,69205],{"class":63}," bits\n",[53,69207,69208,69211,69213,69215,69218,69221,69223,69225,69228],{"class":55,"line":59757},[53,69209,69210],{"class":63}," summary_rows.",[53,69212,1834],{"class":124},[53,69214,128],{"class":63},[53,69216,69217],{"class":124},"summarize_model",[53,69219,69220],{"class":63},"(m, bits, ",[53,69222,68477],{"class":131},[53,69224,12143],{"class":63},[53,69226,69227],{"class":278},"\"epsilons\"",[53,69229,69230],{"class":63},"]))\n",[53,69232,69233],{"class":55,"line":59772},[53,69234,104],{"emptyLinePlaceholder":103},[53,69236,69237,69240,69242,69244,69246],{"class":55,"line":59778},[53,69238,69239],{"class":63},"summary_df ",[53,69241,121],{"class":120},[53,69243,57938],{"class":63},[53,69245,57941],{"class":124},[53,69247,69248],{"class":63},"(summary_rows)\n",[53,69250,69251],{"class":55,"line":59784},[53,69252,104],{"emptyLinePlaceholder":103},[53,69254,69255],{"class":55,"line":59790},[53,69256,69257],{"class":110},"# Add cost estimates\n",[53,69259,69260,69263,69265,69268,69270,69273,69275,69278],{"class":55,"line":59795},[53,69261,69262],{"class":63},"cps ",[53,69264,121],{"class":120},[53,69266,69267],{"class":131}," CONFIG",[53,69269,12143],{"class":63},[53,69271,69272],{"class":278},"\"cost_model\"",[53,69274,1650],{"class":63},[53,69276,69277],{"class":278},"\"default_cost_per_shot_usd\"",[53,69279,1689],{"class":63},[53,69281,69282,69285,69287,69289,69291,69293,69295,69298],{"class":55,"line":59800},[53,69283,69284],{"class":63},"over ",[53,69286,121],{"class":120},[53,69288,69267],{"class":131},[53,69290,12143],{"class":63},[53,69292,69272],{"class":278},[53,69294,1650],{"class":63},[53,69296,69297],{"class":278},"\"fixed_overhead_usd\"",[53,69299,1689],{"class":63},[53,69301,69302,69304,69307,69309,69311,69313,69316,69318,69320],{"class":55,"line":59832},[53,69303,1540],{"class":59},[53,69305,69306],{"class":63}," cps ",[53,69308,1546],{"class":120},[53,69310,21153],{"class":131},[53,69312,22062],{"class":59},[53,69314,69315],{"class":63}," over ",[53,69317,1546],{"class":120},[53,69319,21153],{"class":131},[53,69321,1623],{"class":63},[53,69323,69324,69327,69330,69332,69334],{"class":55,"line":59859},[53,69325,69326],{"class":63}," summary_df[",[53,69328,69329],{"class":278},"\"Illustrative cost (USD)\"",[53,69331,8101],{"class":63},[53,69333,121],{"class":120},[53,69335,2072],{"class":63},[53,69337,69338,69341,69343,69345,69348,69351],{"class":55,"line":59885},[53,69339,69340],{"class":124}," apply_costs",[53,69342,128],{"class":63},[53,69344,1315],{"class":120},[53,69346,69347],{"class":63},"(row[",[53,69349,69350],{"class":278},"\"Shots\"",[53,69352,69353],{"class":63},"]), cps, over)\n",[53,69355,69356,69358,69361,69363,69366,69369],{"class":55,"line":59890},[53,69357,2162],{"class":59},[53,69359,69360],{"class":63}," _, row ",[53,69362,1531],{"class":59},[53,69364,69365],{"class":63}," summary_df.",[53,69367,69368],{"class":124},"iterrows",[53,69370,329],{"class":63},[53,69372,69373],{"class":55,"line":59904},[53,69374,58936],{"class":63},[53,69376,69377],{"class":55,"line":59919},[53,69378,104],{"emptyLinePlaceholder":103},[53,69380,69381],{"class":55,"line":59924},[53,69382,69383],{"class":110},"# ---------- Save artifacts ----------\n",[53,69385,69386],{"class":55,"line":59929},[53,69387,104],{"emptyLinePlaceholder":103},[53,69389,69390],{"class":55,"line":59943},[53,69391,69392],{"class":110},"# Save raw bitstrings and summary CSV\n",[53,69394,69395,69397,69400,69402,69405,69407],{"class":55,"line":59948},[53,69396,1525],{"class":59},[53,69398,69399],{"class":63}," name, arr ",[53,69401,1531],{"class":59},[53,69403,69404],{"class":63}," all_bitstrings.",[53,69406,22502],{"class":124},[53,69408,1825],{"class":63},[53,69410,69411,69414,69416,69418,69421,69423,69426,69429,69431,69434,69436,69439,69441,69443],{"class":55,"line":59954},[53,69412,69413],{"class":63}," out_path ",[53,69415,121],{"class":120},[53,69417,64328],{"class":59},[53,69419,69420],{"class":278},"\"bitstrings_",[53,69422,2705],{"class":131},[53,69424,69425],{"class":63},"name.",[53,69427,69428],{"class":124},"replace",[53,69430,128],{"class":63},[53,69432,69433],{"class":278},"' '",[53,69435,170],{"class":63},[53,69437,69438],{"class":278},"'_'",[53,69440,3544],{"class":63},[53,69442,2767],{"class":131},[53,69444,69445],{"class":278},".csv\"\n",[53,69447,69448,69451,69453,69456,69459,69462,69465,69468,69471,69473,69475],{"class":55,"line":59960},[53,69449,69450],{"class":63}," pd.",[53,69452,57941],{"class":124},[53,69454,69455],{"class":63},"({",[53,69457,69458],{"class":278},"\"bit\"",[53,69460,69461],{"class":63},": arr}).",[53,69463,69464],{"class":124},"to_csv",[53,69466,69467],{"class":63},"(out_path, ",[53,69469,69470],{"class":2050},"index",[53,69472,121],{"class":120},[53,69474,21859],{"class":131},[53,69476,135],{"class":63},[53,69478,69479],{"class":55,"line":59966},[53,69480,104],{"emptyLinePlaceholder":103},[53,69482,69483,69486,69488],{"class":55,"line":59971},[53,69484,69485],{"class":63},"summary_csv_path ",[53,69487,121],{"class":120},[53,69489,69490],{"class":278}," \"summary.csv\"\n",[53,69492,69493,69496,69498,69501,69503,69505,69507],{"class":55,"line":59976},[53,69494,69495],{"class":63},"summary_df.",[53,69497,69464],{"class":124},[53,69499,69500],{"class":63},"(summary_csv_path, ",[53,69502,69470],{"class":2050},[53,69504,121],{"class":120},[53,69506,21859],{"class":131},[53,69508,135],{"class":63},[53,69510,69511],{"class":55,"line":59981},[53,69512,104],{"emptyLinePlaceholder":103},[53,69514,69515],{"class":55,"line":59995},[53,69516,69517],{"class":110},"# Save config JSON\n",[53,69519,69520,69523,69526,69528,69531,69533,69536,69538,69540],{"class":55,"line":60000},[53,69521,69522],{"class":59},"with",[53,69524,69525],{"class":120}," open",[53,69527,128],{"class":63},[53,69529,69530],{"class":278},"\"simulation_config.json\"",[53,69532,170],{"class":63},[53,69534,69535],{"class":278},"\"w\"",[53,69537,203],{"class":63},[53,69539,67],{"class":59},[53,69541,69542],{"class":63}," f:\n",[53,69544,69545,69548,69551,69553,69555,69558,69561,69563,69565],{"class":55,"line":60011},[53,69546,69547],{"class":63}," json.",[53,69549,69550],{"class":124},"dump",[53,69552,128],{"class":63},[53,69554,68477],{"class":131},[53,69556,69557],{"class":63},", f, ",[53,69559,69560],{"class":2050},"indent",[53,69562,121],{"class":120},[53,69564,132],{"class":131},[53,69566,135],{"class":63},[53,69568,69569],{"class":55,"line":60019},[53,69570,104],{"emptyLinePlaceholder":103},[53,69572,69573,69575,69577,69580],{"class":55,"line":60029},[53,69574,8420],{"class":120},[53,69576,128],{"class":63},[53,69578,69579],{"class":278},"\"Saved data to CSV and JSON files.\"",[53,69581,135],{"class":63},[53,69583,69584],{"class":55,"line":60034},[53,69585,104],{"emptyLinePlaceholder":103},[53,69587,69588],{"class":55,"line":60039},[53,69589,69590],{"class":110},"# ---------- Visualizations ----------\n",[53,69592,69593],{"class":55,"line":60044},[53,69594,104],{"emptyLinePlaceholder":103},[53,69596,69597],{"class":55,"line":60051},[53,69598,69599],{"class":110},"# Figure 1: Observed ones fraction with Wilson CI\n",[53,69601,69602,69605,69608,69610,69613],{"class":55,"line":60067},[53,69603,69604],{"class":63},"plt.style.",[53,69606,69607],{"class":124},"use",[53,69609,128],{"class":63},[53,69611,69612],{"class":278},"'seaborn-v0_8-whitegrid'",[53,69614,135],{"class":63},[53,69616,69617,69619,69621,69623,69625,69627,69629,69631,69633,69635],{"class":55,"line":60076},[53,69618,25271],{"class":63},[53,69620,61512],{"class":124},[53,69622,128],{"class":63},[53,69624,58129],{"class":2050},[53,69626,121],{"class":120},[53,69628,128],{"class":63},[53,69630,23700],{"class":131},[53,69632,170],{"class":63},[53,69634,63608],{"class":131},[53,69636,22435],{"class":63},[53,69638,69639,69642,69644,69646,69649,69651,69653],{"class":55,"line":60085},[53,69640,69641],{"class":63},"x ",[53,69643,121],{"class":120},[53,69645,1511],{"class":63},[53,69647,69648],{"class":124},"arange",[53,69650,128],{"class":63},[53,69652,1658],{"class":120},[53,69654,69655],{"class":63},"(summary_df))\n",[53,69657,69658,69661,69663,69666,69669,69671,69674],{"class":55,"line":60094},[53,69659,69660],{"class":63},"y ",[53,69662,121],{"class":120},[53,69664,69665],{"class":63}," summary_df[",[53,69667,69668],{"class":278},"\"Ones fraction (observed)\"",[53,69670,22800],{"class":63},[53,69672,69673],{"class":124},"to_numpy",[53,69675,329],{"class":63},[53,69677,69678],{"class":55,"line":60103},[53,69679,104],{"emptyLinePlaceholder":103},[53,69681,69682],{"class":55,"line":60126},[53,69683,69684],{"class":110},"# --- FIX APPLIED HERE ---\n",[53,69686,69687],{"class":55,"line":60132},[53,69688,69689],{"class":110},"# Take the absolute value to prevent tiny negative numbers from floating-point\n",[53,69691,69692],{"class":55,"line":60137},[53,69693,69694],{"class":110},"# inaccuracies from causing a ValueError in the errorbar plot.\n",[53,69696,69697,69700,69702,69705,69707,69709,69712,69714,69716,69718,69720],{"class":55,"line":60142},[53,69698,69699],{"class":63},"lower_error ",[53,69701,121],{"class":120},[53,69703,69704],{"class":63}," y ",[53,69706,2262],{"class":120},[53,69708,69665],{"class":63},[53,69710,69711],{"class":278},"\"Wilson CI 95",[53,69713,68935],{"class":131},[53,69715,68938],{"class":278},[53,69717,22800],{"class":63},[53,69719,69673],{"class":124},[53,69721,329],{"class":63},[53,69723,69724,69727,69729,69731,69733,69735,69737,69739,69741,69743,69745],{"class":55,"line":60147},[53,69725,69726],{"class":63},"upper_error ",[53,69728,121],{"class":120},[53,69730,69665],{"class":63},[53,69732,69711],{"class":278},[53,69734,68948],{"class":131},[53,69736,68951],{"class":278},[53,69738,22800],{"class":63},[53,69740,69673],{"class":124},[53,69742,1572],{"class":63},[53,69744,2262],{"class":120},[53,69746,69747],{"class":63}," y\n",[53,69749,69750,69753,69755,69757,69759,69762,69764,69767,69769],{"class":55,"line":60157},[53,69751,69752],{"class":63},"yerr ",[53,69754,121],{"class":120},[53,69756,1511],{"class":63},[53,69758,22968],{"class":124},[53,69760,69761],{"class":63},"([np.",[53,69763,64155],{"class":124},[53,69765,69766],{"class":63},"(lower_error), np.",[53,69768,64155],{"class":124},[53,69770,69771],{"class":63},"(upper_error)])\n",[53,69773,69774],{"class":55,"line":60166},[53,69775,69776],{"class":110},"# --- END OF FIX ---\n",[53,69778,69779],{"class":55,"line":60175},[53,69780,104],{"emptyLinePlaceholder":103},[53,69782,69783,69785,69788,69791,69793,69795,69798,69800,69803,69805,69808],{"class":55,"line":60180},[53,69784,25271],{"class":63},[53,69786,69787],{"class":124},"bar",[53,69789,69790],{"class":63},"(x, y, ",[53,69792,64246],{"class":2050},[53,69794,121],{"class":120},[53,69796,69797],{"class":278},"'skyblue'",[53,69799,170],{"class":63},[53,69801,69802],{"class":2050},"edgecolor",[53,69804,121],{"class":120},[53,69806,69807],{"class":278},"'black'",[53,69809,135],{"class":63},[53,69811,69812,69814,69817,69819,69822,69824,69827,69830,69832,69835,69837,69840,69842,69844,69846,69848,69850,69853,69855,69858,69860,69862],{"class":55,"line":60186},[53,69813,25271],{"class":63},[53,69815,69816],{"class":124},"errorbar",[53,69818,69790],{"class":63},[53,69820,69821],{"class":2050},"yerr",[53,69823,121],{"class":120},[53,69825,69826],{"class":63},"yerr, ",[53,69828,69829],{"class":2050},"fmt",[53,69831,121],{"class":120},[53,69833,69834],{"class":278},"\"none\"",[53,69836,170],{"class":63},[53,69838,69839],{"class":2050},"capsize",[53,69841,121],{"class":120},[53,69843,58141],{"class":131},[53,69845,170],{"class":63},[53,69847,64246],{"class":2050},[53,69849,121],{"class":120},[53,69851,69852],{"class":278},"\"black\"",[53,69854,170],{"class":63},[53,69856,69857],{"class":2050},"elinewidth",[53,69859,121],{"class":120},[53,69861,66106],{"class":131},[53,69863,135],{"class":63},[53,69865,69866,69868,69870,69873,69876,69878,69880,69882,69885,69887,69889,69891,69894,69896,69899],{"class":55,"line":60202},[53,69867,25271],{"class":63},[53,69869,25416],{"class":124},[53,69871,69872],{"class":63},"(x, summary_df[",[53,69874,69875],{"class":278},"\"Model\"",[53,69877,22800],{"class":63},[53,69879,24265],{"class":124},[53,69881,13845],{"class":63},[53,69883,69884],{"class":2050},"rotation",[53,69886,121],{"class":120},[53,69888,48484],{"class":131},[53,69890,170],{"class":63},[53,69892,69893],{"class":2050},"ha",[53,69895,121],{"class":120},[53,69897,69898],{"class":278},"\"right\"",[53,69900,135],{"class":63},[53,69902,69903,69905,69907,69909,69912],{"class":55,"line":60213},[53,69904,25271],{"class":63},[53,69906,25476],{"class":124},[53,69908,128],{"class":63},[53,69910,69911],{"class":278},"\"Observed Ones Fraction\"",[53,69913,135],{"class":63},[53,69915,69916,69918,69920,69922,69925,69927,69930,69932,69935],{"class":55,"line":60228},[53,69917,25271],{"class":63},[53,69919,25490],{"class":124},[53,69921,128],{"class":63},[53,69923,69924],{"class":278},"\"Observed Single-Shot Success Rates with 95% Wilson CI\"",[53,69926,170],{"class":63},[53,69928,69929],{"class":2050},"fontsize",[53,69931,121],{"class":120},[53,69933,69934],{"class":131},"14",[53,69936,135],{"class":63},[53,69938,69939,69941,69943],{"class":55,"line":60238},[53,69940,25271],{"class":63},[53,69942,58266],{"class":124},[53,69944,329],{"class":63},[53,69946,69947,69950,69952],{"class":55,"line":60245},[53,69948,69949],{"class":63},"fig1_path ",[53,69951,121],{"class":120},[53,69953,69954],{"class":278}," \"fig_ones_fraction.png\"\n",[53,69956,69957,69959,69962,69965,69967,69969,69971,69973,69976,69978,69981],{"class":55,"line":60254},[53,69958,25271],{"class":63},[53,69960,69961],{"class":124},"savefig",[53,69963,69964],{"class":63},"(fig1_path, ",[53,69966,63613],{"class":2050},[53,69968,121],{"class":120},[53,69970,63618],{"class":131},[53,69972,170],{"class":63},[53,69974,69975],{"class":2050},"bbox_inches",[53,69977,121],{"class":120},[53,69979,69980],{"class":278},"'tight'",[53,69982,135],{"class":63},[53,69984,69985,69987,69989],{"class":55,"line":60259},[53,69986,25271],{"class":63},[53,69988,25508],{"class":124},[53,69990,329],{"class":63},[53,69992,69993],{"class":55,"line":60281},[53,69994,104],{"emptyLinePlaceholder":103},[53,69996,69997],{"class":55,"line":60286},[53,69998,69999],{"class":110},"# Figure 2: Required shots at several ±epsilon targets\n",[53,70001,70002,70004,70006,70008,70010,70012,70014,70016,70018,70020],{"class":55,"line":60303},[53,70003,25271],{"class":63},[53,70005,61512],{"class":124},[53,70007,128],{"class":63},[53,70009,58129],{"class":2050},[53,70011,121],{"class":120},[53,70013,128],{"class":63},[53,70015,23700],{"class":131},[53,70017,170],{"class":63},[53,70019,63608],{"class":131},[53,70021,22435],{"class":63},[53,70023,70024,70027,70029],{"class":55,"line":60320},[53,70025,70026],{"class":63},"width ",[53,70028,121],{"class":120},[53,70030,70031],{"class":131}," 0.2\n",[53,70033,70034,70036,70038,70040,70042,70044,70046,70048,70050],{"class":55,"line":60325},[53,70035,69641],{"class":63},[53,70037,121],{"class":120},[53,70039,1511],{"class":63},[53,70041,69648],{"class":124},[53,70043,128],{"class":63},[53,70045,1658],{"class":120},[53,70047,128],{"class":63},[53,70049,68302],{"class":131},[53,70051,22435],{"class":63},[53,70053,70054,70057,70059,70061,70064,70066,70069,70071,70074],{"class":55,"line":60338},[53,70055,70056],{"class":63},"colors ",[53,70058,121],{"class":120},[53,70060,8322],{"class":63},[53,70062,70063],{"class":278},"'#ff9999'",[53,70065,2714],{"class":63},[53,70067,70068],{"class":278},"'#66b3ff'",[53,70070,2714],{"class":63},[53,70072,70073],{"class":278},"'#99ff99'",[53,70075,1689],{"class":63},[53,70077,70078],{"class":55,"line":60352},[53,70079,104],{"emptyLinePlaceholder":103},[53,70081,70082,70084,70087,70089,70091,70093,70095,70097,70099],{"class":55,"line":60357},[53,70083,1525],{"class":59},[53,70085,70086],{"class":63}," i, eps ",[53,70088,1531],{"class":59},[53,70090,1852],{"class":120},[53,70092,128],{"class":63},[53,70094,68477],{"class":131},[53,70096,12143],{"class":63},[53,70098,69227],{"class":278},[53,70100,70101],{"class":63},"]):\n",[53,70103,70104,70107,70109,70111,70114,70117,70119,70121,70123,70125],{"class":55,"line":60365},[53,70105,70106],{"class":63}," req ",[53,70108,121],{"class":120},[53,70110,8322],{"class":63},[53,70112,70113],{"class":124},"shots_for_epsilon",[53,70115,70116],{"class":63},"(m.satisfaction_prob, eps) ",[53,70118,1525],{"class":59},[53,70120,69166],{"class":63},[53,70122,1531],{"class":59},[53,70124,69171],{"class":131},[53,70126,1689],{"class":63},[53,70128,70129,70132,70134,70137,70139,70141,70143,70146,70148,70150,70152,70155,70157,70160,70163,70165,70167,70169,70171,70173,70176,70178,70180,70182],{"class":55,"line":60370},[53,70130,70131],{"class":63}," plt.",[53,70133,69787],{"class":124},[53,70135,70136],{"class":63},"(x ",[53,70138,2002],{"class":120},[53,70140,8141],{"class":63},[53,70142,2286],{"class":120},[53,70144,70145],{"class":63}," width, req, width, ",[53,70147,20907],{"class":2050},[53,70149,121],{"class":120},[53,70151,22173],{"class":59},[53,70153,70154],{"class":278},"\"±",[53,70156,2705],{"class":131},[53,70158,70159],{"class":63},"eps",[53,70161,70162],{"class":59},":.1%",[53,70164,2767],{"class":131},[53,70166,22176],{"class":278},[53,70168,170],{"class":63},[53,70170,64246],{"class":2050},[53,70172,121],{"class":120},[53,70174,70175],{"class":63},"colors[i], ",[53,70177,69802],{"class":2050},[53,70179,121],{"class":120},[53,70181,69807],{"class":278},[53,70183,135],{"class":63},[53,70185,70186],{"class":55,"line":60375},[53,70187,104],{"emptyLinePlaceholder":103},[53,70189,70190,70192,70194,70196,70198,70201,70203,70205,70207,70209,70211,70213,70215,70217,70219,70221,70223,70225],{"class":55,"line":60385},[53,70191,25271],{"class":63},[53,70193,25416],{"class":124},[53,70195,70136],{"class":63},[53,70197,2002],{"class":120},[53,70199,70200],{"class":63}," width, [m.name ",[53,70202,1525],{"class":59},[53,70204,69166],{"class":63},[53,70206,1531],{"class":59},[53,70208,69171],{"class":131},[53,70210,1655],{"class":63},[53,70212,69884],{"class":2050},[53,70214,121],{"class":120},[53,70216,48484],{"class":131},[53,70218,170],{"class":63},[53,70220,69893],{"class":2050},[53,70222,121],{"class":120},[53,70224,69898],{"class":278},[53,70226,135],{"class":63},[53,70228,70229,70231,70233,70235,70238],{"class":55,"line":60393},[53,70230,25271],{"class":63},[53,70232,25476],{"class":124},[53,70234,128],{"class":63},[53,70236,70237],{"class":278},"\"Required Shots (Log Scale)\"",[53,70239,135],{"class":63},[53,70241,70242,70244,70246,70248,70251,70253,70255,70257,70259],{"class":55,"line":60401},[53,70243,25271],{"class":63},[53,70245,25490],{"class":124},[53,70247,128],{"class":63},[53,70249,70250],{"class":278},"\"Approximate Shots to Estimate Success Rate within ±ε\"",[53,70252,170],{"class":63},[53,70254,69929],{"class":2050},[53,70256,121],{"class":120},[53,70258,69934],{"class":131},[53,70260,135],{"class":63},[53,70262,70263,70265,70268,70270,70273],{"class":55,"line":60412},[53,70264,25271],{"class":63},[53,70266,70267],{"class":124},"yscale",[53,70269,128],{"class":63},[53,70271,70272],{"class":278},"'log'",[53,70274,135],{"class":63},[53,70276,70277,70279,70281,70283,70285,70287,70290],{"class":55,"line":60423},[53,70278,25271],{"class":63},[53,70280,25453],{"class":124},[53,70282,128],{"class":63},[53,70284,25490],{"class":2050},[53,70286,121],{"class":120},[53,70288,70289],{"class":278},"\"Error Tolerance\"",[53,70291,135],{"class":63},[53,70293,70294,70296,70298],{"class":55,"line":60432},[53,70295,25271],{"class":63},[53,70297,58266],{"class":124},[53,70299,329],{"class":63},[53,70301,70302,70305,70307],{"class":55,"line":60438},[53,70303,70304],{"class":63},"fig2_path ",[53,70306,121],{"class":120},[53,70308,70309],{"class":278}," \"fig_required_shots.png\"\n",[53,70311,70312,70314,70316,70319,70321,70323,70325,70327,70329,70331,70333],{"class":55,"line":60448},[53,70313,25271],{"class":63},[53,70315,69961],{"class":124},[53,70317,70318],{"class":63},"(fig2_path, ",[53,70320,63613],{"class":2050},[53,70322,121],{"class":120},[53,70324,63618],{"class":131},[53,70326,170],{"class":63},[53,70328,69975],{"class":2050},[53,70330,121],{"class":120},[53,70332,69980],{"class":278},[53,70334,135],{"class":63},[53,70336,70337,70339,70341],{"class":55,"line":60458},[53,70338,25271],{"class":63},[53,70340,25508],{"class":124},[53,70342,329],{"class":63},[53,70344,70345],{"class":55,"line":60467},[53,70346,104],{"emptyLinePlaceholder":103},[53,70348,70349,70351,70353,70356],{"class":55,"line":60476},[53,70350,8420],{"class":120},[53,70352,128],{"class":63},[53,70354,70355],{"class":278},"\"Generated and saved plots.\"",[53,70357,135],{"class":63},[53,70359,70360],{"class":55,"line":60487},[53,70361,104],{"emptyLinePlaceholder":103},[53,70363,70364],{"class":55,"line":60499},[53,70365,70366],{"class":110},"# ---------- Display interactive table ----------\n",[53,70368,70369],{"class":55,"line":60510},[53,70370,104],{"emptyLinePlaceholder":103},[53,70372,70373,70376,70378,70381,70383,70386],{"class":55,"line":60515},[53,70374,70375],{"class":124},"display",[53,70377,128],{"class":63},[53,70379,70380],{"class":124},"Markdown",[53,70382,128],{"class":63},[53,70384,70385],{"class":278},"\"## Qubit Sampling Budget - Summary\"",[53,70387,22435],{"class":63},[53,70389,70390,70392,70395,70398],{"class":55,"line":60520},[53,70391,70375],{"class":124},[53,70393,70394],{"class":63},"(summary_df.style.",[53,70396,70397],{"class":124},"format",[53,70399,70400],{"class":63},"({\n",[53,70402,70403,70406,70408,70410,70412,70415,70417,70419],{"class":55,"line":60525},[53,70404,70405],{"class":278}," \"Configured p (target)\"",[53,70407,1298],{"class":63},[53,70409,22176],{"class":278},[53,70411,2705],{"class":131},[53,70413,70414],{"class":59},":.2f",[53,70416,2767],{"class":131},[53,70418,22176],{"class":278},[53,70420,1304],{"class":63},[53,70422,70423,70426,70428,70430,70432,70435,70437,70439],{"class":55,"line":60535},[53,70424,70425],{"class":278}," \"Ones fraction (observed)\"",[53,70427,1298],{"class":63},[53,70429,22176],{"class":278},[53,70431,2705],{"class":131},[53,70433,70434],{"class":59},":.4f",[53,70436,2767],{"class":131},[53,70438,22176],{"class":278},[53,70440,1304],{"class":63},[53,70442,70443,70446,70448,70450,70452,70454,70456,70458,70460,70462],{"class":55,"line":60542},[53,70444,70445],{"class":278}," \"Wilson CI 95",[53,70447,68935],{"class":131},[53,70449,68938],{"class":278},[53,70451,1298],{"class":63},[53,70453,22176],{"class":278},[53,70455,2705],{"class":131},[53,70457,70434],{"class":59},[53,70459,2767],{"class":131},[53,70461,22176],{"class":278},[53,70463,1304],{"class":63},[53,70465,70466,70468,70470,70472,70474,70476,70478,70480,70482,70484],{"class":55,"line":60550},[53,70467,70445],{"class":278},[53,70469,68948],{"class":131},[53,70471,68951],{"class":278},[53,70473,1298],{"class":63},[53,70475,22176],{"class":278},[53,70477,2705],{"class":131},[53,70479,70434],{"class":59},[53,70481,2767],{"class":131},[53,70483,22176],{"class":278},[53,70485,1304],{"class":63},[53,70487,70488,70491,70493,70495,70497,70499,70501,70503],{"class":55,"line":60561},[53,70489,70490],{"class":278}," \"CI width (95%)\"",[53,70492,1298],{"class":63},[53,70494,22176],{"class":278},[53,70496,2705],{"class":131},[53,70498,70434],{"class":59},[53,70500,2767],{"class":131},[53,70502,22176],{"class":278},[53,70504,1304],{"class":63},[53,70506,70507,70510,70512,70514,70516,70518,70520],{"class":55,"line":60572},[53,70508,70509],{"class":278}," \"Illustrative cost (USD)\"",[53,70511,1298],{"class":63},[53,70513,61576],{"class":278},[53,70515,2705],{"class":131},[53,70517,70414],{"class":59},[53,70519,2767],{"class":131},[53,70521,61622],{"class":278},[53,70523,70524],{"class":55,"line":60581},[53,70525,70526],{"class":63},"}))\n",[53,70528,70529],{"class":55,"line":60587},[53,70530,104],{"emptyLinePlaceholder":103},[53,70532,70533],{"class":55,"line":60597},[53,70534,104],{"emptyLinePlaceholder":103},[53,70536,70537],{"class":55,"line":60607},[53,70538,70539],{"class":110},"# ---------- Build a ready-to-run Colab notebook ----------\n",[53,70541,70542],{"class":55,"line":60616},[53,70543,104],{"emptyLinePlaceholder":103},[53,70545,70546,70549,70551,70554],{"class":55,"line":60625},[53,70547,70548],{"class":63},"md_intro ",[53,70550,121],{"class":120},[53,70552,70553],{"class":59}," r",[53,70555,70556],{"class":58202},"\"\"\"\n",[53,70558,70559],{"class":55,"line":60634},[53,70560,70561],{"class":110},"# Sampling Budget & Bit-String Analogy for Qubit Classes\n",[53,70563,70564],{"class":55,"line":60646},[53,70565,104],{"emptyLinePlaceholder":103},[53,70567,70568,70570,70573,70575,70578,70580,70582,70584,70587,70589,70592,70594],{"class":55,"line":60658},[53,70569,2272],{"class":131},[53,70571,70572],{"class":58202},"High school level ",[53,70574,128],{"class":131},[53,70576,70577],{"class":58202},"concise",[53,70579,3544],{"class":131},[53,70581,24880],{"class":58202},[53,70583,2272],{"class":131},[53,70585,70586],{"class":58202}," We treat each measurement ",[53,70588,128],{"class":131},[53,70590,70591],{"class":58202},"“single shot”",[53,70593,3544],{"class":131},[53,70595,70596],{"class":58202}," 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",[53,70598,70599],{"class":55,"line":60667},[53,70600,104],{"emptyLinePlaceholder":103},[53,70602,70603,70605,70608,70610,70612,70614,70616,70618,70621,70623,70626,70628,70631,70633,70636,70638,70641,70643,70646,70648,70651,70654,70657,70660,70662,70664,70667,70669,70671,70674,70677,70679],{"class":55,"line":60672},[53,70604,2272],{"class":131},[53,70606,70607],{"class":58202},"Graduate level ",[53,70609,128],{"class":131},[53,70611,70577],{"class":58202},[53,70613,3544],{"class":131},[53,70615,24880],{"class":58202},[53,70617,2272],{"class":131},[53,70619,70620],{"class":58202}," We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ ",[53,70622,128],{"class":131},[53,70624,70625],{"class":58202},"noisy intermediate-scale quantum",[53,70627,3544],{"class":131},[53,70629,70630],{"class":58202},", fault-tolerant ",[53,70632,128],{"class":131},[53,70634,70635],{"class":58202},"logical",[53,70637,3544],{"class":131},[53,70639,70640],{"class":58202},", topologically protected, and ideal. We report Wilson score intervals ",[53,70642,128],{"class":131},[53,70644,70645],{"class":58202},"95%",[53,70647,3544],{"class":131},[53,70649,70650],{"class":58202}," for observed proportions and estimate the required shot budget $n ",[53,70652,70653],{"class":120},"\\a",[53,70655,70656],{"class":58202},"pprox z^2",[53,70658,70659],{"class":120},"\\,",[53,70661,12],{"class":58202},[53,70663,128],{"class":131},[53,70665,70666],{"class":58202},"1-p",[53,70668,3544],{"class":131},[53,70670,193],{"class":58202},[53,70672,70673],{"class":120},"\\e",[53,70675,70676],{"class":58202},"psilon^2$ to bound absolute error by ±$",[53,70678,70673],{"class":120},[53,70680,70681],{"class":58202},"psilon$ at ~95% confidence.\n",[53,70683,70684],{"class":55,"line":60677},[53,70685,70556],{"class":58202},[53,70687,70688],{"class":55,"line":60682},[53,70689,104],{"emptyLinePlaceholder":103},[53,70691,70692,70695,70697,70699],{"class":55,"line":60692},[53,70693,70694],{"class":63},"code_config ",[53,70696,121],{"class":120},[53,70698,70553],{"class":59},[53,70700,70556],{"class":58202},[53,70702,70703],{"class":55,"line":60699},[53,70704,70705],{"class":110},"# @title Configuration\n",[53,70707,70708],{"class":55,"line":60706},[53,70709,70710],{"class":58202},"from dataclasses import dataclass\n",[53,70712,70713],{"class":55,"line":60717},[53,70714,70715],{"class":58202},"from typing import List, Dict\n",[53,70717,70718],{"class":55,"line":60728},[53,70719,70720],{"class":58202},"import numpy as np\n",[53,70722,70723],{"class":55,"line":60737},[53,70724,104],{"emptyLinePlaceholder":103},[53,70726,70727,70730],{"class":55,"line":60743},[53,70728,70729],{"class":58202},"RANDOM_SEED = 579345 ",[53,70731,70732],{"class":110},"# @param {type:\"number\"}\n",[53,70734,70735,70738],{"class":55,"line":60753},[53,70736,70737],{"class":58202},"SHOTS_PER_MODEL = 10000 ",[53,70739,70732],{"class":110},[53,70741,70742,70745,70748],{"class":55,"line":60761},[53,70743,70744],{"class":58202},"EPSILONS = ",[53,70746,70747],{"class":131},"[0.02, 0.01, 0.005]",[53,70749,70750],{"class":110}," # @param\n",[53,70752,70753,70756],{"class":55,"line":60768},[53,70754,70755],{"class":58202},"COST_PER_SHOT_USD = 0.00005 ",[53,70757,70732],{"class":110},[53,70759,70760,70763],{"class":55,"line":60775},[53,70761,70762],{"class":58202},"FIXED_OVERHEAD_USD = 0.10 ",[53,70764,70732],{"class":110},[53,70766,70767],{"class":55,"line":60784},[53,70768,104],{"emptyLinePlaceholder":103},[53,70770,70771,70773,70775,70778],{"class":55,"line":60793},[53,70772,68242],{"class":58202},[53,70774,128],{"class":131},[53,70776,70777],{"class":58202},"frozen=True",[53,70779,135],{"class":131},[53,70781,70782],{"class":55,"line":60802},[53,70783,70784],{"class":58202},"class QubitModel:\n",[53,70786,70787],{"class":55,"line":60811},[53,70788,70789],{"class":58202}," name: str\n",[53,70791,70792],{"class":55,"line":60822},[53,70793,70794],{"class":58202}," satisfaction_prob: float\n",[53,70796,70797],{"class":55,"line":60834},[53,70798,70799],{"class":58202}," description: str\n",[53,70801,70802],{"class":55,"line":60846},[53,70803,104],{"emptyLinePlaceholder":103},[53,70805,70806,70809,70812,70815],{"class":55,"line":60855},[53,70807,70808],{"class":58202},"MODELS: List",[53,70810,70811],{"class":131},"[QubitModel]",[53,70813,70814],{"class":58202}," = ",[53,70816,70817],{"class":131},"[\n",[53,70819,70820],{"class":55,"line":60860},[53,70821,70822],{"class":131}," QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n",[53,70824,70825],{"class":55,"line":60865},[53,70826,70827],{"class":131}," QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n",[53,70829,70830],{"class":55,"line":60870},[53,70831,70832],{"class":131}," QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n",[53,70834,70835],{"class":55,"line":60880},[53,70836,70837],{"class":131}," QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n",[53,70839,70840],{"class":55,"line":60887},[53,70841,70842],{"class":131}," QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n",[53,70844,70845],{"class":55,"line":60894},[53,70846,1689],{"class":131},[53,70848,70849],{"class":55,"line":60906},[53,70850,104],{"emptyLinePlaceholder":103},[53,70852,70853,70856,70858,70861],{"class":55,"line":60917},[53,70854,70855],{"class":58202},"rng = np.random.default_rng",[53,70857,128],{"class":131},[53,70859,70860],{"class":58202},"RANDOM_SEED",[53,70862,135],{"class":131},[53,70864,70865],{"class":55,"line":60926},[53,70866,70556],{"class":58202},[53,70868,70869],{"class":55,"line":60932},[53,70870,104],{"emptyLinePlaceholder":103},[53,70872,70873,70876,70878,70880],{"class":55,"line":60948},[53,70874,70875],{"class":63},"code_lib_and_run ",[53,70877,121],{"class":120},[53,70879,70553],{"class":59},[53,70881,70556],{"class":58202},[53,70883,70884],{"class":55,"line":60953},[53,70885,70886],{"class":110},"# @title Run Simulation & Display Results\n",[53,70888,70889],{"class":55,"line":60962},[53,70890,70891],{"class":58202},"import math\n",[53,70893,70894],{"class":55,"line":60972},[53,70895,70896],{"class":58202},"from typing import Tuple, Dict\n",[53,70898,70899],{"class":55,"line":60981},[53,70900,70901],{"class":58202},"import pandas as pd\n",[53,70903,70904],{"class":55,"line":60990},[53,70905,70720],{"class":58202},[53,70907,70908],{"class":55,"line":60996},[53,70909,70910],{"class":58202},"import matplotlib.pyplot as plt\n",[53,70912,70913],{"class":55,"line":61005},[53,70914,70915],{"class":58202},"from IPython.display import display, Markdown\n",[53,70917,70918],{"class":55,"line":61017},[53,70919,104],{"emptyLinePlaceholder":103},[53,70921,70922,70925,70927,70930,70932,70935,70938],{"class":55,"line":61030},[53,70923,70924],{"class":58202},"def wilson_ci",[53,70926,128],{"class":131},[53,70928,70929],{"class":58202},"k: int, n: int, z: float = 1.96",[53,70931,3544],{"class":131},[53,70933,70934],{"class":58202}," -> Tuple",[53,70936,70937],{"class":131},"[float, float]",[53,70939,1623],{"class":58202},[53,70941,70942,70945,70947,70950],{"class":55,"line":61042},[53,70943,70944],{"class":58202}," if n == 0: return ",[53,70946,128],{"class":131},[53,70948,70949],{"class":58202},"0.0, 1.0",[53,70951,135],{"class":131},[53,70953,70954,70957,70959],{"class":55,"line":61047},[53,70955,70956],{"class":58202}," p_hat, z2_n = k \u002F n, z",[53,70958,2272],{"class":131},[53,70960,70961],{"class":58202},"2 \u002F n\n",[53,70963,70964,70967,70969],{"class":55,"line":61052},[53,70965,70966],{"class":58202}," denom = 1.0 ",[53,70968,2002],{"class":131},[53,70970,70971],{"class":58202}," z2_n\n",[53,70973,70974,70977,70979,70982,70984,70987,70989],{"class":55,"line":61057},[53,70975,70976],{"class":58202}," center = ",[53,70978,128],{"class":131},[53,70980,70981],{"class":58202},"p_hat ",[53,70983,2002],{"class":131},[53,70985,70986],{"class":58202}," z2_n \u002F 2",[53,70988,3544],{"class":131},[53,70990,70991],{"class":58202}," \u002F denom\n",[53,70993,70994,70997,70999,71002,71004,71007,71009,71011,71013,71015,71018,71020,71023,71025,71028,71030,71033,71035,71038,71040,71043,71045,71047,71050],{"class":55,"line":61076},[53,70995,70996],{"class":58202}," span = ",[53,70998,128],{"class":131},[53,71000,71001],{"class":58202},"z ",[53,71003,2286],{"class":131},[53,71005,71006],{"class":58202}," math.sqrt",[53,71008,128],{"class":131},[53,71010,70981],{"class":58202},[53,71012,2286],{"class":131},[53,71014,58540],{"class":131},[53,71016,71017],{"class":58202},"1 - p_hat",[53,71019,3544],{"class":131},[53,71021,71022],{"class":58202}," \u002F n ",[53,71024,2002],{"class":131},[53,71026,71027],{"class":58202}," z",[53,71029,2272],{"class":131},[53,71031,71032],{"class":58202},"2 \u002F ",[53,71034,128],{"class":131},[53,71036,71037],{"class":58202},"4 ",[53,71039,2286],{"class":131},[53,71041,71042],{"class":58202}," n",[53,71044,2272],{"class":131},[53,71046,132],{"class":58202},[53,71048,71049],{"class":131},")))",[53,71051,70991],{"class":58202},[53,71053,71054,71057,71059,71061,71063,71066,71068,71071,71073,71076,71078,71081],{"class":55,"line":61082},[53,71055,71056],{"class":58202}," return ",[53,71058,128],{"class":131},[53,71060,36608],{"class":58202},[53,71062,128],{"class":131},[53,71064,71065],{"class":58202},"0.0, center - span",[53,71067,3544],{"class":131},[53,71069,71070],{"class":58202},", min",[53,71072,128],{"class":131},[53,71074,71075],{"class":58202},"1.0, center ",[53,71077,2002],{"class":131},[53,71079,71080],{"class":58202}," span",[53,71082,22435],{"class":131},[53,71084,71085],{"class":55,"line":61091},[53,71086,104],{"emptyLinePlaceholder":103},[53,71088,71089,71092,71094,71097,71099],{"class":55,"line":61098},[53,71090,71091],{"class":58202},"def shots_for_epsilon",[53,71093,128],{"class":131},[53,71095,71096],{"class":58202},"p: float, epsilon: float = 0.01, z: float = 1.96",[53,71098,3544],{"class":131},[53,71100,71101],{"class":58202}," -> int:\n",[53,71103,71104,71107,71109,71111,71113,71115],{"class":55,"line":61105},[53,71105,71106],{"class":58202}," if p ",[53,71108,2286],{"class":131},[53,71110,58540],{"class":131},[53,71112,70666],{"class":58202},[53,71114,3544],{"class":131},[53,71116,71117],{"class":58202}," == 0: return 1\n",[53,71119,71120,71123,71125,71128,71130,71133,71135,71137,71139,71142,71144,71146,71148,71150,71152,71155,71158,71160,71162,71164,71166],{"class":55,"line":61113},[53,71121,71122],{"class":58202}," return max",[53,71124,128],{"class":131},[53,71126,71127],{"class":58202},"1, int",[53,71129,128],{"class":131},[53,71131,71132],{"class":58202},"math.ceil",[53,71134,63376],{"class":131},[53,71136,214],{"class":58202},[53,71138,2272],{"class":131},[53,71140,71141],{"class":58202},"2 ",[53,71143,2286],{"class":131},[53,71145,68138],{"class":58202},[53,71147,2286],{"class":131},[53,71149,58540],{"class":131},[53,71151,70666],{"class":58202},[53,71153,71154],{"class":131},"))",[53,71156,71157],{"class":58202}," \u002F ",[53,71159,128],{"class":131},[53,71161,68059],{"class":58202},[53,71163,2272],{"class":131},[53,71165,132],{"class":58202},[53,71167,71168],{"class":131},"))))\n",[53,71170,71171],{"class":55,"line":61124},[53,71172,104],{"emptyLinePlaceholder":103},[53,71174,71175,71178,71180,71183,71185],{"class":55,"line":61135},[53,71176,71177],{"class":58202},"def summarize_model",[53,71179,128],{"class":131},[53,71181,71182],{"class":58202},"model, bits: np.ndarray, epsilons",[53,71184,3544],{"class":131},[53,71186,71187],{"class":58202}," -> Dict:\n",[53,71189,71190,71193,71195,71197],{"class":55,"line":61146},[53,71191,71192],{"class":58202}," n = len",[53,71194,128],{"class":131},[53,71196,68742],{"class":58202},[53,71198,135],{"class":131},[53,71200,71201,71204,71206,71208],{"class":55,"line":61151},[53,71202,71203],{"class":58202}," ones = sum",[53,71205,128],{"class":131},[53,71207,68742],{"class":58202},[53,71209,135],{"class":131},[53,71211,71212,71215,71217,71220],{"class":55,"line":61160},[53,71213,71214],{"class":58202}," lo, hi = wilson_ci",[53,71216,128],{"class":131},[53,71218,71219],{"class":58202},"ones, n",[53,71221,135],{"class":131},[53,71223,71224],{"class":55,"line":61167},[53,71225,71226],{"class":58202}," row = {\n",[53,71228,71229,71232,71234,71237,71239],{"class":55,"line":61174},[53,71230,71231],{"class":58202}," \"Model\": model.name, \"p ",[53,71233,128],{"class":131},[53,71235,71236],{"class":58202},"target",[53,71238,3544],{"class":131},[53,71240,71241],{"class":58202},"\": model.satisfaction_prob, \"Shots\": n,\n",[53,71243,71244,71247,71249,71252,71254],{"class":55,"line":61182},[53,71245,71246],{"class":58202}," \"Ones\": ones, \"p ",[53,71248,128],{"class":131},[53,71250,71251],{"class":58202},"observed",[53,71253,3544],{"class":131},[53,71255,71256],{"class":58202},"\": ones \u002F n if n > 0 else 0,\n",[53,71258,71259],{"class":55,"line":61203},[53,71260,71261],{"class":58202}," \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n",[53,71263,71264],{"class":55,"line":61214},[53,71265,59379],{"class":58202},[53,71267,71268],{"class":55,"line":61226},[53,71269,71270],{"class":58202}," for eps in epsilons:\n",[53,71272,71273,71276,71279,71282,71284,71287],{"class":55,"line":61237},[53,71274,71275],{"class":58202}," row",[53,71277,71278],{"class":131},"[f\"Shots for ±{eps:.1%}\"]",[53,71280,71281],{"class":58202}," = shots_for_epsilon",[53,71283,128],{"class":131},[53,71285,71286],{"class":58202},"model.satisfaction_prob, eps",[53,71288,135],{"class":131},[53,71290,71291],{"class":55,"line":61248},[53,71292,71293],{"class":58202}," return row\n",[53,71295,71296],{"class":55,"line":61253},[53,71297,104],{"emptyLinePlaceholder":103},[53,71299,71300],{"class":55,"line":61258},[53,71301,71302],{"class":110},"# --- Run Simulation ---\n",[53,71304,71305,71308],{"class":55,"line":61263},[53,71306,71307],{"class":58202},"summary_rows = ",[53,71309,71310],{"class":131},"[summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\n",[53,71312,71313,71316,71318,71321],{"class":55,"line":61277},[53,71314,71315],{"class":58202},"summary_df = pd.DataFrame",[53,71317,128],{"class":131},[53,71319,71320],{"class":58202},"summary_rows",[53,71322,135],{"class":131},[53,71324,71325],{"class":55,"line":61289},[53,71326,71327],{"class":58202},"if COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n",[53,71329,71330,71333,71336,71339,71341,71344,71347,71349],{"class":55,"line":61295},[53,71331,71332],{"class":58202}," summary_df",[53,71334,71335],{"class":131},"[\"Cost (USD)\"]",[53,71337,71338],{"class":58202}," = FIXED_OVERHEAD_USD ",[53,71340,2002],{"class":131},[53,71342,71343],{"class":58202}," summary_df",[53,71345,71346],{"class":131},"[\"Shots\"]",[53,71348,8124],{"class":131},[53,71350,71351],{"class":58202}," COST_PER_SHOT_USD\n",[53,71353,71354],{"class":55,"line":61300},[53,71355,104],{"emptyLinePlaceholder":103},[53,71357,71358,71360,71362,71364,71366,71368],{"class":55,"line":61308},[53,71359,70375],{"class":58202},[53,71361,128],{"class":131},[53,71363,70380],{"class":58202},[53,71365,128],{"class":131},[53,71367,22176],{"class":58202},[53,71369,71370],{"class":110},"### Simulation Summary\"))\n",[53,71372,71373,71375,71377,71380,71382,71385,71387,71390,71392,71395],{"class":55,"line":61318},[53,71374,70375],{"class":58202},[53,71376,128],{"class":131},[53,71378,71379],{"class":58202},"summary_df.style.format",[53,71381,128],{"class":131},[53,71383,71384],{"class":58202},"precision=4",[53,71386,3544],{"class":131},[53,71388,71389],{"class":58202},".background_gradient",[53,71391,128],{"class":131},[53,71393,71394],{"class":58202},"cmap='viridis', subset=",[53,71396,71397],{"class":131},"['p (observed)', 'CI width']))\n",[53,71399,71400],{"class":55,"line":61323},[53,71401,104],{"emptyLinePlaceholder":103},[53,71403,71404],{"class":55,"line":61335},[53,71405,71406],{"class":110},"# --- Plotting ---\n",[53,71408,71409,71412,71414,71416],{"class":55,"line":61346},[53,71410,71411],{"class":58202},"plt.style.use",[53,71413,128],{"class":131},[53,71415,69612],{"class":58202},[53,71417,135],{"class":131},[53,71419,71420,71423,71425,71428,71430,71433,71435,71438,71440,71443],{"class":55,"line":61356},[53,71421,71422],{"class":58202},"fig, ",[53,71424,128],{"class":131},[53,71426,71427],{"class":58202},"ax1, ax2",[53,71429,3544],{"class":131},[53,71431,71432],{"class":58202}," = plt.subplots",[53,71434,128],{"class":131},[53,71436,71437],{"class":58202},"1, 2, figsize=",[53,71439,128],{"class":131},[53,71441,71442],{"class":58202},"16, 6",[53,71444,22435],{"class":131},[53,71446,71447],{"class":55,"line":61367},[53,71448,104],{"emptyLinePlaceholder":103},[53,71450,71451],{"class":55,"line":61372},[53,71452,71453],{"class":110},"# Plot 1\n",[53,71455,71456,71459,71462,71465],{"class":55,"line":61377},[53,71457,71458],{"class":58202},"y = summary_df",[53,71460,71461],{"class":131},"[\"p (observed)\"]",[53,71463,71464],{"class":58202},".to_numpy",[53,71466,329],{"class":131},[53,71468,71469,71472],{"class":55,"line":61389},[53,71470,71471],{"class":58202},"lower_error = y - summary_df",[53,71473,71474],{"class":131},"[\"CI 95% low\"]\n",[53,71476,71477,71480,71483],{"class":55,"line":61400},[53,71478,71479],{"class":58202},"upper_error = summary_df",[53,71481,71482],{"class":131},"[\"CI 95% high\"]",[53,71484,71485],{"class":58202}," - y\n",[53,71487,71488,71491],{"class":55,"line":61409},[53,71489,71490],{"class":58202},"yerr = np.vstack",[53,71492,71493],{"class":131},"([np.abs(lower_error), np.abs(upper_error)])\n",[53,71495,71496],{"class":55,"line":61420},[53,71497,104],{"emptyLinePlaceholder":103},[53,71499,71500,71503,71505,71508,71511,71514],{"class":55,"line":61425},[53,71501,71502],{"class":58202},"ax1.bar",[53,71504,128],{"class":131},[53,71506,71507],{"class":58202},"summary_df",[53,71509,71510],{"class":131},"[\"Model\"]",[53,71512,71513],{"class":58202},", y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black'",[53,71515,135],{"class":131},[53,71517,71518,71521,71523,71526,71528,71531,71533,71536],{"class":55,"line":61430},[53,71519,71520],{"class":58202},"ax1.set_title",[53,71522,128],{"class":131},[53,71524,71525],{"class":58202},"\"Observed Success Rate ",[53,71527,128],{"class":131},[53,71529,71530],{"class":58202},"95% Wilson CI",[53,71532,3544],{"class":131},[53,71534,71535],{"class":58202},"\", fontsize=14",[53,71537,135],{"class":131},[53,71539,71540,71543,71545,71548],{"class":55,"line":61440},[53,71541,71542],{"class":58202},"ax1.tick_params",[53,71544,128],{"class":131},[53,71546,71547],{"class":58202},"axis='x', rotation=30, labelsize=10",[53,71549,135],{"class":131},[53,71551,71552],{"class":55,"line":61446},[53,71553,104],{"emptyLinePlaceholder":103},[53,71555,71556],{"class":55,"line":61460},[53,71557,71558],{"class":110},"# Plot 2\n",[53,71560,71561],{"class":55,"line":61465},[53,71562,71563],{"class":58202},"width = 0.2\n",[53,71565,71566,71569,71571,71573,71575,71578],{"class":55,"line":61475},[53,71567,71568],{"class":58202},"x = np.arange",[53,71570,128],{"class":131},[53,71572,1658],{"class":58202},[53,71574,128],{"class":131},[53,71576,71577],{"class":58202},"MODELS",[53,71579,22435],{"class":131},[53,71581,71582,71585],{"class":55,"line":61480},[53,71583,71584],{"class":58202},"colors = ",[53,71586,71587],{"class":131},"['#ff9999','#66b3ff','#99ff99']\n",[53,71589,71590,71593,71595,71598,71600],{"class":55,"line":61492},[53,71591,71592],{"class":58202},"for i, eps in enumerate",[53,71594,128],{"class":131},[53,71596,71597],{"class":58202},"EPSILONS",[53,71599,3544],{"class":131},[53,71601,1623],{"class":58202},[53,71603,71604,71607],{"class":55,"line":61497},[53,71605,71606],{"class":58202}," req = summary_df",[53,71608,71609],{"class":131},"[f\"Shots for ±{eps:.1%}\"]\n",[53,71611,71612,71615,71617,71619,71621,71623,71625,71628,71631,71634],{"class":55,"line":61502},[53,71613,71614],{"class":58202}," ax2.bar",[53,71616,128],{"class":131},[53,71618,69641],{"class":58202},[53,71620,2002],{"class":131},[53,71622,8141],{"class":58202},[53,71624,2286],{"class":131},[53,71626,71627],{"class":58202}," width, req, width, label=f\"±{eps:.1%}\", color=colors",[53,71629,71630],{"class":131},"[i]",[53,71632,71633],{"class":58202},", edgecolor='black'",[53,71635,135],{"class":131},[53,71637,71638],{"class":55,"line":61525},[53,71639,104],{"emptyLinePlaceholder":103},[53,71641,71642,71645,71647,71649,71651,71654],{"class":55,"line":61530},[53,71643,71644],{"class":58202},"ax2.set_xticks",[53,71646,128],{"class":131},[53,71648,69641],{"class":58202},[53,71650,2002],{"class":131},[53,71652,71653],{"class":58202}," width, summary_df",[53,71655,71656],{"class":131},"[\"Model\"])\n",[53,71658,71659,71662,71664,71667,71669,71672,71674,71676],{"class":55,"line":61553},[53,71660,71661],{"class":58202},"ax2.set_ylabel",[53,71663,128],{"class":131},[53,71665,71666],{"class":58202},"\"Required Shots ",[53,71668,128],{"class":131},[53,71670,71671],{"class":58202},"Log Scale",[53,71673,3544],{"class":131},[53,71675,22176],{"class":58202},[53,71677,135],{"class":131},[53,71679,71680,71683,71685,71687],{"class":55,"line":61564},[53,71681,71682],{"class":58202},"ax2.set_yscale",[53,71684,128],{"class":131},[53,71686,70272],{"class":58202},[53,71688,135],{"class":131},[53,71690,71691,71694,71696,71699],{"class":55,"line":61603},[53,71692,71693],{"class":58202},"ax2.set_title",[53,71695,128],{"class":131},[53,71697,71698],{"class":58202},"\"Shots Needed for ±ε Tolerance\", fontsize=14",[53,71700,135],{"class":131},[53,71702,71703,71706,71708,71710],{"class":55,"line":61613},[53,71704,71705],{"class":58202},"ax2.tick_params",[53,71707,128],{"class":131},[53,71709,71547],{"class":58202},[53,71711,135],{"class":131},[53,71713,71714,71717,71719,71722],{"class":55,"line":61625},[53,71715,71716],{"class":58202},"ax2.legend",[53,71718,128],{"class":131},[53,71720,71721],{"class":58202},"title=\"Tolerance\"",[53,71723,135],{"class":131},[53,71725,71726],{"class":55,"line":61645},[53,71727,104],{"emptyLinePlaceholder":103},[53,71729,71730,71733],{"class":55,"line":61663},[53,71731,71732],{"class":58202},"plt.tight_layout",[53,71734,329],{"class":131},[53,71736,71737,71740],{"class":55,"line":61668},[53,71738,71739],{"class":58202},"plt.show",[53,71741,329],{"class":131},[53,71743,71744],{"class":55,"line":61673},[53,71745,70556],{"class":58202},[53,71747,71748],{"class":55,"line":61685},[53,71749,104],{"emptyLinePlaceholder":103},[53,71751,71752,71755,71757,71760,71763],{"class":55,"line":61695},[53,71753,71754],{"class":63},"nb 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nbf.v4.",[53,71785,71786],{"class":124},"new_markdown_cell",[53,71788,71789],{"class":63},"(md_intro),\n",[53,71791,71792,71794,71797],{"class":55,"line":61715},[53,71793,71783],{"class":63},[53,71795,71796],{"class":124},"new_code_cell",[53,71798,71799],{"class":63},"(code_config),\n",[53,71801,71802,71804,71806],{"class":55,"line":61723},[53,71803,71783],{"class":63},[53,71805,71796],{"class":124},[53,71807,71808],{"class":63},"(code_lib_and_run),\n",[53,71810,71811],{"class":55,"line":61733},[53,71812,1689],{"class":63},[53,71814,71815],{"class":55,"line":61748},[53,71816,104],{"emptyLinePlaceholder":103},[53,71818,71819,71822,71824],{"class":55,"line":61759},[53,71820,71821],{"class":63},"nb_path ",[53,71823,121],{"class":120},[53,71825,71826],{"class":278}," \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\n",[53,71828,71829,71831,71833,71836,71838,71840,71843,71845,71848,71850,71852],{"class":55,"line":61771},[53,71830,69522],{"class":59},[53,71832,69525],{"class":120},[53,71834,71835],{"class":63},"(nb_path, ",[53,71837,69535],{"class":278},[53,71839,170],{"class":63},[53,71841,71842],{"class":2050},"encoding",[53,71844,121],{"class":120},[53,71846,71847],{"class":278},"\"utf-8\"",[53,71849,203],{"class":63},[53,71851,67],{"class":59},[53,71853,69542],{"class":63},[53,71855,71856,71859,71862],{"class":55,"line":61776},[53,71857,71858],{"class":63}," nbf.",[53,71860,71861],{"class":124},"write",[53,71863,71864],{"class":63},"(nb, f)\n",[53,71866,71867],{"class":55,"line":61783},[53,71868,104],{"emptyLinePlaceholder":103},[53,71870,71871],{"class":55,"line":61792},[53,71872,71873],{"class":110},"# ---------- Final confirmation 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",[53,71957,2705],{"class":131},[53,71959,71960],{"class":63},"summary_csv_path",[53,71962,2767],{"class":131},[53,71964,22176],{"class":278},[53,71966,135],{"class":63},[53,71968,71969,71971,71973,71975,71978,71980,71983,71985,71987,71989,71992,71994,71996],{"class":55,"line":61849},[53,71970,8420],{"class":120},[53,71972,128],{"class":63},[53,71974,22173],{"class":59},[53,71976,71977],{"class":278},"\"Plots saved to: ",[53,71979,2705],{"class":131},[53,71981,71982],{"class":63},"fig1_path",[53,71984,2767],{"class":131},[53,71986,170],{"class":278},[53,71988,2705],{"class":131},[53,71990,71991],{"class":63},"fig2_path",[53,71993,2767],{"class":131},[53,71995,22176],{"class":278},[53,71997,135],{"class":63},[43,71999,72002],{"className":72000,"code":72001,"language":6934,"meta":48},[8426],"Running simulation...\nSaved data to CSV and JSON files.\n",[50,72003,72001],{"__ignoreMap":48},[379,72005],{"alt":58279,"src":72006},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.webp",[379,72008],{"alt":63185,"src":72009},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-02.webp",[43,72011,72014],{"className":72012,"code":72013,"language":6934,"meta":48},[8426],"Generated and saved plots.\n",[50,72015,72013],{"__ignoreMap":48},[43,72017,72020],{"className":72018,"code":72019,"language":6934,"meta":48},[8426],"==================================================\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",[50,72021,72019],{"__ignoreMap":48},[72023,72024],"hr",{},[30,72026,41452],{"id":41451},[27266,72028,72029,72036,72043,72050,72057,72064,72071,72078,72085,72092,72099,72106,72113,72124,72131,72138],{},[998,72030,72031,72032,72035],{},"Agresti, A. and Coull, B.A. (1998) 'Approximate is better than “exact” for interval estimation of binomial proportions', ",[924,72033,72034],{},"The American Statistician",", 52(2), pp. 119-126.",[998,72037,72038,72039,72042],{},"Brown, L.D., Cai, T.T. and DasGupta, A. (2001) 'Interval Estimation for a Binomial Proportion', ",[924,72040,72041],{},"Statistical Science",", 16(2), pp. 101-133.",[998,72044,72045,72046,72049],{},"Cochran, W.G. (1977) ",[924,72047,72048],{},"Sampling Techniques",". 3rd edn. New York: John Wiley & Sons.",[998,72051,72052,72053,72056],{},"Gottesman, D. (2009) 'An introduction to quantum error correction and fault-tolerant quantum computation', in ",[924,72054,72055],{},"Quantum Information Science and Its Contributions to Mathematics, Proceedings of Symposia in Applied Mathematics",", 72, pp. 13-58.",[998,72058,72059,72060,72063],{},"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', ",[924,72061,72062],{},"Nature",", 585, pp. 357–362.",[998,72065,72066,72067,72070],{},"Hunter, J.D. (2007) 'Matplotlib: A 2D Graphics Environment', ",[924,72068,72069],{},"Computing in Science & Engineering",", 9(3), pp. 90-95.",[998,72072,72073,72074,72077],{},"Kitaev, A.Y. (2003) 'Fault-tolerant quantum computation by anyons', ",[924,72075,72076],{},"Annals of Physics",", 303(1), pp. 2-30.",[998,72079,72080,72081,72084],{},"McKinney, W. (2010) 'Data Structures for Statistical Computing in Python', in ",[924,72082,72083],{},"Proceedings of the 9th Python in Science Conference",", pp. 56-61.",[998,72086,72087,72088,72091],{},"Nayak, C., Simon, S.H., Stern, A., Freedman, M. and Das Sarma, S. (2008) 'Non-Abelian anyons and topological quantum computation', ",[924,72089,72090],{},"Reviews of Modern Physics",", 80(3), pp. 1083-1159.",[998,72093,72094,72095,72098],{},"Newman, M. (2013) ",[924,72096,72097],{},"Computational Physics",". CreateSpace Independent Publishing Platform.",[998,72100,72101,72102,72105],{},"Nielsen, M.A. and Chuang, I.L. (2010) ",[924,72103,72104],{},"Quantum Computation and Quantum Information: 10th Anniversary Edition",". Cambridge: Cambridge University Press.",[998,72107,72108,72109,72112],{},"Preskill, J. (2018) 'Quantum Computing in the NISQ era and beyond', ",[924,72110,72111],{},"Quantum",", 2, p. 79.",[998,72114,72115,72116,72119,72120,28],{},"The pandas development team (2020) ",[924,72117,72118],{},"pandas-dev\u002Fpandas: Pandas",". Zenodo. Available at: ",[19,72121,72123],{"href":72122},"https:\u002F\u002Fwww.google.com\u002Fsearch?q=https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134","https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134",[998,72125,72126,72127,72130],{},"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', ",[924,72128,72129],{},"Nature Methods",", 17, pp. 261–272.",[998,72132,72133,72134,72137],{},"Wilde, M.M. (2017) ",[924,72135,72136],{},"Quantum Information Theory",". 2nd edn. Cambridge: Cambridge University Press.",[998,72139,72140,72141,72144],{},"Wilson, E.B. (1927) 'Probable inference, the law of succession, and statistical inference', ",[924,72142,72143],{},"Journal of the American Statistical Association",", 22(158), pp. 209-212.",[30,72146,488],{"id":487},[12,72148,41494,72149,495,72151,28],{},[19,72150,41566],{"href":67582},[19,72152,67530],{"href":72153},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[501,72155,72156],{},"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":48,"searchDepth":73,"depth":73,"links":72158},[72159,72160],{"id":41451,"depth":73,"text":41452},{"id":487,"depth":73,"text":488},[517,518,519,72162],"Sampling budget",[72164],{"username":7,"name":523,"role":524,"bio":525,"links":72165},[72166,72167],{"label":528,"href":529},{"label":531,"href":67582},{"username":7,"name":523,"role":533},"A Sampling-Budget & Bit-String Analogy for Qubit Classifications. 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