[{"data":1,"prerenderedAt":110597},["ShallowReactive",2],{"blog-post-expert-notes\u002Fkrylov-quantum-diagonalization":3,"sibling-dives-expert-notes\u002Fkrylov-quantum-diagonalization":40723,"learn-track-expert-notes\u002Fkrylov-quantum-diagonalization":110596},{"id":4,"title":5,"authors":6,"body":8,"breadcrumb":40691,"builders":40696,"byline":40704,"challenge":40706,"courseAuthor":40706,"courseLead":40706,"dek":40706,"description":40707,"draft":40708,"extension":40709,"eyebrow":40710,"finish":40706,"fork":40706,"hero":40706,"heroAlt":40706,"heroCta":40706,"heroImage":40706,"kind":40711,"lessonCount":40706,"meta":40712,"navigation":227,"newsItems":40706,"next":40706,"ogImage":40706,"order":40706,"outcomes":40706,"path":40713,"publishDate":40714,"readingTime":40715,"related":40716,"relatedProjects":40706,"seo":40717,"stem":40718,"tags":40719,"track":40706,"trackName":40706,"__hash__":40722},"blog\u002Fblog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization.md","Krylov Quantum Diagonalization of Lattice Hamiltonians",[7],"mirko-amico",{"type":9,"value":10,"toc":40662},"minimark",[11,24,29,54,59,62,90,94,97,116,120,1550,1554,1559,1824,2136,2803,3123,3126,3130,3434,3695,3966,4170,4949,5186,5189,5427,5749,5756,6703,6708,7194,7572,7579,7583,7682,8018,8024,8027,8075,8079,8161,8228,8233,8237,8345,8469,8475,8479,8485,9756,10585,11135,11138,12126,12518,12765,13177,13183,13186,13189,13256,13262,13266,13270,13273,13279,13923,16690,16898,18117,19351,19396,19400,19949,20381,20384,20388,20479,20482,20607,20610,20668,20895,20898,20953,20959,20962,20966,20969,21076,21080,21083,21472,21475,21691,21697,21701,22169,22173,22228,22558,22564,22656,23029,23033,23058,23062,23102,23192,23223,23549,23562,23625,23964,23976,24040,24212,24296,24389,24395,24398,24415,24421,24670,24674,24677,25051,25177,25773,25805,26270,26352,26358,26424,27141,27244,27507,27761,28092,28275,28617,29357,29671,31286,31318,32814,33291,33985,34768,35102,35429,35491,35500,35504,35703,36061,36064,36592,36595,36895,36951,37215,37708,37967,38909,39913,40353,40608,40612,40621,40626,40631,40636,40640,40643,40649,40653,40658],[12,13,14],"p",{},[15,16,17,18,23],"em",{},"Republished with the author's permission from ",[19,20,22],"a",{"href":21},"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.",[12,25,26],{},[15,27,28],{},"Usage estimate: 20 minutes on a Heron r2 (NOTE: This is an estimate only. Your runtime might vary.)",[30,31,36],"pre",{"className":32,"code":33,"language":34,"meta":35,"style":35},"language-python shiki shiki-themes one-dark-pro","# This cell is hidden from users – it disables some lint rules\n# ruff: noqa: E402 E722 F601\n","python","",[37,38,39,48],"code",{"__ignoreMap":35},[40,41,44],"span",{"class":42,"line":43},"line",1,[40,45,47],{"class":46},"sV9Aq","# This cell is hidden from users – it disables some lint rules\n",[40,49,51],{"class":42,"line":50},2,[40,52,53],{"class":46},"# ruff: noqa: E402 E722 F601\n",[55,56,58],"h2",{"id":57},"background","Background",[12,60,61],{},"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.",[12,63,64,65,69,70,74,75,79,80,84,85,89],{},"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, ",[19,66,68],{"href":67},"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, ",[19,71,73],{"href":72},"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 ",[19,76,78],{"href":77},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2312.00178","review paper","), the Krylov quantum diagonalization (KQD) algorithm. This algorithm performs well at large scale ",[19,81,83],{"href":82},"#references","[1]"," on existing quantum hardware, shares similar ",[19,86,88],{"href":87},"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.",[55,91,93],{"id":92},"requirements","Requirements",[12,95,96],{},"Before starting this tutorial, be sure you have the following installed:",[98,99,100,109],"ul",{},[101,102,103,104,108],"li",{},"Qiskit SDK v2.0 or later, with ",[19,105,107],{"href":106},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fvisualization","visualization"," support",[101,110,111,112,115],{},"Qiskit Runtime v0.22 or later ( ",[37,113,114],{},"pip install qiskit-ibm-runtime"," )",[55,117,119],{"id":118},"setup","Setup",[30,121,123],{"className":32,"code":122,"language":34,"meta":35,"style":35},"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",[37,124,125,141,153,166,180,193,201,214,222,229,249,254,267,280,293,306,319,332,345,350,355,368,374,385,390,395,400,412,422,430,446,466,484,500,506,512,517,523,529,535,541,547,553,559,565,571,577,583,588,594,600,605,610,615,633,644,660,693,699,727,750,767,776,806,814,834,842,849,869,876,892,897,902,924,929,935,940,951,971,1011,1016,1026,1041,1074,1079,1090,1095,1109,1114,1125,1153,1172,1177,1202,1207,1218,1262,1271,1276,1287,1302,1317,1328,1333,1359,1375,1394,1436,1459,1465,1473,1482,1487,1503,1508,1528,1541],{"__ignoreMap":35},[40,126,127,131,135,138],{"class":42,"line":43},[40,128,130],{"class":129},"seHd6","import",[40,132,134],{"class":133},"sn6KH"," numpy ",[40,136,137],{"class":129},"as",[40,139,140],{"class":133}," np\n",[40,142,143,145,148,150],{"class":42,"line":50},[40,144,130],{"class":129},[40,146,147],{"class":133}," scipy ",[40,149,137],{"class":129},[40,151,152],{"class":133}," sp\n",[40,154,156,158,161,163],{"class":42,"line":155},3,[40,157,130],{"class":129},[40,159,160],{"class":133}," matplotlib.pylab ",[40,162,137],{"class":129},[40,164,165],{"class":133}," plt\n",[40,167,169,172,175,177],{"class":42,"line":168},4,[40,170,171],{"class":129},"from",[40,173,174],{"class":133}," typing ",[40,176,130],{"class":129},[40,178,179],{"class":133}," Union, List\n",[40,181,183,185,188,190],{"class":42,"line":182},5,[40,184,130],{"class":129},[40,186,187],{"class":133}," itertools ",[40,189,137],{"class":129},[40,191,192],{"class":133}," it\n",[40,194,196,198],{"class":42,"line":195},6,[40,197,130],{"class":129},[40,199,200],{"class":133}," copy\n",[40,202,204,206,209,211],{"class":42,"line":203},7,[40,205,171],{"class":129},[40,207,208],{"class":133}," sympy ",[40,210,130],{"class":129},[40,212,213],{"class":133}," Matrix\n",[40,215,217,219],{"class":42,"line":216},8,[40,218,130],{"class":129},[40,220,221],{"class":133}," warnings\n",[40,223,225],{"class":42,"line":224},9,[40,226,228],{"emptyLinePlaceholder":227},true,"\n",[40,230,232,235,239,242,246],{"class":42,"line":231},10,[40,233,234],{"class":133},"warnings.",[40,236,238],{"class":237},"sVbv2","filterwarnings",[40,240,241],{"class":133},"(",[40,243,245],{"class":244},"subq3","\"ignore\"",[40,247,248],{"class":133},")\n",[40,250,252],{"class":42,"line":251},11,[40,253,228],{"emptyLinePlaceholder":227},[40,255,257,259,262,264],{"class":42,"line":256},12,[40,258,171],{"class":129},[40,260,261],{"class":133}," qiskit.quantum_info ",[40,263,130],{"class":129},[40,265,266],{"class":133}," SparsePauliOp, Pauli, StabilizerState\n",[40,268,270,272,275,277],{"class":42,"line":269},13,[40,271,171],{"class":129},[40,273,274],{"class":133}," qiskit.circuit ",[40,276,130],{"class":129},[40,278,279],{"class":133}," Parameter, IfElseOp\n",[40,281,283,285,288,290],{"class":42,"line":282},14,[40,284,171],{"class":129},[40,286,287],{"class":133}," qiskit ",[40,289,130],{"class":129},[40,291,292],{"class":133}," QuantumCircuit, QuantumRegister\n",[40,294,296,298,301,303],{"class":42,"line":295},15,[40,297,171],{"class":129},[40,299,300],{"class":133}," qiskit.circuit.library ",[40,302,130],{"class":129},[40,304,305],{"class":133}," PauliEvolutionGate\n",[40,307,309,311,314,316],{"class":42,"line":308},16,[40,310,171],{"class":129},[40,312,313],{"class":133}," qiskit.synthesis ",[40,315,130],{"class":129},[40,317,318],{"class":133}," LieTrotter\n",[40,320,322,324,327,329],{"class":42,"line":321},17,[40,323,171],{"class":129},[40,325,326],{"class":133}," qiskit.transpiler ",[40,328,130],{"class":129},[40,330,331],{"class":133}," Target, CouplingMap\n",[40,333,335,337,340,342],{"class":42,"line":334},18,[40,336,171],{"class":129},[40,338,339],{"class":133}," qiskit.transpiler.preset_passmanagers ",[40,341,130],{"class":129},[40,343,344],{"class":133}," generate_preset_pass_manager\n",[40,346,348],{"class":42,"line":347},19,[40,349,228],{"emptyLinePlaceholder":227},[40,351,353],{"class":42,"line":352},20,[40,354,228],{"emptyLinePlaceholder":227},[40,356,358,360,363,365],{"class":42,"line":357},21,[40,359,171],{"class":129},[40,361,362],{"class":133}," qiskit_ibm_runtime ",[40,364,130],{"class":129},[40,366,367],{"class":133}," (\n",[40,369,371],{"class":42,"line":370},22,[40,372,373],{"class":133},"    QiskitRuntimeService,\n",[40,375,377,380,382],{"class":42,"line":376},23,[40,378,379],{"class":133},"    EstimatorV2 ",[40,381,137],{"class":129},[40,383,384],{"class":133}," Estimator,\n",[40,386,388],{"class":42,"line":387},24,[40,389,248],{"class":133},[40,391,393],{"class":42,"line":392},25,[40,394,228],{"emptyLinePlaceholder":227},[40,396,398],{"class":42,"line":397},26,[40,399,228],{"emptyLinePlaceholder":227},[40,401,403,406,409],{"class":42,"line":402},27,[40,404,405],{"class":129},"def",[40,407,408],{"class":237}," solve_regularized_gen_eig",[40,410,411],{"class":133},"(\n",[40,413,415,419],{"class":42,"line":414},28,[40,416,418],{"class":417},"sb9H8","    h",[40,420,421],{"class":133},": np.ndarray,\n",[40,423,425,428],{"class":42,"line":424},29,[40,426,427],{"class":417},"    s",[40,429,421],{"class":133},[40,431,433,436,439,443],{"class":42,"line":432},30,[40,434,435],{"class":417},"    threshold",[40,437,438],{"class":133},": ",[40,440,442],{"class":441},"sjrmR","float",[40,444,445],{"class":133},",\n",[40,447,449,452,454,457,460,464],{"class":42,"line":448},31,[40,450,451],{"class":417},"    k",[40,453,438],{"class":133},[40,455,456],{"class":441},"int",[40,458,459],{"class":441}," =",[40,461,463],{"class":462},"sVC51"," 1",[40,465,445],{"class":133},[40,467,469,472,474,477,479,482],{"class":42,"line":468},32,[40,470,471],{"class":417},"    return_dimn",[40,473,438],{"class":133},[40,475,476],{"class":441},"bool",[40,478,459],{"class":441},[40,480,481],{"class":462}," False",[40,483,445],{"class":133},[40,485,487,490,492,495,497],{"class":42,"line":486},33,[40,488,489],{"class":133},") -> Union[",[40,491,442],{"class":441},[40,493,494],{"class":133},", List[",[40,496,442],{"class":441},[40,498,499],{"class":133},"]]:\n",[40,501,503],{"class":42,"line":502},34,[40,504,505],{"class":244},"    \"\"\"\n",[40,507,509],{"class":42,"line":508},35,[40,510,511],{"class":244},"    Method for solving the generalized eigenvalue problem with regularization\n",[40,513,515],{"class":42,"line":514},36,[40,516,228],{"emptyLinePlaceholder":227},[40,518,520],{"class":42,"line":519},37,[40,521,522],{"class":244},"    Args:\n",[40,524,526],{"class":42,"line":525},38,[40,527,528],{"class":244},"        h (numpy.ndarray):\n",[40,530,532],{"class":42,"line":531},39,[40,533,534],{"class":244},"            The effective representation of the matrix in the Krylov subspace\n",[40,536,538],{"class":42,"line":537},40,[40,539,540],{"class":244},"        s (numpy.ndarray):\n",[40,542,544],{"class":42,"line":543},41,[40,545,546],{"class":244},"            The matrix of overlaps between vectors of the Krylov subspace\n",[40,548,550],{"class":42,"line":549},42,[40,551,552],{"class":244},"        threshold (float):\n",[40,554,556],{"class":42,"line":555},43,[40,557,558],{"class":244},"            Cut-off value for the eigenvalue of s\n",[40,560,562],{"class":42,"line":561},44,[40,563,564],{"class":244},"        k (int):\n",[40,566,568],{"class":42,"line":567},45,[40,569,570],{"class":244},"            Number of eigenvalues to return\n",[40,572,574],{"class":42,"line":573},46,[40,575,576],{"class":244},"        return_dimn (bool):\n",[40,578,580],{"class":42,"line":579},47,[40,581,582],{"class":244},"            Whether to return the size of the regularized subspace\n",[40,584,586],{"class":42,"line":585},48,[40,587,228],{"emptyLinePlaceholder":227},[40,589,591],{"class":42,"line":590},49,[40,592,593],{"class":244},"    Returns:\n",[40,595,597],{"class":42,"line":596},50,[40,598,599],{"class":244},"        lowest k-eigenvalue(s) that are the solution of the regularized generalized eigenvalue problem\n",[40,601,603],{"class":42,"line":602},51,[40,604,228],{"emptyLinePlaceholder":227},[40,606,608],{"class":42,"line":607},52,[40,609,228],{"emptyLinePlaceholder":227},[40,611,613],{"class":42,"line":612},53,[40,614,505],{"class":244},[40,616,618,621,624,627,630],{"class":42,"line":617},54,[40,619,620],{"class":133},"    s_vals, s_vecs ",[40,622,623],{"class":441},"=",[40,625,626],{"class":133}," sp.linalg.",[40,628,629],{"class":237},"eigh",[40,631,632],{"class":133},"(s)\n",[40,634,636,639,641],{"class":42,"line":635},55,[40,637,638],{"class":133},"    s_vecs ",[40,640,623],{"class":441},[40,642,643],{"class":133}," s_vecs.T\n",[40,645,647,650,652,655,658],{"class":42,"line":646},56,[40,648,649],{"class":133},"    good_vecs ",[40,651,623],{"class":441},[40,653,654],{"class":133}," np.",[40,656,657],{"class":237},"array",[40,659,411],{"class":133},[40,661,663,666,669,672,675,678,681,684,687,690],{"class":42,"line":662},57,[40,664,665],{"class":133},"        [vec ",[40,667,668],{"class":129},"for",[40,670,671],{"class":133}," val, vec ",[40,673,674],{"class":129},"in",[40,676,677],{"class":441}," zip",[40,679,680],{"class":133},"(s_vals, s_vecs) ",[40,682,683],{"class":129},"if",[40,685,686],{"class":133}," val ",[40,688,689],{"class":441},">",[40,691,692],{"class":133}," threshold]\n",[40,694,696],{"class":42,"line":695},58,[40,697,698],{"class":133},"    )\n",[40,700,702,705,707,710,713,716,719,722,724],{"class":42,"line":701},59,[40,703,704],{"class":133},"    h_reg ",[40,706,623],{"class":441},[40,708,709],{"class":133}," good_vecs.",[40,711,712],{"class":237},"conj",[40,714,715],{"class":133},"() ",[40,717,718],{"class":441},"@",[40,720,721],{"class":133}," h ",[40,723,718],{"class":441},[40,725,726],{"class":133}," good_vecs.T\n",[40,728,730,733,735,737,739,741,743,746,748],{"class":42,"line":729},60,[40,731,732],{"class":133},"    s_reg ",[40,734,623],{"class":441},[40,736,709],{"class":133},[40,738,712],{"class":237},[40,740,715],{"class":133},[40,742,718],{"class":441},[40,744,745],{"class":133}," s ",[40,747,718],{"class":441},[40,749,726],{"class":133},[40,751,753,756,759,762,764],{"class":42,"line":752},61,[40,754,755],{"class":129},"    if",[40,757,758],{"class":133}," k ",[40,760,761],{"class":441},"==",[40,763,463],{"class":462},[40,765,766],{"class":133},":\n",[40,768,770,773],{"class":42,"line":769},62,[40,771,772],{"class":129},"        if",[40,774,775],{"class":133}," return_dimn:\n",[40,777,779,782,784,786,789,792,795,797,800,803],{"class":42,"line":778},63,[40,780,781],{"class":129},"            return",[40,783,626],{"class":133},[40,785,629],{"class":237},[40,787,788],{"class":133},"(h_reg, s_reg)[",[40,790,791],{"class":462},"0",[40,793,794],{"class":133},"][",[40,796,791],{"class":462},[40,798,799],{"class":133},"], ",[40,801,802],{"class":441},"len",[40,804,805],{"class":133},"(good_vecs)\n",[40,807,809,812],{"class":42,"line":808},64,[40,810,811],{"class":129},"        else",[40,813,766],{"class":133},[40,815,817,819,821,823,825,827,829,831],{"class":42,"line":816},65,[40,818,781],{"class":129},[40,820,626],{"class":133},[40,822,629],{"class":237},[40,824,788],{"class":133},[40,826,791],{"class":462},[40,828,794],{"class":133},[40,830,791],{"class":462},[40,832,833],{"class":133},"]\n",[40,835,837,840],{"class":42,"line":836},66,[40,838,839],{"class":129},"    else",[40,841,766],{"class":133},[40,843,845,847],{"class":42,"line":844},67,[40,846,772],{"class":129},[40,848,775],{"class":133},[40,850,852,854,856,858,860,862,865,867],{"class":42,"line":851},68,[40,853,781],{"class":129},[40,855,626],{"class":133},[40,857,629],{"class":237},[40,859,788],{"class":133},[40,861,791],{"class":462},[40,863,864],{"class":133},"][:k], ",[40,866,802],{"class":441},[40,868,805],{"class":133},[40,870,872,874],{"class":42,"line":871},69,[40,873,811],{"class":129},[40,875,766],{"class":133},[40,877,879,881,883,885,887,889],{"class":42,"line":878},70,[40,880,781],{"class":129},[40,882,626],{"class":133},[40,884,629],{"class":237},[40,886,788],{"class":133},[40,888,791],{"class":462},[40,890,891],{"class":133},"][:k]\n",[40,893,895],{"class":42,"line":894},71,[40,896,228],{"emptyLinePlaceholder":227},[40,898,900],{"class":42,"line":899},72,[40,901,228],{"emptyLinePlaceholder":227},[40,903,905,907,910,912,915,918,921],{"class":42,"line":904},73,[40,906,405],{"class":129},[40,908,909],{"class":237}," single_particle_gs",[40,911,241],{"class":133},[40,913,914],{"class":417},"H_op",[40,916,917],{"class":133},", ",[40,919,920],{"class":417},"n_qubits",[40,922,923],{"class":133},"):\n",[40,925,927],{"class":42,"line":926},74,[40,928,505],{"class":244},[40,930,932],{"class":42,"line":931},75,[40,933,934],{"class":244},"    Find the ground state of the single particle(excitation) sector\n",[40,936,938],{"class":42,"line":937},76,[40,939,505],{"class":244},[40,941,943,946,948],{"class":42,"line":942},77,[40,944,945],{"class":133},"    H_x ",[40,947,623],{"class":441},[40,949,950],{"class":133}," []\n",[40,952,954,957,960,962,965,968],{"class":42,"line":953},78,[40,955,956],{"class":129},"    for",[40,958,959],{"class":133}," p, coeff ",[40,961,674],{"class":129},[40,963,964],{"class":133}," H_op.",[40,966,967],{"class":237},"to_list",[40,969,970],{"class":133},"():\n",[40,972,974,977,980,982,985,988,990,993,995,998,1000,1003,1006,1008],{"class":42,"line":973},79,[40,975,976],{"class":133},"        H_x.",[40,978,979],{"class":237},"append",[40,981,241],{"class":133},[40,983,984],{"class":441},"set",[40,986,987],{"class":133},"([i ",[40,989,668],{"class":129},[40,991,992],{"class":133}," i, v ",[40,994,674],{"class":129},[40,996,997],{"class":441}," enumerate",[40,999,241],{"class":133},[40,1001,1002],{"class":237},"Pauli",[40,1004,1005],{"class":133},"(p).x) ",[40,1007,683],{"class":129},[40,1009,1010],{"class":133}," v]))\n",[40,1012,1014],{"class":42,"line":1013},80,[40,1015,228],{"emptyLinePlaceholder":227},[40,1017,1019,1022,1024],{"class":42,"line":1018},81,[40,1020,1021],{"class":133},"    H_z ",[40,1023,623],{"class":441},[40,1025,950],{"class":133},[40,1027,1029,1031,1033,1035,1037,1039],{"class":42,"line":1028},82,[40,1030,956],{"class":129},[40,1032,959],{"class":133},[40,1034,674],{"class":129},[40,1036,964],{"class":133},[40,1038,967],{"class":237},[40,1040,970],{"class":133},[40,1042,1044,1047,1049,1051,1053,1055,1057,1059,1061,1063,1065,1067,1070,1072],{"class":42,"line":1043},83,[40,1045,1046],{"class":133},"        H_z.",[40,1048,979],{"class":237},[40,1050,241],{"class":133},[40,1052,984],{"class":441},[40,1054,987],{"class":133},[40,1056,668],{"class":129},[40,1058,992],{"class":133},[40,1060,674],{"class":129},[40,1062,997],{"class":441},[40,1064,241],{"class":133},[40,1066,1002],{"class":237},[40,1068,1069],{"class":133},"(p).z) ",[40,1071,683],{"class":129},[40,1073,1010],{"class":133},[40,1075,1077],{"class":42,"line":1076},84,[40,1078,228],{"emptyLinePlaceholder":227},[40,1080,1082,1085,1087],{"class":42,"line":1081},85,[40,1083,1084],{"class":133},"    H_c ",[40,1086,623],{"class":441},[40,1088,1089],{"class":133}," H_op.coeffs\n",[40,1091,1093],{"class":42,"line":1092},86,[40,1094,228],{"emptyLinePlaceholder":227},[40,1096,1098,1101,1103,1106],{"class":42,"line":1097},87,[40,1099,1100],{"class":441},"    print",[40,1102,241],{"class":133},[40,1104,1105],{"class":244},"\"n_sys_qubits\"",[40,1107,1108],{"class":133},", n_qubits)\n",[40,1110,1112],{"class":42,"line":1111},88,[40,1113,228],{"emptyLinePlaceholder":227},[40,1115,1117,1120,1122],{"class":42,"line":1116},89,[40,1118,1119],{"class":133},"    n_exc ",[40,1121,623],{"class":441},[40,1123,1124],{"class":462}," 1\n",[40,1126,1128,1131,1133,1136,1139,1142,1145,1148,1150],{"class":42,"line":1127},90,[40,1129,1130],{"class":133},"    sub_dimn ",[40,1132,623],{"class":441},[40,1134,1135],{"class":441}," int",[40,1137,1138],{"class":133},"(sp.special.",[40,1140,1141],{"class":237},"comb",[40,1143,1144],{"class":133},"(n_qubits ",[40,1146,1147],{"class":441},"+",[40,1149,463],{"class":462},[40,1151,1152],{"class":133},", n_exc))\n",[40,1154,1156,1158,1160,1163,1166,1169],{"class":42,"line":1155},91,[40,1157,1100],{"class":441},[40,1159,241],{"class":133},[40,1161,1162],{"class":244},"\"n_exc\"",[40,1164,1165],{"class":133},", n_exc, ",[40,1167,1168],{"class":244},"\", subspace dimension\"",[40,1170,1171],{"class":133},", sub_dimn)\n",[40,1173,1175],{"class":42,"line":1174},92,[40,1176,228],{"emptyLinePlaceholder":227},[40,1178,1180,1183,1185,1187,1190,1193,1197,1200],{"class":42,"line":1179},93,[40,1181,1182],{"class":133},"    few_particle_H ",[40,1184,623],{"class":441},[40,1186,654],{"class":133},[40,1188,1189],{"class":237},"zeros",[40,1191,1192],{"class":133},"((sub_dimn, sub_dimn), ",[40,1194,1196],{"class":1195},"s_ZVi","dtype",[40,1198,1199],{"class":441},"=complex",[40,1201,248],{"class":133},[40,1203,1205],{"class":42,"line":1204},94,[40,1206,228],{"emptyLinePlaceholder":227},[40,1208,1210,1213,1215],{"class":42,"line":1209},95,[40,1211,1212],{"class":133},"    sparse_vecs ",[40,1214,623],{"class":441},[40,1216,1217],{"class":133}," [\n",[40,1219,1221,1224,1227,1229,1232,1234,1237,1240,1242,1245,1247,1249,1251,1254,1257,1259],{"class":42,"line":1220},96,[40,1222,1223],{"class":441},"        set",[40,1225,1226],{"class":133},"(vec) ",[40,1228,668],{"class":129},[40,1230,1231],{"class":133}," vec ",[40,1233,674],{"class":129},[40,1235,1236],{"class":133}," it.",[40,1238,1239],{"class":237},"combinations",[40,1241,241],{"class":133},[40,1243,1244],{"class":441},"range",[40,1246,1144],{"class":133},[40,1248,1147],{"class":441},[40,1250,463],{"class":462},[40,1252,1253],{"class":133},"), ",[40,1255,1256],{"class":1195},"r",[40,1258,623],{"class":441},[40,1260,1261],{"class":133},"n_exc)\n",[40,1263,1265,1268],{"class":42,"line":1264},97,[40,1266,1267],{"class":133},"    ]  ",[40,1269,1270],{"class":46},"# list all of the possible sets of n_exc indices of 1s in n_exc-particle states\n",[40,1272,1274],{"class":42,"line":1273},98,[40,1275,228],{"emptyLinePlaceholder":227},[40,1277,1279,1282,1284],{"class":42,"line":1278},99,[40,1280,1281],{"class":133},"    m ",[40,1283,623],{"class":441},[40,1285,1286],{"class":462}," 0\n",[40,1288,1290,1292,1295,1297,1299],{"class":42,"line":1289},100,[40,1291,956],{"class":129},[40,1293,1294],{"class":133}," i, i_set ",[40,1296,674],{"class":129},[40,1298,997],{"class":441},[40,1300,1301],{"class":133},"(sparse_vecs):\n",[40,1303,1305,1308,1311,1313,1315],{"class":42,"line":1304},101,[40,1306,1307],{"class":129},"        for",[40,1309,1310],{"class":133}," j, j_set ",[40,1312,674],{"class":129},[40,1314,997],{"class":441},[40,1316,1301],{"class":133},[40,1318,1320,1323,1326],{"class":42,"line":1319},102,[40,1321,1322],{"class":133},"            m ",[40,1324,1325],{"class":441},"+=",[40,1327,1124],{"class":462},[40,1329,1331],{"class":42,"line":1330},103,[40,1332,228],{"emptyLinePlaceholder":227},[40,1334,1336,1339,1342,1345,1348,1351,1354,1357],{"class":42,"line":1335},104,[40,1337,1338],{"class":129},"            if",[40,1340,1341],{"class":441}," len",[40,1343,1344],{"class":133},"(i_set.",[40,1346,1347],{"class":237},"symmetric_difference",[40,1349,1350],{"class":133},"(j_set)) ",[40,1352,1353],{"class":441},"\u003C=",[40,1355,1356],{"class":462}," 2",[40,1358,766],{"class":133},[40,1360,1362,1365,1368,1370,1372],{"class":42,"line":1361},105,[40,1363,1364],{"class":129},"                for",[40,1366,1367],{"class":133}," p_x, p_z, coeff ",[40,1369,674],{"class":129},[40,1371,677],{"class":441},[40,1373,1374],{"class":133},"(H_x, H_z, H_c):\n",[40,1376,1378,1381,1384,1386,1389,1391],{"class":42,"line":1377},106,[40,1379,1380],{"class":129},"                    if",[40,1382,1383],{"class":133}," i_set.",[40,1385,1347],{"class":237},[40,1387,1388],{"class":133},"(j_set) ",[40,1390,761],{"class":441},[40,1392,1393],{"class":133}," p_x:\n",[40,1395,1397,1400,1402,1405,1408,1411,1414,1417,1420,1422,1425,1428,1431,1434],{"class":42,"line":1396},107,[40,1398,1399],{"class":133},"                        sgn ",[40,1401,623],{"class":441},[40,1403,1404],{"class":133}," ((",[40,1406,1407],{"class":441},"-",[40,1409,1410],{"class":462},"1",[40,1412,1413],{"class":129},"j",[40,1415,1416],{"class":133},") ",[40,1418,1419],{"class":441},"**",[40,1421,1341],{"class":441},[40,1423,1424],{"class":133},"(p_x.",[40,1426,1427],{"class":237},"intersection",[40,1429,1430],{"class":133},"(p_z))) ",[40,1432,1433],{"class":441},"*",[40,1435,367],{"class":133},[40,1437,1439,1442,1444,1446,1448,1450,1452,1454,1456],{"class":42,"line":1438},108,[40,1440,1441],{"class":133},"                            (",[40,1443,1407],{"class":441},[40,1445,1410],{"class":462},[40,1447,1416],{"class":133},[40,1449,1419],{"class":441},[40,1451,1341],{"class":441},[40,1453,1344],{"class":133},[40,1455,1427],{"class":237},[40,1457,1458],{"class":133},"(p_z))\n",[40,1460,1462],{"class":42,"line":1461},109,[40,1463,1464],{"class":133},"                        )\n",[40,1466,1468,1471],{"class":42,"line":1467},110,[40,1469,1470],{"class":129},"                    else",[40,1472,766],{"class":133},[40,1474,1476,1478,1480],{"class":42,"line":1475},111,[40,1477,1399],{"class":133},[40,1479,623],{"class":441},[40,1481,1286],{"class":462},[40,1483,1485],{"class":42,"line":1484},112,[40,1486,228],{"emptyLinePlaceholder":227},[40,1488,1490,1493,1495,1498,1500],{"class":42,"line":1489},113,[40,1491,1492],{"class":133},"                    few_particle_H[i, j] ",[40,1494,1325],{"class":441},[40,1496,1497],{"class":133}," sgn ",[40,1499,1433],{"class":441},[40,1501,1502],{"class":133}," coeff\n",[40,1504,1506],{"class":42,"line":1505},114,[40,1507,228],{"emptyLinePlaceholder":227},[40,1509,1511,1514,1516,1519,1522,1525],{"class":42,"line":1510},115,[40,1512,1513],{"class":133},"    gs_en ",[40,1515,623],{"class":441},[40,1517,1518],{"class":441}," min",[40,1520,1521],{"class":133},"(np.linalg.",[40,1523,1524],{"class":237},"eigvalsh",[40,1526,1527],{"class":133},"(few_particle_H))\n",[40,1529,1531,1533,1535,1538],{"class":42,"line":1530},116,[40,1532,1100],{"class":441},[40,1534,241],{"class":133},[40,1536,1537],{"class":244},"\"single particle ground state energy: \"",[40,1539,1540],{"class":133},", gs_en)\n",[40,1542,1544,1547],{"class":42,"line":1543},117,[40,1545,1546],{"class":129},"    return",[40,1548,1549],{"class":133}," gs_en\n",[55,1551,1553],{"id":1552},"step-1-map-classical-inputs-to-a-quantum-problem","Step 1: Map classical inputs to a quantum problem",[1555,1556,1558],"h3",{"id":1557},"the-krylov-space","The Krylov space",[12,1560,1561,1562,1655,1656,1685,1686,1715,1716,1775,1776,1823],{},"The Krylov space ",[40,1563,1566,1595],{"className":1564},[1565],"katex",[40,1567,1570],{"className":1568},[1569],"katex-mathml",[1571,1572,1574],"math",{"xmlns":1573},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[1575,1576,1577,1590],"semantics",{},[1578,1579,1580],"mrow",{},[1581,1582,1583,1588],"msup",{},[1584,1585,1587],"mi",{"mathvariant":1586},"script","K",[1584,1589,1256],{},[1591,1592,1594],"annotation",{"encoding":1593},"application\u002Fx-tex","\\mathcal{K}^r",[40,1596,1600],{"className":1597,"ariaHidden":1599},[1598],"katex-html","true",[40,1601,1604,1609],{"className":1602},[1603],"base",[40,1605],{"className":1606,"style":1608},[1607],"strut","height:0.6833em;",[40,1610,1613,1618],{"className":1611},[1612],"mord",[40,1614,1587],{"className":1615,"style":1617},[1612,1616],"mathcal","margin-right:0.0144em;",[40,1619,1622],{"className":1620},[1621],"msupsub",[40,1623,1626],{"className":1624},[1625],"vlist-t",[40,1627,1630],{"className":1628},[1629],"vlist-r",[40,1631,1635],{"className":1632,"style":1634},[1633],"vlist","height:0.6644em;",[40,1636,1638,1643],{"style":1637},"top:-3.063em;margin-right:0.05em;",[40,1639],{"className":1640,"style":1642},[1641],"pstrut","height:2.7em;",[40,1644,1650],{"className":1645},[1646,1647,1648,1649],"sizing","reset-size6","size3","mtight",[40,1651,1256],{"className":1652,"style":1654},[1612,1653,1649],"mathnormal","margin-right:0.0278em;"," of order ",[40,1657,1659,1672],{"className":1658},[1565],[40,1660,1662],{"className":1661},[1569],[1571,1663,1664],{"xmlns":1573},[1575,1665,1666,1670],{},[1578,1667,1668],{},[1584,1669,1256],{},[1591,1671,1256],{"encoding":1593},[40,1673,1675],{"className":1674,"ariaHidden":1599},[1598],[40,1676,1678,1682],{"className":1677},[1603],[40,1679],{"className":1680,"style":1681},[1607],"height:0.4306em;",[40,1683,1256],{"className":1684,"style":1654},[1612,1653]," is the space spanned by vectors obtained by multiplying higher powers of a matrix ",[40,1687,1689,1703],{"className":1688},[1565],[40,1690,1692],{"className":1691},[1569],[1571,1693,1694],{"xmlns":1573},[1575,1695,1696,1701],{},[1578,1697,1698],{},[1584,1699,1700],{},"A",[1591,1702,1700],{"encoding":1593},[40,1704,1706],{"className":1705,"ariaHidden":1599},[1598],[40,1707,1709,1712],{"className":1708},[1603],[40,1710],{"className":1711,"style":1608},[1607],[40,1713,1700],{"className":1714},[1612,1653],", up to ",[40,1717,1719,1740],{"className":1718},[1565],[40,1720,1722],{"className":1721},[1569],[1571,1723,1724],{"xmlns":1573},[1575,1725,1726,1737],{},[1578,1727,1728,1730,1734],{},[1584,1729,1256],{},[1731,1732,1733],"mo",{},"−",[1735,1736,1410],"mn",{},[1591,1738,1739],{"encoding":1593},"r-1",[40,1741,1743,1765],{"className":1742,"ariaHidden":1599},[1598],[40,1744,1746,1750,1753,1758,1762],{"className":1745},[1603],[40,1747],{"className":1748,"style":1749},[1607],"height:0.6667em;vertical-align:-0.0833em;",[40,1751,1256],{"className":1752,"style":1654},[1612,1653],[40,1754],{"className":1755,"style":1757},[1756],"mspace","margin-right:0.2222em;",[40,1759,1733],{"className":1760},[1761],"mbin",[40,1763],{"className":1764,"style":1757},[1756],[40,1766,1768,1772],{"className":1767},[1603],[40,1769],{"className":1770,"style":1771},[1607],"height:0.6444em;",[40,1773,1410],{"className":1774},[1612],", with a reference vector ",[40,1777,1779,1802],{"className":1778},[1565],[40,1780,1782],{"className":1781},[1569],[1571,1783,1784],{"xmlns":1573},[1575,1785,1786,1799],{},[1578,1787,1788,1792,1795],{},[1584,1789,1791],{"mathvariant":1790},"normal","∣",[1584,1793,1794],{},"v",[1731,1796,1798],{"stretchy":1797},"false","⟩",[1591,1800,1801],{"encoding":1593},"\\vert v \\rangle",[40,1803,1805],{"className":1804,"ariaHidden":1599},[1598],[40,1806,1808,1812,1815,1819],{"className":1807},[1603],[40,1809],{"className":1810,"style":1811},[1607],"height:1em;vertical-align:-0.25em;",[40,1813,1791],{"className":1814},[1612],[40,1816,1794],{"className":1817,"style":1818},[1612,1653],"margin-right:0.0359em;",[40,1820,1798],{"className":1821},[1822],"mclose",".",[40,1825,1828],{"className":1826},[1827],"katex-display",[40,1829,1831,1920],{"className":1830},[1565],[40,1832,1834],{"className":1833},[1569],[1571,1835,1837],{"xmlns":1573,"display":1836},"block",[1575,1838,1839,1917],{},[1578,1840,1841,1847,1849],{},[1581,1842,1843,1845],{},[1584,1844,1587],{"mathvariant":1586},[1584,1846,1256],{},[1731,1848,623],{},[1578,1850,1851,1854,1856,1858,1860,1863,1865,1867,1869,1871,1873,1880,1882,1884,1886,1888,1890,1892,1894,1896,1908,1910,1912,1914],{},[1731,1852,1853],{"fence":1599},"{",[1584,1855,1791],{"mathvariant":1790},[1584,1857,1794],{},[1731,1859,1798],{"stretchy":1797},[1731,1861,1862],{"separator":1599},",",[1584,1864,1700],{},[1584,1866,1791],{"mathvariant":1790},[1584,1868,1794],{},[1731,1870,1798],{"stretchy":1797},[1731,1872,1862],{"separator":1599},[1581,1874,1875,1877],{},[1584,1876,1700],{},[1735,1878,1879],{},"2",[1584,1881,1791],{"mathvariant":1790},[1584,1883,1794],{},[1731,1885,1798],{"stretchy":1797},[1731,1887,1862],{"separator":1599},[1584,1889,1823],{"mathvariant":1790},[1584,1891,1823],{"mathvariant":1790},[1584,1893,1823],{"mathvariant":1790},[1731,1895,1862],{"separator":1599},[1581,1897,1898,1900],{},[1584,1899,1700],{},[1578,1901,1902,1904,1906],{},[1584,1903,1256],{},[1731,1905,1733],{},[1735,1907,1410],{},[1584,1909,1791],{"mathvariant":1790},[1584,1911,1794],{},[1731,1913,1798],{"stretchy":1797},[1731,1915,1916],{"fence":1599},"}",[1591,1918,1919],{"encoding":1593},"\\mathcal{K}^r = \\left\\{ \\vert v \\rangle, A \\vert v \\rangle, A^2 \\vert v \\rangle, ..., A^{r-1} \\vert v \\rangle \\right\\}",[40,1921,1923,1971],{"className":1922,"ariaHidden":1599},[1598],[40,1924,1926,1930,1960,1964,1968],{"className":1925},[1603],[40,1927],{"className":1928,"style":1929},[1607],"height:0.7144em;",[40,1931,1933,1936],{"className":1932},[1612],[40,1934,1587],{"className":1935,"style":1617},[1612,1616],[40,1937,1939],{"className":1938},[1621],[40,1940,1942],{"className":1941},[1625],[40,1943,1945],{"className":1944},[1629],[40,1946,1948],{"className":1947,"style":1929},[1633],[40,1949,1951,1954],{"style":1950},"top:-3.113em;margin-right:0.05em;",[40,1952],{"className":1953,"style":1642},[1641],[40,1955,1957],{"className":1956},[1646,1647,1648,1649],[40,1958,1256],{"className":1959,"style":1654},[1612,1653,1649],[40,1961],{"className":1962,"style":1963},[1756],"margin-right:0.2778em;",[40,1965,623],{"className":1966},[1967],"mrel",[40,1969],{"className":1970,"style":1963},[1756],[40,1972,1974,1978],{"className":1973},[1603],[40,1975],{"className":1976,"style":1977},[1607],"height:1.2141em;vertical-align:-0.35em;",[40,1979,1982,1993,1996,1999,2002,2006,2010,2013,2016,2019,2022,2025,2028,2058,2061,2064,2067,2070,2073,2077,2080,2083,2121,2124,2127,2130],{"className":1980},[1981],"minner",[40,1983,1988],{"className":1984,"style":1987},[1985,1986],"mopen","delimcenter","top:0em;",[40,1989,1853],{"className":1990},[1991,1992],"delimsizing","size1",[40,1994,1791],{"className":1995},[1612],[40,1997,1794],{"className":1998,"style":1818},[1612,1653],[40,2000,1798],{"className":2001},[1822],[40,2003,1862],{"className":2004},[2005],"mpunct",[40,2007],{"className":2008,"style":2009},[1756],"margin-right:0.1667em;",[40,2011,1700],{"className":2012},[1612,1653],[40,2014,1791],{"className":2015},[1612],[40,2017,1794],{"className":2018,"style":1818},[1612,1653],[40,2020,1798],{"className":2021},[1822],[40,2023,1862],{"className":2024},[2005],[40,2026],{"className":2027,"style":2009},[1756],[40,2029,2031,2034],{"className":2030},[1612],[40,2032,1700],{"className":2033},[1612,1653],[40,2035,2037],{"className":2036},[1621],[40,2038,2040],{"className":2039},[1625],[40,2041,2043],{"className":2042},[1629],[40,2044,2047],{"className":2045,"style":2046},[1633],"height:0.8641em;",[40,2048,2049,2052],{"style":1950},[40,2050],{"className":2051,"style":1642},[1641],[40,2053,2055],{"className":2054},[1646,1647,1648,1649],[40,2056,1879],{"className":2057},[1612,1649],[40,2059,1791],{"className":2060},[1612],[40,2062,1794],{"className":2063,"style":1818},[1612,1653],[40,2065,1798],{"className":2066},[1822],[40,2068,1862],{"className":2069},[2005],[40,2071],{"className":2072,"style":2009},[1756],[40,2074,2076],{"className":2075},[1612],"...",[40,2078,1862],{"className":2079},[2005],[40,2081],{"className":2082,"style":2009},[1756],[40,2084,2086,2089],{"className":2085},[1612],[40,2087,1700],{"className":2088},[1612,1653],[40,2090,2092],{"className":2091},[1621],[40,2093,2095],{"className":2094},[1625],[40,2096,2098],{"className":2097},[1629],[40,2099,2101],{"className":2100,"style":2046},[1633],[40,2102,2103,2106],{"style":1950},[40,2104],{"className":2105,"style":1642},[1641],[40,2107,2109],{"className":2108},[1646,1647,1648,1649],[40,2110,2112,2115,2118],{"className":2111},[1612,1649],[40,2113,1256],{"className":2114,"style":1654},[1612,1653,1649],[40,2116,1733],{"className":2117},[1761,1649],[40,2119,1410],{"className":2120},[1612,1649],[40,2122,1791],{"className":2123},[1612],[40,2125,1794],{"className":2126,"style":1818},[1612,1653],[40,2128,1798],{"className":2129},[1822],[40,2131,2133],{"className":2132,"style":1987},[1822,1986],[40,2134,1916],{"className":2135},[1991,1992],[12,2137,2138,2139,2167,2168,2198,2199,2279,2280,2308,2309,2416,2417,2487,2488,2516,2517,2618,2619,2622,2623,2651,2652,1823],{},"If the matrix ",[40,2140,2142,2155],{"className":2141},[1565],[40,2143,2145],{"className":2144},[1569],[1571,2146,2147],{"xmlns":1573},[1575,2148,2149,2153],{},[1578,2150,2151],{},[1584,2152,1700],{},[1591,2154,1700],{"encoding":1593},[40,2156,2158],{"className":2157,"ariaHidden":1599},[1598],[40,2159,2161,2164],{"className":2160},[1603],[40,2162],{"className":2163,"style":1608},[1607],[40,2165,1700],{"className":2166},[1612,1653]," is the Hamiltonian ",[40,2169,2171,2185],{"className":2170},[1565],[40,2172,2174],{"className":2173},[1569],[1571,2175,2176],{"xmlns":1573},[1575,2177,2178,2183],{},[1578,2179,2180],{},[1584,2181,2182],{},"H",[1591,2184,2182],{"encoding":1593},[40,2186,2188],{"className":2187,"ariaHidden":1599},[1598],[40,2189,2191,2194],{"className":2190},[1603],[40,2192],{"className":2193,"style":1608},[1607],[40,2195,2182],{"className":2196,"style":2197},[1612,1653],"margin-right:0.0813em;",", we'll refer to the corresponding space as the power Krylov space ",[40,2200,2202,2222],{"className":2201},[1565],[40,2203,2205],{"className":2204},[1569],[1571,2206,2207],{"xmlns":1573},[1575,2208,2209,2219],{},[1578,2210,2211],{},[2212,2213,2214,2216],"msub",{},[1584,2215,1587],{"mathvariant":1586},[1584,2217,2218],{},"P",[1591,2220,2221],{"encoding":1593},"\\mathcal{K}_P",[40,2223,2225],{"className":2224,"ariaHidden":1599},[1598],[40,2226,2228,2232],{"className":2227},[1603],[40,2229],{"className":2230,"style":2231},[1607],"height:0.8333em;vertical-align:-0.15em;",[40,2233,2235,2238],{"className":2234},[1612],[40,2236,1587],{"className":2237,"style":1617},[1612,1616],[40,2239,2241],{"className":2240},[1621],[40,2242,2245,2270],{"className":2243},[1625,2244],"vlist-t2",[40,2246,2248,2265],{"className":2247},[1629],[40,2249,2252],{"className":2250,"style":2251},[1633],"height:0.3283em;",[40,2253,2255,2258],{"style":2254},"top:-2.55em;margin-left:-0.0144em;margin-right:0.05em;",[40,2256],{"className":2257,"style":1642},[1641],[40,2259,2261],{"className":2260},[1646,1647,1648,1649],[40,2262,2218],{"className":2263,"style":2264},[1612,1653,1649],"margin-right:0.1389em;",[40,2266,2269],{"className":2267},[2268],"vlist-s","​",[40,2271,2273],{"className":2272},[1629],[40,2274,2277],{"className":2275,"style":2276},[1633],"height:0.15em;",[40,2278],{},". In the case where ",[40,2281,2283,2296],{"className":2282},[1565],[40,2284,2286],{"className":2285},[1569],[1571,2287,2288],{"xmlns":1573},[1575,2289,2290,2294],{},[1578,2291,2292],{},[1584,2293,1700],{},[1591,2295,1700],{"encoding":1593},[40,2297,2299],{"className":2298,"ariaHidden":1599},[1598],[40,2300,2302,2305],{"className":2301},[1603],[40,2303],{"className":2304,"style":1608},[1607],[40,2306,1700],{"className":2307},[1612,1653]," is the time-evolution operator generated by the Hamiltonian ",[40,2310,2312,2346],{"className":2311},[1565],[40,2313,2315],{"className":2314},[1569],[1571,2316,2317],{"xmlns":1573},[1575,2318,2319,2343],{},[1578,2320,2321,2324,2326],{},[1584,2322,2323],{},"U",[1731,2325,623],{},[1581,2327,2328,2331],{},[1584,2329,2330],{},"e",[1578,2332,2333,2335,2338,2340],{},[1731,2334,1733],{},[1584,2336,2337],{},"i",[1584,2339,2182],{},[1584,2341,2342],{},"t",[1591,2344,2345],{"encoding":1593},"U=e^{-iHt}",[40,2347,2349,2368],{"className":2348,"ariaHidden":1599},[1598],[40,2350,2352,2355,2359,2362,2365],{"className":2351},[1603],[40,2353],{"className":2354,"style":1608},[1607],[40,2356,2323],{"className":2357,"style":2358},[1612,1653],"margin-right:0.109em;",[40,2360],{"className":2361,"style":1963},[1756],[40,2363,623],{"className":2364},[1967],[40,2366],{"className":2367,"style":1963},[1756],[40,2369,2371,2375],{"className":2370},[1603],[40,2372],{"className":2373,"style":2374},[1607],"height:0.8413em;",[40,2376,2378,2381],{"className":2377},[1612],[40,2379,2330],{"className":2380},[1612,1653],[40,2382,2384],{"className":2383},[1621],[40,2385,2387],{"className":2386},[1625],[40,2388,2390],{"className":2389},[1629],[40,2391,2393],{"className":2392,"style":2374},[1633],[40,2394,2395,2398],{"style":1637},[40,2396],{"className":2397,"style":1642},[1641],[40,2399,2401],{"className":2400},[1646,1647,1648,1649],[40,2402,2404,2407,2410,2413],{"className":2403},[1612,1649],[40,2405,1733],{"className":2406},[1612,1649],[40,2408,2337],{"className":2409},[1612,1653,1649],[40,2411,2182],{"className":2412,"style":2197},[1612,1653,1649],[40,2414,2342],{"className":2415},[1612,1653,1649],", we'll refer to the space as the unitary Krylov space ",[40,2418,2420,2438],{"className":2419},[1565],[40,2421,2423],{"className":2422},[1569],[1571,2424,2425],{"xmlns":1573},[1575,2426,2427,2435],{},[1578,2428,2429],{},[2212,2430,2431,2433],{},[1584,2432,1587],{"mathvariant":1586},[1584,2434,2323],{},[1591,2436,2437],{"encoding":1593},"\\mathcal{K}_U",[40,2439,2441],{"className":2440,"ariaHidden":1599},[1598],[40,2442,2444,2447],{"className":2443},[1603],[40,2445],{"className":2446,"style":2231},[1607],[40,2448,2450,2453],{"className":2449},[1612],[40,2451,1587],{"className":2452,"style":1617},[1612,1616],[40,2454,2456],{"className":2455},[1621],[40,2457,2459,2479],{"className":2458},[1625,2244],[40,2460,2462,2476],{"className":2461},[1629],[40,2463,2465],{"className":2464,"style":2251},[1633],[40,2466,2467,2470],{"style":2254},[40,2468],{"className":2469,"style":1642},[1641],[40,2471,2473],{"className":2472},[1646,1647,1648,1649],[40,2474,2323],{"className":2475,"style":2358},[1612,1653,1649],[40,2477,2269],{"className":2478},[2268],[40,2480,2482],{"className":2481},[1629],[40,2483,2485],{"className":2484,"style":2276},[1633],[40,2486],{},". The power Krylov subspace that we use classically cannot be generated directly on a quantum computer as ",[40,2489,2491,2504],{"className":2490},[1565],[40,2492,2494],{"className":2493},[1569],[1571,2495,2496],{"xmlns":1573},[1575,2497,2498,2502],{},[1578,2499,2500],{},[1584,2501,2182],{},[1591,2503,2182],{"encoding":1593},[40,2505,2507],{"className":2506,"ariaHidden":1599},[1598],[40,2508,2510,2513],{"className":2509},[1603],[40,2511],{"className":2512,"style":1608},[1607],[40,2514,2182],{"className":2515,"style":2197},[1612,1653]," is not a unitary operator. Instead, we can use the time-evolution operator ",[40,2518,2520,2550],{"className":2519},[1565],[40,2521,2523],{"className":2522},[1569],[1571,2524,2525],{"xmlns":1573},[1575,2526,2527,2547],{},[1578,2528,2529,2531,2533],{},[1584,2530,2323],{},[1731,2532,623],{},[1581,2534,2535,2537],{},[1584,2536,2330],{},[1578,2538,2539,2541,2543,2545],{},[1731,2540,1733],{},[1584,2542,2337],{},[1584,2544,2182],{},[1584,2546,2342],{},[1591,2548,2549],{"encoding":1593},"U = e^{-iHt}",[40,2551,2553,2571],{"className":2552,"ariaHidden":1599},[1598],[40,2554,2556,2559,2562,2565,2568],{"className":2555},[1603],[40,2557],{"className":2558,"style":1608},[1607],[40,2560,2323],{"className":2561,"style":2358},[1612,1653],[40,2563],{"className":2564,"style":1963},[1756],[40,2566,623],{"className":2567},[1967],[40,2569],{"className":2570,"style":1963},[1756],[40,2572,2574,2577],{"className":2573},[1603],[40,2575],{"className":2576,"style":2374},[1607],[40,2578,2580,2583],{"className":2579},[1612],[40,2581,2330],{"className":2582},[1612,1653],[40,2584,2586],{"className":2585},[1621],[40,2587,2589],{"className":2588},[1625],[40,2590,2592],{"className":2591},[1629],[40,2593,2595],{"className":2594,"style":2374},[1633],[40,2596,2597,2600],{"style":1637},[40,2598],{"className":2599,"style":1642},[1641],[40,2601,2603],{"className":2602},[1646,1647,1648,1649],[40,2604,2606,2609,2612,2615],{"className":2605},[1612,1649],[40,2607,1733],{"className":2608},[1612,1649],[40,2610,2337],{"className":2611},[1612,1653,1649],[40,2613,2182],{"className":2614,"style":2197},[1612,1653,1649],[40,2616,2342],{"className":2617},[1612,1653,1649]," which can be shown to give similar ",[19,2620,2621],{"href":87},"convergence guarantees"," as the power method. Powers of ",[40,2624,2626,2639],{"className":2625},[1565],[40,2627,2629],{"className":2628},[1569],[1571,2630,2631],{"xmlns":1573},[1575,2632,2633,2637],{},[1578,2634,2635],{},[1584,2636,2323],{},[1591,2638,2323],{"encoding":1593},[40,2640,2642],{"className":2641,"ariaHidden":1599},[1598],[40,2643,2645,2648],{"className":2644},[1603],[40,2646],{"className":2647,"style":1608},[1607],[40,2649,2323],{"className":2650,"style":2358},[1612,1653]," then become different time steps ",[40,2653,2655,2697],{"className":2654},[1565],[40,2656,2658],{"className":2657},[1569],[1571,2659,2660],{"xmlns":1573},[1575,2661,2662,2694],{},[1578,2663,2664,2671,2673],{},[1581,2665,2666,2668],{},[1584,2667,2323],{},[1584,2669,2670],{},"k",[1731,2672,623],{},[1581,2674,2675,2677],{},[1584,2676,2330],{},[1578,2678,2679,2681,2683,2685,2687,2689,2691],{},[1731,2680,1733],{},[1584,2682,2337],{},[1584,2684,2182],{},[1731,2686,241],{"stretchy":1797},[1584,2688,2670],{},[1584,2690,2342],{},[1731,2692,2693],{"stretchy":1797},")",[1591,2695,2696],{"encoding":1593},"U^k = e^{-iH(kt)}",[40,2698,2700,2746],{"className":2699,"ariaHidden":1599},[1598],[40,2701,2703,2707,2737,2740,2743],{"className":2702},[1603],[40,2704],{"className":2705,"style":2706},[1607],"height:0.8491em;",[40,2708,2710,2713],{"className":2709},[1612],[40,2711,2323],{"className":2712,"style":2358},[1612,1653],[40,2714,2716],{"className":2715},[1621],[40,2717,2719],{"className":2718},[1625],[40,2720,2722],{"className":2721},[1629],[40,2723,2725],{"className":2724,"style":2706},[1633],[40,2726,2727,2730],{"style":1637},[40,2728],{"className":2729,"style":1642},[1641],[40,2731,2733],{"className":2732},[1646,1647,1648,1649],[40,2734,2670],{"className":2735,"style":2736},[1612,1653,1649],"margin-right:0.0315em;",[40,2738],{"className":2739,"style":1963},[1756],[40,2741,623],{"className":2742},[1967],[40,2744],{"className":2745,"style":1963},[1756],[40,2747,2749,2753],{"className":2748},[1603],[40,2750],{"className":2751,"style":2752},[1607],"height:0.888em;",[40,2754,2756,2759],{"className":2755},[1612],[40,2757,2330],{"className":2758},[1612,1653],[40,2760,2762],{"className":2761},[1621],[40,2763,2765],{"className":2764},[1625],[40,2766,2768],{"className":2767},[1629],[40,2769,2771],{"className":2770,"style":2752},[1633],[40,2772,2773,2776],{"style":1637},[40,2774],{"className":2775,"style":1642},[1641],[40,2777,2779],{"className":2778},[1646,1647,1648,1649],[40,2780,2782,2785,2788,2791,2794,2797,2800],{"className":2781},[1612,1649],[40,2783,1733],{"className":2784},[1612,1649],[40,2786,2337],{"className":2787},[1612,1653,1649],[40,2789,2182],{"className":2790,"style":2197},[1612,1653,1649],[40,2792,241],{"className":2793},[1985,1649],[40,2795,2670],{"className":2796,"style":2736},[1612,1653,1649],[40,2798,2342],{"className":2799},[1612,1653,1649],[40,2801,2693],{"className":2802},[1822,1649],[40,2804,2806],{"className":2805},[1827],[40,2807,2809,2897],{"className":2808},[1565],[40,2810,2812],{"className":2811},[1569],[1571,2813,2814],{"xmlns":1573,"display":1836},[1575,2815,2816,2894],{},[1578,2817,2818,2827,2829],{},[2819,2820,2821,2823,2825],"msubsup",{},[1584,2822,1587],{"mathvariant":1586},[1584,2824,2323],{},[1584,2826,1256],{},[1731,2828,623],{},[1578,2830,2831,2833,2835,2838,2840,2842,2844,2846,2848,2850,2852,2858,2860,2862,2864,2866,2868,2870,2872,2874,2886,2888,2890,2892],{},[1731,2832,1853],{"fence":1599},[1584,2834,1791],{"mathvariant":1790},[1584,2836,2837],{},"ψ",[1731,2839,1798],{"stretchy":1797},[1731,2841,1862],{"separator":1599},[1584,2843,2323],{},[1584,2845,1791],{"mathvariant":1790},[1584,2847,2837],{},[1731,2849,1798],{"stretchy":1797},[1731,2851,1862],{"separator":1599},[1581,2853,2854,2856],{},[1584,2855,2323],{},[1735,2857,1879],{},[1584,2859,1791],{"mathvariant":1790},[1584,2861,2837],{},[1731,2863,1798],{"stretchy":1797},[1731,2865,1862],{"separator":1599},[1584,2867,1823],{"mathvariant":1790},[1584,2869,1823],{"mathvariant":1790},[1584,2871,1823],{"mathvariant":1790},[1731,2873,1862],{"separator":1599},[1581,2875,2876,2878],{},[1584,2877,2323],{},[1578,2879,2880,2882,2884],{},[1584,2881,1256],{},[1731,2883,1733],{},[1735,2885,1410],{},[1584,2887,1791],{"mathvariant":1790},[1584,2889,2837],{},[1731,2891,1798],{"stretchy":1797},[1731,2893,1916],{"fence":1599},[1591,2895,2896],{"encoding":1593},"\\mathcal{K}_U^r = \\left\\{ \\vert \\psi \\rangle, U \\vert \\psi \\rangle, U^2 \\vert \\psi \\rangle, ..., U^{r-1} \\vert \\psi \\rangle \\right\\}",[40,2898,2900,2969],{"className":2899,"ariaHidden":1599},[1598],[40,2901,2903,2907,2960,2963,2966],{"className":2902},[1603],[40,2904],{"className":2905,"style":2906},[1607],"height:0.9614em;vertical-align:-0.247em;",[40,2908,2910,2913],{"className":2909},[1612],[40,2911,1587],{"className":2912,"style":1617},[1612,1616],[40,2914,2916],{"className":2915},[1621],[40,2917,2919,2951],{"className":2918},[1625,2244],[40,2920,2922,2948],{"className":2921},[1629],[40,2923,2925,2937],{"className":2924,"style":1929},[1633],[40,2926,2928,2931],{"style":2927},"top:-2.453em;margin-left:-0.0144em;margin-right:0.05em;",[40,2929],{"className":2930,"style":1642},[1641],[40,2932,2934],{"className":2933},[1646,1647,1648,1649],[40,2935,2323],{"className":2936,"style":2358},[1612,1653,1649],[40,2938,2939,2942],{"style":1950},[40,2940],{"className":2941,"style":1642},[1641],[40,2943,2945],{"className":2944},[1646,1647,1648,1649],[40,2946,1256],{"className":2947,"style":1654},[1612,1653,1649],[40,2949,2269],{"className":2950},[2268],[40,2952,2954],{"className":2953},[1629],[40,2955,2958],{"className":2956,"style":2957},[1633],"height:0.247em;",[40,2959],{},[40,2961],{"className":2962,"style":1963},[1756],[40,2964,623],{"className":2965},[1967],[40,2967],{"className":2968,"style":1963},[1756],[40,2970,2972,2975],{"className":2971},[1603],[40,2973],{"className":2974,"style":1977},[1607],[40,2976,2978,2984,2987,2990,2993,2996,2999,3002,3005,3008,3011,3014,3017,3046,3049,3052,3055,3058,3061,3064,3067,3070,3108,3111,3114,3117],{"className":2977},[1981],[40,2979,2981],{"className":2980,"style":1987},[1985,1986],[40,2982,1853],{"className":2983},[1991,1992],[40,2985,1791],{"className":2986},[1612],[40,2988,2837],{"className":2989,"style":1818},[1612,1653],[40,2991,1798],{"className":2992},[1822],[40,2994,1862],{"className":2995},[2005],[40,2997],{"className":2998,"style":2009},[1756],[40,3000,2323],{"className":3001,"style":2358},[1612,1653],[40,3003,1791],{"className":3004},[1612],[40,3006,2837],{"className":3007,"style":1818},[1612,1653],[40,3009,1798],{"className":3010},[1822],[40,3012,1862],{"className":3013},[2005],[40,3015],{"className":3016,"style":2009},[1756],[40,3018,3020,3023],{"className":3019},[1612],[40,3021,2323],{"className":3022,"style":2358},[1612,1653],[40,3024,3026],{"className":3025},[1621],[40,3027,3029],{"className":3028},[1625],[40,3030,3032],{"className":3031},[1629],[40,3033,3035],{"className":3034,"style":2046},[1633],[40,3036,3037,3040],{"style":1950},[40,3038],{"className":3039,"style":1642},[1641],[40,3041,3043],{"className":3042},[1646,1647,1648,1649],[40,3044,1879],{"className":3045},[1612,1649],[40,3047,1791],{"className":3048},[1612],[40,3050,2837],{"className":3051,"style":1818},[1612,1653],[40,3053,1798],{"className":3054},[1822],[40,3056,1862],{"className":3057},[2005],[40,3059],{"className":3060,"style":2009},[1756],[40,3062,2076],{"className":3063},[1612],[40,3065,1862],{"className":3066},[2005],[40,3068],{"className":3069,"style":2009},[1756],[40,3071,3073,3076],{"className":3072},[1612],[40,3074,2323],{"className":3075,"style":2358},[1612,1653],[40,3077,3079],{"className":3078},[1621],[40,3080,3082],{"className":3081},[1625],[40,3083,3085],{"className":3084},[1629],[40,3086,3088],{"className":3087,"style":2046},[1633],[40,3089,3090,3093],{"style":1950},[40,3091],{"className":3092,"style":1642},[1641],[40,3094,3096],{"className":3095},[1646,1647,1648,1649],[40,3097,3099,3102,3105],{"className":3098},[1612,1649],[40,3100,1256],{"className":3101,"style":1654},[1612,1653,1649],[40,3103,1733],{"className":3104},[1761,1649],[40,3106,1410],{"className":3107},[1612,1649],[40,3109,1791],{"className":3110},[1612],[40,3112,2837],{"className":3113,"style":1818},[1612,1653],[40,3115,1798],{"className":3116},[1822],[40,3118,3120],{"className":3119,"style":1987},[1822,1986],[40,3121,1916],{"className":3122},[1991,1992],[12,3124,3125],{},"See the Appendix for a detailed derivation of how the unitary Krylov space allows to represents low-energy eigenstates accurately.",[1555,3127,3129],{"id":3128},"krylov-quantum-diagonalization-algorithm","Krylov quantum diagonalization algorithm",[12,3131,3132,3133,3161,3162,3231,3232,3301,3302,3372,3373,3433],{},"Given an Hamiltonian ",[40,3134,3136,3149],{"className":3135},[1565],[40,3137,3139],{"className":3138},[1569],[1571,3140,3141],{"xmlns":1573},[1575,3142,3143,3147],{},[1578,3144,3145],{},[1584,3146,2182],{},[1591,3148,2182],{"encoding":1593},[40,3150,3152],{"className":3151,"ariaHidden":1599},[1598],[40,3153,3155,3158],{"className":3154},[1603],[40,3156],{"className":3157,"style":1608},[1607],[40,3159,2182],{"className":3160,"style":2197},[1612,1653]," that we wish to diagonalize, first we consider the corresponding unitary Krylov space ",[40,3163,3165,3182],{"className":3164},[1565],[40,3166,3168],{"className":3167},[1569],[1571,3169,3170],{"xmlns":1573},[1575,3171,3172,3180],{},[1578,3173,3174],{},[2212,3175,3176,3178],{},[1584,3177,1587],{"mathvariant":1586},[1584,3179,2323],{},[1591,3181,2437],{"encoding":1593},[40,3183,3185],{"className":3184,"ariaHidden":1599},[1598],[40,3186,3188,3191],{"className":3187},[1603],[40,3189],{"className":3190,"style":2231},[1607],[40,3192,3194,3197],{"className":3193},[1612],[40,3195,1587],{"className":3196,"style":1617},[1612,1616],[40,3198,3200],{"className":3199},[1621],[40,3201,3203,3223],{"className":3202},[1625,2244],[40,3204,3206,3220],{"className":3205},[1629],[40,3207,3209],{"className":3208,"style":2251},[1633],[40,3210,3211,3214],{"style":2254},[40,3212],{"className":3213,"style":1642},[1641],[40,3215,3217],{"className":3216},[1646,1647,1648,1649],[40,3218,2323],{"className":3219,"style":2358},[1612,1653,1649],[40,3221,2269],{"className":3222},[2268],[40,3224,3226],{"className":3225},[1629],[40,3227,3229],{"className":3228,"style":2276},[1633],[40,3230],{},". The goal is to find a compact representation of the Hamiltonian in ",[40,3233,3235,3252],{"className":3234},[1565],[40,3236,3238],{"className":3237},[1569],[1571,3239,3240],{"xmlns":1573},[1575,3241,3242,3250],{},[1578,3243,3244],{},[2212,3245,3246,3248],{},[1584,3247,1587],{"mathvariant":1586},[1584,3249,2323],{},[1591,3251,2437],{"encoding":1593},[40,3253,3255],{"className":3254,"ariaHidden":1599},[1598],[40,3256,3258,3261],{"className":3257},[1603],[40,3259],{"className":3260,"style":2231},[1607],[40,3262,3264,3267],{"className":3263},[1612],[40,3265,1587],{"className":3266,"style":1617},[1612,1616],[40,3268,3270],{"className":3269},[1621],[40,3271,3273,3293],{"className":3272},[1625,2244],[40,3274,3276,3290],{"className":3275},[1629],[40,3277,3279],{"className":3278,"style":2251},[1633],[40,3280,3281,3284],{"style":2254},[40,3282],{"className":3283,"style":1642},[1641],[40,3285,3287],{"className":3286},[1646,1647,1648,1649],[40,3288,2323],{"className":3289,"style":2358},[1612,1653,1649],[40,3291,2269],{"className":3292},[2268],[40,3294,3296],{"className":3295},[1629],[40,3297,3299],{"className":3298,"style":2276},[1633],[40,3300],{},", which we'll refer to as ",[40,3303,3305,3325],{"className":3304},[1565],[40,3306,3308],{"className":3307},[1569],[1571,3309,3310],{"xmlns":1573},[1575,3311,3312,3322],{},[1578,3313,3314],{},[3315,3316,3317,3319],"mover",{"accent":1599},[1584,3318,2182],{},[1731,3320,3321],{},"~",[1591,3323,3324],{"encoding":1593},"\\tilde{H}",[40,3326,3328],{"className":3327,"ariaHidden":1599},[1598],[40,3329,3331,3335],{"className":3330},[1603],[40,3332],{"className":3333,"style":3334},[1607],"height:0.9202em;",[40,3336,3339],{"className":3337},[1612,3338],"accent",[40,3340,3342],{"className":3341},[1625],[40,3343,3345],{"className":3344},[1629],[40,3346,3348,3358],{"className":3347,"style":3334},[1633],[40,3349,3351,3355],{"style":3350},"top:-3em;",[40,3352],{"className":3353,"style":3354},[1641],"height:3em;",[40,3356,2182],{"className":3357,"style":2197},[1612,1653],[40,3359,3361,3364],{"style":3360},"top:-3.6023em;",[40,3362],{"className":3363,"style":3354},[1641],[40,3365,3369],{"className":3366,"style":3368},[3367],"accent-body","left:-0.1944em;",[40,3370,3321],{"className":3371},[1612],". The matrix elements of ",[40,3374,3376,3393],{"className":3375},[1565],[40,3377,3379],{"className":3378},[1569],[1571,3380,3381],{"xmlns":1573},[1575,3382,3383,3391],{},[1578,3384,3385],{},[3315,3386,3387,3389],{"accent":1599},[1584,3388,2182],{},[1731,3390,3321],{},[1591,3392,3324],{"encoding":1593},[40,3394,3396],{"className":3395,"ariaHidden":1599},[1598],[40,3397,3399,3402],{"className":3398},[1603],[40,3400],{"className":3401,"style":3334},[1607],[40,3403,3405],{"className":3404},[1612,3338],[40,3406,3408],{"className":3407},[1625],[40,3409,3411],{"className":3410},[1629],[40,3412,3414,3422],{"className":3413,"style":3334},[1633],[40,3415,3416,3419],{"style":3350},[40,3417],{"className":3418,"style":3354},[1641],[40,3420,2182],{"className":3421,"style":2197},[1612,1653],[40,3423,3424,3427],{"style":3360},[40,3425],{"className":3426,"style":3354},[1641],[40,3428,3430],{"className":3429,"style":3368},[3367],[40,3431,3321],{"className":3432},[1612],", the projection of the Hamiltonian in the Krylov space, can be calculated by calculating the following expectation values",[40,3435,3437],{"className":3436},[1827],[40,3438,3440,3495],{"className":3439},[1565],[40,3441,3443],{"className":3442},[1569],[1571,3444,3445],{"xmlns":1573,"display":1836},[1575,3446,3447,3492],{},[1578,3448,3449,3465,3467,3470,3476,3478,3480,3482,3488,3490],{},[2212,3450,3451,3457],{},[3315,3452,3453,3455],{"accent":1599},[1584,3454,2182],{},[1731,3456,3321],{},[1578,3458,3459,3462],{},[1584,3460,3461],{},"m",[1584,3463,3464],{},"n",[1731,3466,623],{},[1731,3468,3469],{"stretchy":1797},"⟨",[2212,3471,3472,3474],{},[1584,3473,2837],{},[1584,3475,3461],{},[1584,3477,1791],{"mathvariant":1790},[1584,3479,2182],{},[1584,3481,1791],{"mathvariant":1790},[2212,3483,3484,3486],{},[1584,3485,2837],{},[1584,3487,3464],{},[1731,3489,1798],{"stretchy":1797},[1731,3491,623],{},[1591,3493,3494],{"encoding":1593},"\\tilde{H}_{mn} = \\langle \\psi_m \\vert H \\vert \\psi_n \\rangle =",[40,3496,3498,3587],{"className":3497,"ariaHidden":1599},[1598],[40,3499,3501,3505,3578,3581,3584],{"className":3500},[1603],[40,3502],{"className":3503,"style":3504},[1607],"height:1.0702em;vertical-align:-0.15em;",[40,3506,3508,3539],{"className":3507},[1612],[40,3509,3511],{"className":3510},[1612,3338],[40,3512,3514],{"className":3513},[1625],[40,3515,3517],{"className":3516},[1629],[40,3518,3520,3528],{"className":3519,"style":3334},[1633],[40,3521,3522,3525],{"style":3350},[40,3523],{"className":3524,"style":3354},[1641],[40,3526,2182],{"className":3527,"style":2197},[1612,1653],[40,3529,3530,3533],{"style":3360},[40,3531],{"className":3532,"style":3354},[1641],[40,3534,3536],{"className":3535,"style":3368},[3367],[40,3537,3321],{"className":3538},[1612],[40,3540,3542],{"className":3541},[1621],[40,3543,3545,3570],{"className":3544},[1625,2244],[40,3546,3548,3567],{"className":3547},[1629],[40,3549,3552],{"className":3550,"style":3551},[1633],"height:0.1514em;",[40,3553,3555,3558],{"style":3554},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[40,3556],{"className":3557,"style":1642},[1641],[40,3559,3561],{"className":3560},[1646,1647,1648,1649],[40,3562,3564],{"className":3563},[1612,1649],[40,3565,1735],{"className":3566},[1612,1653,1649],[40,3568,2269],{"className":3569},[2268],[40,3571,3573],{"className":3572},[1629],[40,3574,3576],{"className":3575,"style":2276},[1633],[40,3577],{},[40,3579],{"className":3580,"style":1963},[1756],[40,3582,623],{"className":3583},[1967],[40,3585],{"className":3586,"style":1963},[1756],[40,3588,3590,3593,3596,3637,3640,3643,3646,3686,3689,3692],{"className":3589},[1603],[40,3591],{"className":3592,"style":1811},[1607],[40,3594,3469],{"className":3595},[1985],[40,3597,3599,3602],{"className":3598},[1612],[40,3600,2837],{"className":3601,"style":1818},[1612,1653],[40,3603,3605],{"className":3604},[1621],[40,3606,3608,3629],{"className":3607},[1625,2244],[40,3609,3611,3626],{"className":3610},[1629],[40,3612,3614],{"className":3613,"style":3551},[1633],[40,3615,3617,3620],{"style":3616},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[40,3618],{"className":3619,"style":1642},[1641],[40,3621,3623],{"className":3622},[1646,1647,1648,1649],[40,3624,3461],{"className":3625},[1612,1653,1649],[40,3627,2269],{"className":3628},[2268],[40,3630,3632],{"className":3631},[1629],[40,3633,3635],{"className":3634,"style":2276},[1633],[40,3636],{},[40,3638,1791],{"className":3639},[1612],[40,3641,2182],{"className":3642,"style":2197},[1612,1653],[40,3644,1791],{"className":3645},[1612],[40,3647,3649,3652],{"className":3648},[1612],[40,3650,2837],{"className":3651,"style":1818},[1612,1653],[40,3653,3655],{"className":3654},[1621],[40,3656,3658,3678],{"className":3657},[1625,2244],[40,3659,3661,3675],{"className":3660},[1629],[40,3662,3664],{"className":3663,"style":3551},[1633],[40,3665,3666,3669],{"style":3616},[40,3667],{"className":3668,"style":1642},[1641],[40,3670,3672],{"className":3671},[1646,1647,1648,1649],[40,3673,3464],{"className":3674},[1612,1653,1649],[40,3676,2269],{"className":3677},[2268],[40,3679,3681],{"className":3680},[1629],[40,3682,3684],{"className":3683,"style":2276},[1633],[40,3685],{},[40,3687,1798],{"className":3688},[1822],[40,3690],{"className":3691,"style":1963},[1756],[40,3693,623],{"className":3694},[1967],[40,3696,3698],{"className":3697},[1827],[40,3699,3701,3763],{"className":3700},[1565],[40,3702,3704],{"className":3703},[1569],[1571,3705,3706],{"xmlns":1573,"display":1836},[1575,3707,3708,3760],{},[1578,3709,3710,3712,3714,3716,3718,3734,3736,3754,3756,3758],{},[1731,3711,623],{},[1731,3713,3469],{"stretchy":1797},[1584,3715,2837],{},[1584,3717,1791],{"mathvariant":1790},[1581,3719,3720,3722],{},[1584,3721,2330],{},[1578,3723,3724,3726,3728],{},[1584,3725,2337],{},[1584,3727,2182],{},[2212,3729,3730,3732],{},[1584,3731,2342],{},[1584,3733,3461],{},[1584,3735,2182],{},[1581,3737,3738,3740],{},[1584,3739,2330],{},[1578,3741,3742,3744,3746,3748],{},[1731,3743,1733],{},[1584,3745,2337],{},[1584,3747,2182],{},[2212,3749,3750,3752],{},[1584,3751,2342],{},[1584,3753,3464],{},[1584,3755,1791],{"mathvariant":1790},[1584,3757,2837],{},[1731,3759,1798],{"stretchy":1797},[1591,3761,3762],{"encoding":1593},"= \\langle \\psi \\vert  e^{i H t_m}   H e^{-i H t_n} \\vert \\psi \\rangle",[40,3764,3766,3779],{"className":3765,"ariaHidden":1599},[1598],[40,3767,3769,3773,3776],{"className":3768},[1603],[40,3770],{"className":3771,"style":3772},[1607],"height:0.3669em;",[40,3774,623],{"className":3775},[1967],[40,3777],{"className":3778,"style":1963},[1756],[40,3780,3782,3786,3789,3792,3795,3876,3879,3957,3960,3963],{"className":3781},[1603],[40,3783],{"className":3784,"style":3785},[1607],"height:1.1413em;vertical-align:-0.25em;",[40,3787,3469],{"className":3788},[1985],[40,3790,2837],{"className":3791,"style":1818},[1612,1653],[40,3793,1791],{"className":3794},[1612],[40,3796,3798,3801],{"className":3797},[1612],[40,3799,2330],{"className":3800},[1612,1653],[40,3802,3804],{"className":3803},[1621],[40,3805,3807],{"className":3806},[1625],[40,3808,3810],{"className":3809},[1629],[40,3811,3814],{"className":3812,"style":3813},[1633],"height:0.8913em;",[40,3815,3816,3819],{"style":1950},[40,3817],{"className":3818,"style":1642},[1641],[40,3820,3822],{"className":3821},[1646,1647,1648,1649],[40,3823,3825,3828,3831],{"className":3824},[1612,1649],[40,3826,2337],{"className":3827},[1612,1653,1649],[40,3829,2182],{"className":3830,"style":2197},[1612,1653,1649],[40,3832,3834,3837],{"className":3833},[1612,1649],[40,3835,2342],{"className":3836},[1612,1653,1649],[40,3838,3840],{"className":3839},[1621],[40,3841,3843,3867],{"className":3842},[1625,2244],[40,3844,3846,3864],{"className":3845},[1629],[40,3847,3850],{"className":3848,"style":3849},[1633],"height:0.1645em;",[40,3851,3853,3857],{"style":3852},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[40,3854],{"className":3855,"style":3856},[1641],"height:2.5em;",[40,3858,3861],{"className":3859},[1646,3860,1992,1649],"reset-size3",[40,3862,3461],{"className":3863},[1612,1653,1649],[40,3865,2269],{"className":3866},[2268],[40,3868,3870],{"className":3869},[1629],[40,3871,3874],{"className":3872,"style":3873},[1633],"height:0.143em;",[40,3875],{},[40,3877,2182],{"className":3878,"style":2197},[1612,1653],[40,3880,3882,3885],{"className":3881},[1612],[40,3883,2330],{"className":3884},[1612,1653],[40,3886,3888],{"className":3887},[1621],[40,3889,3891],{"className":3890},[1625],[40,3892,3894],{"className":3893},[1629],[40,3895,3897],{"className":3896,"style":3813},[1633],[40,3898,3899,3902],{"style":1950},[40,3900],{"className":3901,"style":1642},[1641],[40,3903,3905],{"className":3904},[1646,1647,1648,1649],[40,3906,3908,3911,3914,3917],{"className":3907},[1612,1649],[40,3909,1733],{"className":3910},[1612,1649],[40,3912,2337],{"className":3913},[1612,1653,1649],[40,3915,2182],{"className":3916,"style":2197},[1612,1653,1649],[40,3918,3920,3923],{"className":3919},[1612,1649],[40,3921,2342],{"className":3922},[1612,1653,1649],[40,3924,3926],{"className":3925},[1621],[40,3927,3929,3949],{"className":3928},[1625,2244],[40,3930,3932,3946],{"className":3931},[1629],[40,3933,3935],{"className":3934,"style":3849},[1633],[40,3936,3937,3940],{"style":3852},[40,3938],{"className":3939,"style":3856},[1641],[40,3941,3943],{"className":3942},[1646,3860,1992,1649],[40,3944,3464],{"className":3945},[1612,1653,1649],[40,3947,2269],{"className":3948},[2268],[40,3950,3952],{"className":3951},[1629],[40,3953,3955],{"className":3954,"style":3873},[1633],[40,3956],{},[40,3958,1791],{"className":3959},[1612],[40,3961,2837],{"className":3962,"style":1818},[1612,1653],[40,3964,1798],{"className":3965},[1822],[40,3967,3969],{"className":3968},[1827],[40,3970,3972,4035],{"className":3971},[1565],[40,3973,3975],{"className":3974},[1569],[1571,3976,3977],{"xmlns":1573,"display":1836},[1575,3978,3979,4032],{},[1578,3980,3981,3983,3985,3987,3989,4006,4008,4026,4028,4030],{},[1731,3982,623],{},[1731,3984,3469],{"stretchy":1797},[1584,3986,2837],{},[1584,3988,1791],{"mathvariant":1790},[1581,3990,3991,3993],{},[1584,3992,2330],{},[1578,3994,3995,3997,3999,4001,4004],{},[1584,3996,2337],{},[1584,3998,2182],{},[1584,4000,3461],{},[1584,4002,4003],{},"d",[1584,4005,2342],{},[1584,4007,2182],{},[1581,4009,4010,4012],{},[1584,4011,2330],{},[1578,4013,4014,4016,4018,4020,4022,4024],{},[1731,4015,1733],{},[1584,4017,2337],{},[1584,4019,2182],{},[1584,4021,3464],{},[1584,4023,4003],{},[1584,4025,2342],{},[1584,4027,1791],{"mathvariant":1790},[1584,4029,2837],{},[1731,4031,1798],{"stretchy":1797},[1591,4033,4034],{"encoding":1593},"= \\langle \\psi \\vert  e^{i H m dt}   H e^{-i H n dt} \\vert \\psi \\rangle",[40,4036,4038,4050],{"className":4037,"ariaHidden":1599},[1598],[40,4039,4041,4044,4047],{"className":4040},[1603],[40,4042],{"className":4043,"style":3772},[1607],[40,4045,623],{"className":4046},[1967],[40,4048],{"className":4049,"style":1963},[1756],[40,4051,4053,4057,4060,4063,4066,4111,4114,4161,4164,4167],{"className":4052},[1603],[40,4054],{"className":4055,"style":4056},[1607],"height:1.1491em;vertical-align:-0.25em;",[40,4058,3469],{"className":4059},[1985],[40,4061,2837],{"className":4062,"style":1818},[1612,1653],[40,4064,1791],{"className":4065},[1612],[40,4067,4069,4072],{"className":4068},[1612],[40,4070,2330],{"className":4071},[1612,1653],[40,4073,4075],{"className":4074},[1621],[40,4076,4078],{"className":4077},[1625],[40,4079,4081],{"className":4080},[1629],[40,4082,4085],{"className":4083,"style":4084},[1633],"height:0.8991em;",[40,4086,4087,4090],{"style":1950},[40,4088],{"className":4089,"style":1642},[1641],[40,4091,4093],{"className":4092},[1646,1647,1648,1649],[40,4094,4096,4099,4102,4105,4108],{"className":4095},[1612,1649],[40,4097,2337],{"className":4098},[1612,1653,1649],[40,4100,2182],{"className":4101,"style":2197},[1612,1653,1649],[40,4103,3461],{"className":4104},[1612,1653,1649],[40,4106,4003],{"className":4107},[1612,1653,1649],[40,4109,2342],{"className":4110},[1612,1653,1649],[40,4112,2182],{"className":4113,"style":2197},[1612,1653],[40,4115,4117,4120],{"className":4116},[1612],[40,4118,2330],{"className":4119},[1612,1653],[40,4121,4123],{"className":4122},[1621],[40,4124,4126],{"className":4125},[1625],[40,4127,4129],{"className":4128},[1629],[40,4130,4132],{"className":4131,"style":4084},[1633],[40,4133,4134,4137],{"style":1950},[40,4135],{"className":4136,"style":1642},[1641],[40,4138,4140],{"className":4139},[1646,1647,1648,1649],[40,4141,4143,4146,4149,4152,4155,4158],{"className":4142},[1612,1649],[40,4144,1733],{"className":4145},[1612,1649],[40,4147,2337],{"className":4148},[1612,1653,1649],[40,4150,2182],{"className":4151,"style":2197},[1612,1653,1649],[40,4153,3464],{"className":4154},[1612,1653,1649],[40,4156,4003],{"className":4157},[1612,1653,1649],[40,4159,2342],{"className":4160},[1612,1653,1649],[40,4162,1791],{"className":4163},[1612],[40,4165,2837],{"className":4166,"style":1818},[1612,1653],[40,4168,1798],{"className":4169},[1822],[12,4171,4172,4173,4382,4383,4488,4489,4523,4524,4593,4594,4622,4623,4675,4676,4704,4705,4762,4763,4823,4824,4884,4885],{},"Where ",[40,4174,4176,4224],{"className":4175},[1565],[40,4177,4179],{"className":4178},[1569],[1571,4180,4181],{"xmlns":1573},[1575,4182,4183,4221],{},[1578,4184,4185,4187,4193,4195,4197,4215,4217,4219],{},[1584,4186,1791],{"mathvariant":1790},[2212,4188,4189,4191],{},[1584,4190,2837],{},[1584,4192,3464],{},[1731,4194,1798],{"stretchy":1797},[1731,4196,623],{},[1581,4198,4199,4201],{},[1584,4200,2330],{},[1578,4202,4203,4205,4207,4209],{},[1731,4204,1733],{},[1584,4206,2337],{},[1584,4208,2182],{},[2212,4210,4211,4213],{},[1584,4212,2342],{},[1584,4214,3464],{},[1584,4216,1791],{"mathvariant":1790},[1584,4218,2837],{},[1731,4220,1798],{"stretchy":1797},[1591,4222,4223],{"encoding":1593},"\\vert \\psi_n \\rangle = e^{-i H t_n} \\vert \\psi \\rangle",[40,4225,4227,4288],{"className":4226,"ariaHidden":1599},[1598],[40,4228,4230,4233,4236,4276,4279,4282,4285],{"className":4229},[1603],[40,4231],{"className":4232,"style":1811},[1607],[40,4234,1791],{"className":4235},[1612],[40,4237,4239,4242],{"className":4238},[1612],[40,4240,2837],{"className":4241,"style":1818},[1612,1653],[40,4243,4245],{"className":4244},[1621],[40,4246,4248,4268],{"className":4247},[1625,2244],[40,4249,4251,4265],{"className":4250},[1629],[40,4252,4254],{"className":4253,"style":3551},[1633],[40,4255,4256,4259],{"style":3616},[40,4257],{"className":4258,"style":1642},[1641],[40,4260,4262],{"className":4261},[1646,1647,1648,1649],[40,4263,3464],{"className":4264},[1612,1653,1649],[40,4266,2269],{"className":4267},[2268],[40,4269,4271],{"className":4270},[1629],[40,4272,4274],{"className":4273,"style":2276},[1633],[40,4275],{},[40,4277,1798],{"className":4278},[1822],[40,4280],{"className":4281,"style":1963},[1756],[40,4283,623],{"className":4284},[1967],[40,4286],{"className":4287,"style":1963},[1756],[40,4289,4291,4295,4373,4376,4379],{"className":4290},[1603],[40,4292],{"className":4293,"style":4294},[1607],"height:1.0913em;vertical-align:-0.25em;",[40,4296,4298,4301],{"className":4297},[1612],[40,4299,2330],{"className":4300},[1612,1653],[40,4302,4304],{"className":4303},[1621],[40,4305,4307],{"className":4306},[1625],[40,4308,4310],{"className":4309},[1629],[40,4311,4313],{"className":4312,"style":2374},[1633],[40,4314,4315,4318],{"style":1637},[40,4316],{"className":4317,"style":1642},[1641],[40,4319,4321],{"className":4320},[1646,1647,1648,1649],[40,4322,4324,4327,4330,4333],{"className":4323},[1612,1649],[40,4325,1733],{"className":4326},[1612,1649],[40,4328,2337],{"className":4329},[1612,1653,1649],[40,4331,2182],{"className":4332,"style":2197},[1612,1653,1649],[40,4334,4336,4339],{"className":4335},[1612,1649],[40,4337,2342],{"className":4338},[1612,1653,1649],[40,4340,4342],{"className":4341},[1621],[40,4343,4345,4365],{"className":4344},[1625,2244],[40,4346,4348,4362],{"className":4347},[1629],[40,4349,4351],{"className":4350,"style":3849},[1633],[40,4352,4353,4356],{"style":3852},[40,4354],{"className":4355,"style":3856},[1641],[40,4357,4359],{"className":4358},[1646,3860,1992,1649],[40,4360,3464],{"className":4361},[1612,1653,1649],[40,4363,2269],{"className":4364},[2268],[40,4366,4368],{"className":4367},[1629],[40,4369,4371],{"className":4370,"style":3873},[1633],[40,4372],{},[40,4374,1791],{"className":4375},[1612],[40,4377,2837],{"className":4378,"style":1818},[1612,1653],[40,4380,1798],{"className":4381},[1822]," are the vectors of the unitary Krylov space and ",[40,4384,4386,4412],{"className":4385},[1565],[40,4387,4389],{"className":4388},[1569],[1571,4390,4391],{"xmlns":1573},[1575,4392,4393,4409],{},[1578,4394,4395,4401,4403,4405,4407],{},[2212,4396,4397,4399],{},[1584,4398,2342],{},[1584,4400,3464],{},[1731,4402,623],{},[1584,4404,3464],{},[1584,4406,4003],{},[1584,4408,2342],{},[1591,4410,4411],{"encoding":1593},"t_n = n dt",[40,4413,4415,4472],{"className":4414,"ariaHidden":1599},[1598],[40,4416,4418,4422,4463,4466,4469],{"className":4417},[1603],[40,4419],{"className":4420,"style":4421},[1607],"height:0.7651em;vertical-align:-0.15em;",[40,4423,4425,4428],{"className":4424},[1612],[40,4426,2342],{"className":4427},[1612,1653],[40,4429,4431],{"className":4430},[1621],[40,4432,4434,4455],{"className":4433},[1625,2244],[40,4435,4437,4452],{"className":4436},[1629],[40,4438,4440],{"className":4439,"style":3551},[1633],[40,4441,4443,4446],{"style":4442},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[40,4444],{"className":4445,"style":1642},[1641],[40,4447,4449],{"className":4448},[1646,1647,1648,1649],[40,4450,3464],{"className":4451},[1612,1653,1649],[40,4453,2269],{"className":4454},[2268],[40,4456,4458],{"className":4457},[1629],[40,4459,4461],{"className":4460,"style":2276},[1633],[40,4462],{},[40,4464],{"className":4465,"style":1963},[1756],[40,4467,623],{"className":4468},[1967],[40,4470],{"className":4471,"style":1963},[1756],[40,4473,4475,4479,4482,4485],{"className":4474},[1603],[40,4476],{"className":4477,"style":4478},[1607],"height:0.6944em;",[40,4480,3464],{"className":4481},[1612,1653],[40,4483,4003],{"className":4484},[1612,1653],[40,4486,2342],{"className":4487},[1612,1653]," are the multiples of the time step ",[40,4490,4492,4508],{"className":4491},[1565],[40,4493,4495],{"className":4494},[1569],[1571,4496,4497],{"xmlns":1573},[1575,4498,4499,4505],{},[1578,4500,4501,4503],{},[1584,4502,4003],{},[1584,4504,2342],{},[1591,4506,4507],{"encoding":1593},"dt",[40,4509,4511],{"className":4510,"ariaHidden":1599},[1598],[40,4512,4514,4517,4520],{"className":4513},[1603],[40,4515],{"className":4516,"style":4478},[1607],[40,4518,4003],{"className":4519},[1612,1653],[40,4521,2342],{"className":4522},[1612,1653]," 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 ",[40,4525,4527,4544],{"className":4526},[1565],[40,4528,4530],{"className":4529},[1569],[1571,4531,4532],{"xmlns":1573},[1575,4533,4534,4542],{},[1578,4535,4536],{},[2212,4537,4538,4540],{},[1584,4539,1587],{"mathvariant":1586},[1584,4541,2323],{},[1591,4543,2437],{"encoding":1593},[40,4545,4547],{"className":4546,"ariaHidden":1599},[1598],[40,4548,4550,4553],{"className":4549},[1603],[40,4551],{"className":4552,"style":2231},[1607],[40,4554,4556,4559],{"className":4555},[1612],[40,4557,1587],{"className":4558,"style":1617},[1612,1616],[40,4560,4562],{"className":4561},[1621],[40,4563,4565,4585],{"className":4564},[1625,2244],[40,4566,4568,4582],{"className":4567},[1629],[40,4569,4571],{"className":4570,"style":2251},[1633],[40,4572,4573,4576],{"style":2254},[40,4574],{"className":4575,"style":1642},[1641],[40,4577,4579],{"className":4578},[1646,1647,1648,1649],[40,4580,2323],{"className":4581,"style":2358},[1612,1653,1649],[40,4583,2269],{"className":4584},[2268],[40,4586,4588],{"className":4587},[1629],[40,4589,4591],{"className":4590,"style":2276},[1633],[40,4592],{}," has dimension ",[40,4595,4597,4610],{"className":4596},[1565],[40,4598,4600],{"className":4599},[1569],[1571,4601,4602],{"xmlns":1573},[1575,4603,4604,4608],{},[1578,4605,4606],{},[1584,4607,1256],{},[1591,4609,1256],{"encoding":1593},[40,4611,4613],{"className":4612,"ariaHidden":1599},[1598],[40,4614,4616,4619],{"className":4615},[1603],[40,4617],{"className":4618,"style":1681},[1607],[40,4620,1256],{"className":4621,"style":1654},[1612,1653],", the Hamiltonian projected into the subspace will have dimensions ",[40,4624,4626,4645],{"className":4625},[1565],[40,4627,4629],{"className":4628},[1569],[1571,4630,4631],{"xmlns":1573},[1575,4632,4633,4642],{},[1578,4634,4635,4637,4640],{},[1584,4636,1256],{},[1731,4638,4639],{},"×",[1584,4641,1256],{},[1591,4643,4644],{"encoding":1593},"r \\times r",[40,4646,4648,4666],{"className":4647,"ariaHidden":1599},[1598],[40,4649,4651,4654,4657,4660,4663],{"className":4650},[1603],[40,4652],{"className":4653,"style":1749},[1607],[40,4655,1256],{"className":4656,"style":1654},[1612,1653],[40,4658],{"className":4659,"style":1757},[1756],[40,4661,4639],{"className":4662},[1761],[40,4664],{"className":4665,"style":1757},[1756],[40,4667,4669,4672],{"className":4668},[1603],[40,4670],{"className":4671,"style":1681},[1607],[40,4673,1256],{"className":4674,"style":1654},[1612,1653],". With ",[40,4677,4679,4692],{"className":4678},[1565],[40,4680,4682],{"className":4681},[1569],[1571,4683,4684],{"xmlns":1573},[1575,4685,4686,4690],{},[1578,4687,4688],{},[1584,4689,1256],{},[1591,4691,1256],{"encoding":1593},[40,4693,4695],{"className":4694,"ariaHidden":1599},[1598],[40,4696,4698,4701],{"className":4697},[1603],[40,4699],{"className":4700,"style":1681},[1607],[40,4702,1256],{"className":4703,"style":1654},[1612,1653]," small enough (generally ",[40,4706,4708,4730],{"className":4707},[1565],[40,4709,4711],{"className":4710},[1569],[1571,4712,4713],{"xmlns":1573},[1575,4714,4715,4727],{},[1578,4716,4717,4719,4722,4724],{},[1584,4718,1256],{},[1731,4720,4721],{},"\u003C",[1731,4723,4721],{},[1735,4725,4726],{},"100",[1591,4728,4729],{"encoding":1593},"r\u003C\u003C100",[40,4731,4733,4753],{"className":4732,"ariaHidden":1599},[1598],[40,4734,4736,4740,4743,4746,4750],{"className":4735},[1603],[40,4737],{"className":4738,"style":4739},[1607],"height:0.5782em;vertical-align:-0.0391em;",[40,4741,1256],{"className":4742,"style":1654},[1612,1653],[40,4744],{"className":4745,"style":1963},[1756],[40,4747,4749],{"className":4748},[1967],"\u003C\u003C",[40,4751],{"className":4752,"style":1963},[1756],[40,4754,4756,4759],{"className":4755},[1603],[40,4757],{"className":4758,"style":1771},[1607],[40,4760,4726],{"className":4761},[1612]," is sufficient to obtain convergence of estimates of eigenenergies) we can then easily diagonalize the projected Hamiltonian ",[40,4764,4766,4783],{"className":4765},[1565],[40,4767,4769],{"className":4768},[1569],[1571,4770,4771],{"xmlns":1573},[1575,4772,4773,4781],{},[1578,4774,4775],{},[3315,4776,4777,4779],{"accent":1599},[1584,4778,2182],{},[1731,4780,3321],{},[1591,4782,3324],{"encoding":1593},[40,4784,4786],{"className":4785,"ariaHidden":1599},[1598],[40,4787,4789,4792],{"className":4788},[1603],[40,4790],{"className":4791,"style":3334},[1607],[40,4793,4795],{"className":4794},[1612,3338],[40,4796,4798],{"className":4797},[1625],[40,4799,4801],{"className":4800},[1629],[40,4802,4804,4812],{"className":4803,"style":3334},[1633],[40,4805,4806,4809],{"style":3350},[40,4807],{"className":4808,"style":3354},[1641],[40,4810,2182],{"className":4811,"style":2197},[1612,1653],[40,4813,4814,4817],{"style":3360},[40,4815],{"className":4816,"style":3354},[1641],[40,4818,4820],{"className":4819,"style":3368},[3367],[40,4821,3321],{"className":4822},[1612],". However, we cannot directly diagonalize ",[40,4825,4827,4844],{"className":4826},[1565],[40,4828,4830],{"className":4829},[1569],[1571,4831,4832],{"xmlns":1573},[1575,4833,4834,4842],{},[1578,4835,4836],{},[3315,4837,4838,4840],{"accent":1599},[1584,4839,2182],{},[1731,4841,3321],{},[1591,4843,3324],{"encoding":1593},[40,4845,4847],{"className":4846,"ariaHidden":1599},[1598],[40,4848,4850,4853],{"className":4849},[1603],[40,4851],{"className":4852,"style":3334},[1607],[40,4854,4856],{"className":4855},[1612,3338],[40,4857,4859],{"className":4858},[1625],[40,4860,4862],{"className":4861},[1629],[40,4863,4865,4873],{"className":4864,"style":3334},[1633],[40,4866,4867,4870],{"style":3350},[40,4868],{"className":4869,"style":3354},[1641],[40,4871,2182],{"className":4872,"style":2197},[1612,1653],[40,4874,4875,4878],{"style":3360},[40,4876],{"className":4877,"style":3354},[1641],[40,4879,4881],{"className":4880,"style":3368},[3367],[40,4882,3321],{"className":4883},[1612]," because of the non-orthogonality of the Krylov space vectors. We'll have to measure their overlaps and construct a matrix ",[40,4886,4888,4907],{"className":4887},[1565],[40,4889,4891],{"className":4890},[1569],[1571,4892,4893],{"xmlns":1573},[1575,4894,4895,4904],{},[1578,4896,4897],{},[3315,4898,4899,4902],{"accent":1599},[1584,4900,4901],{},"S",[1731,4903,3321],{},[1591,4905,4906],{"encoding":1593},"\\tilde{S}",[40,4908,4910],{"className":4909,"ariaHidden":1599},[1598],[40,4911,4913,4916],{"className":4912},[1603],[40,4914],{"className":4915,"style":3334},[1607],[40,4917,4919],{"className":4918},[1612,3338],[40,4920,4922],{"className":4921},[1625],[40,4923,4925],{"className":4924},[1629],[40,4926,4928,4937],{"className":4927,"style":3334},[1633],[40,4929,4930,4933],{"style":3350},[40,4931],{"className":4932,"style":3354},[1641],[40,4934,4901],{"className":4935,"style":4936},[1612,1653],"margin-right:0.0576em;",[40,4938,4939,4942],{"style":3360},[40,4940],{"className":4941,"style":3354},[1641],[40,4943,4946],{"className":4944,"style":4945},[3367],"left:-0.1667em;",[40,4947,3321],{"className":4948},[1612],[40,4950,4952],{"className":4951},[1827],[40,4953,4955,5001],{"className":4954},[1565],[40,4956,4958],{"className":4957},[1569],[1571,4959,4960],{"xmlns":1573,"display":1836},[1575,4961,4962,4998],{},[1578,4963,4964,4978,4980,4982,4988,4990,4996],{},[2212,4965,4966,4972],{},[3315,4967,4968,4970],{"accent":1599},[1584,4969,4901],{},[1731,4971,3321],{},[1578,4973,4974,4976],{},[1584,4975,3461],{},[1584,4977,3464],{},[1731,4979,623],{},[1731,4981,3469],{"stretchy":1797},[2212,4983,4984,4986],{},[1584,4985,2837],{},[1584,4987,3461],{},[1584,4989,1791],{"mathvariant":1790},[2212,4991,4992,4994],{},[1584,4993,2837],{},[1584,4995,3464],{},[1731,4997,1798],{"stretchy":1797},[1591,4999,5000],{"encoding":1593},"\\tilde{S}_{mn} = \\langle \\psi_m \\vert \\psi_n \\rangle",[40,5002,5004,5091],{"className":5003,"ariaHidden":1599},[1598],[40,5005,5007,5010,5082,5085,5088],{"className":5006},[1603],[40,5008],{"className":5009,"style":3504},[1607],[40,5011,5013,5044],{"className":5012},[1612],[40,5014,5016],{"className":5015},[1612,3338],[40,5017,5019],{"className":5018},[1625],[40,5020,5022],{"className":5021},[1629],[40,5023,5025,5033],{"className":5024,"style":3334},[1633],[40,5026,5027,5030],{"style":3350},[40,5028],{"className":5029,"style":3354},[1641],[40,5031,4901],{"className":5032,"style":4936},[1612,1653],[40,5034,5035,5038],{"style":3360},[40,5036],{"className":5037,"style":3354},[1641],[40,5039,5041],{"className":5040,"style":4945},[3367],[40,5042,3321],{"className":5043},[1612],[40,5045,5047],{"className":5046},[1621],[40,5048,5050,5074],{"className":5049},[1625,2244],[40,5051,5053,5071],{"className":5052},[1629],[40,5054,5056],{"className":5055,"style":3551},[1633],[40,5057,5059,5062],{"style":5058},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[40,5060],{"className":5061,"style":1642},[1641],[40,5063,5065],{"className":5064},[1646,1647,1648,1649],[40,5066,5068],{"className":5067},[1612,1649],[40,5069,1735],{"className":5070},[1612,1653,1649],[40,5072,2269],{"className":5073},[2268],[40,5075,5077],{"className":5076},[1629],[40,5078,5080],{"className":5079,"style":2276},[1633],[40,5081],{},[40,5083],{"className":5084,"style":1963},[1756],[40,5086,623],{"className":5087},[1967],[40,5089],{"className":5090,"style":1963},[1756],[40,5092,5094,5097,5100,5140,5143,5183],{"className":5093},[1603],[40,5095],{"className":5096,"style":1811},[1607],[40,5098,3469],{"className":5099},[1985],[40,5101,5103,5106],{"className":5102},[1612],[40,5104,2837],{"className":5105,"style":1818},[1612,1653],[40,5107,5109],{"className":5108},[1621],[40,5110,5112,5132],{"className":5111},[1625,2244],[40,5113,5115,5129],{"className":5114},[1629],[40,5116,5118],{"className":5117,"style":3551},[1633],[40,5119,5120,5123],{"style":3616},[40,5121],{"className":5122,"style":1642},[1641],[40,5124,5126],{"className":5125},[1646,1647,1648,1649],[40,5127,3461],{"className":5128},[1612,1653,1649],[40,5130,2269],{"className":5131},[2268],[40,5133,5135],{"className":5134},[1629],[40,5136,5138],{"className":5137,"style":2276},[1633],[40,5139],{},[40,5141,1791],{"className":5142},[1612],[40,5144,5146,5149],{"className":5145},[1612],[40,5147,2837],{"className":5148,"style":1818},[1612,1653],[40,5150,5152],{"className":5151},[1621],[40,5153,5155,5175],{"className":5154},[1625,2244],[40,5156,5158,5172],{"className":5157},[1629],[40,5159,5161],{"className":5160,"style":3551},[1633],[40,5162,5163,5166],{"style":3616},[40,5164],{"className":5165,"style":1642},[1641],[40,5167,5169],{"className":5168},[1646,1647,1648,1649],[40,5170,3464],{"className":5171},[1612,1653,1649],[40,5173,2269],{"className":5174},[2268],[40,5176,5178],{"className":5177},[1629],[40,5179,5181],{"className":5180,"style":2276},[1633],[40,5182],{},[40,5184,1798],{"className":5185},[1822],[12,5187,5188],{},"This allows us to solve the eigenvalue problem in a non-orthogonal space (also called generalized eigenvalue problem)",[40,5190,5192],{"className":5191},[1827],[40,5193,5195,5246],{"className":5194},[1565],[40,5196,5198],{"className":5197},[1569],[1571,5199,5200],{"xmlns":1573,"display":1836},[1575,5201,5202,5243],{},[1578,5203,5204,5210,5214,5222,5224,5227,5229,5235,5237],{},[3315,5205,5206,5208],{"accent":1599},[1584,5207,2182],{},[1731,5209,3321],{},[5211,5212,5213],"mtext",{}," ",[3315,5215,5216,5219],{"accent":1599},[1584,5217,5218],{},"c",[1731,5220,5221],{},"⃗",[1731,5223,623],{},[1584,5225,5226],{},"E",[5211,5228,5213],{},[3315,5230,5231,5233],{"accent":1599},[1584,5232,4901],{},[1731,5234,3321],{},[5211,5236,5213],{},[3315,5238,5239,5241],{"accent":1599},[1584,5240,5218],{},[1731,5242,5221],{},[1591,5244,5245],{"encoding":1593},"\\tilde{H} \\ \\vec{c} = E \\ \\tilde{S} \\ \\vec{c}",[40,5247,5249,5346],{"className":5248,"ariaHidden":1599},[1598],[40,5250,5252,5255,5286,5289,5337,5340,5343],{"className":5251},[1603],[40,5253],{"className":5254,"style":3334},[1607],[40,5256,5258],{"className":5257},[1612,3338],[40,5259,5261],{"className":5260},[1625],[40,5262,5264],{"className":5263},[1629],[40,5265,5267,5275],{"className":5266,"style":3334},[1633],[40,5268,5269,5272],{"style":3350},[40,5270],{"className":5271,"style":3354},[1641],[40,5273,2182],{"className":5274,"style":2197},[1612,1653],[40,5276,5277,5280],{"style":3360},[40,5278],{"className":5279,"style":3354},[1641],[40,5281,5283],{"className":5282,"style":3368},[3367],[40,5284,3321],{"className":5285},[1612],[40,5287,5213],{"className":5288},[1756],[40,5290,5292],{"className":5291},[1612,3338],[40,5293,5295],{"className":5294},[1625],[40,5296,5298],{"className":5297},[1629],[40,5299,5302,5310],{"className":5300,"style":5301},[1633],"height:0.714em;",[40,5303,5304,5307],{"style":3350},[40,5305],{"className":5306,"style":3354},[1641],[40,5308,5218],{"className":5309},[1612,1653],[40,5311,5312,5315],{"style":3350},[40,5313],{"className":5314,"style":3354},[1641],[40,5316,5319],{"className":5317,"style":5318},[3367],"left:-0.1799em;",[40,5320,5324],{"className":5321,"style":5323},[5322],"overlay","height:0.714em;width:0.471em;",[5325,5326,5333],"svg",{"xmlns":5327,"width":5328,"height":5329,"style":5330,"viewBox":5331,"preserveAspectRatio":5332},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.471em","0.714em","width:0.471em","0 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can then obtain estimates of the eigenvalues and eigenstates of ",[40,5431,5433,5446],{"className":5432},[1565],[40,5434,5436],{"className":5435},[1569],[1571,5437,5438],{"xmlns":1573},[1575,5439,5440,5444],{},[1578,5441,5442],{},[1584,5443,2182],{},[1591,5445,2182],{"encoding":1593},[40,5447,5449],{"className":5448,"ariaHidden":1599},[1598],[40,5450,5452,5455],{"className":5451},[1603],[40,5453],{"className":5454,"style":1608},[1607],[40,5456,2182],{"className":5457,"style":2197},[1612,1653]," by looking at the ones of ",[40,5460,5462,5479],{"className":5461},[1565],[40,5463,5465],{"className":5464},[1569],[1571,5466,5467],{"xmlns":1573},[1575,5468,5469,5477],{},[1578,5470,5471],{},[3315,5472,5473,5475],{"accent":1599},[1584,5474,2182],{},[1731,5476,3321],{},[1591,5478,3324],{"encoding":1593},[40,5480,5482],{"className":5481,"ariaHidden":1599},[1598],[40,5483,5485,5488],{"className":5484},[1603],[40,5486],{"className":5487,"style":3334},[1607],[40,5489,5491],{"className":5490},[1612,3338],[40,5492,5494],{"className":5493},[1625],[40,5495,5497],{"className":5496},[1629],[40,5498,5500,5508],{"className":5499,"style":3334},[1633],[40,5501,5502,5505],{"style":3350},[40,5503],{"className":5504,"style":3354},[1641],[40,5506,2182],{"className":5507,"style":2197},[1612,1653],[40,5509,5510,5513],{"style":3360},[40,5511],{"className":5512,"style":3354},[1641],[40,5514,5516],{"className":5515,"style":3368},[3367],[40,5517,3321],{"className":5518},[1612],". For example, the estimate of the ground state energy is obtained by taking the smallest eigenvalue ",[40,5521,5523,5536],{"className":5522},[1565],[40,5524,5526],{"className":5525},[1569],[1571,5527,5528],{"xmlns":1573},[1575,5529,5530,5534],{},[1578,5531,5532],{},[1584,5533,5218],{},[1591,5535,5218],{"encoding":1593},[40,5537,5539],{"className":5538,"ariaHidden":1599},[1598],[40,5540,5542,5545],{"className":5541},[1603],[40,5543],{"className":5544,"style":1681},[1607],[40,5546,5218],{"className":5547},[1612,1653]," and the ground state from the corresponding eigenvector ",[40,5550,5552,5570],{"className":5551},[1565],[40,5553,5555],{"className":5554},[1569],[1571,5556,5557],{"xmlns":1573},[1575,5558,5559,5567],{},[1578,5560,5561],{},[3315,5562,5563,5565],{"accent":1599},[1584,5564,5218],{},[1731,5566,5221],{},[1591,5568,5569],{"encoding":1593},"\\vec{c}",[40,5571,5573],{"className":5572,"ariaHidden":1599},[1598],[40,5574,5576,5579],{"className":5575},[1603],[40,5577],{"className":5578,"style":5301},[1607],[40,5580,5582],{"className":5581},[1612,3338],[40,5583,5585],{"className":5584},[1625],[40,5586,5588],{"className":5587},[1629],[40,5589,5591,5599],{"className":5590,"style":5301},[1633],[40,5592,5593,5596],{"style":3350},[40,5594],{"className":5595,"style":3354},[1641],[40,5597,5218],{"className":5598},[1612,1653],[40,5600,5601,5604],{"style":3350},[40,5602],{"className":5603,"style":3354},[1641],[40,5605,5607],{"className":5606,"style":5318},[3367],[40,5608,5610],{"className":5609,"style":5323},[5322],[5325,5611,5612],{"xmlns":5327,"width":5328,"height":5329,"style":5330,"viewBox":5331,"preserveAspectRatio":5332},[5334,5613],{"d":5336},". The coefficients in ",[40,5616,5618,5635],{"className":5617},[1565],[40,5619,5621],{"className":5620},[1569],[1571,5622,5623],{"xmlns":1573},[1575,5624,5625,5633],{},[1578,5626,5627],{},[3315,5628,5629,5631],{"accent":1599},[1584,5630,5218],{},[1731,5632,5221],{},[1591,5634,5569],{"encoding":1593},[40,5636,5638],{"className":5637,"ariaHidden":1599},[1598],[40,5639,5641,5644],{"className":5640},[1603],[40,5642],{"className":5643,"style":5301},[1607],[40,5645,5647],{"className":5646},[1612,3338],[40,5648,5650],{"className":5649},[1625],[40,5651,5653],{"className":5652},[1629],[40,5654,5656,5664],{"className":5655,"style":5301},[1633],[40,5657,5658,5661],{"style":3350},[40,5659],{"className":5660,"style":3354},[1641],[40,5662,5218],{"className":5663},[1612,1653],[40,5665,5666,5669],{"style":3350},[40,5667],{"className":5668,"style":3354},[1641],[40,5670,5672],{"className":5671,"style":5318},[3367],[40,5673,5675],{"className":5674,"style":5323},[5322],[5325,5676,5677],{"xmlns":5327,"width":5328,"height":5329,"style":5330,"viewBox":5331,"preserveAspectRatio":5332},[5334,5678],{"d":5336}," determines the contribution of the different vectors that span ",[40,5681,5683,5700],{"className":5682},[1565],[40,5684,5686],{"className":5685},[1569],[1571,5687,5688],{"xmlns":1573},[1575,5689,5690,5698],{},[1578,5691,5692],{},[2212,5693,5694,5696],{},[1584,5695,1587],{"mathvariant":1586},[1584,5697,2323],{},[1591,5699,2437],{"encoding":1593},[40,5701,5703],{"className":5702,"ariaHidden":1599},[1598],[40,5704,5706,5709],{"className":5705},[1603],[40,5707],{"className":5708,"style":2231},[1607],[40,5710,5712,5715],{"className":5711},[1612],[40,5713,1587],{"className":5714,"style":1617},[1612,1616],[40,5716,5718],{"className":5717},[1621],[40,5719,5721,5741],{"className":5720},[1625,2244],[40,5722,5724,5738],{"className":5723},[1629],[40,5725,5727],{"className":5726,"style":2251},[1633],[40,5728,5729,5732],{"style":2254},[40,5730],{"className":5731,"style":1642},[1641],[40,5733,5735],{"className":5734},[1646,1647,1648,1649],[40,5736,2323],{"className":5737,"style":2358},[1612,1653,1649],[40,5739,2269],{"className":5740},[2268],[40,5742,5744],{"className":5743},[1629],[40,5745,5747],{"className":5746,"style":2276},[1633],[40,5748],{},[12,5750,5751],{},[5752,5753],"img",{"alt":5754,"src":5755},"fig1.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-06.avif",[12,5757,5758,5759,5880,5881,917,5961,6042,6043,917,6130,6214,6215,6275,6276,6358,6359,6438,6439,6521,6522,6550,6551,6702],{},"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 ",[40,5760,5762,5790],{"className":5761},[1565],[40,5763,5765],{"className":5764},[1569],[1571,5766,5767],{"xmlns":1573},[1575,5768,5769,5787],{},[1578,5770,5771],{},[2212,5772,5773,5779],{},[3315,5774,5775,5777],{"accent":1599},[1584,5776,2182],{},[1731,5778,3321],{},[1578,5780,5781,5783,5785],{},[1584,5782,2337],{},[1731,5784,1862],{"separator":1599},[1584,5786,1413],{},[1591,5788,5789],{"encoding":1593},"\\tilde{H}_{i,j}",[40,5791,5793],{"className":5792,"ariaHidden":1599},[1598],[40,5794,5796,5800],{"className":5795},[1603],[40,5797],{"className":5798,"style":5799},[1607],"height:1.2063em;vertical-align:-0.2861em;",[40,5801,5803,5834],{"className":5802},[1612],[40,5804,5806],{"className":5805},[1612,3338],[40,5807,5809],{"className":5808},[1625],[40,5810,5812],{"className":5811},[1629],[40,5813,5815,5823],{"className":5814,"style":3334},[1633],[40,5816,5817,5820],{"style":3350},[40,5818],{"className":5819,"style":3354},[1641],[40,5821,2182],{"className":5822,"style":2197},[1612,1653],[40,5824,5825,5828],{"style":3360},[40,5826],{"className":5827,"style":3354},[1641],[40,5829,5831],{"className":5830,"style":3368},[3367],[40,5832,3321],{"className":5833},[1612],[40,5835,5837],{"className":5836},[1621],[40,5838,5840,5871],{"className":5839},[1625,2244],[40,5841,5843,5868],{"className":5842},[1629],[40,5844,5847],{"className":5845,"style":5846},[1633],"height:0.3117em;",[40,5848,5849,5852],{"style":3554},[40,5850],{"className":5851,"style":1642},[1641],[40,5853,5855],{"className":5854},[1646,1647,1648,1649],[40,5856,5858,5861,5864],{"className":5857},[1612,1649],[40,5859,2337],{"className":5860},[1612,1653,1649],[40,5862,1862],{"className":5863},[2005,1649],[40,5865,1413],{"className":5866,"style":5867},[1612,1653,1649],"margin-right:0.0572em;",[40,5869,2269],{"className":5870},[2268],[40,5872,5874],{"className":5873},[1629],[40,5875,5878],{"className":5876,"style":5877},[1633],"height:0.2861em;",[40,5879],{},", a Hadamard test between the state ",[40,5882,5884,5906],{"className":5883},[1565],[40,5885,5887],{"className":5886},[1569],[1571,5888,5889],{"xmlns":1573},[1575,5890,5891,5903],{},[1578,5892,5893,5895,5901],{},[1584,5894,1791],{"mathvariant":1790},[2212,5896,5897,5899],{},[1584,5898,2837],{},[1584,5900,2337],{},[1731,5902,1798],{"stretchy":1797},[1591,5904,5905],{"encoding":1593},"\\vert \\psi_i \\rangle",[40,5907,5909],{"className":5908,"ariaHidden":1599},[1598],[40,5910,5912,5915,5918,5958],{"className":5911},[1603],[40,5913],{"className":5914,"style":1811},[1607],[40,5916,1791],{"className":5917},[1612],[40,5919,5921,5924],{"className":5920},[1612],[40,5922,2837],{"className":5923,"style":1818},[1612,1653],[40,5925,5927],{"className":5926},[1621],[40,5928,5930,5950],{"className":5929},[1625,2244],[40,5931,5933,5947],{"className":5932},[1629],[40,5934,5936],{"className":5935,"style":5846},[1633],[40,5937,5938,5941],{"style":3616},[40,5939],{"className":5940,"style":1642},[1641],[40,5942,5944],{"className":5943},[1646,1647,1648,1649],[40,5945,2337],{"className":5946},[1612,1653,1649],[40,5948,2269],{"className":5949},[2268],[40,5951,5953],{"className":5952},[1629],[40,5954,5956],{"className":5955,"style":2276},[1633],[40,5957],{},[40,5959,1798],{"className":5960},[1822],[40,5962,5964,5986],{"className":5963},[1565],[40,5965,5967],{"className":5966},[1569],[1571,5968,5969],{"xmlns":1573},[1575,5970,5971,5983],{},[1578,5972,5973,5975,5981],{},[1584,5974,1791],{"mathvariant":1790},[2212,5976,5977,5979],{},[1584,5978,2837],{},[1584,5980,1413],{},[1731,5982,1798],{"stretchy":1797},[1591,5984,5985],{"encoding":1593},"\\vert \\psi_j \\rangle",[40,5987,5989],{"className":5988,"ariaHidden":1599},[1598],[40,5990,5992,5996,5999,6039],{"className":5991},[1603],[40,5993],{"className":5994,"style":5995},[1607],"height:1.0361em;vertical-align:-0.2861em;",[40,5997,1791],{"className":5998},[1612],[40,6000,6002,6005],{"className":6001},[1612],[40,6003,2837],{"className":6004,"style":1818},[1612,1653],[40,6006,6008],{"className":6007},[1621],[40,6009,6011,6031],{"className":6010},[1625,2244],[40,6012,6014,6028],{"className":6013},[1629],[40,6015,6017],{"className":6016,"style":5846},[1633],[40,6018,6019,6022],{"style":3616},[40,6020],{"className":6021,"style":1642},[1641],[40,6023,6025],{"className":6024},[1646,1647,1648,1649],[40,6026,1413],{"className":6027,"style":5867},[1612,1653,1649],[40,6029,2269],{"className":6030},[2268],[40,6032,6034],{"className":6033},[1629],[40,6035,6037],{"className":6036,"style":5877},[1633],[40,6038],{},[40,6040,1798],{"className":6041},[1822]," is carried out. This is highlighted in the figure by the color scheme for the matrix elements and the corresponding ",[40,6044,6046,6070],{"className":6045},[1565],[40,6047,6049],{"className":6048},[1569],[1571,6050,6051],{"xmlns":1573},[1575,6052,6053,6067],{},[1578,6054,6055,6058,6061],{},[5211,6056,6057],{},"Prep",[5211,6059,6060],{},"  ",[2212,6062,6063,6065],{},[1584,6064,2837],{},[1584,6066,2337],{},[1591,6068,6069],{"encoding":1593},"\\text{Prep} \\; \\psi_i",[40,6071,6073],{"className":6072,"ariaHidden":1599},[1598],[40,6074,6076,6080,6087,6090],{"className":6075},[1603],[40,6077],{"className":6078,"style":6079},[1607],"height:0.8889em;vertical-align:-0.1944em;",[40,6081,6084],{"className":6082},[1612,6083],"text",[40,6085,6057],{"className":6086},[1612],[40,6088],{"className":6089,"style":1963},[1756],[40,6091,6093,6096],{"className":6092},[1612],[40,6094,2837],{"className":6095,"style":1818},[1612,1653],[40,6097,6099],{"className":6098},[1621],[40,6100,6102,6122],{"className":6101},[1625,2244],[40,6103,6105,6119],{"className":6104},[1629],[40,6106,6108],{"className":6107,"style":5846},[1633],[40,6109,6110,6113],{"style":3616},[40,6111],{"className":6112,"style":1642},[1641],[40,6114,6116],{"className":6115},[1646,1647,1648,1649],[40,6117,2337],{"className":6118},[1612,1653,1649],[40,6120,2269],{"className":6121},[2268],[40,6123,6125],{"className":6124},[1629],[40,6126,6128],{"className":6127,"style":2276},[1633],[40,6129],{},[40,6131,6133,6155],{"className":6132},[1565],[40,6134,6136],{"className":6135},[1569],[1571,6137,6138],{"xmlns":1573},[1575,6139,6140,6152],{},[1578,6141,6142,6144,6146],{},[5211,6143,6057],{},[5211,6145,6060],{},[2212,6147,6148,6150],{},[1584,6149,2837],{},[1584,6151,1413],{},[1591,6153,6154],{"encoding":1593},"\\text{Prep} \\; \\psi_j",[40,6156,6158],{"className":6157,"ariaHidden":1599},[1598],[40,6159,6161,6165,6171,6174],{"className":6160},[1603],[40,6162],{"className":6163,"style":6164},[1607],"height:0.9805em;vertical-align:-0.2861em;",[40,6166,6168],{"className":6167},[1612,6083],[40,6169,6057],{"className":6170},[1612],[40,6172],{"className":6173,"style":1963},[1756],[40,6175,6177,6180],{"className":6176},[1612],[40,6178,2837],{"className":6179,"style":1818},[1612,1653],[40,6181,6183],{"className":6182},[1621],[40,6184,6186,6206],{"className":6185},[1625,2244],[40,6187,6189,6203],{"className":6188},[1629],[40,6190,6192],{"className":6191,"style":5846},[1633],[40,6193,6194,6197],{"style":3616},[40,6195],{"className":6196,"style":1642},[1641],[40,6198,6200],{"className":6199},[1646,1647,1648,1649],[40,6201,1413],{"className":6202,"style":5867},[1612,1653,1649],[40,6204,2269],{"className":6205},[2268],[40,6207,6209],{"className":6208},[1629],[40,6210,6212],{"className":6211,"style":5877},[1633],[40,6213],{}," 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 ",[40,6216,6218,6235],{"className":6217},[1565],[40,6219,6221],{"className":6220},[1569],[1571,6222,6223],{"xmlns":1573},[1575,6224,6225,6233],{},[1578,6226,6227],{},[3315,6228,6229,6231],{"accent":1599},[1584,6230,2182],{},[1731,6232,3321],{},[1591,6234,3324],{"encoding":1593},[40,6236,6238],{"className":6237,"ariaHidden":1599},[1598],[40,6239,6241,6244],{"className":6240},[1603],[40,6242],{"className":6243,"style":3334},[1607],[40,6245,6247],{"className":6246},[1612,3338],[40,6248,6250],{"className":6249},[1625],[40,6251,6253],{"className":6252},[1629],[40,6254,6256,6264],{"className":6255,"style":3334},[1633],[40,6257,6258,6261],{"style":3350},[40,6259],{"className":6260,"style":3354},[1641],[40,6262,2182],{"className":6263,"style":2197},[1612,1653],[40,6265,6266,6269],{"style":3360},[40,6267],{"className":6268,"style":3354},[1641],[40,6270,6272],{"className":6271,"style":3368},[3367],[40,6273,3321],{"className":6274},[1612],". 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 ",[40,6277,6279,6300],{"className":6278},[1565],[40,6280,6282],{"className":6281},[1569],[1571,6283,6284],{"xmlns":1573},[1575,6285,6286,6298],{},[1578,6287,6288,6290,6292],{},[5211,6289,6057],{},[5211,6291,6060],{},[2212,6293,6294,6296],{},[1584,6295,2837],{},[1584,6297,2337],{},[1591,6299,6069],{"encoding":1593},[40,6301,6303],{"className":6302,"ariaHidden":1599},[1598],[40,6304,6306,6309,6315,6318],{"className":6305},[1603],[40,6307],{"className":6308,"style":6079},[1607],[40,6310,6312],{"className":6311},[1612,6083],[40,6313,6057],{"className":6314},[1612],[40,6316],{"className":6317,"style":1963},[1756],[40,6319,6321,6324],{"className":6320},[1612],[40,6322,2837],{"className":6323,"style":1818},[1612,1653],[40,6325,6327],{"className":6326},[1621],[40,6328,6330,6350],{"className":6329},[1625,2244],[40,6331,6333,6347],{"className":6332},[1629],[40,6334,6336],{"className":6335,"style":5846},[1633],[40,6337,6338,6341],{"style":3616},[40,6339],{"className":6340,"style":1642},[1641],[40,6342,6344],{"className":6343},[1646,1647,1648,1649],[40,6345,2337],{"className":6346},[1612,1653,1649],[40,6348,2269],{"className":6349},[2268],[40,6351,6353],{"className":6352},[1629],[40,6354,6356],{"className":6355,"style":2276},[1633],[40,6357],{}," operation prepares the system qubit in the state ",[40,6360,6362,6383],{"className":6361},[1565],[40,6363,6365],{"className":6364},[1569],[1571,6366,6367],{"xmlns":1573},[1575,6368,6369,6381],{},[1578,6370,6371,6373,6379],{},[1584,6372,1791],{"mathvariant":1790},[2212,6374,6375,6377],{},[1584,6376,2837],{},[1584,6378,2337],{},[1731,6380,1798],{"stretchy":1797},[1591,6382,5905],{"encoding":1593},[40,6384,6386],{"className":6385,"ariaHidden":1599},[1598],[40,6387,6389,6392,6395,6435],{"className":6388},[1603],[40,6390],{"className":6391,"style":1811},[1607],[40,6393,1791],{"className":6394},[1612],[40,6396,6398,6401],{"className":6397},[1612],[40,6399,2837],{"className":6400,"style":1818},[1612,1653],[40,6402,6404],{"className":6403},[1621],[40,6405,6407,6427],{"className":6406},[1625,2244],[40,6408,6410,6424],{"className":6409},[1629],[40,6411,6413],{"className":6412,"style":5846},[1633],[40,6414,6415,6418],{"style":3616},[40,6416],{"className":6417,"style":1642},[1641],[40,6419,6421],{"className":6420},[1646,1647,1648,1649],[40,6422,2337],{"className":6423},[1612,1653,1649],[40,6425,2269],{"className":6426},[2268],[40,6428,6430],{"className":6429},[1629],[40,6431,6433],{"className":6432,"style":2276},[1633],[40,6434],{},[40,6436,1798],{"className":6437},[1822]," controlled by the state of the ancilla qubit (similarly for ",[40,6440,6442,6463],{"className":6441},[1565],[40,6443,6445],{"className":6444},[1569],[1571,6446,6447],{"xmlns":1573},[1575,6448,6449,6461],{},[1578,6450,6451,6453,6455],{},[5211,6452,6057],{},[5211,6454,6060],{},[2212,6456,6457,6459],{},[1584,6458,2837],{},[1584,6460,1413],{},[1591,6462,6154],{"encoding":1593},[40,6464,6466],{"className":6465,"ariaHidden":1599},[1598],[40,6467,6469,6472,6478,6481],{"className":6468},[1603],[40,6470],{"className":6471,"style":6164},[1607],[40,6473,6475],{"className":6474},[1612,6083],[40,6476,6057],{"className":6477},[1612],[40,6479],{"className":6480,"style":1963},[1756],[40,6482,6484,6487],{"className":6483},[1612],[40,6485,2837],{"className":6486,"style":1818},[1612,1653],[40,6488,6490],{"className":6489},[1621],[40,6491,6493,6513],{"className":6492},[1625,2244],[40,6494,6496,6510],{"className":6495},[1629],[40,6497,6499],{"className":6498,"style":5846},[1633],[40,6500,6501,6504],{"style":3616},[40,6502],{"className":6503,"style":1642},[1641],[40,6505,6507],{"className":6506},[1646,1647,1648,1649],[40,6508,1413],{"className":6509,"style":5867},[1612,1653,1649],[40,6511,2269],{"className":6512},[2268],[40,6514,6516],{"className":6515},[1629],[40,6517,6519],{"className":6518,"style":5877},[1633],[40,6520],{},") and the operation ",[40,6523,6525,6538],{"className":6524},[1565],[40,6526,6528],{"className":6527},[1569],[1571,6529,6530],{"xmlns":1573},[1575,6531,6532,6536],{},[1578,6533,6534],{},[1584,6535,2218],{},[1591,6537,2218],{"encoding":1593},[40,6539,6541],{"className":6540,"ariaHidden":1599},[1598],[40,6542,6544,6547],{"className":6543},[1603],[40,6545],{"className":6546,"style":1608},[1607],[40,6548,2218],{"className":6549,"style":2264},[1612,1653]," represents Pauli decomposition of the system Hamiltonian ",[40,6552,6554,6583],{"className":6553},[1565],[40,6555,6557],{"className":6556},[1569],[1571,6558,6559],{"xmlns":1573},[1575,6560,6561,6580],{},[1578,6562,6563,6565,6567,6574],{},[1584,6564,2182],{},[1731,6566,623],{},[2212,6568,6569,6572],{},[1731,6570,6571],{},"∑",[1584,6573,2337],{},[2212,6575,6576,6578],{},[1584,6577,2218],{},[1584,6579,2337],{},[1591,6581,6582],{"encoding":1593},"H = \\sum_i P_i",[40,6584,6586,6604],{"className":6585,"ariaHidden":1599},[1598],[40,6587,6589,6592,6595,6598,6601],{"className":6588},[1603],[40,6590],{"className":6591,"style":1608},[1607],[40,6593,2182],{"className":6594,"style":2197},[1612,1653],[40,6596],{"className":6597,"style":1963},[1756],[40,6599,623],{"className":6600},[1967],[40,6602],{"className":6603,"style":1963},[1756],[40,6605,6607,6611,6658,6661],{"className":6606},[1603],[40,6608],{"className":6609,"style":6610},[1607],"height:1.0497em;vertical-align:-0.2997em;",[40,6612,6615,6621],{"className":6613},[6614],"mop",[40,6616,6571],{"className":6617,"style":6620},[6614,6618,6619],"op-symbol","small-op","position:relative;top:0em;",[40,6622,6624],{"className":6623},[1621],[40,6625,6627,6649],{"className":6626},[1625,2244],[40,6628,6630,6646],{"className":6629},[1629],[40,6631,6634],{"className":6632,"style":6633},[1633],"height:0.162em;",[40,6635,6637,6640],{"style":6636},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[40,6638],{"className":6639,"style":1642},[1641],[40,6641,6643],{"className":6642},[1646,1647,1648,1649],[40,6644,2337],{"className":6645},[1612,1653,1649],[40,6647,2269],{"className":6648},[2268],[40,6650,6652],{"className":6651},[1629],[40,6653,6656],{"className":6654,"style":6655},[1633],"height:0.2997em;",[40,6657],{},[40,6659],{"className":6660,"style":2009},[1756],[40,6662,6664,6667],{"className":6663},[1612],[40,6665,2218],{"className":6666,"style":2264},[1612,1653],[40,6668,6670],{"className":6669},[1621],[40,6671,6673,6694],{"className":6672},[1625,2244],[40,6674,6676,6691],{"className":6675},[1629],[40,6677,6679],{"className":6678,"style":5846},[1633],[40,6680,6682,6685],{"style":6681},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[40,6683],{"className":6684,"style":1642},[1641],[40,6686,6688],{"className":6687},[1646,1647,1648,1649],[40,6689,2337],{"className":6690},[1612,1653,1649],[40,6692,2269],{"className":6693},[2268],[40,6695,6697],{"className":6696},[1629],[40,6698,6700],{"className":6699,"style":2276},[1633],[40,6701],{},". A more detailed derivation of the operations calculated by the Hadamard test is given below.",[6704,6705,6707],"h4",{"id":6706},"define-hamiltonian","Define Hamiltonian",[12,6709,6710,6711,6740,6741],{},"Let's consider the Heisenberg Hamiltonian for ",[40,6712,6714,6728],{"className":6713},[1565],[40,6715,6717],{"className":6716},[1569],[1571,6718,6719],{"xmlns":1573},[1575,6720,6721,6726],{},[1578,6722,6723],{},[1584,6724,6725],{},"N",[1591,6727,6725],{"encoding":1593},[40,6729,6731],{"className":6730,"ariaHidden":1599},[1598],[40,6732,6734,6737],{"className":6733},[1603],[40,6735],{"className":6736,"style":1608},[1607],[40,6738,6725],{"className":6739,"style":2358},[1612,1653]," qubits on a linear chain: ",[40,6742,6744,6820],{"className":6743},[1565],[40,6745,6747],{"className":6746},[1569],[1571,6748,6749],{"xmlns":1573},[1575,6750,6751,6817],{},[1578,6752,6753,6755,6757,6771,6778,6784,6786,6793,6799,6801,6804,6811],{},[1584,6754,2182],{},[1731,6756,623],{},[2819,6758,6759,6761,6769],{},[1731,6760,6571],{},[1578,6762,6763,6765,6767],{},[1584,6764,2337],{},[1731,6766,1862],{"separator":1599},[1584,6768,1413],{},[1584,6770,6725],{},[2212,6772,6773,6776],{},[1584,6774,6775],{},"X",[1584,6777,2337],{},[2212,6779,6780,6782],{},[1584,6781,6775],{},[1584,6783,1413],{},[1731,6785,1147],{},[2212,6787,6788,6791],{},[1584,6789,6790],{},"Y",[1584,6792,2337],{},[2212,6794,6795,6797],{},[1584,6796,6790],{},[1584,6798,1413],{},[1731,6800,1733],{},[1584,6802,6803],{},"J",[2212,6805,6806,6809],{},[1584,6807,6808],{},"Z",[1584,6810,2337],{},[2212,6812,6813,6815],{},[1584,6814,6808],{},[1584,6816,1413],{},[1591,6818,6819],{"encoding":1593},"H= \\sum_{i,j}^N X_i X_j + Y_i Y_j - J Z_i Z_j",[40,6821,6823,6841,7005,7102],{"className":6822,"ariaHidden":1599},[1598],[40,6824,6826,6829,6832,6835,6838],{"className":6825},[1603],[40,6827],{"className":6828,"style":1608},[1607],[40,6830,2182],{"className":6831,"style":2197},[1612,1653],[40,6833],{"className":6834,"style":1963},[1756],[40,6836,623],{"className":6837},[1967],[40,6839],{"className":6840,"style":1963},[1756],[40,6842,6844,6848,6911,6914,6956,6996,6999,7002],{"className":6843},[1603],[40,6845],{"className":6846,"style":6847},[1607],"height:1.417em;vertical-align:-0.4358em;",[40,6849,6851,6854],{"className":6850},[6614],[40,6852,6571],{"className":6853,"style":6620},[6614,6618,6619],[40,6855,6857],{"className":6856},[1621],[40,6858,6860,6902],{"className":6859},[1625,2244],[40,6861,6863,6899],{"className":6862},[1629],[40,6864,6867,6887],{"className":6865,"style":6866},[1633],"height:0.9812em;",[40,6868,6869,6872],{"style":6636},[40,6870],{"className":6871,"style":1642},[1641],[40,6873,6875],{"className":6874},[1646,1647,1648,1649],[40,6876,6878,6881,6884],{"className":6877},[1612,1649],[40,6879,2337],{"className":6880},[1612,1653,1649],[40,6882,1862],{"className":6883},[2005,1649],[40,6885,1413],{"className":6886,"style":5867},[1612,1653,1649],[40,6888,6890,6893],{"style":6889},"top:-3.2029em;margin-right:0.05em;",[40,6891],{"className":6892,"style":1642},[1641],[40,6894,6896],{"className":6895},[1646,1647,1648,1649],[40,6897,6725],{"className":6898,"style":2358},[1612,1653,1649],[40,6900,2269],{"className":6901},[2268],[40,6903,6905],{"className":6904},[1629],[40,6906,6909],{"className":6907,"style":6908},[1633],"height:0.4358em;",[40,6910],{},[40,6912],{"className":6913,"style":2009},[1756],[40,6915,6917,6921],{"className":6916},[1612],[40,6918,6775],{"className":6919,"style":6920},[1612,1653],"margin-right:0.0785em;",[40,6922,6924],{"className":6923},[1621],[40,6925,6927,6948],{"className":6926},[1625,2244],[40,6928,6930,6945],{"className":6929},[1629],[40,6931,6933],{"className":6932,"style":5846},[1633],[40,6934,6936,6939],{"style":6935},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[40,6937],{"className":6938,"style":1642},[1641],[40,6940,6942],{"className":6941},[1646,1647,1648,1649],[40,6943,2337],{"className":6944},[1612,1653,1649],[40,6946,2269],{"className":6947},[2268],[40,6949,6951],{"className":6950},[1629],[40,6952,6954],{"className":6953,"style":2276},[1633],[40,6955],{},[40,6957,6959,6962],{"className":6958},[1612],[40,6960,6775],{"className":6961,"style":6920},[1612,1653],[40,6963,6965],{"className":6964},[1621],[40,6966,6968,6988],{"className":6967},[1625,2244],[40,6969,6971,6985],{"className":6970},[1629],[40,6972,6974],{"className":6973,"style":5846},[1633],[40,6975,6976,6979],{"style":6935},[40,6977],{"className":6978,"style":1642},[1641],[40,6980,6982],{"className":6981},[1646,1647,1648,1649],[40,6983,1413],{"className":6984,"style":5867},[1612,1653,1649],[40,6986,2269],{"className":6987},[2268],[40,6989,6991],{"className":6990},[1629],[40,6992,6994],{"className":6993,"style":5877},[1633],[40,6995],{},[40,6997],{"className":6998,"style":1757},[1756],[40,7000,1147],{"className":7001},[1761],[40,7003],{"className":7004,"style":1757},[1756],[40,7006,7008,7012,7053,7093,7096,7099],{"className":7007},[1603],[40,7009],{"className":7010,"style":7011},[1607],"height:0.9694em;vertical-align:-0.2861em;",[40,7013,7015,7018],{"className":7014},[1612],[40,7016,6790],{"className":7017,"style":1757},[1612,1653],[40,7019,7021],{"className":7020},[1621],[40,7022,7024,7045],{"className":7023},[1625,2244],[40,7025,7027,7042],{"className":7026},[1629],[40,7028,7030],{"className":7029,"style":5846},[1633],[40,7031,7033,7036],{"style":7032},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[40,7034],{"className":7035,"style":1642},[1641],[40,7037,7039],{"className":7038},[1646,1647,1648,1649],[40,7040,2337],{"className":7041},[1612,1653,1649],[40,7043,2269],{"className":7044},[2268],[40,7046,7048],{"className":7047},[1629],[40,7049,7051],{"className":7050,"style":2276},[1633],[40,7052],{},[40,7054,7056,7059],{"className":7055},[1612],[40,7057,6790],{"className":7058,"style":1757},[1612,1653],[40,7060,7062],{"className":7061},[1621],[40,7063,7065,7085],{"className":7064},[1625,2244],[40,7066,7068,7082],{"className":7067},[1629],[40,7069,7071],{"className":7070,"style":5846},[1633],[40,7072,7073,7076],{"style":7032},[40,7074],{"className":7075,"style":1642},[1641],[40,7077,7079],{"className":7078},[1646,1647,1648,1649],[40,7080,1413],{"className":7081,"style":5867},[1612,1653,1649],[40,7083,2269],{"className":7084},[2268],[40,7086,7088],{"className":7087},[1629],[40,7089,7091],{"className":7090,"style":5877},[1633],[40,7092],{},[40,7094],{"className":7095,"style":1757},[1756],[40,7097,1733],{"className":7098},[1761],[40,7100],{"className":7101,"style":1757},[1756],[40,7103,7105,7108,7112,7154],{"className":7104},[1603],[40,7106],{"className":7107,"style":7011},[1607],[40,7109,6803],{"className":7110,"style":7111},[1612,1653],"margin-right:0.0962em;",[40,7113,7115,7119],{"className":7114},[1612],[40,7116,6808],{"className":7117,"style":7118},[1612,1653],"margin-right:0.0715em;",[40,7120,7122],{"className":7121},[1621],[40,7123,7125,7146],{"className":7124},[1625,2244],[40,7126,7128,7143],{"className":7127},[1629],[40,7129,7131],{"className":7130,"style":5846},[1633],[40,7132,7134,7137],{"style":7133},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[40,7135],{"className":7136,"style":1642},[1641],[40,7138,7140],{"className":7139},[1646,1647,1648,1649],[40,7141,2337],{"className":7142},[1612,1653,1649],[40,7144,2269],{"className":7145},[2268],[40,7147,7149],{"className":7148},[1629],[40,7150,7152],{"className":7151,"style":2276},[1633],[40,7153],{},[40,7155,7157,7160],{"className":7156},[1612],[40,7158,6808],{"className":7159,"style":7118},[1612,1653],[40,7161,7163],{"className":7162},[1621],[40,7164,7166,7186],{"className":7165},[1625,2244],[40,7167,7169,7183],{"className":7168},[1629],[40,7170,7172],{"className":7171,"style":5846},[1633],[40,7173,7174,7177],{"style":7133},[40,7175],{"className":7176,"style":1642},[1641],[40,7178,7180],{"className":7179},[1646,1647,1648,1649],[40,7181,1413],{"className":7182,"style":5867},[1612,1653,1649],[40,7184,2269],{"className":7185},[2268],[40,7187,7189],{"className":7188},[1629],[40,7190,7192],{"className":7191,"style":5877},[1633],[40,7193],{},[30,7195,7197],{"className":32,"code":7196,"language":34,"meta":35,"style":35},"# 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",[37,7198,7199,7204,7214,7226,7230,7235,7284,7303,7313,7328,7346,7366,7389,7407,7433,7463,7493,7540,7544,7549,7565],{"__ignoreMap":35},[40,7200,7201],{"class":42,"line":43},[40,7202,7203],{"class":46},"# Define problem Hamiltonian.\n",[40,7205,7206,7209,7211],{"class":42,"line":50},[40,7207,7208],{"class":133},"n_qubits ",[40,7210,623],{"class":441},[40,7212,7213],{"class":462}," 30\n",[40,7215,7216,7219,7221,7223],{"class":42,"line":155},[40,7217,7218],{"class":133},"J ",[40,7220,623],{"class":441},[40,7222,463],{"class":462},[40,7224,7225],{"class":46},"  # coupling strength for ZZ interaction\n",[40,7227,7228],{"class":42,"line":168},[40,7229,228],{"emptyLinePlaceholder":227},[40,7231,7232],{"class":42,"line":182},[40,7233,7234],{"class":46},"# Define the Hamiltonian:\n",[40,7236,7237,7240,7242,7245,7248,7251,7253,7256,7258,7261,7263,7266,7268,7271,7274,7277,7279,7281],{"class":42,"line":195},[40,7238,7239],{"class":133},"H_int ",[40,7241,623],{"class":441},[40,7243,7244],{"class":133}," [[",[40,7246,7247],{"class":244},"\"I\"",[40,7249,7250],{"class":133},"] ",[40,7252,1433],{"class":441},[40,7254,7255],{"class":133}," n_qubits ",[40,7257,668],{"class":129},[40,7259,7260],{"class":133}," _ ",[40,7262,674],{"class":129},[40,7264,7265],{"class":441}," range",[40,7267,241],{"class":133},[40,7269,7270],{"class":462},"3",[40,7272,7273],{"class":441}," *",[40,7275,7276],{"class":133}," (n_qubits ",[40,7278,1407],{"class":441},[40,7280,463],{"class":462},[40,7282,7283],{"class":133},"))]\n",[40,7285,7286,7288,7291,7293,7295,7297,7299,7301],{"class":42,"line":203},[40,7287,668],{"class":129},[40,7289,7290],{"class":133}," i ",[40,7292,674],{"class":129},[40,7294,7265],{"class":441},[40,7296,1144],{"class":133},[40,7298,1407],{"class":441},[40,7300,463],{"class":462},[40,7302,923],{"class":133},[40,7304,7305,7308,7310],{"class":42,"line":216},[40,7306,7307],{"class":133},"    H_int[i][i] ",[40,7309,623],{"class":441},[40,7311,7312],{"class":244}," \"Z\"\n",[40,7314,7315,7318,7320,7322,7324,7326],{"class":42,"line":224},[40,7316,7317],{"class":133},"    H_int[i][i ",[40,7319,1147],{"class":441},[40,7321,463],{"class":462},[40,7323,7250],{"class":133},[40,7325,623],{"class":441},[40,7327,7312],{"class":244},[40,7329,7330,7332,7334,7336,7338,7340,7342,7344],{"class":42,"line":231},[40,7331,668],{"class":129},[40,7333,7290],{"class":133},[40,7335,674],{"class":129},[40,7337,7265],{"class":441},[40,7339,1144],{"class":133},[40,7341,1407],{"class":441},[40,7343,463],{"class":462},[40,7345,923],{"class":133},[40,7347,7348,7351,7353,7355,7358,7361,7363],{"class":42,"line":251},[40,7349,7350],{"class":133},"    H_int[n_qubits ",[40,7352,1407],{"class":441},[40,7354,463],{"class":462},[40,7356,7357],{"class":441}," +",[40,7359,7360],{"class":133}," i][i] ",[40,7362,623],{"class":441},[40,7364,7365],{"class":244}," \"X\"\n",[40,7367,7368,7370,7372,7374,7376,7379,7381,7383,7385,7387],{"class":42,"line":256},[40,7369,7350],{"class":133},[40,7371,1407],{"class":441},[40,7373,463],{"class":462},[40,7375,7357],{"class":441},[40,7377,7378],{"class":133}," i][i ",[40,7380,1147],{"class":441},[40,7382,463],{"class":462},[40,7384,7250],{"class":133},[40,7386,623],{"class":441},[40,7388,7365],{"class":244},[40,7390,7391,7393,7395,7397,7399,7401,7403,7405],{"class":42,"line":269},[40,7392,668],{"class":129},[40,7394,7290],{"class":133},[40,7396,674],{"class":129},[40,7398,7265],{"class":441},[40,7400,1144],{"class":133},[40,7402,1407],{"class":441},[40,7404,463],{"class":462},[40,7406,923],{"class":133},[40,7408,7409,7412,7414,7416,7418,7420,7422,7424,7426,7428,7430],{"class":42,"line":282},[40,7410,7411],{"class":133},"    H_int[",[40,7413,1879],{"class":462},[40,7415,7273],{"class":441},[40,7417,7276],{"class":133},[40,7419,1407],{"class":441},[40,7421,463],{"class":462},[40,7423,1416],{"class":133},[40,7425,1147],{"class":441},[40,7427,7360],{"class":133},[40,7429,623],{"class":441},[40,7431,7432],{"class":244}," \"Y\"\n",[40,7434,7435,7437,7439,7441,7443,7445,7447,7449,7451,7453,7455,7457,7459,7461],{"class":42,"line":295},[40,7436,7411],{"class":133},[40,7438,1879],{"class":462},[40,7440,7273],{"class":441},[40,7442,7276],{"class":133},[40,7444,1407],{"class":441},[40,7446,463],{"class":462},[40,7448,1416],{"class":133},[40,7450,1147],{"class":441},[40,7452,7378],{"class":133},[40,7454,1147],{"class":441},[40,7456,463],{"class":462},[40,7458,7250],{"class":133},[40,7460,623],{"class":441},[40,7462,7432],{"class":244},[40,7464,7465,7467,7469,7472,7475,7477,7480,7483,7485,7488,7490],{"class":42,"line":308},[40,7466,7239],{"class":133},[40,7468,623],{"class":441},[40,7470,7471],{"class":133}," [",[40,7473,7474],{"class":244},"\"\"",[40,7476,1823],{"class":133},[40,7478,7479],{"class":237},"join",[40,7481,7482],{"class":133},"(term) ",[40,7484,668],{"class":129},[40,7486,7487],{"class":133}," term ",[40,7489,674],{"class":129},[40,7491,7492],{"class":133}," H_int]\n",[40,7494,7495,7498,7500,7503,7505,7508,7511,7513,7516,7518,7520,7522,7525,7528,7530,7532,7534,7536,7538],{"class":42,"line":321},[40,7496,7497],{"class":133},"H_tot ",[40,7499,623],{"class":441},[40,7501,7502],{"class":133}," [(term, J) ",[40,7504,683],{"class":129},[40,7506,7507],{"class":133}," term.",[40,7509,7510],{"class":237},"count",[40,7512,241],{"class":133},[40,7514,7515],{"class":244},"\"Z\"",[40,7517,1416],{"class":133},[40,7519,761],{"class":441},[40,7521,1356],{"class":462},[40,7523,7524],{"class":129}," else",[40,7526,7527],{"class":133}," (term, ",[40,7529,1410],{"class":462},[40,7531,1416],{"class":133},[40,7533,668],{"class":129},[40,7535,7487],{"class":133},[40,7537,674],{"class":129},[40,7539,7492],{"class":133},[40,7541,7542],{"class":42,"line":334},[40,7543,228],{"emptyLinePlaceholder":227},[40,7545,7546],{"class":42,"line":347},[40,7547,7548],{"class":46},"# Get operator\n",[40,7550,7551,7554,7556,7559,7562],{"class":42,"line":352},[40,7552,7553],{"class":133},"H_op ",[40,7555,623],{"class":441},[40,7557,7558],{"class":133}," SparsePauliOp.",[40,7560,7561],{"class":237},"from_list",[40,7563,7564],{"class":133},"(H_tot)\n",[40,7566,7567,7570],{"class":42,"line":357},[40,7568,7569],{"class":441},"print",[40,7571,7564],{"class":133},[30,7573,7577],{"className":7574,"code":7576,"language":6083,"meta":35},[7575],"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",[37,7578,7576],{"__ignoreMap":35},[6704,7580,7582],{"id":7581},"set-parameters-for-the-algorithm","Set parameters for the algorithm",[12,7584,7585,7586,7588,7589,7592,7593,7647,7648,7681],{},"We heuristically choose a value for the time-step ",[37,7587,4507],{}," (based on upper bounds on the Hamiltonian norm). Ref ",[19,7590,7591],{"href":82},"[2]","  showed that a sufficiently small timestep is ",[40,7594,7596,7624],{"className":7595},[1565],[40,7597,7599],{"className":7598},[1569],[1571,7600,7601],{"xmlns":1573},[1575,7602,7603,7621],{},[1578,7604,7605,7608,7611,7613,7615,7617,7619],{},[1584,7606,7607],{},"π",[1584,7609,7610],{"mathvariant":1790},"\u002F",[1584,7612,1791],{"mathvariant":1790},[1584,7614,1791],{"mathvariant":1790},[1584,7616,2182],{},[1584,7618,1791],{"mathvariant":1790},[1584,7620,1791],{"mathvariant":1790},[1591,7622,7623],{"encoding":1593},"\\pi\u002F\\vert \\vert H \\vert \\vert",[40,7625,7627],{"className":7626,"ariaHidden":1599},[1598],[40,7628,7630,7633,7636,7640,7643],{"className":7629},[1603],[40,7631],{"className":7632,"style":1811},[1607],[40,7634,7607],{"className":7635,"style":1818},[1612,1653],[40,7637,7639],{"className":7638},[1612],"\u002F∣∣",[40,7641,2182],{"className":7642,"style":2197},[1612,1653],[40,7644,7646],{"className":7645},[1612],"∣∣",", 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 ",[40,7649,7651,7666],{"className":7650},[1565],[40,7652,7654],{"className":7653},[1569],[1571,7655,7656],{"xmlns":1573},[1575,7657,7658,7664],{},[1578,7659,7660,7662],{},[1584,7661,4003],{},[1584,7663,2342],{},[1591,7665,4507],{"encoding":1593},[40,7667,7669],{"className":7668,"ariaHidden":1599},[1598],[40,7670,7672,7675,7678],{"className":7671},[1603],[40,7673],{"className":7674,"style":4478},[1607],[40,7676,4003],{"className":7677},[1612,1653],[40,7679,2342],{"className":7680},[1612,1653]," to be too small leads to worse conditioning of the Krylov subspace, since the Krylov basis vectors differ less from timestep to timestep.",[30,7683,7685],{"className":32,"code":7684,"language":34,"meta":35,"style":35},"# 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",[37,7686,7687,7692,7706,7719,7739,7753,7766,7778,7787,7829,7834,7843,7881,7901,7908,7916,7929,7941,7960,7974,7978,7983,8013],{"__ignoreMap":35},[40,7688,7689],{"class":42,"line":43},[40,7690,7691],{"class":46},"# Get Hamiltonian restricted to single-particle states\n",[40,7693,7694,7697,7699,7701,7703],{"class":42,"line":50},[40,7695,7696],{"class":133},"single_particle_H ",[40,7698,623],{"class":441},[40,7700,654],{"class":133},[40,7702,1189],{"class":237},[40,7704,7705],{"class":133},"((n_qubits, n_qubits))\n",[40,7707,7708,7710,7712,7714,7716],{"class":42,"line":155},[40,7709,668],{"class":129},[40,7711,7290],{"class":133},[40,7713,674],{"class":129},[40,7715,7265],{"class":441},[40,7717,7718],{"class":133},"(n_qubits):\n",[40,7720,7721,7723,7726,7728,7730,7733,7735,7737],{"class":42,"line":168},[40,7722,956],{"class":129},[40,7724,7725],{"class":133}," j ",[40,7727,674],{"class":129},[40,7729,7265],{"class":441},[40,7731,7732],{"class":133},"(i ",[40,7734,1147],{"class":441},[40,7736,463],{"class":462},[40,7738,923],{"class":133},[40,7740,7741,7743,7745,7747,7749,7751],{"class":42,"line":182},[40,7742,1307],{"class":129},[40,7744,959],{"class":133},[40,7746,674],{"class":129},[40,7748,964],{"class":133},[40,7750,967],{"class":237},[40,7752,970],{"class":133},[40,7754,7755,7758,7760,7763],{"class":42,"line":195},[40,7756,7757],{"class":133},"            p_x ",[40,7759,623],{"class":441},[40,7761,7762],{"class":237}," Pauli",[40,7764,7765],{"class":133},"(p).x\n",[40,7767,7768,7771,7773,7775],{"class":42,"line":203},[40,7769,7770],{"class":133},"            p_z ",[40,7772,623],{"class":441},[40,7774,7762],{"class":237},[40,7776,7777],{"class":133},"(p).z\n",[40,7779,7780,7782,7785],{"class":42,"line":216},[40,7781,1338],{"class":129},[40,7783,7784],{"class":441}," all",[40,7786,411],{"class":133},[40,7788,7789,7792,7794,7797,7799,7802,7804,7807,7809,7812,7815,7817,7820,7822,7824,7826],{"class":42,"line":224},[40,7790,7791],{"class":133},"                p_x[k] ",[40,7793,761],{"class":441},[40,7795,7796],{"class":133}," ((i ",[40,7798,761],{"class":441},[40,7800,7801],{"class":133}," k) ",[40,7803,1147],{"class":441},[40,7805,7806],{"class":133}," (j ",[40,7808,761],{"class":441},[40,7810,7811],{"class":133}," k)) ",[40,7813,7814],{"class":441},"%",[40,7816,1356],{"class":462},[40,7818,7819],{"class":129}," for",[40,7821,758],{"class":133},[40,7823,674],{"class":129},[40,7825,7265],{"class":441},[40,7827,7828],{"class":133},"(n_qubits)\n",[40,7830,7831],{"class":42,"line":231},[40,7832,7833],{"class":133},"            ):\n",[40,7835,7836,7839,7841],{"class":42,"line":251},[40,7837,7838],{"class":133},"                sgn ",[40,7840,623],{"class":441},[40,7842,367],{"class":133},[40,7844,7845,7848,7850,7852,7854,7856,7858,7861,7864,7867,7870,7872,7874,7876,7878],{"class":42,"line":256},[40,7846,7847],{"class":133},"                    (",[40,7849,1407],{"class":441},[40,7851,1410],{"class":462},[40,7853,1413],{"class":129},[40,7855,1416],{"class":133},[40,7857,1419],{"class":441},[40,7859,7860],{"class":441}," sum",[40,7862,7863],{"class":133},"(p_z[k] ",[40,7865,7866],{"class":129},"and",[40,7868,7869],{"class":133}," p_x[k] ",[40,7871,668],{"class":129},[40,7873,758],{"class":133},[40,7875,674],{"class":129},[40,7877,7265],{"class":441},[40,7879,7880],{"class":133},"(n_qubits))\n",[40,7882,7883,7886,7888,7890,7892,7894,7896,7898],{"class":42,"line":269},[40,7884,7885],{"class":133},"                ) ",[40,7887,1433],{"class":441},[40,7889,1404],{"class":133},[40,7891,1407],{"class":441},[40,7893,1410],{"class":462},[40,7895,1416],{"class":133},[40,7897,1419],{"class":441},[40,7899,7900],{"class":133}," p_z[i])\n",[40,7902,7903,7906],{"class":42,"line":282},[40,7904,7905],{"class":129},"            else",[40,7907,766],{"class":133},[40,7909,7910,7912,7914],{"class":42,"line":295},[40,7911,7838],{"class":133},[40,7913,623],{"class":441},[40,7915,1286],{"class":462},[40,7917,7918,7921,7923,7925,7927],{"class":42,"line":308},[40,7919,7920],{"class":133},"            single_particle_H[i, j] ",[40,7922,1325],{"class":441},[40,7924,1497],{"class":133},[40,7926,1433],{"class":441},[40,7928,1502],{"class":133},[40,7930,7931,7933,7935,7937,7939],{"class":42,"line":321},[40,7932,668],{"class":129},[40,7934,7290],{"class":133},[40,7936,674],{"class":129},[40,7938,7265],{"class":441},[40,7940,7718],{"class":133},[40,7942,7943,7945,7947,7949,7951,7953,7955,7957],{"class":42,"line":334},[40,7944,956],{"class":129},[40,7946,7725],{"class":133},[40,7948,674],{"class":129},[40,7950,7265],{"class":441},[40,7952,7732],{"class":133},[40,7954,1147],{"class":441},[40,7956,463],{"class":462},[40,7958,7959],{"class":133},", n_qubits):\n",[40,7961,7962,7965,7967,7969,7971],{"class":42,"line":347},[40,7963,7964],{"class":133},"        single_particle_H[i, j] ",[40,7966,623],{"class":441},[40,7968,654],{"class":133},[40,7970,712],{"class":237},[40,7972,7973],{"class":133},"(single_particle_H[j, i])\n",[40,7975,7976],{"class":42,"line":352},[40,7977,228],{"emptyLinePlaceholder":227},[40,7979,7980],{"class":42,"line":357},[40,7981,7982],{"class":46},"# Set dt according to spectral norm\n",[40,7984,7985,7988,7990,7993,7995,7998,8001,8004,8007,8009,8011],{"class":42,"line":370},[40,7986,7987],{"class":133},"dt ",[40,7989,623],{"class":441},[40,7991,7992],{"class":133}," np.pi ",[40,7994,7610],{"class":441},[40,7996,7997],{"class":133}," np.linalg.",[40,7999,8000],{"class":237},"norm",[40,8002,8003],{"class":133},"(single_particle_H, ",[40,8005,8006],{"class":1195},"ord",[40,8008,623],{"class":441},[40,8010,1879],{"class":462},[40,8012,248],{"class":133},[40,8014,8015],{"class":42,"line":376},[40,8016,8017],{"class":133},"dt\n",[30,8019,8022],{"className":8020,"code":8021,"language":6083,"meta":35},[7575],"np.float64(0.10833078115826875)\n",[37,8023,8021],{"__ignoreMap":35},[12,8025,8026],{},"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.",[30,8028,8030],{"className":32,"code":8029,"language":34,"meta":35,"style":35},"# 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",[37,8031,8032,8037,8050,8060],{"__ignoreMap":35},[40,8033,8034],{"class":42,"line":43},[40,8035,8036],{"class":46},"# Set parameters for quantum Krylov algorithm\n",[40,8038,8039,8042,8044,8047],{"class":42,"line":50},[40,8040,8041],{"class":133},"krylov_dim ",[40,8043,623],{"class":441},[40,8045,8046],{"class":462}," 5",[40,8048,8049],{"class":46},"  # size of Krylov subspace\n",[40,8051,8052,8055,8057],{"class":42,"line":155},[40,8053,8054],{"class":133},"num_trotter_steps ",[40,8056,623],{"class":441},[40,8058,8059],{"class":462}," 6\n",[40,8061,8062,8065,8067,8070,8072],{"class":42,"line":168},[40,8063,8064],{"class":133},"dt_circ ",[40,8066,623],{"class":441},[40,8068,8069],{"class":133}," dt ",[40,8071,7610],{"class":441},[40,8073,8074],{"class":133}," num_trotter_steps\n",[6704,8076,8078],{"id":8077},"state-preparation","State preparation",[12,8080,8081,8082,8121,8122,8160],{},"Pick a reference state ",[40,8083,8085,8103],{"className":8084},[1565],[40,8086,8088],{"className":8087},[1569],[1571,8089,8090],{"xmlns":1573},[1575,8091,8092,8100],{},[1578,8093,8094,8096,8098],{},[1584,8095,1791],{"mathvariant":1790},[1584,8097,2837],{},[1731,8099,1798],{"stretchy":1797},[1591,8101,8102],{"encoding":1593},"\\vert \\psi \\rangle",[40,8104,8106],{"className":8105,"ariaHidden":1599},[1598],[40,8107,8109,8112,8115,8118],{"className":8108},[1603],[40,8110],{"className":8111,"style":1811},[1607],[40,8113,1791],{"className":8114},[1612],[40,8116,2837],{"className":8117,"style":1818},[1612,1653],[40,8119,1798],{"className":8120},[1822]," that has some overlap with the ground state. For this Hamiltonian, We use the a state with an excitation in the middle qubit ",[40,8123,8125,8144],{"className":8124},[1565],[40,8126,8128],{"className":8127},[1569],[1571,8129,8130],{"xmlns":1573},[1575,8131,8132,8141],{},[1578,8133,8134,8136,8139],{},[1584,8135,1791],{"mathvariant":1790},[1735,8137,8138],{},"00..010...00",[1731,8140,1798],{"stretchy":1797},[1591,8142,8143],{"encoding":1593},"\\vert 00..010...00 \\rangle",[40,8145,8147],{"className":8146,"ariaHidden":1599},[1598],[40,8148,8150,8153,8157],{"className":8149},[1603],[40,8151],{"className":8152,"style":1811},[1607],[40,8154,8156],{"className":8155},[1612],"∣00..010...00",[40,8158,1798],{"className":8159},[1822]," as our reference state.",[30,8162,8164],{"className":32,"code":8163,"language":34,"meta":35,"style":35},"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",[37,8165,8166,8178,8204],{"__ignoreMap":35},[40,8167,8168,8171,8173,8176],{"class":42,"line":43},[40,8169,8170],{"class":133},"qc_state_prep ",[40,8172,623],{"class":441},[40,8174,8175],{"class":237}," QuantumCircuit",[40,8177,7828],{"class":133},[40,8179,8180,8183,8186,8188,8190,8192,8194,8196,8198,8200,8202],{"class":42,"line":50},[40,8181,8182],{"class":133},"qc_state_prep.",[40,8184,8185],{"class":237},"x",[40,8187,241],{"class":133},[40,8189,456],{"class":441},[40,8191,1144],{"class":133},[40,8193,7610],{"class":441},[40,8195,1356],{"class":462},[40,8197,1416],{"class":133},[40,8199,1147],{"class":441},[40,8201,463],{"class":462},[40,8203,248],{"class":133},[40,8205,8206,8208,8211,8213,8216,8218,8221,8223,8226],{"class":42,"line":155},[40,8207,8182],{"class":133},[40,8209,8210],{"class":237},"draw",[40,8212,241],{"class":133},[40,8214,8215],{"class":244},"\"mpl\"",[40,8217,917],{"class":133},[40,8219,8220],{"class":1195},"scale",[40,8222,623],{"class":441},[40,8224,8225],{"class":462},"0.5",[40,8227,248],{"class":133},[8229,8230],"blog-figure",{"alt":8231,"src":8232},"Output of the previous code cell","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-01.avif",[6704,8234,8236],{"id":8235},"time-evolution","Time evolution",[12,8238,8239,8240,8340,8341,1823],{},"We can realize the time-evolution operator generated by a given Hamiltonian: ",[40,8241,8243,8272],{"className":8242},[1565],[40,8244,8246],{"className":8245},[1569],[1571,8247,8248],{"xmlns":1573},[1575,8249,8250,8270],{},[1578,8251,8252,8254,8256],{},[1584,8253,2323],{},[1731,8255,623],{},[1581,8257,8258,8260],{},[1584,8259,2330],{},[1578,8261,8262,8264,8266,8268],{},[1731,8263,1733],{},[1584,8265,2337],{},[1584,8267,2182],{},[1584,8269,2342],{},[1591,8271,2345],{"encoding":1593},[40,8273,8275,8293],{"className":8274,"ariaHidden":1599},[1598],[40,8276,8278,8281,8284,8287,8290],{"className":8277},[1603],[40,8279],{"className":8280,"style":1608},[1607],[40,8282,2323],{"className":8283,"style":2358},[1612,1653],[40,8285],{"className":8286,"style":1963},[1756],[40,8288,623],{"className":8289},[1967],[40,8291],{"className":8292,"style":1963},[1756],[40,8294,8296,8299],{"className":8295},[1603],[40,8297],{"className":8298,"style":2374},[1607],[40,8300,8302,8305],{"className":8301},[1612],[40,8303,2330],{"className":8304},[1612,1653],[40,8306,8308],{"className":8307},[1621],[40,8309,8311],{"className":8310},[1625],[40,8312,8314],{"className":8313},[1629],[40,8315,8317],{"className":8316,"style":2374},[1633],[40,8318,8319,8322],{"style":1637},[40,8320],{"className":8321,"style":1642},[1641],[40,8323,8325],{"className":8324},[1646,1647,1648,1649],[40,8326,8328,8331,8334,8337],{"className":8327},[1612,1649],[40,8329,1733],{"className":8330},[1612,1649],[40,8332,2337],{"className":8333},[1612,1653,1649],[40,8335,2182],{"className":8336,"style":2197},[1612,1653,1649],[40,8338,2342],{"className":8339},[1612,1653,1649]," via the ",[19,8342,8344],{"href":8343},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fqiskit.synthesis.LieTrotter","Lie-Trotter approximation",[30,8346,8348],{"className":32,"code":8347,"language":34,"meta":35,"style":35},"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",[37,8349,8350,8367,8371,8376,8388,8419,8423,8427,8439,8451],{"__ignoreMap":35},[40,8351,8352,8355,8357,8360,8362,8365],{"class":42,"line":43},[40,8353,8354],{"class":133},"t ",[40,8356,623],{"class":441},[40,8358,8359],{"class":237}," Parameter",[40,8361,241],{"class":133},[40,8363,8364],{"class":244},"\"t\"",[40,8366,248],{"class":133},[40,8368,8369],{"class":42,"line":50},[40,8370,228],{"emptyLinePlaceholder":227},[40,8372,8373],{"class":42,"line":155},[40,8374,8375],{"class":46},"## Create the time-evo op circuit\n",[40,8377,8378,8381,8383,8386],{"class":42,"line":168},[40,8379,8380],{"class":133},"evol_gate ",[40,8382,623],{"class":441},[40,8384,8385],{"class":237}," PauliEvolutionGate",[40,8387,411],{"class":133},[40,8389,8390,8393,8396,8398,8401,8404,8406,8409,8411,8414,8416],{"class":42,"line":182},[40,8391,8392],{"class":133},"    H_op, ",[40,8394,8395],{"class":1195},"time",[40,8397,623],{"class":441},[40,8399,8400],{"class":133},"t, ",[40,8402,8403],{"class":1195},"synthesis",[40,8405,623],{"class":441},[40,8407,8408],{"class":237},"LieTrotter",[40,8410,241],{"class":133},[40,8412,8413],{"class":1195},"reps",[40,8415,623],{"class":441},[40,8417,8418],{"class":133},"num_trotter_steps)\n",[40,8420,8421],{"class":42,"line":195},[40,8422,248],{"class":133},[40,8424,8425],{"class":42,"line":203},[40,8426,228],{"emptyLinePlaceholder":227},[40,8428,8429,8432,8434,8437],{"class":42,"line":216},[40,8430,8431],{"class":133},"qr ",[40,8433,623],{"class":441},[40,8435,8436],{"class":237}," QuantumRegister",[40,8438,7828],{"class":133},[40,8440,8441,8444,8446,8448],{"class":42,"line":224},[40,8442,8443],{"class":133},"qc_evol ",[40,8445,623],{"class":441},[40,8447,8175],{"class":237},[40,8449,8450],{"class":133},"(qr)\n",[40,8452,8453,8456,8458,8461,8464,8466],{"class":42,"line":231},[40,8454,8455],{"class":133},"qc_evol.",[40,8457,979],{"class":237},[40,8459,8460],{"class":133},"(evol_gate, ",[40,8462,8463],{"class":1195},"qargs",[40,8465,623],{"class":441},[40,8467,8468],{"class":133},"qr)\n",[30,8470,8473],{"className":8471,"code":8472,"language":6083,"meta":35},[7575],"\u003Cqiskit.circuit.instructionset.InstructionSet at 0x11eef9be0>\n",[37,8474,8472],{"__ignoreMap":35},[6704,8476,8478],{"id":8477},"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 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vectors of the unitary Krylov space, with 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= e^{-i H k dt } \\vert \\psi \\rangle = e^{-i H k dt } U_{\\psi} \\vert 0 \\rangle^N",[40,10261,10263,10325,10397],{"className":10262,"ariaHidden":1599},[1598],[40,10264,10266,10269,10272,10313,10316,10319,10322],{"className":10265},[1603],[40,10267],{"className":10268,"style":1811},[1607],[40,10270,1791],{"className":10271},[1612],[40,10273,10275,10278],{"className":10274},[1612],[40,10276,2837],{"className":10277,"style":1818},[1612,1653],[40,10279,10281],{"className":10280},[1621],[40,10282,10284,10305],{"className":10283},[1625,2244],[40,10285,10287,10302],{"className":10286},[1629],[40,10288,10291],{"className":10289,"style":10290},[1633],"height:0.3361em;",[40,10292,10293,10296],{"style":3616},[40,10294],{"className":10295,"style":1642},[1641],[40,10297,10299],{"className":10298},[1646,1647,1648,1649],[40,10300,2670],{"className":10301,"style":2736},[1612,1653,1649],[40,10303,2269],{"className":10304},[2268],[40,10306,10308],{"className":10307},[1629],[40,10309,10311],{"className":10310,"style":2276},[1633],[40,10312],{},[40,10314,1798],{"className":10315},[1822],[40,10317],{"className":10318,"style":1963},[1756],[40,10320,623],{"className":10321},[1967],[40,10323],{"className":10324,"style":1963},[1756],[40,10326,10328,10332,10379,10382,10385,10388,10391,10394],{"className":10327},[1603],[40,10329],{"className":10330,"style":10331},[1607],"height:1.0991em;vertical-align:-0.25em;",[40,10333,10335,10338],{"className":10334},[1612],[40,10336,2330],{"className":10337},[1612,1653],[40,10339,10341],{"className":10340},[1621],[40,10342,10344],{"className":10343},[1625],[40,10345,10347],{"className":10346},[1629],[40,10348,10350],{"className":10349,"style":2706},[1633],[40,10351,10352,10355],{"style":1637},[40,10353],{"className":10354,"style":1642},[1641],[40,10356,10358],{"className":10357},[1646,1647,1648,1649],[40,10359,10361,10364,10367,10370,10373,10376],{"className":10360},[1612,1649],[40,10362,1733],{"className":10363},[1612,1649],[40,10365,2337],{"className":10366},[1612,1653,1649],[40,10368,2182],{"className":10369,"style":2197},[1612,1653,1649],[40,10371,2670],{"className":10372,"style":2736},[1612,1653,1649],[40,10374,4003],{"className":10375},[1612,1653,1649],[40,10377,2342],{"className":10378},[1612,1653,1649],[40,10380,1791],{"className":10381},[1612],[40,10383,2837],{"className":10384,"style":1818},[1612,1653],[40,10386,1798],{"className":10387},[1822],[40,10389],{"className":10390,"style":1963},[1756],[40,10392,623],{"className":10393},[1967],[40,10395],{"className":10396,"style":1963},[1756],[40,10398,10400,10404,10451,10495,10498],{"className":10399},[1603],[40,10401],{"className":10402,"style":10403},[1607],"height:1.1352em;vertical-align:-0.2861em;",[40,10405,10407,10410],{"className":10406},[1612],[40,10408,2330],{"className":10409},[1612,1653],[40,10411,10413],{"className":10412},[1621],[40,10414,10416],{"className":10415},[1625],[40,10417,10419],{"className":10418},[1629],[40,10420,10422],{"className":10421,"style":2706},[1633],[40,10423,10424,10427],{"style":1637},[40,10425],{"className":10426,"style":1642},[1641],[40,10428,10430],{"className":10429},[1646,1647,1648,1649],[40,10431,10433,10436,10439,10442,10445,10448],{"className":10432},[1612,1649],[40,10434,1733],{"className":10435},[1612,1649],[40,10437,2337],{"className":10438},[1612,1653,1649],[40,10440,2182],{"className":10441,"style":2197},[1612,1653,1649],[40,10443,2670],{"className":10444,"style":2736},[1612,1653,1649],[40,10446,4003],{"className":10447},[1612,1653,1649],[40,10449,2342],{"className":10450},[1612,1653,1649],[40,10452,10454,10457],{"className":10453},[1612],[40,10455,2323],{"className":10456,"style":2358},[1612,1653],[40,10458,10460],{"className":10459},[1621],[40,10461,10463,10487],{"className":10462},[1625,2244],[40,10464,10466,10484],{"className":10465},[1629],[40,10467,10469],{"className":10468,"style":10290},[1633],[40,10470,10472,10475],{"style":10471},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[40,10473],{"className":10474,"style":1642},[1641],[40,10476,10478],{"className":10477},[1646,1647,1648,1649],[40,10479,10481],{"className":10480},[1612,1649],[40,10482,2837],{"className":10483,"style":1818},[1612,1653,1649],[40,10485,2269],{"className":10486},[2268],[40,10488,10490],{"className":10489},[1629],[40,10491,10493],{"className":10492,"style":5877},[1633],[40,10494],{},[40,10496,8782],{"className":10497},[1612],[40,10499,10501,10504],{"className":10500},[1822],[40,10502,1798],{"className":10503},[1822],[40,10505,10507],{"className":10506},[1621],[40,10508,10510],{"className":10509},[1625],[40,10511,10513],{"className":10512},[1629],[40,10514,10516],{"className":10515,"style":2374},[1633],[40,10517,10518,10521],{"style":1637},[40,10519],{"className":10520,"style":1642},[1641],[40,10522,10524],{"className":10523},[1646,1647,1648,1649],[40,10525,6725],{"className":10526,"style":2358},[1612,1653,1649],". To measure ",[40,10529,10531,10544],{"className":10530},[1565],[40,10532,10534],{"className":10533},[1569],[1571,10535,10536],{"xmlns":1573},[1575,10537,10538,10542],{},[1578,10539,10540],{},[1584,10541,6775],{},[1591,10543,6775],{"encoding":1593},[40,10545,10547],{"className":10546,"ariaHidden":1599},[1598],[40,10548,10550,10553],{"className":10549},[1603],[40,10551],{"className":10552,"style":1608},[1607],[40,10554,6775],{"className":10555,"style":6920},[1612,1653],", first apply ",[40,10558,10560,10573],{"className":10559},[1565],[40,10561,10563],{"className":10562},[1569],[1571,10564,10565],{"xmlns":1573},[1575,10566,10567,10571],{},[1578,10568,10569],{},[1584,10570,2182],{},[1591,10572,2182],{"encoding":1593},[40,10574,10576],{"className":10575,"ariaHidden":1599},[1598],[40,10577,10579,10582],{"className":10578},[1603],[40,10580],{"className":10581,"style":1608},[1607],[40,10583,2182],{"className":10584,"style":2197},[1612,1653],[40,10586,10588],{"className":10587},[1827],[40,10589,10591,10697],{"className":10590},[1565],[40,10592,10594],{"className":10593},[1569],[1571,10595,10596],{"xmlns":1573,"display":1836},[1575,10597,10598,10694],{},[8499,10599,10600],{"rowspacing":8501,"columnspacing":8502},[8504,10601,10602],{},[8507,10603,10604],{},[8510,10605,10606],{"scriptlevel":791,"displaystyle":1599},[1578,10607,10608,10610,10612,10618,10620,10622,10624,10626,10628,10634,10636,10638,10640,10642,10648,10650,10652,10654,10660,10662,10664,10666,10668,10670,10676,10678,10680,10682,10684,10690,10692],{},[1731,10609,8534],{},[1756,10611],{"width":8502},[8538,10613,10614,10616],{},[1735,10615,1410],{},[1735,10617,1879],{},[1584,10619,1791],{"mathvariant":1790},[1735,10621,791],{},[1731,10623,1798],{"stretchy":1797},[1731,10625,241],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[1584,10627,1791],{"mathvariant":1790},[2212,10629,10630,10632],{},[1584,10631,2837],{},[1584,10633,1413],{},[1731,10635,1798],{"stretchy":1797},[1731,10637,1147],{},[1584,10639,2218],{},[1584,10641,1791],{"mathvariant":1790},[2212,10643,10644,10646],{},[1584,10645,2837],{},[1584,10647,2337],{},[1731,10649,1798],{"stretchy":1797},[1731,10651,2693],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[1731,10653,1147],{},[8538,10655,10656,10658],{},[1735,10657,1410],{},[1735,10659,1879],{},[1584,10661,1791],{"mathvariant":1790},[1735,10663,1410],{},[1731,10665,1798],{"stretchy":1797},[1731,10667,241],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[1584,10669,1791],{"mathvariant":1790},[2212,10671,10672,10674],{},[1584,10673,2837],{},[1584,10675,1413],{},[1731,10677,1798],{"stretchy":1797},[1731,10679,1733],{},[1584,10681,2218],{},[1584,10683,1791],{"mathvariant":1790},[2212,10685,10686,10688],{},[1584,10687,2837],{},[1584,10689,2337],{},[1731,10691,1798],{"stretchy":1797},[1731,10693,2693],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[1591,10695,10696],{"encoding":1593},"\\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*}",[40,10698,10700],{"className":10699,"ariaHidden":1599},[1598],[40,10701,10703,10707],{"className":10702},[1603],[40,10704],{"className":10705,"style":10706},[1607],"height:2.0074em;vertical-align:-0.7537em;",[40,10708,10710],{"className":10709},[1612],[40,10711,10713],{"className":10712},[8499],[40,10714,10716],{"className":10715},[8758],[40,10717,10719,11126],{"className":10718},[1625,2244],[40,10720,10722,11123],{"className":10721},[1629],[40,10723,10726],{"className":10724,"style":10725},[1633],"height:1.2537em;",[40,10727,10729,10732],{"style":10728},"top:-3.2537em;",[40,10730],{"className":10731,"style":8775},[1641],[40,10733,10735,10738,10741,10744,10808,10811,10814,10820,10823,10863,10866,10869,10872,10875,10878,10881,10921,10924,10930,10933,10936,10939,11001,11004,11007,11013,11016,11056,11059,11062,11065,11068,11071,11074,11114,11117],{"className":10734},[1612],[40,10736,8534],{"className":10737},[1967],[40,10739],{"className":10740,"style":8821},[1756],[40,10742],{"className":10743,"style":1963},[1756],[40,10745,10747,10750,10805],{"className":10746},[1612],[40,10748],{"className":10749},[1985,8840],[40,10751,10753],{"className":10752},[8538],[40,10754,10756,10796],{"className":10755},[1625,2244],[40,10757,10759,10793],{"className":10758},[1629],[40,10760,10762,10774,10782],{"className":10761,"style":8853},[1633],[40,10763,10765,10768],{"style":10764},"top:-2.314em;",[40,10766],{"className":10767,"style":3354},[1641],[40,10769,10771],{"className":10770},[1612],[40,10772,1879],{"className":10773},[1612],[40,10775,10776,10779],{"style":8925},[40,10777],{"className":10778,"style":3354},[1641],[40,10780],{"className":10781,"style":8933},[8932],[40,10783,10784,10787],{"style":8936},[40,10785],{"className":10786,"style":3354},[1641],[40,10788,10790],{"className":10789},[1612],[40,10791,1410],{"className":10792},[1612],[40,10794,2269],{"className":10795},[2268],[40,10797,10799],{"className":10798},[1629],[40,10800,10803],{"className":10801,"style":10802},[1633],"height:0.686em;",[40,10804],{},[40,10806],{"className":10807},[1822,8840],[40,10809,8782],{"className":10810},[1612],[40,10812,1798],{"className":10813},[1822],[40,10815,10817],{"className":10816},[1612],[40,10818,241],{"className":10819},[1991,8967],[40,10821,1791],{"className":10822},[1612],[40,10824,10826,10829],{"className":10825},[1612],[40,10827,2837],{"className":10828,"style":1818},[1612,1653],[40,10830,10832],{"className":10831},[1621],[40,10833,10835,10855],{"className":10834},[1625,2244],[40,10836,10838,10852],{"className":10837},[1629],[40,10839,10841],{"className":10840,"style":5846},[1633],[40,10842,10843,10846],{"style":3616},[40,10844],{"className":10845,"style":1642},[1641],[40,10847,10849],{"className":10848},[1646,1647,1648,1649],[40,10850,1413],{"className":10851,"style":5867},[1612,1653,1649],[40,10853,2269],{"className":10854},[2268],[40,10856,10858],{"className":10857},[1629],[40,10859,10861],{"className":10860,"style":5877},[1633],[40,10862],{},[40,10864,1798],{"className":10865},[1822],[40,10867],{"className":10868,"style":1757},[1756],[40,10870,1147],{"className":10871},[1761],[40,10873],{"className":10874,"style":1757},[1756],[40,10876,2218],{"className":10877,"style":2264},[1612,1653],[40,10879,1791],{"className":10880},[1612],[40,10882,10884,10887],{"className":10883},[1612],[40,10885,2837],{"className":10886,"style":1818},[1612,1653],[40,10888,10890],{"className":10889},[1621],[40,10891,10893,10913],{"className":10892},[1625,2244],[40,10894,10896,10910],{"className":10895},[1629],[40,10897,10899],{"className":10898,"style":5846},[1633],[40,10900,10901,10904],{"style":3616},[40,10902],{"className":10903,"style":1642},[1641],[40,10905,10907],{"className":10906},[1646,1647,1648,1649],[40,10908,2337],{"className":10909},[1612,1653,1649],[40,10911,2269],{"className":10912},[2268],[40,10914,10916],{"className":10915},[1629],[40,10917,10919],{"className":10918,"style":2276},[1633],[40,10920],{},[40,10922,1798],{"className":10923},[1822],[40,10925,10927],{"className":10926},[1612],[40,10928,2693],{"className":10929},[1991,8967],[40,10931],{"className":10932,"style":1757},[1756],[40,10934,1147],{"className":10935},[1761],[40,10937],{"className":10938,"style":1757},[1756],[40,10940,10942,10945,10998],{"className":10941},[1612],[40,10943],{"className":10944},[1985,8840],[40,10946,10948],{"className":10947},[8538],[40,10949,10951,10990],{"className":10950},[1625,2244],[40,10952,10954,10987],{"className":10953},[1629],[40,10955,10957,10968,10976],{"className":10956,"style":8853},[1633],[40,10958,10959,10962],{"style":10764},[40,10960],{"className":10961,"style":3354},[1641],[40,10963,10965],{"className":10964},[1612],[40,10966,1879],{"className":10967},[1612],[40,10969,10970,10973],{"style":8925},[40,10971],{"className":10972,"style":3354},[1641],[40,10974],{"className":10975,"style":8933},[8932],[40,10977,10978,10981],{"style":8936},[40,10979],{"className":10980,"style":3354},[1641],[40,10982,10984],{"className":10983},[1612],[40,10985,1410],{"className":10986},[1612],[40,10988,2269],{"className":10989},[2268],[40,10991,10993],{"className":10992},[1629],[40,10994,10996],{"className":10995,"style":10802},[1633],[40,10997],{},[40,10999],{"className":11000},[1822,8840],[40,11002,8986],{"className":11003},[1612],[40,11005,1798],{"className":11006},[1822],[40,11008,11010],{"className":11009},[1612],[40,11011,241],{"className":11012},[1991,8967],[40,11014,1791],{"className":11015},[1612],[40,11017,11019,11022],{"className":11018},[1612],[40,11020,2837],{"className":11021,"style":1818},[1612,1653],[40,11023,11025],{"className":11024},[1621],[40,11026,11028,11048],{"className":11027},[1625,2244],[40,11029,11031,11045],{"className":11030},[1629],[40,11032,11034],{"className":11033,"style":5846},[1633],[40,11035,11036,11039],{"style":3616},[40,11037],{"className":11038,"style":1642},[1641],[40,11040,11042],{"className":11041},[1646,1647,1648,1649],[40,11043,1413],{"className":11044,"style":5867},[1612,1653,1649],[40,11046,2269],{"className":11047},[2268],[40,11049,11051],{"className":11050},[1629],[40,11052,11054],{"className":11053,"style":5877},[1633],[40,11055],{},[40,11057,1798],{"className":11058},[1822],[40,11060],{"className":11061,"style":1757},[1756],[40,11063,1733],{"className":11064},[1761],[40,11066],{"className":11067,"style":1757},[1756],[40,11069,2218],{"className":11070,"style":2264},[1612,1653],[40,11072,1791],{"className":11073},[1612],[40,11075,11077,11080],{"className":11076},[1612],[40,11078,2837],{"className":11079,"style":1818},[1612,1653],[40,11081,11083],{"className":11082},[1621],[40,11084,11086,11106],{"className":11085},[1625,2244],[40,11087,11089,11103],{"className":11088},[1629],[40,11090,11092],{"className":11091,"style":5846},[1633],[40,11093,11094,11097],{"style":3616},[40,11095],{"className":11096,"style":1642},[1641],[40,11098,11100],{"className":11099},[1646,1647,1648,1649],[40,11101,2337],{"className":11102},[1612,1653,1649],[40,11104,2269],{"className":11105},[2268],[40,11107,11109],{"className":11108},[1629],[40,11110,11112],{"className":11111,"style":2276},[1633],[40,11113],{},[40,11115,1798],{"className":11116},[1822],[40,11118,11120],{"className":11119},[1612],[40,11121,2693],{"className":11122},[1991,8967],[40,11124,2269],{"className":11125},[2268],[40,11127,11129],{"className":11128},[1629],[40,11130,11133],{"className":11131,"style":11132},[1633],"height:0.7537em;",[40,11134],{},[12,11136,11137],{},"... then measure:",[40,11139,11141],{"className":11140},[1827],[40,11142,11144,11326],{"className":11143},[1565],[40,11145,11147],{"className":11146},[1569],[1571,11148,11149],{"xmlns":1573,"display":1836},[1575,11150,11151,11323],{},[8499,11152,11153],{"rowspacing":8501,"columnspacing":8502},[8504,11154,11155],{},[8507,11156,11157],{},[8510,11158,11159],{"scriptlevel":791,"displaystyle":1599},[8499,11160,11164,11272],{"rowspacing":11161,"columnalign":11162,"columnspacing":11163},"0.25em","right left","0em",[8504,11165,11166,11183],{},[8507,11167,11168],{},[8510,11169,11170],{"scriptlevel":791,"displaystyle":1599},[1578,11171,11172,11175,11177,11179,11181],{},[1731,11173,11174],{},"⇒",[1756,11176],{"width":8502},[1731,11178,3469],{"stretchy":1797},[1584,11180,6775],{},[1731,11182,1798],{"stretchy":1797},[8507,11184,11185],{},[8510,11186,11187],{"scriptlevel":791,"displaystyle":1599},[1578,11188,11189,11191,11193,11200,11203,11206,11208,11214,11216,11218,11220,11222,11228,11230,11236,11238,11240,11242,11248,11250,11252,11254,11256,11262,11264,11270],{},[1578,11190],{},[1731,11192,623],{},[8538,11194,11195,11197],{},[1735,11196,1410],{},[1735,11198,11199],{},"4",[1731,11201,241],{"fence":1797,"stretchy":1599,"minsize":11202,"maxsize":11202},"3em",[1731,11204,11205],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},"∥",[1584,11207,1791],{"mathvariant":1790},[2212,11209,11210,11212],{},[1584,11211,2837],{},[1584,11213,1413],{},[1731,11215,1798],{"stretchy":1797},[1731,11217,1147],{},[1584,11219,2218],{},[1584,11221,1791],{"mathvariant":1790},[2212,11223,11224,11226],{},[1584,11225,2837],{},[1584,11227,2337],{},[1731,11229,1798],{"stretchy":1797},[1581,11231,11232,11234],{},[1731,11233,11205],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[1735,11235,1879],{},[1731,11237,1733],{},[1731,11239,11205],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[1584,11241,1791],{"mathvariant":1790},[2212,11243,11244,11246],{},[1584,11245,2837],{},[1584,11247,1413],{},[1731,11249,1798],{"stretchy":1797},[1731,11251,1733],{},[1584,11253,2218],{},[1584,11255,1791],{"mathvariant":1790},[2212,11257,11258,11260],{},[1584,11259,2837],{},[1584,11261,2337],{},[1731,11263,1798],{"stretchy":1797},[1581,11265,11266,11268],{},[1731,11267,11205],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[1735,11269,1879],{},[1731,11271,2693],{"fence":1797,"stretchy":1599,"minsize":11202,"maxsize":11202},[8504,11273,11274,11280],{},[8507,11275,11276],{},[8510,11277,11278],{"scriptlevel":791,"displaystyle":1599},[1578,11279],{},[8507,11281,11282],{},[8510,11283,11284],{"scriptlevel":791,"displaystyle":1599},[1578,11285,11286,11288,11290,11293,11296,11298,11304,11306,11308,11310,11316,11318,11321],{},[1578,11287],{},[1731,11289,623],{},[5211,11291,11292],{},"Re",[1731,11294,11295],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},"[",[1731,11297,3469],{"stretchy":1797},[2212,11299,11300,11302],{},[1584,11301,2837],{},[1584,11303,1413],{},[1584,11305,1791],{"mathvariant":1790},[1584,11307,2218],{},[1584,11309,1791],{"mathvariant":1790},[2212,11311,11312,11314],{},[1584,11313,2837],{},[1584,11315,2337],{},[1731,11317,1798],{"stretchy":1797},[1731,11319,11320],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},"]",[1584,11322,1823],{"mathvariant":1790},[1591,11324,11325],{"encoding":1593},"\\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*}",[40,11327,11329],{"className":11328,"ariaHidden":1599},[1598],[40,11330,11332,11336],{"className":11331},[1603],[40,11333],{"className":11334,"style":11335},[1607],"height:5.1em;vertical-align:-2.3em;",[40,11337,11339],{"className":11338},[1612],[40,11340,11342],{"className":11341},[8499],[40,11343,11345],{"className":11344},[8758],[40,11346,11348,12118],{"className":11347},[1625,2244],[40,11349,11351,12115],{"className":11350},[1629],[40,11352,11355],{"className":11353,"style":11354},[1633],"height:2.8em;",[40,11356,11358,11362],{"style":11357},"top:-4.8em;",[40,11359],{"className":11360,"style":11361},[1641],"height:4.8em;",[40,11363,11365],{"className":11364},[1612],[40,11366,11368],{"className":11367},[1612],[40,11369,11371,11432],{"className":11370},[8499],[40,11372,11375],{"className":11373},[11374],"col-align-r",[40,11376,11378,11423],{"className":11377},[1625,2244],[40,11379,11381,11420],{"className":11380},[1629],[40,11382,11384,11411],{"className":11383,"style":11354},[1633],[40,11385,11386,11390],{"style":11357},[40,11387],{"className":11388,"style":11389},[1641],"height:3.75em;",[40,11391,11393,11396,11399,11402,11405,11408],{"className":11392},[1612],[40,11394,11174],{"className":11395},[1967],[40,11397],{"className":11398,"style":8821},[1756],[40,11400],{"className":11401,"style":1963},[1756],[40,11403,3469],{"className":11404},[1985],[40,11406,6775],{"className":11407,"style":6920},[1612,1653],[40,11409,1798],{"className":11410},[1822],[40,11412,11414,11417],{"style":11413},"top:-2.1em;",[40,11415],{"className":11416,"style":11389},[1641],[40,11418],{"className":11419},[1612],[40,11421,2269],{"className":11422},[2268],[40,11424,11426],{"className":11425},[1629],[40,11427,11430],{"className":11428,"style":11429},[1633],"height:2.3em;",[40,11431],{},[40,11433,11436],{"className":11434},[11435],"col-align-l",[40,11437,11439,12107],{"className":11438},[1625,2244],[40,11440,11442,12104],{"className":11441},[1629],[40,11443,11445,11968],{"className":11444,"style":11354},[1633],[40,11446,11447,11450],{"style":11357},[40,11448],{"className":11449,"style":11389},[1641],[40,11451,11453,11456,11459,11462,11465,11527,11534,11580,11583,11623,11626,11629,11632,11635,11638,11641,11681,11684,11749,11752,11755,11758,11795,11798,11838,11841,11844,11847,11850,11853,11856,11896,11899,11962],{"className":11452},[1612],[40,11454],{"className":11455},[1612],[40,11457],{"className":11458,"style":1963},[1756],[40,11460,623],{"className":11461},[1967],[40,11463],{"className":11464,"style":1963},[1756],[40,11466,11468,11471,11524],{"className":11467},[1612],[40,11469],{"className":11470},[1985,8840],[40,11472,11474],{"className":11473},[8538],[40,11475,11477,11516],{"className":11476},[1625,2244],[40,11478,11480,11513],{"className":11479},[1629],[40,11481,11483,11494,11502],{"className":11482,"style":8853},[1633],[40,11484,11485,11488],{"style":10764},[40,11486],{"className":11487,"style":3354},[1641],[40,11489,11491],{"className":11490},[1612],[40,11492,11199],{"className":11493},[1612],[40,11495,11496,11499],{"style":8925},[40,11497],{"className":11498,"style":3354},[1641],[40,11500],{"className":11501,"style":8933},[8932],[40,11503,11504,11507],{"style":8936},[40,11505],{"className":11506,"style":3354},[1641],[40,11508,11510],{"className":11509},[1612],[40,11511,1410],{"className":11512},[1612],[40,11514,2269],{"className":11515},[2268],[40,11517,11519],{"className":11518},[1629],[40,11520,11522],{"className":11521,"style":10802},[1633],[40,11523],{},[40,11525],{"className":11526},[1822,8840],[40,11528,11530],{"className":11529},[1612],[40,11531,241],{"className":11532},[1991,11533],"size4",[40,11535,11537],{"className":11536},[1612],[40,11538,11541],{"className":11539},[1991,11540],"mult",[40,11542,11544,11571],{"className":11543},[1625,2244],[40,11545,11547,11568],{"className":11546},[1629],[40,11548,11551],{"className":11549,"style":11550},[1633],"height:1.15em;",[40,11552,11554,11558],{"style":11553},"top:-3.15em;",[40,11555],{"className":11556,"style":11557},[1641],"height:3.8em;",[40,11559,11561],{"style":11560},"width:0.556em;height:1.8em;",[5325,11562,11565],{"xmlns":5327,"width":11563,"height":8549,"viewBox":11564},"0.556em","0 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",[40,12130,12132,12219],{"className":12131},[1565],[40,12133,12135],{"className":12134},[1569],[1571,12136,12137],{"xmlns":1573},[1575,12138,12139,12216],{},[1578,12140,12141,12143,12145,12147,12150,12156,12158,12160,12162,12164,12166,12168,12170,12172,12174,12176,12178,12180,12182,12188,12190,12192,12194,12200,12202,12204,12206,12208,12210,12212,12214],{},[1584,12142,1791],{"mathvariant":1790},[1584,12144,19],{},[1731,12146,1147],{},[1584,12148,12149],{},"b",[1581,12151,12152,12154],{},[1584,12153,11205],{"mathvariant":1790},[1735,12155,1879],{},[1731,12157,623],{},[1731,12159,3469],{"stretchy":1797},[1584,12161,19],{},[1731,12163,1147],{},[1584,12165,12149],{},[1584,12167,1791],{"mathvariant":1790},[1584,12169,19],{},[1731,12171,1147],{},[1584,12173,12149],{},[1731,12175,1798],{"stretchy":1797},[1731,12177,623],{},[1584,12179,11205],{"mathvariant":1790},[1584,12181,19],{},[1581,12183,12184,12186],{},[1584,12185,11205],{"mathvariant":1790},[1735,12187,1879],{},[1731,12189,1147],{},[1584,12191,11205],{"mathvariant":1790},[1584,12193,12149],{},[1581,12195,12196,12198],{},[1584,12197,11205],{"mathvariant":1790},[1735,12199,1879],{},[1731,12201,1147],{},[1735,12203,1879],{},[5211,12205,11292],{},[1731,12207,3469],{"stretchy":1797},[1584,12209,19],{},[1584,12211,1791],{"mathvariant":1790},[1584,12213,12149],{},[1731,12215,1798],{"stretchy":1797},[1591,12217,12218],{"encoding":1593},"|a + b\\|^2 = \\langle a + b | a + b \\rangle = \\|a\\|^2 + \\|b\\|^2 + 2\\text{Re}\\langle a | b \\rangle",[40,12220,12222,12243,12292,12313,12337,12358,12408,12458],{"className":12221,"ariaHidden":1599},[1598],[40,12223,12225,12228,12231,12234,12237,12240],{"className":12224},[1603],[40,12226],{"className":12227,"style":1811},[1607],[40,12229,1791],{"className":12230},[1612],[40,12232,19],{"className":12233},[1612,1653],[40,12235],{"className":12236,"style":1757},[1756],[40,12238,1147],{"className":12239},[1761],[40,12241],{"className":12242,"style":1757},[1756],[40,12244,12246,12250,12253,12283,12286,12289],{"className":12245},[1603],[40,12247],{"className":12248,"style":12249},[1607],"height:1.0641em;vertical-align:-0.25em;",[40,12251,12149],{"className":12252},[1612,1653],[40,12254,12256,12259],{"className":12255},[1612],[40,12257,11205],{"className":12258},[1612],[40,12260,12262],{"className":12261},[1621],[40,12263,12265],{"className":12264},[1625],[40,12266,12268],{"className":12267},[1629],[40,12269,12272],{"className":12270,"style":12271},[1633],"height:0.8141em;",[40,12273,12274,12277],{"style":1637},[40,12275],{"className":12276,"style":1642},[1641],[40,12278,12280],{"className":12279},[1646,1647,1648,1649],[40,12281,1879],{"className":12282},[1612,1649],[40,12284],{"className":12285,"style":1963},[1756],[40,12287,623],{"className":12288},[1967],[40,12290],{"className":12291,"style":1963},[1756],[40,12293,12295,12298,12301,12304,12307,12310],{"className":12294},[1603],[40,12296],{"className":12297,"style":1811},[1607],[40,12299,3469],{"className":12300},[1985],[40,12302,19],{"className":12303},[1612,1653],[40,12305],{"className":12306,"style":1757},[1756],[40,12308,1147],{"className":12309},[1761],[40,12311],{"className":12312,"style":1757},[1756],[40,12314,12316,12319,12322,12325,12328,12331,12334],{"className":12315},[1603],[40,12317],{"className":12318,"style":1811},[1607],[40,12320,12149],{"className":12321},[1612,1653],[40,12323,1791],{"className":12324},[1612],[40,12326,19],{"className":12327},[1612,1653],[40,12329],{"className":12330,"style":1757},[1756],[40,12332,1147],{"className":12333},[1761],[40,12335],{"className":12336,"style":1757},[1756],[40,12338,12340,12343,12346,12349,12352,12355],{"className":12339},[1603],[40,12341],{"className":12342,"style":1811},[1607],[40,12344,12149],{"className":12345},[1612,1653],[40,12347,1798],{"className":12348},[1822],[40,12350],{"className":12351,"style":1963},[1756],[40,12353,623],{"className":12354},[1967],[40,12356],{"className":12357,"style":1963},[1756],[40,12359,12361,12364,12367,12370,12399,12402,12405],{"className":12360},[1603],[40,12362],{"className":12363,"style":12249},[1607],[40,12365,11205],{"className":12366},[1612],[40,12368,19],{"className":12369},[1612,1653],[40,12371,12373,12376],{"className":12372},[1612],[40,12374,11205],{"className":12375},[1612],[40,12377,12379],{"className":12378},[1621],[40,12380,12382],{"className":12381},[1625],[40,12383,12385],{"className":12384},[1629],[40,12386,12388],{"className":12387,"style":12271},[1633],[40,12389,12390,12393],{"style":1637},[40,12391],{"className":12392,"style":1642},[1641],[40,12394,12396],{"className":12395},[1646,1647,1648,1649],[40,12397,1879],{"className":12398},[1612,1649],[40,12400],{"className":12401,"style":1757},[1756],[40,12403,1147],{"className":12404},[1761],[40,12406],{"className":12407,"style":1757},[1756],[40,12409,12411,12414,12417,12420,12449,12452,12455],{"className":12410},[1603],[40,12412],{"className":12413,"style":12249},[1607],[40,12415,11205],{"className":12416},[1612],[40,12418,12149],{"className":12419},[1612,1653],[40,12421,12423,12426],{"className":12422},[1612],[40,12424,11205],{"className":12425},[1612],[40,12427,12429],{"className":12428},[1621],[40,12430,12432],{"className":12431},[1625],[40,12433,12435],{"className":12434},[1629],[40,12436,12438],{"className":12437,"style":12271},[1633],[40,12439,12440,12443],{"style":1637},[40,12441],{"className":12442,"style":1642},[1641],[40,12444,12446],{"className":12445},[1646,1647,1648,1649],[40,12447,1879],{"className":12448},[1612,1649],[40,12450],{"className":12451,"style":1757},[1756],[40,12453,1147],{"className":12454},[1761],[40,12456],{"className":12457,"style":1757},[1756],[40,12459,12461,12464,12467,12473,12476,12479,12482,12485],{"className":12460},[1603],[40,12462],{"className":12463,"style":1811},[1607],[40,12465,1879],{"className":12466},[1612],[40,12468,12470],{"className":12469},[1612,6083],[40,12471,11292],{"className":12472},[1612],[40,12474,3469],{"className":12475},[1985],[40,12477,19],{"className":12478},[1612,1653],[40,12480,1791],{"className":12481},[1612],[40,12483,12149],{"className":12484},[1612,1653],[40,12486,1798],{"className":12487},[1822],". Similarly, measuring ",[40,12490,12492,12505],{"className":12491},[1565],[40,12493,12495],{"className":12494},[1569],[1571,12496,12497],{"xmlns":1573},[1575,12498,12499,12503],{},[1578,12500,12501],{},[1584,12502,6790],{},[1591,12504,6790],{"encoding":1593},[40,12506,12508],{"className":12507,"ariaHidden":1599},[1598],[40,12509,12511,12514],{"className":12510},[1603],[40,12512],{"className":12513,"style":1608},[1607],[40,12515,6790],{"className":12516,"style":1757},[1612,1653]," yields",[40,12519,12521],{"className":12520},[1827],[40,12522,12524,12583],{"className":12523},[1565],[40,12525,12527],{"className":12526},[1569],[1571,12528,12529],{"xmlns":1573,"display":1836},[1575,12530,12531,12580],{},[8499,12532,12533],{"rowspacing":8501,"columnspacing":8502},[8504,12534,12535],{},[8507,12536,12537],{},[8510,12538,12539],{"scriptlevel":791,"displaystyle":1599},[1578,12540,12541,12543,12545,12547,12549,12552,12554,12556,12562,12564,12566,12568,12574,12576,12578],{},[1731,12542,3469],{"stretchy":1797},[1584,12544,6790],{},[1731,12546,1798],{"stretchy":1797},[1731,12548,623],{},[5211,12550,12551],{},"Im",[1731,12553,11295],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[1731,12555,3469],{"stretchy":1797},[2212,12557,12558,12560],{},[1584,12559,2837],{},[1584,12561,1413],{},[1584,12563,1791],{"mathvariant":1790},[1584,12565,2218],{},[1584,12567,1791],{"mathvariant":1790},[2212,12569,12570,12572],{},[1584,12571,2837],{},[1584,12573,2337],{},[1731,12575,1798],{"stretchy":1797},[1731,12577,11320],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[1584,12579,1823],{"mathvariant":1790},[1591,12581,12582],{"encoding":1593},"\\begin{equation*}\n    \\langle Y\\rangle = \\text{Im}\\Big[\\langle\\psi_j| P|\\psi_i\\rangle\\Big].\n\\end{equation*}",[40,12584,12586],{"className":12585,"ariaHidden":1599},[1598],[40,12587,12589,12593],{"className":12588},[1603],[40,12590],{"className":12591,"style":12592},[1607],"height:1.8em;vertical-align:-0.65em;",[40,12594,12596],{"className":12595},[1612],[40,12597,12599],{"className":12598},[8499],[40,12600,12602],{"className":12601},[8758],[40,12603,12605,12757],{"className":12604},[1625,2244],[40,12606,12608,12754],{"className":12607},[1629],[40,12609,12611],{"className":12610,"style":11550},[1633],[40,12612,12613,12617],{"style":11553},[40,12614],{"className":12615,"style":12616},[1641],"height:3.15em;",[40,12618,12620,12623,12626,12629,12632,12635,12638,12644,12650,12653,12693,12696,12699,12702,12742,12745,12751],{"className":12619},[1612],[40,12621,3469],{"className":12622},[1985],[40,12624,6790],{"className":12625,"style":1757},[1612,1653],[40,12627,1798],{"className":12628},[1822],[40,12630],{"className":12631,"style":1963},[1756],[40,12633,623],{"className":12634},[1967],[40,12636],{"className":12637,"style":1963},[1756],[40,12639,12641],{"className":12640},[1612,6083],[40,12642,12551],{"className":12643},[1612],[40,12645,12647],{"className":12646},[1612],[40,12648,11295],{"className":12649},[1991,8967],[40,12651,3469],{"className":12652},[1985],[40,12654,12656,12659],{"className":12655},[1612],[40,12657,2837],{"className":12658,"style":1818},[1612,1653],[40,12660,12662],{"className":12661},[1621],[40,12663,12665,12685],{"className":12664},[1625,2244],[40,12666,12668,12682],{"className":12667},[1629],[40,12669,12671],{"className":12670,"style":5846},[1633],[40,12672,12673,12676],{"style":3616},[40,12674],{"className":12675,"style":1642},[1641],[40,12677,12679],{"className":12678},[1646,1647,1648,1649],[40,12680,1413],{"className":12681,"style":5867},[1612,1653,1649],[40,12683,2269],{"className":12684},[2268],[40,12686,12688],{"className":12687},[1629],[40,12689,12691],{"className":12690,"style":5877},[1633],[40,12692],{},[40,12694,1791],{"className":12695},[1612],[40,12697,2218],{"className":12698,"style":2264},[1612,1653],[40,12700,1791],{"className":12701},[1612],[40,12703,12705,12708],{"className":12704},[1612],[40,12706,2837],{"className":12707,"style":1818},[1612,1653],[40,12709,12711],{"className":12710},[1621],[40,12712,12714,12734],{"className":12713},[1625,2244],[40,12715,12717,12731],{"className":12716},[1629],[40,12718,12720],{"className":12719,"style":5846},[1633],[40,12721,12722,12725],{"style":3616},[40,12723],{"className":12724,"style":1642},[1641],[40,12726,12728],{"className":12727},[1646,1647,1648,1649],[40,12729,2337],{"className":12730},[1612,1653,1649],[40,12732,2269],{"className":12733},[2268],[40,12735,12737],{"className":12736},[1629],[40,12738,12740],{"className":12739,"style":2276},[1633],[40,12741],{},[40,12743,1798],{"className":12744},[1822],[40,12746,12748],{"className":12747},[1612],[40,12749,11320],{"className":12750},[1991,8967],[40,12752,1823],{"className":12753},[1612],[40,12755,2269],{"className":12756},[2268],[40,12758,12760],{"className":12759},[1629],[40,12761,12763],{"className":12762,"style":11577},[1633],[40,12764],{},[30,12766,12768],{"className":32,"code":12767,"language":34,"meta":35,"style":35},"## 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",[37,12769,12770,12774,12784,12809,12813,12817,12822,12833,12857,12861,12876,12880,12885,12896,12908,12929,12942,12958,12970,12985,13007,13011,13016,13041,13045,13050,13066,13077,13091,13116,13128,13132,13138,13143,13147],{"__ignoreMap":35},[40,12771,12772],{"class":42,"line":43},[40,12773,8375],{"class":46},[40,12775,12776,12778,12780,12782],{"class":42,"line":50},[40,12777,8380],{"class":133},[40,12779,623],{"class":441},[40,12781,8385],{"class":237},[40,12783,411],{"class":133},[40,12785,12786,12788,12790,12792,12795,12797,12799,12801,12803,12805,12807],{"class":42,"line":155},[40,12787,8392],{"class":133},[40,12789,8395],{"class":1195},[40,12791,623],{"class":441},[40,12793,12794],{"class":133},"dt, ",[40,12796,8403],{"class":1195},[40,12798,623],{"class":441},[40,12800,8408],{"class":237},[40,12802,241],{"class":133},[40,12804,8413],{"class":1195},[40,12806,623],{"class":441},[40,12808,8418],{"class":133},[40,12810,12811],{"class":42,"line":168},[40,12812,248],{"class":133},[40,12814,12815],{"class":42,"line":182},[40,12816,228],{"emptyLinePlaceholder":227},[40,12818,12819],{"class":42,"line":195},[40,12820,12821],{"class":46},"## Create the time-evo op dagger circuit\n",[40,12823,12824,12827,12829,12831],{"class":42,"line":203},[40,12825,12826],{"class":133},"evol_gate_d ",[40,12828,623],{"class":441},[40,12830,8385],{"class":237},[40,12832,411],{"class":133},[40,12834,12835,12837,12839,12841,12843,12845,12847,12849,12851,12853,12855],{"class":42,"line":216},[40,12836,8392],{"class":133},[40,12838,8395],{"class":1195},[40,12840,623],{"class":441},[40,12842,12794],{"class":133},[40,12844,8403],{"class":1195},[40,12846,623],{"class":441},[40,12848,8408],{"class":237},[40,12850,241],{"class":133},[40,12852,8413],{"class":1195},[40,12854,623],{"class":441},[40,12856,8418],{"class":133},[40,12858,12859],{"class":42,"line":224},[40,12860,248],{"class":133},[40,12862,12863,12865,12867,12870,12873],{"class":42,"line":231},[40,12864,12826],{"class":133},[40,12866,623],{"class":441},[40,12868,12869],{"class":133}," evol_gate_d.",[40,12871,12872],{"class":237},"inverse",[40,12874,12875],{"class":133},"()\n",[40,12877,12878],{"class":42,"line":251},[40,12879,228],{"emptyLinePlaceholder":227},[40,12881,12882],{"class":42,"line":256},[40,12883,12884],{"class":46},"# Put pieces together\n",[40,12886,12887,12890,12892,12894],{"class":42,"line":269},[40,12888,12889],{"class":133},"qc_reg ",[40,12891,623],{"class":441},[40,12893,8436],{"class":237},[40,12895,7828],{"class":133},[40,12897,12898,12901,12903,12905],{"class":42,"line":282},[40,12899,12900],{"class":133},"qc_temp ",[40,12902,623],{"class":441},[40,12904,8175],{"class":237},[40,12906,12907],{"class":133},"(qc_reg)\n",[40,12909,12910,12913,12916,12919,12922,12924,12927],{"class":42,"line":295},[40,12911,12912],{"class":133},"qc_temp.",[40,12914,12915],{"class":237},"compose",[40,12917,12918],{"class":133},"(qc_state_prep, ",[40,12920,12921],{"class":1195},"inplace",[40,12923,623],{"class":441},[40,12925,12926],{"class":462},"True",[40,12928,248],{"class":133},[40,12930,12931,12933,12935,12937,12939],{"class":42,"line":308},[40,12932,668],{"class":129},[40,12934,7260],{"class":133},[40,12936,674],{"class":129},[40,12938,7265],{"class":441},[40,12940,12941],{"class":133},"(num_trotter_steps):\n",[40,12943,12944,12947,12949,12951,12953,12955],{"class":42,"line":321},[40,12945,12946],{"class":133},"    qc_temp.",[40,12948,979],{"class":237},[40,12950,8460],{"class":133},[40,12952,8463],{"class":1195},[40,12954,623],{"class":441},[40,12956,12957],{"class":133},"qc_reg)\n",[40,12959,12960,12962,12964,12966,12968],{"class":42,"line":334},[40,12961,668],{"class":129},[40,12963,7260],{"class":133},[40,12965,674],{"class":129},[40,12967,7265],{"class":441},[40,12969,12941],{"class":133},[40,12971,12972,12974,12976,12979,12981,12983],{"class":42,"line":347},[40,12973,12946],{"class":133},[40,12975,979],{"class":237},[40,12977,12978],{"class":133},"(evol_gate_d, ",[40,12980,8463],{"class":1195},[40,12982,623],{"class":441},[40,12984,12957],{"class":133},[40,12986,12987,12989,12991,12994,12996,12999,13001,13003,13005],{"class":42,"line":352},[40,12988,12912],{"class":133},[40,12990,12915],{"class":237},[40,12992,12993],{"class":133},"(qc_state_prep.",[40,12995,12872],{"class":237},[40,12997,12998],{"class":133},"(), ",[40,13000,12921],{"class":1195},[40,13002,623],{"class":441},[40,13004,12926],{"class":462},[40,13006,248],{"class":133},[40,13008,13009],{"class":42,"line":357},[40,13010,228],{"emptyLinePlaceholder":227},[40,13012,13013],{"class":42,"line":370},[40,13014,13015],{"class":46},"# Create controlled version of the circuit\n",[40,13017,13018,13021,13023,13026,13029,13032,13035,13037,13039],{"class":42,"line":376},[40,13019,13020],{"class":133},"controlled_U ",[40,13022,623],{"class":441},[40,13024,13025],{"class":133}," qc_temp.",[40,13027,13028],{"class":237},"to_gate",[40,13030,13031],{"class":133},"().",[40,13033,13034],{"class":237},"control",[40,13036,241],{"class":133},[40,13038,1410],{"class":462},[40,13040,248],{"class":133},[40,13042,13043],{"class":42,"line":387},[40,13044,228],{"emptyLinePlaceholder":227},[40,13046,13047],{"class":42,"line":392},[40,13048,13049],{"class":46},"# Create hadamard test circuit for real part\n",[40,13051,13052,13054,13056,13058,13060,13062,13064],{"class":42,"line":397},[40,13053,8431],{"class":133},[40,13055,623],{"class":441},[40,13057,8436],{"class":237},[40,13059,1144],{"class":133},[40,13061,1147],{"class":441},[40,13063,463],{"class":462},[40,13065,248],{"class":133},[40,13067,13068,13071,13073,13075],{"class":42,"line":402},[40,13069,13070],{"class":133},"qc_real ",[40,13072,623],{"class":441},[40,13074,8175],{"class":237},[40,13076,8450],{"class":133},[40,13078,13079,13082,13085,13087,13089],{"class":42,"line":414},[40,13080,13081],{"class":133},"qc_real.",[40,13083,13084],{"class":237},"h",[40,13086,241],{"class":133},[40,13088,791],{"class":462},[40,13090,248],{"class":133},[40,13092,13093,13095,13097,13100,13103,13105,13107,13109,13111,13113],{"class":42,"line":424},[40,13094,13081],{"class":133},[40,13096,979],{"class":237},[40,13098,13099],{"class":133},"(controlled_U, ",[40,13101,13102],{"class":441},"list",[40,13104,241],{"class":133},[40,13106,1244],{"class":441},[40,13108,1144],{"class":133},[40,13110,1147],{"class":441},[40,13112,463],{"class":462},[40,13114,13115],{"class":133},")))\n",[40,13117,13118,13120,13122,13124,13126],{"class":42,"line":432},[40,13119,13081],{"class":133},[40,13121,13084],{"class":237},[40,13123,241],{"class":133},[40,13125,791],{"class":462},[40,13127,248],{"class":133},[40,13129,13130],{"class":42,"line":448},[40,13131,228],{"emptyLinePlaceholder":227},[40,13133,13134,13136],{"class":42,"line":468},[40,13135,7569],{"class":441},[40,13137,411],{"class":133},[40,13139,13140],{"class":42,"line":486},[40,13141,13142],{"class":244},"    \"Circuit for calculating the real part of the overlap in S via Hadamard test\"\n",[40,13144,13145],{"class":42,"line":502},[40,13146,248],{"class":133},[40,13148,13149,13151,13153,13155,13157,13159,13162,13165,13167,13169,13171,13173,13175],{"class":42,"line":508},[40,13150,13081],{"class":133},[40,13152,8210],{"class":237},[40,13154,241],{"class":133},[40,13156,8215],{"class":244},[40,13158,917],{"class":133},[40,13160,13161],{"class":1195},"fold",[40,13163,13164],{"class":441},"=-",[40,13166,1410],{"class":462},[40,13168,917],{"class":133},[40,13170,8220],{"class":1195},[40,13172,623],{"class":441},[40,13174,8225],{"class":462},[40,13176,248],{"class":133},[30,13178,13181],{"className":13179,"code":13180,"language":6083,"meta":35},[7575],"Circuit for calculating the real part of the overlap in S via Hadamard test\n",[37,13182,13180],{"__ignoreMap":35},[8229,13184],{"alt":8231,"src":13185},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-02.avif",[12,13187,13188],{},"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)",[30,13190,13192],{"className":32,"code":13191,"language":34,"meta":35,"style":35},"print(\n    \"Number of layers of 2Q operations\",\n    qc_real.decompose(reps=2).depth(lambda x: x[0].num_qubits == 2),\n)\n",[37,13193,13194,13200,13207,13252],{"__ignoreMap":35},[40,13195,13196,13198],{"class":42,"line":43},[40,13197,7569],{"class":441},[40,13199,411],{"class":133},[40,13201,13202,13205],{"class":42,"line":50},[40,13203,13204],{"class":244},"    \"Number of layers of 2Q operations\"",[40,13206,445],{"class":133},[40,13208,13209,13212,13215,13217,13219,13221,13223,13226,13229,13231,13234,13237,13240,13242,13245,13247,13249],{"class":42,"line":155},[40,13210,13211],{"class":133},"    qc_real.",[40,13213,13214],{"class":237},"decompose",[40,13216,241],{"class":133},[40,13218,8413],{"class":1195},[40,13220,623],{"class":441},[40,13222,1879],{"class":462},[40,13224,13225],{"class":133},").",[40,13227,13228],{"class":237},"depth",[40,13230,241],{"class":133},[40,13232,13233],{"class":129},"lambda",[40,13235,13236],{"class":417}," x",[40,13238,13239],{"class":133},": x[",[40,13241,791],{"class":462},[40,13243,13244],{"class":133},"].num_qubits ",[40,13246,761],{"class":441},[40,13248,1356],{"class":462},[40,13250,13251],{"class":133},"),\n",[40,13253,13254],{"class":42,"line":168},[40,13255,248],{"class":133},[30,13257,13260],{"className":13258,"code":13259,"language":6083,"meta":35},[7575],"Number of layers of 2Q operations 112753\n",[37,13261,13259],{"__ignoreMap":35},[55,13263,13265],{"id":13264},"step-2-optimize-problem-for-quantum-hardware-execution","Step 2: Optimize problem for quantum hardware execution",[1555,13267,13269],{"id":13268},"efficient-hadamard-test","Efficient Hadamard test",[12,13271,13272],{},"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:",[12,13274,13275],{},[5752,13276],{"alt":13277,"src":13278},"fig3.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-08.avif",[12,13280,13281,13282,13353,13354,13420,13421,13449,13450,13515,13516,13558,13559,13838,13839,13880,13881,13922],{},"Assume we can classically calculate ",[40,13283,13285,13303],{"className":13284},[1565],[40,13286,13288],{"className":13287},[1569],[1571,13289,13290],{"xmlns":1573},[1575,13291,13292,13300],{},[1578,13293,13294],{},[2212,13295,13296,13298],{},[1584,13297,5226],{},[1735,13299,791],{},[1591,13301,13302],{"encoding":1593},"E_0",[40,13304,13306],{"className":13305,"ariaHidden":1599},[1598],[40,13307,13309,13312],{"className":13308},[1603],[40,13310],{"className":13311,"style":2231},[1607],[40,13313,13315,13318],{"className":13314},[1612],[40,13316,5226],{"className":13317,"style":4936},[1612,1653],[40,13319,13321],{"className":13320},[1621],[40,13322,13324,13345],{"className":13323},[1625,2244],[40,13325,13327,13342],{"className":13326},[1629],[40,13328,13331],{"className":13329,"style":13330},[1633],"height:0.3011em;",[40,13332,13333,13336],{"style":5058},[40,13334],{"className":13335,"style":1642},[1641],[40,13337,13339],{"className":13338},[1646,1647,1648,1649],[40,13340,791],{"className":13341},[1612,1649],[40,13343,2269],{"className":13344},[2268],[40,13346,13348],{"className":13347},[1629],[40,13349,13351],{"className":13350,"style":2276},[1633],[40,13352],{},", the eigenvalue of ",[40,13355,13357,13379],{"className":13356},[1565],[40,13358,13360],{"className":13359},[1569],[1571,13361,13362],{"xmlns":1573},[1575,13363,13364,13376],{},[1578,13365,13366,13368,13370],{},[1584,13367,1791],{"mathvariant":1790},[1735,13369,791],{},[1581,13371,13372,13374],{},[1731,13373,1798],{"stretchy":1797},[1584,13375,6725],{},[1591,13377,13378],{"encoding":1593},"|0\\rangle^N",[40,13380,13382],{"className":13381,"ariaHidden":1599},[1598],[40,13383,13385,13388,13391],{"className":13384},[1603],[40,13386],{"className":13387,"style":4294},[1607],[40,13389,8782],{"className":13390},[1612],[40,13392,13394,13397],{"className":13393},[1822],[40,13395,1798],{"className":13396},[1822],[40,13398,13400],{"className":13399},[1621],[40,13401,13403],{"className":13402},[1625],[40,13404,13406],{"className":13405},[1629],[40,13407,13409],{"className":13408,"style":2374},[1633],[40,13410,13411,13414],{"style":1637},[40,13412],{"className":13413,"style":1642},[1641],[40,13415,13417],{"className":13416},[1646,1647,1648,1649],[40,13418,6725],{"className":13419,"style":2358},[1612,1653,1649]," under the Hamiltonian ",[40,13422,13424,13437],{"className":13423},[1565],[40,13425,13427],{"className":13426},[1569],[1571,13428,13429],{"xmlns":1573},[1575,13430,13431,13435],{},[1578,13432,13433],{},[1584,13434,2182],{},[1591,13436,2182],{"encoding":1593},[40,13438,13440],{"className":13439,"ariaHidden":1599},[1598],[40,13441,13443,13446],{"className":13442},[1603],[40,13444],{"className":13445,"style":1608},[1607],[40,13447,2182],{"className":13448,"style":2197},[1612,1653],".\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 ",[40,13451,13453,13474],{"className":13452},[1565],[40,13454,13456],{"className":13455},[1569],[1571,13457,13458],{"xmlns":1573},[1575,13459,13460,13472],{},[1578,13461,13462,13464,13466],{},[1584,13463,1791],{"mathvariant":1790},[1735,13465,791],{},[1581,13467,13468,13470],{},[1731,13469,1798],{"stretchy":1797},[1584,13471,6725],{},[1591,13473,13378],{"encoding":1593},[40,13475,13477],{"className":13476,"ariaHidden":1599},[1598],[40,13478,13480,13483,13486],{"className":13479},[1603],[40,13481],{"className":13482,"style":4294},[1607],[40,13484,8782],{"className":13485},[1612],[40,13487,13489,13492],{"className":13488},[1822],[40,13490,1798],{"className":13491},[1822],[40,13493,13495],{"className":13494},[1621],[40,13496,13498],{"className":13497},[1625],[40,13499,13501],{"className":13500},[1629],[40,13502,13504],{"className":13503,"style":2374},[1633],[40,13505,13506,13509],{"style":1637},[40,13507],{"className":13508,"style":1642},[1641],[40,13510,13512],{"className":13511},[1646,1647,1648,1649],[40,13513,6725],{"className":13514,"style":2358},[1612,1653,1649]," 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 ",[40,13517,13519,13537],{"className":13518},[1565],[40,13520,13522],{"className":13521},[1569],[1571,13523,13524],{"xmlns":1573},[1575,13525,13526,13534],{},[1578,13527,13528,13530,13532],{},[5211,13529,6057],{},[5211,13531,6060],{},[1584,13533,2837],{},[1591,13535,13536],{"encoding":1593},"\\text{Prep} \\; \\psi",[40,13538,13540],{"className":13539,"ariaHidden":1599},[1598],[40,13541,13543,13546,13552,13555],{"className":13542},[1603],[40,13544],{"className":13545,"style":6079},[1607],[40,13547,13549],{"className":13548},[1612,6083],[40,13550,6057],{"className":13551},[1612],[40,13553],{"className":13554,"style":1963},[1756],[40,13556,2837],{"className":13557,"style":1818},[1612,1653],", prepares the desired reference state ",[40,13560,13562,13640],{"className":13561},[1565],[40,13563,13565],{"className":13564},[1569],[1571,13566,13567],{"xmlns":1573},[1575,13568,13569,13637],{},[1578,13570,13571,13587,13589,13591,13593,13595,13603,13605,13623,13629],{},[13572,13573,13574,13576,13585],"mpadded",{},[1584,13575,1791],{"mathvariant":1790},[1578,13577,13578,13580,13583],{},[1584,13579,12],{},[1584,13581,13582],{},"s",[1584,13584,2337],{},[1731,13586,1798],{"stretchy":1797},[1731,13588,623],{},[5211,13590,6057],{},[5211,13592,6060],{},[1584,13594,2837],{},[13572,13596,13597,13599,13601],{},[1584,13598,1791],{"mathvariant":1790},[1735,13600,791],{},[1731,13602,1798],{"stretchy":1797},[1731,13604,623],{},[1581,13606,13607,13609],{},[1584,13608,2330],{},[1578,13610,13611,13613,13615,13617,13619,13621],{},[1731,13612,1733],{},[1584,13614,2337],{},[1584,13616,2182],{},[1735,13618,791],{},[1584,13620,4003],{},[1584,13622,2342],{},[2212,13624,13625,13627],{},[1584,13626,2323],{},[1584,13628,2837],{},[13572,13630,13631,13633,13635],{},[1584,13632,1791],{"mathvariant":1790},[1735,13634,791],{},[1731,13636,1798],{"stretchy":1797},[1591,13638,13639],{"encoding":1593},"\\ket{psi} = \\text{Prep} \\; \\psi \\ket{0} = e^{-i H 0 dt} U_{\\psi} \\ket{0}",[40,13641,13643,13679,13724],{"className":13642,"ariaHidden":1599},[1598],[40,13644,13646,13649,13670,13673,13676],{"className":13645},[1603],[40,13647],{"className":13648,"style":1811},[1607],[40,13650,13652,13655,13667],{"className":13651},[1981],[40,13653,1791],{"className":13654},[1612],[40,13656,13658,13661,13664],{"className":13657},[1612],[40,13659,12],{"className":13660},[1612,1653],[40,13662,13582],{"className":13663},[1612,1653],[40,13665,2337],{"className":13666},[1612,1653],[40,13668,1798],{"className":13669},[1822],[40,13671],{"className":13672,"style":1963},[1756],[40,13674,623],{"className":13675},[1967],[40,13677],{"className":13678,"style":1963},[1756],[40,13680,13682,13685,13691,13694,13697,13700,13715,13718,13721],{"className":13681},[1603],[40,13683],{"className":13684,"style":1811},[1607],[40,13686,13688],{"className":13687},[1612,6083],[40,13689,6057],{"className":13690},[1612],[40,13692],{"className":13693,"style":1963},[1756],[40,13695,2837],{"className":13696,"style":1818},[1612,1653],[40,13698],{"className":13699,"style":2009},[1756],[40,13701,13703,13706,13712],{"className":13702},[1981],[40,13704,1791],{"className":13705},[1612],[40,13707,13709],{"className":13708},[1612],[40,13710,791],{"className":13711},[1612],[40,13713,1798],{"className":13714},[1822],[40,13716],{"className":13717,"style":1963},[1756],[40,13719,623],{"className":13720},[1967],[40,13722],{"className":13723,"style":1963},[1756],[40,13725,13727,13730,13777,13820,13823],{"className":13726},[1603],[40,13728],{"className":13729,"style":10403},[1607],[40,13731,13733,13736],{"className":13732},[1612],[40,13734,2330],{"className":13735},[1612,1653],[40,13737,13739],{"className":13738},[1621],[40,13740,13742],{"className":13741},[1625],[40,13743,13745],{"className":13744},[1629],[40,13746,13748],{"className":13747,"style":2706},[1633],[40,13749,13750,13753],{"style":1637},[40,13751],{"className":13752,"style":1642},[1641],[40,13754,13756],{"className":13755},[1646,1647,1648,1649],[40,13757,13759,13762,13765,13768,13771,13774],{"className":13758},[1612,1649],[40,13760,1733],{"className":13761},[1612,1649],[40,13763,2337],{"className":13764},[1612,1653,1649],[40,13766,2182],{"className":13767,"style":2197},[1612,1653,1649],[40,13769,791],{"className":13770},[1612,1649],[40,13772,4003],{"className":13773},[1612,1653,1649],[40,13775,2342],{"className":13776},[1612,1653,1649],[40,13778,13780,13783],{"className":13779},[1612],[40,13781,2323],{"className":13782,"style":2358},[1612,1653],[40,13784,13786],{"className":13785},[1621],[40,13787,13789,13812],{"className":13788},[1625,2244],[40,13790,13792,13809],{"className":13791},[1629],[40,13793,13795],{"className":13794,"style":10290},[1633],[40,13796,13797,13800],{"style":10471},[40,13798],{"className":13799,"style":1642},[1641],[40,13801,13803],{"className":13802},[1646,1647,1648,1649],[40,13804,13806],{"className":13805},[1612,1649],[40,13807,2837],{"className":13808,"style":1818},[1612,1653,1649],[40,13810,2269],{"className":13811},[2268],[40,13813,13815],{"className":13814},[1629],[40,13816,13818],{"className":13817,"style":5877},[1633],[40,13819],{},[40,13821],{"className":13822,"style":2009},[1756],[40,13824,13826,13829,13835],{"className":13825},[1981],[40,13827,1791],{"className":13828},[1612],[40,13830,13832],{"className":13831},[1612],[40,13833,791],{"className":13834},[1612],[40,13836,1798],{"className":13837},[1822],", for example, to prepare the HF state for chemistry ",[40,13840,13842,13859],{"className":13841},[1565],[40,13843,13845],{"className":13844},[1569],[1571,13846,13847],{"xmlns":1573},[1575,13848,13849,13857],{},[1578,13850,13851,13853,13855],{},[5211,13852,6057],{},[5211,13854,6060],{},[1584,13856,2837],{},[1591,13858,13536],{"encoding":1593},[40,13860,13862],{"className":13861,"ariaHidden":1599},[1598],[40,13863,13865,13868,13874,13877],{"className":13864},[1603],[40,13866],{"className":13867,"style":6079},[1607],[40,13869,13871],{"className":13870},[1612,6083],[40,13872,6057],{"className":13873},[1612],[40,13875],{"className":13876,"style":1963},[1756],[40,13878,2837],{"className":13879,"style":1818},[1612,1653]," would be a product of single-qubit NOTs, so controlled-",[40,13882,13884,13901],{"className":13883},[1565],[40,13885,13887],{"className":13886},[1569],[1571,13888,13889],{"xmlns":1573},[1575,13890,13891,13899],{},[1578,13892,13893,13895,13897],{},[5211,13894,6057],{},[5211,13896,6060],{},[1584,13898,2837],{},[1591,13900,13536],{"encoding":1593},[40,13902,13904],{"className":13903,"ariaHidden":1599},[1598],[40,13905,13907,13910,13916,13919],{"className":13906},[1603],[40,13908],{"className":13909,"style":6079},[1607],[40,13911,13913],{"className":13912},[1612,6083],[40,13914,6057],{"className":13915},[1612],[40,13917],{"className":13918,"style":1963},[1756],[40,13920,2837],{"className":13921,"style":1818},[1612,1653]," is just a product of CNOTs.\nThen the circuit above implements the following state prior to measurement:",[40,13924,13926],{"className":13925},[1827],[40,13927,13929,14549],{"className":13928},[1565],[40,13930,13932],{"className":13931},[1569],[1571,13933,13934],{"xmlns":1573,"display":1836},[1575,13935,13936,14546],{},[8499,13937,13938],{"rowspacing":8501,"columnspacing":8502},[8504,13939,13940,13944,14539,14542],{},[8507,13941],{"className":13942},[13943],"mtr-glue",[8507,13945,13946],{},[8510,13947,13948],{"scriptlevel":791,"displaystyle":1599},[8499,13949,13950,14054,14121,14208,14291,14409],{"rowspacing":11161,"columnalign":11162,"columnspacing":11163},[8504,13951,13952,13990],{},[8507,13953,13954],{},[8510,13955,13956],{"scriptlevel":791,"displaystyle":1599},[1578,13957,13958,13966,13978],{},[13572,13959,13960,13962,13964],{},[1584,13961,1791],{"mathvariant":1790},[1735,13963,791],{},[1731,13965,1798],{"stretchy":1797},[1581,13967,13968,13976],{},[13572,13969,13970,13972,13974],{},[1584,13971,1791],{"mathvariant":1790},[1735,13973,791],{},[1731,13975,1798],{"stretchy":1797},[1584,13977,6725],{},[3315,13979,13980,13984],{},[1731,13981,13983],{"stretchy":1599,"minsize":13982},"3.0em","→",[13572,13985,13988],{"width":13986,"lspace":13987},"+0.6em","0.3em",[1584,13989,2182],{},[8507,13991,13992],{},[8510,13993,13994],{"scriptlevel":791,"displaystyle":1599},[1578,13995,13996,13998,14006],{},[1578,13997],{},[8538,13999,14000,14002],{},[1735,14001,1410],{},[8543,14003,14004],{},[1735,14005,1879],{},[1578,14007,14008,14010,14018,14030,14032,14040,14052],{},[1731,14009,241],{"fence":1599},[13572,14011,14012,14014,14016],{},[1584,14013,1791],{"mathvariant":1790},[1735,14015,791],{},[1731,14017,1798],{"stretchy":1797},[1581,14019,14020,14028],{},[13572,14021,14022,14024,14026],{},[1584,14023,1791],{"mathvariant":1790},[1735,14025,791],{},[1731,14027,1798],{"stretchy":1797},[1584,14029,6725],{},[1731,14031,1147],{},[13572,14033,14034,14036,14038],{},[1584,14035,1791],{"mathvariant":1790},[1735,14037,1410],{},[1731,14039,1798],{"stretchy":1797},[1581,14041,14042,14050],{},[13572,14043,14044,14046,14048],{},[1584,14045,1791],{"mathvariant":1790},[1735,14047,791],{},[1731,14049,1798],{"stretchy":1797},[1584,14051,6725],{},[1731,14053,2693],{"fence":1599},[8504,14055,14056,14069],{},[8507,14057,14058],{},[8510,14059,14060],{"scriptlevel":791,"displaystyle":1599},[3315,14061,14062,14064],{},[1731,14063,13983],{"stretchy":1599,"minsize":13982},[13572,14065,14066],{"width":13986,"lspace":13987},[5211,14067,14068],{},"1-ctrl-init",[8507,14070,14071],{},[8510,14072,14073],{"scriptlevel":791,"displaystyle":1599},[1578,14074,14075,14077,14085],{},[1578,14076],{},[8538,14078,14079,14081],{},[1735,14080,1410],{},[8543,14082,14083],{},[1735,14084,1879],{},[1578,14086,14087,14089,14091,14093,14095,14097,14099,14105,14107,14109,14111,14113,14115,14117,14119],{},[1731,14088,241],{"fence":1599},[1584,14090,1791],{"mathvariant":1790},[1735,14092,791],{},[1731,14094,1798],{"stretchy":1797},[1584,14096,1791],{"mathvariant":1790},[1735,14098,791],{},[1581,14100,14101,14103],{},[1731,14102,1798],{"stretchy":1797},[1584,14104,6725],{},[1731,14106,1147],{},[1584,14108,1791],{"mathvariant":1790},[1735,14110,1410],{},[1731,14112,1798],{"stretchy":1797},[1584,14114,1791],{"mathvariant":1790},[1584,14116,2837],{},[1731,14118,1798],{"stretchy":1797},[1731,14120,2693],{"fence":1599},[8504,14122,14123,14135],{},[8507,14124,14125],{},[8510,14126,14127],{"scriptlevel":791,"displaystyle":1599},[3315,14128,14129,14131],{},[1731,14130,13983],{"stretchy":1599,"minsize":13982},[13572,14132,14133],{"width":13986,"lspace":13987},[1584,14134,2323],{},[8507,14136,14137],{},[8510,14138,14139],{"scriptlevel":791,"displaystyle":1599},[1578,14140,14141,14143,14151],{},[1578,14142],{},[8538,14144,14145,14147],{},[1735,14146,1410],{},[8543,14148,14149],{},[1735,14150,1879],{},[1578,14152,14153,14155,14166,14174,14186,14188,14196,14198,14206],{},[1731,14154,241],{"fence":1599},[1581,14156,14157,14159],{},[1584,14158,2330],{},[1578,14160,14161,14163],{},[1584,14162,2337],{},[1584,14164,14165],{},"ϕ",[13572,14167,14168,14170,14172],{},[1584,14169,1791],{"mathvariant":1790},[1735,14171,791],{},[1731,14173,1798],{"stretchy":1797},[1581,14175,14176,14184],{},[13572,14177,14178,14180,14182],{},[1584,14179,1791],{"mathvariant":1790},[1735,14181,791],{},[1731,14183,1798],{"stretchy":1797},[1584,14185,6725],{},[1731,14187,1147],{},[13572,14189,14190,14192,14194],{},[1584,14191,1791],{"mathvariant":1790},[1735,14193,1410],{},[1731,14195,1798],{"stretchy":1797},[1584,14197,2323],{},[13572,14199,14200,14202,14204],{},[1584,14201,1791],{"mathvariant":1790},[1584,14203,2837],{},[1731,14205,1798],{"stretchy":1797},[1731,14207,2693],{"fence":1599},[8504,14209,14210,14223],{},[8507,14211,14212],{},[8510,14213,14214],{"scriptlevel":791,"displaystyle":1599},[3315,14215,14216,14218],{},[1731,14217,13983],{"stretchy":1599,"minsize":13982},[13572,14219,14220],{"width":13986,"lspace":13987},[5211,14221,14222],{},"0-ctrl-init",[8507,14224,14225],{},[8510,14226,14227],{"scriptlevel":791,"displaystyle":1599},[1578,14228,14229,14231,14239],{},[1578,14230],{},[8538,14232,14233,14235],{},[1735,14234,1410],{},[8543,14236,14237],{},[1735,14238,1879],{},[1578,14240,14241,14243,14253,14261,14269,14271,14279,14281,14289],{},[1731,14242,241],{"fence":1599},[1581,14244,14245,14247],{},[1584,14246,2330],{},[1578,14248,14249,14251],{},[1584,14250,2337],{},[1584,14252,14165],{},[13572,14254,14255,14257,14259],{},[1584,14256,1791],{"mathvariant":1790},[1735,14258,791],{},[1731,14260,1798],{"stretchy":1797},[13572,14262,14263,14265,14267],{},[1584,14264,1791],{"mathvariant":1790},[1584,14266,2837],{},[1731,14268,1798],{"stretchy":1797},[1731,14270,1147],{},[13572,14272,14273,14275,14277],{},[1584,14274,1791],{"mathvariant":1790},[1735,14276,1410],{},[1731,14278,1798],{"stretchy":1797},[1584,14280,2323],{},[13572,14282,14283,14285,14287],{},[1584,14284,1791],{"mathvariant":1790},[1584,14286,2837],{},[1731,14288,1798],{"stretchy":1797},[1731,14290,2693],{"fence":1599},[8504,14292,14293,14299],{},[8507,14294,14295],{},[8510,14296,14297],{"scriptlevel":791,"displaystyle":1599},[1731,14298,623],{"lspace":11163,"rspace":11163},[8507,14300,14301],{},[8510,14302,14303],{"scriptlevel":791,"displaystyle":1599},[1578,14304,14305,14307,14313],{},[1578,14306],{},[8538,14308,14309,14311],{},[1735,14310,1410],{},[1735,14312,1879],{},[1578,14314,14315,14317,14325,14361,14363,14371,14407],{},[1731,14316,241],{"fence":1599},[13572,14318,14319,14321,14323],{},[1584,14320,1791],{"mathvariant":1790},[1731,14322,1147],{"lspace":11163,"rspace":11163},[1731,14324,1798],{"stretchy":1797},[1578,14326,14327,14329,14339,14347,14349,14351,14359],{},[1731,14328,241],{"fence":1599},[1581,14330,14331,14333],{},[1584,14332,2330],{},[1578,14334,14335,14337],{},[1584,14336,2337],{},[1584,14338,14165],{},[13572,14340,14341,14343,14345],{},[1584,14342,1791],{"mathvariant":1790},[1584,14344,2837],{},[1731,14346,1798],{"stretchy":1797},[1731,14348,1147],{},[1584,14350,2323],{},[13572,14352,14353,14355,14357],{},[1584,14354,1791],{"mathvariant":1790},[1584,14356,2837],{},[1731,14358,1798],{"stretchy":1797},[1731,14360,2693],{"fence":1599},[1731,14362,1147],{},[13572,14364,14365,14367,14369],{},[1584,14366,1791],{"mathvariant":1790},[1731,14368,1733],{"lspace":11163,"rspace":11163},[1731,14370,1798],{"stretchy":1797},[1578,14372,14373,14375,14385,14393,14395,14397,14405],{},[1731,14374,241],{"fence":1599},[1581,14376,14377,14379],{},[1584,14378,2330],{},[1578,14380,14381,14383],{},[1584,14382,2337],{},[1584,14384,14165],{},[13572,14386,14387,14389,14391],{},[1584,14388,1791],{"mathvariant":1790},[1584,14390,2837],{},[1731,14392,1798],{"stretchy":1797},[1731,14394,1733],{},[1584,14396,2323],{},[13572,14398,14399,14401,14403],{},[1584,14400,1791],{"mathvariant":1790},[1584,14402,2837],{},[1731,14404,1798],{"stretchy":1797},[1731,14406,2693],{"fence":1599},[1731,14408,2693],{"fence":1599},[8504,14410,14411,14417],{},[8507,14412,14413],{},[8510,14414,14415],{"scriptlevel":791,"displaystyle":1599},[1731,14416,623],{"lspace":11163,"rspace":11163},[8507,14418,14419],{},[8510,14420,14421],{"scriptlevel":791,"displaystyle":1599},[1578,14422,14423,14425,14431],{},[1578,14424],{},[8538,14426,14427,14429],{},[1735,14428,1410],{},[1735,14430,1879],{},[1578,14432,14433,14435,14447,14485,14487,14499,14537],{},[1731,14434,241],{"fence":1599},[13572,14436,14437,14439,14445],{},[1584,14438,1791],{"mathvariant":1790},[1578,14440,14441,14443],{},[1731,14442,1147],{},[1584,14444,2337],{},[1731,14446,1798],{"stretchy":1797},[1578,14448,14449,14451,14461,14469,14471,14473,14475,14483],{},[1731,14450,241],{"fence":1599},[1581,14452,14453,14455],{},[1584,14454,2330],{},[1578,14456,14457,14459],{},[1584,14458,2337],{},[1584,14460,14165],{},[13572,14462,14463,14465,14467],{},[1584,14464,1791],{"mathvariant":1790},[1584,14466,2837],{},[1731,14468,1798],{"stretchy":1797},[1731,14470,1733],{},[1584,14472,2337],{},[1584,14474,2323],{},[13572,14476,14477,14479,14481],{},[1584,14478,1791],{"mathvariant":1790},[1584,14480,2837],{},[1731,14482,1798],{"stretchy":1797},[1731,14484,2693],{"fence":1599},[1731,14486,1147],{},[13572,14488,14489,14491,14497],{},[1584,14490,1791],{"mathvariant":1790},[1578,14492,14493,14495],{},[1731,14494,1733],{},[1584,14496,2337],{},[1731,14498,1798],{"stretchy":1797},[1578,14500,14501,14503,14513,14521,14523,14525,14527,14535],{},[1731,14502,241],{"fence":1599},[1581,14504,14505,14507],{},[1584,14506,2330],{},[1578,14508,14509,14511],{},[1584,14510,2337],{},[1584,14512,14165],{},[13572,14514,14515,14517,14519],{},[1584,14516,1791],{"mathvariant":1790},[1584,14518,2837],{},[1731,14520,1798],{"stretchy":1797},[1731,14522,1147],{},[1584,14524,2337],{},[1584,14526,2323],{},[13572,14528,14529,14531,14533],{},[1584,14530,1791],{"mathvariant":1790},[1584,14532,2837],{},[1731,14534,1798],{"stretchy":1797},[1731,14536,2693],{"fence":1599},[1731,14538,2693],{"fence":1599},[8507,14540],{"className":14541},[13943],[8507,14543],{"className":14544},[14545],"mml-eqn-num",[1591,14547,14548],{"encoding":1593},"\\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}",[40,14550,14552,16657],{"className":14551,"ariaHidden":1599},[1598],[40,14553,14555,14559],{"className":14554},[1603],[40,14556],{"className":14557,"style":14558},[1607],"height:14.5206em;vertical-align:-7.0103em;",[40,14560,14562],{"className":14561},[8499],[40,14563,14565],{"className":14564},[8758],[40,14566,14568,16649],{"className":14567},[1625,2244],[40,14569,14571,16646],{"className":14570},[1629],[40,14572,14575],{"className":14573,"style":14574},[1633],"height:7.5103em;",[40,14576,14578,14582],{"style":14577},"top:-9.5103em;",[40,14579],{"className":14580,"style":14581},[1641],"height:9.5103em;",[40,14583,14585],{"className":14584},[1612],[40,14586,14588],{"className":14587},[1612],[40,14589,14591,14954],{"className":14590},[8499],[40,14592,14594],{"className":14593},[11374],[40,14595,14597,14945],{"className":14596},[1625,2244],[40,14598,14600,14942],{"className":14599},[1629],[40,14601,14603,14734,14797,14856,14918,14930],{"className":14602,"style":14574},[1633],[40,14604,14605,14608],{"style":14577},[40,14606],{"className":14607,"style":8775},[1641],[40,14609,14611,14626,14629,14670,14673],{"className":14610},[1612],[40,14612,14614,14617,14623],{"className":14613},[1981],[40,14615,1791],{"className":14616},[1612],[40,14618,14620],{"className":14619},[1612],[40,14621,791],{"className":14622},[1612],[40,14624,1798],{"className":14625},[1822],[40,14627],{"className":14628,"style":2009},[1756],[40,14630,14632,14647],{"className":14631},[1981],[40,14633,14635,14638,14644],{"className":14634},[1981],[40,14636,1791],{"className":14637},[1612],[40,14639,14641],{"className":14640},[1612],[40,14642,791],{"className":14643},[1612],[40,14645,1798],{"className":14646},[1822],[40,14648,14650],{"className":14649},[1621],[40,14651,14653],{"className":14652},[1625],[40,14654,14656],{"className":14655},[1629],[40,14657,14659],{"className":14658,"style":6866},[1633],[40,14660,14661,14664],{"style":6889},[40,14662],{"className":14663,"style":1642},[1641],[40,14665,14667],{"className":14666},[1646,1647,1648,1649],[40,14668,6725],{"className":14669,"style":2358},[1612,1653,1649],[40,14671],{"className":14672,"style":1963},[1756],[40,14674,14677],{"className":14675},[1967,14676],"x-arrow",[40,14678,14680,14725],{"className":14679},[1625,2244],[40,14681,14683,14722],{"className":14682},[1629],[40,14684,14687,14703],{"className":14685,"style":14686},[1633],"height:1.1003em;",[40,14688,14690,14693],{"style":14689},"top:-3.322em;",[40,14691],{"className":14692,"style":1642},[1641],[40,14694,14697],{"className":14695},[1646,1647,1648,1649,14696],"x-arrow-pad",[40,14698,14700],{"className":14699},[1612,1649],[40,14701,2182],{"className":14702,"style":2197},[1612,1653,1649],[40,14704,14707,14710],{"className":14705,"style":14706},[8880],"top:-2.689em;",[40,14708],{"className":14709,"style":1642},[1641],[40,14711,14714],{"className":14712,"style":14713},[8900],"height:0.522em;min-width:1.469em;",[5325,14715,14719],{"xmlns":5327,"width":8904,"height":14716,"viewBox":14717,"preserveAspectRatio":14718},"0.522em","0 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we have used the classical simulable phase shift ",[40,16694,16696,16746],{"className":16695},[1565],[40,16697,16699],{"className":16698},[1569],[1571,16700,16701],{"xmlns":1573},[1575,16702,16703,16743],{},[1578,16704,16705,16707,16719,16721,16731],{},[1584,16706,2323],{},[1581,16708,16709,16717],{},[13572,16710,16711,16713,16715],{},[1584,16712,1791],{"mathvariant":1790},[1735,16714,791],{},[1731,16716,1798],{"stretchy":1797},[1584,16718,6725],{},[1731,16720,623],{},[1581,16722,16723,16725],{},[1584,16724,2330],{},[1578,16726,16727,16729],{},[1584,16728,2337],{},[1584,16730,14165],{},[1581,16732,16733,16741],{},[13572,16734,16735,16737,16739],{},[1584,16736,1791],{"mathvariant":1790},[1735,16738,791],{},[1731,16740,1798],{"stretchy":1797},[1584,16742,6725],{},[1591,16744,16745],{"encoding":1593}," U\\ket{0}^N = e^{i\\phi}\\ket{0}^N",[40,16747,16749,16812],{"className":16748,"ariaHidden":1599},[1598],[40,16750,16752,16756,16759,16762,16803,16806,16809],{"className":16751},[1603],[40,16753],{"className":16754,"style":16755},[1607],"height:1.2312em;vertical-align:-0.25em;",[40,16757,2323],{"className":16758,"style":2358},[1612,1653],[40,16760],{"className":16761,"style":2009},[1756],[40,16763,16765,16780],{"className":16764},[1981],[40,16766,16768,16771,16777],{"className":16767},[1981],[40,16769,1791],{"className":16770},[1612],[40,16772,16774],{"className":16773},[1612],[40,16775,791],{"className":16776},[1612],[40,16778,1798],{"className":16779},[1822],[40,16781,16783],{"className":16782},[1621],[40,16784,16786],{"className":16785},[1625],[40,16787,16789],{"className":16788},[1629],[40,16790,16792],{"className":16791,"style":6866},[1633],[40,16793,16794,16797],{"style":6889},[40,16795],{"className":16796,"style":1642},[1641],[40,16798,16800],{"className":16799},[1646,1647,1648,1649],[40,16801,6725],{"className":16802,"style":2358},[1612,1653,1649],[40,16804],{"className":16805,"style":1963},[1756],[40,16807,623],{"className":16808},[1967],[40,16810],{"className":16811,"style":1963},[1756],[40,16813,16815,16818,16853,16856],{"className":16814},[1603],[40,16816],{"className":16817,"style":16755},[1607],[40,16819,16821,16824],{"className":16820},[1612],[40,16822,2330],{"className":16823},[1612,1653],[40,16825,16827],{"className":16826},[1621],[40,16828,16830],{"className":16829},[1625],[40,16831,16833],{"className":16832},[1629],[40,16834,16836],{"className":16835,"style":2706},[1633],[40,16837,16838,16841],{"style":1637},[40,16839],{"className":16840,"style":1642},[1641],[40,16842,16844],{"className":16843},[1646,1647,1648,1649],[40,16845,16847,16850],{"className":16846},[1612,1649],[40,16848,2337],{"className":16849},[1612,1653,1649],[40,16851,14165],{"className":16852},[1612,1653,1649],[40,16854],{"className":16855,"style":2009},[1756],[40,16857,16859,16874],{"className":16858},[1981],[40,16860,16862,16865,16871],{"className":16861},[1981],[40,16863,1791],{"className":16864},[1612],[40,16866,16868],{"className":16867},[1612],[40,16869,791],{"className":16870},[1612],[40,16872,1798],{"className":16873},[1822],[40,16875,16877],{"className":16876},[1621],[40,16878,16880],{"className":16879},[1625],[40,16881,16883],{"className":16882},[1629],[40,16884,16886],{"className":16885,"style":6866},[1633],[40,16887,16888,16891],{"style":6889},[40,16889],{"className":16890,"style":1642},[1641],[40,16892,16894],{"className":16893},[1646,1647,1648,1649],[40,16895,6725],{"className":16896,"style":2358},[1612,1653,1649]," in the third line. Therefore the expectation values are obtained as",[40,16899,16901],{"className":16900},[1827],[40,16902,16904,17214],{"className":16903},[1565],[40,16905,16907],{"className":16906},[1569],[1571,16908,16909],{"xmlns":1573,"display":1836},[1575,16910,16911,17211],{},[8499,16912,16913],{"rowspacing":8501,"columnspacing":8502},[8504,16914,16915,16918,17205,17208],{},[8507,16916],{"className":16917},[13943],[8507,16919,16920],{},[8510,16921,16922],{"scriptlevel":791,"displaystyle":1599},[8499,16923,16924,17042,17145],{"rowspacing":11161,"columnalign":11162,"columnspacing":11163},[8504,16925,16926,16943],{},[8507,16927,16928],{},[8510,16929,16930],{"scriptlevel":791,"displaystyle":1599},[1578,16931,16932,16934,16936,16939,16941],{},[1731,16933,3469],{"stretchy":1797},[1584,16935,6775],{},[1731,16937,16938],{},"⊗",[1584,16940,2218],{},[1731,16942,1798],{"stretchy":1797},[8507,16944,16945],{},[8510,16946,16947],{"scriptlevel":791,"displaystyle":1599},[1578,16948,16949,16951,16953,16959,16961,17004,17006],{},[1578,16950],{},[1731,16952,623],{},[8538,16954,16955,16957],{},[1735,16956,1410],{},[1735,16958,11199],{},[1731,16960,241],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[1578,16962,16963,16965,16977,16985,16987,16995,17002],{},[1731,16964,241],{"fence":1599},[1581,16966,16967,16969],{},[1584,16968,2330],{},[1578,16970,16971,16973,16975],{},[1731,16972,1733],{},[1584,16974,2337],{},[1584,16976,14165],{},[13572,16978,16979,16981,16983],{},[1731,16980,3469],{"stretchy":1797},[1584,16982,2837],{},[1584,16984,1791],{"mathvariant":1790},[1731,16986,1147],{},[13572,16988,16989,16991,16993],{},[1731,16990,3469],{"stretchy":1797},[1584,16992,2837],{},[1584,16994,1791],{"mathvariant":1790},[1581,16996,16997,16999],{},[1584,16998,2323],{},[1731,17000,17001],{},"†",[1731,17003,2693],{"fence":1599},[1584,17005,2218],{},[1578,17007,17008,17010,17020,17028,17030,17032,17040],{},[1731,17009,241],{"fence":1599},[1581,17011,17012,17014],{},[1584,17013,2330],{},[1578,17015,17016,17018],{},[1584,17017,2337],{},[1584,17019,14165],{},[13572,17021,17022,17024,17026],{},[1584,17023,1791],{"mathvariant":1790},[1584,17025,2837],{},[1731,17027,1798],{"stretchy":1797},[1731,17029,1147],{},[1584,17031,2323],{},[13572,17033,17034,17036,17038],{},[1584,17035,1791],{"mathvariant":1790},[1584,17037,2837],{},[1731,17039,1798],{"stretchy":1797},[1731,17041,2693],{"fence":1599},[8504,17043,17044,17050],{},[8507,17045,17046],{},[8510,17047,17048],{"scriptlevel":791,"displaystyle":1599},[1578,17049],{},[8507,17051,17052],{},[8510,17053,17054],{"scriptlevel":791,"displaystyle":1599},[1578,17055,17056,17058,17061,17063,17105,17107,17143],{},[1578,17057],{},[1756,17059],{"width":17060},"2em",[1731,17062,1733],{},[1578,17064,17065,17067,17079,17087,17089,17097,17103],{},[1731,17066,241],{"fence":1599},[1581,17068,17069,17071],{},[1584,17070,2330],{},[1578,17072,17073,17075,17077],{},[1731,17074,1733],{},[1584,17076,2337],{},[1584,17078,14165],{},[13572,17080,17081,17083,17085],{},[1731,17082,3469],{"stretchy":1797},[1584,17084,2837],{},[1584,17086,1791],{"mathvariant":1790},[1731,17088,1733],{},[13572,17090,17091,17093,17095],{},[1731,17092,3469],{"stretchy":1797},[1584,17094,2837],{},[1584,17096,1791],{"mathvariant":1790},[1581,17098,17099,17101],{},[1584,17100,2323],{},[1731,17102,17001],{},[1731,17104,2693],{"fence":1599},[1584,17106,2218],{},[1578,17108,17109,17111,17121,17129,17131,17133,17141],{},[1731,17110,241],{"fence":1599},[1581,17112,17113,17115],{},[1584,17114,2330],{},[1578,17116,17117,17119],{},[1584,17118,2337],{},[1584,17120,14165],{},[13572,17122,17123,17125,17127],{},[1584,17124,1791],{"mathvariant":1790},[1584,17126,2837],{},[1731,17128,1798],{"stretchy":1797},[1731,17130,1733],{},[1584,17132,2323],{},[13572,17134,17135,17137,17139],{},[1584,17136,1791],{"mathvariant":1790},[1584,17138,2837],{},[1731,17140,1798],{"stretchy":1797},[1731,17142,2693],{"fence":1599},[1731,17144,2693],{"fence":1797,"stretchy":1599,"minsize":8549,"maxsize":8549},[8504,17146,17147,17153],{},[8507,17148,17149],{},[8510,17150,17151],{"scriptlevel":791,"displaystyle":1599},[1578,17152],{},[8507,17154,17155],{},[8510,17156,17157],{"scriptlevel":791,"displaystyle":1599},[1578,17158,17159,17161,17163,17165,17203],{},[1578,17160],{},[1731,17162,623],{},[5211,17164,11292],{},[1578,17166,17167,17169,17181,17189,17191,17193,17201],{},[1731,17168,11295],{"fence":1599},[1581,17170,17171,17173],{},[1584,17172,2330],{},[1578,17174,17175,17177,17179],{},[1731,17176,1733],{},[1584,17178,2337],{},[1584,17180,14165],{},[13572,17182,17183,17185,17187],{},[1731,17184,3469],{"stretchy":1797},[1584,17186,2837],{},[1584,17188,1791],{"mathvariant":1790},[1584,17190,2218],{},[1584,17192,2323],{},[13572,17194,17195,17197,17199],{},[1584,17196,1791],{"mathvariant":1790},[1584,17198,2837],{},[1731,17200,1798],{"stretchy":1797},[1731,17202,11320],{"fence":1599},[1731,17204,1862],{"separator":1599},[8507,17206],{"className":17207},[13943],[8507,17209],{"className":17210},[14545],[1591,17212,17213],{"encoding":1593},"\\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}",[40,17215,17217,18086],{"className":17216,"ariaHidden":1599},[1598],[40,17218,17220,17224],{"className":17219},[1603],[40,17221],{"className":17222,"style":17223},[1607],"height:5.6666em;vertical-align:-2.5833em;",[40,17225,17227],{"className":17226},[8499],[40,17228,17230],{"className":17229},[8758],[40,17231,17233,18078],{"className":17232},[1625,2244],[40,17234,17236,18075],{"className":17235},[1629],[40,17237,17240],{"className":17238,"style":17239},[1633],"height:3.0833em;",[40,17241,17243,17247],{"style":17242},"top:-5.0833em;",[40,17244],{"className":17245,"style":17246},[1641],"height:5.0833em;",[40,17248,17250],{"className":17249},[1612],[40,17251,17253],{"className":17252},[1612],[40,17254,17256,17327],{"className":17255},[8499],[40,17257,17259],{"className":17258},[11374],[40,17260,17262,17318],{"className":17261},[1625,2244],[40,17263,17265,17315],{"className":17264},[1629],[40,17266,17268,17297,17306],{"className":17267,"style":17239},[1633],[40,17269,17270,17273],{"style":17242},[40,17271],{"className":17272,"style":8775},[1641],[40,17274,17276,17279,17282,17285,17288,17291,17294],{"className":17275},[1612],[40,17277,3469],{"className":17278},[1985],[40,17280,6775],{"className":17281,"style":6920},[1612,1653],[40,17283],{"className":17284,"style":1757},[1756],[40,17286,16938],{"className":17287},[1761],[40,17289],{"className":17290,"style":1757},[1756],[40,17292,2218],{"className":17293,"style":2264},[1612,1653],[40,17295,1798],{"className":17296},[1822],[40,17298,17300,17303],{"style":17299},"top:-2.9473em;",[40,17301],{"className":17302,"style":8775},[1641],[40,17304],{"className":17305},[1612],[40,17307,17309,17312],{"style":17308},"top:-1.0982em;",[40,17310],{"className":17311,"style":8775},[1641],[40,17313],{"className":17314},[1612],[40,17316,2269],{"className":17317},[2268],[40,17319,17321],{"className":17320},[1629],[40,17322,17325],{"className":17323,"style":17324},[1633],"height:2.5833em;",[40,17326],{},[40,17328,17330],{"className":17329},[11435],[40,17331,17333,18067],{"className":17332},[1625,2244],[40,17334,17336,18064],{"className":17335},[1629],[40,17337,17339,17664,17931],{"className":17338,"style":17239},[1633],[40,17340,17341,17344],{"style":17242},[40,17342],{"className":17343,"style":8775},[1641],[40,17345,17347,17350,17353,17356,17359,17421,17427,17430,17557,17560,17563,17566],{"className":17346},[1612],[40,17348],{"className":17349},[1612],[40,17351],{"className":17352,"style":1963},[1756],[40,17354,623],{"className":17355},[1967],[40,17357],{"className":17358,"style":1963},[1756],[40,17360,17362,17365,17418],{"className":17361},[1612],[40,17363],{"className":17364},[1985,8840],[40,17366,17368],{"className":17367},[8538],[40,17369,17371,17410],{"className":17370},[1625,2244],[40,17372,17374,17407],{"className":17373},[1629],[40,17375,17377,17388,17396],{"className":17376,"style":8853},[1633],[40,17378,17379,17382],{"style":10764},[40,17380],{"className":17381,"style":3354},[1641],[40,17383,17385],{"className":17384},[1612],[40,17386,11199],{"className":17387},[1612],[40,17389,17390,17393],{"style":8925},[40,17391],{"className":17392,"style":3354},[1641],[40,17394],{"className":17395,"style":8933},[8932],[40,17397,17398,17401],{"style":8936},[40,17399],{"className":17400,"style":3354},[1641],[40,17402,17404],{"className":17403},[1612],[40,17405,1410],{"className":17406},[1612],[40,17408,2269],{"className":17409},[2268],[40,17411,17413],{"className":17412},[1629],[40,17414,17416],{"className":17415,"style":10802},[1633],[40,17417],{},[40,17419],{"className":17420},[1822,8840],[40,17422,17424],{"className":17423},[1612],[40,17425,241],{"className":17426},[1991,8967],[40,17428],{"className":17429,"style":2009},[1756],[40,17431,17433,17439,17477,17480,17495,17498,17501,17504,17519,17522,17551],{"className":17432},[1981],[40,17434,17436],{"className":17435,"style":1987},[1985,1986],[40,17437,241],{"className":17438},[1991,1992],[40,17440,17442,17445],{"className":17441},[1612],[40,17443,2330],{"className":17444},[1612,1653],[40,17446,17448],{"className":17447},[1621],[40,17449,17451],{"className":17450},[1625],[40,17452,17454],{"className":17453},[1629],[40,17455,17457],{"className":17456,"style":4084},[1633],[40,17458,17459,17462],{"style":1950},[40,17460],{"className":17461,"style":1642},[1641],[40,17463,17465],{"className":17464},[1646,1647,1648,1649],[40,17466,17468,17471,17474],{"className":17467},[1612,1649],[40,17469,1733],{"className":17470},[1612,1649],[40,17472,2337],{"className":17473},[1612,1653,1649],[40,17475,14165],{"className":17476},[1612,1653,1649],[40,17478],{"className":17479,"style":2009},[1756],[40,17481,17483,17486,17492],{"className":17482},[1981],[40,17484,3469],{"className":17485},[1985],[40,17487,17489],{"className":17488},[1612],[40,17490,2837],{"className":17491,"style":1818},[1612,1653],[40,17493,1791],{"className":17494},[1612],[40,17496],{"className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splaystyle":1599},[1578,18382,18383,18385,18387,18389,18427],{},[1578,18384],{},[1731,18386,623],{},[5211,18388,12551],{},[1578,18390,18391,18393,18405,18413,18415,18417,18425],{},[1731,18392,11295],{"fence":1599},[1581,18394,18395,18397],{},[1584,18396,2330],{},[1578,18398,18399,18401,18403],{},[1731,18400,1733],{},[1584,18402,2337],{},[1584,18404,14165],{},[13572,18406,18407,18409,18411],{},[1731,18408,3469],{"stretchy":1797},[1584,18410,2837],{},[1584,18412,1791],{"mathvariant":1790},[1584,18414,2218],{},[1584,18416,2323],{},[13572,18418,18419,18421,18423],{},[1584,18420,1791],{"mathvariant":1790},[1584,18422,2837],{},[1731,18424,1798],{"stretchy":1797},[1731,18426,11320],{"fence":1599},[1584,18428,1823],{"mathvariant":1790},[8507,18430],{"className":18431},[13943],[8507,18433],{"className":18434},[14545],[1591,18436,18437],{"encoding":1593},"\\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}",[40,18439,18441,19320],{"className":18440,"ariaHidden":1599},[1598],[40,18442,18444,18447],{"className":18443},[1603],[40,18445],{"className":18446,"style":17223},[1607],[40,18448,18450],{"className":18449},[8499],[40,18451,18453],{"className":18452},[8758],[40,18454,18456,19312],{"className":18455},[1625,2244],[40,18457,18459,19309],{"className":18458},[1629],[40,18460,18462],{"className":18461,"style":17239},[1633],[40,18463,18464,18467],{"style":17242},[40,18465],{"className":18466,"style":17246},[1641],[40,18468,18470],{"className":18469},[1612],[40,18471,18473],{"className":18472},[1612],[40,18474,18476,18544],{"className":18475},[8499],[40,18477,18479],{"className":18478},[11374],[40,18480,18482,18536],{"className":18481},[1625,2244],[40,18483,18485,18533],{"className":18484},[1629],[40,18486,18488,18517,18525],{"className":18487,"style":17239},[1633],[40,18489,18490,18493],{"style":17242},[40,18491],{"className":18492,"style":8775},[1641],[40,18494,18496,18499,18502,18505,18508,18511,18514],{"className":18495},[1612],[40,18497,3469],{"className":18498},[1985],[40,18500,6790],{"className":18501,"style":1757},[1612,1653],[40,18503],{"className":18504,"style":1757},[1756],[40,18506,16938],{"className":18507},[1761],[40,18509],{"className":18510,"style":1757},[1756],[40,18512,2218],{"className":18513,"style":2264},[1612,1653],[40,18515,1798],{"className":18516},[1822],[40,18518,18519,18522],{"style":17299},[40,18520],{"className":18521,"style":8775},[1641],[40,18523],{"className":18524},[1612],[40,18526,18527,18530],{"style":17308},[40,18528],{"className":18529,"style":8775},[1641],[40,18531],{"className":18532},[1612],[40,18534,2269],{"className":18535},[2268],[40,18537,18539],{"className":18538},[1629],[40,18540,18542],{"className":18541,"style":17324},[1633],[40,18543],{},[40,18545,18547],{"className":18546},[11435],[40,18548,18550,19301],{"className":18549},[1625,2244],[40,18551,18553,19298],{"className":18552},[1629],[40,18554,18556,18890,19165],{"className":18555,"style":17239},[1633],[40,18557,18558,18561],{"style":17242},[40,18559],{"className":18560,"style":8775},[1641],[40,18562,18564,18567,18570,18573,18576,18638,18644,18647,18780,18783,18786,18789],{"className":18563},[1612],[40,18565],{"className":18566},[1612],[40,18568],{"className":18569,"style":1963},[1756],[40,18571,623],{"className":18572},[1967],[40,18574],{"className":18575,"style":1963},[1756],[40,18577,18579,18582,18635],{"className":18578},[1612],[40,18580],{"className":18581},[1985,8840],[40,18583,18585],{"className":18584},[8538],[40,18586,18588,18627],{"className":18587},[1625,2244],[40,18589,18591,18624],{"className":18590},[1629],[40,18592,18594,18605,18613],{"className":18593,"style":8853},[1633],[40,18595,18596,18599],{"style":10764},[40,18597],{"className":18598,"style":3354},[1641],[40,18600,18602],{"className":18601},[1612],[40,18603,11199],{"className":18604},[1612],[40,18606,18607,18610],{"style":8925},[40,18608],{"className":18609,"style":3354},[1641],[40,18611],{"className":18612,"style":8933},[8932],[40,18614,18615,18618],{"style":8936},[40,18616],{"className":18617,"style":3354},[1641],[40,18619,18621],{"className":18620},[1612],[40,18622,1410],{"className":18623},[1612],[40,18625,2269],{"className":18626},[2268],[40,18628,18630],{"className":18629},[1629],[40,18631,18633],{"className":18632,"style":10802},[1633],[40,18634],{},[40,18636],{"className":18637},[1822,8840],[40,18639,18641],{"className":18640},[1612],[40,18642,241],{"className":18643},[1991,8967],[40,18645],{"className":18646,"style":2009},[1756],[40,18648,18650,18656,18694,18697,18712,18715,18718,18721,18724,18727,18742,18745,18774],{"className":18649},[1981],[40,18651,18653],{"className":18652,"style":1987},[1985,1986],[40,18654,241],{"className":18655},[1991,1992],[40,18657,18659,18662],{"className":18658},[1612],[40,18660,2330],{"className":18661},[1612,1653],[40,18663,18665],{"className":18664},[1621],[40,18666,18668],{"className":18667},[1625],[40,18669,18671],{"className":18670},[1629],[40,18672,18674],{"className":18673,"style":4084},[1633],[40,18675,18676,18679],{"style":1950},[40,18677],{"className":18678,"style":1642},[1641],[40,18680,18682],{"className":18681},[1646,1647,1648,1649],[40,18683,18685,18688,18691],{"className":18684},[1612,1649],[40,18686,1733],{"className":18687},[1612,1649],[40,18689,2337],{"className":18690},[1612,1653,1649],[40,18692,14165],{"className":18693},[1612,1653,1649],[40,18695],{"className":18696,"style":2009},[1756],[40,18698,18700,18703,18709],{"className":18699},[1981],[40,18701,3469],{"className":18702},[1985],[40,18704,18706],{"className":18705},[1612],[40,18707,2837],{"className":18708,"style":1818},[1612,1653],[40,18710,1791],{"className":18711},[1612],[40,18713],{"className":18714,"style":1757},[1756],[40,18716,1147],{"className":18717},[1761],[40,18719],{"className":18720,"style":1757},[1756],[40,18722,2337],{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},[1991,1992],[40,19293],{"className":19294,"style":2009},[1756],[40,19296,1823],{"className":19297},[1612],[40,19299,2269],{"className":19300},[2268],[40,19302,19304],{"className":19303},[1629],[40,19305,19307],{"className":19306,"style":17324},[1633],[40,19308],{},[40,19310,2269],{"className":19311},[2268],[40,19313,19315],{"className":19314},[1629],[40,19316,19318],{"className":19317,"style":17324},[1633],[40,19319],{},[40,19321,19323],{"className":19322},[16660],[40,19324,19326,19343],{"className":19325},[1625,2244],[40,19327,19329,19340],{"className":19328},[1629],[40,19330,19332],{"className":19331,"style":17239},[1633],[40,19333,19334,19337],{"style":17242},[40,19335],{"className":19336,"style":17246},[1641],[40,19338],{"className":19339},[16678],[40,19341,2269],{"className":19342},[2268],[40,19344,19346],{"className":19345},[1629],[40,19347,19349],{"className":19348,"style":17324},[1633],[40,19350],{},[12,19352,19353,19354,19395],{},"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 ",[40,19355,19357,19374],{"className":19356},[1565],[40,19358,19360],{"className":19359},[1569],[1571,19361,19362],{"xmlns":1573},[1575,19363,19364,19372],{},[1578,19365,19366,19368,19370],{},[5211,19367,6057],{},[5211,19369,6060],{},[1584,19371,2837],{},[1591,19373,13536],{"encoding":1593},[40,19375,19377],{"className":19376,"ariaHidden":1599},[1598],[40,19378,19380,19383,19389,19392],{"className":19379},[1603],[40,19381],{"className":19382,"style":6079},[1607],[40,19384,19386],{"className":19385},[1612,6083],[40,19387,6057],{"className":19388},[1612],[40,19390],{"className":19391,"style":1963},[1756],[40,19393,2837],{"className":19394,"style":1818},[1612,1653]," and not controlled time evolutions. Reframing our calculation as above will allow us to greatly reduce the depth of the resulting circuits.",[1555,19397,19399],{"id":19398},"decompose-time-evolution-operator-with-trotter-decomposition","Decompose time-evolution operator with Trotter decomposition",[12,19401,19402,19403,917,19484,917,19565,19647,19648,19677,19678,19802,19803,19860,19861,19894,19895,19944,19945,19948],{},"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 ",[40,19404,19406,19429],{"className":19405},[1565],[40,19407,19409],{"className":19408},[1569],[1571,19410,19411],{"xmlns":1573},[1575,19412,19413,19426],{},[1578,19414,19415],{},[2212,19416,19417,19420],{},[1584,19418,19419],{},"R",[1578,19421,19422,19424],{},[1584,19423,8185],{},[1584,19425,8185],{},[1591,19427,19428],{"encoding":1593},"R_{xx}",[40,19430,19432],{"className":19431,"ariaHidden":1599},[1598],[40,19433,19435,19438],{"className":19434},[1603],[40,19436],{"className":19437,"style":2231},[1607],[40,19439,19441,19445],{"className":19440},[1612],[40,19442,19419],{"className":19443,"style":19444},[1612,1653],"margin-right:0.0077em;",[40,19446,19448],{"className":19447},[1621],[40,19449,19451,19476],{"className":19450},[1625,2244],[40,19452,19454,19473],{"className":19453},[1629],[40,19455,19457],{"className":19456,"style":3551},[1633],[40,19458,19460,19463],{"style":19459},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[40,19461],{"className":19462,"style":1642},[1641],[40,19464,19466],{"className":19465},[1646,1647,1648,1649],[40,19467,19469],{"className":19468},[1612,1649],[40,19470,19472],{"className":19471},[1612,1653,1649],"xx",[40,19474,2269],{"className":19475},[2268],[40,19477,19479],{"className":19478},[1629],[40,19480,19482],{"className":19481,"style":2276},[1633],[40,19483],{},[40,19485,19487,19510],{"className":19486},[1565],[40,19488,19490],{"className":19489},[1569],[1571,19491,19492],{"xmlns":1573},[1575,19493,19494,19507],{},[1578,19495,19496],{},[2212,19497,19498,19500],{},[1584,19499,19419],{},[1578,19501,19502,19505],{},[1584,19503,19504],{},"y",[1584,19506,19504],{},[1591,19508,19509],{"encoding":1593},"R_{yy}",[40,19511,19513],{"className":19512,"ariaHidden":1599},[1598],[40,19514,19516,19519],{"className":19515},[1603],[40,19517],{"className":19518,"style":7011},[1607],[40,19520,19522,19525],{"className":19521},[1612],[40,19523,19419],{"className":19524,"style":19444},[1612,1653],[40,19526,19528],{"className":19527},[1621],[40,19529,19531,19557],{"className":19530},[1625,2244],[40,19532,19534,19554],{"className":19533},[1629],[40,19535,19537],{"className":19536,"style":3551},[1633],[40,19538,19539,19542],{"style":19459},[40,19540],{"className":19541,"style":1642},[1641],[40,19543,19545],{"className":19544},[1646,1647,1648,1649],[40,19546,19548,19551],{"className":19547},[1612,1649],[40,19549,19504],{"className":19550,"style":1818},[1612,1653,1649],[40,19552,19504],{"className":19553,"style":1818},[1612,1653,1649],[40,19555,2269],{"className":19556},[2268],[40,19558,19560],{"className":19559},[1629],[40,19561,19563],{"className":19562,"style":5877},[1633],[40,19564],{},[40,19566,19568,19591],{"className":19567},[1565],[40,19569,19571],{"className":19570},[1569],[1571,19572,19573],{"xmlns":1573},[1575,19574,19575,19588],{},[1578,19576,19577],{},[2212,19578,19579,19581],{},[1584,19580,19419],{},[1578,19582,19583,19586],{},[1584,19584,19585],{},"z",[1584,19587,19585],{},[1591,19589,19590],{"encoding":1593},"R_{zz}",[40,19592,19594],{"className":19593,"ariaHidden":1599},[1598],[40,19595,19597,19600],{"className":19596},[1603],[40,19598],{"className":19599,"style":2231},[1607],[40,19601,19603,19606],{"className":19602},[1612],[40,19604,19419],{"className":19605,"style":19444},[1612,1653],[40,19607,19609],{"className":19608},[1621],[40,19610,19612,19639],{"className":19611},[1625,2244],[40,19613,19615,19636],{"className":19614},[1629],[40,19616,19618],{"className":19617,"style":3551},[1633],[40,19619,19620,19623],{"style":19459},[40,19621],{"className":19622,"style":1642},[1641],[40,19624,19626],{"className":19625},[1646,1647,1648,1649],[40,19627,19629,19633],{"className":19628},[1612,1649],[40,19630,19585],{"className":19631,"style":19632},[1612,1653,1649],"margin-right:0.044em;",[40,19634,19585],{"className":19635,"style":19632},[1612,1653,1649],[40,19637,2269],{"className":19638},[2268],[40,19640,19642],{"className":19641},[1629],[40,19643,19645],{"className":19644,"style":2276},[1633],[40,19646],{}," with a parametrized angle ",[40,19649,19651,19664],{"className":19650},[1565],[40,19652,19654],{"className":19653},[1569],[1571,19655,19656],{"xmlns":1573},[1575,19657,19658,19662],{},[1578,19659,19660],{},[1584,19661,2342],{},[1591,19663,2342],{"encoding":1593},[40,19665,19667],{"className":19666,"ariaHidden":1599},[1598],[40,19668,19670,19674],{"className":19669},[1603],[40,19671],{"className":19672,"style":19673},[1607],"height:0.6151em;",[40,19675,2342],{"className":19676},[1612,1653]," which correspond to the approximate implementation of ",[40,19679,19681,19725],{"className":19680},[1565],[40,19682,19684],{"className":19683},[1569],[1571,19685,19686],{"xmlns":1573},[1575,19687,19688,19722],{},[1578,19689,19690],{},[1581,19691,19692,19694],{},[1584,19693,2330],{},[1578,19695,19696,19698,19700,19702,19704,19706,19708,19710,19712,19714,19716,19718,19720],{},[1731,19697,1733],{},[1584,19699,2337],{},[1731,19701,241],{"stretchy":1797},[1584,19703,6775],{},[1584,19705,6775],{},[1731,19707,1147],{},[1584,19709,6790],{},[1584,19711,6790],{},[1731,19713,1147],{},[1584,19715,6808],{},[1584,19717,6808],{},[1731,19719,2693],{"stretchy":1797},[1584,19721,2342],{},[1591,19723,19724],{"encoding":1593},"e^{-i (XX + YY + ZZ) t}",[40,19726,19728],{"className":19727,"ariaHidden":1599},[1598],[40,19729,19731,19734],{"className":19730},[1603],[40,19732],{"className":19733,"style":2752},[1607],[40,19735,19737,19740],{"className":19736},[1612],[40,19738,2330],{"className":19739},[1612,1653],[40,19741,19743],{"className":19742},[1621],[40,19744,19746],{"className":19745},[1625],[40,19747,19749],{"className":19748},[1629],[40,19750,19752],{"className":19751,"style":2752},[1633],[40,19753,19754,19757],{"style":1637},[40,19755],{"className":19756,"style":1642},[1641],[40,19758,19760],{"className":19759},[1646,1647,1648,1649],[40,19761,19763,19766,19769,19772,19775,19778,19781,19784,19787,19790,19793,19796,19799],{"className":19762},[1612,1649],[40,19764,1733],{"className":19765},[1612,1649],[40,19767,2337],{"className":19768},[1612,1653,1649],[40,19770,241],{"className":19771},[1985,1649],[40,19773,6775],{"className":19774,"style":6920},[1612,1653,1649],[40,19776,6775],{"className":19777,"style":6920},[1612,1653,1649],[40,19779,1147],{"className":19780},[1761,1649],[40,19782,6790],{"className":19783,"style":1757},[1612,1653,1649],[40,19785,6790],{"className":19786,"style":1757},[1612,1653,1649],[40,19788,1147],{"className":19789},[1761,1649],[40,19791,6808],{"className":19792,"style":7118},[1612,1653,1649],[40,19794,6808],{"className":19795,"style":7118},[1612,1653,1649],[40,19797,2693],{"className":19798},[1822,1649],[40,19800,2342],{"className":19801},[1612,1653,1649],". 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 ",[40,19804,19806,19827],{"className":19805},[1565],[40,19807,19809],{"className":19808},[1569],[1571,19810,19811],{"xmlns":1573},[1575,19812,19813,19824],{},[1578,19814,19815,19817,19820,19822],{},[1735,19816,1879],{},[1731,19818,19819],{},"∗",[1584,19821,4003],{},[1584,19823,2342],{},[1591,19825,19826],{"encoding":1593},"2*dt",[40,19828,19830,19848],{"className":19829,"ariaHidden":1599},[1598],[40,19831,19833,19836,19839,19842,19845],{"className":19832},[1603],[40,19834],{"className":19835,"style":1771},[1607],[40,19837,1879],{"className":19838},[1612],[40,19840],{"className":19841,"style":1757},[1756],[40,19843,19819],{"className":19844},[1761],[40,19846],{"className":19847,"style":1757},[1756],[40,19849,19851,19854,19857],{"className":19850},[1603],[40,19852],{"className":19853,"style":4478},[1607],[40,19855,4003],{"className":19856},[1612,1653],[40,19858,2342],{"className":19859},[1612,1653]," to achieve a time-evolution of ",[40,19862,19864,19879],{"className":19863},[1565],[40,19865,19867],{"className":19866},[1569],[1571,19868,19869],{"xmlns":1573},[1575,19870,19871,19877],{},[1578,19872,19873,19875],{},[1584,19874,4003],{},[1584,19876,2342],{},[1591,19878,4507],{"encoding":1593},[40,19880,19882],{"className":19881,"ariaHidden":1599},[1598],[40,19883,19885,19888,19891],{"className":19884},[1603],[40,19886],{"className":19887,"style":4478},[1607],[40,19889,4003],{"className":19890},[1612,1653],[40,19892,2342],{"className":19893},[1612,1653],". 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 ",[40,19896,19898,19920],{"className":19897},[1565],[40,19899,19901],{"className":19900},[1569],[1571,19902,19903],{"xmlns":1573},[1575,19904,19905,19917],{},[1578,19906,19907,19909,19911,19913,19915],{},[1584,19908,4901],{},[1584,19910,2323],{},[1731,19912,241],{"stretchy":1797},[1735,19914,1879],{},[1731,19916,2693],{"stretchy":1797},[1591,19918,19919],{"encoding":1593},"SU(2)",[40,19921,19923],{"className":19922,"ariaHidden":1599},[1598],[40,19924,19926,19929,19932,19935,19938,19941],{"className":19925},[1603],[40,19927],{"className":19928,"style":1811},[1607],[40,19930,4901],{"className":19931,"style":4936},[1612,1653],[40,19933,2323],{"className":19934,"style":2358},[1612,1653],[40,19936,241],{"className":19937},[1985],[40,19939,1879],{"className":19940},[1612],[40,19942,2693],{"className":19943},[1822]," unitary. This gives a much shallower circuit than what is obtained using the generic ",[37,19946,19947],{},"PauliEvolutionGate()"," functionality.",[30,19950,19952],{"className":32,"code":19951,"language":34,"meta":35,"style":35},"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",[37,19953,19954,19968,19972,19977,19992,20011,20028,20045,20070,20074,20083,20120,20154,20162,20166,20176,20187,20201,20213,20223,20242,20251,20266,20270,20280,20293,20311,20326,20332,20345,20349],{"__ignoreMap":35},[40,19955,19956,19958,19960,19962,19964,19966],{"class":42,"line":43},[40,19957,8354],{"class":133},[40,19959,623],{"class":441},[40,19961,8359],{"class":237},[40,19963,241],{"class":133},[40,19965,8364],{"class":244},[40,19967,248],{"class":133},[40,19969,19970],{"class":42,"line":50},[40,19971,228],{"emptyLinePlaceholder":227},[40,19973,19974],{"class":42,"line":155},[40,19975,19976],{"class":46},"# Create instruction for rotation about XX+YY-ZZ:\n",[40,19978,19979,19982,19984,19986,19988,19990],{"class":42,"line":168},[40,19980,19981],{"class":133},"Rxyz_circ ",[40,19983,623],{"class":441},[40,19985,8175],{"class":237},[40,19987,241],{"class":133},[40,19989,1879],{"class":462},[40,19991,248],{"class":133},[40,19993,19994,19997,20000,20003,20005,20007,20009],{"class":42,"line":182},[40,19995,19996],{"class":133},"Rxyz_circ.",[40,19998,19999],{"class":237},"rxx",[40,20001,20002],{"class":133},"(t, ",[40,20004,791],{"class":462},[40,20006,917],{"class":133},[40,20008,1410],{"class":462},[40,20010,248],{"class":133},[40,20012,20013,20015,20018,20020,20022,20024,20026],{"class":42,"line":195},[40,20014,19996],{"class":133},[40,20016,20017],{"class":237},"ryy",[40,20019,20002],{"class":133},[40,20021,791],{"class":462},[40,20023,917],{"class":133},[40,20025,1410],{"class":462},[40,20027,248],{"class":133},[40,20029,20030,20032,20035,20037,20039,20041,20043],{"class":42,"line":203},[40,20031,19996],{"class":133},[40,20033,20034],{"class":237},"rzz",[40,20036,20002],{"class":133},[40,20038,791],{"class":462},[40,20040,917],{"class":133},[40,20042,1410],{"class":462},[40,20044,248],{"class":133},[40,20046,20047,20050,20052,20055,20058,20060,20063,20065,20068],{"class":42,"line":216},[40,20048,20049],{"class":133},"Rxyz_instr ",[40,20051,623],{"class":441},[40,20053,20054],{"class":133}," Rxyz_circ.",[40,20056,20057],{"class":237},"to_instruction",[40,20059,241],{"class":133},[40,20061,20062],{"class":1195},"label",[40,20064,623],{"class":441},[40,20066,20067],{"class":244},"\"RXX+YY+ZZ\"",[40,20069,248],{"class":133},[40,20071,20072],{"class":42,"line":224},[40,20073,228],{"emptyLinePlaceholder":227},[40,20075,20076,20079,20081],{"class":42,"line":231},[40,20077,20078],{"class":133},"interaction_list ",[40,20080,623],{"class":441},[40,20082,1217],{"class":133},[40,20084,20085,20088,20090,20092,20094,20096,20098,20100,20102,20104,20106,20109,20111,20113,20115,20117],{"class":42,"line":251},[40,20086,20087],{"class":133},"    [[i, i ",[40,20089,1147],{"class":441},[40,20091,463],{"class":462},[40,20093,7250],{"class":133},[40,20095,668],{"class":129},[40,20097,7290],{"class":133},[40,20099,674],{"class":129},[40,20101,7265],{"class":441},[40,20103,241],{"class":133},[40,20105,791],{"class":462},[40,20107,20108],{"class":133},", n_qubits ",[40,20110,1407],{"class":441},[40,20112,463],{"class":462},[40,20114,917],{"class":133},[40,20116,1879],{"class":462},[40,20118,20119],{"class":133},")],\n",[40,20121,20122,20124,20126,20128,20130,20132,20134,20136,20138,20140,20142,20144,20146,20148,20150,20152],{"class":42,"line":256},[40,20123,20087],{"class":133},[40,20125,1147],{"class":441},[40,20127,463],{"class":462},[40,20129,7250],{"class":133},[40,20131,668],{"class":129},[40,20133,7290],{"class":133},[40,20135,674],{"class":129},[40,20137,7265],{"class":441},[40,20139,241],{"class":133},[40,20141,1410],{"class":462},[40,20143,20108],{"class":133},[40,20145,1407],{"class":441},[40,20147,463],{"class":462},[40,20149,917],{"class":133},[40,20151,1879],{"class":462},[40,20153,20119],{"class":133},[40,20155,20156,20159],{"class":42,"line":269},[40,20157,20158],{"class":133},"]  ",[40,20160,20161],{"class":46},"# linear chain\n",[40,20163,20164],{"class":42,"line":282},[40,20165,228],{"emptyLinePlaceholder":227},[40,20167,20168,20170,20172,20174],{"class":42,"line":295},[40,20169,8431],{"class":133},[40,20171,623],{"class":441},[40,20173,8436],{"class":237},[40,20175,7828],{"class":133},[40,20177,20178,20181,20183,20185],{"class":42,"line":308},[40,20179,20180],{"class":133},"trotter_step_circ ",[40,20182,623],{"class":441},[40,20184,8175],{"class":237},[40,20186,8450],{"class":133},[40,20188,20189,20191,20194,20196,20198],{"class":42,"line":321},[40,20190,668],{"class":129},[40,20192,20193],{"class":133}," i, color ",[40,20195,674],{"class":129},[40,20197,997],{"class":441},[40,20199,20200],{"class":133},"(interaction_list):\n",[40,20202,20203,20205,20208,20210],{"class":42,"line":334},[40,20204,956],{"class":129},[40,20206,20207],{"class":133}," interaction ",[40,20209,674],{"class":129},[40,20211,20212],{"class":133}," color:\n",[40,20214,20215,20218,20220],{"class":42,"line":347},[40,20216,20217],{"class":133},"        trotter_step_circ.",[40,20219,979],{"class":237},[40,20221,20222],{"class":133},"(Rxyz_instr, interaction)\n",[40,20224,20225,20227,20229,20231,20233,20236,20238,20240],{"class":42,"line":352},[40,20226,755],{"class":129},[40,20228,7290],{"class":133},[40,20230,4721],{"class":441},[40,20232,1341],{"class":441},[40,20234,20235],{"class":133},"(interaction_list) ",[40,20237,1407],{"class":441},[40,20239,463],{"class":462},[40,20241,766],{"class":133},[40,20243,20244,20246,20249],{"class":42,"line":357},[40,20245,20217],{"class":133},[40,20247,20248],{"class":237},"barrier",[40,20250,12875],{"class":133},[40,20252,20253,20256,20258,20261,20264],{"class":42,"line":370},[40,20254,20255],{"class":133},"reverse_trotter_step_circ ",[40,20257,623],{"class":441},[40,20259,20260],{"class":133}," trotter_step_circ.",[40,20262,20263],{"class":237},"reverse_ops",[40,20265,12875],{"class":133},[40,20267,20268],{"class":42,"line":376},[40,20269,228],{"emptyLinePlaceholder":227},[40,20271,20272,20274,20276,20278],{"class":42,"line":387},[40,20273,8443],{"class":133},[40,20275,623],{"class":441},[40,20277,8175],{"class":237},[40,20279,8450],{"class":133},[40,20281,20282,20284,20287,20289,20291],{"class":42,"line":392},[40,20283,668],{"class":129},[40,20285,20286],{"class":133}," step ",[40,20288,674],{"class":129},[40,20290,7265],{"class":441},[40,20292,12941],{"class":133},[40,20294,20295,20297,20299,20301,20303,20306,20309],{"class":42,"line":397},[40,20296,755],{"class":129},[40,20298,20286],{"class":133},[40,20300,7814],{"class":441},[40,20302,1356],{"class":462},[40,20304,20305],{"class":441}," ==",[40,20307,20308],{"class":462}," 0",[40,20310,766],{"class":133},[40,20312,20313,20316,20318,20321,20323],{"class":42,"line":402},[40,20314,20315],{"class":133},"        qc_evol ",[40,20317,623],{"class":441},[40,20319,20320],{"class":133}," qc_evol.",[40,20322,12915],{"class":237},[40,20324,20325],{"class":133},"(trotter_step_circ)\n",[40,20327,20328,20330],{"class":42,"line":414},[40,20329,839],{"class":129},[40,20331,766],{"class":133},[40,20333,20334,20336,20338,20340,20342],{"class":42,"line":424},[40,20335,20315],{"class":133},[40,20337,623],{"class":441},[40,20339,20320],{"class":133},[40,20341,12915],{"class":237},[40,20343,20344],{"class":133},"(reverse_trotter_step_circ)\n",[40,20346,20347],{"class":42,"line":432},[40,20348,228],{"emptyLinePlaceholder":227},[40,20350,20351,20353,20355,20357,20359,20361,20363,20365,20367,20369,20371,20373,20375,20377,20379],{"class":42,"line":448},[40,20352,8455],{"class":133},[40,20354,13214],{"class":237},[40,20356,13031],{"class":133},[40,20358,8210],{"class":237},[40,20360,241],{"class":133},[40,20362,8215],{"class":244},[40,20364,917],{"class":133},[40,20366,13161],{"class":1195},[40,20368,13164],{"class":441},[40,20370,1410],{"class":462},[40,20372,917],{"class":133},[40,20374,8220],{"class":1195},[40,20376,623],{"class":441},[40,20378,8225],{"class":462},[40,20380,248],{"class":133},[8229,20382],{"alt":8231,"src":20383},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-03.avif",[1555,20385,20387],{"id":20386},"use-an-optimized-circuit-for-state-preparation","Use an optimized circuit for state preparation",[30,20389,20391],{"className":32,"code":20390,"language":34,"meta":35,"style":35},"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",[37,20392,20393,20402,20423,20440,20451],{"__ignoreMap":35},[40,20394,20395,20398,20400],{"class":42,"line":43},[40,20396,20397],{"class":133},"control ",[40,20399,623],{"class":441},[40,20401,1286],{"class":462},[40,20403,20404,20407,20409,20411,20413,20415,20417,20419,20421],{"class":42,"line":50},[40,20405,20406],{"class":133},"excitation ",[40,20408,623],{"class":441},[40,20410,1135],{"class":441},[40,20412,1144],{"class":133},[40,20414,7610],{"class":441},[40,20416,1356],{"class":462},[40,20418,1416],{"class":133},[40,20420,1147],{"class":441},[40,20422,1124],{"class":462},[40,20424,20425,20428,20430,20432,20434,20436,20438],{"class":42,"line":155},[40,20426,20427],{"class":133},"controlled_state_prep ",[40,20429,623],{"class":441},[40,20431,8175],{"class":237},[40,20433,1144],{"class":133},[40,20435,1147],{"class":441},[40,20437,463],{"class":462},[40,20439,248],{"class":133},[40,20441,20442,20445,20448],{"class":42,"line":168},[40,20443,20444],{"class":133},"controlled_state_prep.",[40,20446,20447],{"class":237},"cx",[40,20449,20450],{"class":133},"(control, excitation)\n",[40,20452,20453,20455,20457,20459,20461,20463,20465,20467,20469,20471,20473,20475,20477],{"class":42,"line":182},[40,20454,20444],{"class":133},[40,20456,8210],{"class":237},[40,20458,241],{"class":133},[40,20460,8215],{"class":244},[40,20462,917],{"class":133},[40,20464,13161],{"class":1195},[40,20466,13164],{"class":441},[40,20468,1410],{"class":462},[40,20470,917],{"class":133},[40,20472,8220],{"class":1195},[40,20474,623],{"class":441},[40,20476,8225],{"class":462},[40,20478,248],{"class":133},[8229,20480],{"alt":8231,"src":20481},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-04.avif",[1555,20483,20485,20486,9846,20546,20606],{"id":20484},"template-circuits-for-calculating-matrix-elements-of-stildess-and-htildehh-via-hadamard-test","Template circuits for calculating matrix elements of ",[40,20487,20489,20506],{"className":20488},[1565],[40,20490,20492],{"className":20491},[1569],[1571,20493,20494],{"xmlns":1573},[1575,20495,20496,20504],{},[1578,20497,20498],{},[3315,20499,20500,20502],{"accent":1599},[1584,20501,4901],{},[1731,20503,3321],{},[1591,20505,4906],{"encoding":1593},[40,20507,20509],{"className":20508,"ariaHidden":1599},[1598],[40,20510,20512,20515],{"className":20511},[1603],[40,20513],{"className":20514,"style":3334},[1607],[40,20516,20518],{"className":20517},[1612,3338],[40,20519,20521],{"className":20520},[1625],[40,20522,20524],{"className":20523},[1629],[40,20525,20527,20535],{"className":20526,"style":3334},[1633],[40,20528,20529,20532],{"style":3350},[40,20530],{"className":20531,"style":3354},[1641],[40,20533,4901],{"className":20534,"style":4936},[1612,1653],[40,20536,20537,20540],{"style":3360},[40,20538],{"className":20539,"style":3354},[1641],[40,20541,20543],{"className":20542,"style":4945},[3367],[40,20544,3321],{"className":20545},[1612],[40,20547,20549,20566],{"className":20548},[1565],[40,20550,20552],{"className":20551},[1569],[1571,20553,20554],{"xmlns":1573},[1575,20555,20556,20564],{},[1578,20557,20558],{},[3315,20559,20560,20562],{"accent":1599},[1584,20561,2182],{},[1731,20563,3321],{},[1591,20565,3324],{"encoding":1593},[40,20567,20569],{"className":20568,"ariaHidden":1599},[1598],[40,20570,20572,20575],{"className":20571},[1603],[40,20573],{"className":20574,"style":3334},[1607],[40,20576,20578],{"className":20577},[1612,3338],[40,20579,20581],{"className":20580},[1625],[40,20582,20584],{"className":20583},[1629],[40,20585,20587,20595],{"className":20586,"style":3334},[1633],[40,20588,20589,20592],{"style":3350},[40,20590],{"className":20591,"style":3354},[1641],[40,20593,2182],{"className":20594,"style":2197},[1612,1653],[40,20596,20597,20600],{"style":3360},[40,20598],{"className":20599,"style":3354},[1641],[40,20601,20603],{"className":20602,"style":3368},[3367],[40,20604,3321],{"className":20605},[1612]," via Hadamard test",[12,20608,20609],{},"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.",[30,20611,20613],{"className":32,"code":20612,"language":34,"meta":35,"style":35},"# Parameters for the template circuits\nparameters = []\nfor idx in range(1, krylov_dim):\n    parameters.append(2 * dt_circ * (idx))\n",[37,20614,20615,20620,20629,20647],{"__ignoreMap":35},[40,20616,20617],{"class":42,"line":43},[40,20618,20619],{"class":46},"# Parameters for the template circuits\n",[40,20621,20622,20625,20627],{"class":42,"line":50},[40,20623,20624],{"class":133},"parameters ",[40,20626,623],{"class":441},[40,20628,950],{"class":133},[40,20630,20631,20633,20636,20638,20640,20642,20644],{"class":42,"line":155},[40,20632,668],{"class":129},[40,20634,20635],{"class":133}," idx ",[40,20637,674],{"class":129},[40,20639,7265],{"class":441},[40,20641,241],{"class":133},[40,20643,1410],{"class":462},[40,20645,20646],{"class":133},", krylov_dim):\n",[40,20648,20649,20652,20654,20656,20658,20660,20663,20665],{"class":42,"line":168},[40,20650,20651],{"class":133},"    parameters.",[40,20653,979],{"class":237},[40,20655,241],{"class":133},[40,20657,1879],{"class":462},[40,20659,7273],{"class":441},[40,20661,20662],{"class":133}," dt_circ ",[40,20664,1433],{"class":441},[40,20666,20667],{"class":133}," (idx))\n",[30,20669,20671],{"className":32,"code":20670,"language":34,"meta":35,"style":35},"# 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",[37,20672,20673,20678,20694,20705,20718,20750,20758,20793,20801,20813,20821,20851,20855,20867,20871],{"__ignoreMap":35},[40,20674,20675],{"class":42,"line":43},[40,20676,20677],{"class":46},"# Create modified hadamard test circuit\n",[40,20679,20680,20682,20684,20686,20688,20690,20692],{"class":42,"line":50},[40,20681,8431],{"class":133},[40,20683,623],{"class":441},[40,20685,8436],{"class":237},[40,20687,1144],{"class":133},[40,20689,1147],{"class":441},[40,20691,463],{"class":462},[40,20693,248],{"class":133},[40,20695,20696,20699,20701,20703],{"class":42,"line":155},[40,20697,20698],{"class":133},"qc ",[40,20700,623],{"class":441},[40,20702,8175],{"class":237},[40,20704,8450],{"class":133},[40,20706,20707,20710,20712,20714,20716],{"class":42,"line":168},[40,20708,20709],{"class":133},"qc.",[40,20711,13084],{"class":237},[40,20713,241],{"class":133},[40,20715,791],{"class":462},[40,20717,248],{"class":133},[40,20719,20720,20722,20724,20727,20729,20731,20733,20735,20737,20739,20742,20744,20746,20748],{"class":42,"line":182},[40,20721,20709],{"class":133},[40,20723,12915],{"class":237},[40,20725,20726],{"class":133},"(controlled_state_prep, ",[40,20728,13102],{"class":441},[40,20730,241],{"class":133},[40,20732,1244],{"class":441},[40,20734,1144],{"class":133},[40,20736,1147],{"class":441},[40,20738,463],{"class":462},[40,20740,20741],{"class":133},")), ",[40,20743,12921],{"class":1195},[40,20745,623],{"class":441},[40,20747,12926],{"class":462},[40,20749,248],{"class":133},[40,20751,20752,20754,20756],{"class":42,"line":195},[40,20753,20709],{"class":133},[40,20755,20248],{"class":237},[40,20757,12875],{"class":133},[40,20759,20760,20762,20764,20767,20769,20771,20773,20775,20777,20779,20781,20783,20785,20787,20789,20791],{"class":42,"line":203},[40,20761,20709],{"class":133},[40,20763,12915],{"class":237},[40,20765,20766],{"class":133},"(qc_evol, ",[40,20768,13102],{"class":441},[40,20770,241],{"class":133},[40,20772,1244],{"class":441},[40,20774,241],{"class":133},[40,20776,1410],{"class":462},[40,20778,20108],{"class":133},[40,20780,1147],{"class":441},[40,20782,463],{"class":462},[40,20784,20741],{"class":133},[40,20786,12921],{"class":1195},[40,20788,623],{"class":441},[40,20790,12926],{"class":462},[40,20792,248],{"class":133},[40,20794,20795,20797,20799],{"class":42,"line":216},[40,20796,20709],{"class":133},[40,20798,20248],{"class":237},[40,20800,12875],{"class":133},[40,20802,20803,20805,20807,20809,20811],{"class":42,"line":224},[40,20804,20709],{"class":133},[40,20806,8185],{"class":237},[40,20808,241],{"class":133},[40,20810,791],{"class":462},[40,20812,248],{"class":133},[40,20814,20815,20817,20819],{"class":42,"line":231},[40,20816,20709],{"class":133},[40,20818,12915],{"class":237},[40,20820,411],{"class":133},[40,20822,20823,20826,20828,20830,20832,20834,20836,20838,20840,20842,20844,20846,20848],{"class":42,"line":251},[40,20824,20825],{"class":133},"    controlled_state_prep.",[40,20827,12872],{"class":237},[40,20829,12998],{"class":133},[40,20831,13102],{"class":441},[40,20833,241],{"class":133},[40,20835,1244],{"class":441},[40,20837,1144],{"class":133},[40,20839,1147],{"class":441},[40,20841,463],{"class":462},[40,20843,20741],{"class":133},[40,20845,12921],{"class":1195},[40,20847,623],{"class":441},[40,20849,20850],{"class":462},"True\n",[40,20852,20853],{"class":42,"line":256},[40,20854,248],{"class":133},[40,20856,20857,20859,20861,20863,20865],{"class":42,"line":269},[40,20858,20709],{"class":133},[40,20860,8185],{"class":237},[40,20862,241],{"class":133},[40,20864,791],{"class":462},[40,20866,248],{"class":133},[40,20868,20869],{"class":42,"line":282},[40,20870,228],{"emptyLinePlaceholder":227},[40,20872,20873,20875,20877,20879,20881,20883,20885,20887,20889,20891,20893],{"class":42,"line":295},[40,20874,20709],{"class":133},[40,20876,13214],{"class":237},[40,20878,13031],{"class":133},[40,20880,8210],{"class":237},[40,20882,241],{"class":133},[40,20884,8215],{"class":244},[40,20886,917],{"class":133},[40,20888,13161],{"class":1195},[40,20890,13164],{"class":441},[40,20892,1410],{"class":462},[40,20894,248],{"class":133},[8229,20896],{"alt":8231,"src":20897},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-05.avif",[30,20899,20901],{"className":32,"code":20900,"language":34,"meta":35,"style":35},"print(\n    \"The optimized circuit has 2Q gates depth: \",\n    qc.decompose().decompose().depth(lambda x: x[0].num_qubits == 2),\n)\n",[37,20902,20903,20909,20916,20949],{"__ignoreMap":35},[40,20904,20905,20907],{"class":42,"line":43},[40,20906,7569],{"class":441},[40,20908,411],{"class":133},[40,20910,20911,20914],{"class":42,"line":50},[40,20912,20913],{"class":244},"    \"The optimized circuit has 2Q gates depth: \"",[40,20915,445],{"class":133},[40,20917,20918,20921,20923,20925,20927,20929,20931,20933,20935,20937,20939,20941,20943,20945,20947],{"class":42,"line":155},[40,20919,20920],{"class":133},"    qc.",[40,20922,13214],{"class":237},[40,20924,13031],{"class":133},[40,20926,13214],{"class":237},[40,20928,13031],{"class":133},[40,20930,13228],{"class":237},[40,20932,241],{"class":133},[40,20934,13233],{"class":129},[40,20936,13236],{"class":417},[40,20938,13239],{"class":133},[40,20940,791],{"class":462},[40,20942,13244],{"class":133},[40,20944,761],{"class":441},[40,20946,1356],{"class":462},[40,20948,13251],{"class":133},[40,20950,20951],{"class":42,"line":168},[40,20952,248],{"class":133},[30,20954,20957],{"className":20955,"code":20956,"language":6083,"meta":35},[7575],"The optimized circuit has 2Q gates depth:  74\n",[37,20958,20956],{"__ignoreMap":35},[12,20960,20961],{},"We have considerably reduced the depth of the Hadamard test with a combination of Trotter approximation and uncontrolled unitaries",[55,20963,20965],{"id":20964},"step-3-execute-using-qiskit-primitives","Step 3: Execute using Qiskit primitives",[12,20967,20968],{},"Instantiate the backend and set runtime parameters",[30,20970,20972],{"className":32,"code":20971,"language":34,"meta":35,"style":35},"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",[37,20973,20974,20986,21020,21026,21040,21048,21069],{"__ignoreMap":35},[40,20975,20976,20979,20981,20984],{"class":42,"line":43},[40,20977,20978],{"class":133},"service ",[40,20980,623],{"class":441},[40,20982,20983],{"class":237}," QiskitRuntimeService",[40,20985,12875],{"class":133},[40,20987,20988,20991,20993,20996,20999,21001,21004,21006,21008,21010,21013,21015,21018],{"class":42,"line":50},[40,20989,20990],{"class":133},"backend ",[40,20992,623],{"class":441},[40,20994,20995],{"class":133}," service.",[40,20997,20998],{"class":237},"least_busy",[40,21000,241],{"class":133},[40,21002,21003],{"class":1195},"operational",[40,21005,623],{"class":441},[40,21007,12926],{"class":462},[40,21009,917],{"class":133},[40,21011,21012],{"class":1195},"simulator",[40,21014,623],{"class":441},[40,21016,21017],{"class":462},"False",[40,21019,248],{"class":133},[40,21021,21022,21024],{"class":42,"line":155},[40,21023,683],{"class":129},[40,21025,367],{"class":133},[40,21027,21028,21031,21034,21037],{"class":42,"line":168},[40,21029,21030],{"class":244},"    \"if_else\"",[40,21032,21033],{"class":129}," not",[40,21035,21036],{"class":129}," in",[40,21038,21039],{"class":133}," backend.target.operation_names\n",[40,21041,21042,21045],{"class":42,"line":182},[40,21043,21044],{"class":133},"):  ",[40,21046,21047],{"class":46},"# Needed as \"op_name\" could be \"if_else\"\n",[40,21049,21050,21053,21056,21059,21062,21064,21067],{"class":42,"line":195},[40,21051,21052],{"class":133},"    backend.target.",[40,21054,21055],{"class":237},"add_instruction",[40,21057,21058],{"class":133},"(IfElseOp, ",[40,21060,21061],{"class":1195},"name",[40,21063,623],{"class":441},[40,21065,21066],{"class":244},"\"if_else\"",[40,21068,248],{"class":133},[40,21070,21071,21073],{"class":42,"line":203},[40,21072,7569],{"class":441},[40,21074,21075],{"class":133},"(backend.name)\n",[1555,21077,21079],{"id":21078},"transpiling-to-a-qpu","Transpiling to a QPU",[12,21081,21082],{},"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",[30,21084,21086],{"className":32,"code":21085,"language":34,"meta":35,"style":35},"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",[37,21087,21088,21098,21122,21141,21145,21161,21175,21191,21206,21231,21243,21254,21261,21272,21279,21284,21288,21298,21320,21347,21361,21371,21381,21387,21396,21402,21406,21410,21414,21427,21442,21458,21468],{"__ignoreMap":35},[40,21089,21090,21093,21095],{"class":42,"line":43},[40,21091,21092],{"class":133},"target ",[40,21094,623],{"class":441},[40,21096,21097],{"class":133}," backend.target\n",[40,21099,21100,21103,21105,21108,21111,21113,21116,21118,21120],{"class":42,"line":50},[40,21101,21102],{"class":133},"cmap ",[40,21104,623],{"class":441},[40,21106,21107],{"class":133}," target.",[40,21109,21110],{"class":237},"build_coupling_map",[40,21112,241],{"class":133},[40,21114,21115],{"class":1195},"filter_idle_qubits",[40,21117,623],{"class":441},[40,21119,12926],{"class":462},[40,21121,248],{"class":133},[40,21123,21124,21127,21129,21132,21135,21138],{"class":42,"line":155},[40,21125,21126],{"class":133},"cmap_list ",[40,21128,623],{"class":441},[40,21130,21131],{"class":441}," list",[40,21133,21134],{"class":133},"(cmap.",[40,21136,21137],{"class":237},"get_edges",[40,21139,21140],{"class":133},"())\n",[40,21142,21143],{"class":42,"line":168},[40,21144,228],{"emptyLinePlaceholder":227},[40,21146,21147,21150,21152,21155,21158],{"class":42,"line":182},[40,21148,21149],{"class":133},"cust_cmap_list ",[40,21151,623],{"class":441},[40,21153,21154],{"class":133}," copy.",[40,21156,21157],{"class":237},"deepcopy",[40,21159,21160],{"class":133},"(cmap_list)\n",[40,21162,21163,21165,21168,21170,21172],{"class":42,"line":195},[40,21164,668],{"class":129},[40,21166,21167],{"class":133}," q ",[40,21169,674],{"class":129},[40,21171,7265],{"class":441},[40,21173,21174],{"class":133},"(target.num_qubits):\n",[40,21176,21177,21180,21182,21185,21188],{"class":42,"line":203},[40,21178,21179],{"class":133},"    meas_err ",[40,21181,623],{"class":441},[40,21183,21184],{"class":133}," target[",[40,21186,21187],{"class":244},"\"measure\"",[40,21189,21190],{"class":133},"][(q,)].error\n",[40,21192,21193,21196,21198,21201,21203],{"class":42,"line":216},[40,21194,21195],{"class":133},"    t2 ",[40,21197,623],{"class":441},[40,21199,21200],{"class":133}," target.qubit_properties[q].t2 ",[40,21202,1433],{"class":441},[40,21204,21205],{"class":462}," 1e6\n",[40,21207,21208,21210,21213,21215,21218,21221,21224,21226,21229],{"class":42,"line":224},[40,21209,755],{"class":129},[40,21211,21212],{"class":133}," meas_err ",[40,21214,689],{"class":441},[40,21216,21217],{"class":462}," 0.02",[40,21219,21220],{"class":129}," or",[40,21222,21223],{"class":133}," t2 ",[40,21225,4721],{"class":441},[40,21227,21228],{"class":462}," 100",[40,21230,766],{"class":133},[40,21232,21233,21235,21238,21240],{"class":42,"line":231},[40,21234,1307],{"class":129},[40,21236,21237],{"class":133}," q_pair ",[40,21239,674],{"class":129},[40,21241,21242],{"class":133}," cmap_list:\n",[40,21244,21245,21247,21249,21251],{"class":42,"line":251},[40,21246,1338],{"class":129},[40,21248,21167],{"class":133},[40,21250,674],{"class":129},[40,21252,21253],{"class":133}," q_pair:\n",[40,21255,21256,21259],{"class":42,"line":256},[40,21257,21258],{"class":129},"                try",[40,21260,766],{"class":133},[40,21262,21263,21266,21269],{"class":42,"line":269},[40,21264,21265],{"class":133},"                    cust_cmap_list.",[40,21267,21268],{"class":237},"remove",[40,21270,21271],{"class":133},"(q_pair)\n",[40,21273,21274,21277],{"class":42,"line":282},[40,21275,21276],{"class":129},"                except",[40,21278,766],{"class":133},[40,21280,21281],{"class":42,"line":295},[40,21282,21283],{"class":129},"                    continue\n",[40,21285,21286],{"class":42,"line":308},[40,21287,228],{"emptyLinePlaceholder":227},[40,21289,21290,21292,21294,21296],{"class":42,"line":321},[40,21291,668],{"class":129},[40,21293,21167],{"class":133},[40,21295,674],{"class":129},[40,21297,21242],{"class":133},[40,21299,21300,21303,21305,21307,21310,21313,21316,21318],{"class":42,"line":334},[40,21301,21302],{"class":133},"    op_name ",[40,21304,623],{"class":441},[40,21306,21131],{"class":441},[40,21308,21309],{"class":133},"(target.",[40,21311,21312],{"class":237},"operation_names_for_qargs",[40,21314,21315],{"class":133},"(q))[",[40,21317,791],{"class":462},[40,21319,833],{"class":133},[40,21321,21322,21325,21327,21329,21332,21335,21337,21340,21342,21344],{"class":42,"line":347},[40,21323,21324],{"class":133},"    twoq_gate_err ",[40,21326,623],{"class":441},[40,21328,21184],{"class":133},[40,21330,21331],{"class":129},"f",[40,21333,21334],{"class":244},"\"",[40,21336,1853],{"class":462},[40,21338,21339],{"class":133},"op_name",[40,21341,1916],{"class":462},[40,21343,21334],{"class":244},[40,21345,21346],{"class":133},"][q].error\n",[40,21348,21349,21351,21354,21356,21359],{"class":42,"line":352},[40,21350,755],{"class":129},[40,21352,21353],{"class":133}," twoq_gate_err ",[40,21355,689],{"class":441},[40,21357,21358],{"class":462}," 0.005",[40,21360,766],{"class":133},[40,21362,21363,21365,21367,21369],{"class":42,"line":357},[40,21364,1307],{"class":129},[40,21366,21237],{"class":133},[40,21368,674],{"class":129},[40,21370,21242],{"class":133},[40,21372,21373,21375,21377,21379],{"class":42,"line":370},[40,21374,1338],{"class":129},[40,21376,21167],{"class":133},[40,21378,761],{"class":441},[40,21380,21253],{"class":133},[40,21382,21383,21385],{"class":42,"line":376},[40,21384,21258],{"class":129},[40,21386,766],{"class":133},[40,21388,21389,21391,21393],{"class":42,"line":387},[40,21390,21265],{"class":133},[40,21392,21268],{"class":237},[40,21394,21395],{"class":133},"(q)\n",[40,21397,21398,21400],{"class":42,"line":392},[40,21399,21276],{"class":129},[40,21401,766],{"class":133},[40,21403,21404],{"class":42,"line":397},[40,21405,21283],{"class":129},[40,21407,21408],{"class":42,"line":402},[40,21409,228],{"emptyLinePlaceholder":227},[40,21411,21412],{"class":42,"line":414},[40,21413,228],{"emptyLinePlaceholder":227},[40,21415,21416,21419,21421,21424],{"class":42,"line":424},[40,21417,21418],{"class":133},"cust_cmap ",[40,21420,623],{"class":441},[40,21422,21423],{"class":237}," CouplingMap",[40,21425,21426],{"class":133},"(cust_cmap_list)\n",[40,21428,21429,21432,21434,21437,21440],{"class":42,"line":432},[40,21430,21431],{"class":133},"cust_target ",[40,21433,623],{"class":441},[40,21435,21436],{"class":133}," Target.",[40,21438,21439],{"class":237},"from_configuration",[40,21441,411],{"class":133},[40,21443,21444,21447,21449,21452,21455],{"class":42,"line":448},[40,21445,21446],{"class":1195},"    basis_gates",[40,21448,623],{"class":441},[40,21450,21451],{"class":133},"backend.",[40,21453,21454],{"class":237},"configuration",[40,21456,21457],{"class":133},"().basis_gates,\n",[40,21459,21460,21463,21465],{"class":42,"line":468},[40,21461,21462],{"class":1195},"    coupling_map",[40,21464,623],{"class":441},[40,21466,21467],{"class":133},"cust_cmap,\n",[40,21469,21470],{"class":42,"line":486},[40,21471,248],{"class":133},[12,21473,21474],{},"Then transpile the virtual circuit to the best physical layout in this new target",[30,21476,21478],{"className":32,"code":21477,"language":34,"meta":35,"style":35},"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",[37,21479,21480,21492,21504,21515,21525,21534,21538,21542,21558,21562,21595,21611,21617,21624,21631,21636,21641,21664,21677,21682,21687],{"__ignoreMap":35},[40,21481,21482,21485,21487,21489],{"class":42,"line":43},[40,21483,21484],{"class":133},"basis_gates ",[40,21486,623],{"class":441},[40,21488,21131],{"class":441},[40,21490,21491],{"class":133},"(target.operation_names)\n",[40,21493,21494,21497,21499,21502],{"class":42,"line":50},[40,21495,21496],{"class":133},"pm ",[40,21498,623],{"class":441},[40,21500,21501],{"class":237}," generate_preset_pass_manager",[40,21503,411],{"class":133},[40,21505,21506,21509,21511,21513],{"class":42,"line":155},[40,21507,21508],{"class":1195},"    optimization_level",[40,21510,623],{"class":441},[40,21512,7270],{"class":462},[40,21514,445],{"class":133},[40,21516,21517,21520,21522],{"class":42,"line":168},[40,21518,21519],{"class":1195},"    target",[40,21521,623],{"class":441},[40,21523,21524],{"class":133},"cust_target,\n",[40,21526,21527,21529,21531],{"class":42,"line":182},[40,21528,21446],{"class":1195},[40,21530,623],{"class":441},[40,21532,21533],{"class":133},"basis_gates,\n",[40,21535,21536],{"class":42,"line":195},[40,21537,248],{"class":133},[40,21539,21540],{"class":42,"line":203},[40,21541,228],{"emptyLinePlaceholder":227},[40,21543,21544,21547,21549,21552,21555],{"class":42,"line":216},[40,21545,21546],{"class":133},"qc_trans ",[40,21548,623],{"class":441},[40,21550,21551],{"class":133}," pm.",[40,21553,21554],{"class":237},"run",[40,21556,21557],{"class":133},"(qc)\n",[40,21559,21560],{"class":42,"line":224},[40,21561,228],{"emptyLinePlaceholder":227},[40,21563,21564,21566,21568,21571,21574,21576,21578,21580,21582,21584,21586,21588,21590,21592],{"class":42,"line":231},[40,21565,7569],{"class":441},[40,21567,241],{"class":133},[40,21569,21570],{"class":244},"\"depth\"",[40,21572,21573],{"class":133},", qc_trans.",[40,21575,13228],{"class":237},[40,21577,241],{"class":133},[40,21579,13233],{"class":129},[40,21581,13236],{"class":417},[40,21583,13239],{"class":133},[40,21585,791],{"class":462},[40,21587,13244],{"class":133},[40,21589,761],{"class":441},[40,21591,1356],{"class":462},[40,21593,21594],{"class":133},"))\n",[40,21596,21597,21599,21601,21604,21606,21609],{"class":42,"line":251},[40,21598,7569],{"class":441},[40,21600,241],{"class":133},[40,21602,21603],{"class":244},"\"num 2q ops\"",[40,21605,21573],{"class":133},[40,21607,21608],{"class":237},"count_ops",[40,21610,21140],{"class":133},[40,21612,21613,21615],{"class":42,"line":256},[40,21614,7569],{"class":441},[40,21616,411],{"class":133},[40,21618,21619,21622],{"class":42,"line":269},[40,21620,21621],{"class":244},"    \"physical qubits\"",[40,21623,445],{"class":133},[40,21625,21626,21629],{"class":42,"line":282},[40,21627,21628],{"class":441},"    sorted",[40,21630,411],{"class":133},[40,21632,21633],{"class":42,"line":295},[40,21634,21635],{"class":133},"        [\n",[40,21637,21638],{"class":42,"line":308},[40,21639,21640],{"class":133},"            idx\n",[40,21642,21643,21646,21649,21651,21654,21657,21659,21662],{"class":42,"line":321},[40,21644,21645],{"class":129},"            for",[40,21647,21648],{"class":133}," idx, qb ",[40,21650,674],{"class":129},[40,21652,21653],{"class":133}," qc_trans.layout.initial_layout.",[40,21655,21656],{"class":237},"get_physical_bits",[40,21658,13031],{"class":133},[40,21660,21661],{"class":237},"items",[40,21663,12875],{"class":133},[40,21665,21666,21668,21671,21674],{"class":42,"line":334},[40,21667,1338],{"class":129},[40,21669,21670],{"class":133}," qb._register.name ",[40,21672,21673],{"class":441},"!=",[40,21675,21676],{"class":244}," \"ancilla\"\n",[40,21678,21679],{"class":42,"line":347},[40,21680,21681],{"class":133},"        ]\n",[40,21683,21684],{"class":42,"line":352},[40,21685,21686],{"class":133},"    ),\n",[40,21688,21689],{"class":42,"line":357},[40,21690,248],{"class":133},[30,21692,21695],{"className":21693,"code":21694,"language":6083,"meta":35},[7575],"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",[37,21696,21694],{"__ignoreMap":35},[1555,21698,21700],{"id":21699},"create-pubs-for-execution-with-estimator","Create PUBs for execution with Estimator",[30,21702,21704],{"className":32,"code":21703,"language":34,"meta":35,"style":35},"# 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",[37,21705,21706,21711,21730,21747,21751,21763,21768,21776,21788,21792,21805,21819,21824,21831,21845,21853,21863,21870,21883,21887,21897,21901,21905,21910,21915,21924,21938,21943,21969,21992,22002,22011,22015,22024,22028,22037,22049,22061,22074,22089,22093,22103,22107,22111,22116,22131,22135,22139,22144,22149,22154],{"__ignoreMap":35},[40,21707,21708],{"class":42,"line":43},[40,21709,21710],{"class":46},"# Define observables to measure for S\n",[40,21712,21713,21716,21718,21721,21723,21726,21728],{"class":42,"line":50},[40,21714,21715],{"class":133},"observable_S_real ",[40,21717,623],{"class":441},[40,21719,21720],{"class":244}," \"I\"",[40,21722,7273],{"class":441},[40,21724,21725],{"class":133}," (n_qubits) ",[40,21727,1147],{"class":441},[40,21729,7365],{"class":244},[40,21731,21732,21735,21737,21739,21741,21743,21745],{"class":42,"line":155},[40,21733,21734],{"class":133},"observable_S_imag ",[40,21736,623],{"class":441},[40,21738,21720],{"class":244},[40,21740,7273],{"class":441},[40,21742,21725],{"class":133},[40,21744,1147],{"class":441},[40,21746,7432],{"class":244},[40,21748,21749],{"class":42,"line":168},[40,21750,228],{"emptyLinePlaceholder":227},[40,21752,21753,21756,21758,21761],{"class":42,"line":182},[40,21754,21755],{"class":133},"observable_op_real ",[40,21757,623],{"class":441},[40,21759,21760],{"class":237}," SparsePauliOp",[40,21762,411],{"class":133},[40,21764,21765],{"class":42,"line":195},[40,21766,21767],{"class":133},"    observable_S_real\n",[40,21769,21770,21773],{"class":42,"line":203},[40,21771,21772],{"class":133},")  ",[40,21774,21775],{"class":46},"# define a sparse pauli operator for the observable\n",[40,21777,21778,21781,21783,21785],{"class":42,"line":216},[40,21779,21780],{"class":133},"observable_op_imag ",[40,21782,623],{"class":441},[40,21784,21760],{"class":237},[40,21786,21787],{"class":133},"(observable_S_imag)\n",[40,21789,21790],{"class":42,"line":224},[40,21791,228],{"emptyLinePlaceholder":227},[40,21793,21794,21797,21799,21802],{"class":42,"line":231},[40,21795,21796],{"class":133},"layout ",[40,21798,623],{"class":441},[40,21800,21801],{"class":133}," qc_trans.layout  ",[40,21803,21804],{"class":46},"# get layout of transpiled circuit\n",[40,21806,21807,21809,21811,21814,21817],{"class":42,"line":251},[40,21808,21755],{"class":133},[40,21810,623],{"class":441},[40,21812,21813],{"class":133}," observable_op_real.",[40,21815,21816],{"class":237},"apply_layout",[40,21818,411],{"class":133},[40,21820,21821],{"class":42,"line":256},[40,21822,21823],{"class":133},"    layout\n",[40,21825,21826,21828],{"class":42,"line":269},[40,21827,21772],{"class":133},[40,21829,21830],{"class":46},"# apply physical layout to the observable\n",[40,21832,21833,21835,21837,21840,21842],{"class":42,"line":282},[40,21834,21780],{"class":133},[40,21836,623],{"class":441},[40,21838,21839],{"class":133}," observable_op_imag.",[40,21841,21816],{"class":237},[40,21843,21844],{"class":133},"(layout)\n",[40,21846,21847,21849,21851],{"class":42,"line":295},[40,21848,21715],{"class":133},[40,21850,623],{"class":441},[40,21852,367],{"class":133},[40,21854,21855,21858,21861],{"class":42,"line":308},[40,21856,21857],{"class":133},"    observable_op_real.paulis.",[40,21859,21860],{"class":237},"to_labels",[40,21862,12875],{"class":133},[40,21864,21865,21867],{"class":42,"line":321},[40,21866,21772],{"class":133},[40,21868,21869],{"class":46},"# get the label of the physical observable\n",[40,21871,21872,21874,21876,21879,21881],{"class":42,"line":334},[40,21873,21734],{"class":133},[40,21875,623],{"class":441},[40,21877,21878],{"class":133}," observable_op_imag.paulis.",[40,21880,21860],{"class":237},[40,21882,12875],{"class":133},[40,21884,21885],{"class":42,"line":347},[40,21886,228],{"emptyLinePlaceholder":227},[40,21888,21889,21892,21894],{"class":42,"line":352},[40,21890,21891],{"class":133},"observables_S ",[40,21893,623],{"class":441},[40,21895,21896],{"class":133}," [[observable_S_real], [observable_S_imag]]\n",[40,21898,21899],{"class":42,"line":357},[40,21900,228],{"emptyLinePlaceholder":227},[40,21902,21903],{"class":42,"line":370},[40,21904,228],{"emptyLinePlaceholder":227},[40,21906,21907],{"class":42,"line":376},[40,21908,21909],{"class":46},"# Define observables to measure for H\n",[40,21911,21912],{"class":42,"line":387},[40,21913,21914],{"class":46},"# Hamiltonian terms to measure\n",[40,21916,21917,21920,21922],{"class":42,"line":392},[40,21918,21919],{"class":133},"observable_list ",[40,21921,623],{"class":441},[40,21923,950],{"class":133},[40,21925,21926,21928,21931,21933,21935],{"class":42,"line":397},[40,21927,668],{"class":129},[40,21929,21930],{"class":133}," pauli, coeff ",[40,21932,674],{"class":129},[40,21934,677],{"class":441},[40,21936,21937],{"class":133},"(H_op.paulis, H_op.coeffs):\n",[40,21939,21940],{"class":42,"line":402},[40,21941,21942],{"class":46},"    # print(pauli)\n",[40,21944,21945,21948,21950,21953,21955,21957,21960,21963,21965,21967],{"class":42,"line":414},[40,21946,21947],{"class":133},"    observable_H_real ",[40,21949,623],{"class":441},[40,21951,21952],{"class":133}," pauli[::",[40,21954,1407],{"class":441},[40,21956,1410],{"class":462},[40,21958,21959],{"class":133},"].",[40,21961,21962],{"class":237},"to_label",[40,21964,715],{"class":133},[40,21966,1147],{"class":441},[40,21968,7365],{"class":244},[40,21970,21971,21974,21976,21978,21980,21982,21984,21986,21988,21990],{"class":42,"line":424},[40,21972,21973],{"class":133},"    observable_H_imag ",[40,21975,623],{"class":441},[40,21977,21952],{"class":133},[40,21979,1407],{"class":441},[40,21981,1410],{"class":462},[40,21983,21959],{"class":133},[40,21985,21962],{"class":237},[40,21987,715],{"class":133},[40,21989,1147],{"class":441},[40,21991,7432],{"class":244},[40,21993,21994,21997,21999],{"class":42,"line":432},[40,21995,21996],{"class":133},"    observable_list.",[40,21998,979],{"class":237},[40,22000,22001],{"class":133},"([observable_H_real])\n",[40,22003,22004,22006,22008],{"class":42,"line":448},[40,22005,21996],{"class":133},[40,22007,979],{"class":237},[40,22009,22010],{"class":133},"([observable_H_imag])\n",[40,22012,22013],{"class":42,"line":468},[40,22014,228],{"emptyLinePlaceholder":227},[40,22016,22017,22019,22021],{"class":42,"line":486},[40,22018,21796],{"class":133},[40,22020,623],{"class":441},[40,22022,22023],{"class":133}," qc_trans.layout\n",[40,22025,22026],{"class":42,"line":502},[40,22027,228],{"emptyLinePlaceholder":227},[40,22029,22030,22033,22035],{"class":42,"line":508},[40,22031,22032],{"class":133},"observable_trans_list ",[40,22034,623],{"class":441},[40,22036,950],{"class":133},[40,22038,22039,22041,22044,22046],{"class":42,"line":514},[40,22040,668],{"class":129},[40,22042,22043],{"class":133}," observable ",[40,22045,674],{"class":129},[40,22047,22048],{"class":133}," observable_list:\n",[40,22050,22051,22054,22056,22058],{"class":42,"line":519},[40,22052,22053],{"class":133},"    observable_op ",[40,22055,623],{"class":441},[40,22057,21760],{"class":237},[40,22059,22060],{"class":133},"(observable)\n",[40,22062,22063,22065,22067,22070,22072],{"class":42,"line":525},[40,22064,22053],{"class":133},[40,22066,623],{"class":441},[40,22068,22069],{"class":133}," observable_op.",[40,22071,21816],{"class":237},[40,22073,21844],{"class":133},[40,22075,22076,22079,22081,22084,22086],{"class":42,"line":531},[40,22077,22078],{"class":133},"    observable_trans_list.",[40,22080,979],{"class":237},[40,22082,22083],{"class":133},"([observable_op.paulis.",[40,22085,21860],{"class":237},[40,22087,22088],{"class":133},"()])\n",[40,22090,22091],{"class":42,"line":537},[40,22092,228],{"emptyLinePlaceholder":227},[40,22094,22095,22098,22100],{"class":42,"line":543},[40,22096,22097],{"class":133},"observables_H ",[40,22099,623],{"class":441},[40,22101,22102],{"class":133}," observable_trans_list\n",[40,22104,22105],{"class":42,"line":549},[40,22106,228],{"emptyLinePlaceholder":227},[40,22108,22109],{"class":42,"line":555},[40,22110,228],{"emptyLinePlaceholder":227},[40,22112,22113],{"class":42,"line":561},[40,22114,22115],{"class":46},"# Define a sweep over parameter values\n",[40,22117,22118,22121,22123,22125,22128],{"class":42,"line":567},[40,22119,22120],{"class":133},"params ",[40,22122,623],{"class":441},[40,22124,654],{"class":133},[40,22126,22127],{"class":237},"vstack",[40,22129,22130],{"class":133},"(parameters).T\n",[40,22132,22133],{"class":42,"line":573},[40,22134,228],{"emptyLinePlaceholder":227},[40,22136,22137],{"class":42,"line":579},[40,22138,228],{"emptyLinePlaceholder":227},[40,22140,22141],{"class":42,"line":585},[40,22142,22143],{"class":46},"# Estimate the expectation value for all combinations of\n",[40,22145,22146],{"class":42,"line":590},[40,22147,22148],{"class":46},"# observables and parameter values, where the pub result will have\n",[40,22150,22151],{"class":42,"line":596},[40,22152,22153],{"class":46},"# shape (# observables, # parameter values).\n",[40,22155,22156,22159,22161,22164,22166],{"class":42,"line":602},[40,22157,22158],{"class":133},"pub ",[40,22160,623],{"class":441},[40,22162,22163],{"class":133}," (qc_trans, observables_S ",[40,22165,1147],{"class":441},[40,22167,22168],{"class":133}," observables_H, params)\n",[1555,22170,22172],{"id":22171},"run-circuits","Run circuits",[12,22174,22175,22176,22227],{},"Circuits for ",[40,22177,22179,22197],{"className":22178},[1565],[40,22180,22182],{"className":22181},[1569],[1571,22183,22184],{"xmlns":1573},[1575,22185,22186,22194],{},[1578,22187,22188,22190,22192],{},[1584,22189,2342],{},[1731,22191,623],{},[1735,22193,791],{},[1591,22195,22196],{"encoding":1593},"t=0",[40,22198,22200,22218],{"className":22199,"ariaHidden":1599},[1598],[40,22201,22203,22206,22209,22212,22215],{"className":22202},[1603],[40,22204],{"className":22205,"style":19673},[1607],[40,22207,2342],{"className":22208},[1612,1653],[40,22210],{"className":22211,"style":1963},[1756],[40,22213,623],{"className":22214},[1967],[40,22216],{"className":22217,"style":1963},[1756],[40,22219,22221,22224],{"className":22220},[1603],[40,22222],{"className":22223,"style":1771},[1607],[40,22225,791],{"className":22226},[1612]," are classically calculable",[30,22229,22231],{"className":32,"code":22230,"language":34,"meta":35,"style":35},"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",[37,22232,22233,22254,22258,22262,22267,22285,22305,22309,22324,22343,22347,22351,22356,22377,22381,22390,22409,22414,22429,22453,22457,22472,22494,22498,22519,22523,22528,22543,22547,22551],{"__ignoreMap":35},[40,22234,22235,22238,22240,22243,22246,22249,22251],{"class":42,"line":43},[40,22236,22237],{"class":133},"qc_cliff ",[40,22239,623],{"class":441},[40,22241,22242],{"class":133}," qc.",[40,22244,22245],{"class":237},"assign_parameters",[40,22247,22248],{"class":133},"({t: ",[40,22250,791],{"class":462},[40,22252,22253],{"class":133},"})\n",[40,22255,22256],{"class":42,"line":50},[40,22257,228],{"emptyLinePlaceholder":227},[40,22259,22260],{"class":42,"line":155},[40,22261,228],{"emptyLinePlaceholder":227},[40,22263,22264],{"class":42,"line":168},[40,22265,22266],{"class":46},"# Get expectation values from experiment\n",[40,22268,22269,22272,22274,22277,22280,22283],{"class":42,"line":182},[40,22270,22271],{"class":133},"S_expval_real ",[40,22273,623],{"class":441},[40,22275,22276],{"class":237}," StabilizerState",[40,22278,22279],{"class":133},"(qc_cliff).",[40,22281,22282],{"class":237},"expectation_value",[40,22284,411],{"class":133},[40,22286,22287,22290,22292,22294,22296,22298,22300,22303],{"class":42,"line":195},[40,22288,22289],{"class":237},"    Pauli",[40,22291,241],{"class":133},[40,22293,7247],{"class":244},[40,22295,7273],{"class":441},[40,22297,21725],{"class":133},[40,22299,1147],{"class":441},[40,22301,22302],{"class":244}," \"X\"",[40,22304,248],{"class":133},[40,22306,22307],{"class":42,"line":203},[40,22308,248],{"class":133},[40,22310,22311,22314,22316,22318,22320,22322],{"class":42,"line":216},[40,22312,22313],{"class":133},"S_expval_imag ",[40,22315,623],{"class":441},[40,22317,22276],{"class":237},[40,22319,22279],{"class":133},[40,22321,22282],{"class":237},[40,22323,411],{"class":133},[40,22325,22326,22328,22330,22332,22334,22336,22338,22341],{"class":42,"line":224},[40,22327,22289],{"class":237},[40,22329,241],{"class":133},[40,22331,7247],{"class":244},[40,22333,7273],{"class":441},[40,22335,21725],{"class":133},[40,22337,1147],{"class":441},[40,22339,22340],{"class":244}," \"Y\"",[40,22342,248],{"class":133},[40,22344,22345],{"class":42,"line":231},[40,22346,248],{"class":133},[40,22348,22349],{"class":42,"line":251},[40,22350,228],{"emptyLinePlaceholder":227},[40,22352,22353],{"class":42,"line":256},[40,22354,22355],{"class":46},"# Get expectation values\n",[40,22357,22358,22361,22363,22366,22368,22370,22372,22374],{"class":42,"line":269},[40,22359,22360],{"class":133},"S_expval ",[40,22362,623],{"class":441},[40,22364,22365],{"class":133}," S_expval_real ",[40,22367,1147],{"class":441},[40,22369,463],{"class":462},[40,22371,1413],{"class":129},[40,22373,7273],{"class":441},[40,22375,22376],{"class":133}," S_expval_imag\n",[40,22378,22379],{"class":42,"line":282},[40,22380,228],{"emptyLinePlaceholder":227},[40,22382,22383,22386,22388],{"class":42,"line":295},[40,22384,22385],{"class":133},"H_expval ",[40,22387,623],{"class":441},[40,22389,1286],{"class":462},[40,22391,22392,22394,22397,22399,22401,22403,22406],{"class":42,"line":308},[40,22393,668],{"class":129},[40,22395,22396],{"class":133}," obs_idx, (pauli, coeff) ",[40,22398,674],{"class":129},[40,22400,997],{"class":441},[40,22402,241],{"class":133},[40,22404,22405],{"class":441},"zip",[40,22407,22408],{"class":133},"(H_op.paulis, H_op.coeffs)):\n",[40,22410,22411],{"class":42,"line":321},[40,22412,22413],{"class":46},"    # Get expectation values from experiment\n",[40,22415,22416,22419,22421,22423,22425,22427],{"class":42,"line":334},[40,22417,22418],{"class":133},"    expval_real ",[40,22420,623],{"class":441},[40,22422,22276],{"class":237},[40,22424,22279],{"class":133},[40,22426,22282],{"class":237},[40,22428,411],{"class":133},[40,22430,22431,22434,22437,22439,22441,22443,22445,22447,22449,22451],{"class":42,"line":347},[40,22432,22433],{"class":237},"        Pauli",[40,22435,22436],{"class":133},"(pauli[::",[40,22438,1407],{"class":441},[40,22440,1410],{"class":462},[40,22442,21959],{"class":133},[40,22444,21962],{"class":237},[40,22446,715],{"class":133},[40,22448,1147],{"class":441},[40,22450,22302],{"class":244},[40,22452,248],{"class":133},[40,22454,22455],{"class":42,"line":352},[40,22456,698],{"class":133},[40,22458,22459,22462,22464,22466,22468,22470],{"class":42,"line":357},[40,22460,22461],{"class":133},"    expval_imag ",[40,22463,623],{"class":441},[40,22465,22276],{"class":237},[40,22467,22279],{"class":133},[40,22469,22282],{"class":237},[40,22471,411],{"class":133},[40,22473,22474,22476,22478,22480,22482,22484,22486,22488,22490,22492],{"class":42,"line":370},[40,22475,22433],{"class":237},[40,22477,22436],{"class":133},[40,22479,1407],{"class":441},[40,22481,1410],{"class":462},[40,22483,21959],{"class":133},[40,22485,21962],{"class":237},[40,22487,715],{"class":133},[40,22489,1147],{"class":441},[40,22491,22340],{"class":244},[40,22493,248],{"class":133},[40,22495,22496],{"class":42,"line":376},[40,22497,698],{"class":133},[40,22499,22500,22503,22505,22508,22510,22512,22514,22516],{"class":42,"line":387},[40,22501,22502],{"class":133},"    expval ",[40,22504,623],{"class":441},[40,22506,22507],{"class":133}," expval_real ",[40,22509,1147],{"class":441},[40,22511,463],{"class":462},[40,22513,1413],{"class":129},[40,22515,7273],{"class":441},[40,22517,22518],{"class":133}," expval_imag\n",[40,22520,22521],{"class":42,"line":392},[40,22522,228],{"emptyLinePlaceholder":227},[40,22524,22525],{"class":42,"line":397},[40,22526,22527],{"class":46},"    # Fill-in matrix elements\n",[40,22529,22530,22533,22535,22538,22540],{"class":42,"line":402},[40,22531,22532],{"class":133},"    H_expval ",[40,22534,1325],{"class":441},[40,22536,22537],{"class":133}," coeff ",[40,22539,1433],{"class":441},[40,22541,22542],{"class":133}," expval\n",[40,22544,22545],{"class":42,"line":414},[40,22546,228],{"emptyLinePlaceholder":227},[40,22548,22549],{"class":42,"line":424},[40,22550,228],{"emptyLinePlaceholder":227},[40,22552,22553,22555],{"class":42,"line":432},[40,22554,7569],{"class":441},[40,22556,22557],{"class":133},"(H_expval)\n",[30,22559,22562],{"className":22560,"code":22561,"language":6083,"meta":35},[7575],"(25+0j)\n",[37,22563,22561],{"__ignoreMap":35},[12,22565,22566,22567,9846,22595,22655],{},"Execute circuits for ",[40,22568,22570,22583],{"className":22569},[1565],[40,22571,22573],{"className":22572},[1569],[1571,22574,22575],{"xmlns":1573},[1575,22576,22577,22581],{},[1578,22578,22579],{},[1584,22580,4901],{},[1591,22582,4901],{"encoding":1593},[40,22584,22586],{"className":22585,"ariaHidden":1599},[1598],[40,22587,22589,22592],{"className":22588},[1603],[40,22590],{"className":22591,"style":1608},[1607],[40,22593,4901],{"className":22594,"style":4936},[1612,1653],[40,22596,22598,22615],{"className":22597},[1565],[40,22599,22601],{"className":22600},[1569],[1571,22602,22603],{"xmlns":1573},[1575,22604,22605,22613],{},[1578,22606,22607],{},[3315,22608,22609,22611],{"accent":1599},[1584,22610,2182],{},[1731,22612,3321],{},[1591,22614,3324],{"encoding":1593},[40,22616,22618],{"className":22617,"ariaHidden":1599},[1598],[40,22619,22621,22624],{"className":22620},[1603],[40,22622],{"className":22623,"style":3334},[1607],[40,22625,22627],{"className":22626},[1612,3338],[40,22628,22630],{"className":22629},[1625],[40,22631,22633],{"className":22632},[1629],[40,22634,22636,22644],{"className":22635,"style":3334},[1633],[40,22637,22638,22641],{"style":3350},[40,22639],{"className":22640,"style":3354},[1641],[40,22642,2182],{"className":22643,"style":2197},[1612,1653],[40,22645,22646,22649],{"style":3360},[40,22647],{"className":22648,"style":3354},[1641],[40,22650,22652],{"className":22651,"style":3368},[3367],[40,22653,3321],{"className":22654},[1612]," with Estimator",[30,22657,22659],{"className":32,"code":22658,"language":34,"meta":35,"style":35},"# 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",[37,22660,22661,22666,22676,22685,22695,22718,22745,22749,22753,22763,22778,22789,22797,22805,22813,22818,22829,22843,22850,22862,22870,22876,22882,22886,22892,22904,22921,22925,22930,22943,22951,22958,22970,22974,22978,23006,23010,23014],{"__ignoreMap":35},[40,22662,22663],{"class":42,"line":43},[40,22664,22665],{"class":46},"# Experiment options\n",[40,22667,22668,22671,22673],{"class":42,"line":50},[40,22669,22670],{"class":133},"num_randomizations ",[40,22672,623],{"class":441},[40,22674,22675],{"class":462}," 300\n",[40,22677,22678,22681,22683],{"class":42,"line":155},[40,22679,22680],{"class":133},"num_randomizations_learning ",[40,22682,623],{"class":441},[40,22684,7213],{"class":462},[40,22686,22687,22690,22692],{"class":42,"line":168},[40,22688,22689],{"class":133},"shots_per_randomization ",[40,22691,623],{"class":441},[40,22693,22694],{"class":462}," 100\n",[40,22696,22697,22700,22702,22704,22706,22708,22711,22713,22716],{"class":42,"line":182},[40,22698,22699],{"class":133},"noise_factors ",[40,22701,623],{"class":441},[40,22703,7471],{"class":133},[40,22705,1410],{"class":462},[40,22707,917],{"class":133},[40,22709,22710],{"class":462},"1.2",[40,22712,917],{"class":133},[40,22714,22715],{"class":462},"1.4",[40,22717,833],{"class":133},[40,22719,22720,22723,22725,22727,22729,22731,22733,22735,22738,22740,22743],{"class":42,"line":195},[40,22721,22722],{"class":133},"learning_pair_depths ",[40,22724,623],{"class":441},[40,22726,7471],{"class":133},[40,22728,791],{"class":462},[40,22730,917],{"class":133},[40,22732,11199],{"class":462},[40,22734,917],{"class":133},[40,22736,22737],{"class":462},"24",[40,22739,917],{"class":133},[40,22741,22742],{"class":462},"48",[40,22744,833],{"class":133},[40,22746,22747],{"class":42,"line":203},[40,22748,228],{"emptyLinePlaceholder":227},[40,22750,22751],{"class":42,"line":216},[40,22752,228],{"emptyLinePlaceholder":227},[40,22754,22755,22758,22760],{"class":42,"line":224},[40,22756,22757],{"class":133},"experimental_opts ",[40,22759,623],{"class":441},[40,22761,22762],{"class":133}," {}\n",[40,22764,22765,22768,22771,22773,22775],{"class":42,"line":231},[40,22766,22767],{"class":133},"experimental_opts[",[40,22769,22770],{"class":244},"\"resilience\"",[40,22772,7250],{"class":133},[40,22774,623],{"class":441},[40,22776,22777],{"class":133}," {\n",[40,22779,22780,22783,22785,22787],{"class":42,"line":251},[40,22781,22782],{"class":244},"    \"measure_mitigation\"",[40,22784,438],{"class":133},[40,22786,12926],{"class":462},[40,22788,445],{"class":133},[40,22790,22791,22794],{"class":42,"line":256},[40,22792,22793],{"class":244},"    \"measure_noise_learning\"",[40,22795,22796],{"class":133},": {\n",[40,22798,22799,22802],{"class":42,"line":269},[40,22800,22801],{"class":244},"        \"num_randomizations\"",[40,22803,22804],{"class":133},": num_randomizations_learning,\n",[40,22806,22807,22810],{"class":42,"line":282},[40,22808,22809],{"class":244},"        \"shots_per_randomization\"",[40,22811,22812],{"class":133},": shots_per_randomization,\n",[40,22814,22815],{"class":42,"line":295},[40,22816,22817],{"class":133},"    },\n",[40,22819,22820,22823,22825,22827],{"class":42,"line":308},[40,22821,22822],{"class":244},"    \"zne_mitigation\"",[40,22824,438],{"class":133},[40,22826,12926],{"class":462},[40,22828,445],{"class":133},[40,22830,22831,22834,22837,22840],{"class":42,"line":321},[40,22832,22833],{"class":244},"    \"zne\"",[40,22835,22836],{"class":133},": {",[40,22838,22839],{"class":244},"\"noise_factors\"",[40,22841,22842],{"class":133},": noise_factors},\n",[40,22844,22845,22848],{"class":42,"line":334},[40,22846,22847],{"class":244},"    \"layer_noise_learning\"",[40,22849,22796],{"class":133},[40,22851,22852,22855,22857,22860],{"class":42,"line":347},[40,22853,22854],{"class":244},"        \"max_layers_to_learn\"",[40,22856,438],{"class":133},[40,22858,22859],{"class":462},"10",[40,22861,445],{"class":133},[40,22863,22864,22867],{"class":42,"line":352},[40,22865,22866],{"class":244},"        \"layer_pair_depths\"",[40,22868,22869],{"class":133},": learning_pair_depths,\n",[40,22871,22872,22874],{"class":42,"line":357},[40,22873,22809],{"class":244},[40,22875,22812],{"class":133},[40,22877,22878,22880],{"class":42,"line":370},[40,22879,22801],{"class":244},[40,22881,22804],{"class":133},[40,22883,22884],{"class":42,"line":376},[40,22885,22817],{"class":133},[40,22887,22888,22890],{"class":42,"line":387},[40,22889,22833],{"class":244},[40,22891,22796],{"class":133},[40,22893,22894,22897,22899,22902],{"class":42,"line":392},[40,22895,22896],{"class":244},"        \"amplifier\"",[40,22898,438],{"class":133},[40,22900,22901],{"class":244},"\"pea\"",[40,22903,445],{"class":133},[40,22905,22906,22909,22912,22914,22916,22918],{"class":42,"line":397},[40,22907,22908],{"class":244},"        \"extrapolated_noise_factors\"",[40,22910,22911],{"class":133},": [",[40,22913,791],{"class":462},[40,22915,7250],{"class":133},[40,22917,1147],{"class":441},[40,22919,22920],{"class":133}," noise_factors,\n",[40,22922,22923],{"class":42,"line":402},[40,22924,22817],{"class":133},[40,22926,22927],{"class":42,"line":414},[40,22928,22929],{"class":133},"}\n",[40,22931,22932,22934,22937,22939,22941],{"class":42,"line":424},[40,22933,22767],{"class":133},[40,22935,22936],{"class":244},"\"twirling\"",[40,22938,7250],{"class":133},[40,22940,623],{"class":441},[40,22942,22777],{"class":133},[40,22944,22945,22948],{"class":42,"line":432},[40,22946,22947],{"class":244},"    \"num_randomizations\"",[40,22949,22950],{"class":133},": num_randomizations,\n",[40,22952,22953,22956],{"class":42,"line":448},[40,22954,22955],{"class":244},"    \"shots_per_randomization\"",[40,22957,22812],{"class":133},[40,22959,22960,22963,22965,22968],{"class":42,"line":468},[40,22961,22962],{"class":244},"    \"strategy\"",[40,22964,438],{"class":133},[40,22966,22967],{"class":244},"\"all\"",[40,22969,445],{"class":133},[40,22971,22972],{"class":42,"line":486},[40,22973,22929],{"class":133},[40,22975,22976],{"class":42,"line":502},[40,22977,228],{"emptyLinePlaceholder":227},[40,22979,22980,22983,22985,22988,22990,22993,22995,22998,23001,23003],{"class":42,"line":508},[40,22981,22982],{"class":133},"estimator ",[40,22984,623],{"class":441},[40,22986,22987],{"class":237}," Estimator",[40,22989,241],{"class":133},[40,22991,22992],{"class":1195},"mode",[40,22994,623],{"class":441},[40,22996,22997],{"class":133},"backend, ",[40,22999,23000],{"class":1195},"options",[40,23002,623],{"class":441},[40,23004,23005],{"class":133},"experimental_opts)\n",[40,23007,23008],{"class":42,"line":514},[40,23009,228],{"emptyLinePlaceholder":227},[40,23011,23012],{"class":42,"line":519},[40,23013,228],{"emptyLinePlaceholder":227},[40,23015,23016,23019,23021,23024,23026],{"class":42,"line":525},[40,23017,23018],{"class":133},"job ",[40,23020,623],{"class":441},[40,23022,23023],{"class":133}," estimator.",[40,23025,21554],{"class":237},[40,23027,23028],{"class":133},"([pub])\n",[55,23030,23032],{"id":23031},"step-4-post-process-and-return-result-in-desired-classical-format","Step 4: Post-process and return result in desired classical format",[30,23034,23036],{"className":32,"code":23035,"language":34,"meta":35,"style":35},"results = job.result()[0]\n",[37,23037,23038],{"__ignoreMap":35},[40,23039,23040,23043,23045,23048,23051,23054,23056],{"class":42,"line":43},[40,23041,23042],{"class":133},"results ",[40,23044,623],{"class":441},[40,23046,23047],{"class":133}," job.",[40,23049,23050],{"class":237},"result",[40,23052,23053],{"class":133},"()[",[40,23055,791],{"class":462},[40,23057,833],{"class":133},[1555,23059,23061],{"id":23060},"calculate-effective-hamiltonian-and-overlap-matrices","Calculate Effective Hamiltonian and Overlap matrices",[12,23063,23064,23065,23101],{},"First calculate the phase accumulated by the ",[40,23066,23068,23086],{"className":23067},[1565],[40,23069,23071],{"className":23070},[1569],[1571,23072,23073],{"xmlns":1573},[1575,23074,23075,23083],{},[1578,23076,23077,23079,23081],{},[1584,23078,1791],{"mathvariant":1790},[1735,23080,791],{},[1731,23082,1798],{"stretchy":1797},[1591,23084,23085],{"encoding":1593},"\\vert 0 \\rangle",[40,23087,23089],{"className":23088,"ariaHidden":1599},[1598],[40,23090,23092,23095,23098],{"className":23091},[1603],[40,23093],{"className":23094,"style":1811},[1607],[40,23096,8782],{"className":23097},[1612],[40,23099,1798],{"className":23100},[1822]," state during the uncontrolled time evolution",[30,23103,23105],{"className":32,"code":23104,"language":34,"meta":35,"style":35},"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",[37,23106,23107,23116,23171,23188],{"__ignoreMap":35},[40,23108,23109,23112,23114],{"class":42,"line":43},[40,23110,23111],{"class":133},"prefactors ",[40,23113,623],{"class":441},[40,23115,1217],{"class":133},[40,23117,23118,23121,23124,23126,23128,23130,23132,23134,23136,23139,23141,23144,23146,23148,23150,23152,23154,23157,23159,23162,23164,23166,23168],{"class":42,"line":50},[40,23119,23120],{"class":133},"    np.",[40,23122,23123],{"class":237},"exp",[40,23125,241],{"class":133},[40,23127,1407],{"class":441},[40,23129,1410],{"class":462},[40,23131,1413],{"class":129},[40,23133,7273],{"class":441},[40,23135,7860],{"class":441},[40,23137,23138],{"class":133},"([c ",[40,23140,668],{"class":129},[40,23142,23143],{"class":133}," p, c ",[40,23145,674],{"class":129},[40,23147,964],{"class":133},[40,23149,967],{"class":237},[40,23151,715],{"class":133},[40,23153,683],{"class":129},[40,23155,23156],{"class":244}," \"Z\"",[40,23158,21036],{"class":129},[40,23160,23161],{"class":133}," p]) ",[40,23163,1433],{"class":441},[40,23165,7290],{"class":133},[40,23167,1433],{"class":441},[40,23169,23170],{"class":133}," dt)\n",[40,23172,23173,23175,23177,23179,23181,23183,23185],{"class":42,"line":155},[40,23174,956],{"class":129},[40,23176,7290],{"class":133},[40,23178,674],{"class":129},[40,23180,7265],{"class":441},[40,23182,241],{"class":133},[40,23184,1410],{"class":462},[40,23186,23187],{"class":133},", krylov_dim)\n",[40,23189,23190],{"class":42,"line":168},[40,23191,833],{"class":133},[12,23193,23194,23195],{},"Once we have the results of the circuit executions we can post-process the data to calculate the matrix elements of ",[40,23196,23198,23211],{"className":23197},[1565],[40,23199,23201],{"className":23200},[1569],[1571,23202,23203],{"xmlns":1573},[1575,23204,23205,23209],{},[1578,23206,23207],{},[1584,23208,4901],{},[1591,23210,4901],{"encoding":1593},[40,23212,23214],{"className":23213,"ariaHidden":1599},[1598],[40,23215,23217,23220],{"className":23216},[1603],[40,23218],{"className":23219,"style":1608},[1607],[40,23221,4901],{"className":23222,"style":4936},[1612,1653],[30,23224,23226],{"className":32,"code":23225,"language":34,"meta":35,"style":35},"# 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",[37,23227,23228,23233,23253,23273,23277,23282,23301,23305,23323,23328,23335,23351,23355,23361,23365,23370,23388,23408,23412,23431,23435,23439,23459,23463,23468,23498,23510,23525,23531],{"__ignoreMap":35},[40,23229,23230],{"class":42,"line":43},[40,23231,23232],{"class":46},"# Assemble S, the overlap matrix of dimension D:\n",[40,23234,23235,23238,23240,23242,23244,23247,23249,23251],{"class":42,"line":50},[40,23236,23237],{"class":133},"S_first_row ",[40,23239,623],{"class":441},[40,23241,654],{"class":133},[40,23243,1189],{"class":237},[40,23245,23246],{"class":133},"(krylov_dim, ",[40,23248,1196],{"class":1195},[40,23250,1199],{"class":441},[40,23252,248],{"class":133},[40,23254,23255,23258,23260,23262,23264,23266,23268,23270],{"class":42,"line":155},[40,23256,23257],{"class":133},"S_first_row[",[40,23259,791],{"class":462},[40,23261,7250],{"class":133},[40,23263,623],{"class":441},[40,23265,463],{"class":462},[40,23267,7357],{"class":441},[40,23269,20308],{"class":462},[40,23271,23272],{"class":129},"j\n",[40,23274,23275],{"class":42,"line":168},[40,23276,228],{"emptyLinePlaceholder":227},[40,23278,23279],{"class":42,"line":182},[40,23280,23281],{"class":46},"# Add in ancilla-only measurements:\n",[40,23283,23284,23286,23288,23290,23292,23295,23297,23299],{"class":42,"line":195},[40,23285,668],{"class":129},[40,23287,7290],{"class":133},[40,23289,674],{"class":129},[40,23291,7265],{"class":441},[40,23293,23294],{"class":133},"(krylov_dim ",[40,23296,1407],{"class":441},[40,23298,463],{"class":462},[40,23300,923],{"class":133},[40,23302,23303],{"class":42,"line":203},[40,23304,22413],{"class":46},[40,23306,23307,23309,23311,23314,23316,23318,23320],{"class":42,"line":216},[40,23308,22418],{"class":133},[40,23310,623],{"class":441},[40,23312,23313],{"class":133}," results.data.evs[",[40,23315,791],{"class":462},[40,23317,794],{"class":133},[40,23319,791],{"class":462},[40,23321,23322],{"class":133},"][\n",[40,23324,23325],{"class":42,"line":224},[40,23326,23327],{"class":133},"        i\n",[40,23329,23330,23332],{"class":42,"line":231},[40,23331,1267],{"class":133},[40,23333,23334],{"class":46},"# automatic extrapolated evs if ZNE is used\n",[40,23336,23337,23339,23341,23343,23345,23347,23349],{"class":42,"line":251},[40,23338,22461],{"class":133},[40,23340,623],{"class":441},[40,23342,23313],{"class":133},[40,23344,1410],{"class":462},[40,23346,794],{"class":133},[40,23348,791],{"class":462},[40,23350,23322],{"class":133},[40,23352,23353],{"class":42,"line":256},[40,23354,23327],{"class":133},[40,23356,23357,23359],{"class":42,"line":269},[40,23358,1267],{"class":133},[40,23360,23334],{"class":46},[40,23362,23363],{"class":42,"line":282},[40,23364,228],{"emptyLinePlaceholder":227},[40,23366,23367],{"class":42,"line":295},[40,23368,23369],{"class":46},"    # Get expectation values\n",[40,23371,23372,23374,23376,23378,23380,23382,23384,23386],{"class":42,"line":308},[40,23373,22502],{"class":133},[40,23375,623],{"class":441},[40,23377,22507],{"class":133},[40,23379,1147],{"class":441},[40,23381,463],{"class":462},[40,23383,1413],{"class":129},[40,23385,7273],{"class":441},[40,23387,22518],{"class":133},[40,23389,23390,23393,23395,23397,23399,23401,23404,23406],{"class":42,"line":321},[40,23391,23392],{"class":133},"    S_first_row[i ",[40,23394,1147],{"class":441},[40,23396,463],{"class":462},[40,23398,7250],{"class":133},[40,23400,1325],{"class":441},[40,23402,23403],{"class":133}," prefactors[i] ",[40,23405,1433],{"class":441},[40,23407,22542],{"class":133},[40,23409,23410],{"class":42,"line":334},[40,23411,228],{"emptyLinePlaceholder":227},[40,23413,23414,23417,23419,23422,23425,23428],{"class":42,"line":347},[40,23415,23416],{"class":133},"S_first_row_list ",[40,23418,623],{"class":441},[40,23420,23421],{"class":133}," S_first_row.",[40,23423,23424],{"class":237},"tolist",[40,23426,23427],{"class":133},"()  ",[40,23429,23430],{"class":46},"# for saving purposes\n",[40,23432,23433],{"class":42,"line":352},[40,23434,228],{"emptyLinePlaceholder":227},[40,23436,23437],{"class":42,"line":357},[40,23438,228],{"emptyLinePlaceholder":227},[40,23440,23441,23444,23446,23448,23450,23453,23455,23457],{"class":42,"line":370},[40,23442,23443],{"class":133},"S_circ ",[40,23445,623],{"class":441},[40,23447,654],{"class":133},[40,23449,1189],{"class":237},[40,23451,23452],{"class":133},"((krylov_dim, krylov_dim), ",[40,23454,1196],{"class":1195},[40,23456,1199],{"class":441},[40,23458,248],{"class":133},[40,23460,23461],{"class":42,"line":376},[40,23462,228],{"emptyLinePlaceholder":227},[40,23464,23465],{"class":42,"line":387},[40,23466,23467],{"class":46},"# Distribute entries from first row across matrix:\n",[40,23469,23470,23472,23475,23477,23479,23482,23484,23486,23489,23492,23494,23496],{"class":42,"line":392},[40,23471,668],{"class":129},[40,23473,23474],{"class":133}," i, j ",[40,23476,674],{"class":129},[40,23478,1236],{"class":133},[40,23480,23481],{"class":237},"product",[40,23483,241],{"class":133},[40,23485,1244],{"class":441},[40,23487,23488],{"class":133},"(krylov_dim), ",[40,23490,23491],{"class":1195},"repeat",[40,23493,623],{"class":441},[40,23495,1879],{"class":462},[40,23497,923],{"class":133},[40,23499,23500,23502,23504,23507],{"class":42,"line":397},[40,23501,755],{"class":129},[40,23503,7290],{"class":133},[40,23505,23506],{"class":441},">=",[40,23508,23509],{"class":133}," j:\n",[40,23511,23512,23515,23517,23520,23522],{"class":42,"line":402},[40,23513,23514],{"class":133},"        S_circ[j, i] ",[40,23516,623],{"class":441},[40,23518,23519],{"class":133}," S_first_row[i ",[40,23521,1407],{"class":441},[40,23523,23524],{"class":133}," j]\n",[40,23526,23527,23529],{"class":42,"line":414},[40,23528,839],{"class":129},[40,23530,766],{"class":133},[40,23532,23533,23535,23537,23539,23541,23544,23546],{"class":42,"line":424},[40,23534,23514],{"class":133},[40,23536,623],{"class":441},[40,23538,654],{"class":133},[40,23540,712],{"class":237},[40,23542,23543],{"class":133},"(S_first_row[j ",[40,23545,1407],{"class":441},[40,23547,23548],{"class":133}," i])\n",[30,23550,23552],{"className":32,"code":23551,"language":34,"meta":35,"style":35},"Matrix(S_circ)\n",[37,23553,23554],{"__ignoreMap":35},[40,23555,23556,23559],{"class":42,"line":43},[40,23557,23558],{"class":237},"Matrix",[40,23560,23561],{"class":133},"(S_circ)\n",[12,23563,23564,23565],{},"And the matrix elements of ",[40,23566,23568,23585],{"className":23567},[1565],[40,23569,23571],{"className":23570},[1569],[1571,23572,23573],{"xmlns":1573},[1575,23574,23575,23583],{},[1578,23576,23577],{},[3315,23578,23579,23581],{"accent":1599},[1584,23580,2182],{},[1731,23582,3321],{},[1591,23584,3324],{"encoding":1593},[40,23586,23588],{"className":23587,"ariaHidden":1599},[1598],[40,23589,23591,23594],{"className":23590},[1603],[40,23592],{"className":23593,"style":3334},[1607],[40,23595,23597],{"className":23596},[1612,3338],[40,23598,23600],{"className":23599},[1625],[40,23601,23603],{"className":23602},[1629],[40,23604,23606,23614],{"className":23605,"style":3334},[1633],[40,23607,23608,23611],{"style":3350},[40,23609],{"className":23610,"style":3354},[1641],[40,23612,2182],{"className":23613,"style":2197},[1612,1653],[40,23615,23616,23619],{"style":3360},[40,23617],{"className":23618,"style":3354},[1641],[40,23620,23622],{"className":23621,"style":3368},[3367],[40,23623,3321],{"className":23624},[1612],[30,23626,23628],{"className":32,"code":23627,"language":34,"meta":35,"style":35},"# 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",[37,23629,23630,23634,23653,23667,23671,23687,23692,23710,23715,23739,23744,23751,23781,23785,23791,23795,23800,23819,23842,23846,23860,23864,23883,23887,23891,23917,23927,23941,23947],{"__ignoreMap":35},[40,23631,23632],{"class":42,"line":43},[40,23633,23232],{"class":46},[40,23635,23636,23639,23641,23643,23645,23647,23649,23651],{"class":42,"line":50},[40,23637,23638],{"class":133},"H_first_row ",[40,23640,623],{"class":441},[40,23642,654],{"class":133},[40,23644,1189],{"class":237},[40,23646,23246],{"class":133},[40,23648,1196],{"class":1195},[40,23650,1199],{"class":441},[40,23652,248],{"class":133},[40,23654,23655,23658,23660,23662,23664],{"class":42,"line":155},[40,23656,23657],{"class":133},"H_first_row[",[40,23659,791],{"class":462},[40,23661,7250],{"class":133},[40,23663,623],{"class":441},[40,23665,23666],{"class":133}," H_expval\n",[40,23668,23669],{"class":42,"line":168},[40,23670,228],{"emptyLinePlaceholder":227},[40,23672,23673,23675,23677,23679,23681,23683,23685],{"class":42,"line":182},[40,23674,668],{"class":129},[40,23676,22396],{"class":133},[40,23678,674],{"class":129},[40,23680,997],{"class":441},[40,23682,241],{"class":133},[40,23684,22405],{"class":441},[40,23686,22408],{"class":133},[40,23688,23689],{"class":42,"line":195},[40,23690,23691],{"class":46},"    # Add in ancilla-only measurements:\n",[40,23693,23694,23696,23698,23700,23702,23704,23706,23708],{"class":42,"line":203},[40,23695,956],{"class":129},[40,23697,7290],{"class":133},[40,23699,674],{"class":129},[40,23701,7265],{"class":441},[40,23703,23294],{"class":133},[40,23705,1407],{"class":441},[40,23707,463],{"class":462},[40,23709,923],{"class":133},[40,23711,23712],{"class":42,"line":216},[40,23713,23714],{"class":46},"        # Get expectation values from experiment\n",[40,23716,23717,23720,23722,23724,23726,23728,23730,23732,23735,23737],{"class":42,"line":224},[40,23718,23719],{"class":133},"        expval_real ",[40,23721,623],{"class":441},[40,23723,23313],{"class":133},[40,23725,1879],{"class":462},[40,23727,7357],{"class":441},[40,23729,1356],{"class":462},[40,23731,7273],{"class":441},[40,23733,23734],{"class":133}," obs_idx][",[40,23736,791],{"class":462},[40,23738,23322],{"class":133},[40,23740,23741],{"class":42,"line":231},[40,23742,23743],{"class":133},"            i\n",[40,23745,23746,23749],{"class":42,"line":251},[40,23747,23748],{"class":133},"        ]  ",[40,23750,23334],{"class":46},[40,23752,23753,23756,23758,23760,23762,23764,23766,23768,23771,23773,23775,23777,23779],{"class":42,"line":256},[40,23754,23755],{"class":133},"        expval_imag ",[40,23757,623],{"class":441},[40,23759,23313],{"class":133},[40,23761,1879],{"class":462},[40,23763,7357],{"class":441},[40,23765,1356],{"class":462},[40,23767,7273],{"class":441},[40,23769,23770],{"class":133}," obs_idx ",[40,23772,1147],{"class":441},[40,23774,463],{"class":462},[40,23776,794],{"class":133},[40,23778,791],{"class":462},[40,23780,23322],{"class":133},[40,23782,23783],{"class":42,"line":269},[40,23784,23743],{"class":133},[40,23786,23787,23789],{"class":42,"line":282},[40,23788,23748],{"class":133},[40,23790,23334],{"class":46},[40,23792,23793],{"class":42,"line":295},[40,23794,228],{"emptyLinePlaceholder":227},[40,23796,23797],{"class":42,"line":308},[40,23798,23799],{"class":46},"        # Get expectation values\n",[40,23801,23802,23805,23807,23809,23811,23813,23815,23817],{"class":42,"line":321},[40,23803,23804],{"class":133},"        expval ",[40,23806,623],{"class":441},[40,23808,22507],{"class":133},[40,23810,1147],{"class":441},[40,23812,463],{"class":462},[40,23814,1413],{"class":129},[40,23816,7273],{"class":441},[40,23818,22518],{"class":133},[40,23820,23821,23824,23826,23828,23830,23832,23834,23836,23838,23840],{"class":42,"line":334},[40,23822,23823],{"class":133},"        H_first_row[i ",[40,23825,1147],{"class":441},[40,23827,463],{"class":462},[40,23829,7250],{"class":133},[40,23831,1325],{"class":441},[40,23833,23403],{"class":133},[40,23835,1433],{"class":441},[40,23837,22537],{"class":133},[40,23839,1433],{"class":441},[40,23841,22542],{"class":133},[40,23843,23844],{"class":42,"line":347},[40,23845,228],{"emptyLinePlaceholder":227},[40,23847,23848,23851,23853,23856,23858],{"class":42,"line":352},[40,23849,23850],{"class":133},"H_first_row_list ",[40,23852,623],{"class":441},[40,23854,23855],{"class":133}," H_first_row.",[40,23857,23424],{"class":237},[40,23859,12875],{"class":133},[40,23861,23862],{"class":42,"line":357},[40,23863,228],{"emptyLinePlaceholder":227},[40,23865,23866,23869,23871,23873,23875,23877,23879,23881],{"class":42,"line":370},[40,23867,23868],{"class":133},"H_eff_circ ",[40,23870,623],{"class":441},[40,23872,654],{"class":133},[40,23874,1189],{"class":237},[40,23876,23452],{"class":133},[40,23878,1196],{"class":1195},[40,23880,1199],{"class":441},[40,23882,248],{"class":133},[40,23884,23885],{"class":42,"line":376},[40,23886,228],{"emptyLinePlaceholder":227},[40,23888,23889],{"class":42,"line":387},[40,23890,23467],{"class":46},[40,23892,23893,23895,23897,23899,23901,23903,23905,23907,23909,23911,23913,23915],{"class":42,"line":392},[40,23894,668],{"class":129},[40,23896,23474],{"class":133},[40,23898,674],{"class":129},[40,23900,1236],{"class":133},[40,23902,23481],{"class":237},[40,23904,241],{"class":133},[40,23906,1244],{"class":441},[40,23908,23488],{"class":133},[40,23910,23491],{"class":1195},[40,23912,623],{"class":441},[40,23914,1879],{"class":462},[40,23916,923],{"class":133},[40,23918,23919,23921,23923,23925],{"class":42,"line":397},[40,23920,755],{"class":129},[40,23922,7290],{"class":133},[40,23924,23506],{"class":441},[40,23926,23509],{"class":133},[40,23928,23929,23932,23934,23937,23939],{"class":42,"line":402},[40,23930,23931],{"class":133},"        H_eff_circ[j, i] ",[40,23933,623],{"class":441},[40,23935,23936],{"class":133}," H_first_row[i ",[40,23938,1407],{"class":441},[40,23940,23524],{"class":133},[40,23942,23943,23945],{"class":42,"line":414},[40,23944,839],{"class":129},[40,23946,766],{"class":133},[40,23948,23949,23951,23953,23955,23957,23960,23962],{"class":42,"line":424},[40,23950,23931],{"class":133},[40,23952,623],{"class":441},[40,23954,654],{"class":133},[40,23956,712],{"class":237},[40,23958,23959],{"class":133},"(H_first_row[j ",[40,23961,1407],{"class":441},[40,23963,23548],{"class":133},[30,23965,23967],{"className":32,"code":23966,"language":34,"meta":35,"style":35},"Matrix(H_eff_circ)\n",[37,23968,23969],{"__ignoreMap":35},[40,23970,23971,23973],{"class":42,"line":43},[40,23972,23558],{"class":237},[40,23974,23975],{"class":133},"(H_eff_circ)\n",[12,23977,23978,23979,24039],{},"Finally, we can solve the generalized eigenvalue problem for ",[40,23980,23982,23999],{"className":23981},[1565],[40,23983,23985],{"className":23984},[1569],[1571,23986,23987],{"xmlns":1573},[1575,23988,23989,23997],{},[1578,23990,23991],{},[3315,23992,23993,23995],{"accent":1599},[1584,23994,2182],{},[1731,23996,3321],{},[1591,23998,3324],{"encoding":1593},[40,24000,24002],{"className":24001,"ariaHidden":1599},[1598],[40,24003,24005,24008],{"className":24004},[1603],[40,24006],{"className":24007,"style":3334},[1607],[40,24009,24011],{"className":24010},[1612,3338],[40,24012,24014],{"className":24013},[1625],[40,24015,24017],{"className":24016},[1629],[40,24018,24020,24028],{"className":24019,"style":3334},[1633],[40,24021,24022,24025],{"style":3350},[40,24023],{"className":24024,"style":3354},[1641],[40,24026,2182],{"className":24027,"style":2197},[1612,1653],[40,24029,24030,24033],{"style":3360},[40,24031],{"className":24032,"style":3354},[1641],[40,24034,24036],{"className":24035,"style":3368},[3367],[40,24037,3321],{"className":24038},[1612],":",[12,24041,24042],{},[40,24043,24045,24081],{"className":24044},[1565],[40,24046,24048],{"className":24047},[1569],[1571,24049,24050],{"xmlns":1573},[1575,24051,24052,24078],{},[1578,24053,24054,24060,24066,24068,24070,24072],{},[3315,24055,24056,24058],{"accent":1599},[1584,24057,2182],{},[1731,24059,3321],{},[3315,24061,24062,24064],{"accent":1599},[1584,24063,5218],{},[1731,24065,5221],{},[1731,24067,623],{},[1584,24069,5218],{},[1584,24071,4901],{},[3315,24073,24074,24076],{"accent":1599},[1584,24075,5218],{},[1731,24077,5221],{},[1591,24079,24080],{"encoding":1593},"\\tilde{H} \\vec{c} = c S \\vec{c}",[40,24082,24084,24165],{"className":24083,"ariaHidden":1599},[1598],[40,24085,24087,24090,24121,24156,24159,24162],{"className":24086},[1603],[40,24088],{"className":24089,"style":3334},[1607],[40,24091,24093],{"className":24092},[1612,3338],[40,24094,24096],{"className":24095},[1625],[40,24097,24099],{"className":24098},[1629],[40,24100,24102,24110],{"className":24101,"style":3334},[1633],[40,24103,24104,24107],{"style":3350},[40,24105],{"className":24106,"style":3354},[1641],[40,24108,2182],{"className":24109,"style":2197},[1612,1653],[40,24111,24112,24115],{"style":3360},[40,24113],{"className":24114,"style":3354},[1641],[40,24116,24118],{"className":24117,"style":3368},[3367],[40,24119,3321],{"className":24120},[1612],[40,24122,24124],{"className":24123},[1612,3338],[40,24125,24127],{"className":24126},[1625],[40,24128,24130],{"className":24129},[1629],[40,24131,24133,24141],{"className":24132,"style":5301},[1633],[40,24134,24135,24138],{"style":3350},[40,24136],{"className":24137,"style":3354},[1641],[40,24139,5218],{"className":24140},[1612,1653],[40,24142,24143,24146],{"style":3350},[40,24144],{"className":24145,"style":3354},[1641],[40,24147,24149],{"className":24148,"style":5318},[3367],[40,24150,24152],{"className":24151,"style":5323},[5322],[5325,24153,24154],{"xmlns":5327,"width":5328,"height":5329,"style":5330,"viewBox":5331,"preserveAspectRatio":5332},[5334,24155],{"d":5336},[40,24157],{"className":24158,"style":1963},[1756],[40,24160,623],{"className":24161},[1967],[40,24163],{"className":24164,"style":1963},[1756],[40,24166,24168,24171,24174,24177],{"className":24167},[1603],[40,24169],{"className":24170,"style":5301},[1607],[40,24172,5218],{"className":24173},[1612,1653],[40,24175,4901],{"className":24176,"style":4936},[1612,1653],[40,24178,24180],{"className":24179},[1612,3338],[40,24181,24183],{"className":24182},[1625],[40,24184,24186],{"className":24185},[1629],[40,24187,24189,24197],{"className":24188,"style":5301},[1633],[40,24190,24191,24194],{"style":3350},[40,24192],{"className":24193,"style":3354},[1641],[40,24195,5218],{"className":24196},[1612,1653],[40,24198,24199,24202],{"style":3350},[40,24200],{"className":24201,"style":3354},[1641],[40,24203,24205],{"className":24204,"style":5318},[3367],[40,24206,24208],{"className":24207,"style":5323},[5322],[5325,24209,24210],{"xmlns":5327,"width":5328,"height":5329,"style":5330,"viewBox":5331,"preserveAspectRatio":5332},[5334,24211],{"d":5336},[12,24213,24214,24215],{},"and get an estimate of the ground state energy ",[40,24216,24218,24242],{"className":24217},[1565],[40,24219,24221],{"className":24220},[1569],[1571,24222,24223],{"xmlns":1573},[1575,24224,24225,24239],{},[1578,24226,24227],{},[2212,24228,24229,24231],{},[1584,24230,5218],{},[1578,24232,24233,24235,24237],{},[1584,24234,3461],{},[1584,24236,2337],{},[1584,24238,3464],{},[1591,24240,24241],{"encoding":1593},"c_{min}",[40,24243,24245],{"className":24244,"ariaHidden":1599},[1598],[40,24246,24248,24252],{"className":24247},[1603],[40,24249],{"className":24250,"style":24251},[1607],"height:0.5806em;vertical-align:-0.15em;",[40,24253,24255,24258],{"className":24254},[1612],[40,24256,5218],{"className":24257},[1612,1653],[40,24259,24261],{"className":24260},[1621],[40,24262,24264,24288],{"className":24263},[1625,2244],[40,24265,24267,24285],{"className":24266},[1629],[40,24268,24270],{"className":24269,"style":5846},[1633],[40,24271,24272,24275],{"style":4442},[40,24273],{"className":24274,"style":1642},[1641],[40,24276,24278],{"className":24277},[1646,1647,1648,1649],[40,24279,24281],{"className":24280},[1612,1649],[40,24282,24284],{"className":24283},[1612,1653,1649],"min",[40,24286,2269],{"className":24287},[2268],[40,24289,24291],{"className":24290},[1629],[40,24292,24294],{"className":24293,"style":2276},[1633],[40,24295],{},[30,24297,24299],{"className":32,"code":24298,"language":34,"meta":35,"style":35},"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",[37,24300,24301,24310,24334,24339,24350,24363,24367,24377],{"__ignoreMap":35},[40,24302,24303,24306,24308],{"class":42,"line":43},[40,24304,24305],{"class":133},"gnd_en_circ_est_list ",[40,24307,623],{"class":441},[40,24309,950],{"class":133},[40,24311,24312,24314,24317,24319,24321,24323,24325,24328,24330,24332],{"class":42,"line":50},[40,24313,668],{"class":129},[40,24315,24316],{"class":133}," d ",[40,24318,674],{"class":129},[40,24320,7265],{"class":441},[40,24322,241],{"class":133},[40,24324,1410],{"class":462},[40,24326,24327],{"class":133},", krylov_dim ",[40,24329,1147],{"class":441},[40,24331,463],{"class":462},[40,24333,923],{"class":133},[40,24335,24336],{"class":42,"line":155},[40,24337,24338],{"class":46},"    # Solve generalized eigenvalue problem for different size of the Krylov space\n",[40,24340,24341,24344,24346,24348],{"class":42,"line":168},[40,24342,24343],{"class":133},"    gnd_en_circ_est ",[40,24345,623],{"class":441},[40,24347,408],{"class":237},[40,24349,411],{"class":133},[40,24351,24352,24355,24358,24360],{"class":42,"line":182},[40,24353,24354],{"class":133},"        H_eff_circ[:d, :d], S_circ[:d, :d], ",[40,24356,24357],{"class":1195},"threshold",[40,24359,623],{"class":441},[40,24361,24362],{"class":462},"9e-1\n",[40,24364,24365],{"class":42,"line":195},[40,24366,698],{"class":133},[40,24368,24369,24372,24374],{"class":42,"line":203},[40,24370,24371],{"class":133},"    gnd_en_circ_est_list.",[40,24373,979],{"class":237},[40,24375,24376],{"class":133},"(gnd_en_circ_est)\n",[40,24378,24379,24381,24383,24386],{"class":42,"line":216},[40,24380,1100],{"class":441},[40,24382,241],{"class":133},[40,24384,24385],{"class":244},"\"The estimated ground state energy is: \"",[40,24387,24388],{"class":133},", gnd_en_circ_est)\n",[30,24390,24393],{"className":24391,"code":24392,"language":6083,"meta":35},[7575],"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",[37,24394,24392],{"__ignoreMap":35},[12,24396,24397],{},"For a single-particle sector, we can efficiently calculate the ground state of this sector of the Hamiltonian classically",[30,24399,24401],{"className":32,"code":24400,"language":34,"meta":35,"style":35},"gs_en = single_particle_gs(H_op, n_qubits)\n",[37,24402,24403],{"__ignoreMap":35},[40,24404,24405,24408,24410,24412],{"class":42,"line":43},[40,24406,24407],{"class":133},"gs_en ",[40,24409,623],{"class":441},[40,24411,909],{"class":237},[40,24413,24414],{"class":133},"(H_op, n_qubits)\n",[30,24416,24419],{"className":24417,"code":24418,"language":6083,"meta":35},[7575],"n_sys_qubits 30\nn_exc 1 , subspace dimension 31\nsingle particle ground state energy:  21.021912418526906\n",[37,24420,24418],{"__ignoreMap":35},[30,24422,24424],{"className":32,"code":24423,"language":34,"meta":35,"style":35},"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",[37,24425,24426,24436,24453,24458,24470,24482,24494,24498,24506,24522,24532,24543,24554,24565,24569,24606,24615,24629,24643,24652,24657,24661],{"__ignoreMap":35},[40,24427,24428,24431,24434],{"class":42,"line":43},[40,24429,24430],{"class":133},"plt.",[40,24432,24433],{"class":237},"plot",[40,24435,411],{"class":133},[40,24437,24438,24441,24443,24445,24447,24449,24451],{"class":42,"line":50},[40,24439,24440],{"class":441},"    range",[40,24442,241],{"class":133},[40,24444,1410],{"class":462},[40,24446,24327],{"class":133},[40,24448,1147],{"class":441},[40,24450,463],{"class":462},[40,24452,13251],{"class":133},[40,24454,24455],{"class":42,"line":155},[40,24456,24457],{"class":133},"    gnd_en_circ_est_list,\n",[40,24459,24460,24463,24465,24468],{"class":42,"line":168},[40,24461,24462],{"class":1195},"    color",[40,24464,623],{"class":441},[40,24466,24467],{"class":244},"\"blue\"",[40,24469,445],{"class":133},[40,24471,24472,24475,24477,24480],{"class":42,"line":182},[40,24473,24474],{"class":1195},"    linestyle",[40,24476,623],{"class":441},[40,24478,24479],{"class":244},"\"-.\"",[40,24481,445],{"class":133},[40,24483,24484,24487,24489,24492],{"class":42,"line":195},[40,24485,24486],{"class":1195},"    label",[40,24488,623],{"class":441},[40,24490,24491],{"class":244},"\"KQD estimate\"",[40,24493,445],{"class":133},[40,24495,24496],{"class":42,"line":203},[40,24497,248],{"class":133},[40,24499,24500,24502,24504],{"class":42,"line":216},[40,24501,24430],{"class":133},[40,24503,24433],{"class":237},[40,24505,411],{"class":133},[40,24507,24508,24510,24512,24514,24516,24518,24520],{"class":42,"line":224},[40,24509,24440],{"class":441},[40,24511,241],{"class":133},[40,24513,1410],{"class":462},[40,24515,24327],{"class":133},[40,24517,1147],{"class":441},[40,24519,463],{"class":462},[40,24521,13251],{"class":133},[40,24523,24524,24527,24529],{"class":42,"line":231},[40,24525,24526],{"class":133},"    [gs_en] ",[40,24528,1433],{"class":441},[40,24530,24531],{"class":133}," krylov_dim,\n",[40,24533,24534,24536,24538,24541],{"class":42,"line":251},[40,24535,24462],{"class":1195},[40,24537,623],{"class":441},[40,24539,24540],{"class":244},"\"red\"",[40,24542,445],{"class":133},[40,24544,24545,24547,24549,24552],{"class":42,"line":256},[40,24546,24474],{"class":1195},[40,24548,623],{"class":441},[40,24550,24551],{"class":244},"\"-\"",[40,24553,445],{"class":133},[40,24555,24556,24558,24560,24563],{"class":42,"line":269},[40,24557,24486],{"class":1195},[40,24559,623],{"class":441},[40,24561,24562],{"class":244},"\"exact\"",[40,24564,445],{"class":133},[40,24566,24567],{"class":42,"line":282},[40,24568,248],{"class":133},[40,24570,24571,24573,24576,24578,24580,24582,24584,24586,24588,24590,24592,24594,24596,24598,24600,24602,24604],{"class":42,"line":295},[40,24572,24430],{"class":133},[40,24574,24575],{"class":237},"xticks",[40,24577,241],{"class":133},[40,24579,1244],{"class":441},[40,24581,241],{"class":133},[40,24583,1410],{"class":462},[40,24585,24327],{"class":133},[40,24587,1147],{"class":441},[40,24589,463],{"class":462},[40,24591,1253],{"class":133},[40,24593,1244],{"class":441},[40,24595,241],{"class":133},[40,24597,1410],{"class":462},[40,24599,24327],{"class":133},[40,24601,1147],{"class":441},[40,24603,463],{"class":462},[40,24605,21594],{"class":133},[40,24607,24608,24610,24613],{"class":42,"line":308},[40,24609,24430],{"class":133},[40,24611,24612],{"class":237},"legend",[40,24614,12875],{"class":133},[40,24616,24617,24619,24622,24624,24627],{"class":42,"line":321},[40,24618,24430],{"class":133},[40,24620,24621],{"class":237},"xlabel",[40,24623,241],{"class":133},[40,24625,24626],{"class":244},"\"Krylov space dimension\"",[40,24628,248],{"class":133},[40,24630,24631,24633,24636,24638,24641],{"class":42,"line":334},[40,24632,24430],{"class":133},[40,24634,24635],{"class":237},"ylabel",[40,24637,241],{"class":133},[40,24639,24640],{"class":244},"\"Energy\"",[40,24642,248],{"class":133},[40,24644,24645,24647,24650],{"class":42,"line":347},[40,24646,24430],{"class":133},[40,24648,24649],{"class":237},"title",[40,24651,411],{"class":133},[40,24653,24654],{"class":42,"line":352},[40,24655,24656],{"class":244},"    \"Estimating Ground state energy with Krylov Quantum Diagonalization\"\n",[40,24658,24659],{"class":42,"line":357},[40,24660,248],{"class":133},[40,24662,24663,24665,24668],{"class":42,"line":370},[40,24664,24430],{"class":133},[40,24666,24667],{"class":237},"show",[40,24669,12875],{"class":133},[55,24671,24673],{"id":24672},"appendix-krylov-subspace-from-real-time-evolutions","Appendix: Krylov subspace from real time-evolutions",[12,24675,24676],{},"The unitary Krylov space is defined as",[40,24678,24680],{"className":24679},[1827],[40,24681,24683,24791],{"className":24682},[1565],[40,24684,24686],{"className":24685},[1569],[1571,24687,24688],{"xmlns":1573,"display":1836},[1575,24689,24690,24788],{},[1578,24691,24692,24698,24700,24702,24704,24706,24708,24710,24712,24714,24716],{},[2212,24693,24694,24696],{},[1584,24695,1587],{"mathvariant":1586},[1584,24697,2323],{},[1731,24699,241],{"stretchy":1797},[1584,24701,2182],{},[1731,24703,1862],{"separator":1599},[1584,24705,1791],{"mathvariant":1790},[1584,24707,2837],{},[1731,24709,1798],{"stretchy":1797},[1731,24711,2693],{"stretchy":1797},[1731,24713,623],{},[5211,24715,40],{},[1578,24717,24718,24720,24722,24724,24726,24728,24747,24749,24751,24753,24755,24758,24760,24780,24782,24784,24786],{},[1731,24719,1853],{"fence":1599},[1584,24721,1791],{"mathvariant":1790},[1584,24723,2837],{},[1731,24725,1798],{"stretchy":1797},[1731,24727,1862],{"separator":1599},[1581,24729,24730,24732],{},[1584,24731,2330],{},[1578,24733,24734,24736,24738,24740,24743,24745],{},[1731,24735,1733],{},[1584,24737,2337],{},[1584,24739,2182],{},[5211,24741,24742],{}," ",[1584,24744,4003],{},[1584,24746,2342],{},[1584,24748,1791],{"mathvariant":1790},[1584,24750,2837],{},[1731,24752,1798],{"stretchy":1797},[1731,24754,1862],{"separator":1599},[1731,24756,24757],{},"…",[1731,24759,1862],{"separator":1599},[1581,24761,24762,24764],{},[1584,24763,2330],{},[1578,24765,24766,24768,24770,24772,24774,24776,24778],{},[1731,24767,1733],{},[1584,24769,2337],{},[1584,24771,1256],{},[1584,24773,2182],{},[5211,24775,24742],{},[1584,24777,4003],{},[1584,24779,2342],{},[1584,24781,1791],{"mathvariant":1790},[1584,24783,2837],{},[1731,24785,1798],{"stretchy":1797},[1731,24787,1916],{"fence":1599},[1591,24789,24790],{"encoding":1593},"\\mathcal{K}_U(H, |\\psi\\rangle) = \\text{span}\\left\\{ |\\psi\\rangle,  e^{-iH\\,dt} |\\psi\\rangle, \\dots, e^{-irH\\,dt} |\\psi\\rangle \\right\\}",[40,24792,24794,24871],{"className":24793,"ariaHidden":1599},[1598],[40,24795,24797,24800,24840,24843,24846,24849,24852,24855,24858,24862,24865,24868],{"className":24796},[1603],[40,24798],{"className":24799,"style":1811},[1607],[40,24801,24803,24806],{"className":24802},[1612],[40,24804,1587],{"className":24805,"style":1617},[1612,1616],[40,24807,24809],{"className":24808},[1621],[40,24810,24812,24832],{"className":24811},[1625,2244],[40,24813,24815,24829],{"className":24814},[1629],[40,24816,24818],{"className":24817,"style":2251},[1633],[40,24819,24820,24823],{"style":2254},[40,24821],{"className":24822,"style":1642},[1641],[40,24824,24826],{"className":24825},[1646,1647,1648,1649],[40,24827,2323],{"className":24828,"style":2358},[1612,1653,1649],[40,24830,2269],{"className":24831},[2268],[40,24833,24835],{"className":24834},[1629],[40,24836,24838],{"className":24837,"style":2276},[1633],[40,24839],{},[40,24841,241],{"className":24842},[1985],[40,24844,2182],{"className":24845,"style":2197},[1612,1653],[40,24847,1862],{"className":24848},[2005],[40,24850],{"className":24851,"style":2009},[1756],[40,24853,1791],{"className":24854},[1612],[40,24856,2837],{"className":24857,"style":1818},[1612,1653],[40,24859,24861],{"className":24860},[1822],"⟩)",[40,24863],{"className":24864,"style":1963},[1756],[40,24866,623],{"className":24867},[1967],[40,24869],{"className":24870,"style":1963},[1756],[40,24872,24874,24878,24884,24887],{"className":24873},[1603],[40,24875],{"className":24876,"style":24877},[1607],"height:1.2491em;vertical-align:-0.35em;",[40,24879,24881],{"className":24880},[1612,6083],[40,24882,40],{"className":24883},[1612],[40,24885],{"className":24886,"style":2009},[1756],[40,24888,24890,24896,24899,24902,24905,24908,24911,24959,24962,24965,24968,24971,24974,24977,24980,24983,24986,25036,25039,25042,25045],{"className":24889},[1981],[40,24891,24893],{"className":24892,"style":1987},[1985,1986],[40,24894,1853],{"className":24895},[1991,1992],[40,24897,1791],{"className":24898},[1612],[40,24900,2837],{"className":24901,"style":1818},[1612,1653],[40,24903,1798],{"className":24904},[1822],[40,24906,1862],{"className":24907},[2005],[40,24909],{"className":24910,"style":2009},[1756],[40,24912,24914,24917],{"className":24913},[1612],[40,24915,2330],{"className":24916},[1612,1653],[40,24918,24920],{"className":24919},[1621],[40,24921,24923],{"className":24922},[1625],[40,24924,24926],{"className":24925},[1629],[40,24927,24929],{"className":24928,"style":4084},[1633],[40,24930,24931,24934],{"style":1950},[40,24932],{"className":24933,"style":1642},[1641],[40,24935,24937],{"className":24936},[1646,1647,1648,1649],[40,24938,24940,24943,24946,24949,24953,24956],{"className":24939},[1612,1649],[40,24941,1733],{"className":24942},[1612,1649],[40,24944,2337],{"className":24945},[1612,1653,1649],[40,24947,2182],{"className":24948,"style":2197},[1612,1653,1649],[40,24950],{"className":24951,"style":24952},[1756,1649],"margin-right:0.1952em;",[40,24954,4003],{"className":24955},[1612,1653,1649],[40,24957,2342],{"className":24958},[1612,1653,1649],[40,24960,1791],{"className":24961},[1612],[40,24963,2837],{"className":24964,"style":1818},[1612,1653],[40,24966,1798],{"className":24967},[1822],[40,24969,1862],{"className":24970},[2005],[40,24972],{"className":24973,"style":2009},[1756],[40,24975,24757],{"className":24976},[1981],[40,24978],{"className":24979,"style":2009},[1756],[40,24981,1862],{"className":24982},[2005],[40,24984],{"className":24985,"style":2009},[1756],[40,24987,24989,24992],{"className":24988},[1612],[40,24990,2330],{"className":24991},[1612,1653],[40,24993,24995],{"className":24994},[1621],[40,24996,24998],{"className":24997},[1625],[40,24999,25001],{"className":25000},[1629],[40,25002,25004],{"className":25003,"style":4084},[1633],[40,25005,25006,25009],{"style":1950},[40,25007],{"className":25008,"style":1642},[1641],[40,25010,25012],{"className":25011},[1646,1647,1648,1649],[40,25013,25015,25018,25021,25024,25027,25030,25033],{"className":25014},[1612,1649],[40,25016,1733],{"className":25017},[1612,1649],[40,25019,2337],{"className":25020},[1612,1653,1649],[40,25022,1256],{"className":25023,"style":1654},[1612,1653,1649],[40,25025,2182],{"className":25026,"style":2197},[1612,1653,1649],[40,25028],{"className":25029,"style":24952},[1756,1649],[40,25031,4003],{"className":25032},[1612,1653,1649],[40,25034,2342],{"className":25035},[1612,1653,1649],[40,25037,1791],{"className":25038},[1612],[40,25040,2837],{"className":25041,"style":1818},[1612,1653],[40,25043,1798],{"className":25044},[1822],[40,25046,25048],{"className":25047,"style":1987},[1822,1986],[40,25049,1916],{"className":25050},[1991,1992],[12,25052,25053,25054,25087,25088,25116,25117,25176],{},"for some timestep ",[40,25055,25057,25072],{"className":25056},[1565],[40,25058,25060],{"className":25059},[1569],[1571,25061,25062],{"xmlns":1573},[1575,25063,25064,25070],{},[1578,25065,25066,25068],{},[1584,25067,4003],{},[1584,25069,2342],{},[1591,25071,4507],{"encoding":1593},[40,25073,25075],{"className":25074,"ariaHidden":1599},[1598],[40,25076,25078,25081,25084],{"className":25077},[1603],[40,25079],{"className":25080,"style":4478},[1607],[40,25082,4003],{"className":25083},[1612,1653],[40,25085,2342],{"className":25086},[1612,1653]," that we will determine later. Temporarily assume ",[40,25089,25091,25104],{"className":25090},[1565],[40,25092,25094],{"className":25093},[1569],[1571,25095,25096],{"xmlns":1573},[1575,25097,25098,25102],{},[1578,25099,25100],{},[1584,25101,1256],{},[1591,25103,1256],{"encoding":1593},[40,25105,25107],{"className":25106,"ariaHidden":1599},[1598],[40,25108,25110,25113],{"className":25109},[1603],[40,25111],{"className":25112,"style":1681},[1607],[40,25114,1256],{"className":25115,"style":1654},[1612,1653]," is even: then define ",[40,25118,25120,25142],{"className":25119},[1565],[40,25121,25123],{"className":25122},[1569],[1571,25124,25125],{"xmlns":1573},[1575,25126,25127,25139],{},[1578,25128,25129,25131,25133,25135,25137],{},[1584,25130,4003],{},[1731,25132,623],{},[1584,25134,1256],{},[1584,25136,7610],{"mathvariant":1790},[1735,25138,1879],{},[1591,25140,25141],{"encoding":1593},"d=r\u002F2",[40,25143,25145,25163],{"className":25144,"ariaHidden":1599},[1598],[40,25146,25148,25151,25154,25157,25160],{"className":25147},[1603],[40,25149],{"className":25150,"style":4478},[1607],[40,25152,4003],{"className":25153},[1612,1653],[40,25155],{"className":25156,"style":1963},[1756],[40,25158,623],{"className":25159},[1967],[40,25161],{"className":25162,"style":1963},[1756],[40,25164,25166,25169,25172],{"className":25165},[1603],[40,25167],{"className":25168,"style":1811},[1607],[40,25170,1256],{"className":25171,"style":1654},[1612,1653],[40,25173,25175],{"className":25174},[1612],"\u002F2",". Notice that when we project the Hamiltonian into the Krylov space above, it is indistinguishable from the Krylov space",[40,25178,25180],{"className":25179},[1827],[40,25181,25183,25361],{"className":25182},[1565],[40,25184,25186],{"className":25185},[1569],[1571,25187,25188],{"xmlns":1573,"display":1836},[1575,25189,25190,25358],{},[1578,25191,25192,25198,25200,25202,25204,25206,25208,25210,25212,25214,25216,25356],{},[2212,25193,25194,25196],{},[1584,25195,1587],{"mathvariant":1586},[1584,25197,2323],{},[1731,25199,241],{"stretchy":1797},[1584,25201,2182],{},[1731,25203,1862],{"separator":1599},[1584,25205,1791],{"mathvariant":1790},[1584,25207,2837],{},[1731,25209,1798],{"stretchy":1797},[1731,25211,2693],{"stretchy":1797},[1731,25213,623],{},[5211,25215,40],{},[1578,25217,25218,25220,25242,25244,25246,25248,25250,25276,25278,25280,25282,25284,25286,25288,25316,25318,25320,25322,25324,25348,25350,25352,25354],{},[1731,25219,1853],{"fence":1599},[1581,25221,25222,25224],{},[1584,25223,2330],{},[1578,25225,25226,25228,25230,25232,25234,25236,25238,25240],{},[1584,25227,2337],{},[5211,25229,24742],{},[1584,25231,4003],{},[5211,25233,24742],{},[1584,25235,2182],{},[5211,25237,24742],{},[1584,25239,4003],{},[1584,25241,2342],{},[1584,25243,1791],{"mathvariant":1790},[1584,25245,2837],{},[1731,25247,1798],{"stretchy":1797},[1731,25249,1862],{"separator":1599},[1581,25251,25252,25254],{},[1584,25253,2330],{},[1578,25255,25256,25258,25260,25262,25264,25266,25268,25270,25272,25274],{},[1584,25257,2337],{},[1731,25259,241],{"stretchy":1797},[1584,25261,4003],{},[1731,25263,1733],{},[1735,25265,1410],{},[1731,25267,2693],{"stretchy":1797},[1584,25269,2182],{},[5211,25271,24742],{},[1584,25273,4003],{},[1584,25275,2342],{},[1584,25277,1791],{"mathvariant":1790},[1584,25279,2837],{},[1731,25281,1798],{"stretchy":1797},[1731,25283,1862],{"separator":1599},[1731,25285,24757],{},[1731,25287,1862],{"separator":1599},[1581,25289,25290,25292],{},[1584,25291,2330],{},[1578,25293,25294,25296,25298,25300,25302,25304,25306,25308,25310,25312,25314],{},[1731,25295,1733],{},[1584,25297,2337],{},[1731,25299,241],{"stretchy":1797},[1584,25301,4003],{},[1731,25303,1733],{},[1735,25305,1410],{},[1731,25307,2693],{"stretchy":1797},[1584,25309,2182],{},[5211,25311,24742],{},[1584,25313,4003],{},[1584,25315,2342],{},[1584,25317,1791],{"mathvariant":1790},[1584,25319,2837],{},[1731,25321,1798],{"stretchy":1797},[1731,25323,1862],{"separator":1599},[1581,25325,25326,25328],{},[1584,25327,2330],{},[1578,25329,25330,25332,25334,25336,25338,25340,25342,25344,25346],{},[1731,25331,1733],{},[1584,25333,2337],{},[5211,25335,24742],{},[1584,25337,4003],{},[5211,25339,24742],{},[1584,25341,2182],{},[5211,25343,24742],{},[1584,25345,4003],{},[1584,25347,2342],{},[1584,25349,1791],{"mathvariant":1790},[1584,25351,2837],{},[1731,25353,1798],{"stretchy":1797},[1731,25355,1916],{"fence":1599},[1731,25357,1862],{"separator":1599},[1591,25359,25360],{"encoding":1593},"\\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\\},",[40,25362,25364,25440],{"className":25363,"ariaHidden":1599},[1598],[40,25365,25367,25370,25410,25413,25416,25419,25422,25425,25428,25431,25434,25437],{"className":25366},[1603],[40,25368],{"className":25369,"style":1811},[1607],[40,25371,25373,25376],{"className":25372},[1612],[40,25374,1587],{"className":25375,"style":1617},[1612,1616],[40,25377,25379],{"className":25378},[1621],[40,25380,25382,25402],{"className":25381},[1625,2244],[40,25383,25385,25399],{"className":25384},[1629],[40,25386,25388],{"className":25387,"style":2251},[1633],[40,25389,25390,25393],{"style":2254},[40,25391],{"className":25392,"style":1642},[1641],[40,25394,25396],{"className":25395},[1646,1647,1648,1649],[40,25397,2323],{"className":25398,"style":2358},[1612,1653,1649],[40,25400,2269],{"className":25401},[2268],[40,25403,25405],{"className":25404},[1629],[40,25406,25408],{"className":25407,"style":2276},[1633],[40,25409],{},[40,25411,241],{"className":25412},[1985],[40,25414,2182],{"className":25415,"style":2197},[1612,1653],[40,25417,1862],{"className":25418},[2005],[40,25420],{"className":25421,"style":2009},[1756],[40,25423,1791],{"className":25424},[1612],[40,25426,2837],{"className":25427,"style":1818},[1612,1653],[40,25429,24861],{"className":25430},[1822],[40,25432],{"className":25433,"style":1963},[1756],[40,25435,623],{"className":25436},[1967],[40,25438],{"className":25439,"style":1963},[1756],[40,25441,25443,25446,25452,25455,25767,25770],{"className":25442},[1603],[40,25444],{"className":25445,"style":12592},[1607],[40,25447,25449],{"className":25448},[1612,6083],[40,25450,40],{"className":25451},[1612],[40,25453],{"className":25454,"style":2009},[1756],[40,25456,25458,25464,25517,25520,25523,25526,25529,25532,25592,25595,25598,25601,25604,25607,25610,25613,25616,25619,25681,25684,25687,25690,25693,25696,25752,25755,25758,25761],{"className":25457},[1981],[40,25459,25461],{"className":25460,"style":1987},[1985,1986],[40,25462,1853],{"className":25463},[1991,8967],[40,25465,25467,25470],{"className":25466},[1612],[40,25468,2330],{"className":25469},[1612,1653],[40,25471,25473],{"className":25472},[1621],[40,25474,25476],{"className":25475},[1625],[40,25477,25479],{"className":25478},[1629],[40,25480,25482],{"className":25481,"style":4084},[1633],[40,25483,25484,25487],{"style":1950},[40,25485],{"className":25486,"style":1642},[1641],[40,25488,25490],{"className":25489},[1646,1647,1648,1649],[40,25491,25493,25496,25499,25502,25505,25508,25511,25514],{"className":25492},[1612,1649],[40,25494,2337],{"className":25495},[1612,1653,1649],[40,25497],{"className":25498,"style":24952},[1756,1649],[40,25500,4003],{"className":25501},[1612,1653,1649],[40,25503],{"className":25504,"style":24952},[1756,1649],[40,25506,2182],{"className":25507,"style":2197},[1612,1653,1649],[40,25509],{"className":25510,"style":24952},[1756,1649],[40,25512,4003],{"className":25513},[1612,1653,1649],[40,25515,2342],{"className":25516},[1612,1653,1649],[40,25518,1791],{"className":25519},[1612],[40,25521,2837],{"className":25522,"style":1818},[1612,1653],[40,25524,1798],{"className":25525},[1822],[40,25527,1862],{"className":25528},[2005],[40,25530],{"className":25531,"style":2009},[1756],[40,25533,25535,25538],{"className":25534},[1612],[40,25536,2330],{"className":25537},[1612,1653],[40,25539,25541],{"className":25540},[1621],[40,25542,25544],{"className":25543},[1625],[40,25545,25547],{"className":25546},[1629],[40,25548,25551],{"className":25549,"style":25550},[1633],"height:0.938em;",[40,25552,25553,25556],{"style":1950},[40,25554],{"className":25555,"style":1642},[1641],[40,25557,25559],{"className":25558},[1646,1647,1648,1649],[40,25560,25562,25565,25568,25571,25574,25577,25580,25583,25586,25589],{"className":25561},[1612,1649],[40,25563,2337],{"className":25564},[1612,1653,1649],[40,25566,241],{"className":25567},[1985,1649],[40,25569,4003],{"className":25570},[1612,1653,1649],[40,25572,1733],{"className":25573},[1761,1649],[40,25575,1410],{"className":25576},[1612,1649],[40,25578,2693],{"className":25579},[1822,1649],[40,25581,2182],{"className":25582,"style":2197},[1612,1653,1649],[40,25584],{"className":25585,"style":24952},[1756,1649],[40,25587,4003],{"className":25588},[1612,1653,1649],[40,25590,2342],{"className":25591},[1612,1653,1649],[40,25593,1791],{"className":25594},[1612],[40,25596,2837],{"className":25597,"style":1818},[1612,1653],[40,25599,1798],{"className":25600},[1822],[40,25602,1862],{"className":25603},[2005],[40,25605],{"className":25606,"style":2009},[1756],[40,25608,24757],{"className":25609},[1981],[40,25611],{"className":25612,"style":2009},[1756],[40,25614,1862],{"className":25615},[2005],[40,25617],{"className":25618,"style":2009},[1756],[40,25620,25622,25625],{"className":25621},[1612],[40,25623,2330],{"className":25624},[1612,1653],[40,25626,25628],{"className":25627},[1621],[40,25629,25631],{"className":25630},[1625],[40,25632,25634],{"className":25633},[1629],[40,25635,25637],{"className":25636,"style":25550},[1633],[40,25638,25639,25642],{"style":1950},[40,25640],{"className":25641,"style":1642},[1641],[40,25643,25645],{"className":25644},[1646,1647,1648,1649],[40,25646,25648,25651,25654,25657,25660,25663,25666,25669,25672,25675,25678],{"className":25647},[1612,1649],[40,25649,1733],{"className":25650},[1612,1649],[40,25652,2337],{"className":25653},[1612,1653,1649],[40,25655,241],{"className":25656},[1985,1649],[40,25658,4003],{"className":25659},[1612,1653,1649],[40,25661,1733],{"className":25662},[1761,1649],[40,25664,1410],{"className":25665},[1612,1649],[40,25667,2693],{"className":25668},[1822,1649],[40,25670,2182],{"className":25671,"style":2197},[1612,1653,1649],[40,25673],{"className":25674,"style":24952},[1756,1649],[40,25676,4003],{"className":25677},[1612,1653,1649],[40,25679,2342],{"className":25680},[1612,1653,1649],[40,25682,1791],{"className":25683},[1612],[40,25685,2837],{"className":25686,"style":1818},[1612,1653],[40,25688,1798],{"className":25689},[1822],[40,25691,1862],{"className":25692},[2005],[40,25694],{"className":25695,"style":2009},[1756],[40,25697,25699,25702],{"className":25698},[1612],[40,25700,2330],{"className":25701},[1612,1653],[40,25703,25705],{"className":25704},[1621],[40,25706,25708],{"className":25707},[1625],[40,25709,25711],{"className":25710},[1629],[40,25712,25714],{"className":25713,"style":4084},[1633],[40,25715,25716,25719],{"style":1950},[40,25717],{"className":25718,"style":1642},[1641],[40,25720,25722],{"className":25721},[1646,1647,1648,1649],[40,25723,25725,25728,25731,25734,25737,25740,25743,25746,25749],{"className":25724},[1612,1649],[40,25726,1733],{"className":25727},[1612,1649],[40,25729,2337],{"className":25730},[1612,1653,1649],[40,25732],{"className":25733,"style":24952},[1756,1649],[40,25735,4003],{"className":25736},[1612,1653,1649],[40,25738],{"className":25739,"style":24952},[1756,1649],[40,25741,2182],{"className":25742,"style":2197},[1612,1653,1649],[40,25744],{"className":25745,"style":24952},[1756,1649],[40,25747,4003],{"className":25748},[1612,1653,1649],[40,25750,2342],{"className":25751},[1612,1653,1649],[40,25753,1791],{"className":25754},[1612],[40,25756,2837],{"className":25757,"style":1818},[1612,1653],[40,25759,1798],{"className":25760},[1822],[40,25762,25764],{"className":25763,"style":1987},[1822,1986],[40,25765,1916],{"className":25766},[1991,8967],[40,25768],{"className":25769,"style":2009},[1756],[40,25771,1862],{"className":25772},[2005],[12,25774,25775,25776,25804],{},"that is, where all the time-evolutions are shifted backward by ",[40,25777,25779,25792],{"className":25778},[1565],[40,25780,25782],{"className":25781},[1569],[1571,25783,25784],{"xmlns":1573},[1575,25785,25786,25790],{},[1578,25787,25788],{},[1584,25789,4003],{},[1591,25791,4003],{"encoding":1593},[40,25793,25795],{"className":25794,"ariaHidden":1599},[1598],[40,25796,25798,25801],{"className":25797},[1603],[40,25799],{"className":25800,"style":4478},[1607],[40,25802,4003],{"className":25803},[1612,1653]," timesteps.\nThe reason it is indistinguishable is because the matrix elements",[40,25806,25808],{"className":25807},[1827],[40,25809,25811,25943],{"className":25810},[1565],[40,25812,25814],{"className":25813},[1569],[1571,25815,25816],{"xmlns":1573,"display":1836},[1575,25817,25818,25940],{},[1578,25819,25820,25836,25838,25840,25842,25844,25866,25868,25892,25894,25896,25898,25900,25902,25904,25906,25908,25934,25936,25938],{},[2212,25821,25822,25828],{},[3315,25823,25824,25826],{"accent":1599},[1584,25825,2182],{},[1731,25827,3321],{},[1578,25829,25830,25832,25834],{},[1584,25831,1413],{},[1731,25833,1862],{"separator":1599},[1584,25835,2670],{},[1731,25837,623],{},[1731,25839,3469],{"stretchy":1797},[1584,25841,2837],{},[1584,25843,1791],{"mathvariant":1790},[1581,25845,25846,25848],{},[1584,25847,2330],{},[1578,25849,25850,25852,25854,25856,25858,25860,25862,25864],{},[1584,25851,2337],{},[5211,25853,24742],{},[1584,25855,1413],{},[5211,25857,24742],{},[1584,25859,2182],{},[5211,25861,24742],{},[1584,25863,4003],{},[1584,25865,2342],{},[1584,25867,2182],{},[1581,25869,25870,25872],{},[1584,25871,2330],{},[1578,25873,25874,25876,25878,25880,25882,25884,25886,25888,25890],{},[1731,25875,1733],{},[1584,25877,2337],{},[5211,25879,24742],{},[1584,25881,2670],{},[5211,25883,24742],{},[1584,25885,2182],{},[5211,25887,24742],{},[1584,25889,4003],{},[1584,25891,2342],{},[1584,25893,1791],{"mathvariant":1790},[1584,25895,2837],{},[1731,25897,1798],{"stretchy":1797},[1731,25899,623],{},[1731,25901,3469],{"stretchy":1797},[1584,25903,2837],{},[1584,25905,1791],{"mathvariant":1790},[1584,25907,2182],{},[1581,25909,25910,25912],{},[1584,25911,2330],{},[1578,25913,25914,25916,25918,25920,25922,25924,25926,25928,25930,25932],{},[1584,25915,2337],{},[1731,25917,241],{"stretchy":1797},[1584,25919,1413],{},[1731,25921,1733],{},[1584,25923,2670],{},[1731,25925,2693],{"stretchy":1797},[1584,25927,2182],{},[5211,25929,24742],{},[1584,25931,4003],{},[1584,25933,2342],{},[1584,25935,1791],{"mathvariant":1790},[1584,25937,2837],{},[1731,25939,1798],{"stretchy":1797},[1591,25941,25942],{"encoding":1593},"\\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",[40,25944,25946,26038,26183],{"className":25945,"ariaHidden":1599},[1598],[40,25947,25949,25952,26029,26032,26035],{"className":25948},[1603],[40,25950],{"className":25951,"style":5799},[1607],[40,25953,25955,25986],{"className":25954},[1612],[40,25956,25958],{"className":25957},[1612,3338],[40,25959,25961],{"className":25960},[1625],[40,25962,25964],{"className":25963},[1629],[40,25965,25967,25975],{"className":25966,"style":3334},[1633],[40,25968,25969,25972],{"style":3350},[40,25970],{"className":25971,"style":3354},[1641],[40,25973,2182],{"className":25974,"style":2197},[1612,1653],[40,25976,25977,25980],{"style":3360},[40,25978],{"className":25979,"style":3354},[1641],[40,25981,25983],{"className":25982,"style":3368},[3367],[40,25984,3321],{"className":25985},[1612],[40,25987,25989],{"className":25988},[1621],[40,25990,25992,26021],{"className":25991},[1625,2244],[40,25993,25995,26018],{"className":25994},[1629],[40,25996,25998],{"className":25997,"style":10290},[1633],[40,25999,26000,26003],{"style":3554},[40,26001],{"className":26002,"style":1642},[1641],[40,26004,26006],{"className":26005},[1646,1647,1648,1649],[40,26007,26009,26012,26015],{"className":26008},[1612,1649],[40,26010,1413],{"className":26011,"style":5867},[1612,1653,1649],[40,26013,1862],{"className":26014},[2005,1649],[40,26016,2670],{"className":26017,"style":2736},[1612,1653,1649],[40,26019,2269],{"className":26020},[2268],[40,26022,26024],{"className":26023},[1629],[40,26025,26027],{"className":26026,"style":5877},[1633],[40,26028],{},[40,26030],{"className":26031,"style":1963},[1756],[40,26033,623],{"className":26034},[1967],[40,26036],{"className":26037,"style":1963},[1756],[40,26039,26041,26044,26047,26050,26053,26106,26109,26165,26168,26171,26174,26177,26180],{"className":26040},[1603],[40,26042],{"className":26043,"style":4056},[1607],[40,26045,3469],{"className":26046},[1985],[40,26048,2837],{"className":26049,"style":1818},[1612,1653],[40,26051,1791],{"className":26052},[1612],[40,26054,26056,26059],{"className":26055},[1612],[40,26057,2330],{"className":26058},[1612,1653],[40,26060,26062],{"className":26061},[1621],[40,26063,26065],{"className":26064},[1625],[40,26066,26068],{"className":26067},[1629],[40,26069,26071],{"className":26070,"style":4084},[1633],[40,26072,26073,26076],{"style":1950},[40,26074],{"className":26075,"style":1642},[1641],[40,26077,26079],{"className":26078},[1646,1647,1648,1649],[40,26080,26082,26085,26088,26091,26094,26097,26100,26103],{"className":26081},[1612,1649],[40,26083,2337],{"className":26084},[1612,1653,1649],[40,26086],{"className":26087,"style":24952},[1756,1649],[40,26089,1413],{"className":26090,"style":5867},[1612,1653,1649],[40,26092],{"className":26093,"style":24952},[1756,1649],[40,26095,2182],{"className":26096,"style":2197},[1612,1653,1649],[40,26098],{"className":26099,"style":24952},[1756,1649],[40,26101,4003],{"className":26102},[1612,1653,1649],[40,26104,2342],{"className":26105},[1612,1653,1649],[40,26107,2182],{"className":26108,"style":2197},[1612,1653],[40,26110,26112,26115],{"className":26111},[1612],[40,26113,2330],{"className":26114},[1612,1653],[40,26116,26118],{"className":26117},[1621],[40,26119,26121],{"className":26120},[1625],[40,26122,26124],{"className":26123},[1629],[40,26125,26127],{"className":26126,"style":4084},[1633],[40,26128,26129,26132],{"style":1950},[40,26130],{"className":26131,"style":1642},[1641],[40,26133,26135],{"className":26134},[1646,1647,1648,1649],[40,26136,26138,26141,26144,26147,26150,26153,26156,26159,26162],{"className":26137},[1612,1649],[40,26139,1733],{"className":26140},[1612,1649],[40,26142,2337],{"className":26143},[1612,1653,1649],[40,26145],{"className":26146,"style":24952},[1756,1649],[40,26148,2670],{"className":26149,"style":2736},[1612,1653,1649],[40,26151],{"className":26152,"style":24952},[1756,1649],[40,26154,2182],{"className":26155,"style":2197},[1612,1653,1649],[40,26157],{"className":26158,"style":24952},[1756,1649],[40,26160,4003],{"className":26161},[1612,1653,1649],[40,26163,2342],{"className":26164},[1612,1653,1649],[40,26166,1791],{"className":26167},[1612],[40,26169,2837],{"className":26170,"style":1818},[1612,1653],[40,26172,1798],{"className":26173},[1822],[40,26175],{"className":26176,"style":1963},[1756],[40,26178,623],{"className":26179},[1967],[40,26181],{"className":26182,"style":1963},[1756],[40,26184,26186,26190,26193,26196,26199,26202,26261,26264,26267],{"className":26185},[1603],[40,26187],{"className":26188,"style":26189},[1607],"height:1.188em;vertical-align:-0.25em;",[40,26191,3469],{"className":26192},[1985],[40,26194,2837],{"className":26195,"style":1818},[1612,1653],[40,26197,1791],{"className":26198},[1612],[40,26200,2182],{"className":26201,"style":2197},[1612,1653],[40,26203,26205,26208],{"className":26204},[1612],[40,26206,2330],{"className":26207},[1612,1653],[40,26209,26211],{"className":26210},[1621],[40,26212,26214],{"className":26213},[1625],[40,26215,26217],{"className":26216},[1629],[40,26218,26220],{"className":26219,"style":25550},[1633],[40,26221,26222,26225],{"style":1950},[40,26223],{"className":26224,"style":1642},[1641],[40,26226,26228],{"className":26227},[1646,1647,1648,1649],[40,26229,26231,26234,26237,26240,26243,26246,26249,26252,26255,26258],{"className":26230},[1612,1649],[40,26232,2337],{"className":26233},[1612,1653,1649],[40,26235,241],{"className":26236},[1985,1649],[40,26238,1413],{"className":26239,"style":5867},[1612,1653,1649],[40,26241,1733],{"className":26242},[1761,1649],[40,26244,2670],{"className":26245,"style":2736},[1612,1653,1649],[40,26247,2693],{"className":26248},[1822,1649],[40,26250,2182],{"className":26251,"style":2197},[1612,1653,1649],[40,26253],{"className":26254,"style":24952},[1756,1649],[40,26256,4003],{"className":26257},[1612,1653,1649],[40,26259,2342],{"className":26260},[1612,1653,1649],[40,26262,1791],{"className":26263},[1612],[40,26265,2837],{"className":26266,"style":1818},[1612,1653],[40,26268,1798],{"className":26269},[1822],[12,26271,26272,26273,26301,26302,1823],{},"are invariant under overall shifts of the evolution time, since the time-evolutions commute with the Hamiltonian. For odd ",[40,26274,26276,26289],{"className":26275},[1565],[40,26277,26279],{"className":26278},[1569],[1571,26280,26281],{"xmlns":1573},[1575,26282,26283,26287],{},[1578,26284,26285],{},[1584,26286,1256],{},[1591,26288,1256],{"encoding":1593},[40,26290,26292],{"className":26291,"ariaHidden":1599},[1598],[40,26293,26295,26298],{"className":26294},[1603],[40,26296],{"className":26297,"style":1681},[1607],[40,26299,1256],{"className":26300,"style":1654},[1612,1653],", we can use the analysis for ",[40,26303,26305,26322],{"className":26304},[1565],[40,26306,26308],{"className":26307},[1569],[1571,26309,26310],{"xmlns":1573},[1575,26311,26312,26320],{},[1578,26313,26314,26316,26318],{},[1584,26315,1256],{},[1731,26317,1733],{},[1735,26319,1410],{},[1591,26321,1739],{"encoding":1593},[40,26323,26325,26343],{"className":26324,"ariaHidden":1599},[1598],[40,26326,26328,26331,26334,26337,26340],{"className":26327},[1603],[40,26329],{"className":26330,"style":1749},[1607],[40,26332,1256],{"className":26333,"style":1654},[1612,1653],[40,26335],{"className":26336,"style":1757},[1756],[40,26338,1733],{"className":26339},[1761],[40,26341],{"className":26342,"style":1757},[1756],[40,26344,26346,26349],{"className":26345},[1603],[40,26347],{"className":26348,"style":1771},[1607],[40,26350,1410],{"className":26351},[1612],[12,26353,26354,26355,24039],{},"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 ",[19,26356,26357],{"href":82},"[3]",[12,26359,26360,26364,26365,26394,26395,26423],{},[26361,26362,26363],"strong",{},"Claim 1:"," there exists a function ",[40,26366,26368,26381],{"className":26367},[1565],[40,26369,26371],{"className":26370},[1569],[1571,26372,26373],{"xmlns":1573},[1575,26374,26375,26379],{},[1578,26376,26377],{},[1584,26378,21331],{},[1591,26380,21331],{"encoding":1593},[40,26382,26384],{"className":26383,"ariaHidden":1599},[1598],[40,26385,26387,26390],{"className":26386},[1603],[40,26388],{"className":26389,"style":6079},[1607],[40,26391,21331],{"className":26392,"style":26393},[1612,1653],"margin-right:0.1076em;"," such that for energies ",[40,26396,26398,26411],{"className":26397},[1565],[40,26399,26401],{"className":26400},[1569],[1571,26402,26403],{"xmlns":1573},[1575,26404,26405,26409],{},[1578,26406,26407],{},[1584,26408,5226],{},[1591,26410,5226],{"encoding":1593},[40,26412,26414],{"className":26413,"ariaHidden":1599},[1598],[40,26415,26417,26420],{"className":26416},[1603],[40,26418],{"className":26419,"style":1608},[1607],[40,26421,5226],{"className":26422,"style":4936},[1612,1653]," in the spectral range of the Hamiltonian (that is, between the ground state energy and the maximum energy)...",[26425,26426,26427,26536,26844],"ol",{},[101,26428,26429],{},[40,26430,26432,26460],{"className":26431},[1565],[40,26433,26435],{"className":26434},[1569],[1571,26436,26437],{"xmlns":1573},[1575,26438,26439,26457],{},[1578,26440,26441,26443,26445,26451,26453,26455],{},[1584,26442,21331],{},[1731,26444,241],{"stretchy":1797},[2212,26446,26447,26449],{},[1584,26448,5226],{},[1735,26450,791],{},[1731,26452,2693],{"stretchy":1797},[1731,26454,623],{},[1735,26456,1410],{},[1591,26458,26459],{"encoding":1593},"f(E_0)=1",[40,26461,26463,26527],{"className":26462,"ariaHidden":1599},[1598],[40,26464,26466,26469,26472,26475,26515,26518,26521,26524],{"className":26465},[1603],[40,26467],{"className":26468,"style":1811},[1607],[40,26470,21331],{"className":26471,"style":26393},[1612,1653],[40,26473,241],{"className":26474},[1985],[40,26476,26478,26481],{"className":26477},[1612],[40,26479,5226],{"className":26480,"style":4936},[1612,1653],[40,26482,26484],{"className":26483},[1621],[40,26485,26487,26507],{"className":26486},[1625,2244],[40,26488,26490,26504],{"className":26489},[1629],[40,26491,26493],{"className":26492,"style":13330},[1633],[40,26494,26495,26498],{"style":5058},[40,26496],{"className":26497,"style":1642},[1641],[40,26499,26501],{"className":26500},[1646,1647,1648,1649],[40,26502,791],{"className":26503},[1612,1649],[40,26505,2269],{"className":26506},[2268],[40,26508,26510],{"className":26509},[1629],[40,26511,26513],{"className":26512,"style":2276},[1633],[40,26514],{},[40,26516,2693],{"className":26517},[1822],[40,26519],{"className":26520,"style":1963},[1756],[40,26522,623],{"className":26523},[1967],[40,26525],{"className":26526,"style":1963},[1756],[40,26528,26530,26533],{"className":26529},[1603],[40,26531],{"className":26532,"style":1771},[1607],[40,26534,1410],{"className":26535},[1612],[101,26537,26538,26698,26699,26727,26728,26773,26774,26843],{},[40,26539,26541,26591],{"className":26540},[1565],[40,26542,26544],{"className":26543},[1569],[1571,26545,26546],{"xmlns":1573},[1575,26547,26548,26588],{},[1578,26549,26550,26552,26554,26556,26558,26560,26562,26565,26567],{},[1584,26551,1791],{"mathvariant":1790},[1584,26553,21331],{},[1731,26555,241],{"stretchy":1797},[1584,26557,5226],{},[1731,26559,2693],{"stretchy":1797},[1584,26561,1791],{"mathvariant":1790},[1731,26563,26564],{},"≤",[1735,26566,1879],{},[1581,26568,26569,26582],{},[1578,26570,26571,26573,26575,26577,26580],{},[1731,26572,241],{"fence":1599},[1735,26574,1410],{},[1731,26576,1147],{},[1584,26578,26579],{},"δ",[1731,26581,2693],{"fence":1599},[1578,26583,26584,26586],{},[1731,26585,1733],{},[1584,26587,4003],{},[1591,26589,26590],{"encoding":1593},"|f(E)|\\le2\\left(1 + \\delta\\right)^{-d}",[40,26592,26594,26627],{"className":26593,"ariaHidden":1599},[1598],[40,26595,26597,26600,26603,26606,26609,26612,26615,26618,26621,26624],{"className":26596},[1603],[40,26598],{"className":26599,"style":1811},[1607],[40,26601,1791],{"className":26602},[1612],[40,26604,21331],{"className":26605,"style":26393},[1612,1653],[40,26607,241],{"className":26608},[1985],[40,26610,5226],{"className":26611,"style":4936},[1612,1653],[40,26613,2693],{"className":26614},[1822],[40,26616,1791],{"className":26617},[1612],[40,26619],{"className":26620,"style":1963},[1756],[40,26622,26564],{"className":26623},[1967],[40,26625],{"className":26626,"style":1963},[1756],[40,26628,26630,26634,26637,26640],{"className":26629},[1603],[40,26631],{"className":26632,"style":26633},[1607],"height:1.239em;vertical-align:-0.25em;",[40,26635,1879],{"className":26636},[1612],[40,26638],{"className":26639,"style":2009},[1756],[40,26641,26643,26668],{"className":26642},[1981],[40,26644,26646,26649,26652,26655,26658,26661,26665],{"className":26645},[1981],[40,26647,241],{"className":26648,"style":1987},[1985,1986],[40,26650,1410],{"className":26651},[1612],[40,26653],{"className":26654,"style":1757},[1756],[40,26656,1147],{"className":26657},[1761],[40,26659],{"className":26660,"style":1757},[1756],[40,26662,26579],{"className":26663,"style":26664},[1612,1653],"margin-right:0.0379em;",[40,26666,2693],{"className":26667,"style":1987},[1822,1986],[40,26669,26671],{"className":26670},[1621],[40,26672,26674],{"className":26673},[1625],[40,26675,26677],{"className":26676},[1629],[40,26678,26681],{"className":26679,"style":26680},[1633],"height:0.989em;",[40,26682,26683,26686],{"style":6889},[40,26684],{"className":26685,"style":1642},[1641],[40,26687,26689],{"className":26688},[1646,1647,1648,1649],[40,26690,26692,26695],{"className":26691},[1612,1649],[40,26693,1733],{"className":26694},[1612,1649],[40,26696,4003],{"className":26697},[1612,1653,1649]," for all values of ",[40,26700,26702,26715],{"className":26701},[1565],[40,26703,26705],{"className":26704},[1569],[1571,26706,26707],{"xmlns":1573},[1575,26708,26709,26713],{},[1578,26710,26711],{},[1584,26712,5226],{},[1591,26714,5226],{"encoding":1593},[40,26716,26718],{"className":26717,"ariaHidden":1599},[1598],[40,26719,26721,26724],{"className":26720},[1603],[40,26722],{"className":26723,"style":1608},[1607],[40,26725,5226],{"className":26726,"style":4936},[1612,1653]," that lie ",[40,26729,26731,26748],{"className":26730},[1565],[40,26732,26734],{"className":26733},[1569],[1571,26735,26736],{"xmlns":1573},[1575,26737,26738,26745],{},[1578,26739,26740,26743],{},[1731,26741,26742],{},"≥",[1584,26744,26579],{},[1591,26746,26747],{"encoding":1593},"\\ge\\delta",[40,26749,26751,26764],{"className":26750,"ariaHidden":1599},[1598],[40,26752,26754,26758,26761],{"className":26753},[1603],[40,26755],{"className":26756,"style":26757},[1607],"height:0.7719em;vertical-align:-0.136em;",[40,26759,26742],{"className":26760},[1967],[40,26762],{"className":26763,"style":1963},[1756],[40,26765,26767,26770],{"className":26766},[1603],[40,26768],{"className":26769,"style":4478},[1607],[40,26771,26579],{"className":26772,"style":26664},[1612,1653]," away from ",[40,26775,26777,26794],{"className":26776},[1565],[40,26778,26780],{"className":26779},[1569],[1571,26781,26782],{"xmlns":1573},[1575,26783,26784,26792],{},[1578,26785,26786],{},[2212,26787,26788,26790],{},[1584,26789,5226],{},[1735,26791,791],{},[1591,26793,13302],{"encoding":1593},[40,26795,26797],{"className":26796,"ariaHidden":1599},[1598],[40,26798,26800,26803],{"className":26799},[1603],[40,26801],{"className":26802,"style":2231},[1607],[40,26804,26806,26809],{"className":26805},[1612],[40,26807,5226],{"className":26808,"style":4936},[1612,1653],[40,26810,26812],{"className":26811},[1621],[40,26813,26815,26835],{"className":26814},[1625,2244],[40,26816,26818,26832],{"className":26817},[1629],[40,26819,26821],{"className":26820,"style":13330},[1633],[40,26822,26823,26826],{"style":5058},[40,26824],{"className":26825,"style":1642},[1641],[40,26827,26829],{"className":26828},[1646,1647,1648,1649],[40,26830,791],{"className":26831},[1612,1649],[40,26833,2269],{"className":26834},[2268],[40,26836,26838],{"className":26837},[1629],[40,26839,26841],{"className":26840,"style":2276},[1633],[40,26842],{},", that is, it is exponentially suppressed",[101,26845,26846,26890,26891,26978,26979],{},[40,26847,26849,26869],{"className":26848},[1565],[40,26850,26852],{"className":26851},[1569],[1571,26853,26854],{"xmlns":1573},[1575,26855,26856,26866],{},[1578,26857,26858,26860,26862,26864],{},[1584,26859,21331],{},[1731,26861,241],{"stretchy":1797},[1584,26863,5226],{},[1731,26865,2693],{"stretchy":1797},[1591,26867,26868],{"encoding":1593},"f(E)",[40,26870,26872],{"className":26871,"ariaHidden":1599},[1598],[40,26873,26875,26878,26881,26884,26887],{"className":26874},[1603],[40,26876],{"className":26877,"style":1811},[1607],[40,26879,21331],{"className":26880,"style":26393},[1612,1653],[40,26882,241],{"className":26883},[1985],[40,26885,5226],{"className":26886,"style":4936},[1612,1653],[40,26888,2693],{"className":26889},[1822]," is a linear combination of ",[40,26892,26894,26924],{"className":26893},[1565],[40,26895,26897],{"className":26896},[1569],[1571,26898,26899],{"xmlns":1573},[1575,26900,26901,26921],{},[1578,26902,26903],{},[1581,26904,26905,26907],{},[1584,26906,2330],{},[1578,26908,26909,26911,26913,26915,26917,26919],{},[1584,26910,2337],{},[1584,26912,1413],{},[1584,26914,5226],{},[5211,26916,24742],{},[1584,26918,4003],{},[1584,26920,2342],{},[1591,26922,26923],{"encoding":1593},"e^{ijE\\,dt}",[40,26925,26927],{"className":26926,"ariaHidden":1599},[1598],[40,26928,26930,26933],{"className":26929},[1603],[40,26931],{"className":26932,"style":2706},[1607],[40,26934,26936,26939],{"className":26935},[1612],[40,26937,2330],{"className":26938},[1612,1653],[40,26940,26942],{"className":26941},[1621],[40,26943,26945],{"className":26944},[1625],[40,26946,26948],{"className":26947},[1629],[40,26949,26951],{"className":26950,"style":2706},[1633],[40,26952,26953,26956],{"style":1637},[40,26954],{"className":26955,"style":1642},[1641],[40,26957,26959],{"className":26958},[1646,1647,1648,1649],[40,26960,26962,26966,26969,26972,26975],{"className":26961},[1612,1649],[40,26963,26965],{"className":26964,"style":5867},[1612,1653,1649],"ij",[40,26967,5226],{"className":26968,"style":4936},[1612,1653,1649],[40,26970],{"className":26971,"style":24952},[1756,1649],[40,26973,4003],{"className":26974},[1612,1653,1649],[40,26976,2342],{"className":26977},[1612,1653,1649]," for ",[40,26980,26982,27032],{"className":26981},[1565],[40,26983,26985],{"className":26984},[1569],[1571,26986,26987],{"xmlns":1573},[1575,26988,26989,27029],{},[1578,26990,26991,26993,26995,26997,26999,27001,27003,27005,27007,27009,27011,27013,27015,27017,27019,27021,27023,27025,27027],{},[1584,26992,1413],{},[1731,26994,623],{},[1731,26996,1733],{},[1584,26998,4003],{},[1731,27000,1862],{"separator":1599},[1731,27002,1733],{},[1584,27004,4003],{},[1731,27006,1147],{},[1735,27008,1410],{},[1731,27010,1862],{"separator":1599},[1584,27012,1823],{"mathvariant":1790},[1584,27014,1823],{"mathvariant":1790},[1584,27016,1823],{"mathvariant":1790},[1731,27018,1862],{"separator":1599},[1584,27020,4003],{},[1731,27022,1733],{},[1735,27024,1410],{},[1731,27026,1862],{"separator":1599},[1584,27028,4003],{},[1591,27030,27031],{"encoding":1593},"j=-d,-d+1,...,d-1,d",[40,27033,27035,27054,27087,27123],{"className":27034,"ariaHidden":1599},[1598],[40,27036,27038,27042,27045,27048,27051],{"className":27037},[1603],[40,27039],{"className":27040,"style":27041},[1607],"height:0.854em;vertical-align:-0.1944em;",[40,27043,1413],{"className":27044,"style":5867},[1612,1653],[40,27046],{"className":27047,"style":1963},[1756],[40,27049,623],{"className":27050},[1967],[40,27052],{"className":27053,"style":1963},[1756],[40,27055,27057,27060,27063,27066,27069,27072,27075,27078,27081,27084],{"className":27056},[1603],[40,27058],{"className":27059,"style":6079},[1607],[40,27061,1733],{"className":27062},[1612],[40,27064,4003],{"className":27065},[1612,1653],[40,27067,1862],{"className":27068},[2005],[40,27070],{"className":27071,"style":2009},[1756],[40,27073,1733],{"className":27074},[1612],[40,27076,4003],{"className":27077},[1612,1653],[40,27079],{"className":27080,"style":1757},[1756],[40,27082,1147],{"className":27083},[1761],[40,27085],{"className":27086,"style":1757},[1756],[40,27088,27090,27093,27096,27099,27102,27105,27108,27111,27114,27117,27120],{"className":27089},[1603],[40,27091],{"className":27092,"style":6079},[1607],[40,27094,1410],{"className":27095},[1612],[40,27097,1862],{"className":27098},[2005],[40,27100],{"className":27101,"style":2009},[1756],[40,27103,2076],{"className":27104},[1612],[40,27106,1862],{"className":27107},[2005],[40,27109],{"className":27110,"style":2009},[1756],[40,27112,4003],{"className":27113},[1612,1653],[40,27115],{"className":27116,"style":1757},[1756],[40,27118,1733],{"className":27119},[1761],[40,27121],{"className":27122,"style":1757},[1756],[40,27124,27126,27129,27132,27135,27138],{"className":27125},[1603],[40,27127],{"className":27128,"style":6079},[1607],[40,27130,1410],{"className":27131},[1612],[40,27133,1862],{"className":27134},[2005],[40,27136],{"className":27137,"style":2009},[1756],[40,27139,4003],{"className":27140},[1612,1653],[12,27142,27143,27144,27203,27204,27243],{},"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 ",[40,27145,27147,27173],{"className":27146},[1565],[40,27148,27150],{"className":27149},[1569],[1571,27151,27152],{"xmlns":1573},[1575,27153,27154,27170],{},[1578,27155,27156,27158,27160,27162,27164,27166,27168],{},[1584,27157,21331],{},[1731,27159,241],{"stretchy":1797},[1584,27161,2182],{},[1731,27163,2693],{"stretchy":1797},[1584,27165,1791],{"mathvariant":1790},[1584,27167,2837],{},[1731,27169,1798],{"stretchy":1797},[1591,27171,27172],{"encoding":1593},"f(H)|\\psi\\rangle",[40,27174,27176],{"className":27175,"ariaHidden":1599},[1598],[40,27177,27179,27182,27185,27188,27191,27194,27197,27200],{"className":27178},[1603],[40,27180],{"className":27181,"style":1811},[1607],[40,27183,21331],{"className":27184,"style":26393},[1612,1653],[40,27186,241],{"className":27187},[1985],[40,27189,2182],{"className":27190,"style":2197},[1612,1653],[40,27192,2693],{"className":27193},[1822],[40,27195,1791],{"className":27196},[1612],[40,27198,2837],{"className":27199,"style":1818},[1612,1653],[40,27201,1798],{"className":27202},[1822],". This is our low-energy state. To see why, write ",[40,27205,27207,27225],{"className":27206},[1565],[40,27208,27210],{"className":27209},[1569],[1571,27211,27212],{"xmlns":1573},[1575,27213,27214,27222],{},[1578,27215,27216,27218,27220],{},[1584,27217,1791],{"mathvariant":1790},[1584,27219,2837],{},[1731,27221,1798],{"stretchy":1797},[1591,27223,27224],{"encoding":1593},"|\\psi\\rangle",[40,27226,27228],{"className":27227,"ariaHidden":1599},[1598],[40,27229,27231,27234,27237,27240],{"className":27230},[1603],[40,27232],{"className":27233,"style":1811},[1607],[40,27235,1791],{"className":27236},[1612],[40,27238,2837],{"className":27239,"style":1818},[1612,1653],[40,27241,1798],{"className":27242},[1822]," in the energy eigenbasis:",[40,27245,27247],{"className":27246},[1827],[40,27248,27250,27304],{"className":27249},[1565],[40,27251,27253],{"className":27252},[1569],[1571,27254,27255],{"xmlns":1573,"display":1836},[1575,27256,27257,27301],{},[1578,27258,27259,27261,27263,27265,27267,27282,27289,27291,27297,27299],{},[1584,27260,1791],{"mathvariant":1790},[1584,27262,2837],{},[1731,27264,1798],{"stretchy":1797},[1731,27266,623],{},[27268,27269,27270,27272,27280],"munderover",{},[1731,27271,6571],{},[1578,27273,27274,27276,27278],{},[1584,27275,2670],{},[1731,27277,623],{},[1735,27279,791],{},[1584,27281,6725],{},[2212,27283,27284,27287],{},[1584,27285,27286],{},"γ",[1584,27288,2670],{},[1584,27290,1791],{"mathvariant":1790},[2212,27292,27293,27295],{},[1584,27294,5226],{},[1584,27296,2670],{},[1731,27298,1798],{"stretchy":1797},[1731,27300,1862],{"separator":1599},[1591,27302,27303],{"encoding":1593},"|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_k|E_k\\rangle,",[40,27305,27307,27331],{"className":27306,"ariaHidden":1599},[1598],[40,27308,27310,27313,27316,27319,27322,27325,27328],{"className":27309},[1603],[40,27311],{"className":27312,"style":1811},[1607],[40,27314,1791],{"className":27315},[1612],[40,27317,2837],{"className":27318,"style":1818},[1612,1653],[40,27320,1798],{"className":27321},[1822],[40,27323],{"className":27324,"style":1963},[1756],[40,27326,623],{"className":27327},[1967],[40,27329],{"className":27330,"style":1963},[1756],[40,27332,27334,27338,27413,27416,27458,27461,27501,27504],{"className":27333},[1603],[40,27335],{"className":27336,"style":27337},[1607],"height:3.1304em;vertical-align:-1.3021em;",[40,27339,27342],{"className":27340},[6614,27341],"op-limits",[40,27343,27345,27404],{"className":27344},[1625,2244],[40,27346,27348,27401],{"className":27347},[1629],[40,27349,27352,27374,27386],{"className":27350,"style":27351},[1633],"height:1.8283em;",[40,27353,27355,27359],{"style":27354},"top:-1.8479em;margin-left:0em;",[40,27356],{"className":27357,"style":27358},[1641],"height:3.05em;",[40,27360,27362],{"className":27361},[1646,1647,1648,1649],[40,27363,27365,27368,27371],{"className":27364},[1612,1649],[40,27366,2670],{"className":27367,"style":2736},[1612,1653,1649],[40,27369,623],{"className":27370},[1967,1649],[40,27372,791],{"className":27373},[1612,1649],[40,27375,27377,27380],{"style":27376},"top:-3.05em;",[40,27378],{"className":27379,"style":27358},[1641],[40,27381,27382],{},[40,27383,6571],{"className":27384},[6614,6618,27385],"large-op",[40,27387,27389,27392],{"style":27388},"top:-4.3em;margin-left:0em;",[40,27390],{"className":27391,"style":27358},[1641],[40,27393,27395],{"className":27394},[1646,1647,1648,1649],[40,27396,27398],{"className":27397},[1612,1649],[40,27399,6725],{"className":27400,"style":2358},[1612,1653,1649],[40,27402,2269],{"className":27403},[2268],[40,27405,27407],{"className":27406},[1629],[40,27408,27411],{"className":27409,"style":27410},[1633],"height:1.3021em;",[40,27412],{},[40,27414],{"className":27415,"style":2009},[1756],[40,27417,27419,27423],{"className":27418},[1612],[40,27420,27286],{"className":27421,"style":27422},[1612,1653],"margin-right:0.0556em;",[40,27424,27426],{"className":27425},[1621],[40,27427,27429,27450],{"className":27428},[1625,2244],[40,27430,27432,27447],{"className":27431},[1629],[40,27433,27435],{"className":27434,"style":10290},[1633],[40,27436,27438,27441],{"style":27437},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[40,27439],{"className":27440,"style":1642},[1641],[40,27442,27444],{"className":27443},[1646,1647,1648,1649],[40,27445,2670],{"className":27446,"style":2736},[1612,1653,1649],[40,27448,2269],{"className":27449},[2268],[40,27451,27453],{"className":27452},[1629],[40,27454,27456],{"className":27455,"style":2276},[1633],[40,27457],{},[40,27459,1791],{"className":27460},[1612],[40,27462,27464,27467],{"className":27463},[1612],[40,27465,5226],{"className":27466,"style":4936},[1612,1653],[40,27468,27470],{"className":27469},[1621],[40,27471,27473,27493],{"className":27472},[1625,2244],[40,27474,27476,27490],{"className":27475},[1629],[40,27477,27479],{"className":27478,"style":10290},[1633],[40,27480,27481,27484],{"style":5058},[40,27482],{"className":27483,"style":1642},[1641],[40,27485,27487],{"className":27486},[1646,1647,1648,1649],[40,27488,2670],{"className":27489,"style":2736},[1612,1653,1649],[40,27491,2269],{"className":27492},[2268],[40,27494,27496],{"className":27495},[1629],[40,27497,27499],{"className":27498,"style":2276},[1633],[40,27500],{},[40,27502,1798],{"className":27503},[1822],[40,27505,1862],{"className":27506},[2005],[12,27508,27509,27510,27590,27591,27662,27663,27701,27702,27760],{},"where ",[40,27511,27513,27535],{"className":27512},[1565],[40,27514,27516],{"className":27515},[1569],[1571,27517,27518],{"xmlns":1573},[1575,27519,27520,27532],{},[1578,27521,27522,27524,27530],{},[1584,27523,1791],{"mathvariant":1790},[2212,27525,27526,27528],{},[1584,27527,5226],{},[1584,27529,2670],{},[1731,27531,1798],{"stretchy":1797},[1591,27533,27534],{"encoding":1593},"|E_k\\rangle",[40,27536,27538],{"className":27537,"ariaHidden":1599},[1598],[40,27539,27541,27544,27547,27587],{"className":27540},[1603],[40,27542],{"className":27543,"style":1811},[1607],[40,27545,1791],{"className":27546},[1612],[40,27548,27550,27553],{"className":27549},[1612],[40,27551,5226],{"className":27552,"style":4936},[1612,1653],[40,27554,27556],{"className":27555},[1621],[40,27557,27559,27579],{"className":27558},[1625,2244],[40,27560,27562,27576],{"className":27561},[1629],[40,27563,27565],{"className":27564,"style":10290},[1633],[40,27566,27567,27570],{"style":5058},[40,27568],{"className":27569,"style":1642},[1641],[40,27571,27573],{"className":27572},[1646,1647,1648,1649],[40,27574,2670],{"className":27575,"style":2736},[1612,1653,1649],[40,27577,2269],{"className":27578},[2268],[40,27580,27582],{"className":27581},[1629],[40,27583,27585],{"className":27584,"style":2276},[1633],[40,27586],{},[40,27588,1798],{"className":27589},[1822]," is the kth energy eigenstate and ",[40,27592,27594,27612],{"className":27593},[1565],[40,27595,27597],{"className":27596},[1569],[1571,27598,27599],{"xmlns":1573},[1575,27600,27601,27609],{},[1578,27602,27603],{},[2212,27604,27605,27607],{},[1584,27606,27286],{},[1584,27608,2670],{},[1591,27610,27611],{"encoding":1593},"\\gamma_k",[40,27613,27615],{"className":27614,"ariaHidden":1599},[1598],[40,27616,27618,27622],{"className":27617},[1603],[40,27619],{"className":27620,"style":27621},[1607],"height:0.625em;vertical-align:-0.1944em;",[40,27623,27625,27628],{"className":27624},[1612],[40,27626,27286],{"className":27627,"style":27422},[1612,1653],[40,27629,27631],{"className":27630},[1621],[40,27632,27634,27654],{"className":27633},[1625,2244],[40,27635,27637,27651],{"className":27636},[1629],[40,27638,27640],{"className":27639,"style":10290},[1633],[40,27641,27642,27645],{"style":27437},[40,27643],{"className":27644,"style":1642},[1641],[40,27646,27648],{"className":27647},[1646,1647,1648,1649],[40,27649,2670],{"className":27650,"style":2736},[1612,1653,1649],[40,27652,2269],{"className":27653},[2268],[40,27655,27657],{"className":27656},[1629],[40,27658,27660],{"className":27659,"style":2276},[1633],[40,27661],{}," is its amplitude in the initial state ",[40,27664,27666,27683],{"className":27665},[1565],[40,27667,27669],{"className":27668},[1569],[1571,27670,27671],{"xmlns":1573},[1575,27672,27673,27681],{},[1578,27674,27675,27677,27679],{},[1584,27676,1791],{"mathvariant":1790},[1584,27678,2837],{},[1731,27680,1798],{"stretchy":1797},[1591,27682,27224],{"encoding":1593},[40,27684,27686],{"className":27685,"ariaHidden":1599},[1598],[40,27687,27689,27692,27695,27698],{"className":27688},[1603],[40,27690],{"className":27691,"style":1811},[1607],[40,27693,1791],{"className":27694},[1612],[40,27696,2837],{"className":27697,"style":1818},[1612,1653],[40,27699,1798],{"className":27700},[1822],". Expressed in terms of this, ",[40,27703,27705,27730],{"className":27704},[1565],[40,27706,27708],{"className":27707},[1569],[1571,27709,27710],{"xmlns":1573},[1575,27711,27712,27728],{},[1578,27713,27714,27716,27718,27720,27722,27724,27726],{},[1584,27715,21331],{},[1731,27717,241],{"stretchy":1797},[1584,27719,2182],{},[1731,27721,2693],{"stretchy":1797},[1584,27723,1791],{"mathvariant":1790},[1584,27725,2837],{},[1731,27727,1798],{"stretchy":1797},[1591,27729,27172],{"encoding":1593},[40,27731,27733],{"className":27732,"ariaHidden":1599},[1598],[40,27734,27736,27739,27742,27745,27748,27751,27754,27757],{"className":27735},[1603],[40,27737],{"className":27738,"style":1811},[1607],[40,27740,21331],{"className":27741,"style":26393},[1612,1653],[40,27743,241],{"className":27744},[1985],[40,27746,2182],{"className":27747,"style":2197},[1612,1653],[40,27749,2693],{"className":27750},[1822],[40,27752,1791],{"className":27753},[1612],[40,27755,2837],{"className":27756,"style":1818},[1612,1653],[40,27758,1798],{"className":27759},[1822]," is given by",[40,27762,27764],{"className":27763},[1827],[40,27765,27767,27839],{"className":27766},[1565],[40,27768,27770],{"className":27769},[1569],[1571,27771,27772],{"xmlns":1573,"display":1836},[1575,27773,27774,27836],{},[1578,27775,27776,27778,27780,27782,27784,27786,27788,27790,27792,27806,27812,27814,27816,27822,27824,27826,27832,27834],{},[1584,27777,21331],{},[1731,27779,241],{"stretchy":1797},[1584,27781,2182],{},[1731,27783,2693],{"stretchy":1797},[1584,27785,1791],{"mathvariant":1790},[1584,27787,2837],{},[1731,27789,1798],{"stretchy":1797},[1731,27791,623],{},[27268,27793,27794,27796,27804],{},[1731,27795,6571],{},[1578,27797,27798,27800,27802],{},[1584,27799,2670],{},[1731,27801,623],{},[1735,27803,791],{},[1584,27805,6725],{},[2212,27807,27808,27810],{},[1584,27809,27286],{},[1584,27811,2670],{},[1584,27813,21331],{},[1731,27815,241],{"stretchy":1797},[2212,27817,27818,27820],{},[1584,27819,5226],{},[1584,27821,2670],{},[1731,27823,2693],{"stretchy":1797},[1584,27825,1791],{"mathvariant":1790},[2212,27827,27828,27830],{},[1584,27829,5226],{},[1584,27831,2670],{},[1731,27833,1798],{"stretchy":1797},[1731,27835,1862],{"separator":1599},[1591,27837,27838],{"encoding":1593},"f(H)|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_kf(E_k)|E_k\\rangle,",[40,27840,27842,27878],{"className":27841,"ariaHidden":1599},[1598],[40,27843,27845,27848,27851,27854,27857,27860,27863,27866,27869,27872,27875],{"className":27844},[1603],[40,27846],{"className":27847,"style":1811},[1607],[40,27849,21331],{"className":27850,"style":26393},[1612,1653],[40,27852,241],{"className":27853},[1985],[40,27855,2182],{"className":27856,"style":2197},[1612,1653],[40,27858,2693],{"className":27859},[1822],[40,27861,1791],{"className":27862},[1612],[40,27864,2837],{"className":27865,"style":1818},[1612,1653],[40,27867,1798],{"className":27868},[1822],[40,27870],{"className":27871,"style":1963},[1756],[40,27873,623],{"className":27874},[1967],[40,27876],{"className":27877,"style":1963},[1756],[40,27879,27881,27884,27951,27954,27994,27997,28000,28040,28043,28046,28086,28089],{"className":27880},[1603],[40,27882],{"className":27883,"style":27337},[1607],[40,27885,27887],{"className":27886},[6614,27341],[40,27888,27890,27943],{"className":27889},[1625,2244],[40,27891,27893,27940],{"className":27892},[1629],[40,27894,27896,27916,27926],{"className":27895,"style":27351},[1633],[40,27897,27898,27901],{"style":27354},[40,27899],{"className":27900,"style":27358},[1641],[40,27902,27904],{"className":27903},[1646,1647,1648,1649],[40,27905,27907,27910,27913],{"className":27906},[1612,1649],[40,27908,2670],{"className":27909,"style":2736},[1612,1653,1649],[40,27911,623],{"className":27912},[1967,1649],[40,27914,791],{"className":27915},[1612,1649],[40,27917,27918,27921],{"style":27376},[40,27919],{"className":27920,"style":27358},[1641],[40,27922,27923],{},[40,27924,6571],{"className":27925},[6614,6618,27385],[40,27927,27928,27931],{"style":27388},[40,27929],{"className":27930,"style":27358},[1641],[40,27932,27934],{"className":27933},[1646,1647,1648,1649],[40,27935,27937],{"className":27936},[1612,1649],[40,27938,6725],{"className":27939,"style":2358},[1612,1653,1649],[40,27941,2269],{"className":27942},[2268],[40,27944,27946],{"className":27945},[1629],[40,27947,27949],{"className":27948,"style":27410},[1633],[40,27950],{},[40,27952],{"className":27953,"style":2009},[1756],[40,27955,27957,27960],{"className":27956},[1612],[40,27958,27286],{"className":27959,"style":27422},[1612,1653],[40,27961,27963],{"className":27962},[1621],[40,27964,27966,27986],{"className":27965},[1625,2244],[40,27967,27969,27983],{"className":27968},[1629],[40,27970,27972],{"className":27971,"style":10290},[1633],[40,27973,27974,27977],{"style":27437},[40,27975],{"className":27976,"style":1642},[1641],[40,27978,27980],{"className":27979},[1646,1647,1648,1649],[40,27981,2670],{"className":27982,"style":2736},[1612,1653,1649],[40,27984,2269],{"className":27985},[2268],[40,27987,27989],{"className":27988},[1629],[40,27990,27992],{"className":27991,"style":2276},[1633],[40,27993],{},[40,27995,21331],{"className":27996,"style":26393},[1612,1653],[40,27998,241],{"className":27999},[1985],[40,28001,28003,28006],{"className":28002},[1612],[40,28004,5226],{"className":28005,"style":4936},[1612,1653],[40,28007,28009],{"className":28008},[1621],[40,28010,28012,28032],{"className":28011},[1625,2244],[40,28013,28015,28029],{"className":28014},[1629],[40,28016,28018],{"className":28017,"style":10290},[1633],[40,28019,28020,28023],{"style":5058},[40,28021],{"className":28022,"style":1642},[1641],[40,28024,28026],{"className":28025},[1646,1647,1648,1649],[40,28027,2670],{"className":28028,"style":2736},[1612,1653,1649],[40,28030,2269],{"className":28031},[2268],[40,28033,28035],{"className":28034},[1629],[40,28036,28038],{"className":28037,"style":2276},[1633],[40,28039],{},[40,28041,2693],{"className":28042},[1822],[40,28044,1791],{"className":28045},[1612],[40,28047,28049,28052],{"className":28048},[1612],[40,28050,5226],{"className":28051,"style":4936},[1612,1653],[40,28053,28055],{"className":28054},[1621],[40,28056,28058,28078],{"className":28057},[1625,2244],[40,28059,28061,28075],{"className":28060},[1629],[40,28062,28064],{"className":28063,"style":10290},[1633],[40,28065,28066,28069],{"style":5058},[40,28067],{"className":28068,"style":1642},[1641],[40,28070,28072],{"className":28071},[1646,1647,1648,1649],[40,28073,2670],{"className":28074,"style":2736},[1612,1653,1649],[40,28076,2269],{"className":28077},[2268],[40,28079,28081],{"className":28080},[1629],[40,28082,28084],{"className":28083,"style":2276},[1633],[40,28085],{},[40,28087,1798],{"className":28088},[1822],[40,28090,1862],{"className":28091},[2005],[12,28093,28094,28095,28123,28124,28194,28195,28274],{},"using the fact that we can replace ",[40,28096,28098,28111],{"className":28097},[1565],[40,28099,28101],{"className":28100},[1569],[1571,28102,28103],{"xmlns":1573},[1575,28104,28105,28109],{},[1578,28106,28107],{},[1584,28108,2182],{},[1591,28110,2182],{"encoding":1593},[40,28112,28114],{"className":28113,"ariaHidden":1599},[1598],[40,28115,28117,28120],{"className":28116},[1603],[40,28118],{"className":28119,"style":1608},[1607],[40,28121,2182],{"className":28122,"style":2197},[1612,1653]," by ",[40,28125,28127,28145],{"className":28126},[1565],[40,28128,28130],{"className":28129},[1569],[1571,28131,28132],{"xmlns":1573},[1575,28133,28134,28142],{},[1578,28135,28136],{},[2212,28137,28138,28140],{},[1584,28139,5226],{},[1584,28141,2670],{},[1591,28143,28144],{"encoding":1593},"E_k",[40,28146,28148],{"className":28147,"ariaHidden":1599},[1598],[40,28149,28151,28154],{"className":28150},[1603],[40,28152],{"className":28153,"style":2231},[1607],[40,28155,28157,28160],{"className":28156},[1612],[40,28158,5226],{"className":28159,"style":4936},[1612,1653],[40,28161,28163],{"className":28162},[1621],[40,28164,28166,28186],{"className":28165},[1625,2244],[40,28167,28169,28183],{"className":28168},[1629],[40,28170,28172],{"className":28171,"style":10290},[1633],[40,28173,28174,28177],{"style":5058},[40,28175],{"className":28176,"style":1642},[1641],[40,28178,28180],{"className":28179},[1646,1647,1648,1649],[40,28181,2670],{"className":28182,"style":2736},[1612,1653,1649],[40,28184,2269],{"className":28185},[2268],[40,28187,28189],{"className":28188},[1629],[40,28190,28192],{"className":28191,"style":2276},[1633],[40,28193],{}," when it acts on the eigenstate ",[40,28196,28198,28219],{"className":28197},[1565],[40,28199,28201],{"className":28200},[1569],[1571,28202,28203],{"xmlns":1573},[1575,28204,28205,28217],{},[1578,28206,28207,28209,28215],{},[1584,28208,1791],{"mathvariant":1790},[2212,28210,28211,28213],{},[1584,28212,5226],{},[1584,28214,2670],{},[1731,28216,1798],{"stretchy":1797},[1591,28218,27534],{"encoding":1593},[40,28220,28222],{"className":28221,"ariaHidden":1599},[1598],[40,28223,28225,28228,28231,28271],{"className":28224},[1603],[40,28226],{"className":28227,"style":1811},[1607],[40,28229,1791],{"className":28230},[1612],[40,28232,28234,28237],{"className":28233},[1612],[40,28235,5226],{"className":28236,"style":4936},[1612,1653],[40,28238,28240],{"className":28239},[1621],[40,28241,28243,28263],{"className":28242},[1625,2244],[40,28244,28246,28260],{"className":28245},[1629],[40,28247,28249],{"className":28248,"style":10290},[1633],[40,28250,28251,28254],{"style":5058},[40,28252],{"className":28253,"style":1642},[1641],[40,28255,28257],{"className":28256},[1646,1647,1648,1649],[40,28258,2670],{"className":28259,"style":2736},[1612,1653,1649],[40,28261,2269],{"className":28262},[2268],[40,28264,28266],{"className":28265},[1629],[40,28267,28269],{"className":28268,"style":2276},[1633],[40,28270],{},[40,28272,1798],{"className":28273},[1822],". The energy error of this state is therefore",[40,28276,28278],{"className":28277},[1827],[40,28279,28281,28370],{"className":28280},[1565],[40,28282,28284],{"className":28283},[1569],[1571,28285,28286],{"xmlns":1573,"display":1836},[1575,28287,28288,28367],{},[1578,28289,28290,28293,28295],{},[5211,28291,28292],{},"energy error",[1731,28294,623],{},[8538,28296,28297,28341],{},[1578,28298,28299,28301,28303,28305,28307,28309,28311,28313,28315,28317,28319,28325,28327,28329,28331,28333,28335,28337,28339],{},[1731,28300,3469],{"stretchy":1797},[1584,28302,2837],{},[1584,28304,1791],{"mathvariant":1790},[1584,28306,21331],{},[1731,28308,241],{"stretchy":1797},[1584,28310,2182],{},[1731,28312,2693],{"stretchy":1797},[1731,28314,241],{"stretchy":1797},[1584,28316,2182],{},[1731,28318,1733],{},[2212,28320,28321,28323],{},[1584,28322,5226],{},[1735,28324,791],{},[1731,28326,2693],{"stretchy":1797},[1584,28328,21331],{},[1731,28330,241],{"stretchy":1797},[1584,28332,2182],{},[1731,28334,2693],{"stretchy":1797},[1584,28336,1791],{"mathvariant":1790},[1584,28338,2837],{},[1731,28340,1798],{"stretchy":1797},[1578,28342,28343,28345,28347,28349,28351,28353,28355,28361,28363,28365],{},[1731,28344,3469],{"stretchy":1797},[1584,28346,2837],{},[1584,28348,1791],{"mathvariant":1790},[1584,28350,21331],{},[1731,28352,241],{"stretchy":1797},[1584,28354,2182],{},[1581,28356,28357,28359],{},[1731,28358,2693],{"stretchy":1797},[1735,28360,1879],{},[1584,28362,1791],{"mathvariant":1790},[1584,28364,2837],{},[1731,28366,1798],{"stretchy":1797},[1591,28368,28369],{"encoding":1593},"\\text{energy error} = \\frac{\\langle\\psi|f(H)(H-E_0)f(H)|\\psi\\rangle}{\\langle\\psi|f(H)^2|\\psi\\rangle}",[40,28371,28373,28394],{"className":28372,"ariaHidden":1599},[1598],[40,28374,28376,28379,28385,28388,28391],{"className":28375},[1603],[40,28377],{"className":28378,"style":27621},[1607],[40,28380,28382],{"className":28381},[1612,6083],[40,28383,28292],{"className":28384},[1612],[40,28386],{"className":28387,"style":1963},[1756],[40,28389,623],{"className":28390},[1967],[40,28392],{"className":28393,"style":1963},[1756],[40,28395,28397,28401],{"className":28396},[1603],[40,28398],{"className":28399,"style":28400},[1607],"height:2.363em;vertical-align:-0.936em;",[40,28402,28404,28407,28614],{"className":28403},[1612],[40,28405],{"className":28406},[1985,8840],[40,28408,28410],{"className":28409},[8538],[40,28411,28413,28605],{"className":28412},[1625,2244],[40,28414,28416,28602],{"className":28415},[1629],[40,28417,28420,28486,28494],{"className":28418,"style":28419},[1633],"height:1.427em;",[40,28421,28422,28425],{"style":10764},[40,28423],{"className":28424,"style":3354},[1641],[40,28426,28428,28431,28434,28437,28440,28443,28446,28477,28480,28483],{"className":28427},[1612],[40,28429,3469],{"className":28430},[1985],[40,28432,2837],{"className":28433,"style":1818},[1612,1653],[40,28435,1791],{"className":28436},[1612],[40,28438,21331],{"className":28439,"style":26393},[1612,1653],[40,28441,241],{"className":28442},[1985],[40,28444,2182],{"className":28445,"style":2197},[1612,1653],[40,28447,28449,28452],{"className":28448},[1822],[40,28450,2693],{"className":28451},[1822],[40,28453,28455],{"className":28454},[1621],[40,28456,28458],{"className":28457},[1625],[40,28459,28461],{"className":28460},[1629],[40,28462,28465],{"className":28463,"style":28464},[1633],"height:0.7401em;",[40,28466,28468,28471],{"style":28467},"top:-2.989em;margin-right:0.05em;",[40,28469],{"className":28470,"style":1642},[1641],[40,28472,28474],{"className":28473},[1646,1647,1648,1649],[40,28475,1879],{"className":28476},[1612,1649],[40,28478,1791],{"className":28479},[1612],[40,28481,2837],{"className":28482,"style":1818},[1612,1653],[40,28484,1798],{"className":28485},[1822],[40,28487,28488,28491],{"style":8925},[40,28489],{"className":28490,"style":3354},[1641],[40,28492],{"className":28493,"style":8933},[8932],[40,28495,28496,28499],{"style":8936},[40,28497],{"className":28498,"style":3354},[1641],[40,28500,28502,28505,28508,28511,28514,28517,28520,28523,28526,28529,28532,28535,28538,28578,28581,28584,28587,28590,28593,28596,28599],{"className":28501},[1612],[40,28503,3469],{"className":28504},[1985],[40,28506,2837],{"className":28507,"style":1818},[1612,1653],[40,28509,1791],{"className":28510},[1612],[40,28512,21331],{"className":28513,"style":26393},[1612,1653],[40,28515,241],{"className":28516},[1985],[40,28518,2182],{"className":28519,"style":2197},[1612,1653],[40,28521,2693],{"className":28522},[1822],[40,28524,241],{"className":28525},[1985],[40,28527,2182],{"className":28528,"style":2197},[1612,1653],[40,28530],{"className":28531,"style":1757},[1756],[40,28533,1733],{"className":28534},[1761],[40,28536],{"className":28537,"style":1757},[1756],[40,28539,28541,28544],{"className":28540},[1612],[40,28542,5226],{"className":28543,"style":4936},[1612,1653],[40,28545,28547],{"className":28546},[1621],[40,28548,28550,28570],{"className":28549},[1625,2244],[40,28551,28553,28567],{"className":28552},[1629],[40,28554,28556],{"className":28555,"style":13330},[1633],[40,28557,28558,28561],{"style":5058},[40,28559],{"className":28560,"style":1642},[1641],[40,28562,28564],{"className":28563},[1646,1647,1648,1649],[40,28565,791],{"className":28566},[1612,1649],[40,28568,2269],{"className":28569},[2268],[40,28571,28573],{"className":28572},[1629],[40,28574,28576],{"className":28575,"style":2276},[1633],[40,28577],{},[40,28579,2693],{"className":28580},[1822],[40,28582,21331],{"className":28583,"style":26393},[1612,1653],[40,28585,241],{"className":28586},[1985],[40,28588,2182],{"className":28589,"style":2197},[1612,1653],[40,28591,2693],{"className":28592},[1822],[40,28594,1791],{"className":28595},[1612],[40,28597,2837],{"className":28598,"style":1818},[1612,1653],[40,28600,1798],{"className":28601},[1822],[40,28603,2269],{"className":28604},[2268],[40,28606,28608],{"className":28607},[1629],[40,28609,28612],{"className":28610,"style":28611},[1633],"height:0.936em;",[40,28613],{},[40,28615],{"className":28616},[1822,8840],[40,28618,28620],{"className":28619},[1827],[40,28621,28623,28751],{"className":28622},[1565],[40,28624,28626],{"className":28625},[1569],[1571,28627,28628],{"xmlns":1573,"display":1836},[1575,28629,28630,28748],{},[1578,28631,28632,28634,28746],{},[1731,28633,623],{},[8538,28635,28636,28700],{},[1578,28637,28638,28652,28654,28660,28666,28668,28670,28676,28682,28684,28690,28692,28698],{},[27268,28639,28640,28642,28650],{},[1731,28641,6571],{},[1578,28643,28644,28646,28648],{},[1584,28645,2670],{},[1731,28647,623],{},[1735,28649,79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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}.",[40,28752,28754,28766],{"className":28753,"ariaHidden":1599},[1598],[40,28755,28757,28760,28763],{"className":28756},[1603],[40,28758],{"className":28759,"style":3772},[1607],[40,28761,623],{"className":28762},[1967],[40,28764],{"className":28765,"style":1963},[1756],[40,28767,28769,28773,29354],{"className":28768},[1603],[40,28770],{"className":28771,"style":28772},[1607],"height:2.8419em;vertical-align:-1.1709em;",[40,28774,28776,28779,29351],{"className":28775},[1612],[40,28777],{"className":28778},[1985,8840],[40,28780,28782],{"className":28781},[8538],[40,28783,28785,29342],{"className":28784},[1625,2244],[40,28786,28788,29339],{"className":28787},[1629],[40,28789,28792,29014,29022],{"className":28790,"style":28791},[1633],"height:1.6709em;",[40,28793,28795,28798],{"style":28794},"top:-2.1288em;",[40,28796],{"className":28797,"style":3354},[1641],[40,28799,28801,28864,28867,28870,28910,28939,28942,28945,28985],{"className":28800},[1612],[40,28802,28804,28807],{"className":28803},[6614],[40,28805,6571],{"className":28806,"style":6620},[6614,6618,6619],[40,28808,28810],{"className":28809},[1621],[40,28811,28813,28856],{"className":28812},[1625,2244],[40,28814,28816,28853],{"className":28815},[1629],[40,28817,28819,28839],{"className":28818,"style":6866},[1633],[40,28820,28821,28824],{"style":6636},[40,28822],{"className":28823,"style":1642},[1641],[40,28825,28827],{"className":28826},[1646,1647,1648,1649],[40,28828,28830,28833,28836],{"className":28829},[1612,1649],[40,28831,2670],{"className":28832,"style":2736},[1612,1653,1649],[40,28834,623],{"className":28835},[1967,1649],[40,28837,791],{"className":28838},[1612,1649],[40,28840,28841,28844],{"style":6889},[40,28842],{"className":28843,"style":1642},[1641],[40,28845,28847],{"className":28846},[1646,1647,1648,1649],[40,28848,28850],{"className":28849},[1612,1649],[40,28851,6725],{"className":28852,"style":2358},[1612,1653,1649],[40,28854,2269],{"className":28855},[2268],[40,28857,28859],{"className":28858},[1629],[40,28860,28862],{"className":28861,"style":6655},[1633],[40,28863],{},[40,28865],{"className":28866,"style":2009},[1756],[40,28868,1791],{"className":28869},[1612],[40,28871,28873,28876],{"className":28872},[1612],[40,28874,27286],{"className":28875,"style":27422},[1612,1653],[40,28877,28879],{"className":28878},[1621],[40,28880,28882,28902],{"className":28881},[1625,2244],[40,28883,28885,28899],{"className":28884},[1629],[40,28886,28888],{"className":28887,"style":10290},[1633],[40,28889,28890,28893],{"style":27437},[40,28891],{"className":28892,"style":1642},[1641],[40,28894,28896],{"className":28895},[1646,1647,1648,1649],[40,28897,2670],{"className":28898,"style":2736},[1612,1653,1649],[40,28900,2269],{"className":28901},[2268],[40,28903,28905],{"className":28904},[1629],[40,28906,28908],{"className":28907,"style":2276},[1633],[40,28909],{},[40,28911,28913,28916],{"className":28912},[1612],[40,28914,1791],{"className":28915},[1612],[40,28917,28919],{"className":28918},[1621],[40,28920,28922],{"className":28921},[1625],[40,28923,28925],{"className":28924},[1629],[40,28926,28928],{"className":28927,"style":28464},[1633],[40,28929,28930,28933],{"style":28467},[40,28931],{"className":28932,"style":1642},[1641],[40,28934,28936],{"className":28935},[1646,1647,1648,1649],[40,28937,1879],{"className":28938},[1612,1649],[40,28940,21331],{"className":28941,"style":26393},[1612,1653],[40,28943,241],{"className":28944},[1985],[40,28946,28948,28951],{"className":28947},[1612],[40,28949,5226],{"className":28950,"style":4936},[1612,1653],[40,28952,28954],{"className":28953},[1621],[40,28955,28957,28977],{"className":28956},[1625,2244],[40,28958,28960,28974],{"className":28959},[1629],[40,28961,28963],{"className":28962,"style":10290},[1633],[40,28964,28965,28968],{"style":5058},[40,28966],{"className":28967,"style":1642},[1641],[40,28969,28971],{"className":28970},[1646,1647,1648,1649],[40,28972,2670],{"className":28973,"style":2736},[1612,1653,1649],[40,28975,2269],{"className":28976},[2268],[40,28978,28980],{"className":28979},[1629],[40,28981,28983],{"className":28982,"style":2276},[1633],[40,28984],{},[40,28986,28988,28991],{"className":28987},[1822],[40,28989,2693],{"className":28990},[1822],[40,28992,28994],{"className":28993},[1621],[40,28995,28997],{"className":28996},[1625],[40,28998,29000],{"className":28999},[1629],[40,29001,29003],{"className":29002,"style":28464},[1633],[40,29004,29005,29008],{"style":28467},[40,29006],{"className":29007,"style":1642},[1641],[40,29009,29011],{"className":29010},[1646,1647,1648,1649],[40,29012,1879],{"className":29013},[1612,1649],[40,29015,29016,29019],{"style":8925},[40,29017],{"className":29018,"style":3354},[1641],[40,29020],{"className":29021,"style":8933},[8932],[40,29023,29025,29028],{"style":29024},"top:-3.6897em;",[40,29026],{"className":29027,"style":3354},[1641],[40,29029,29031,29094,29097,29100,29140,291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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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can upper bound the first term by ",[40,31290,31292,31306],{"className":31291},[1565],[40,31293,31295],{"className":31294},[1569],[1571,31296,31297],{"xmlns":1573},[1575,31298,31299,31303],{},[1578,31300,31301],{},[1584,31302,26579],{},[1591,31304,31305],{"encoding":1593},"\\delta",[40,31307,31309],{"className":31308,"ariaHidden":1599},[1598],[40,31310,31312,31315],{"className":31311},[1603],[40,31313],{"className":31314,"style":4478},[1607],[40,31316,26579],{"className":31317,"style":26664},[1612,1653],[40,31319,31321],{"className":31320},[1827],[40,31322,31324,31572],{"className":31323},[1565],[40,31325,31327],{"className":31326},[1569],[1571,31328,31329],{"xmlns":1573,"display":1836},[1575,31330,31331,31569],{},[1578,31332,31333,31455,31457,31563,31565,31567],{},[8538,31334,31335,31409],{},[1578,31336,31337,31361,31363,31369,31375,31377,31379,31385,31391,31393,31399,31401,31407],{},[29695,31338,31339,31341],{},[1731,31340,6571],{},[1578,31342,31343,31349,31351,31357,31359],{},[2212,31344,31345,31347],{},[1584,31346,5226],{},[1584,31348,2670],{},[1731,31350,26564],{},[2212,31352,31353,31355],{},[1584,31354,5226],{},[1735,31356,791],{},[1731,31358,1147],{},[1584,31360,26579],{},[1584,31362,1791],{"mathvariant":1790},[2212,31364,31365,31367],{},[1584,31366,27286],{},[1584,31368,2670],{},[1581,31370,31371,31373],{},[1584,31372,1791],{"mathvariant":1790},[1735,31374,1879],{},[1584,31376,21331],{},[1731,31378,241],{"stretchy":1797},[2212,31380,31381,31383],{},[1584,31382,5226],{},[1584,31384,2670],{},[1581,31386,31387,31389],{},[1731,31388,2693],{"stretchy":1797},[1735,31390,1879],{},[1731,31392,241],{"stretchy":1797},[2212,31394,31395,31397],{},[1584,31396,5226],{},[1584,31398,2670],{},[1731,31400,1733],{},[2212,31402,31403,31405],{},[1584,31404,5226],{},[1735,31406,791],{},[1731,31408,2693],{"stretchy":1797},[1578,31410,31411,31425,31427,31433,31439,31441,31443,31449],{},[27268,31412,31413,31415,31423],{},[1731,31414,6571],{},[1578,31416,31417,31419,31421],{},[1584,31418,2670],{},[1731,31420,623],{},[1735,31422,791],{},[1584,31424,6725],{},[1584,31426,1791],{"mathvariant":1790},[2212,31428,31429,31431],{},[1584,31430,27286],{},[1584,31432,2670],{},[1581,31434,31435,31437],{},[1584,31436,1791],{"mathvariant":1790},[1735,31438,1879],{},[1584,31440,21331],{},[1731,31442,241],{"stretchy":1797},[2212,31444,31445,31447],{},[1584,31446,5226],{},[1584,31448,2670],{},[1581,31450,31451,31453],{},[1731,31452,2693],{"stretchy":1797},[1735,31454,1879],{},[1731,31456,4721],{},[8538,31458,31459,31517],{},[1578,31460,31461,31463,31487,31489,31495,31501,31503,31505,31511],{},[1584,31462,26579],{},[29695,31464,31465,31467],{},[1731,31466,6571],{},[1578,31468,31469,31475,31477,31483,31485],{},[2212,31470,31471,31473],{},[1584,31472,5226],{},[1584,31474,2670],{},[1731,31476,26564],{},[2212,31478,31479,31481],{},[1584,31480,5226],{},[1735,31482,791],{},[1731,31484,1147],{},[1584,31486,26579],{},[1584,31488,1791],{"mathvariant":1790},[2212,31490,31491,31493],{},[1584,31492,27286],{},[1584,31494,2670],{},[1581,31496,31497,31499],{},[1584,31498,1791],{"mathvariant":1790},[1735,31500,1879],{},[1584,31502,21331],{},[1731,31504,241],{"stretchy":1797},[2212,31506,31507,31509],{},[1584,31508,5226],{},[1584,31510,2670],{},[1581,31512,31513,31515],{},[1731,31514,2693],{"stretchy":1797},[1735,31516,1879],{},[1578,31518,31519,31533,31535,31541,31547,31549,31551,31557],{},[27268,31520,31521,31523,31531],{},[1731,31522,6571],{},[1578,31524,31525,31527,31529],{},[1584,31526,2670],{},[1731,31528,623],{},[1735,31530,791],{},[1584,31532,6725],{},[1584,31534,1791],{"mathvariant":1790},[2212,31536,31537,31539],{},[1584,31538,27286],{},[1584,31540,2670],{},[1581,31542,31543,31545],{},[1584,31544,1791],{"mathvariant":1790},[1735,31546,1879],{},[1584,31548,21331],{},[1731,31550,241],{"stretchy":1797},[2212,31552,31553,31555],{},[1584,31554,5226],{},[1584,31556,2670],{},[1581,31558,31559,31561],{},[1731,31560,2693],{"stretchy":1797},[1735,31562,1879],{},[1731,31564,26564],{},[1584,31566,26579],{},[1731,31568,1862],{"separator":1599},[1591,31570,31571],{"encoding":1593},"\\frac{\\sum_{E_k\\le E_0+\\delta}|\\gamma_k|^2f(E_k)^2(E_k-E_0)}{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2} \u003C \\frac{\\delta\\sum_{E_k\\le E_0+\\delta}|\\gamma_k|^2f(E_k)^2}{\\sum_{k=0}^{N}|\\gamma_k|^2f(E_k)^2} \\le 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the first step follows because 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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 ",[40,33043,33045,33071],{"className":33044},[1565],[40,33046,33048],{"className":33047},[1569],[1571,33049,33050],{"xmlns":1573},[1575,33051,33052,33068],{},[1578,33053,33054,33056,33062],{},[1584,33055,1791],{"mathvariant":1790},[2212,33057,33058,33060],{},[1584,33059,27286],{},[1735,33061,791],{},[1581,33063,33064,33066],{},[1584,33065,1791],{"mathvariant":1790},[1735,33067,1879],{},[1591,33069,33070],{"encoding":1593},"|\\gamma_0|^2",[40,33072,33074],{"className":33073,"ariaHidden":1599},[1598],[40,33075,33077,33080,33083,33123],{"className":33076},[1603],[40,33078],{"className":33079,"style":12249},[1607],[40,33081,1791],{"className":33082},[1612],[40,33084,33086,33089],{"className":33085},[1612],[40,33087,27286],{"className":33088,"style":27422},[1612,1653],[40,33090,33092],{"className":33091},[1621],[40,33093,33095,33115],{"className":33094},[1625,2244],[40,33096,33098,33112],{"className":33097},[1629],[40,33099,33101],{"className":33100,"style":13330},[1633],[40,33102,33103,33106],{"style":27437},[40,33104],{"className":33105,"style":1642},[1641],[40,33107,33109],{"className":33108},[1646,1647,1648,1649],[40,33110,791],{"className":33111},[1612,1649],[40,33113,2269],{"className":33114},[2268],[40,33116,33118],{"className":33117},[1629],[40,33119,33121],{"className":33120,"style":2276},[1633],[40,33122],{},[40,33124,33126,33129],{"className":33125},[1612],[40,33127,1791],{"className":33128},[1612],[40,33130,33132],{"className":33131},[1621],[40,33133,33135],{"className":33134},[1625],[40,33136,33138],{"className":33137},[1629],[40,33139,33141],{"className":33140,"style":12271},[1633],[40,33142,33143,33146],{"style":1637},[40,33144],{"className":33145,"style":1642},[1641],[40,33147,33149],{"className":33148},[1646,1647,1648,1649],[40,33150,1879],{"className":33151},[1612,1649],", since ",[40,33154,33156,33188],{"className":33155},[1565],[40,33157,33159],{"className":33158},[1569],[1571,33160,33161],{"xmlns":1573},[1575,33162,33163,33185],{},[1578,33164,33165,33167,33169,33175,33181,33183],{},[1584,33166,21331],{},[1731,33168,241],{"stretchy":1797},[2212,33170,33171,33173],{},[1584,33172,5226],{},[1735,33174,791],{},[1581,33176,33177,33179],{},[1731,33178,2693],{"stretchy":1797},[1735,33180,1879],{},[1731,33182,623],{},[1735,33184,1410],{},[1591,33186,33187],{"encoding":1593},"f(E_0)^2=1",[40,33189,33191,33281],{"className":33190,"ariaHidden":1599},[1598],[40,33192,33194,33197,33200,33203,33243,33272,33275,33278],{"className":33193},[1603],[40,33195],{"className":33196,"style":12249},[1607],[40,33198,21331],{"className":33199,"style":26393},[1612,1653],[40,33201,241],{"className":33202},[1985],[40,33204,33206,33209],{"className":33205},[1612],[40,33207,5226],{"className":33208,"style":4936},[1612,1653],[40,33210,33212],{"className":33211},[1621],[40,33213,33215,33235],{"className":33214},[1625,2244],[40,33216,33218,33232],{"className":33217},[1629],[40,33219,33221],{"className":33220,"style":13330},[1633],[40,33222,33223,33226],{"style":5058},[40,33224],{"className":33225,"style":1642},[1641],[40,33227,33229],{"className":33228},[1646,1647,1648,1649],[40,33230,791],{"className":33231},[1612,1649],[40,33233,2269],{"className":33234},[2268],[40,33236,33238],{"className":33237},[1629],[40,33239,33241],{"className":33240,"style":2276},[1633],[40,33242],{},[40,33244,33246,33249],{"className":33245},[1822],[40,33247,2693],{"className":33248},[1822],[40,33250,33252],{"className":33251},[1621],[40,33253,33255],{"className":33254},[1625],[40,33256,33258],{"className":33257},[1629],[40,33259,33261],{"className":33260,"style":12271},[1633],[40,33262,33263,33266],{"style":1637},[40,33264],{"className":33265,"style":1642},[1641],[40,33267,33269],{"className":33268},[1646,1647,1648,1649],[40,33270,1879],{"className":33271},[1612,1649],[40,33273],{"className":33274,"style":1963},[1756],[40,33276,623],{"className":33277},[1967],[40,33279],{"className":33280,"style":1963},[1756],[40,33282,33284,33287],{"className":33283},[1603],[40,33285],{"className":33286,"style":1771},[1607],[40,33288,1410],{"className":33289},[1612],": adding everything back together, this gives",[40,33292,33294],{"className":33293},[1827],[40,33295,33297,33411],{"className":33296},[1565],[40,33298,33300],{"className":33299},[1569],[1571,33301,33302],{"xmlns":1573,"display":1836},[1575,33303,33304,33408],{},[1578,33305,33306,33308,33310,33312,33314,33334,33358,33360,33366,33372,33374,33376,33382,33388,33390,33396,33398,33404,33406],{},[5211,33307,28292],{},[1731,33309,26564],{},[1584,33311,26579],{},[1731,33313,1147],{},[8538,33315,33316,33318],{},[1735,33317,1410],{},[1578,33319,33320,33322,33328],{},[1584,33321,1791],{"mathvariant":1790},[2212,33323,33324,33326],{},[1584,33325,27286],{},[1735,33327,791],{},[1581,33329,33330,33332],{},[1584,33331,1791],{"mathvariant":1790},[1735,33333,1879],{},[29695,33335,33336,33338],{},[1731,33337,6571],{},[1578,33339,33340,33346,33348,33354,33356],{},[2212,33341,33342,33344],{},[1584,33343,5226],{},[1584,33345,2670],{},[1731,33347,689],{},[2212,33349,33350,33352],{},[1584,33351,5226],{},[1735,33353,791],{},[1731,33355,1147],{},[1584,33357,26579],{},[1584,33359,1791],{"mathvariant":1790},[2212,33361,33362,33364],{},[1584,33363,27286],{},[1584,33365,2670],{},[1581,33367,33368,33370],{},[1584,33369,1791],{"mathvariant":1790},[1735,33371,1879],{},[1584,33373,21331],{},[1731,33375,241],{"stretchy":1797},[2212,33377,33378,33380],{},[1584,33379,5226],{},[1584,33381,2670],{},[1581,33383,33384,33386],{},[1731,33385,2693],{"stretchy":1797},[1735,33387,1879],{},[1731,33389,241],{"stretchy":1797},[2212,33391,33392,33394],{},[1584,33393,5226],{},[1584,33395,2670],{},[1731,33397,1733],{},[2212,33399,33400,33402],{},[1584,33401,5226],{},[1735,33403,791],{},[1731,33405,2693],{"stretchy":1797},[1584,33407,1823],{"mathvariant":1790},[1591,33409,33410],{"encoding":1593},"\\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).",[40,33412,33414,33436,33455,33933],{"className":33413,"ariaHidden":1599},[1598],[40,33415,33417,33421,33427,33430,33433],{"className":33416},[1603],[40,33418],{"className":33419,"style":33420},[1607],"height:0.8304em;vertical-align:-0.1944em;",[40,33422,33424],{"className":33423},[1612,6083],[40,33425,28292],{"className":33426},[1612],[40,33428],{"className":33429,"style":1963},[1756],[40,33431,26564],{"className":33432},[1967],[40,33434],{"className":33435,"style":1963},[1756],[40,33437,33439,33443,33446,33449,33452],{"className":33438},[1603],[40,33440],{"className":33441,"style":33442},[1607],"height:0.7778em;vertical-align:-0.0833em;",[40,33444,26579],{"className":33445,"style":26664},[1612,1653],[40,33447],{"className":33448,"style":1757},[1756],[40,33450,1147],{"className":33451},[1761],[40,33453],{"className":33454,"style":1757},[1756],[40,33456,33458,33462,33593,33596,33731,33734,33737,33777,33806,33809,33812,33852,33881,33884,33924,33927,33930],{"className":33457},[1603],[40,33459],{"className":33460,"style":33461},[1607],"height:2.7294em;vertical-align:-1.408em;",[40,33463,33465,33468,33590],{"className":33464},[1612],[40,33466],{"className":33467},[1985,8840],[40,33469,33471],{"className":33470},[8538],[40,33472,33474,33582],{"className":33473},[1625,2244],[40,33475,33477,33579],{"className":33476},[1629],[40,33478,33480,33560,33568],{"className":33479,"style":8853},[1633],[40,33481,33482,33485],{"style":10764},[40,33483],{"className":33484,"style":3354},[1641],[40,33486,33488,33491,33531],{"className":33487},[1612],[40,33489,1791],{"className":33490},[1612],[40,33492,33494,33497],{"className":33493},[1612],[40,33495,27286],{"className":33496,"style":27422},[1612,1653],[40,33498,33500],{"className":33499},[1621],[40,33501,33503,33523],{"className":33502},[1625,2244],[40,33504,33506,33520],{"className":33505},[1629],[40,33507,33509],{"className":33508,"style":13330},[1633],[40,33510,33511,33514],{"style":27437},[40,33512],{"className":33513,"style":1642},[1641],[40,33515,33517],{"className":33516},[1646,1647,1648,1649],[40,33518,791],{"className":33519},[1612,1649],[40,33521,2269],{"className":33522},[2268],[40,33524,33526],{"className":33525},[1629],[40,33527,33529],{"className":33528,"style":2276},[1633],[40,33530],{},[40,33532,33534,33537],{"className":33533},[1612],[40,33535,1791],{"className":33536},[1612],[40,33538,33540],{"className":33539},[1621],[40,33541,33543],{"className":33542},[1625],[40,33544,33546],{"className":33545},[1629],[40,33547,33549],{"className":33548,"style":28464},[1633],[40,33550,33551,33554],{"style":28467},[40,33552],{"className":33553,"style":1642},[1641],[40,33555,33557],{"className":33556},[1646,1647,1648,1649],[40,33558,1879],{"className":33559},[1612,1649],[40,33561,33562,33565],{"style":8925},[40,33563],{"className":33564,"style":3354},[1641],[40,33566],{"className":33567,"style":8933},[8932],[40,33569,33570,33573],{"style":8936},[40,33571],{"className":33572,"style":3354},[1641],[40,33574,33576],{"className":33575},[1612],[40,33577,1410],{"className":33578},[1612],[40,33580,2269],{"className":33581},[2268],[40,33583,33585],{"className":33584},[1629],[40,33586,33588],{"className":33587,"style":28611},[1633],[40,33589],{},[40,33591],{"className":33592},[1822,8840],[40,33594],{"className":33595,"style":2009},[1756],[40,33597,33599],{"className":33598},[6614,27341],[40,33600,33602,33722],{"className":33601},[1625,2244],[40,33603,33605,33719],{"className":33604},[1629],[40,33606,33609,33709],{"className":33607,"style":33608},[1633],"height:1.05em;",[40,33610,33611,33614],{"style":27354},[40,33612],{"className":33613,"style":27358},[1641],[40,33615,33617],{"className":33616},[1646,1647,1648,1649],[40,33618,33620,33660,33663,33703,33706],{"className":33619},[1612,1649],[40,33621,33623,33626],{"className":33622},[1612,1649],[40,33624,5226],{"className":33625,"style":4936},[1612,1653,1649],[40,33627,33629],{"className":33628},[1621],[40,33630,33632,33652],{"className":33631},[1625,2244],[40,33633,33635,33649],{"className":33634},[1629],[40,33636,33638],{"className":33637,"style":30278},[1633],[40,33639,33640,33643],{"style":30281},[40,33641],{"className":33642,"style":3856},[1641],[40,33644,33646],{"className":33645},[1646,3860,1992,1649],[40,33647,2670],{"className":33648,"style":2736},[1612,1653,1649],[40,33650,2269],{"className":33651},[2268],[40,33653,33655],{"className":33654},[1629],[40,33656,33658],{"className":33657,"style":30300},[1633],[40,33659],{},[40,33661,689],{"className":33662},[1967,1649],[40,33664,33666,33669],{"className":33665},[1612,1649],[40,33667,5226],{"className":33668,"style":4936},[1612,1653,1649],[40,33670,33672],{"className":33671},[1621],[40,33673,33675,33695],{"className":33674},[1625,2244],[40,33676,33678,33692],{"className":33677},[1629],[40,33679,33681],{"className":33680,"style":30324},[1633],[40,33682,33683,33686],{"style":30327},[40,33684],{"className":33685,"style":3856},[1641],[40,33687,33689],{"className":33688},[1646,3860,1992,1649],[40,33690,791],{"className":33691},[1612,1649],[40,33693,2269],{"className":33694},[2268],[40,33696,33698],{"className":33697},[1629],[40,33699,33701],{"className":33700,"style":3873},[1633],[40,33702],{},[40,33704,1147],{"className":33705},[1761,1649],[40,33707,26579],{"className":33708,"style":26664},[1612,1653,1649],[40,33710,33711,33714],{"style":27376},[40,33712],{"className":33713,"style":27358},[1641],[40,33715,33716],{},[40,33717,6571],{"className":33718},[6614,6618,27385],[40,33720,2269],{"className":33721},[2268],[40,33723,33725],{"className":33724},[1629],[40,33726,33729],{"className":33727,"style":33728},[1633],"height:1.408em;",[40,33730],{},[40,33732],{"className":33733,"style":2009},[1756],[40,33735,1791],{"className":33736},[1612],[40,33738,33740,33743],{"className":33739},[1612],[40,33741,27286],{"className":33742,"style":27422},[1612,1653],[40,33744,33746],{"className":33745},[1621],[40,33747,33749,33769],{"className":33748},[1625,2244],[40,33750,33752,33766],{"className":33751},[1629],[40,33753,33755],{"className":33754,"style":10290},[1633],[40,33756,33757,33760],{"style":27437},[40,33758],{"className":33759,"style":1642},[1641],[40,33761,33763],{"className":33762},[1646,1647,1648,1649],[40,33764,2670],{"className":33765,"style":2736},[1612,1653,1649],[40,33767,2269],{"className":33768},[2268],[40,33770,33772],{"className":33771},[1629],[40,33773,33775],{"className":33774,"style":2276},[1633],[40,33776],{},[40,33778,33780,33783],{"className":33779},[1612],[40,33781,1791],{"className":33782},[1612],[40,33784,33786],{"className":33785},[1621],[40,33787,33789],{"className":33788},[1625],[40,33790,33792],{"className":33791},[1629],[40,33793,33795],{"className":33794,"style":2046},[1633],[40,33796,33797,33800],{"style":1950},[40,33798],{"className":33799,"style":1642},[1641],[40,33801,33803],{"className":33802},[1646,1647,1648,1649],[40,33804,1879],{"className":33805},[1612,1649],[40,33807,21331],{"className":33808,"style":26393},[1612,1653],[40,33810,241],{"className":33811},[1985],[40,33813,33815,33818],{"className":33814},[1612],[40,33816,5226],{"className":33817,"style":4936},[1612,1653],[40,33819,33821],{"className":33820},[1621],[40,33822,33824,33844],{"className":33823},[1625,2244],[40,33825,33827,33841],{"className":33826},[1629],[40,33828,33830],{"className":33829,"style":10290},[1633],[40,33831,33832,33835],{"style":5058},[40,33833],{"className":33834,"style":1642},[1641],[40,33836,33838],{"className":33837},[1646,1647,1648,1649],[40,33839,2670],{"className":33840,"style":2736},[1612,1653,1649],[40,33842,2269],{"className":33843},[2268],[40,33845,33847],{"className":33846},[1629],[40,33848,33850],{"className":33849,"style":2276},[1633],[40,33851],{},[40,33853,33855,33858],{"className":33854},[1822],[40,33856,2693],{"className":33857},[1822],[40,33859,33861],{"className":33860},[1621],[40,33862,33864],{"className":33863},[1625],[40,33865,33867],{"className":33866},[1629],[40,33868,33870],{"className":33869,"style":2046},[1633],[40,33871,33872,33875],{"style":1950},[40,33873],{"className":33874,"style":1642},[1641],[40,33876,33878],{"className":33877},[1646,1647,1648,1649],[40,33879,1879],{"className":33880},[1612,1649],[40,33882,241],{"className":33883},[1985],[40,33885,33887,33890],{"className":33886},[1612],[40,33888,5226],{"className":33889,"style":4936},[1612,1653],[40,33891,33893],{"className":33892},[1621],[40,33894,33896,33916],{"className":33895},[1625,2244],[40,33897,33899,33913],{"className":33898},[1629],[40,33900,33902],{"className":33901,"style":10290},[1633],[40,33903,33904,33907],{"style":5058},[40,33905],{"className":33906,"style":1642},[1641],[40,33908,33910],{"className":33909},[1646,1647,1648,1649],[40,33911,2670],{"className":33912,"style":2736},[1612,1653,1649],[40,33914,2269],{"className":33915},[2268],[40,33917,33919],{"className":33918},[1629],[40,33920,33922],{"className":33921,"style":2276},[1633],[40,33923],{},[40,33925],{"className":33926,"style":1757},[1756],[40,33928,1733],{"className":33929},[1761],[40,33931],{"className":33932,"style":1757},[1756],[40,33934,33936,33939,33979,33982],{"className":33935},[1603],[40,33937],{"className":33938,"style":1811},[1607],[40,33940,33942,33945],{"className":33941},[1612],[40,33943,5226],{"className":33944,"style":4936},[1612,1653],[40,33946,33948],{"className":33947},[1621],[40,33949,33951,33971],{"className":33950},[1625,2244],[40,33952,33954,33968],{"className":33953},[1629],[40,33955,33957],{"className":33956,"style":13330},[1633],[40,33958,33959,33962],{"style":5058},[40,33960],{"className":33961,"style":1642},[1641],[40,33963,33965],{"className":33964},[1646,1647,1648,1649],[40,33966,791],{"className":33967},[1612,1649],[40,33969,2269],{"className":33970},[2268],[40,33972,33974],{"className":33973},[1629],[40,33975,33977],{"className":33976,"style":2276},[1633],[40,33978],{},[40,33980,2693],{"className":33981},[1822],[40,33983,1823],{"className":33984},[1612],[12,33986,33987,33988,34057,34058,34086,34087,34308,34309,34477,34478,34767],{},"To simplify what is left, notice that for all these ",[40,33989,33991,34008],{"className":33990},[1565],[40,33992,33994],{"className":33993},[1569],[1571,33995,33996],{"xmlns":1573},[1575,33997,33998,34006],{},[1578,33999,34000],{},[2212,34001,34002,34004],{},[1584,34003,5226],{},[1584,34005,2670],{},[1591,34007,28144],{"encoding":1593},[40,34009,34011],{"className":34010,"ariaHidden":1599},[1598],[40,34012,34014,34017],{"className":34013},[1603],[40,34015],{"className":34016,"style":2231},[1607],[40,34018,34020,34023],{"className":34019},[1612],[40,34021,5226],{"className":34022,"style":4936},[1612,1653],[40,34024,34026],{"className":34025},[1621],[40,34027,34029,34049],{"className":34028},[1625,2244],[40,34030,34032,34046],{"className":34031},[1629],[40,34033,34035],{"className":34034,"style":10290},[1633],[40,34036,34037,34040],{"style":5058},[40,34038],{"className":34039,"style":1642},[1641],[40,34041,34043],{"className":34042},[1646,1647,1648,1649],[40,34044,2670],{"className":34045,"style":2736},[1612,1653,1649],[40,34047,2269],{"className":34048},[2268],[40,34050,34052],{"className":34051},[1629],[40,34053,34055],{"className":34054,"style":2276},[1633],[40,34056],{},", by the definition of ",[40,34059,34061,34074],{"className":34060},[1565],[40,34062,34064],{"className":34063},[1569],[1571,34065,34066],{"xmlns":1573},[1575,34067,34068,34072],{},[1578,34069,34070],{},[1584,34071,21331],{},[1591,34073,21331],{"encoding":1593},[40,34075,34077],{"className":34076,"ariaHidden":1599},[1598],[40,34078,34080,34083],{"className":34079},[1603],[40,34081],{"className":34082,"style":6079},[1607],[40,34084,21331],{"className":34085,"style":26393},[1612,1653]," we know that ",[40,34088,34090,34144],{"className":34089},[1565],[40,34091,34093],{"className":34092},[1569],[1571,34094,34095],{"xmlns":1573},[1575,34096,34097,34141],{},[1578,34098,34099,34101,34103,34109,34115,34117,34119],{},[1584,34100,21331],{},[1731,34102,241],{"stretchy":1797},[2212,34104,34105,34107],{},[1584,34106,5226],{},[1584,34108,2670],{},[1581,34110,34111,34113],{},[1731,34112,2693],{"stretchy":1797},[1735,34114,1879],{},[1731,34116,26564],{},[1735,34118,11199],{},[1581,34120,34121,34133],{},[1578,34122,34123,34125,34127,34129,34131],{},[1731,34124,241],{"fence":1599},[1735,34126,1410],{},[1731,34128,1147],{},[1584,34130,26579],{},[1731,34132,2693],{"fence":1599},[1578,34134,34135,34137,34139],{},[1731,34136,1733],{},[1735,34138,1879],{},[1584,34140,4003],{},[1591,34142,34143],{"encoding":1593},"f(E_k)^2 \\le 4\\left(1 + \\delta\\right)^{-2d}",[40,34145,34147,34237],{"className":34146,"ariaHidden":1599},[1598],[40,34148,34150,34153,34156,34159,34199,34228,34231,34234],{"className":34149},[1603],[40,34151],{"className":34152,"style":12249},[1607],[40,34154,21331],{"className":34155,"style":26393},[1612,1653],[40,34157,241],{"className":34158},[1985],[40,34160,34162,34165],{"className":34161},[1612],[40,34163,5226],{"className":34164,"style":4936},[1612,1653],[40,34166,34168],{"className":34167},[1621],[40,34169,34171,34191],{"className":34170},[1625,2244],[40,34172,34174,34188],{"className":34173},[1629],[40,34175,34177],{"className":34176,"style":10290},[1633],[40,34178,34179,34182],{"style":5058},[40,34180],{"className":34181,"style":1642},[1641],[40,34183,34185],{"className":34184},[1646,1647,1648,1649],[40,34186,2670],{"className":34187,"style":2736},[1612,1653,1649],[40,34189,2269],{"className":34190},[2268],[40,34192,34194],{"className":34193},[1629],[40,34195,34197],{"className":34196,"style":2276},[1633],[40,34198],{},[40,34200,34202,34205],{"className":34201},[1822],[40,34203,2693],{"className":34204},[1822],[40,34206,34208],{"className":34207},[1621],[40,34209,34211],{"className":34210},[1625],[40,34212,34214],{"className":34213},[1629],[40,34215,34217],{"className":34216,"style":12271},[1633],[40,34218,34219,34222],{"style":1637},[40,34220],{"className":34221,"style":1642},[1641],[40,34223,34225],{"className":34224},[1646,1647,1648,1649],[40,34226,1879],{"className":34227},[1612,1649],[40,34229],{"className":34230,"style":1963},[1756],[40,34232,26564],{"className":34233},[1967],[40,34235],{"className":34236,"style":1963},[1756],[40,34238,34240,34243,34246,34249],{"className":34239},[1603],[40,34241],{"className":34242,"style":26633},[1607],[40,34244,11199],{"className":34245},[1612],[40,34247],{"className":34248,"style":2009},[1756],[40,34250,34252,34276],{"className":34251},[1981],[40,34253,34255,34258,34261,34264,34267,34270,34273],{"className":34254},[1981],[40,34256,241],{"className":34257,"style":1987},[1985,1986],[40,34259,1410],{"className":34260},[1612],[40,34262],{"className":34263,"style":1757},[1756],[40,34265,1147],{"className":34266},[1761],[40,34268],{"className":34269,"style":1757},[1756],[40,34271,26579],{"className":34272,"style":26664},[1612,1653],[40,34274,2693],{"className":34275,"style":1987},[1822,1986],[40,34277,34279],{"className":34278},[1621],[40,34280,34282],{"className":34281},[1625],[40,34283,34285],{"className":34284},[1629],[40,34286,34288],{"className":34287,"style":26680},[1633],[40,34289,34290,34293],{"style":6889},[40,34291],{"className":34292,"style":1642},[1641],[40,34294,34296],{"className":34295},[1646,1647,1648,1649],[40,34297,34299,34302,34305],{"className":34298},[1612,1649],[40,34300,1733],{"className":34301},[1612,1649],[40,34303,1879],{"className":34304},[1612,1649],[40,34306,4003],{"className":34307},[1612,1653,1649],". Additionally upper bounding ",[40,34310,34312,34348],{"className":34311},[1565],[40,34313,34315],{"className":34314},[1569],[1571,34316,34317],{"xmlns":1573},[1575,34318,34319,34345],{},[1578,34320,34321,34327,34329,34335,34337,34339,34341,34343],{},[2212,34322,34323,34325],{},[1584,34324,5226],{},[1584,34326,2670],{},[1731,34328,1733],{},[2212,34330,34331,34333],{},[1584,34332,5226],{},[1735,34334,791],{},[1731,34336,4721],{},[1735,34338,1879],{},[1584,34340,11205],{"mathvariant":1790},[1584,34342,2182],{},[1584,34344,11205],{"mathvariant":1790},[1591,34346,34347],{"encoding":1593},"E_k-E_0\u003C2\\|H\\|",[40,34349,34351,34406,34461],{"className":34350,"ariaHidden":1599},[1598],[40,34352,34354,34357,34397,34400,34403],{"className":34353},[1603],[40,34355],{"className":34356,"style":2231},[1607],[40,34358,34360,34363],{"className":34359},[1612],[40,34361,5226],{"className":34362,"style":4936},[1612,1653],[40,34364,34366],{"className":34365},[1621],[40,34367,34369,34389],{"className":34368},[1625,2244],[40,34370,34372,34386],{"className":34371},[1629],[40,34373,34375],{"className":34374,"style":10290},[1633],[40,34376,34377,34380],{"style":5058},[40,34378],{"className":34379,"style":1642},[1641],[40,34381,34383],{"className":34382},[1646,1647,1648,1649],[40,34384,2670],{"className":34385,"style":2736},[1612,1653,1649],[40,34387,2269],{"className":34388},[2268],[40,34390,34392],{"className":34391},[1629],[40,34393,34395],{"className":34394,"style":2276},[1633],[40,34396],{},[40,34398],{"className":34399,"style":1757},[1756],[40,34401,1733],{"className":34402},[1761],[40,34404],{"className":34405,"style":1757},[1756],[40,34407,34409,34412,34452,34455,34458],{"className":34408},[1603],[40,34410],{"className":34411,"style":2231},[1607],[40,34413,34415,34418],{"className":34414},[1612],[40,34416,5226],{"className":34417,"style":4936},[1612,1653],[40,34419,34421],{"className":34420},[1621],[40,34422,34424,34444],{"className":34423},[1625,2244],[40,34425,34427,34441],{"className":34426},[1629],[40,34428,34430],{"className":34429,"style":13330},[1633],[40,34431,34432,34435],{"style":5058},[40,34433],{"className":34434,"style":1642},[1641],[40,34436,34438],{"className":34437},[1646,1647,1648,1649],[40,34439,791],{"className":34440},[1612,1649],[40,34442,2269],{"className":34443},[2268],[40,34445,34447],{"className":34446},[1629],[40,34448,34450],{"className":34449,"style":2276},[1633],[40,34451],{},[40,34453],{"className":34454,"style":1963},[1756],[40,34456,4721],{"className":34457},[1967],[40,34459],{"className":34460,"style":1963},[1756],[40,34462,34464,34467,34471,34474],{"className":34463},[1603],[40,34465],{"className":34466,"style":1811},[1607],[40,34468,34470],{"className":34469},[1612],"2∥",[40,34472,2182],{"className":34473,"style":2197},[1612,1653],[40,34475,11205],{"className":34476},[1612]," and upper bounding ",[40,34479,34481,34535],{"className":34480},[1565],[40,34482,34484],{"className":34483},[1569],[1571,34485,34486],{"xmlns":1573},[1575,34487,34488,34532],{},[1578,34489,34490,34514,34516,34522,34528,34530],{},[2212,34491,34492,34494],{},[1731,34493,6571],{},[1578,34495,34496,34502,34504,34510,34512],{},[2212,34497,34498,34500],{},[1584,34499,5226],{},[1584,34501,2670],{},[1731,34503,689],{},[2212,34505,34506,34508],{},[1584,34507,5226],{},[1735,34509,791],{},[1731,34511,1147],{},[1584,34513,26579],{},[1584,34515,1791],{"mathvariant":1790},[2212,34517,34518,34520],{},[1584,34519,27286],{},[1584,34521,2670],{},[1581,34523,34524,34526],{},[1584,34525,1791],{"mathvariant":1790},[1735,34527,1879],{},[1731,34529,4721],{},[1735,34531,1410],{},[1591,34533,34534],{"encoding":1593},"\\sum_{E_k>E_0+\\delta}|\\gamma_k|^2\u003C1",[40,34536,34538,34758],{"className":34537,"ariaHidden":1599},[1598],[40,34539,34541,34545,34674,34677,34680,34720,34749,34752,34755],{"className":34540},[1603],[40,34542],{"className":34543,"style":34544},[1607],"height:1.2197em;vertical-align:-0.4056em;",[40,34546,34548,34551],{"className":34547},[6614],[40,34549,6571],{"className":34550,"style":6620},[6614,6618,6619],[40,34552,34554],{"className":34553},[1621],[40,34555,34557,34666],{"className":34556},[1625,2244],[40,34558,34560,34663],{"className":34559},[1629],[40,34561,34563],{"className":34562,"style":30248},[1633],[40,34564,34565,34568],{"style":6636},[40,34566],{"className":34567,"style":1642},[1641],[40,34569,34571],{"className":34570},[1646,1647,1648,1649],[40,34572,34574,34614,34617,34657,34660],{"className":34573},[1612,1649],[40,34575,34577,34580],{"className":34576},[1612,1649],[40,34578,5226],{"className":34579,"style":4936},[1612,1653,1649],[40,34581,34583],{"className":34582},[1621],[40,34584,34586,34606],{"className":34585},[1625,2244],[40,34587,34589,34603],{"className":34588},[1629],[40,34590,34592],{"className":34591,"style":30278},[1633],[40,34593,34594,34597],{"style":30281},[40,34595],{"className":34596,"style":3856},[1641],[40,34598,34600],{"className":34599},[1646,3860,1992,1649],[40,34601,2670],{"className":34602,"style":2736},[1612,1653,1649],[40,34604,2269],{"className":34605},[2268],[40,34607,34609],{"className":34608},[1629],[40,34610,34612],{"className":34611,"style":30300},[1633],[40,34613],{},[40,34615,689],{"className":34616},[1967,1649],[40,34618,34620,34623],{"className":34619},[1612,1649],[40,34621,5226],{"className":34622,"style":4936},[1612,1653,1649],[40,34624,34626],{"className":34625},[1621],[40,34627,34629,34649],{"className":34628},[1625,2244],[40,34630,34632,34646],{"className":34631},[1629],[40,34633,34635],{"className":34634,"style":30324},[1633],[40,34636,34637,34640],{"style":30327},[40,34638],{"className":34639,"style":3856},[1641],[40,34641,34643],{"className":34642},[1646,3860,1992,1649],[40,34644,791],{"className":34645},[1612,1649],[40,34647,2269],{"className":34648},[2268],[40,34650,34652],{"className":34651},[1629],[40,34653,34655],{"className":34654,"style":3873},[1633],[40,34656],{},[40,34658,1147],{"className":34659},[1761,1649],[40,34661,26579],{"className":34662,"style":26664},[1612,1653,1649],[40,34664,2269],{"className":34665},[2268],[40,34667,34669],{"className":34668},[1629],[40,34670,34672],{"className":34671,"style":30363},[1633],[40,34673],{},[40,34675],{"className":34676,"style":2009},[1756],[40,34678,1791],{"className":34679},[1612],[40,34681,34683,34686],{"className":34682},[1612],[40,34684,27286],{"className":34685,"style":27422},[1612,1653],[40,34687,34689],{"className":34688},[1621],[40,34690,34692,34712],{"className":34691},[1625,2244],[40,34693,34695,34709],{"className":34694},[1629],[40,34696,34698],{"className":34697,"style":10290},[1633],[40,34699,34700,34703],{"style":27437},[40,34701],{"className":34702,"style":1642},[1641],[40,34704,34706],{"className":34705},[1646,1647,1648,1649],[40,34707,2670],{"className":34708,"style":2736},[1612,1653,1649],[40,34710,2269],{"className":34711},[2268],[40,34713,34715],{"className":34714},[1629],[40,34716,34718],{"className":34717,"style":2276},[1633],[40,34719],{},[40,34721,34723,34726],{"className":34722},[1612],[40,34724,1791],{"className":34725},[1612],[40,34727,34729],{"className":34728},[1621],[40,34730,34732],{"className":34731},[1625],[40,34733,34735],{"className":34734},[1629],[40,34736,34738],{"className":34737,"style":12271},[1633],[40,34739,34740,34743],{"style":1637},[40,34741],{"className":34742,"style":1642},[1641],[40,34744,34746],{"className":34745},[1646,1647,1648,1649],[40,34747,1879],{"className":34748},[1612,1649],[40,34750],{"className":34751,"style":1963},[1756],[40,34753,4721],{"className":34754},[1967],[40,34756],{"className":34757,"style":1963},[1756],[40,34759,34761,34764],{"className":34760},[1603],[40,34762],{"className":34763,"style":1771},[1607],[40,34765,1410],{"className":34766},[1612]," gives",[40,34769,34771],{"className":34770},[1827],[40,34772,34774,34845],{"className":34773},[1565],[40,34775,34777],{"className":34776},[1569],[1571,34778,34779],{"xmlns":1573,"display":1836},[1575,34780,34781,34842],{},[1578,34782,34783,34785,34787,34789,34791,34812,34814,34816,34818,34840],{},[5211,34784,28292],{},[1731,34786,26564],{},[1584,34788,26579],{},[1731,34790,1147],{},[8538,34792,34793,34796],{},[1735,34794,34795],{},"8",[1578,34797,34798,34800,34806],{},[1584,34799,1791],{"mathvariant":1790},[2212,34801,34802,34804],{},[1584,34803,27286],{},[1735,34805,791],{},[1581,34807,34808,34810],{},[1584,34809,1791],{"mathvariant":1790},[1735,34811,1879],{},[1584,34813,11205],{"mathvariant":1790},[1584,34815,2182],{},[1584,34817,11205],{"mathvariant":1790},[1581,34819,34820,34832],{},[1578,34821,34822,34824,34826,34828,34830],{},[1731,34823,241],{"fence":1599},[1735,34825,1410],{},[1731,34827,1147],{},[1584,34829,26579],{},[1731,34831,2693],{"fence":1599},[1578,34833,34834,34836,34838],{},[1731,34835,1733],{},[1735,34837,1879],{},[1584,34839,4003],{},[1584,34841,1823],{"mathvariant":1790},[1591,34843,34844],{"encoding":1593},"\\text{energy error} \\le \\delta + \\frac{8}{|\\gamma_0|^2}\\|H\\|\\left(1 + \\delta\\right)^{-2d}.",[40,34846,34848,34869,34887],{"className":34847,"ariaHidden":1599},[1598],[40,34849,34851,34854,34860,34863,34866],{"className":34850},[1603],[40,34852],{"className":34853,"style":33420},[1607],[40,34855,34857],{"className":34856},[1612,6083],[40,34858,28292],{"className":34859},[1612],[40,34861],{"className":34862,"style":1963},[1756],[40,34864,26564],{"className":34865},[1967],[40,34867],{"className":34868,"style":1963},[1756],[40,34870,34872,34875,34878,34881,34884],{"className":34871},[1603],[40,34873],{"className":34874,"style":33442},[1607],[40,34876,26579],{"className":34877,"style":26664},[1612,1653],[40,34879],{"className":34880,"style":1757},[1756],[40,34882,1147],{"className":34883},[1761],[40,34885],{"className":34886,"style":1757},[1756],[40,34888,34890,34894,35025,35028,35031,35034,35037,35096,35099],{"className":34889},[1603],[40,34891],{"className":34892,"style":34893},[1607],"height:2.2574em;vertical-align:-0.936em;",[40,34895,34897,34900,35022],{"className":34896},[1612],[40,34898],{"className":34899},[1985,8840],[40,34901,34903],{"className":34902},[8538],[40,34904,34906,35014],{"className":34905},[1625,2244],[40,34907,34909,35011],{"className":34908},[1629],[40,34910,34912,34992,35000],{"className":34911,"style":8853},[1633],[40,34913,34914,34917],{"style":10764},[40,34915],{"className":34916,"style":3354},[1641],[40,34918,34920,34923,34963],{"className":34919},[1612],[40,34921,1791],{"className":34922},[1612],[40,34924,34926,34929],{"className":34925},[1612],[40,34927,27286],{"className":34928,"style":27422},[1612,1653],[40,34930,34932],{"className":34931},[1621],[40,34933,34935,34955],{"className":34934},[1625,2244],[40,34936,34938,34952],{"className":34937},[1629],[40,34939,34941],{"className":34940,"style":13330},[1633],[40,34942,34943,34946],{"style":27437},[40,34944],{"className":34945,"style":1642},[1641],[40,34947,34949],{"className":34948},[1646,1647,1648,1649],[40,34950,791],{"className":34951},[1612,1649],[40,34953,2269],{"className":34954},[2268],[40,34956,34958],{"className":34957},[1629],[40,34959,34961],{"className":34960,"style":2276},[1633],[40,34962],{},[40,34964,34966,34969],{"className":34965},[1612],[40,34967,1791],{"className":34968},[1612],[40,34970,34972],{"className":34971},[1621],[40,34973,34975],{"className":34974},[1625],[40,34976,34978],{"className":34977},[1629],[40,34979,34981],{"className":34980,"style":28464},[1633],[40,34982,34983,34986],{"style":28467},[40,34984],{"className":34985,"style":1642},[1641],[40,34987,34989],{"className":34988},[1646,1647,1648,1649],[40,34990,1879],{"className":34991},[1612,1649],[40,34993,34994,34997],{"style":8925},[40,34995],{"className":34996,"style":3354},[1641],[40,34998],{"className":34999,"style":8933},[8932],[40,35001,35002,35005],{"style":8936},[40,35003],{"className":35004,"style":3354},[1641],[40,35006,35008],{"className":35007},[1612],[40,35009,34795],{"className":35010},[1612],[40,35012,2269],{"className":35013},[2268],[40,35015,35017],{"className":35016},[1629],[40,35018,35020],{"className":35019,"style":28611},[1633],[40,35021],{},[40,35023],{"className":35024},[1822,8840],[40,35026,11205],{"className":35027},[1612],[40,35029,2182],{"className":35030,"style":2197},[1612,1653],[40,35032,11205],{"className":35033},[1612],[40,35035],{"className":35036,"style":2009},[1756],[40,35038,35040,35064],{"className":35039},[1981],[40,35041,35043,35046,35049,35052,35055,35058,35061],{"className":35042},[1981],[40,35044,241],{"className":35045,"style":1987},[1985,1986],[40,35047,1410],{"className":35048},[1612],[40,35050],{"className":35051,"style":1757},[1756],[40,35053,1147],{"className":35054},[1761],[40,35056],{"className":35057,"style":1757},[1756],[40,35059,26579],{"className":35060,"style":26664},[1612,1653],[40,35062,2693],{"className":35063,"style":1987},[1822,1986],[40,35065,35067],{"className":35066},[1621],[40,35068,35070],{"className":35069},[1625],[40,35071,35073],{"className":35072},[1629],[40,35074,35076],{"className":35075,"style":26680},[1633],[40,35077,35078,35081],{"style":6889},[40,35079],{"className":35080,"style":1642},[1641],[40,35082,35084],{"className":35083},[1646,1647,1648,1649],[40,35085,35087,35090,35093],{"className":35086},[1612,1649],[40,35088,1733],{"className":35089},[1612,1649],[40,35091,1879],{"className":35092},[1612,1649],[40,35094,4003],{"className":35095},[1612,1653,1649],[40,35097],{"className":35098,"style":2009},[1756],[40,35100,1823],{"className":35101},[1612],[12,35103,35104,35105,35157,35158,35186,35187,35243,35244,35399,35400,35428],{},"This holds for any ",[40,35106,35108,35126],{"className":35107},[1565],[40,35109,35111],{"className":35110},[1569],[1571,35112,35113],{"xmlns":1573},[1575,35114,35115,35123],{},[1578,35116,35117,35119,35121],{},[1584,35118,26579],{},[1731,35120,689],{},[1735,35122,791],{},[1591,35124,35125],{"encoding":1593},"\\delta>0",[40,35127,35129,35148],{"className":35128,"ariaHidden":1599},[1598],[40,35130,35132,35136,35139,35142,35145],{"className":35131},[1603],[40,35133],{"className":35134,"style":35135},[1607],"height:0.7335em;vertical-align:-0.0391em;",[40,35137,26579],{"className":35138,"style":26664},[1612,1653],[40,35140],{"className":35141,"style":1963},[1756],[40,35143,689],{"className":35144},[1967],[40,35146],{"className":35147,"style":1963},[1756],[40,35149,35151,35154],{"className":35150},[1603],[40,35152],{"className":35153,"style":1771},[1607],[40,35155,791],{"className":35156},[1612],", so if we set ",[40,35159,35161,35174],{"className":35160},[1565],[40,35162,35164],{"className":35163},[1569],[1571,35165,35166],{"xmlns":1573},[1575,35167,35168,35172],{},[1578,35169,35170],{},[1584,35171,26579],{},[1591,35173,31305],{"encoding":1593},[40,35175,35177],{"className":35176,"ariaHidden":1599},[1598],[40,35178,35180,35183],{"className":35179},[1603],[40,35181],{"className":35182,"style":4478},[1607],[40,35184,26579],{"className":35185,"style":26664},[1612,1653]," equal to our goal error, then the error bound above converges towards that exponentially with the Krylov dimension ",[40,35188,35190,35210],{"className":35189},[1565],[40,35191,35193],{"className":35192},[1569],[1571,35194,35195],{"xmlns":1573},[1575,35196,35197,35207],{},[1578,35198,35199,35201,35203,35205],{},[1735,35200,1879],{},[1584,35202,4003],{},[1731,35204,623],{},[1584,35206,1256],{},[1591,35208,35209],{"encoding":1593},"2d=r",[40,35211,35213,35234],{"className":35212,"ariaHidden":1599},[1598],[40,35214,35216,35219,35222,35225,35228,35231],{"className":35215},[1603],[40,35217],{"className":35218,"style":4478},[1607],[40,35220,1879],{"className":35221},[1612],[40,35223,4003],{"className":35224},[1612,1653],[40,35226],{"className":35227,"style":1963},[1756],[40,35229,623],{"className":35230},[1967],[40,35232],{"className":35233,"style":1963},[1756],[40,35235,35237,35240],{"className":35236},[1603],[40,35238],{"className":35239,"style":1681},[1607],[40,35241,1256],{"className":35242,"style":1654},[1612,1653],". Also note that if ",[40,35245,35247,35277],{"className":35246},[1565],[40,35248,35250],{"className":35249},[1569],[1571,35251,35252],{"xmlns":1573},[1575,35253,35254,35274],{},[1578,35255,35256,35258,35260,35266,35268],{},[1584,35257,26579],{},[1731,35259,4721],{},[2212,35261,35262,35264],{},[1584,35263,5226],{},[1735,35265,1410],{},[1731,35267,1733],{},[2212,35269,35270,35272],{},[1584,35271,5226],{},[1735,35273,791],{},[1591,35275,35276],{"encoding":1593},"\\delta\u003CE_1-E_0",[40,35278,35280,35298,35353],{"className":35279,"ariaHidden":1599},[1598],[40,35281,35283,35286,35289,35292,35295],{"className":35282},[1603],[40,35284],{"className":35285,"style":35135},[1607],[40,35287,26579],{"className":35288,"style":26664},[1612,1653],[40,35290],{"className":35291,"style":1963},[1756],[40,35293,4721],{"className":35294},[1967],[40,35296],{"className":35297,"style":1963},[1756],[40,35299,35301,35304,35344,35347,35350],{"className":35300},[1603],[40,35302],{"className":35303,"style":2231},[1607],[40,35305,35307,35310],{"className":35306},[1612],[40,35308,5226],{"className":35309,"style":4936},[1612,1653],[40,35311,35313],{"className":35312},[1621],[40,35314,35316,35336],{"className":35315},[1625,2244],[40,35317,35319,35333],{"className":35318},[1629],[40,35320,35322],{"className":35321,"style":13330},[1633],[40,35323,35324,35327],{"style":5058},[40,35325],{"className":35326,"style":1642},[1641],[40,35328,35330],{"className":35329},[1646,1647,1648,1649],[40,35331,1410],{"className":35332},[1612,1649],[40,35334,2269],{"className":35335},[2268],[40,35337,35339],{"className":35338},[1629],[40,35340,35342],{"className":35341,"style":2276},[1633],[40,35343],{},[40,35345],{"className":35346,"style":1757},[1756],[40,35348,1733],{"className":35349},[1761],[40,35351],{"className":35352,"style":1757},[1756],[40,35354,35356,35359],{"className":35355},[1603],[40,35357],{"className":35358,"style":2231},[1607],[40,35360,35362,35365],{"className":35361},[1612],[40,35363,5226],{"className":35364,"style":4936},[1612,1653],[40,35366,35368],{"className":35367},[1621],[40,35369,35371,35391],{"className":35370},[1625,2244],[40,35372,35374,35388],{"className":35373},[1629],[40,35375,35377],{"className":35376,"style":13330},[1633],[40,35378,35379,35382],{"style":5058},[40,35380],{"className":35381,"style":1642},[1641],[40,35383,35385],{"className":35384},[1646,1647,1648,1649],[40,35386,791],{"className":35387},[1612,1649],[40,35389,2269],{"className":35390},[2268],[40,35392,35394],{"className":35393},[1629],[40,35395,35397],{"className":35396,"style":2276},[1633],[40,35398],{}," then the ",[40,35401,35403,35416],{"className":35402},[1565],[40,35404,35406],{"className":35405},[1569],[1571,35407,35408],{"xmlns":1573},[1575,35409,35410,35414],{},[1578,35411,35412],{},[1584,35413,26579],{},[1591,35415,31305],{"encoding":1593},[40,35417,35419],{"className":35418,"ariaHidden":1599},[1598],[40,35420,35422,35425],{"className":35421},[1603],[40,35423],{"className":35424,"style":4478},[1607],[40,35426,26579],{"className":35427,"style":26664},[1612,1653]," term actually goes away entirely in the above bound.",[12,35430,35431,35432,35490],{},"To complete the argument, we first note that the above is just the energy error of the particular state ",[40,35433,35435,35460],{"className":35434},[1565],[40,35436,35438],{"className":35437},[1569],[1571,35439,35440],{"xmlns":1573},[1575,35441,35442,35458],{},[1578,35443,35444,35446,35448,35450,35452,35454,35456],{},[1584,35445,21331],{},[1731,35447,241],{"stretchy":1797},[1584,35449,2182],{},[1731,35451,2693],{"stretchy":1797},[1584,35453,1791],{"mathvariant":1790},[1584,35455,2837],{},[1731,35457,1798],{"stretchy":1797},[1591,35459,27172],{"encoding":1593},[40,35461,35463],{"className":35462,"ariaHidden":1599},[1598],[40,35464,35466,35469,35472,35475,35478,35481,35484,35487],{"className":35465},[1603],[40,35467],{"className":35468,"style":1811},[1607],[40,35470,21331],{"className":35471,"style":26393},[1612,1653],[40,35473,241],{"className":35474},[1985],[40,35476,2182],{"className":35477,"style":2197},[1612,1653],[40,35479,2693],{"className":35480},[1822],[40,35482,1791],{"className":35483},[1612],[40,35485,2837],{"className":35486,"style":1818},[1612,1653],[40,35488,1798],{"className":35489},[1822],", 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.",[12,35492,35493,35494,9846,35496,35499],{},"A similar analysis as the above can be carried out that additionally accounts for noise and the thresholding procedure discussed in the notebook. See ",[19,35495,7591],{"href":82},[19,35497,35498],{"href":82},"[4]"," for this analysis.",[55,35501,35503],{"id":35502},"appendix-proof-of-claim-1","Appendix: proof of Claim 1",[12,35505,35506,35507,35509,35510,35584,35585,35673,35674,35702],{},"The following is mostly derived from ",[19,35508,26357],{"href":82},", Theorem 3.1: let ",[40,35511,35513,35535],{"className":35512},[1565],[40,35514,35516],{"className":35515},[1569],[1571,35517,35518],{"xmlns":1573},[1575,35519,35520,35532],{},[1578,35521,35522,35524,35526,35528,35530],{},[1735,35523,791],{},[1731,35525,4721],{},[1584,35527,19],{},[1731,35529,4721],{},[1584,35531,12149],{},[1591,35533,35534],{"encoding":1593},"0 \u003C a \u003C b",[40,35536,35538,35557,35575],{"className":35537,"ariaHidden":1599},[1598],[40,35539,35541,35545,35548,35551,35554],{"className":35540},[1603],[40,35542],{"className":35543,"style":35544},[1607],"height:0.6835em;vertical-align:-0.0391em;",[40,35546,791],{"className":35547},[1612],[40,35549],{"className":35550,"style":1963},[1756],[40,35552,4721],{"className":35553},[1967],[40,35555],{"className":35556,"style":1963},[1756],[40,35558,35560,35563,35566,35569,35572],{"className":35559},[1603],[40,35561],{"className":35562,"style":4739},[1607],[40,35564,19],{"className":35565},[1612,1653],[40,35567],{"className":35568,"style":1963},[1756],[40,35570,4721],{"className":35571},[1967],[40,35573],{"className":35574,"style":1963},[1756],[40,35576,35578,35581],{"className":35577},[1603],[40,35579],{"className":35580,"style":4478},[1607],[40,35582,12149],{"className":35583},[1612,1653]," and let ",[40,35586,35588,35609],{"className":35587},[1565],[40,35589,35591],{"className":35590},[1569],[1571,35592,35593],{"xmlns":1573},[1575,35594,35595,35606],{},[1578,35596,35597],{},[2819,35598,35599,35602,35604],{},[1584,35600,35601],{"mathvariant":1790},"Π",[1584,35603,4003],{},[1731,35605,19819],{},[1591,35607,35608],{"encoding":1593},"\\Pi^*_d",[40,35610,35612],{"className":35611,"ariaHidden":1599},[1598],[40,35613,35615,35619],{"className":35614},[1603],[40,35616],{"className":35617,"style":35618},[1607],"height:0.9718em;vertical-align:-0.2831em;",[40,35620,35622,35625],{"className":35621},[1612],[40,35623,35601],{"className":35624},[1612],[40,35626,35628],{"className":35627},[1621],[40,35629,35631,35664],{"className":35630},[1625,2244],[40,35632,35634,35661],{"className":35633},[1629],[40,35635,35638,35650],{"className":35636,"style":35637},[1633],"height:0.6887em;",[40,35639,35641,35644],{"style":35640},"top:-2.4169em;margin-left:0em;margin-right:0.05em;",[40,35642],{"className":35643,"style":1642},[1641],[40,35645,35647],{"className":35646},[1646,1647,1648,1649],[40,35648,4003],{"className":35649},[1612,1653,1649],[40,35651,35652,35655],{"style":1637},[40,35653],{"className":35654,"style":1642},[1641],[40,35656,35658],{"className":35657},[1646,1647,1648,1649],[40,35659,19819],{"className":35660},[1761,1649],[40,35662,2269],{"className":35663},[2268],[40,35665,35667],{"className":35666},[1629],[40,35668,35671],{"className":35669,"style":35670},[1633],"height:0.2831em;",[40,35672],{}," be the space of residual polynomials (polynomials whose value at 0 is 1) of degree at most ",[40,35675,35677,35690],{"className":35676},[1565],[40,35678,35680],{"className":35679},[1569],[1571,35681,35682],{"xmlns":1573},[1575,35683,35684,35688],{},[1578,35685,35686],{},[1584,35687,4003],{},[1591,35689,4003],{"encoding":1593},[40,35691,35693],{"className":35692,"ariaHidden":1599},[1598],[40,35694,35696,35699],{"className":35695},[1603],[40,35697],{"className":35698,"style":4478},[1607],[40,35700,4003],{"className":35701},[1612,1653],". The solution to",[40,35704,35706],{"className":35705},[1827],[40,35707,35709,35803],{"className":35708},[1565],[40,35710,35712],{"className":35711},[1569],[1571,35713,35714],{"xmlns":1573,"display":1836},[1575,35715,35716,35800],{},[1578,35717,35718,35721,35723,35725,35727,35729,35731,35733,35735,35737,35761,35786,35788,35790,35792,35794,35796,35798],{},[1584,35719,35720],{},"β",[1731,35722,241],{"stretchy":1797},[1584,35724,19],{},[1731,35726,1862],{"separator":1599},[1584,35728,12149],{},[1731,35730,1862],{"separator":1599},[1584,35732,4003],{},[1731,35734,2693],{"stretchy":1797},[1731,35736,623],{},[29695,35738,35739,35746],{},[1578,35740,35741,35743],{},[1584,35742,24284],{},[1731,35744,35745],{},"⁡",[1578,35747,35748,35750,35753],{},[1584,35749,12],{},[1731,35751,35752],{},"∈",[2819,35754,35755,35757,35759],{},[1584,35756,35601],{"mathvariant":1790},[1584,35758,4003],{},[1731,35760,19819],{},[29695,35762,35763,35770],{},[1578,35764,35765,35768],{},[1584,35766,35767],{},"max",[1731,35769,35745],{},[1578,35771,35772,35774,35776,35778,35780,35782,35784],{},[1584,35773,8185],{},[1731,35775,35752],{},[1731,35777,11295],{"stretchy":1797},[1584,35779,19],{},[1731,35781,1862],{"separator":1599},[1584,35783,12149],{},[1731,35785,11320],{"stretchy":1797},[1584,35787,1791],{"mathvariant":1790},[1584,35789,12],{},[1731,35791,241],{"stretchy":1797},[1584,35793,8185],{},[1731,35795,2693],{"stretchy":1797},[1584,35797,1791],{"mathvariant":1790},[1756,35799],{"width":8502},[1591,35801,35802],{"encoding":1593},"\\beta(a, b, d) = \\min_{p \\in \\Pi^*_d} \\max_{x \\in [a, b]} |p(x)| \\quad",[40,35804,35806,35852],{"className":35805,"ariaHidden":1599},[1598],[40,35807,35809,35812,35816,35819,35822,35825,35828,35831,35834,35837,35840,35843,35846,35849],{"className":35808},[1603],[40,35810],{"className":35811,"style":1811},[1607],[40,35813,35720],{"className":35814,"style":35815},[1612,1653],"margin-right:0.0528em;",[40,35817,241],{"className":35818},[1985],[40,35820,19],{"className":35821},[1612,1653],[40,35823,1862],{"className":35824},[2005],[40,35826],{"className":35827,"style":2009},[1756],[40,35829,12149],{"className":35830},[1612,1653],[40,35832,1862],{"className":35833},[2005],[40,35835],{"className":35836,"style":2009},[1756],[40,35838,4003],{"className":35839},[1612,1653],[40,35841,2693],{"className":35842},[1822],[40,35844],{"className":35845,"style":1963},[1756],[40,35847,623],{"className":35848},[1967],[40,35850],{"className":35851,"style":1963},[1756],[40,35853,35855,35859,35967,35970,36037,36040,36043,36046,36049,36052,36055,36058],{"className":35854},[1603],[40,35856],{"className":35857,"style":35858},[1607],"height:1.7374em;vertical-align:-0.9874em;",[40,35860,35862],{"className":35861},[6614,27341],[40,35863,35865,35958],{"className":35864},[1625,2244],[40,35866,35868,35955],{"className":35867},[1629],[40,35869,35872,35945],{"className":35870,"style":35871},[1633],"height:0.6679em;",[40,35873,35875,35878],{"style":35874},"top:-2.3557em;margin-left:0em;",[40,35876],{"className":35877,"style":3354},[1641],[40,35879,35881],{"className":35880},[1646,1647,1648,1649],[40,35882,35884,35887,35890],{"className":35883},[1612,1649],[40,35885,12],{"className":35886},[1612,1653,1649],[40,35888,35752],{"className":35889},[1967,1649],[40,35891,35893,35896],{"className":35892},[1612,1649],[40,35894,35601],{"className":35895},[1612,1649],[40,35897,35899],{"className":35898},[1621],[40,35900,35902,35936],{"className":35901},[1625,2244],[40,35903,35905,35933],{"className":35904},[1629],[40,35906,35909,35921],{"className":35907,"style":35908},[1633],"height:0.6771em;",[40,35910,35912,35915],{"style":35911},"top:-2.1528em;margin-left:0em;margin-right:0.0714em;",[40,35913],{"className":35914,"style":3856},[1641],[40,35916,35918],{"className":35917},[1646,3860,1992,1649],[40,35919,4003],{"className":35920},[1612,1653,1649],[40,35922,35924,35927],{"style":35923},"top:-2.8448em;margin-right:0.0714em;",[40,35925],{"className":35926,"style":3856},[1641],[40,35928,35930],{"className":35929},[1646,3860,1992,1649],[40,35931,19819],{"className":35932},[1761,1649],[40,35934,2269],{"className":35935},[2268],[40,35937,35939],{"className":35938},[1629],[40,35940,35943],{"className":35941,"style":35942},[1633],"height:0.3472em;",[40,35944],{},[40,35946,35947,35950],{"style":3350},[40,35948],{"className":35949,"style":3354},[1641],[40,35951,35952],{},[40,35953,24284],{"className":35954},[6614],[40,35956,2269],{"className":35957},[2268],[40,35959,35961],{"className":35960},[1629],[40,35962,35965],{"className":35963,"style":35964},[1633],"height:0.9874em;",[40,35966],{},[40,35968],{"className":35969,"style":2009},[1756],[40,35971,35973],{"className":35972},[6614,27341],[40,35974,35976,36028],{"className":35975},[1625,2244],[40,35977,35979,36025],{"className":35978},[1629],[40,35980,35982,36015],{"className":35981,"style":1681},[1633],[40,35983,35985,35988],{"style":35984},"top:-2.309em;margin-left:0em;",[40,35986],{"className":35987,"style":3354},[1641],[40,35989,35991],{"className":35990},[1646,1647,1648,1649],[40,35992,35994,35997,36000,36003,36006,36009,36012],{"className":35993},[1612,1649],[40,35995,8185],{"className":35996},[1612,1653,1649],[40,35998,35752],{"className":35999},[1967,1649],[40,36001,11295],{"className":36002},[1985,1649],[40,36004,19],{"className":36005},[1612,1653,1649],[40,36007,1862],{"className":36008},[2005,1649],[40,36010,12149],{"className":36011},[1612,1653,1649],[40,36013,11320],{"className":36014},[1822,1649],[40,36016,36017,36020],{"style":3350},[40,36018],{"className":36019,"style":3354},[1641],[40,36021,36022],{},[40,36023,35767],{"className":36024},[6614],[40,36026,2269],{"className":36027},[2268],[40,36029,36031],{"className":36030},[1629],[40,36032,36035],{"className":36033,"style":36034},[1633],"height:0.966em;",[40,36036],{},[40,36038],{"className":36039,"style":2009},[1756],[40,36041,1791],{"className":36042},[1612],[40,36044,12],{"className":36045},[1612,1653],[40,36047,241],{"className":36048},[1985],[40,36050,8185],{"className":36051},[1612,1653],[40,36053,2693],{"className":36054},[1822],[40,36056,1791],{"className":36057},[1612],[40,36059],{"className":36060,"style":8821},[1756],[12,36062,36063],{},"is",[40,36065,36067],{"className":36066},[1827],[40,36068,36070,36173],{"className":36069},[1565],[40,36071,36073],{"className":36072},[1569],[1571,36074,36075],{"xmlns":1573,"display":1836},[1575,36076,36077,36170],{},[1578,36078,36079,36085,36087,36089,36091,36093,36166,36168],{},[1581,36080,36081,36083],{},[1584,36082,12],{},[1731,36084,19819],{},[1731,36086,241],{"stretchy":1797},[1584,36088,8185],{},[1731,36090,2693],{"stretchy":1797},[1731,36092,623],{},[8538,36094,36095,36134],{},[1578,36096,36097,36104],{},[2212,36098,36099,36102],{},[1584,36100,36101],{},"T",[1584,36103,4003],{},[1578,36105,36106,36108,36132],{},[1731,36107,241],{"fence":1599},[8538,36109,36110,36124],{},[1578,36111,36112,36114,36116,36118,36120,36122],{},[1584,36113,12149],{},[1731,36115,1147],{},[1584,36117,19],{},[1731,36119,1733],{},[1735,36121,1879],{},[1584,36123,8185],{},[1578,36125,36126,36128,36130],{},[1584,36127,12149],{},[1731,36129,1733],{},[1584,36131,19],{},[1731,36133,2693],{"fence":1599},[1578,36135,36136,36142],{},[2212,36137,36138,36140],{},[1584,36139,36101],{},[1584,36141,4003],{},[1578,36143,36144,36146,36164],{},[1731,36145,241],{"fence":1599},[8538,36147,36148,36156],{},[1578,36149,36150,36152,36154],{},[1584,36151,12149],{},[1731,36153,1147],{},[1584,36155,19],{},[1578,36157,36158,36160,36162],{},[1584,36159,12149],{},[1731,36161,1733],{},[1584,36163,19],{},[1731,36165,2693],{"fence":1599},[1731,36167,1862],{"separator":1599},[1756,36169],{"width":8502},[1591,36171,36172],{"encoding":1593},"p^*(x) = \\frac{T_d\\left(\\frac{b + a - 2x}{b - a}\\right)}{T_d\\left(\\frac{b + a}{b - a}\\right)}, \\quad",[40,36174,36176,36230],{"className":36175,"ariaHidden":1599},[1598],[40,36177,36179,36182,36212,36215,36218,36221,36224,36227],{"className":36178},[1603],[40,36180],{"className":36181,"style":1811},[1607],[40,36183,36185,36188],{"className":36184},[1612],[40,36186,12],{"className":36187},[1612,1653],[40,36189,36191],{"className":36190},[1621],[40,36192,36194],{"className":36193},[1625],[40,36195,36197],{"className":36196},[1629],[40,36198,36201],{"className":36199,"style":36200},[1633],"height:0.7387em;",[40,36202,36203,36206],{"style":1950},[40,36204],{"className":36205,"style":1642},[1641],[40,36207,36209],{"className":36208},[1646,1647,1648,1649],[40,36210,19819],{"className":36211},[1761,1649],[40,36213,241],{"className":36214},[1985],[40,36216,8185],{"className":36217},[1612,1653],[40,36219,2693],{"className":36220},[1822],[40,36222],{"className":36223,"style":1963},[1756],[40,36225,623],{"className":36226},[1967],[40,36228],{"className":36229,"style":1963},[1756],[40,36231,36233,36237,36586,36589],{"className":36232},[1603],[40,36234],{"className":36235,"style":36236},[1607],"height:2.8469em;vertical-align:-1.1734em;",[40,36238,36240,36243,36583],{"className":36239},[1612],[40,36241],{"className":36242},[1985,8840],[40,36244,36246],{"className":36245},[8538],[40,36247,36249,36574],{"className":36248},[1625,2244],[40,36250,36252,36571],{"className":36251},[1629],[40,36253,36256,36407,36415],{"className":36254,"style":36255},[1633],"height:1.6734em;",[40,36257,36259,36262],{"style":36258},"top:-2.2299em;",[40,36260],{"className":36261,"style":3354},[1641],[40,36263,36265,36305,36308],{"className":36264},[1612],[40,36266,36268,36271],{"className":36267},[1612],[40,36269,36101],{"className":36270,"style":2264},[1612,1653],[40,36272,36274],{"className":36273},[1621],[40,36275,36277,36297],{"className":36276},[1625,2244],[40,36278,36280,36294],{"className":36279},[1629],[40,36281,36283],{"className":36282,"style":10290},[1633],[40,36284,36285,36288],{"style":6681},[40,36286],{"className":36287,"style":1642},[1641],[40,36289,36291],{"className":36290},[1646,1647,1648,1649],[40,36292,4003],{"className":36293},[1612,1653,1649],[40,36295,2269],{"className":36296},[2268],[40,36298,36300],{"className":36299},[1629],[40,36301,36303],{"className":36302,"style":2276},[1633],[40,36304],{},[40,36306],{"className":36307,"style":2009},[1756],[40,36309,36311,36317,36401],{"className":36310},[1981],[40,36312,36314],{"className":36313,"style":1987},[1985,1986],[40,36315,241],{"className":36316},[1991,1992],[40,36318,36320,36323,36398],{"className":36319},[1612],[40,36321],{"className":36322},[1985,8840],[40,36324,36326],{"className":36325},[8538],[40,36327,36329,36389],{"className":36328},[1625,2244],[40,36330,36332,36386],{"className":36331},[1629],[40,36333,36336,36357,36365],{"className":36334,"style":36335},[1633],"height:0.8801em;",[40,36337,36339,36342],{"style":36338},"top:-2.655em;",[40,36340],{"className":36341,"style":3354},[1641],[40,36343,36345],{"className":36344},[1646,1647,1648,1649],[40,36346,36348,36351,36354],{"className":36347},[1612,1649],[40,36349,12149],{"className":36350},[1612,1653,1649],[40,36352,1733],{"className":36353},[1761,1649],[40,36355,19],{"className":36356},[1612,1653,1649],[40,36358,36359,36362],{"style":8925},[40,36360],{"className":36361,"style":3354},[1641],[40,36363],{"className":36364,"style":8933},[8932],[40,36366,36368,36371],{"style":36367},"top:-3.394em;",[40,36369],{"className":36370,"style":3354},[1641],[40,36372,36374],{"className":36373},[1646,1647,1648,1649],[40,36375,36377,36380,36383],{"className":36376},[1612,1649],[40,36378,12149],{"className":36379},[1612,1653,1649],[40,36381,1147],{"className":36382},[1761,1649],[40,36384,19],{"className":36385},[1612,1653,1649],[40,36387,2269],{"className":36388},[2268],[40,36390,36392],{"className":36391},[1629],[40,36393,36396],{"className":36394,"style":36395},[1633],"height:0.4033em;",[40,36397],{},[40,36399],{"className":36400},[1822,8840],[40,36402,36404],{"className":36403,"style":1987},[1822,1986],[40,36405,2693],{"className":36406},[1991,1992],[40,36408,36409,36412],{"style":8925},[40,36410],{"className":36411,"style":3354},[1641],[40,36413],{"className":36414,"style":8933},[8932],[40,36416,36418,36421],{"style":36417},"top:-3.7933em;",[40,36419],{"className":36420,"style":3354},[1641],[40,36422,36424,36464,36467],{"className":36423},[1612],[40,36425,36427,36430],{"className":36426},[1612],[40,36428,36101],{"className":36429,"style":2264},[1612,1653],[40,36431,36433],{"className":36432},[1621],[40,36434,36436,36456],{"className":36435},[1625,2244],[40,36437,36439,36453],{"className":36438},[1629],[40,36440,36442],{"className":36441,"style":10290},[1633],[40,36443,36444,36447],{"style":6681},[40,36445],{"className":36446,"style":1642},[1641],[40,36448,36450],{"className":36449},[1646,1647,1648,1649],[40,36451,4003],{"className":36452},[1612,1653,1649],[40,36454,2269],{"className":36455},[2268],[40,36457,36459],{"className":36458},[1629],[40,36460,36462],{"className":36461,"style":2276},[1633],[40,36463],{},[40,36465],{"className":36466,"style":2009},[1756],[40,36468,36470,36476,36565],{"className":36469},[1981],[40,36471,36473],{"className":36472,"style":1987},[1985,1986],[40,36474,241],{"className":36475},[1991,1992],[40,36477,36479,36482,36562],{"className":36478},[1612],[40,36480],{"className":36481},[1985,8840],[40,36483,36485],{"className":36484},[8538],[40,36486,36488,36554],{"className":36487},[1625,2244],[40,36489,36491,36551],{"className":36490},[1629],[40,36492,36494,36514,36522],{"className":36493,"style":36335},[1633],[40,36495,36496,36499],{"style":36338},[40,36497],{"className":36498,"style":3354},[1641],[40,36500,36502],{"className":36501},[1646,1647,1648,1649],[40,36503,36505,36508,36511],{"className":36504},[1612,1649],[40,36506,12149],{"className":36507},[1612,1653,1649],[40,36509,1733],{"className":36510},[1761,1649],[40,36512,19],{"className":36513},[1612,1653,1649],[40,36515,36516,36519],{"style":8925},[40,36517],{"className":36518,"style":3354},[1641],[40,36520],{"className":36521,"style":8933},[8932],[40,36523,36524,36527],{"style":36367},[40,36525],{"className":36526,"style":3354},[1641],[40,36528,36530],{"className":36529},[1646,1647,1648,1649],[40,36531,36533,36536,36539,36542,36545,36548],{"className":36532},[1612,1649],[40,36534,12149],{"className":36535},[1612,1653,1649],[40,36537,1147],{"className":36538},[1761,1649],[40,36540,19],{"className":36541},[1612,1653,1649],[40,36543,1733],{"className":36544},[1761,1649],[40,36546,1879],{"className":36547},[1612,1649],[40,36549,8185],{"className":36550},[1612,1653,1649],[40,36552,2269],{"className":36553},[2268],[40,36555,36557],{"className":36556},[1629],[40,36558,36560],{"className":36559,"style":36395},[1633],[40,36561],{},[40,36563],{"className":36564},[1822,8840],[40,36566,36568],{"className":36567,"style":1987},[1822,1986],[40,36569,2693],{"className":36570},[1991,1992],[40,36572,2269],{"className":36573},[2268],[40,36575,36577],{"className":36576},[1629],[40,36578,36581],{"className":36579,"style":36580},[1633],"height:1.1734em;",[40,36582],{},[40,36584],{"className":36585},[1822,8840],[40,36587,1862],{"className":36588},[2005],[40,36590],{"className":36591,"style":8821},[1756],[12,36593,36594],{},"and the corresponding minimal value is",[40,36596,36598],{"className":36597},[1827],[40,36599,36601,36669],{"className":36600},[1565],[40,36602,36604],{"className":36603},[1569],[1571,36605,36606],{"xmlns":1573,"display":1836},[1575,36607,36608,36666],{},[1578,36609,36610,36612,36614,36616,36618,36620,36622,36624,36626,36628,36640,36664],{},[1584,36611,35720],{},[1731,36613,241],{"stretchy":1797},[1584,36615,19],{},[1731,36617,1862],{"separator":1599},[1584,36619,12149],{},[1731,36621,1862],{"separator":1599},[1584,36623,4003],{},[1731,36625,2693],{"stretchy":1797},[1731,36627,623],{},[2819,36629,36630,36632,36634],{},[1584,36631,36101],{},[1584,36633,4003],{},[1578,36635,36636,36638],{},[1731,36637,1733],{},[1735,36639,1410],{},[1578,36641,36642,36644,36662],{},[1731,36643,241],{"fence":1599},[8538,36645,36646,36654],{},[1578,36647,36648,36650,36652],{},[1584,36649,12149],{},[1731,36651,1147],{},[1584,36653,19],{},[1578,36655,36656,36658,36660],{},[1584,36657,12149],{},[1731,36659,1733],{},[1584,36661,19],{},[1731,36663,2693],{"fence":1599},[1584,36665,1823],{"mathvariant":1790},[1591,36667,36668],{"encoding":1593},"\\beta(a, b, d) = T_d^{-1}\\left(\\frac{b + a}{b - a}\\right).",[40,36670,36672,36717],{"className":36671,"ariaHidden":1599},[1598],[40,36673,36675,36678,36681,36684,36687,36690,36693,36696,36699,36702,36705,36708,36711,36714],{"className":36674},[1603],[40,36676],{"className":36677,"style":1811},[1607],[40,36679,35720],{"className":36680,"style":35815},[1612,1653],[40,36682,241],{"className":36683},[1985],[40,36685,19],{"className":36686},[1612,1653],[40,36688,1862],{"className":36689},[2005],[40,36691],{"className":36692,"style":2009},[1756],[40,36694,12149],{"className":36695},[1612,1653],[40,36697,1862],{"className":36698},[2005],[40,36700],{"className":36701,"style":2009},[1756],[40,36703,4003],{"className":36704},[1612,1653],[40,36706,2693],{"className":36707},[1822],[40,36709],{"className":36710,"style":1963},[1756],[40,36712,623],{"className":36713},[1967],[40,36715],{"className":36716,"style":1963},[1756],[40,36718,36720,36724,36783,36786,36889,36892],{"className":36719},[1603],[40,36721],{"className":36722,"style":36723},[1607],"height:2.4em;vertical-align:-0.95em;",[40,36725,36727,36730],{"className":36726},[1612],[40,36728,36101],{"className":36729,"style":2264},[1612,1653],[40,36731,36733],{"className":36732},[1621],[40,36734,36736,36774],{"className":36735},[1625,2244],[40,36737,36739,36771],{"className":36738},[1629],[40,36740,36742,36754],{"className":36741,"style":2046},[1633],[40,36743,36745,36748],{"style":36744},"top:-2.4086em;margin-left:-0.1389em;margin-right:0.05em;",[40,36746],{"className":36747,"style":1642},[1641],[40,36749,36751],{"className":36750},[1646,1647,1648,1649],[40,36752,4003],{"className":36753},[1612,1653,1649],[40,36755,36756,36759],{"style":1950},[40,36757],{"className":36758,"style":1642},[1641],[40,36760,36762],{"className":36761},[1646,1647,1648,1649],[40,36763,36765,36768],{"className":36764},[1612,1649],[40,36766,1733],{"className":36767},[1612,1649],[40,36769,1410],{"className":36770},[1612,1649],[40,36772,2269],{"className":36773},[2268],[40,36775,36777],{"className":36776},[1629],[40,36778,36781],{"className":36779,"style":36780},[1633],"height:0.2914em;",[40,36782],{},[40,36784],{"className":36785,"style":2009},[1756],[40,36787,36789,36795,36883],{"className":36788},[1981],[40,36790,36792],{"className":36791,"style":1987},[1985,1986],[40,36793,241],{"className":36794},[1991,1648],[40,36796,36798,36801,36880],{"className":36797},[1612],[40,36799],{"className":36800},[1985,8840],[40,36802,36804],{"className":36803},[8538],[40,36805,36807,36871],{"className":36806},[1625,2244],[40,36808,36810,36868],{"className":36809},[1629],[40,36811,36814,36837,36845],{"className":36812,"style":36813},[1633],"height:1.3714em;",[40,36815,36816,36819],{"style":10764},[40,36817],{"className":36818,"style":3354},[1641],[40,36820,36822,36825,36828,36831,36834],{"className":36821},[1612],[40,36823,12149],{"className":36824},[1612,1653],[40,36826],{"className":36827,"style":1757},[1756],[40,36829,1733],{"className":36830},[1761],[40,36832],{"className":36833,"style":1757},[1756],[40,36835,19],{"className":36836},[1612,1653],[40,36838,36839,36842],{"style":8925},[40,36840],{"className":36841,"style":3354},[1641],[40,36843],{"className":36844,"style":8933},[8932],[40,36846,36847,36850],{"style":8936},[40,36848],{"className":36849,"style":3354},[1641],[40,36851,36853,36856,36859,36862,36865],{"className":36852},[1612],[40,36854,12149],{"className":36855},[1612,1653],[40,36857],{"className":36858,"style":1757},[1756],[40,36860,1147],{"className":36861},[1761],[40,36863],{"className":36864,"style":1757},[1756],[40,36866,19],{"className":36867},[1612,1653],[40,36869,2269],{"className":36870},[2268],[40,36872,36874],{"className":36873},[1629],[40,36875,36878],{"className":36876,"style":36877},[1633],"height:0.7693em;",[40,36879],{},[40,36881],{"className":36882},[1822,8840],[40,36884,36886],{"className":36885,"style":1987},[1822,1986],[40,36887,2693],{"className":36888},[1991,1648],[40,36890],{"className":36891,"style":2009},[1756],[40,36893,1823],{"className":36894},[1612],[12,36896,36897,36898,36950],{},"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 ",[40,36899,36901,36923],{"className":36900},[1565],[40,36902,36904],{"className":36903},[1569],[1571,36905,36906],{"xmlns":1573},[1575,36907,36908,36920],{},[1578,36909,36910,36912,36914,36916,36918],{},[1731,36911,11295],{"stretchy":1797},[1735,36913,791],{},[1731,36915,1862],{"separator":1599},[1735,36917,1410],{},[1731,36919,11320],{"stretchy":1797},[1591,36921,36922],{"encoding":1593},"[0,1]",[40,36924,36926],{"className":36925,"ariaHidden":1599},[1598],[40,36927,36929,36932,36935,36938,36941,36944,36947],{"className":36928},[1603],[40,36930],{"className":36931,"style":1811},[1607],[40,36933,11295],{"className":36934},[1985],[40,36936,791],{"className":36937},[1612],[40,36939,1862],{"className":36940},[2005],[40,36942],{"className":36943,"style":2009},[1756],[40,36945,1410],{"className":36946},[1612],[40,36948,11320],{"className":36949},[1822],": define",[40,36952,36954],{"className":36953},[1827],[40,36955,36957,37020],{"className":36956},[1565],[40,36958,36960],{"className":36959},[1569],[1571,36961,36962],{"xmlns":1573,"display":1836},[1575,36963,36964,37017],{},[1578,36965,36966,36969,36971,36973,36975,36977,37015],{},[1584,36967,36968],{},"g",[1731,36970,241],{"stretchy":1797},[1584,36972,5226],{},[1731,36974,2693],{"stretchy":1797},[1731,36976,623],{},[8538,36978,36979,37013],{},[1578,36980,36981,36983,36985,36988,36990,36993,36995,36997,36999,37005,37007,37009,37011],{},[1735,36982,1410],{},[1731,36984,1733],{},[1584,36986,36987],{},"cos",[1731,36989,35745],{},[1731,36991,241],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},"1.2em",[1731,36994,241],{"stretchy":1797},[1584,36996,5226],{},[1731,36998,1733],{},[2212,37000,37001,37003],{},[1584,37002,5226],{},[1735,37004,791],{},[1731,37006,2693],{"stretchy":1797},[1584,37008,4003],{},[1584,37010,2342],{},[1731,37012,2693],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1735,37014,1879],{},[1731,37016,1862],{"separator":1599},[1591,37018,37019],{"encoding":1593},"g(E) = \\frac{1-\\cos\\big((E-E_0)dt\\big)}{2},",[40,37021,37023,37050],{"className":37022,"ariaHidden":1599},[1598],[40,37024,37026,37029,37032,37035,37038,37041,37044,37047],{"className":37025},[1603],[40,37027],{"className":37028,"style":1811},[1607],[40,37030,36968],{"className":37031,"style":1818},[1612,1653],[40,37033,241],{"className":37034},[1985],[40,37036,5226],{"className":37037,"style":4936},[1612,1653],[40,37039,2693],{"className":37040},[1822],[40,37042],{"className":37043,"style":1963},[1756],[40,37045,623],{"className":37046},[1967],[40,37048],{"className":37049,"style":1963},[1756],[40,37051,37053,37057,37212],{"className":37052},[1603],[40,37054],{"className":37055,"style":37056},[1607],"height:2.276em;vertical-align:-0.686em;",[40,37058,37060,37063,37209],{"className":37059},[1612],[40,37061],{"className":37062},[1985,8840],[40,37064,37066],{"className":37065},[8538],[40,37067,37069,37201],{"className":37068},[1625,2244],[40,37070,37072,37198],{"className":37071},[1629],[40,37073,37076,37087,37095],{"className":37074,"style":37075},[1633],"height:1.59em;",[40,37077,37078,37081],{"style":10764},[40,37079],{"className":37080,"style":3354},[1641],[40,37082,37084],{"className":37083},[1612],[40,37085,1879],{"className":37086},[1612],[40,37088,37089,37092],{"style":8925},[40,37090],{"className":37091,"style":3354},[1641],[40,37093],{"className":37094,"style":8933},[8932],[40,37096,37098,37101],{"style":37097},"top:-3.74em;",[40,37099],{"className":37100,"style":3354},[1641],[40,37102,37104,37107,37110,37113,37116,37119,37122,37128,37131,37134,37137,37140,37143,37183,37186,37189,37192],{"className":37103},[1612],[40,37105,1410],{"className":37106},[1612],[40,37108],{"className":37109,"style":1757},[1756],[40,37111,1733],{"className":37112},[1761],[40,37114],{"className":37115,"style":1757},[1756],[40,37117,36987],{"className":37118},[6614],[40,37120],{"className":37121,"style":2009},[1756],[40,37123,37125],{"className":37124},[1612],[40,37126,241],{"className":37127},[1991,1992],[40,37129,241],{"className":37130},[1985],[40,37132,5226],{"className":37133,"style":4936},[1612,1653],[40,37135],{"className":37136,"style":1757},[1756],[40,37138,1733],{"className":37139},[1761],[40,37141],{"className":37142,"style":1757},[1756],[40,37144,37146,37149],{"className":37145},[1612],[40,37147,5226],{"className":37148,"style":4936},[1612,1653],[40,37150,37152],{"className":37151},[1621],[40,37153,37155,37175],{"className":37154},[1625,2244],[40,37156,37158,37172],{"className":37157},[1629],[40,37159,37161],{"className":37160,"style":13330},[1633],[40,37162,37163,37166],{"style":5058},[40,37164],{"className":37165,"style":1642},[1641],[40,37167,37169],{"className":37168},[1646,1647,1648,1649],[40,37170,791],{"className":37171},[1612,1649],[40,37173,2269],{"className":37174},[2268],[40,37176,37178],{"className":37177},[1629],[40,37179,37181],{"className":37180,"style":2276},[1633],[40,37182],{},[40,37184,2693],{"className":37185},[1822],[40,37187,4003],{"className":37188},[1612,1653],[40,37190,2342],{"className":37191},[1612,1653],[40,37193,37195],{"className":37194},[1612],[40,37196,2693],{"className":37197},[1991,1992],[40,37199,2269],{"className":37200},[2268],[40,37202,37204],{"className":37203},[1629],[40,37205,37207],{"className":37206,"style":10802},[1633],[40,37208],{},[40,37210],{"className":37211},[1822,8840],[40,37213,1862],{"className":37214},[2005],[12,37216,27509,37217,37250,37251,37457,37458,9846,37565,37609,37610,37638,37639,1823],{},[40,37218,37220,37235],{"className":37219},[1565],[40,37221,37223],{"className":37222},[1569],[1571,37224,37225],{"xmlns":1573},[1575,37226,37227,37233],{},[1578,37228,37229,37231],{},[1584,37230,4003],{},[1584,37232,2342],{},[1591,37234,4507],{"encoding":1593},[40,37236,37238],{"className":37237,"ariaHidden":1599},[1598],[40,37239,37241,37244,37247],{"className":37240},[1603],[40,37242],{"className":37243,"style":4478},[1607],[40,37245,4003],{"className":37246},[1612,1653],[40,37248,2342],{"className":37249},[1612,1653]," is a timestep such that ",[40,37252,37254,37298],{"className":37253},[1565],[40,37255,37257],{"className":37256},[1569],[1571,37258,37259],{"xmlns":1573},[1575,37260,37261,37295],{},[1578,37262,37263,37265,37267,37269,37275,37277,37279,37281,37287,37289,37291,37293],{},[1731,37264,1733],{},[1584,37266,7607],{},[1731,37268,4721],{},[2212,37270,37271,37273],{},[1584,37272,5226],{},[1735,37274,791],{},[1584,37276,4003],{},[1584,37278,2342],{},[1731,37280,4721],{},[2212,37282,37283,37285],{},[1584,37284,5226],{},[5211,37286,35767],{},[1584,37288,4003],{},[1584,37290,2342],{},[1731,37292,4721],{},[1584,37294,7607],{},[1591,37296,37297],{"encoding":1593},"-\\pi \u003C E_0dt \u003C E_\\text{max}dt \u003C \\pi",[40,37299,37301,37322,37384,37448],{"className":37300,"ariaHidden":1599},[1598],[40,37302,37304,37307,37310,37313,37316,37319],{"className":37303},[1603],[40,37305],{"className":37306,"style":1749},[1607],[40,37308,1733],{"className":37309},[1612],[40,37311,7607],{"className":37312,"style":1818},[1612,1653],[40,37314],{"className":37315,"style":1963},[1756],[40,37317,4721],{"className":37318},[1967],[40,37320],{"className":37321,"style":1963},[1756],[40,37323,37325,37329,37369,37372,37375,37378,37381],{"className":37324},[1603],[40,37326],{"className":37327,"style":37328},[1607],"height:0.8444em;vertical-align:-0.15em;",[40,37330,37332,37335],{"className":37331},[1612],[40,37333,5226],{"className":37334,"style":4936},[1612,1653],[40,37336,37338],{"className":37337},[1621],[40,37339,37341,37361],{"className":37340},[1625,2244],[40,37342,37344,37358],{"className":37343},[1629],[40,37345,37347],{"className":37346,"style":13330},[1633],[40,37348,37349,37352],{"style":5058},[40,37350],{"className":37351,"style":1642},[1641],[40,37353,37355],{"className":37354},[1646,1647,1648,1649],[40,37356,791],{"className":37357},[1612,1649],[40,37359,2269],{"className":37360},[2268],[40,37362,37364],{"className":37363},[1629],[40,37365,37367],{"className":37366,"style":2276},[1633],[40,37368],{},[40,37370,4003],{"className":37371},[1612,1653],[40,37373,2342],{"className":37374},[1612,1653],[40,37376],{"className":37377,"style":1963},[1756],[40,37379,4721],{"className":37380},[1967],[40,37382],{"className":37383,"style":1963},[1756],[40,37385,37387,37390,37433,37436,37439,37442,37445],{"className":37386},[1603],[40,37388],{"className":37389,"style":37328},[1607],[40,37391,37393,37396],{"className":37392},[1612],[40,37394,5226],{"className":37395,"style":4936},[1612,1653],[40,37397,37399],{"className":37398},[1621],[40,37400,37402,37425],{"className":37401},[1625,2244],[40,37403,37405,37422],{"className":37404},[1629],[40,37406,37408],{"className":37407,"style":3551},[1633],[40,37409,37410,37413],{"style":5058},[40,37411],{"className":37412,"style":1642},[1641],[40,37414,37416],{"className":37415},[1646,1647,1648,1649],[40,37417,37419],{"className":37418},[1612,6083,1649],[40,37420,35767],{"className":37421},[1612,1649],[40,37423,2269],{"className":37424},[2268],[40,37426,37428],{"className":37427},[1629],[40,37429,37431],{"className":37430,"style":2276},[1633],[40,37432],{},[40,37434,4003],{"className":37435},[1612,1653],[40,37437,2342],{"className":37438},[1612,1653],[40,37440],{"className":37441,"style":1963},[1756],[40,37443,4721],{"className":37444},[1967],[40,37446],{"className":37447,"style":1963},[1756],[40,37449,37451,37454],{"className":37450},[1603],[40,37452],{"className":37453,"style":1681},[1607],[40,37455,7607],{"className":37456,"style":1818},[1612,1653],".\nNotice that ",[40,37459,37461,37489],{"className":37460},[1565],[40,37462,37464],{"className":37463},[1569],[1571,37465,37466],{"xmlns":1573},[1575,37467,37468,37486],{},[1578,37469,37470,37472,37474,37480,37482,37484],{},[1584,37471,36968],{},[1731,37473,241],{"stretchy":1797},[2212,37475,37476,37478],{},[1584,37477,5226],{},[1735,37479,791],{},[1731,37481,2693],{"stretchy":1797},[1731,37483,623],{},[1735,37485,791],{},[1591,37487,37488],{"encoding":1593},"g(E_0)=0",[40,37490,37492,37556],{"className":37491,"ariaHidden":1599},[1598],[40,37493,37495,37498,37501,37504,37544,37547,37550,37553],{"className":37494},[1603],[40,37496],{"className":37497,"style":1811},[1607],[40,37499,36968],{"className":37500,"style":1818},[1612,1653],[40,37502,241],{"className":37503},[1985],[40,37505,37507,37510],{"className":37506},[1612],[40,37508,5226],{"className":37509,"style":4936},[1612,1653],[40,37511,37513],{"className":37512},[1621],[40,37514,37516,37536],{"className":37515},[1625,2244],[40,37517,37519,37533],{"className":37518},[1629],[40,37520,37522],{"className":37521,"style":13330},[1633],[40,37523,37524,37527],{"style":5058},[40,37525],{"className":37526,"style":1642},[1641],[40,37528,37530],{"className":37529},[1646,1647,1648,1649],[40,37531,791],{"className":37532},[1612,1649],[40,37534,2269],{"className":37535},[2268],[40,37537,37539],{"className":37538},[1629],[40,37540,37542],{"className":37541,"style":2276},[1633],[40,37543],{},[40,37545,2693],{"className":37546},[1822],[40,37548],{"className":37549,"style":1963},[1756],[40,37551,623],{"className":37552},[1967],[40,37554],{"className":37555,"style":1963},[1756],[40,37557,37559,37562],{"className":37558},[1603],[40,37560],{"className":37561,"style":1771},[1607],[40,37563,791],{"className":37564},[1612],[40,37566,37568,37588],{"className":37567},[1565],[40,37569,37571],{"className":37570},[1569],[1571,37572,37573],{"xmlns":1573},[1575,37574,37575,37585],{},[1578,37576,37577,37579,37581,37583],{},[1584,37578,36968],{},[1731,37580,241],{"stretchy":1797},[1584,37582,5226],{},[1731,37584,2693],{"stretchy":1797},[1591,37586,37587],{"encoding":1593},"g(E)",[40,37589,37591],{"className":37590,"ariaHidden":1599},[1598],[40,37592,37594,37597,37600,37603,37606],{"className":37593},[1603],[40,37595],{"className":37596,"style":1811},[1607],[40,37598,36968],{"className":37599,"style":1818},[1612,1653],[40,37601,241],{"className":37602},[1985],[40,37604,5226],{"className":37605,"style":4936},[1612,1653],[40,37607,2693],{"className":37608},[1822]," grows as ",[40,37611,37613,37626],{"className":37612},[1565],[40,37614,37616],{"className":37615},[1569],[1571,37617,37618],{"xmlns":1573},[1575,37619,37620,37624],{},[1578,37621,37622],{},[1584,37623,5226],{},[1591,37625,5226],{"encoding":1593},[40,37627,37629],{"className":37628,"ariaHidden":1599},[1598],[40,37630,37632,37635],{"className":37631},[1603],[40,37633],{"className":37634,"style":1608},[1607],[40,37636,5226],{"className":37637,"style":4936},[1612,1653]," moves away from ",[40,37640,37642,37659],{"className":37641},[1565],[40,37643,37645],{"className":37644},[1569],[1571,37646,37647],{"xmlns":1573},[1575,37648,37649,37657],{},[1578,37650,37651],{},[2212,37652,37653,37655],{},[1584,37654,5226],{},[1735,37656,791],{},[1591,37658,13302],{"encoding":1593},[40,37660,37662],{"className":37661,"ariaHidden":1599},[1598],[40,37663,37665,37668],{"className":37664},[1603],[40,37666],{"className":37667,"style":2231},[1607],[40,37669,37671,37674],{"className":37670},[1612],[40,37672,5226],{"className":37673,"style":4936},[1612,1653],[40,37675,37677],{"className":37676},[1621],[40,37678,37680,37700],{"className":37679},[1625,2244],[40,37681,37683,37697],{"className":37682},[1629],[40,37684,37686],{"className":37685,"style":13330},[1633],[40,37687,37688,37691],{"style":5058},[40,37689],{"className":37690,"style":1642},[1641],[40,37692,37694],{"className":37693},[1646,1647,1648,1649],[40,37695,791],{"className":37696},[1612,1649],[40,37698,2269],{"className":37699},[2268],[40,37701,37703],{"className":37702},[1629],[40,37704,37706],{"className":37705,"style":2276},[1633],[40,37707],{},[12,37709,37710,37711,37785,37786,917,37915,37966],{},"Now using the polynomial ",[40,37712,37714,37738],{"className":37713},[1565],[40,37715,37717],{"className":37716},[1569],[1571,37718,37719],{"xmlns":1573},[1575,37720,37721,37735],{},[1578,37722,37723,37729,37731,37733],{},[1581,37724,37725,37727],{},[1584,37726,12],{},[1731,37728,19819],{},[1731,37730,241],{"stretchy":1797},[1584,37732,8185],{},[1731,37734,2693],{"stretchy":1797},[1591,37736,37737],{"encoding":1593},"p^*(x)",[40,37739,37741],{"className":37740,"ariaHidden":1599},[1598],[40,37742,37744,37747,37776,37779,37782],{"className":37743},[1603],[40,37745],{"className":37746,"style":1811},[1607],[40,37748,37750,37753],{"className":37749},[1612],[40,37751,12],{"className":37752},[1612,1653],[40,37754,37756],{"className":37755},[1621],[40,37757,37759],{"className":37758},[1625],[40,37760,37762],{"className":37761},[1629],[40,37763,37765],{"className":37764,"style":35637},[1633],[40,37766,37767,37770],{"style":1637},[40,37768],{"className":37769,"style":1642},[1641],[40,37771,37773],{"className":37772},[1646,1647,1648,1649],[40,37774,19819],{"className":37775},[1761,1649],[40,37777,241],{"className":37778},[1985],[40,37780,8185],{"className":37781},[1612,1653],[40,37783,2693],{"className":37784},[1822]," with the parameters a, b, d set to ",[40,37787,37789,37821],{"className":37788},[1565],[40,37790,37792],{"className":37791},[1569],[1571,37793,37794],{"xmlns":1573},[1575,37795,37796,37818],{},[1578,37797,37798,37800,37802,37804,37806,37812,37814,37816],{},[1584,37799,19],{},[1731,37801,623],{},[1584,37803,36968],{},[1731,37805,241],{"stretchy":1797},[2212,37807,37808,37810],{},[1584,37809,5226],{},[1735,37811,791],{},[1731,37813,1147],{},[1584,37815,26579],{},[1731,37817,2693],{"stretchy":1797},[1591,37819,37820],{"encoding":1593},"a = g(E_0 + \\delta)",[40,37822,37824,37842,37903],{"className":37823,"ariaHidden":1599},[1598],[40,37825,37827,37830,37833,37836,37839],{"className":37826},[1603],[40,37828],{"className":37829,"style":1681},[1607],[40,37831,19],{"className":37832},[1612,1653],[40,37834],{"className":37835,"style":1963},[1756],[40,37837,623],{"className":37838},[1967],[40,37840],{"className":37841,"style":1963},[1756],[40,37843,37845,37848,37851,37854,37894,37897,37900],{"className":37844},[1603],[40,37846],{"className":37847,"style":1811},[1607],[40,37849,36968],{"className":37850,"style":1818},[1612,1653],[40,37852,241],{"className":37853},[1985],[40,37855,37857,37860],{"className":37856},[1612],[40,37858,5226],{"className":37859,"style":4936},[1612,1653],[40,37861,37863],{"className":37862},[1621],[40,37864,37866,37886],{"className":37865},[1625,2244],[40,37867,37869,37883],{"className":37868},[1629],[40,37870,37872],{"className":37871,"style":13330},[1633],[40,37873,37874,37877],{"style":5058},[40,37875],{"className":37876,"style":1642},[1641],[40,37878,37880],{"className":37879},[1646,1647,1648,1649],[40,37881,791],{"className":37882},[1612,1649],[40,37884,2269],{"className":37885},[2268],[40,37887,37889],{"className":37888},[1629],[40,37890,37892],{"className":37891,"style":2276},[1633],[40,37893],{},[40,37895],{"className":37896,"style":1757},[1756],[40,37898,1147],{"className":37899},[1761],[40,37901],{"className":37902,"style":1757},[1756],[40,37904,37906,37909,37912],{"className":37905},[1603],[40,37907],{"className":37908,"style":1811},[1607],[40,37910,26579],{"className":37911,"style":26664},[1612,1653],[40,37913,2693],{"className":37914},[1822],[40,37916,37918,37936],{"className":37917},[1565],[40,37919,37921],{"className":37920},[1569],[1571,37922,37923],{"xmlns":1573},[1575,37924,37925,37933],{},[1578,37926,37927,37929,37931],{},[1584,37928,12149],{},[1731,37930,623],{},[1735,37932,1410],{},[1591,37934,37935],{"encoding":1593},"b = 1",[40,37937,37939,37957],{"className":37938,"ariaHidden":1599},[1598],[40,37940,37942,37945,37948,37951,37954],{"className":37941},[1603],[40,37943],{"className":37944,"style":4478},[1607],[40,37946,12149],{"className":37947},[1612,1653],[40,37949],{"className":37950,"style":1963},[1756],[40,37952,623],{"className":37953},[1967],[40,37955],{"className":37956,"style":1963},[1756],[40,37958,37960,37963],{"className":37959},[1603],[40,37961],{"className":37962,"style":1771},[1607],[40,37964,1410],{"className":37965},[1612],", and d = int(r\u002F2), we define the function:",[40,37968,37970],{"className":37969},[1827],[40,37971,37973,38175],{"className":37972},[1565],[40,37974,37976],{"className":37975},[1569],[1571,37977,37978],{"xmlns":1573,"display":1836},[1575,37979,37980,38172],{},[1578,37981,37982,37984,37986,37988,37990,37992,37998,38012,38014],{},[1584,37983,21331],{},[1731,37985,241],{"stretchy":1797},[1584,37987,5226],{},[1731,37989,2693],{"stretchy":1797},[1731,37991,623],{},[1581,37993,37994,37996],{},[1584,37995,12],{},[1731,37997,19819],{},[1578,37999,38000,38002,38004,38006,38008,38010],{},[1731,38001,241],{"fence":1599},[1584,38003,36968],{},[1731,38005,241],{"stretchy":1797},[1584,38007,5226],{},[1731,38009,2693],{"stretchy":1797},[1731,38011,2693],{"fence":1599},[1731,38013,623],{},[8538,38015,38016,38106],{},[1578,38017,38018,38024],{},[2212,38019,38020,38022],{},[1584,38021,36101],{},[1584,38023,4003],{},[1578,38025,38026,38028,38030,38032,38034,38104],{},[1731,38027,241],{"fence":1599},[1735,38029,1410],{},[1731,38031,1147],{},[1735,38033,1879],{},[8538,38035,38036,38082],{},[1578,38037,38038,38040,38042,38044,38046,38048,38050,38056,38058,38060,38062,38064,38066,38068,38070,38072,38074,38076,38078,38080],{},[1584,38039,36987],{},[1731,38041,35745],{},[1731,38043,241],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1731,38045,241],{"stretchy":1797},[1584,38047,5226],{},[1731,38049,1733],{},[2212,38051,38052,38054],{},[1584,38053,5226],{},[1735,38055,791],{},[1731,38057,2693],{"stretchy":1797},[1584,38059,4003],{},[1584,38061,2342],{},[1731,38063,2693],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1731,38065,1733],{},[1584,38067,36987],{},[1731,38069,35745],{},[1731,38071,241],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1584,38073,26579],{},[5211,38075,24742],{},[1584,38077,4003],{},[1584,38079,2342],{},[1731,38081,2693],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1578,38083,38084,38086,38088,38090,38092,38094,38096,38098,38100,38102],{},[1735,38085,1410],{},[1731,38087,1147],{},[1584,38089,36987],{},[1731,38091,35745],{},[1731,38093,241],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1584,38095,26579],{},[5211,38097,24742],{},[1584,38099,4003],{},[1584,38101,2342],{},[1731,38103,2693],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1731,38105,2693],{"fence":1599},[1578,38107,38108,38114],{},[2212,38109,38110,38112],{},[1584,38111,36101],{},[1584,38113,4003],{},[1578,38115,38116,38118,38120,38122,38124,38170],{},[1731,38117,241],{"fence":1599},[1735,38119,1410],{},[1731,38121,1147],{},[1735,38123,1879],{},[8538,38125,38126,38148],{},[1578,38127,38128,38130,38132,38134,38136,38138,38140,38142,38144,38146],{},[1735,38129,1410],{},[1731,38131,1733],{},[1584,38133,36987],{},[1731,38135,35745],{},[1731,38137,241],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1584,38139,26579],{},[5211,38141,24742],{},[1584,38143,4003],{},[1584,38145,2342],{},[1731,38147,2693],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1578,38149,38150,38152,38154,38156,38158,38160,38162,38164,38166,38168],{},[1735,38151,1410],{},[1731,38153,1147],{},[1584,38155,36987],{},[1731,38157,35745],{},[1731,38159,241],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1584,38161,26579],{},[5211,38163,24742],{},[1584,38165,4003],{},[1584,38167,2342],{},[1731,38169,2693],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1731,38171,2693],{"fence":1599},[1591,38173,38174],{"encoding":1593},"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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is the ground state energy. We can see by inserting ",[40,38982,38984,39038],{"className":38983},[1565],[40,38985,38987],{"className":38986},[1569],[1571,38988,38989],{"xmlns":1573},[1575,38990,38991,39035],{},[1578,38992,38993,38995,38997,38999,39001,39003,39005],{},[1584,38994,36987],{},[1731,38996,35745],{},[1731,38998,241],{"stretchy":1797},[1584,39000,8185],{},[1731,39002,2693],{"stretchy":1797},[1731,39004,623],{},[8538,39006,39007,39033],{},[1578,39008,39009,39019,39021],{},[1581,39010,39011,39013],{},[1584,39012,2330],{},[1578,39014,39015,39017],{},[1584,39016,2337],{},[1584,39018,8185],{},[1731,39020,1147],{},[1581,39022,39023,39025],{},[1584,39024,2330],{},[1578,39026,39027,39029,39031],{},[1731,39028,1733],{},[1584,39030,2337],{},[1584,39032,8185],{},[1735,39034,1879],{},[1591,39036,39037],{"encoding":1593},"\\cos(x)=\\frac{e^{ix}+e^{-ix}}{2}",[40,39039,39041,39068],{"className":39040,"ariaHidden":1599},[1598],[40,39042,39044,39047,39050,39053,39056,39059,39062,39065],{"className":39043},[1603],[40,39045],{"className":39046,"style":1811},[1607],[40,39048,36987],{"className":39049},[6614],[40,39051,241],{"className":39052},[1985],[40,39054,8185],{"className":39055},[1612,1653],[40,39057,2693],{"className":39058},[1822],[40,39060],{"className":39061,"style":1963},[1756],[40,39063,623],{"className":39064},[1967],[40,39066],{"className":39067,"style":1963},[1756],[40,39069,39071,39075],{"className":39070},[1603],[40,39072],{"className":39073,"style":39074},[1607],"height:1.3705em;vertical-align:-0.345em;",[40,39076,39078,39081,39217],{"className":39077},[1612],[40,39079],{"className":39080},[1985,8840],[40,39082,39084],{"className":39083},[8538],[40,39085,39087,39208],{"className":39086},[1625,2244],[40,39088,39090,39205],{"className":39089},[1629],[40,39091,39094,39108,39116],{"className":39092,"style":39093},[1633],"height:1.0255em;",[40,39095,39096,39099],{"style":36338},[40,39097],{"className":39098,"style":3354},[1641],[40,39100,39102],{"className":39101},[1646,1647,1648,1649],[40,39103,39105],{"className":39104},[1612,1649],[40,39106,1879],{"className":39107},[1612,1649],[40,39109,39110,39113],{"style":8925},[40,39111],{"className":39112,"style":3354},[1641],[40,39114],{"className":39115,"style":8933},[8932],[40,39117,39118,39121],{"style":36367},[40,39119],{"className":39120,"style":3354},[1641],[40,39122,39124],{"className":39123},[1646,1647,1648,1649],[40,39125,39127,39164,39167],{"className":39126},[1612,1649],[40,39128,39130,39133],{"className":39129},[1612,1649],[40,39131,2330],{"className":39132},[1612,1653,1649],[40,39134,39136],{"className":39135},[1621],[40,39137,39139],{"className":39138},[1625],[40,39140,39142],{"className":39141},[1629],[40,39143,39146],{"className":39144,"style":39145},[1633],"height:0.9021em;",[40,39147,39149,39152],{"style":39148},"top:-2.931em;margin-right:0.0714em;",[40,39150],{"className":39151,"style":3856},[1641],[40,39153,39155],{"className":39154},[1646,3860,1992,1649],[40,39156,39158,39161],{"className":39157},[1612,1649],[40,39159,2337],{"className":39160},[1612,1653,1649],[40,39162,8185],{"className":39163},[1612,1653,1649],[40,39165,1147],{"className":39166},[1761,1649],[40,39168,39170,39173],{"className":39169},[1612,1649],[40,39171,2330],{"className":39172},[1612,1653,1649],[40,39174,39176],{"className":39175},[1621],[40,39177,39179],{"className":39178},[1625],[40,39180,39182],{"className":39181},[1629],[40,39183,39185],{"className":39184,"style":39145},[1633],[40,39186,39187,39190],{"style":39148},[40,39188],{"className":39189,"style":3856},[1641],[40,39191,39193],{"className":39192},[1646,3860,1992,1649],[40,39194,39196,39199,39202],{"className":39195},[1612,1649],[40,39197,1733],{"className":39198},[1612,1649],[40,39200,2337],{"className":39201},[1612,1653,1649],[40,39203,8185],{"className":39204},[1612,1653,1649],[40,39206,2269],{"className":39207},[2268],[40,39209,39211],{"className":39210},[1629],[40,39212,39215],{"className":39213,"style":39214},[1633],"height:0.345em;",[40,39216],{},[40,39218],{"className":39219},[1822,8840]," that ",[40,39222,39224,39243],{"className":39223},[1565],[40,39225,39227],{"className":39226},[1569],[1571,39228,39229],{"xmlns":1573},[1575,39230,39231,39241],{},[1578,39232,39233,39235,39237,39239],{},[1584,39234,21331],{},[1731,39236,241],{"stretchy":1797},[1584,39238,5226],{},[1731,39240,2693],{"stretchy":1797},[1591,39242,26868],{"encoding":1593},[40,39244,39246],{"className":39245,"ariaHidden":1599},[1598],[40,39247,39249,39252,39255,39258,39261],{"className":39248},[1603],[40,39250],{"className":39251,"style":1811},[1607],[40,39253,21331],{"className":39254,"style":26393},[1612,1653],[40,39256,241],{"className":39257},[1985],[40,39259,5226],{"className":39260,"style":4936},[1612,1653],[40,39262,2693],{"className":39263},[1822]," is a trigonometric polynomial of degree ",[40,39266,39268,39281],{"className":39267},[1565],[40,39269,39271],{"className":39270},[1569],[1571,39272,39273],{"xmlns":1573},[1575,39274,39275,39279],{},[1578,39276,39277],{},[1584,39278,4003],{},[1591,39280,4003],{"encoding":1593},[40,39282,39284],{"className":39283,"ariaHidden":1599},[1598],[40,39285,39287,39290],{"className":39286},[1603],[40,39288],{"className":39289,"style":4478},[1607],[40,39291,4003],{"className":39292},[1612,1653],", that is, a linear combination of ",[40,39295,39297,39326],{"className":39296},[1565],[40,39298,39300],{"className":39299},[1569],[1571,39301,39302],{"xmlns":1573},[1575,39303,39304,39324],{},[1578,39305,39306],{},[1581,39307,39308,39310],{},[1584,39309,2330],{},[1578,39311,39312,39314,39316,39318,39320,39322],{},[1584,39313,2337],{},[1584,39315,1413],{},[1584,39317,5226],{},[5211,39319,24742],{},[1584,39321,4003],{},[1584,39323,2342],{},[1591,39325,26923],{"encoding":1593},[40,39327,39329],{"className":39328,"ariaHidden":1599},[1598],[40,39330,39332,39335],{"className":39331},[1603],[40,39333],{"className":39334,"style":2706},[1607],[40,39336,39338,39341],{"className":39337},[1612],[40,39339,2330],{"className":39340},[1612,1653],[40,39342,39344],{"className":39343},[1621],[40,39345,39347],{"className":39346},[1625],[40,39348,39350],{"className":39349},[1629],[40,39351,39353],{"className":39352,"style":2706},[1633],[40,39354,39355,39358],{"style":1637},[40,39356],{"className":39357,"style":1642},[1641],[40,39359,39361],{"className":39360},[1646,1647,1648,1649],[40,39362,39364,39367,39370,39373,39376],{"className":39363},[1612,1649],[40,39365,26965],{"className":39366,"style":5867},[1612,1653,1649],[40,39368,5226],{"className":39369,"style":4936},[1612,1653,1649],[40,39371],{"className":39372,"style":24952},[1756,1649],[40,39374,4003],{"className":39375},[1612,1653,1649],[40,39377,2342],{"className":39378},[1612,1653,1649],[40,39380,39382,39431],{"className":39381},[1565],[40,39383,39385],{"className":39384},[1569],[1571,39386,39387],{"xmlns":1573},[1575,39388,39389,39429],{},[1578,39390,39391,39393,39395,39397,39399,39401,39403,39405,39407,39409,39411,39413,39415,39417,39419,39421,39423,39425,39427],{},[1584,39392,1413],{},[1731,39394,623],{},[1731,39396,1733],{},[1584,39398,4003],{},[1731,39400,1862],{"separator":1599},[1731,39402,1733],{},[1584,39404,4003],{},[1731,39406,1147],{},[1735,39408,1410],{},[1731,39410,1862],{"separator":1599},[1584,39412,1823],{"mathvariant":1790},[1584,39414,1823],{"mathvariant":1790},[1584,39416,1823],{"mathvariant":1790},[1731,39418,1862],{"separator":1599},[1584,39420,4003],{},[1731,39422,1733],{},[1735,39424,1410],{},[1731,39426,1862],{"separator":1599},[1584,39428,4003],{},[1591,39430,27031],{"encoding":1593},[40,39432,39434,39452,39485,39521],{"className":39433,"ariaHidden":1599},[1598],[40,39435,39437,39440,39443,39446,39449],{"className":39436},[1603],[40,39438],{"className":39439,"style":27041},[1607],[40,39441,1413],{"className":39442,"style":5867},[1612,1653],[40,39444],{"className":39445,"style":1963},[1756],[40,39447,623],{"className":39448},[1967],[40,39450],{"className":39451,"style":1963},[1756],[40,39453,39455,39458,39461,39464,39467,39470,39473,39476,39479,39482],{"className":39454},[1603],[40,39456],{"className":39457,"style":6079},[1607],[40,39459,1733],{"className":39460},[1612],[40,39462,4003],{"className":39463},[1612,1653],[40,39465,1862],{"className":39466},[2005],[40,39468],{"className":39469,"style":2009},[1756],[40,39471,1733],{"className":39472},[1612],[40,39474,4003],{"className":39475},[1612,1653],[40,39477],{"className":39478,"style":1757},[1756],[40,39480,1147],{"className":39481},[1761],[40,39483],{"className":39484,"style":1757},[1756],[40,39486,39488,39491,39494,39497,39500,39503,39506,39509,39512,39515,39518],{"className":39487},[1603],[40,39489],{"className":39490,"style":6079},[1607],[40,39492,1410],{"className":39493},[1612],[40,39495,1862],{"className":39496},[2005],[40,39498],{"className":39499,"style":2009},[1756],[40,39501,2076],{"className":39502},[1612],[40,39504,1862],{"className":39505},[2005],[40,39507],{"className":39508,"style":2009},[1756],[40,39510,4003],{"className":39511},[1612,1653],[40,39513],{"className":39514,"style":1757},[1756],[40,39516,1733],{"className":39517},[1761],[40,39519],{"className":39520,"style":1757},[1756],[40,39522,39524,39527,39530,39533,39536],{"className":39523},[1603],[40,39525],{"className":39526,"style":6079},[1607],[40,39528,1410],{"className":39529},[1612],[40,39531,1862],{"className":39532},[2005],[40,39534],{"className":39535,"style":2009},[1756],[40,39537,4003],{"className":39538},[1612,1653],". Furthermore, from the definition of ",[40,39541,39543,39566],{"className":39542},[1565],[40,39544,39546],{"className":39545},[1569],[1571,39547,39548],{"xmlns":1573},[1575,39549,39550,39564],{},[1578,39551,39552,39558,39560,39562],{},[1581,39553,39554,39556],{},[1584,39555,12],{},[1731,39557,19819],{},[1731,39559,241],{"stretchy":1797},[1584,39561,8185],{},[1731,39563,2693],{"stretchy":1797},[1591,39565,37737],{"encoding":1593},[40,39567,39569],{"className":39568,"ariaHidden":1599},[1598],[40,39570,39572,39575,39604,39607,39610],{"className":39571},[1603],[40,39573],{"className":39574,"style":1811},[1607],[40,39576,39578,39581],{"className":39577},[1612],[40,39579,12],{"className":39580},[1612,1653],[40,39582,39584],{"className":39583},[1621],[40,39585,39587],{"className":39586},[1625],[40,39588,39590],{"className":39589},[1629],[40,39591,39593],{"className":39592,"style":35637},[1633],[40,39594,39595,39598],{"style":1637},[40,39596],{"className":39597,"style":1642},[1641],[40,39599,39601],{"className":39600},[1646,1647,1648,1649],[40,39602,19819],{"className":39603},[1761,1649],[40,39605,241],{"className":39606},[1985],[40,39608,8185],{"className":39609},[1612,1653],[40,39611,2693],{"className":39612},[1822]," above we have that ",[40,39615,39617,39655],{"className":39616},[1565],[40,39618,39620],{"className":39619},[1569],[1571,39621,39622],{"xmlns":1573},[1575,39623,39624,39652],{},[1578,39625,39626,39628,39630,39636,39638,39640,39642,39644,39646,39648,39650],{},[1584,39627,21331],{},[1731,39629,241],{"stretchy":1797},[2212,39631,39632,39634],{},[1584,39633,5226],{},[1735,39635,791],{},[1731,39637,2693],{"stretchy":1797},[1731,39639,623],{},[1584,39641,12],{},[1731,39643,241],{"stretchy":1797},[1735,39645,791],{},[1731,39647,2693],{"stretchy":1797},[1731,39649,623],{},[1735,39651,1410],{},[1591,39653,39654],{"encoding":1593},"f(E_0)=p(0)=1",[40,39656,39658,39722,39749],{"className":39657,"ariaHidden":1599},[1598],[40,39659,39661,39664,39667,39670,39710,39713,39716,39719],{"className":39660},[1603],[40,39662],{"className":39663,"style":1811},[1607],[40,39665,21331],{"className":39666,"style":26393},[1612,1653],[40,39668,241],{"className":39669},[1985],[40,39671,39673,39676],{"className":39672},[1612],[40,39674,5226],{"className":39675,"style":4936},[1612,1653],[40,39677,39679],{"className":39678},[1621],[40,39680,39682,39702],{"className":39681},[1625,2244],[40,39683,39685,39699],{"className":39684},[1629],[40,39686,39688],{"className":39687,"style":13330},[1633],[40,39689,39690,39693],{"style":5058},[40,39691],{"className":39692,"style":1642},[1641],[40,39694,39696],{"className":39695},[1646,1647,1648,1649],[40,39697,791],{"className":39698},[1612,1649],[40,39700,2269],{"className":39701},[2268],[40,39703,39705],{"className":39704},[1629],[40,39706,39708],{"className":39707,"style":2276},[1633],[40,39709],{},[40,39711,2693],{"className":39712},[1822],[40,39714],{"className":39715,"style":1963},[1756],[40,39717,623],{"className":39718},[1967],[40,39720],{"className":39721,"style":1963},[1756],[40,39723,39725,39728,39731,39734,39737,39740,39743,39746],{"className":39724},[1603],[40,39726],{"className":39727,"style":1811},[1607],[40,39729,12],{"className":39730},[1612,1653],[40,39732,241],{"className":39733},[1985],[40,39735,791],{"className":39736},[1612],[40,39738,2693],{"className":39739},[1822],[40,39741],{"className":39742,"style":1963},[1756],[40,39744,623],{"className":39745},[1967],[40,39747],{"className":39748,"style":1963},[1756],[40,39750,39752,39755],{"className":39751},[1603],[40,39753],{"className":39754,"style":1771},[1607],[40,39756,1410],{"className":39757},[1612]," and for any ",[40,39760,39762,39775],{"className":39761},[1565],[40,39763,39765],{"className":39764},[1569],[1571,39766,39767],{"xmlns":1573},[1575,39768,39769,39773],{},[1578,39770,39771],{},[1584,39772,5226],{},[1591,39774,5226],{"encoding":1593},[40,39776,39778],{"className":39777,"ariaHidden":1599},[1598],[40,39779,39781,39784],{"className":39780},[1603],[40,39782],{"className":39783,"style":1608},[1607],[40,39785,5226],{"className":39786,"style":4936},[1612,1653]," in the spectral range such that ",[40,39789,39791,39821],{"className":39790},[1565],[40,39792,39794],{"className":39793},[1569],[1571,39795,39796],{"xmlns":1573},[1575,39797,39798,39818],{},[1578,39799,39800,39802,39804,39806,39812,39814,39816],{},[1584,39801,1791],{"mathvariant":1790},[1584,39803,5226],{},[1731,39805,1733],{},[2212,39807,39808,39810],{},[1584,39809,5226],{},[1735,39811,791],{},[1584,39813,1791],{"mathvariant":1790},[1731,39815,689],{},[1584,39817,26579],{},[1591,39819,39820],{"encoding":1593},"\\vert E-E_0 \\vert > \\delta",[40,39822,39824,39845,39903],{"className":39823,"ariaHidden":1599},[1598],[40,39825,39827,39830,39833,39836,39839,39842],{"className":39826},[1603],[40,39828],{"className":39829,"style":1811},[1607],[40,39831,1791],{"className":39832},[1612],[40,39834,5226],{"className":39835,"style":4936},[1612,1653],[40,39837],{"className":39838,"style":1757},[1756],[40,39840,1733],{"className":39841},[1761],[40,39843],{"className":39844,"style":1757},[1756],[40,39846,39848,39851,39891,39894,39897,39900],{"className":39847},[1603],[40,39849],{"className":39850,"style":1811},[1607],[40,39852,39854,39857],{"className":39853},[1612],[40,39855,5226],{"className":39856,"style":4936},[1612,1653],[40,39858,39860],{"className":39859},[1621],[40,39861,39863,39883],{"className":39862},[1625,2244],[40,39864,39866,39880],{"className":39865},[1629],[40,39867,39869],{"className":39868,"style":13330},[1633],[40,39870,39871,39874],{"style":5058},[40,39872],{"className":39873,"style":1642},[1641],[40,39875,39877],{"className":39876},[1646,1647,1648,1649],[40,39878,791],{"className":39879},[1612,1649],[40,39881,2269],{"className":39882},[2268],[40,39884,39886],{"className":39885},[1629],[40,39887,39889],{"className":39888,"style":2276},[1633],[40,39890],{},[40,39892,1791],{"className":39893},[1612],[40,39895],{"className":39896,"style":1963},[1756],[40,39898,689],{"className":39899},[1967],[40,39901],{"className":39902,"style":1963},[1756],[40,39904,39906,39909],{"className":39905},[1603],[40,39907],{"className":39908,"style":4478},[1607],[40,39910,26579],{"className":39911,"style":26664},[1612,1653]," we have",[40,39914,39916],{"className":39915},[1827],[40,39917,39919,40033],{"className":39918},[1565],[40,39920,39922],{"className":39921},[1569],[1571,39923,39924],{"xmlns":1573,"display":1836},[1575,39925,39926,40030],{},[1578,39927,39928,39930,39932,39934,39936,39938,39940,39942,39944,39946,39948,39950,39952,39954,39956,39958,39960,39972],{},[1584,39929,1791],{"mathvariant":1790},[1584,39931,21331],{},[1731,39933,241],{"stretchy":1797},[1584,39935,5226],{},[1731,39937,2693],{"stretchy":1797},[1584,39939,1791],{"mathvariant":1790},[1731,39941,26564],{},[1584,39943,35720],{},[1731,39945,241],{"stretchy":1797},[1584,39947,19],{},[1731,39949,1862],{"separator":1599},[1584,39951,12149],{},[1731,39953,1862],{"separator":1599},[1584,39955,4003],{},[1731,39957,2693],{"stretchy":1797},[1731,39959,623],{},[2819,39961,39962,39964,39966],{},[1584,39963,36101],{},[1584,39965,4003],{},[1578,39967,39968,39970],{},[1731,39969,1733],{},[1735,39971,1410],{},[1578,39973,39974,39976,39978,39980,39982,40028],{},[1731,39975,241],{"fence":1599},[1735,39977,1410],{},[1731,39979,1147],{},[1735,39981,1879],{},[8538,39983,39984,40006],{},[1578,39985,39986,39988,39990,39992,39994,39996,39998,40000,40002,40004],{},[1735,39987,1410],{},[1731,39989,1733],{},[1584,39991,36987],{},[1731,39993,35745],{},[1731,39995,241],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1584,39997,26579],{},[5211,39999,24742],{},[1584,40001,4003],{},[1584,40003,2342],{},[1731,40005,2693],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1578,40007,40008,40010,40012,40014,40016,40018,40020,40022,40024,40026],{},[1735,40009,1410],{},[1731,40011,1147],{},[1584,40013,36987],{},[1731,40015,35745],{},[1731,40017,241],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1584,40019,26579],{},[5211,40021,24742],{},[1584,40023,4003],{},[1584,40025,2342],{},[1731,40027,2693],{"fence":1797,"stretchy":1599,"minsize":36992,"maxsize":36992},[1731,40029,2693],{"fence":1599},[1591,40031,40032],{"encoding":1593},"|f(E)| \\le \\beta(a, b, d) = T_d^{-1}\\left(1 + 2\\frac{1-\\cos\\big(\\delta\\,dt\\big)}{1 + \\cos\\big(\\delta\\,dt\\big)}\\right)",[40,40034,40036,40069,40114],{"className":40035,"ariaHidden":1599},[1598],[40,40037,40039,40042,40045,40048,40051,40054,40057,40060,40063,40066],{"className":40038},[1603],[40,40040],{"className":40041,"style":1811},[1607],[40,40043,1791],{"className":40044},[1612],[40,40046,21331],{"className":40047,"style":26393},[1612,1653],[40,40049,241],{"className":40050},[1985],[40,40052,5226],{"className":40053,"style":4936},[1612,1653],[40,40055,2693],{"className":40056},[1822],[40,40058,1791],{"className":40059},[1612],[40,40061],{"className":40062,"style":1963},[1756],[40,40064,26564],{"className":40065},[1967],[40,40067],{"className":40068,"style":1963},[1756],[40,40070,40072,40075,40078,40081,40084,40087,40090,40093,40096,40099,40102,40105,40108,40111],{"className":40071},[1603],[40,40073],{"className":40074,"style":1811},[1607],[40,40076,35720],{"className":40077,"style":35815},[1612,1653],[40,40079,241],{"className":40080},[1985],[40,40082,19],{"className":40083},[1612,1653],[40,40085,1862],{"className":40086},[2005],[40,40088],{"className":40089,"style":2009},[1756],[40,40091,12149],{"className":40092},[1612,1653],[40,40094,1862],{"className":40095},[2005],[40,40097],{"className":40098,"style":2009},[1756],[40,40100,4003],{"className":40101},[1612,1653],[40,40103,2693],{"className":40104},[1822],[40,40106],{"className":40107,"style":1963},[1756],[40,40109,623],{"className":40110},[1967],[40,40112],{"className":40113,"style":1963},[1756],[40,40115,40117,40121,40178,40181],{"className":40116},[1603],[40,40118],{"className":40119,"style":40120},[1607],"height:3em;vertical-align:-1.25em;",[40,40122,40124,40127],{"className":40123},[1612],[40,40125,36101],{"className":40126,"style":2264},[1612,1653],[40,40128,40130],{"className":40129},[1621],[40,40131,40133,40170],{"className":40132},[1625,2244],[40,40134,40136,40167],{"className":40135},[1629],[40,40137,40139,40150],{"className":40138,"style":2046},[1633],[40,40140,40141,40144],{"style":36744},[40,40142],{"className":40143,"style":1642},[1641],[40,40145,40147],{"className":40146},[1646,1647,1648,1649],[40,40148,4003],{"className":40149},[1612,1653,1649],[40,40151,40152,40155],{"style":1950},[40,40153],{"className":40154,"style":1642},[1641],[40,40156,40158],{"className":40157},[1646,1647,1648,1649],[40,40159,40161,40164],{"className":40160},[1612,1649],[40,40162,1733],{"className":40163},[1612,1649],[40,40165,1410],{"className":40166},[1612,1649],[40,40168,2269],{"className":40169},[2268],[40,40171,40173],{"className":40172},[1629],[40,40174,40176],{"className":40175,"style":36780},[1633],[40,40177],{},[40,40179],{"className":40180,"style":2009},[1756],[40,40182,40184,40190,40193,40196,40199,40202,40205,40347],{"className":40183},[1981],[40,40185,40187],{"className":40186,"style":1987},[1985,1986],[40,40188,241],{"className":40189},[1991,11533],[40,40191,1410],{"className":40192},[1612],[40,40194],{"className":40195,"style":1757},[1756],[40,40197,1147],{"className":40198},[1761],[40,40200],{"className":40201,"style":1757},[1756],[40,40203,1879],{"className":40204},[1612],[40,40206,40208,40211,40344],{"className":40207},[1612],[40,40209],{"className":40210},[1985,8840],[40,40212,40214],{"className":40213},[8538],[40,40215,40217,40335],{"className":40216},[1625,2244],[40,40218,40220,40332],{"className":40219},[1629],[40,40221,40223,40274,40282],{"className":40222,"style":37075},[1633],[40,40224,40226,40229],{"style":40225},"top:-2.26em;",[40,40227],{"className":40228,"style":3354},[1641],[40,40230,40232,40235,40238,40241,40244,40247,40250,40256,40259,40262,40265,40268],{"className":40231},[1612],[40,40233,1410],{"className":40234},[1612],[40,40236],{"className":40237,"style":1757},[1756],[40,40239,1147],{"className":40240},[1761],[40,40242],{"className":40243,"style":1757},[1756],[40,40245,36987],{"className":40246},[6614],[40,40248],{"className":40249,"style":2009},[1756],[40,40251,40253],{"className":40252},[1612],[40,40254,241],{"className":40255},[1991,1992],[40,40257,26579],{"className":40258,"style":26664},[1612,1653],[40,40260],{"className":40261,"style":2009},[1756],[40,40263,4003],{"className":40264},[1612,1653],[40,40266,2342],{"className":40267},[1612,1653],[40,40269,40271],{"className":40270},[1612],[40,40272,2693],{"className":40273},[1991,1992],[40,40275,40276,40279],{"style":8925},[40,40277],{"className":40278,"style":3354},[1641],[40,40280],{"className":40281,"style":8933},[8932],[40,40283,40284,40287],{"style":37097},[40,40285],{"className":40286,"style":3354},[1641],[40,40288,40290,40293,40296,40299,40302,40305,40308,40314,40317,40320,40323,40326],{"className":40289},[1612],[40,40291,1410],{"className":40292},[1612],[40,40294],{"className":40295,"style":1757},[1756],[40,40297,1733],{"className":40298},[1761],[40,40300],{"className":40301,"style":1757},[1756],[40,40303,36987],{"className":40304},[6614],[40,40306],{"className":40307,"style":2009},[1756],[40,40309,40311],{"className":40310},[1612],[40,40312,2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2\\left(1 + \\delta\\right)^{-d} = 2\\left(1 + \\delta\\right)^{-\\lfloor k\u002F2\\rfloor}.",[40,40432,40434,40446,40523],{"className":40433,"ariaHidden":1599},[1598],[40,40435,40437,40440,40443],{"className":40436},[1603],[40,40438],{"className":40439,"style":26757},[1607],[40,40441,26564],{"className":40442},[1967],[40,40444],{"className":40445,"style":1963},[1756],[40,40447,40449,40452,40455,40458,40514,40517,40520],{"className":40448},[1603],[40,40450],{"className":40451,"style":26633},[1607],[40,40453,1879],{"className":40454},[1612],[40,40456],{"className":40457,"style":2009},[1756],[40,40459,40461,40485],{"className":40460},[1981],[40,40462,40464,40467,40470,40473,40476,40479,40482],{"className":40463},[1981],[40,40465,241],{"className":40466,"style":1987},[1985,1986],[40,40468,1410],{"className":40469},[1612],[40,40471],{"className":40472,"style":1757},[1756],[40,40474,1147],{"className":40475},[1761],[40,40477],{"className":40478,"style":1757},[1756],[40,40480,26579],{"className":40481,"style":26664},[1612,1653],[40,40483,2693],{"className":40484,"style":1987},[1822,1986],[40,40486,40488],{"className":40487},[1621],[40,40489,40491],{"className":40490},[1625],[40,40492,40494],{"className":40493},[1629],[40,40495,40497],{"className":40496,"style":26680},[1633],[40,40498,40499,40502],{"style":6889},[40,40500],{"className":40501,"style":1642},[1641],[40,40503,40505],{"className":40504},[1646,1647,1648,1649],[40,40506,40508,40511],{"className":40507},[1612,1649],[40,40509,1733],{"className":40510},[1612,1649],[40,40512,4003],{"className":40513},[1612,1653,1649],[40,40515],{"className":40516,"style":1963},[1756],[40,40518,623],{"className":40519},[1967],[40,40521],{"className":40522,"style":1963},[1756],[40,40524,40526,40530,40533,40536,40602,40605],{"className":40525},[1603],[40,40527],{"className":40528,"style":40529},[1607],"height:1.2779em;vertical-align:-0.25em;",[40,40531,1879],{"className":40532},[1612],[40,40534],{"className":40535,"style":2009},[1756],[40,40537,40539,40563],{"className":40538},[1981],[40,40540,40542,40545,40548,40551,40554,40557,40560],{"className":40541},[1981],[40,40543,241],{"className":40544,"style":1987},[1985,1986],[40,40546,1410],{"className":40547},[1612],[40,40549],{"className":40550,"style":1757},[1756],[40,40552,1147],{"className":40553},[1761],[40,40555],{"className":40556,"style":1757},[1756],[40,40558,26579],{"className":40559,"style":26664},[1612,1653],[40,40561,2693],{"className":40562,"style":1987},[1822,1986],[40,40564,40566],{"className":40565},[1621],[40,40567,40569],{"className":40568},[1625],[40,40570,40572],{"className":40571},[1629],[40,40573,40576],{"className":40574,"style":40575},[1633],"height:1.0279em;",[40,40577,40578,40581],{"style":6889},[40,40579],{"className":40580,"style":1642},[1641],[40,40582,40584],{"className":40583},[1646,1647,1648,1649],[40,40585,40587,40590,40593,40596,40599],{"className":40586},[1612,1649],[40,40588,1733],{"className":40589},[1612,1649],[40,40591,40416],{"className":40592},[1985,1649],[40,40594,2670],{"className":40595,"style":2736},[1612,1653,1649],[40,40597,25175],{"className":40598},[1612,1649],[40,40600,40425],{"className":40601},[1822,1649],[40,40603],{"className":40604,"style":2009},[1756],[40,40606,1823],{"className":40607},[1612],[55,40609,40611],{"id":40610},"references","References",[12,40613,40614,40616,40617],{},[40,40615,1410],{}," N. Yoshioka, M. Amico, W. Kirby et al. \"Diagonalization of large many-body Hamiltonians on a quantum processor\". ",[19,40618,40620],{"href":40619},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.14431","arXiv:2407.14431",[12,40622,40623,40625],{},[40,40624,1879],{}," 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).",[12,40627,40628,40630],{},[40,40629,7270],{}," Å. Björck. \"Numerical methods in matrix computations\". Texts in Applied Mathematics. Springer International Publishing. (2014).",[12,40632,40633,40635],{},[40,40634,11199],{}," William Kirby. \"Analysis of quantum Krylov algorithms with errors\". Quantum 8, 1457 (2024).",[55,40637,40639],{"id":40638},"tutorial-survey","Tutorial survey",[12,40641,40642],{},"Please take this short survey to provide feedback on this tutorial. Your insights will help us improve our content offerings and user experience.",[12,40644,40645],{},[19,40646,40648],{"href":40647},"https:\u002F\u002Fyour.feedback.ibm.com\u002Fjfe\u002Fform\u002FSV_82nennpKIjjD8rQ","Link to survey",[55,40650,40652],{"id":40651},"run-it-yourself","Run it yourself",[12,40654,40655,40656,1823],{},"Get ",[19,40657,22],{"href":21},[40659,40660,40661],"style",{},"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":35,"searchDepth":50,"depth":50,"links":40663},[40664,40665,40666,40667,40671,40678,40683,40686,40687,40688,40689,40690],{"id":57,"depth":50,"text":58},{"id":92,"depth":50,"text":93},{"id":118,"depth":50,"text":119},{"id":1552,"depth":50,"text":1553,"children":40668},[40669,40670],{"id":1557,"depth":155,"text":1558},{"id":3128,"depth":155,"text":3129},{"id":13264,"depth":50,"text":13265,"children":40672},[40673,40674,40675,40676],{"id":13268,"depth":155,"text":13269},{"id":19398,"depth":155,"text":19399},{"id":20386,"depth":155,"text":20387},{"id":20484,"depth":155,"text":40677},"Template circuits for calculating matrix elements of S~\\tilde{S}S~ and H~\\tilde{H}H~ via Hadamard test",{"id":20964,"depth":50,"text":20965,"children":40679},[40680,40681,40682],{"id":21078,"depth":155,"text":21079},{"id":21699,"depth":155,"text":21700},{"id":22171,"depth":155,"text":22172},{"id":23031,"depth":50,"text":23032,"children":40684},[40685],{"id":23060,"depth":155,"text":23061},{"id":24672,"depth":50,"text":24673},{"id":35502,"depth":50,"text":35503},{"id":40610,"depth":50,"text":40611},{"id":40638,"depth":50,"text":40639},{"id":40651,"depth":50,"text":40652},[40692,40693,40694,40695],"Qollab","Blog","Expert notes","Krylov diagonalization",[40697],{"username":7,"name":40698,"role":40699,"bio":40700,"links":40701},"Mirko Amico","Researcher, Extropic · formerly IBM Research","A researcher at Extropic, previously at IBM Research, and a Qiskit Pulse expert.",[40702],{"label":40703,"href":21},"Original tutorial ↗",{"username":7,"name":40698,"role":40705},"Researcher, Extropic",null,"Implement the Krylov Quantum Diagonalization Algorithm (KQD) within the context of Qiskit patterns.",false,"md","Deep dive · Algorithms","article",{},"\u002Fblog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization","2026-07-15","15 min read",[],{"title":5,"description":40707},"blog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization",[40720,40721],"algorithms","tutorial","h_4WJ9crI4tBC8SOea5gUgWXCAFpFws4fEwgpxBRWsU",[40724,41214,79970,105987],{"id":40725,"title":40726,"authors":40727,"body":40729,"breadcrumb":41165,"builders":41167,"byline":41178,"challenge":40706,"courseAuthor":40706,"courseLead":40706,"dek":41180,"description":41181,"draft":40708,"extension":40709,"eyebrow":41182,"finish":40706,"fork":40706,"hero":40706,"heroAlt":40706,"heroCta":40706,"heroImage":40706,"kind":40711,"lessonCount":40706,"meta":41183,"navigation":227,"newsItems":40706,"next":40706,"ogImage":41184,"order":40706,"outcomes":40706,"path":41185,"publishDate":40714,"readingTime":41186,"related":41187,"relatedProjects":41188,"seo":41208,"stem":41210,"tags":41211,"track":40706,"trackName":40706,"__hash__":41213},"blog\u002Fblog\u002Fexpert-notes\u002F2-qubit-state-visualization.md","2-qubit states, visualized 8 ways",[40728],"onri-jay-benally",{"type":9,"value":40730,"toc":41153},[40731,40734,40746,40750,40753,41021,41024,41028,41031,41037,41040,41046,41050,41053,41059,41063,41066,41072,41076,41079,41085,41089,41092,41096,41099,41105,41109,41112,41118,41122,41125,41131,41137,41139,41150],[12,40732,40733],{},"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.",[12,40735,40736,40737,40741,40742,1823],{},"Everything below comes from Onri's runnable notebook, shared openly on ",[19,40738,40740],{"href":40739},"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 ",[19,40743,40745],{"href":40744},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002F2_Qubit_State_Visualization_in_Python.ipynb","Google Colab",[55,40747,40749],{"id":40748},"one-circuit-two-states","One circuit, two states",[12,40751,40752],{},"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:",[40754,40755,40758],"code-block",{"name":40756,"tag":40757},"2_qubit_state_visualization.ipynb","Python",[30,40759,40761],{"className":32,"code":40760,"language":34,"meta":35,"style":35},"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",[37,40762,40763,40773,40784,40795,40799,40804,40818,40834,40854,40878,40893,40897,40909,40921,40925,40930,40943,40954,40966,40979,40986,40997,41009],{"__ignoreMap":35},[40,40764,40765,40767,40769,40771],{"class":42,"line":43},[40,40766,130],{"class":129},[40,40768,134],{"class":133},[40,40770,137],{"class":129},[40,40772,140],{"class":133},[40,40774,40775,40777,40779,40781],{"class":42,"line":50},[40,40776,171],{"class":129},[40,40778,287],{"class":133},[40,40780,130],{"class":129},[40,40782,40783],{"class":133}," QuantumCircuit\n",[40,40785,40786,40788,40790,40792],{"class":42,"line":155},[40,40787,171],{"class":129},[40,40789,261],{"class":133},[40,40791,130],{"class":129},[40,40793,40794],{"class":133}," Statevector, DensityMatrix\n",[40,40796,40797],{"class":42,"line":168},[40,40798,228],{"emptyLinePlaceholder":227},[40,40800,40801],{"class":42,"line":182},[40,40802,40803],{"class":46},"# Create a circuit with 2 qubits\n",[40,40805,40806,40808,40810,40812,40814,40816],{"class":42,"line":195},[40,40807,20698],{"class":133},[40,40809,623],{"class":441},[40,40811,8175],{"class":237},[40,40813,241],{"class":133},[40,40815,1879],{"class":462},[40,40817,248],{"class":133},[40,40819,40820,40822,40824,40826,40828,40831],{"class":42,"line":203},[40,40821,20709],{"class":133},[40,40823,13084],{"class":237},[40,40825,241],{"class":133},[40,40827,791],{"class":462},[40,40829,40830],{"class":133},")             ",[40,40832,40833],{"class":46},"# Hadamard on q0\n",[40,40835,40836,40838,40840,40842,40844,40846,40848,40851],{"class":42,"line":216},[40,40837,20709],{"class":133},[40,40839,20447],{"class":237},[40,40841,241],{"class":133},[40,40843,791],{"class":462},[40,40845,917],{"class":133},[40,40847,1410],{"class":462},[40,40849,40850],{"class":133},")         ",[40,40852,40853],{"class":46},"# CNOT with q0 as control, q1 as target\n",[40,40855,40856,40858,40861,40864,40866,40869,40871,40873,40875],{"class":42,"line":224},[40,40857,20709],{"class":133},[40,40859,40860],{"class":237},"ry",[40,40862,40863],{"class":133},"(np.pi ",[40,40865,7610],{"class":441},[40,40867,40868],{"class":462}," 4",[40,40870,917],{"class":133},[40,40872,1410],{"class":462},[40,40874,1416],{"class":133},[40,40876,40877],{"class":46},"# Y-rotation of pi\u002F4 on q1\n",[40,40879,40880,40882,40884,40886,40888,40890],{"class":42,"line":231},[40,40881,20709],{"class":133},[40,40883,19585],{"class":237},[40,40885,241],{"class":133},[40,40887,1410],{"class":462},[40,40889,40830],{"class":133},[40,40891,40892],{"class":46},"# Phase flip (Z gate) on q1\n",[40,40894,40895],{"class":42,"line":251},[40,40896,228],{"emptyLinePlaceholder":227},[40,40898,40899,40902,40904,40907],{"class":42,"line":256},[40,40900,40901],{"class":133},"final_statevector ",[40,40903,623],{"class":441},[40,40905,40906],{"class":237}," Statevector",[40,40908,21557],{"class":133},[40,40910,40911,40914,40916,40919],{"class":42,"line":269},[40,40912,40913],{"class":133},"final_density_matrix ",[40,40915,623],{"class":441},[40,40917,40918],{"class":237}," DensityMatrix",[40,40920,21557],{"class":133},[40,40922,40923],{"class":42,"line":282},[40,40924,228],{"emptyLinePlaceholder":227},[40,40926,40927],{"class":42,"line":295},[40,40928,40929],{"class":46},"# Generate all 8 plots for the final state\n",[40,40931,40932,40935,40938,40941],{"class":42,"line":308},[40,40933,40934],{"class":237},"plot_bloch_multivector_viz",[40,40936,40937],{"class":133},"(final_density_matrix, ",[40,40939,40940],{"class":244},"\"Final Entangled State\"",[40,40942,248],{"class":133},[40,40944,40945,40948,40950,40952],{"class":42,"line":321},[40,40946,40947],{"class":237},"plot_q_sphere_viz",[40,40949,40937],{"class":133},[40,40951,40940],{"class":244},[40,40953,248],{"class":133},[40,40955,40956,40959,40961,40964],{"class":42,"line":334},[40,40957,40958],{"class":237},"plot_steering_ellipsoid",[40,40960,40937],{"class":133},[40,40962,40963],{"class":244},"\"Steering Ellipsoid for Final State\"",[40,40965,248],{"class":133},[40,40967,40968,40971,40974,40977],{"class":42,"line":347},[40,40969,40970],{"class":237},"visualize_schmidt_decomposition",[40,40972,40973],{"class":133},"(final_statevector, ",[40,40975,40976],{"class":244},"\"Schmidt Decomposition for Final State\"",[40,40978,248],{"class":133},[40,40980,40981,40984],{"class":42,"line":352},[40,40982,40983],{"class":237},"explain_toroidal_geometry",[40,40985,12875],{"class":133},[40,40987,40988,40991,40993,40995],{"class":42,"line":357},[40,40989,40990],{"class":237},"plot_correlation_invariants",[40,40992,40937],{"class":133},[40,40994,40940],{"class":244},[40,40996,248],{"class":133},[40,40998,40999,41002,41004,41007],{"class":42,"line":370},[40,41000,41001],{"class":237},"plot_wigner_function",[40,41003,40937],{"class":133},[40,41005,41006],{"class":244},"\"Wigner Functions for Final State\"",[40,41008,248],{"class":133},[40,41010,41011,41014,41016,41019],{"class":42,"line":376},[40,41012,41013],{"class":237},"plot_density_matrix",[40,41015,40937],{"class":133},[40,41017,41018],{"class":244},"\"Density Matrix for Final State\"",[40,41020,248],{"class":133},[12,41022,41023],{},"Every method below shows the entangled state; where the contrast with |00⟩ is the whole point, both are shown.",[55,41025,41027],{"id":41026},"_1-two-bloch-spheres-one-giveaway","1. Two Bloch spheres, one giveaway",[12,41029,41030],{},"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.",[8229,41032],{"alt":41033,"caption":41034,"no":41035,"src":41036},"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",[12,41038,41039],{},"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.",[8229,41041],{"alt":41042,"caption":41043,"no":41044,"src":41045},"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",[55,41047,41049],{"id":41048},"_2-the-q-sphere","2. The Q-sphere",[12,41051,41052],{},"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.",[8229,41054],{"alt":41055,"caption":41056,"no":41057,"src":41058},"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",[55,41060,41062],{"id":41061},"_3-the-steering-ellipsoid","3. The steering ellipsoid",[12,41064,41065],{},"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.",[8229,41067],{"alt":41068,"caption":41069,"no":41070,"src":41071},"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",[55,41073,41075],{"id":41074},"_4-schmidt-decomposition","4. Schmidt decomposition",[12,41077,41078],{},"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.",[8229,41080],{"alt":41081,"caption":41082,"no":41083,"src":41084},"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",[55,41086,41088],{"id":41087},"_5-toroidal-geometry","5. Toroidal geometry",[12,41090,41091],{},"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.",[55,41093,41095],{"id":41094},"_6-correlation-invariants","6. Correlation invariants",[12,41097,41098],{},"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.",[8229,41100],{"alt":41101,"caption":41102,"no":41103,"src":41104},"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",[55,41106,41108],{"id":41107},"_7-the-wigner-function","7. The Wigner function",[12,41110,41111],{},"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.",[8229,41113],{"alt":41114,"caption":41115,"no":41116,"src":41117},"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",[55,41119,41121],{"id":41120},"_8-the-density-matrix","8. The density matrix",[12,41123,41124],{},"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.",[8229,41126],{"alt":41127,"caption":41128,"no":41129,"src":41130},"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",[8229,41132],{"alt":41133,"caption":41134,"no":41135,"src":41136},"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",[55,41138,40652],{"id":40651},[12,41140,41141,41142,41145,41146,41149],{},"Every figure above comes out of one script you can run top to bottom: ",[19,41143,41144],{"href":40739},"open the notebook on GitHub"," or ",[19,41147,41148],{"href":40744},"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.",[40659,41151,41152],{},"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":35,"searchDepth":50,"depth":50,"links":41154},[41155,41156,41157,41158,41159,41160,41161,41162,41163,41164],{"id":40748,"depth":50,"text":40749},{"id":41026,"depth":50,"text":41027},{"id":41048,"depth":50,"text":41049},{"id":41061,"depth":50,"text":41062},{"id":41074,"depth":50,"text":41075},{"id":41087,"depth":50,"text":41088},{"id":41094,"depth":50,"text":41095},{"id":41107,"depth":50,"text":41108},{"id":41120,"depth":50,"text":41121},{"id":40651,"depth":50,"text":40652},[40692,40693,40694,41166],"2-qubit states",[41168],{"username":40728,"name":41169,"role":41170,"bio":41171,"links":41172},"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.",[41173,41176],{"label":41174,"href":41175},"GitHub ↗","https:\u002F\u002Fgithub.com\u002FOJB-Quantum",{"label":41177,"href":40739},"Original notebook ↗",{"username":40728,"name":41169,"role":41179},"Quantum Hardware Engineer, IBM","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.","Deep dive · Visualization",{},"\u002F_content\u002Fimages\u002F2-qubit-state-visualization\u002Fbloch-entangled.png","\u002Fblog\u002Fexpert-notes\u002F2-qubit-state-visualization","6 min read",[],[41189,41196,41203],{"username":41190,"project":41191,"title":41192,"category":41193,"thumb":41194,"to":41195},"q-inho","qave","QAVE","Visualization","\u002F_content\u002Fimages\u002Fqave\u002Fthumbnail.webp","\u002Fexplore\u002Fqave",{"username":41197,"project":41198,"title":41199,"category":41200,"thumb":41201,"to":41202},"AmberPincar","quantum-garden","Quantum Garden","Art","\u002F_content\u002Fimages\u002Fquantum-garden\u002Fthumbnail.webp","\u002Fexplore\u002Fquantum-garden",{"username":41204,"project":41205,"title":41206,"category":41200,"thumb":41207},"xinyi","quantum-butterfly-field","Quantum Butterfly Field","\u002F_content\u002Fimages\u002Fquantum-butterfly-field\u002Fthumbnail.webp",{"description":41209,"title":40726},"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 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Oscillations: Visualizing Excited-State 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original notebook"," and shared under CC-BY-4.0. The text, code, and figures below are Onri's, as published.",[12,79986,79987],{},"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",[40,79989,79991],{"className":79990},[1827],[40,79992,79994,80025],{"className":79993},[1565],[40,79995,79997],{"className":79996},[1569],[1571,79998,79999],{"xmlns":1573,"display":1836},[1575,80000,80001,80022],{},[1578,80002,80003,80009,80011,80013,80015,80018,80020],{},[2212,80004,80005,80007],{},[1584,80006,2218],{},[1584,80008,2330],{},[1731,80010,241],{"stretchy":1797},[1584,80012,21331],{},[1731,80014,1862],{"separator":1599},[1584,80016,80017],{},"τ",[1731,80019,2693],{"stretchy":1797},[1731,80021,1862],{"separator":1599},[1591,80023,80024],{"encoding":1593},"P_e(f,\\tau),",[40,80026,80028],{"className":80027,"ariaHidden":1599},[1598],[40,80029,80031,80034,80074,80077,80080,80083,80086,80090,80093],{"className":80030},[1603],[40,80032],{"className":80033,"style":1811},[1607],[40,80035,80037,80040],{"className":80036},[1612],[40,80038,2218],{"className":80039,"style":2264},[1612,1653],[40,80041,80043],{"className":80042},[1621],[40,80044,80046,80066],{"className":80045},[1625,2244],[40,80047,80049,80063],{"className":80048},[1629],[40,80050,80052],{"className":80051,"style":3551},[1633],[40,80053,80054,80057],{"style":6681},[40,80055],{"className":80056,"style":1642},[1641],[40,80058,80060],{"className":80059},[1646,1647,1648,1649],[40,80061,2330],{"className":80062},[1612,1653,1649],[40,80064,2269],{"className":80065},[2268],[40,80067,80069],{"className":80068},[1629],[40,80070,80072],{"className":80071,"style":2276},[1633],[40,80073],{},[40,80075,241],{"className":80076},[1985],[40,80078,21331],{"className":80079,"style":26393},[1612,1653],[40,80081,1862],{"className":80082},[2005],[40,80084],{"className":80085,"style":2009},[1756],[40,80087,80017],{"className":80088,"style":80089},[1612,1653],"margin-right:0.1132em;",[40,80091,2693],{"className":80092},[1822],[40,80094,1862],{"className":80095},[2005],[12,80097,27509,80098,80126,80127,80156],{},[40,80099,80101,80114],{"className":80100},[1565],[40,80102,80104],{"className":80103},[1569],[1571,80105,80106],{"xmlns":1573},[1575,80107,80108,80112],{},[1578,80109,80110],{},[1584,80111,21331],{},[1591,80113,21331],{"encoding":1593},[40,80115,80117],{"className":80116,"ariaHidden":1599},[1598],[40,80118,80120,80123],{"className":80119},[1603],[40,80121],{"className":80122,"style":6079},[1607],[40,80124,21331],{"className":80125,"style":26393},[1612,1653]," denotes the applied drive frequency and ",[40,80128,80130,80144],{"className":80129},[1565],[40,80131,80133],{"className":80132},[1569],[1571,80134,80135],{"xmlns":1573},[1575,80136,80137,80141],{},[1578,80138,80139],{},[1584,80140,80017],{},[1591,80142,80143],{"encoding":1593},"\\tau",[40,80145,80147],{"className":80146,"ariaHidden":1599},[1598],[40,80148,80150,80153],{"className":80149},[1603],[40,80151],{"className":80152,"style":1681},[1607],[40,80154,80017],{"className":80155,"style":80089},[1612,1653]," 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.",[55,80158,80160],{"id":80159},"nomenclature-and-definitions","Nomenclature and Definitions",[12,80162,80163],{},"The following table defines the physical and mathematical quantities that recur throughout the visualization workflow.",[80165,80166,80167,80180],"table",{},[80168,80169,80170],"thead",{},[80171,80172,80173,80177],"tr",{},[80174,80175,80176],"th",{},"Symbol or Term",[80174,80178,80179],{},"Meaning",[80181,80182,80183,80230,80276,80322,80449,80526,80604,80639,80860,80957,80994,81040,81191,81268,81493,81528,81621,81629],"tbody",{},[80171,80184,80185,80227],{},[80186,80187,80188],"td",{},[40,80189,80191,80209],{"className":80190},[1565],[40,80192,80194],{"className":80193},[1569],[1571,80195,80196],{"xmlns":1573},[1575,80197,80198,80206],{},[1578,80199,80200,80202,80204],{},[1731,80201,1791],{"stretchy":1797},[1584,80203,36968],{},[1731,80205,1798],{"stretchy":1797},[1591,80207,80208],{"encoding":1593},"\\lvert g\\rangle",[40,80210,80212],{"className":80211,"ariaHidden":1599},[1598],[40,80213,80215,80218,80221,80224],{"className":80214},[1603],[40,80216],{"className":80217,"style":1811},[1607],[40,80219,1791],{"className":80220},[1985],[40,80222,36968],{"className":80223,"style":1818},[1612,1653],[40,80225,1798],{"className":80226},[1822],[80186,80228,80229],{},"Ground state of the qubit",[80171,80231,80232,80273],{},[80186,80233,80234],{},[40,80235,80237,80255],{"className":80236},[1565],[40,80238,80240],{"className":80239},[1569],[1571,80241,80242],{"xmlns":1573},[1575,80243,80244,80252],{},[1578,80245,80246,80248,80250],{},[1731,80247,1791],{"stretchy":1797},[1584,80249,2330],{},[1731,80251,1798],{"stretchy":1797},[1591,80253,80254],{"encoding":1593},"\\lvert e\\rangle",[40,80256,80258],{"className":80257,"ariaHidden":1599},[1598],[40,80259,80261,80264,80267,80270],{"className":80260},[1603],[40,80262],{"className":80263,"style":1811},[1607],[40,80265,1791],{"className":80266},[1985],[40,80268,2330],{"className":80269},[1612,1653],[40,80271,1798],{"className":80272},[1822],[80186,80274,80275],{},"Excited state of the qubit",[80171,80277,80278,80319],{},[80186,80279,80280],{},[40,80281,80283,80301],{"className":80282},[1565],[40,80284,80286],{"className":80285},[1569],[1571,80287,80288],{"xmlns":1573},[1575,80289,80290,80298],{},[1578,80291,80292,80294,80296],{},[1731,80293,1791],{"stretchy":1797},[1584,80295,2837],{},[1731,80297,1798],{"stretchy":1797},[1591,80299,80300],{"encoding":1593},"\\lvert \\psi\\rangle",[40,80302,80304],{"className":80303,"ariaHidden":1599},[1598],[40,80305,80307,80310,80313,80316],{"className":80306},[1603],[40,80308],{"className":80309,"style":1811},[1607],[40,80311,1791],{"className":80312},[1985],[40,80314,2837],{"className":80315,"style":1818},[1612,1653],[40,80317,1798],{"className":80318},[1822],[80186,80320,80321],{},"Quantum state vector in a two-dimensional Hilbert space",[80171,80323,80324,80370],{},[80186,80325,80326],{},[40,80327,80329,80348],{"className":80328},[1565],[40,80330,80332],{"className":80331},[1569],[1571,80333,80334],{"xmlns":1573},[1575,80335,80336,80345],{},[1578,80337,80338,80341,80343],{},[1584,80339,80340],{},"α",[1731,80342,1862],{"separator":1599},[1584,80344,35720],{},[1591,80346,80347],{"encoding":1593},"\\alpha,\\beta",[40,80349,80351],{"className":80350,"ariaHidden":1599},[1598],[40,80352,80354,80357,80361,80364,80367],{"className":80353},[1603],[40,80355],{"className":80356,"style":6079},[1607],[40,80358,80340],{"className":80359,"style":80360},[1612,1653],"margin-right:0.0037em;",[40,80362,1862],{"className":80363},[2005],[40,80365],{"className":80366,"style":2009},[1756],[40,80368,35720],{"className":80369,"style":35815},[1612,1653],[80186,80371,80372,80373,9846,80411],{},"Complex probability amplitudes for ",[40,80374,80376,80393],{"className":80375},[1565],[40,80377,80379],{"className":80378},[1569],[1571,80380,80381],{"xmlns":1573},[1575,80382,80383,80391],{},[1578,80384,80385,80387,80389],{},[1731,80386,1791],{"stretchy":1797},[1584,80388,36968],{},[1731,80390,1798],{"stretchy":1797},[1591,80392,80208],{"encoding":1593},[40,80394,80396],{"className":80395,"ariaHidden":1599},[1598],[40,80397,80399,80402,80405,80408],{"className":80398},[1603],[40,80400],{"className":80401,"style":1811},[1607],[40,80403,1791],{"className":80404},[1985],[40,80406,36968],{"className":80407,"style":1818},[1612,1653],[40,80409,1798],{"className":80410},[1822],[40,80412,80414,80431],{"className":80413},[1565],[40,80415,80417],{"className":80416},[1569],[1571,80418,80419],{"xmlns":1573},[1575,80420,80421,80429],{},[1578,80422,80423,80425,80427],{},[1731,80424,1791],{"stretchy":1797},[1584,80426,2330],{},[1731,80428,1798],{"stretchy":1797},[1591,80430,80254],{"encoding":1593},[40,80432,80434],{"className":80433,"ariaHidden":1599},[1598],[40,80435,80437,80440,80443,80446],{"className":80436},[1603],[40,80438],{"className":80439,"style":1811},[1607],[40,80441,1791],{"className":80442},[1985],[40,80444,2330],{"className":80445},[1612,1653],[40,80447,1798],{"className":80448},[1822],[80171,80450,80451,80523],{},[80186,80452,80453],{},[40,80454,80456,80474],{"className":80455},[1565],[40,80457,80459],{"className":80458},[1569],[1571,80460,80461],{"xmlns":1573},[1575,80462,80463,80471],{},[1578,80464,80465],{},[2212,80466,80467,80469],{},[1584,80468,2218],{},[1584,80470,2330],{},[1591,80472,80473],{"encoding":1593},"P_e",[40,80475,80477],{"className":80476,"ariaHidden":1599},[1598],[40,80478,80480,80483],{"className":80479},[1603],[40,80481],{"className":80482,"style":2231},[1607],[40,80484,80486,80489],{"className":80485},[1612],[40,80487,2218],{"className":80488,"style":2264},[1612,1653],[40,80490,80492],{"className":80491},[1621],[40,80493,80495,80515],{"className":80494},[1625,2244],[40,80496,80498,80512],{"className":80497},[1629],[40,80499,80501],{"className":80500,"style":3551},[1633],[40,80502,80503,80506],{"style":6681},[40,80504],{"className":80505,"style":1642},[1641],[40,80507,80509],{"className":80508},[1646,1647,1648,1649],[40,80510,2330],{"className":80511},[1612,1653,1649],[40,80513,2269],{"className":80514},[2268],[40,80516,80518],{"className":80517},[1629],[40,80519,80521],{"className":80520,"style":2276},[1633],[40,80522],{},[80186,80524,80525],{},"Probability of measuring the qubit in the excited state",[80171,80527,80528,80601],{},[80186,80529,80530],{},[40,80531,80533,80551],{"className":80532},[1565],[40,80534,80536],{"className":80535},[1569],[1571,80537,80538],{"xmlns":1573},[1575,80539,80540,80548],{},[1578,80541,80542],{},[2212,80543,80544,80546],{},[1584,80545,21331],{},[1735,80547,791],{},[1591,80549,80550],{"encoding":1593},"f_0",[40,80552,80554],{"className":80553,"ariaHidden":1599},[1598],[40,80555,80557,80560],{"className":80556},[1603],[40,80558],{"className":80559,"style":6079},[1607],[40,80561,80563,80566],{"className":80562},[1612],[40,80564,21331],{"className":80565,"style":26393},[1612,1653],[40,80567,80569],{"className":80568},[1621],[40,80570,80572,80593],{"className":80571},[1625,2244],[40,80573,80575,80590],{"className":80574},[1629],[40,80576,80578],{"className":80577,"style":13330},[1633],[40,80579,80581,80584],{"style":80580},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[40,80582],{"className":80583,"style":1642},[1641],[40,80585,80587],{"className":80586},[1646,1647,1648,1649],[40,80588,791],{"className":80589},[1612,1649],[40,80591,2269],{"className":80592},[2268],[40,80594,80596],{"className":80595},[1629],[40,80597,80599],{"className":80598,"style":2276},[1633],[40,80600],{},[80186,80602,80603],{},"Natural transition frequency of the qubit",[80171,80605,80606,80636],{},[80186,80607,80608],{},[40,80609,80611,80624],{"className":80610},[1565],[40,80612,80614],{"className":80613},[1569],[1571,80615,80616],{"xmlns":1573},[1575,80617,80618,80622],{},[1578,80619,80620],{},[1584,80621,21331],{},[1591,80623,21331],{"encoding":1593},[40,80625,80627],{"className":80626,"ariaHidden":1599},[1598],[40,80628,80630,80633],{"className":80629},[1603],[40,80631],{"className":80632,"style":6079},[1607],[40,80634,21331],{"className":80635,"style":26393},[1612,1653],[80186,80637,80638],{},"Applied drive frequency",[80171,80640,80641,80714],{},[80186,80642,80643],{},[40,80644,80646,80665],{"className":80645},[1565],[40,80647,80649],{"className":80648},[1569],[1571,80650,80651],{"xmlns":1573},[1575,80652,80653,80662],{},[1578,80654,80655],{},[2212,80656,80657,80660],{},[1584,80658,80659],{},"ω",[1735,80661,791],{},[1591,80663,80664],{"encoding":1593},"\\omega_0",[40,80666,80668],{"className":80667,"ariaHidden":1599},[1598],[40,80669,80671,80674],{"className":80670},[1603],[40,80672],{"className":80673,"style":24251},[1607],[40,80675,80677,80680],{"className":80676},[1612],[40,80678,80659],{"className":80679,"style":1818},[1612,1653],[40,80681,80683],{"className":80682},[1621],[40,80684,80686,80706],{"className":80685},[1625,2244],[40,80687,80689,80703],{"className":80688},[1629],[40,80690,80692],{"className":80691,"style":13330},[1633],[40,80693,80694,80697],{"style":3616},[40,80695],{"className":80696,"style":1642},[1641],[40,80698,80700],{"className":80699},[1646,1647,1648,1649],[40,80701,791],{"className":80702},[1612,1649],[40,80704,2269],{"className":80705},[2268],[40,80707,80709],{"className":80708},[1629],[40,80710,80712],{"className":80711,"style":2276},[1633],[40,80713],{},[80186,80715,80716,80717],{},"Angular transition frequency, ",[40,80718,80720,80750],{"className":80719},[1565],[40,80721,80723],{"className":80722},[1569],[1571,80724,80725],{"xmlns":1573},[1575,80726,80727,80747],{},[1578,80728,80729,80735,80737,80739,80741],{},[2212,80730,80731,80733],{},[1584,80732,80659],{},[1735,80734,791],{},[1731,80736,623],{},[1735,80738,1879],{},[1584,80740,7607],{},[2212,80742,80743,80745],{},[1584,80744,21331],{},[1735,80746,791],{},[1591,80748,80749],{"encoding":1593},"\\omega_0=2\\pi f_0",[40,80751,80753,80808],{"className":80752,"ariaHidden":1599},[1598],[40,80754,80756,80759,80799,80802,80805],{"className":80755},[1603],[40,80757],{"className":80758,"style":24251},[1607],[40,80760,80762,80765],{"className":80761},[1612],[40,80763,80659],{"className":80764,"style":1818},[1612,1653],[40,80766,80768],{"className":80767},[1621],[40,80769,80771,80791],{"className":80770},[1625,2244],[40,80772,80774,80788],{"className":80773},[1629],[40,80775,80777],{"className":80776,"style":13330},[1633],[40,80778,80779,80782],{"style":3616},[40,80780],{"className":80781,"style":1642},[1641],[40,80783,80785],{"className":80784},[1646,1647,1648,1649],[40,80786,791],{"className":80787},[1612,1649],[40,80789,2269],{"className":80790},[2268],[40,80792,80794],{"className":80793},[1629],[40,80795,80797],{"className":80796,"style":2276},[1633],[40,80798],{},[40,80800],{"className":80801,"style":1963},[1756],[40,80803,623],{"className":80804},[1967],[40,80806],{"className":80807,"style":1963},[1756],[40,80809,80811,80814,80817,80820],{"className":80810},[1603],[40,80812],{"className":80813,"style":6079},[1607],[40,80815,1879],{"className":80816},[1612],[40,80818,7607],{"className":80819,"style":1818},[1612,1653],[40,80821,80823,80826],{"className":80822},[1612],[40,80824,21331],{"className":80825,"style":26393},[1612,1653],[40,80827,80829],{"className":80828},[1621],[40,80830,80832,80852],{"className":80831},[1625,2244],[40,80833,80835,80849],{"className":80834},[1629],[40,80836,80838],{"className":80837,"style":13330},[1633],[40,80839,80840,80843],{"style":80580},[40,80841],{"className":80842,"style":1642},[1641],[40,80844,80846],{"className":80845},[1646,1647,1648,1649],[40,80847,791],{"className":80848},[1612,1649],[40,80850,2269],{"className":80851},[2268],[40,80853,80855],{"className":80854},[1629],[40,80856,80858],{"className":80857,"style":2276},[1633],[40,80859],{},[80171,80861,80862,80893],{},[80186,80863,80864],{},[40,80865,80867,80881],{"className":80866},[1565],[40,80868,80870],{"className":80869},[1569],[1571,80871,80872],{"xmlns":1573},[1575,80873,80874,80878],{},[1578,80875,80876],{},[1584,80877,80659],{},[1591,80879,80880],{"encoding":1593},"\\omega",[40,80882,80884],{"className":80883,"ariaHidden":1599},[1598],[40,80885,80887,80890],{"className":80886},[1603],[40,80888],{"className":80889,"style":1681},[1607],[40,80891,80659],{"className":80892,"style":1818},[1612,1653],[80186,80894,80895,80896],{},"Angular drive frequency, ",[40,80897,80899,80921],{"className":80898},[1565],[40,80900,80902],{"className":80901},[1569],[1571,80903,80904],{"xmlns":1573},[1575,80905,80906,80918],{},[1578,80907,80908,80910,80912,80914,80916],{},[1584,80909,80659],{},[1731,80911,623],{},[1735,80913,1879],{},[1584,80915,7607],{},[1584,80917,21331],{},[1591,80919,80920],{"encoding":1593},"\\omega=2\\pi f",[40,80922,80924,80942],{"className":80923,"ariaHidden":1599},[1598],[40,80925,80927,80930,80933,80936,80939],{"className":80926},[1603],[40,80928],{"className":80929,"style":1681},[1607],[40,80931,80659],{"className":80932,"style":1818},[1612,1653],[40,80934],{"className":80935,"style":1963},[1756],[40,80937,623],{"className":80938},[1967],[40,80940],{"className":80941,"style":1963},[1756],[40,80943,80945,80948,80951,80954],{"className":80944},[1603],[40,80946],{"className":80947,"style":6079},[1607],[40,80949,1879],{"className":80950},[1612],[40,80952,7607],{"className":80953,"style":1818},[1612,1653],[40,80955,21331],{"className":80956,"style":26393},[1612,1653],[80171,80958,80959,80991],{},[80186,80960,80961],{},[40,80962,80964,80979],{"className":80963},[1565],[40,80965,80967],{"className":80966},[1569],[1571,80968,80969],{"xmlns":1573},[1575,80970,80971,80976],{},[1578,80972,80973],{},[1584,80974,80975],{"mathvariant":1790},"Ω",[1591,80977,80978],{"encoding":1593},"\\Omega",[40,80980,80982],{"className":80981,"ariaHidden":1599},[1598],[40,80983,80985,80988],{"className":80984},[1603],[40,80986],{"className":80987,"style":1608},[1607],[40,80989,80975],{"className":80990},[1612],[80186,80992,80993],{},"On-resonance Rabi angular frequency",[80171,80995,80996,81037],{},[80186,80997,80998],{},[40,80999,81001,81021],{"className":81000},[1565],[40,81002,81004],{"className":81003},[1569],[1571,81005,81006],{"xmlns":1573},[1575,81007,81008,81018],{},[1578,81009,81010,81012,81014,81016],{},[1584,81011,80975],{"mathvariant":1790},[1584,81013,7610],{"mathvariant":1790},[1735,81015,1879],{},[1584,81017,7607],{},[1591,81019,81020],{"encoding":1593},"\\Omega\u002F2\\pi",[40,81022,81024],{"className":81023,"ariaHidden":1599},[1598],[40,81025,81027,81030,81034],{"className":81026},[1603],[40,81028],{"className":81029,"style":1811},[1607],[40,81031,81033],{"className":81032},[1612],"Ω\u002F2",[40,81035,7607],{"className":81036,"style":1818},[1612,1653],[80186,81038,81039],{},"On-resonance Rabi frequency in ordinary frequency units",[80171,81041,81042,81074],{},[80186,81043,81044],{},[40,81045,81047,81062],{"className":81046},[1565],[40,81048,81050],{"className":81049},[1569],[1571,81051,81052],{"xmlns":1573},[1575,81053,81054,81059],{},[1578,81055,81056],{},[1584,81057,81058],{"mathvariant":1790},"Δ",[1591,81060,81061],{"encoding":1593},"\\Delta",[40,81063,81065],{"className":81064,"ariaHidden":1599},[1598],[40,81066,81068,81071],{"className":81067},[1603],[40,81069],{"className":81070,"style":1608},[1607],[40,81072,81058],{"className":81073},[1612],[80186,81075,81076,81077],{},"Drive detuning, ",[40,81078,81080,81106],{"className":81079},[1565],[40,81081,81083],{"className":81082},[1569],[1571,81084,81085],{"xmlns":1573},[1575,81086,81087,81103],{},[1578,81088,81089,81091,81093,81095,81097],{},[1584,81090,81058],{"mathvariant":1790},[1731,81092,623],{},[1584,81094,80659],{},[1731,81096,1733],{},[2212,81098,81099,81101],{},[1584,81100,80659],{},[1735,81102,791],{},[1591,81104,81105],{"encoding":1593},"\\Delta=\\omega-\\omega_0",[40,81107,81109,81127,81145],{"className":81108,"ariaHidden":1599},[1598],[40,81110,81112,81115,81118,81121,81124],{"className":81111},[1603],[40,81113],{"className":81114,"style":1608},[1607],[40,81116,81058],{"className":81117},[1612],[40,81119],{"className":81120,"style":1963},[1756],[40,81122,623],{"className":81123},[1967],[40,81125],{"className":81126,"style":1963},[1756],[40,81128,81130,81133,81136,81139,81142],{"className":81129},[1603],[40,81131],{"className":81132,"style":1749},[1607],[40,81134,80659],{"className":81135,"style":1818},[1612,1653],[40,81137],{"className":81138,"style":1757},[1756],[40,81140,1733],{"className":81141},[1761],[40,81143],{"className":81144,"style":1757},[1756],[40,81146,81148,81151],{"className":81147},[1603],[40,81149],{"className":81150,"style":24251},[1607],[40,81152,81154,81157],{"className":81153},[1612],[40,81155,80659],{"className":81156,"style":1818},[1612,1653],[40,81158,81160],{"className":81159},[1621],[40,81161,81163,81183],{"className":81162},[1625,2244],[40,81164,81166,81180],{"className":81165},[1629],[40,81167,81169],{"className":81168,"style":13330},[1633],[40,81170,81171,81174],{"style":3616},[40,81172],{"className":81173,"style":1642},[1641],[40,81175,81177],{"className":81176},[1646,1647,1648,1649],[40,81178,791],{"className":81179},[1612,1649],[40,81181,2269],{"className":81182},[2268],[40,81184,81186],{"className":81185},[1629],[40,81187,81189],{"className":81188,"style":2276},[1633],[40,81190],{},[80171,81192,81193,81265],{},[80186,81194,81195],{},[40,81196,81198,81216],{"className":81197},[1565],[40,81199,81201],{"className":81200},[1569],[1571,81202,81203],{"xmlns":1573},[1575,81204,81205,81213],{},[1578,81206,81207],{},[2212,81208,81209,81211],{},[1584,81210,80975],{"mathvariant":1790},[1584,81212,19419],{},[1591,81214,81215],{"encoding":1593},"\\Omega_R",[40,81217,81219],{"className":81218,"ariaHidden":1599},[1598],[40,81220,81222,81225],{"className":81221},[1603],[40,81223],{"className":81224,"style":2231},[1607],[40,81226,81228,81231],{"className":81227},[1612],[40,81229,80975],{"className":81230},[1612],[40,81232,81234],{"className":81233},[1621],[40,81235,81237,81257],{"className":81236},[1625,2244],[40,81238,81240,81254],{"className":81239},[1629],[40,81241,81243],{"className":81242,"style":2251},[1633],[40,81244,81245,81248],{"style":4442},[40,81246],{"className":81247,"style":1642},[1641],[40,81249,81251],{"className":81250},[1646,1647,1648,1649],[40,81252,19419],{"className":81253,"style":19444},[1612,1653,1649],[40,81255,2269],{"className":81256},[2268],[40,81258,81260],{"className":81259},[1629],[40,81261,81263],{"className":81262,"style":2276},[1633],[40,81264],{},[80186,81266,81267],{},"Generalized Rabi angular frequency",[80171,81269,81270,81345],{},[80186,81271,81272],{},[40,81273,81275,81294],{"className":81274},[1565],[40,81276,81278],{"className":81277},[1569],[1571,81279,81280],{"xmlns":1573},[1575,81281,81282,81291],{},[1578,81283,81284],{},[2212,81285,81286,81289],{},[1584,81287,81288],{},"ν",[1584,81290,19419],{},[1591,81292,81293],{"encoding":1593},"\\nu_R",[40,81295,81297],{"className":81296,"ariaHidden":1599},[1598],[40,81298,81300,81303],{"className":81299},[1603],[40,81301],{"className":81302,"style":24251},[1607],[40,81304,81306,81310],{"className":81305},[1612],[40,81307,81288],{"className":81308,"style":81309},[1612,1653],"margin-right:0.0637em;",[40,81311,81313],{"className":81312},[1621],[40,81314,81316,81337],{"className":81315},[1625,2244],[40,81317,81319,81334],{"className":81318},[1629],[40,81320,81322],{"className":81321,"style":2251},[1633],[40,81323,81325,81328],{"style":81324},"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;",[40,81326],{"className":81327,"style":1642},[1641],[40,81329,81331],{"className":81330},[1646,1647,1648,1649],[40,81332,19419],{"className":81333,"style":19444},[1612,1653,1649],[40,81335,2269],{"className":81336},[2268],[40,81338,81340],{"className":81339},[1629],[40,81341,81343],{"className":81342,"style":2276},[1633],[40,81344],{},[80186,81346,81347,81348],{},"Generalized Rabi frequency, ",[40,81349,81351,81383],{"className":81350},[1565],[40,81352,81354],{"className":81353},[1569],[1571,81355,81356],{"xmlns":1573},[1575,81357,81358,81380],{},[1578,81359,81360,81366,81368,81374,81376,81378],{},[2212,81361,81362,81364],{},[1584,81363,81288],{},[1584,81365,19419],{},[1731,81367,623],{},[2212,81369,81370,81372],{},[1584,81371,80975],{"mathvariant":1790},[1584,81373,19419],{},[1584,81375,7610],{"mathvariant":1790},[1735,81377,1879],{},[1584,81379,7607],{},[1591,81381,81382],{"encoding":1593},"\\nu_R=\\Omega_R\u002F2\\pi",[40,81384,81386,81441],{"className":81385,"ariaHidden":1599},[1598],[40,81387,81389,81392,81432,81435,81438],{"className":81388},[1603],[40,81390],{"className":81391,"style":24251},[1607],[40,81393,81395,81398],{"className":81394},[1612],[40,81396,81288],{"className":81397,"style":81309},[1612,1653],[40,81399,81401],{"className":81400},[1621],[40,81402,81404,81424],{"className":81403},[1625,2244],[40,81405,81407,81421],{"className":81406},[1629],[40,81408,81410],{"className":81409,"style":2251},[1633],[40,81411,81412,81415],{"style":81324},[40,81413],{"className":81414,"style":1642},[1641],[40,81416,81418],{"className":81417},[1646,1647,1648,1649],[40,81419,19419],{"className":81420,"style":19444},[1612,1653,1649],[40,81422,2269],{"className":81423},[2268],[40,81425,81427],{"className":81426},[1629],[40,81428,81430],{"className":81429,"style":2276},[1633],[40,81431],{},[40,81433],{"className":81434,"style":1963},[1756],[40,81436,623],{"className":81437},[1967],[40,81439],{"className":81440,"style":1963},[1756],[40,81442,81444,81447,81487,81490],{"className":81443},[1603],[40,81445],{"className":81446,"style":1811},[1607],[40,81448,81450,81453],{"className":81449},[1612],[40,81451,80975],{"className":81452},[1612],[40,81454,81456],{"className":81455},[1621],[40,81457,81459,81479],{"className":81458},[1625,2244],[40,81460,81462,81476],{"className":81461},[1629],[40,81463,81465],{"className":81464,"style":2251},[1633],[40,81466,81467,81470],{"style":4442},[40,81468],{"className":81469,"style":1642},[1641],[40,81471,81473],{"className":81472},[1646,1647,1648,1649],[40,81474,19419],{"className":81475,"style":19444},[1612,1653,1649],[40,81477,2269],{"className":81478},[2268],[40,81480,81482],{"className":81481},[1629],[40,81483,81485],{"className":81484,"style":2276},[1633],[40,81486],{},[40,81488,25175],{"className":81489},[1612],[40,81491,7607],{"className":81492,"style":1818},[1612,1653],[80171,81494,81495,81525],{},[80186,81496,81497],{},[40,81498,81500,81513],{"className":81499},[1565],[40,81501,81503],{"className":81502},[1569],[1571,81504,81505],{"xmlns":1573},[1575,81506,81507,81511],{},[1578,81508,81509],{},[1584,81510,80017],{},[1591,81512,80143],{"encoding":1593},[40,81514,81516],{"className":81515,"ariaHidden":1599},[1598],[40,81517,81519,81522],{"className":81518},[1603],[40,81520],{"className":81521,"style":1681},[1607],[40,81523,80017],{"className":81524,"style":80089},[1612,1653],[80186,81526,81527],{},"Pulse duration",[80171,81529,81530,81618],{},[80186,81531,81532],{},[40,81533,81535,81555],{"className":81534},[1565],[40,81536,81538],{"className":81537},[1569],[1571,81539,81540],{"xmlns":1573},[1575,81541,81542,81552],{},[1578,81543,81544],{},[2819,81545,81546,81548,81550],{},[1584,81547,36101],{},[1735,81549,1879],{},[1731,81551,19819],{},[1591,81553,81554],{"encoding":1593},"T_2^\\ast",[40,81556,81558],{"className":81557,"ariaHidden":1599},[1598],[40,81559,81561,81565],{"className":81560},[1603],[40,81562],{"className":81563,"style":81564},[1607],"height:0.9368em;vertical-align:-0.2481em;",[40,81566,81568,81571],{"className":81567},[1612],[40,81569,36101],{"className":81570,"style":2264},[1612,1653],[40,81572,81574],{"className":81573},[1621],[40,81575,81577,81609],{"className":81576},[1625,2244],[40,81578,81580,81606],{"className":81579},[1629],[40,81581,81583,81595],{"className":81582,"style":35637},[1633],[40,81584,81586,81589],{"style":81585},"top:-2.4519em;margin-left:-0.1389em;margin-right:0.05em;",[40,81587],{"className":81588,"style":1642},[1641],[40,81590,81592],{"className":81591},[1646,1647,1648,1649],[40,81593,1879],{"className":81594},[1612,1649],[40,81596,81597,81600],{"style":1637},[40,81598],{"className":81599,"style":1642},[1641],[40,81601,81603],{"className":81602},[1646,1647,1648,1649],[40,81604,19819],{"className":81605},[1761,1649],[40,81607,2269],{"className":81608},[2268],[40,81610,81612],{"className":81611},[1629],[40,81613,81616],{"className":81614,"style":81615},[1633],"height:0.2481em;",[40,81617],{},[80186,81619,81620],{},"Phenomenological dephasing time used as an optional damping parameter",[80171,81622,81623,81626],{},[80186,81624,81625],{},"RWA",[80186,81627,81628],{},"Rotating-wave approximation",[80171,81630,81631,81634],{},[80186,81632,81633],{},"FFT",[80186,81635,81636],{},"Fast Fourier transform",[55,81638,81640],{"id":81639},"physical-model","Physical Model",[12,81642,81643,81644,81682,81683,81721],{},"A qubit can be modeled as a two-level quantum system with a ground state ",[40,81645,81647,81664],{"className":81646},[1565],[40,81648,81650],{"className":81649},[1569],[1571,81651,81652],{"xmlns":1573},[1575,81653,81654,81662],{},[1578,81655,81656,81658,81660],{},[1731,81657,1791],{"stretchy":1797},[1584,81659,36968],{},[1731,81661,1798],{"stretchy":1797},[1591,81663,80208],{"encoding":1593},[40,81665,81667],{"className":81666,"ariaHidden":1599},[1598],[40,81668,81670,81673,81676,81679],{"className":81669},[1603],[40,81671],{"className":81672,"style":1811},[1607],[40,81674,1791],{"className":81675},[1985],[40,81677,36968],{"className":81678,"style":1818},[1612,1653],[40,81680,1798],{"className":81681},[1822]," and an excited state ",[40,81684,81686,81703],{"className":81685},[1565],[40,81687,81689],{"className":81688},[1569],[1571,81690,81691],{"xmlns":1573},[1575,81692,81693,81701],{},[1578,81694,81695,81697,81699],{},[1731,81696,1791],{"stretchy":1797},[1584,81698,2330],{},[1731,81700,1798],{"stretchy":1797},[1591,81702,80254],{"encoding":1593},[40,81704,81706],{"className":81705,"ariaHidden":1599},[1598],[40,81707,81709,81712,81715,81718],{"className":81708},[1603],[40,81710],{"className":81711,"style":1811},[1607],[40,81713,1791],{"className":81714},[1985],[40,81716,2330],{"className":81717},[1612,1653],[40,81719,1798],{"className":81720},[1822],". In the absence of a microwave drive, the energy splitting is",[40,81723,81725],{"className":81724},[1827],[40,81726,81728,81767],{"className":81727},[1565],[40,81729,81731],{"className":81730},[1569],[1571,81732,81733],{"xmlns":1573,"display":1836},[1575,81734,81735,81764],{},[1578,81736,81737,81743,81745,81751,81753,81756,81762],{},[2212,81738,81739,81741],{},[1584,81740,5226],{},[1584,81742,2330],{},[1731,81744,1733],{},[2212,81746,81747,81749],{},[1584,81748,5226],{},[1584,81750,36968],{},[1731,81752,623],{},[1584,81754,81755],{"mathvariant":1790},"ℏ",[2212,81757,81758,81760],{},[1584,81759,80659],{},[1735,81761,791],{},[1731,81763,1862],{"separator":1599},[1591,81765,81766],{"encoding":1593},"E_e-E_g=\\hbar\\omega_0,",[40,81768,81770,81825,81880],{"className":81769,"ariaHidden":1599},[1598],[40,81771,81773,81776,81816,81819,81822],{"className":81772},[1603],[40,81774],{"className":81775,"style":2231},[1607],[40,81777,81779,81782],{"className":81778},[1612],[40,81780,5226],{"className":81781,"style":4936},[1612,1653],[40,81783,81785],{"className":81784},[1621],[40,81786,81788,81808],{"className":81787},[1625,2244],[40,81789,81791,81805],{"className":81790},[1629],[40,81792,81794],{"className":81793,"style":3551},[1633],[40,81795,81796,81799],{"style":5058},[40,81797],{"className":81798,"style":1642},[1641],[40,81800,81802],{"className":81801},[1646,1647,1648,1649],[40,81803,2330],{"className":81804},[1612,1653,1649],[40,81806,2269],{"className":81807},[2268],[40,81809,81811],{"className":81810},[1629],[40,81812,81814],{"className":81813,"style":2276},[1633],[40,81815],{},[40,81817],{"className":81818,"style":1757},[1756],[40,81820,1733],{"className":81821},[1761],[40,81823],{"className":81824,"style":1757},[1756],[40,81826,81828,81831,81871,81874,81877],{"className":81827},[1603],[40,81829],{"className":81830,"style":7011},[1607],[40,81832,81834,81837],{"className":81833},[1612],[40,81835,5226],{"className":81836,"style":4936},[1612,1653],[40,81838,81840],{"className":81839},[1621],[40,81841,81843,81863],{"className":81842},[1625,2244],[40,81844,81846,81860],{"className":81845},[1629],[40,81847,81849],{"className":81848,"style":3551},[1633],[40,81850,81851,81854],{"style":5058},[40,81852],{"className":81853,"style":1642},[1641],[40,81855,81857],{"className":81856},[1646,1647,1648,1649],[40,81858,36968],{"className":81859,"style":1818},[1612,1653,1649],[40,81861,2269],{"className":81862},[2268],[40,81864,81866],{"className":81865},[1629],[40,81867,81869],{"className":81868,"style":5877},[1633],[40,81870],{},[40,81872],{"className":81873,"style":1963},[1756],[40,81875,623],{"className":81876},[1967],[40,81878],{"className":81879,"style":1963},[1756],[40,81881,81883,81887,81890,81930],{"className":81882},[1603],[40,81884],{"className":81885,"style":81886},[1607],"height:0.8833em;vertical-align:-0.1944em;",[40,81888,81755],{"className":81889},[1612],[40,81891,81893,81896],{"className":81892},[1612],[40,81894,80659],{"className":81895,"style":1818},[1612,1653],[40,81897,81899],{"className":81898},[1621],[40,81900,81902,81922],{"className":81901},[1625,2244],[40,81903,81905,81919],{"className":81904},[1629],[40,81906,81908],{"className":81907,"style":13330},[1633],[40,81909,81910,81913],{"style":3616},[40,81911],{"className":81912,"style":1642},[1641],[40,81914,81916],{"className":81915},[1646,1647,1648,1649],[40,81917,791],{"className":81918},[1612,1649],[40,81920,2269],{"className":81921},[2268],[40,81923,81925],{"className":81924},[1629],[40,81926,81928],{"className":81927,"style":2276},[1633],[40,81929],{},[40,81931,1862],{"className":81932},[2005],[12,81934,81935],{},"where the angular transition frequency is",[40,81937,81939],{"className":81938},[1827],[40,81940,81942,81974],{"className":81941},[1565],[40,81943,81945],{"className":81944},[1569],[1571,81946,81947],{"xmlns":1573,"display":1836},[1575,81948,81949,81971],{},[1578,81950,81951,81957,81959,81961,81963,81969],{},[2212,81952,81953,81955],{},[1584,81954,80659],{},[1735,81956,791],{},[1731,81958,623],{},[1735,81960,1879],{},[1584,81962,7607],{},[2212,81964,81965,81967],{},[1584,81966,21331],{},[1735,81968,791],{},[1584,81970,1823],{"mathvariant":1790},[1591,81972,81973],{"encoding":1593},"\\omega_0=2\\pi f_0.",[40,81975,81977,82032],{"className":81976,"ariaHidden":1599},[1598],[40,81978,81980,81983,82023,82026,82029],{"className":81979},[1603],[40,81981],{"className":81982,"style":24251},[1607],[40,81984,81986,81989],{"className":81985},[1612],[40,81987,80659],{"className":81988,"style":1818},[1612,1653],[40,81990,81992],{"className":81991},[1621],[40,81993,81995,82015],{"className":81994},[1625,2244],[40,81996,81998,82012],{"className":81997},[1629],[40,81999,82001],{"className":82000,"style":13330},[1633],[40,82002,82003,82006],{"style":3616},[40,82004],{"className":82005,"style":1642},[1641],[40,82007,82009],{"className":82008},[1646,1647,1648,1649],[40,82010,791],{"className":82011},[1612,1649],[40,82013,2269],{"className":82014},[2268],[40,82016,82018],{"className":82017},[1629],[40,82019,82021],{"className":82020,"style":2276},[1633],[40,82022],{},[40,82024],{"className":82025,"style":1963},[1756],[40,82027,623],{"className":82028},[1967],[40,82030],{"className":82031,"style":1963},[1756],[40,82033,82035,82038,82041,82044,82084],{"className":82034},[1603],[40,82036],{"className":82037,"style":6079},[1607],[40,82039,1879],{"className":82040},[1612],[40,82042,7607],{"className":82043,"style":1818},[1612,1653],[40,82045,82047,82050],{"className":82046},[1612],[40,82048,21331],{"className":82049,"style":26393},[1612,1653],[40,82051,82053],{"className":82052},[1621],[40,82054,82056,82076],{"className":82055},[1625,2244],[40,82057,82059,82073],{"className":82058},[1629],[40,82060,82062],{"className":82061,"style":13330},[1633],[40,82063,82064,82067],{"style":80580},[40,82065],{"className":82066,"style":1642},[1641],[40,82068,82070],{"className":82069},[1646,1647,1648,1649],[40,82071,791],{"className":82072},[1612,1649],[40,82074,2269],{"className":82075},[2268],[40,82077,82079],{"className":82078},[1629],[40,82080,82082],{"className":82081,"style":2276},[1633],[40,82083],{},[40,82085,1823],{"className":82086},[1612],[12,82088,82089],{},"The default configuration uses a transition frequency of",[40,82091,82093],{"className":82092},[1827],[40,82094,82096,82132],{"className":82095},[1565],[40,82097,82099],{"className":82098},[1569],[1571,82100,82101],{"xmlns":1573,"display":1836},[1575,82102,82103,82129],{},[1578,82104,82105,82111,82113,82116,82118,82127],{},[2212,82106,82107,82109],{},[1584,82108,21331],{},[1735,82110,791],{},[1731,82112,623],{},[1735,82114,82115],{},"5.000",[5211,82117,5213],{},[1578,82119,82120,82123,82125],{},[1584,82121,82122],{"mathvariant":1790},"G",[1584,82124,2182],{"mathvariant":1790},[1584,82126,19585],{"mathvariant":1790},[1584,82128,1823],{"mathvariant":1790},[1591,82130,82131],{"encoding":1593},"f_0=5.000~\\mathrm{GHz}.",[40,82133,82135,82190],{"className":82134,"ariaHidden":1599},[1598],[40,82136,82138,82141,82181,82184,82187],{"className":82137},[1603],[40,82139],{"className":82140,"style":6079},[1607],[40,82142,82144,82147],{"className":82143},[1612],[40,82145,21331],{"className":82146,"style":26393},[1612,1653],[40,82148,82150],{"className":82149},[1621],[40,82151,82153,82173],{"className":82152},[1625,2244],[40,82154,82156,82170],{"className":82155},[1629],[40,82157,82159],{"className":82158,"style":13330},[1633],[40,82160,82161,82164],{"style":80580},[40,82162],{"className":82163,"style":1642},[1641],[40,82165,82167],{"className":82166},[1646,1647,1648,1649],[40,82168,791],{"className":82169},[1612,1649],[40,82171,2269],{"className":82172},[2268],[40,82174,82176],{"className":82175},[1629],[40,82177,82179],{"className":82178,"style":2276},[1633],[40,82180],{},[40,82182],{"className":82183,"style":1963},[1756],[40,82185,623],{"className":82186},[1967],[40,82188],{"className":82189,"style":1963},[1756],[40,82191,82193,82196,82199,82203,82211],{"className":82192},[1603],[40,82194],{"className":82195,"style":1608},[1607],[40,82197,82115],{"className":82198},[1612],[40,82200,5213],{"className":82201},[1756,82202],"nobreak",[40,82204,82206],{"className":82205},[1612],[40,82207,82210],{"className":82208},[1612,82209],"mathrm","GHz",[40,82212,1823],{"className":82213},[1612],[12,82215,82216,82217,82245,82246,9846,82284,82322],{},"Applying a microwave drive with frequency ",[40,82218,82220,82233],{"className":82219},[1565],[40,82221,82223],{"className":82222},[1569],[1571,82224,82225],{"xmlns":1573},[1575,82226,82227,82231],{},[1578,82228,82229],{},[1584,82230,21331],{},[1591,82232,21331],{"encoding":1593},[40,82234,82236],{"className":82235,"ariaHidden":1599},[1598],[40,82237,82239,82242],{"className":82238},[1603],[40,82240],{"className":82241,"style":6079},[1607],[40,82243,21331],{"className":82244,"style":26393},[1612,1653]," induces coherent rotations between ",[40,82247,82249,82266],{"className":82248},[1565],[40,82250,82252],{"className":82251},[1569],[1571,82253,82254],{"xmlns":1573},[1575,82255,82256,82264],{},[1578,82257,82258,82260,82262],{},[1731,82259,1791],{"stretchy":1797},[1584,82261,36968],{},[1731,82263,1798],{"stretchy":1797},[1591,82265,80208],{"encoding":1593},[40,82267,82269],{"className":82268,"ariaHidden":1599},[1598],[40,82270,82272,82275,82278,82281],{"className":82271},[1603],[40,82273],{"className":82274,"style":1811},[1607],[40,82276,1791],{"className":82277},[1985],[40,82279,36968],{"className":82280,"style":1818},[1612,1653],[40,82282,1798],{"className":82283},[1822],[40,82285,82287,82304],{"className":82286},[1565],[40,82288,82290],{"className":82289},[1569],[1571,82291,82292],{"xmlns":1573},[1575,82293,82294,82302],{},[1578,82295,82296,82298,82300],{},[1731,82297,1791],{"stretchy":1797},[1584,82299,2330],{},[1731,82301,1798],{"stretchy":1797},[1591,82303,80254],{"encoding":1593},[40,82305,82307],{"className":82306,"ariaHidden":1599},[1598],[40,82308,82310,82313,82316,82319],{"className":82309},[1603],[40,82311],{"className":82312,"style":1811},[1607],[40,82314,1791],{"className":82315},[1985],[40,82317,2330],{"className":82318},[1612,1653],[40,82320,1798],{"className":82321},[1822],". Exact resonance occurs when",[40,82324,82326],{"className":82325},[1827],[40,82327,82329,82353],{"className":82328},[1565],[40,82330,82332],{"className":82331},[1569],[1571,82333,82334],{"xmlns":1573,"display":1836},[1575,82335,82336,82350],{},[1578,82337,82338,82340,82342,82348],{},[1584,82339,21331],{},[1731,82341,623],{},[2212,82343,82344,82346],{},[1584,82345,21331],{},[1735,82347,791],{},[1584,82349,1823],{"mathvariant":1790},[1591,82351,82352],{"encoding":1593},"f=f_0.",[40,82354,82356,82374],{"className":82355,"ariaHidden":1599},[1598],[40,82357,82359,82362,82365,82368,82371],{"className":82358},[1603],[40,82360],{"className":82361,"style":6079},[1607],[40,82363,21331],{"className":82364,"style":26393},[1612,1653],[40,82366],{"className":82367,"style":1963},[1756],[40,82369,623],{"className":82370},[1967],[40,82372],{"className":82373,"style":1963},[1756],[40,82375,82377,82380,82420],{"className":82376},[1603],[40,82378],{"className":82379,"style":6079},[1607],[40,82381,82383,82386],{"className":82382},[1612],[40,82384,21331],{"className":82385,"style":26393},[1612,1653],[40,82387,82389],{"className":82388},[1621],[40,82390,82392,82412],{"className":82391},[1625,2244],[40,82393,82395,82409],{"className":82394},[1629],[40,82396,82398],{"className":82397,"style":13330},[1633],[40,82399,82400,82403],{"style":80580},[40,82401],{"className":82402,"style":1642},[1641],[40,82404,82406],{"className":82405},[1646,1647,1648,1649],[40,82407,791],{"className":82408},[1612,1649],[40,82410,2269],{"className":82411},[2268],[40,82413,82415],{"className":82414},[1629],[40,82416,82418],{"className":82417,"style":2276},[1633],[40,82419],{},[40,82421,1823],{"className":82422},[1612],[12,82424,82425],{},"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.",[55,82427,82429],{"id":82428},"mathematical-foundation","Mathematical Foundation",[12,82431,82432],{},"The quantum state is a normalized vector in a two-dimensional Hilbert space,",[40,82434,82436],{"className":82435},[1827],[40,82437,82439,82479],{"className":82438},[1565],[40,82440,82442],{"className":82441},[1569],[1571,82443,82444],{"xmlns":1573,"display":1836},[1575,82445,82446,82476],{},[1578,82447,82448,82450,82452,82454,82456,82458,82460,82462,82464,82466,82468,82470,82472,82474],{},[1731,82449,1791],{"stretchy":1797},[1584,82451,2837],{},[1731,82453,1798],{"stretchy":1797},[1731,82455,623],{},[1584,82457,80340],{},[1731,82459,1791],{"stretchy":1797},[1584,82461,36968],{},[1731,82463,1798],{"stretchy":1797},[1731,82465,1147],{},[1584,82467,35720],{},[1731,82469,1791],{"stretchy":1797},[1584,82471,2330],{},[1731,82473,1798],{"stretchy":1797},[1731,82475,1862],{"separator":1599},[1591,82477,82478],{"encoding":1593},"\\lvert \\psi\\rangle=\\alpha\\lvert g\\rangle+\\beta\\lvert e\\rangle,",[40,82480,82482,82506,82533],{"className":82481,"ariaHidden":1599},[1598],[40,82483,82485,82488,82491,82494,82497,82500,82503],{"className":82484},[1603],[40,82486],{"className":82487,"style":1811},[1607],[40,82489,1791],{"className":82490},[1985],[40,82492,2837],{"className":82493,"style":1818},[1612,1653],[40,82495,1798],{"className":82496},[1822],[40,82498],{"className":82499,"style":1963},[1756],[40,82501,623],{"className":82502},[1967],[40,82504],{"className":82505,"style":1963},[1756],[40,82507,82509,82512,82515,82518,82521,82524,82527,82530],{"className":82508},[1603],[40,82510],{"className":82511,"style":1811},[1607],[40,82513,80340],{"className":82514,"style":80360},[1612,1653],[40,82516,1791],{"className":82517},[1985],[40,82519,36968],{"className":82520,"style":1818},[1612,1653],[40,82522,1798],{"className":82523},[1822],[40,82525],{"className":82526,"style":1757},[1756],[40,82528,1147],{"className":82529},[1761],[40,82531],{"className":82532,"style":1757},[1756],[40,82534,82536,82539,82542,82545,82548,82551],{"className":82535},[1603],[40,82537],{"className":82538,"style":1811},[1607],[40,82540,35720],{"className":82541,"style":35815},[1612,1653],[40,82543,1791],{"className":82544},[1985],[40,82546,2330],{"className":82547},[1612,1653],[40,82549,1798],{"className":82550},[1822],[40,82552,1862],{"className":82553},[2005],[12,82555,27509,82556,9846,82585,82614],{},[40,82557,82559,82573],{"className":82558},[1565],[40,82560,82562],{"className":82561},[1569],[1571,82563,82564],{"xmlns":1573},[1575,82565,82566,82570],{},[1578,82567,82568],{},[1584,82569,80340],{},[1591,82571,82572],{"encoding":1593},"\\alpha",[40,82574,82576],{"className":82575,"ariaHidden":1599},[1598],[40,82577,82579,82582],{"className":82578},[1603],[40,82580],{"className":82581,"style":1681},[1607],[40,82583,80340],{"className":82584,"style":80360},[1612,1653],[40,82586,82588,82602],{"className":82587},[1565],[40,82589,82591],{"className":82590},[1569],[1571,82592,82593],{"xmlns":1573},[1575,82594,82595,82599],{},[1578,82596,82597],{},[1584,82598,35720],{},[1591,82600,82601],{"encoding":1593},"\\beta",[40,82603,82605],{"className":82604,"ariaHidden":1599},[1598],[40,82606,82608,82611],{"className":82607},[1603],[40,82609],{"className":82610,"style":6079},[1607],[40,82612,35720],{"className":82613,"style":35815},[1612,1653]," are complex probability amplitudes. Normalization requires",[40,82616,82618],{"className":82617},[1827],[40,82619,82621,82660],{"className":82620},[1565],[40,82622,82624],{"className":82623},[1569],[1571,82625,82626],{"xmlns":1573,"display":1836},[1575,82627,82628,82657],{},[1578,82629,82630,82632,82634,82640,82642,82644,82646,82652,82654],{},[1584,82631,1791],{"mathvariant":1790},[1584,82633,80340],{},[1581,82635,82636,82638],{},[1584,82637,1791],{"mathvariant":1790},[1735,82639,1879],{},[1731,82641,1147],{},[1584,82643,1791],{"mathvariant":1790},[1584,82645,35720],{},[1581,82647,82648,82650],{},[1584,82649,1791],{"mathvariant":1790},[1735,82651,1879],{},[1731,82653,623],{},[1735,82655,82656],{},"1.",[1591,82658,82659],{"encoding":1593},"|\\alpha|^2+|\\beta|^2=1.",[40,82661,82663,82714,82764],{"className":82662,"ariaHidden":1599},[1598],[40,82664,82666,82670,82673,82676,82705,82708,82711],{"className":82665},[1603],[40,82667],{"className":82668,"style":82669},[1607],"height:1.1141em;vertical-align:-0.25em;",[40,82671,1791],{"className":82672},[1612],[40,82674,80340],{"className":82675,"style":80360},[1612,1653],[40,82677,82679,82682],{"className":82678},[1612],[40,82680,1791],{"className":82681},[1612],[40,82683,82685],{"className":82684},[1621],[40,82686,82688],{"className":82687},[1625],[40,82689,82691],{"className":82690},[1629],[40,82692,82694],{"className":82693,"style":2046},[1633],[40,82695,82696,82699],{"style":1950},[40,82697],{"className":82698,"style":1642},[1641],[40,82700,82702],{"className":82701},[1646,1647,1648,1649],[40,82703,1879],{"className":82704},[1612,1649],[40,82706],{"className":82707,"style":1757},[1756],[40,82709,1147],{"className":82710},[1761],[40,82712],{"className":82713,"style":1757},[1756],[40,82715,82717,82720,82723,82726,82755,82758,82761],{"className":82716},[1603],[40,82718],{"className":82719,"style":82669},[1607],[40,82721,1791],{"className":82722},[1612],[40,82724,35720],{"className":82725,"style":35815},[1612,1653],[40,82727,82729,82732],{"className":82728},[1612],[40,82730,1791],{"className":82731},[1612],[40,82733,82735],{"className":82734},[1621],[40,82736,82738],{"className":82737},[1625],[40,82739,82741],{"className":82740},[1629],[40,82742,82744],{"className":82743,"style":2046},[1633],[40,82745,82746,82749],{"style":1950},[40,82747],{"className":82748,"style":1642},[1641],[40,82750,82752],{"className":82751},[1646,1647,1648,1649],[40,82753,1879],{"className":82754},[1612,1649],[40,82756],{"className":82757,"style":1963},[1756],[40,82759,623],{"className":82760},[1967],[40,82762],{"className":82763,"style":1963},[1756],[40,82765,82767,82770],{"className":82766},[1603],[40,82768],{"className":82769,"style":1771},[1607],[40,82771,82656],{"className":82772},[1612],[12,82774,82775],{},"The Born rule gives the probability of measuring the excited state,",[40,82777,82779],{"className":82778},[1827],[40,82780,82782,82834],{"className":82781},[1565],[40,82783,82785],{"className":82784},[1569],[1571,82786,82787],{"xmlns":1573,"display":1836},[1575,82788,82789,82831],{},[1578,82790,82791,82797,82799,82801,82803,82805,82807,82809,82811,82817,82819,82821,82823,82829],{},[2212,82792,82793,82795],{},[1584,82794,2218],{},[1584,82796,2330],{},[1731,82798,623],{},[1584,82800,1791],{"mathvariant":1790},[1731,82802,3469],{"stretchy":1797},[1584,82804,2330],{},[1731,82806,1791],{"stretchy":1797},[1584,82808,2837],{},[1731,82810,1798],{"stretchy":1797},[1581,82812,82813,82815],{},[1584,82814,1791],{"mathvariant":1790},[1735,82816,1879],{},[1731,82818,623],{},[1584,82820,1791],{"mathvariant":1790},[1584,82822,35720],{},[1581,82824,82825,82827],{},[1584,82826,1791],{"mathvariant":1790},[1735,82828,1879],{},[1584,82830,1823],{"mathvariant":1790},[1591,82832,82833],{"encoding":1593},"P_e=|\\langle e\\lvert\\psi\\rangle|^2=|\\beta|^2.",[40,82835,82837,82892,82954],{"className":82836,"ariaHidden":1599},[1598],[40,82838,82840,82843,82883,82886,82889],{"className":82839},[1603],[40,82841],{"className":82842,"style":2231},[1607],[40,82844,82846,82849],{"className":82845},[1612],[40,82847,2218],{"className":82848,"style":2264},[1612,1653],[40,82850,82852],{"className":82851},[1621],[40,82853,82855,82875],{"className":82854},[1625,2244],[40,82856,82858,82872],{"className":82857},[1629],[40,82859,82861],{"className":82860,"style":3551},[1633],[40,82862,82863,82866],{"style":6681},[40,82864],{"className":82865,"style":1642},[1641],[40,82867,82869],{"className":82868},[1646,1647,1648,1649],[40,82870,2330],{"className":82871},[1612,1653,1649],[40,82873,2269],{"className":82874},[2268],[40,82876,82878],{"className":82877},[1629],[40,82879,82881],{"className":82880,"style":2276},[1633],[40,82882],{},[40,82884],{"className":82885,"style":1963},[1756],[40,82887,623],{"className":82888},[1967],[40,82890],{"className":82891,"style":1963},[1756],[40,82893,82895,82898,82901,82904,82907,82910,82913,82916,82945,82948,82951],{"className":82894},[1603],[40,82896],{"className":82897,"style":82669},[1607],[40,82899,1791],{"className":82900},[1612],[40,82902,3469],{"className":82903},[1985],[40,82905,2330],{"className":82906},[1612,1653],[40,82908,1791],{"className":82909},[1985],[40,82911,2837],{"className":82912,"style":1818},[1612,1653],[40,82914,1798],{"className":82915},[1822],[40,82917,82919,82922],{"className":82918},[1612],[40,82920,1791],{"className":82921},[1612],[40,82923,82925],{"className":82924},[1621],[40,82926,82928],{"className":82927},[1625],[40,82929,82931],{"className":82930},[1629],[40,82932,82934],{"className":82933,"style":2046},[1633],[40,82935,82936,82939],{"style":1950},[40,82937],{"className":82938,"style":1642},[1641],[40,82940,82942],{"className":82941},[1646,1647,1648,1649],[40,82943,1879],{"className":82944},[1612,1649],[40,82946],{"className":82947,"style":1963},[1756],[40,82949,623],{"className":82950},[1967],[40,82952],{"className":82953,"style":1963},[1756],[40,82955,82957,82960,82963,82966,82995],{"className":82956},[1603],[40,82958],{"className":82959,"style":82669},[1607],[40,82961,1791],{"className":82962},[1612],[40,82964,35720],{"className":82965,"style":35815},[1612,1653],[40,82967,82969,82972],{"className":82968},[1612],[40,82970,1791],{"className":82971},[1612],[40,82973,82975],{"className":82974},[1621],[40,82976,82978],{"className":82977},[1625],[40,82979,82981],{"className":82980},[1629],[40,82982,82984],{"className":82983,"style":2046},[1633],[40,82985,82986,82989],{"style":1950},[40,82987],{"className":82988,"style":1642},[1641],[40,82990,82992],{"className":82991},[1646,1647,1648,1649],[40,82993,1879],{"className":82994},[1612,1649],[40,82996,1823],{"className":82997},[1612],[12,82999,83000],{},"Pauli matrices then provide a natural operator basis for the two-level system,",[40,83002,83004],{"className":83003},[1827],[40,83005,83007,83171],{"className":83006},[1565],[40,83008,83010],{"className":83009},[1569],[1571,83011,83012],{"xmlns":1573,"display":1836},[1575,83013,83014,83168],{},[1578,83015,83016,83023,83025,83062,83064,83066,83072,83074,83114,83116,83118,83124,83126,83166],{},[2212,83017,83018,83021],{},[1584,83019,83020],{},"σ",[1584,83022,8185],{},[1731,83024,623],{},[1578,83026,83027,83029,83060],{},[1731,83028,11295],{"fence":1599},[8499,83030,83032,83046],{"rowspacing":8501,"columnalign":83031,"columnspacing":8502},"center center",[8504,83033,83034,83040],{},[8507,83035,83036],{},[8510,83037,83038],{"scriptlevel":791,"displaystyle":1797},[1735,83039,791],{},[8507,83041,83042],{},[8510,83043,83044],{"scriptlevel":791,"displaystyle":1797},[1735,83045,1410],{},[8504,83047,83048,83054],{},[8507,83049,83050],{},[8510,83051,83052],{"scriptlevel":791,"displaystyle":1797},[1735,83053,1410],{},[8507,83055,83056],{},[8510,83057,83058],{"scriptlevel":791,"displaystyle":1797},[1735,83059,791],{},[1731,83061,11320],{"fence":1599},[1731,83063,1862],{"separator":1599},[1756,83065],{"width":17060},[2212,83067,83068,83070],{},[1584,83069,83020],{},[1584,83071,19504],{},[1731,83073,623],{},[1578,83075,83076,83078,83112],{},[1731,83077,11295],{"fence":1599},[8499,83079,83080,83098],{"rowspacing":8501,"columnalign":83031,"columnspacing":8502},[8504,83081,83082,83088],{},[8507,83083,83084],{},[8510,83085,83086],{"scriptlevel":791,"displaystyle":1797},[1735,83087,791],{},[8507,83089,83090],{},[8510,83091,83092],{"scriptlevel":791,"displaystyle":1797},[1578,83093,83094,83096],{},[1731,83095,1733],{},[1584,83097,2337],{},[8504,83099,83100,83106],{},[8507,83101,83102],{},[8510,83103,83104],{"scriptlevel":791,"displaystyle":1797},[1584,83105,2337],{},[8507,83107,83108],{},[8510,83109,83110],{"scriptlevel":791,"displaystyle":1797},[1735,83111,791],{},[1731,83113,11320],{"fence":1599},[1731,83115,1862],{"separator":1599},[1756,83117],{"width":17060},[2212,83119,83120,83122],{},[1584,83121,83020],{},[1584,83123,19585],{},[1731,83125,623],{},[1578,83127,83128,83130,83164],{},[1731,83129,11295],{"fence":1599},[8499,83131,83132,83146],{"rowspacing":8501,"columnalign":83031,"columnspacing":8502},[8504,83133,83134,83140],{},[8507,83135,83136],{},[8510,83137,83138],{"scriptlevel":791,"displaystyle":1797},[1735,83139,1410],{},[8507,83141,83142],{},[8510,83143,83144],{"scriptlevel":791,"displaystyle":1797},[1735,83145,791],{},[8504,83147,83148,83154],{},[8507,83149,83150],{},[8510,83151,83152],{"scriptlevel":791,"displaystyle":1797},[1735,83153,791],{},[8507,83155,83156],{},[8510,83157,83158],{"scriptlevel":791,"displaystyle":1797},[1578,83159,83160,83162],{},[1731,83161,1733],{},[1735,83163,1410],{},[1731,83165,11320],{"fence":1599},[1584,83167,1823],{"mathvariant":1790},[1591,83169,83170],{"encoding":1593},"\\sigma_x=\\begin{bmatrix}0 & 1 \\\\ 1 & 0\\end{bmatrix}, \\qquad\n\\sigma_y=\\begin{bmatrix}0 & -i \\\\ i & 0\\end{bmatrix}, \\qquad\n\\sigma_z=\\begin{bmatrix}1 & 0 \\\\ 0 & -1\\end{bmatrix}.",[40,83172,83174,83229,83419,83606],{"className":83173,"ariaHidden":1599},[1598],[40,83175,83177,83180,83220,83223,83226],{"className":83176},[1603],[40,83178],{"className":83179,"style":24251},[1607],[40,83181,83183,83186],{"className":83182},[1612],[40,83184,83020],{"className":83185,"style":1818},[1612,1653],[40,83187,83189],{"className":83188},[1621],[40,83190,83192,83212],{"className":83191},[1625,2244],[40,83193,83195,83209],{"className":83194},[1629],[40,83196,83198],{"className":83197,"style":3551},[1633],[40,83199,83200,83203],{"style":3616},[40,83201],{"className":83202,"style":1642},[1641],[40,83204,83206],{"className":83205},[1646,1647,1648,1649],[40,83207,8185],{"className":83208},[1612,1653,1649],[40,83210,2269],{"className":83211},[2268],[40,83213,83215],{"className":83214},[1629],[40,83216,83218],{"className":83217,"style":2276},[1633],[40,83219],{},[40,83221],{"className":83222,"style":1963},[1756],[40,83224,623],{"className":83225},[1967],[40,83227],{"className":83228,"style":1963},[1756],[40,83230,83232,83235,83358,83361,83364,83367,83370,83410,83413,83416],{"className":83231},[1603],[40,83233],{"className":83234,"style":36723},[1607],[40,83236,83238,83244,83352],{"className":83237},[1981],[40,83239,83241],{"className":83240,"style":1987},[1985,1986],[40,83242,11295],{"className":83243},[1991,1648],[40,83245,83247],{"className":83246},[1612],[40,83248,83250,83299,83304,83307],{"className":83249},[8499],[40,83251,83253],{"className":83252},[8758],[40,83254,83256,83290],{"className":83255},[1625,2244],[40,83257,83259,83287],{"className":83258},[1629],[40,83260,83263,83275],{"className":83261,"style":83262},[1633],"height:1.45em;",[40,83264,83266,83269],{"style":83265},"top:-3.61em;",[40,83267],{"className":83268,"style":3354},[1641],[40,83270,83272],{"className":83271},[1612],[40,83273,791],{"className":83274},[1612],[40,83276,83278,83281],{"style":83277},"top:-2.41em;",[40,83279],{"className":83280,"style":3354},[1641],[40,83282,83284],{"className":83283},[1612],[40,83285,1410],{"className":83286},[1612],[40,83288,2269],{"className":83289},[2268],[40,83291,83293],{"className":83292},[1629],[40,83294,83297],{"className":83295,"style":83296},[1633],"height:0.95em;",[40,83298],{},[40,83300],{"className":83301,"style":83303},[83302],"arraycolsep","width:0.5em;",[40,83305],{"className":83306,"style":83303},[83302],[40,83308,83310],{"className":83309},[8758],[40,83311,83313,83344],{"className":83312},[1625,2244],[40,83314,83316,83341],{"className":83315},[1629],[40,83317,83319,83330],{"className":83318,"style":83262},[1633],[40,83320,83321,83324],{"style":83265},[40,83322],{"className":83323,"style":3354},[1641],[40,83325,83327],{"className":83326},[1612],[40,83328,1410],{"className":83329},[1612],[40,83331,83332,83335],{"style":83277},[40,83333],{"className":83334,"style":3354},[1641],[40,83336,83338],{"className":83337},[1612],[40,83339,791],{"className":83340},[1612],[40,83342,2269],{"className":83343},[2268],[40,83345,83347],{"className":83346},[1629],[40,83348,83350],{"className":83349,"style":83296},[1633],[40,83351],{},[40,83353,83355],{"className":83354,"style":1987},[1822,1986],[40,83356,11320],{"className":83357},[1991,1648],[40,83359],{"className":83360,"style":2009},[1756],[40,83362,1862],{"className":83363},[2005],[40,83365],{"className":83366,"style":17678},[1756],[40,83368],{"className":83369,"style":2009},[1756],[40,83371,83373,83376],{"className":83372},[1612],[40,83374,83020],{"className":83375,"style":1818},[1612,1653],[40,83377,83379],{"className":83378},[1621],[40,83380,83382,83402],{"className":83381},[1625,2244],[40,83383,83385,83399],{"className":83384},[1629],[40,83386,83388],{"className":83387,"style":3551},[1633],[40,83389,83390,83393],{"style":3616},[40,83391],{"className":83392,"style":1642},[1641],[40,83394,83396],{"className":83395},[1646,1647,1648,1649],[40,83397,19504],{"className":83398,"style":1818},[1612,1653,1649],[40,83400,2269],{"className":83401},[2268],[40,83403,83405],{"className":83404},[1629],[40,83406,83408],{"className":83407,"style":5877},[1633],[40,83409],{},[40,83411],{"className":83412,"style":1963},[1756],[40,83414,623],{"className":83415},[1967],[40,83417],{"className":83418,"style":1963},[1756],[40,83420,83422,83425,83545,83548,83551,83554,83557,83597,83600,83603],{"className":83421},[1603],[40,83423],{"className":83424,"style":36723},[1607],[40,83426,83428,83434,83539],{"className":83427},[1981],[40,83429,83431],{"className":83430,"style":1987},[1985,1986],[40,83432,11295],{"className":83433},[1991,1648],[40,83435,83437],{"className":83436},[1612],[40,83438,83440,83485,83488,83491],{"className":83439},[8499],[40,83441,83443],{"className":83442},[8758],[40,83444,83446,83477],{"className":83445},[1625,2244],[40,83447,83449,83474],{"className":83448},[1629],[40,83450,83452,83463],{"className":83451,"style":83262},[1633],[40,83453,83454,83457],{"style":83265},[40,83455],{"className":83456,"style":3354},[1641],[40,83458,83460],{"className":83459},[1612],[40,83461,791],{"className":83462},[1612],[40,83464,83465,83468],{"style":83277},[40,83466],{"className":83467,"style":3354},[1641],[40,83469,83471],{"className":83470},[1612],[40,83472,2337],{"className":83473},[1612,1653],[40,83475,2269],{"className":83476},[2268],[40,83478,83480],{"className":83479},[1629],[40,83481,83483],{"className":83482,"style":83296},[1633],[40,83484],{},[40,83486],{"className":83487,"style":83303},[83302],[40,83489],{"className":83490,"style":83303},[83302],[40,83492,83494],{"className":83493},[8758],[40,83495,83497,83531],{"className":83496},[1625,2244],[40,83498,83500,83528],{"className":83499},[1629],[40,83501,83503,83517],{"className":83502,"style":83262},[1633],[40,83504,83505,83508],{"style":83265},[40,83506],{"className":83507,"style":3354},[1641],[40,83509,83511,83514],{"className":83510},[1612],[40,83512,1733],{"className":83513},[1612],[40,83515,2337],{"className":83516},[1612,1653],[40,83518,83519,83522],{"style":83277},[40,83520],{"className":83521,"style":3354},[1641],[40,83523,83525],{"className":83524},[1612],[40,83526,791],{"className":83527},[1612],[40,83529,2269],{"className":83530},[2268],[40,83532,83534],{"className":83533},[1629],[40,83535,83537],{"className":83536,"style":83296},[1633],[40,83538],{},[40,83540,83542],{"className":83541,"style":1987},[1822,1986],[40,83543,11320],{"className":83544},[1991,1648],[40,83546],{"className":83547,"style":2009},[1756],[40,83549,1862],{"className":83550},[2005],[40,83552],{"className":83553,"style":17678},[1756],[40,83555],{"className":83556,"style":2009},[1756],[40,83558,83560,83563],{"className":83559},[1612],[40,83561,83020],{"className":83562,"style":1818},[1612,1653],[40,83564,83566],{"className":83565},[1621],[40,83567,83569,83589],{"className":83568},[1625,2244],[40,83570,83572,83586],{"className":83571},[1629],[40,83573,83575],{"className":83574,"style":3551},[1633],[40,83576,83577,83580],{"style":3616},[40,83578],{"className":83579,"style":1642},[1641],[40,83581,83583],{"className":83582},[1646,1647,1648,1649],[40,83584,19585],{"className":83585,"style":19632},[1612,1653,1649],[40,83587,2269],{"className":83588},[2268],[40,83590,83592],{"className":83591},[1629],[40,83593,83595],{"className":83594,"style":2276},[1633],[40,83596],{},[40,83598],{"className":83599,"style":1963},[1756],[40,83601,623],{"className":83602},[1967],[40,83604],{"className":83605,"style":1963},[1756],[40,83607,83609,83612,83732,83735],{"className":83608},[1603],[40,83610],{"className":83611,"style":36723},[1607],[40,83613,83615,83621,83726],{"className":83614},[1981],[40,83616,83618],{"className":83617,"style":1987},[1985,1986],[40,83619,11295],{"className":83620},[1991,1648],[40,83622,83624],{"className":83623},[1612],[40,83625,83627,83672,83675,83678],{"className":83626},[8499],[40,83628,83630],{"className":83629},[8758],[40,83631,83633,83664],{"className":83632},[1625,2244],[40,83634,83636,83661],{"className":83635},[1629],[40,83637,83639,83650],{"className":83638,"style":83262},[1633],[40,83640,83641,83644],{"style":83265},[40,83642],{"className":83643,"style":3354},[1641],[40,83645,83647],{"className":83646},[1612],[40,83648,1410],{"className":83649},[1612],[40,83651,83652,83655],{"style":83277},[40,83653],{"className":83654,"style":3354},[1641],[40,83656,83658],{"className":83657},[1612],[40,83659,791],{"className":83660},[1612],[40,83662,2269],{"className":83663},[2268],[40,83665,83667],{"className":83666},[1629],[40,83668,83670],{"className":83669,"style":83296},[1633],[40,83671],{},[40,83673],{"className":83674,"style":83303},[83302],[40,83676],{"className":83677,"style":83303},[83302],[40,83679,83681],{"className":83680},[8758],[40,83682,83684,83718],{"className":83683},[1625,2244],[40,83685,83687,83715],{"className":83686},[1629],[40,83688,83690,83701],{"className":83689,"style":83262},[1633],[40,83691,83692,83695],{"style":83265},[40,83693],{"className":83694,"style":3354},[1641],[40,83696,83698],{"className":83697},[1612],[40,83699,791],{"className":83700},[1612],[40,83702,83703,83706],{"style":83277},[40,83704],{"className":83705,"style":3354},[1641],[40,83707,83709,83712],{"className":83708},[1612],[40,83710,1733],{"className":83711},[1612],[40,83713,1410],{"className":83714},[1612],[40,83716,2269],{"className":83717},[2268],[40,83719,83721],{"className":83720},[1629],[40,83722,83724],{"className":83723,"style":83296},[1633],[40,83725],{},[40,83727,83729],{"className":83728,"style":1987},[1822,1986],[40,83730,11320],{"className":83731},[1991,1648],[40,83733],{"className":83734,"style":2009},[1756],[40,83736,1823],{"className":83737},[1612],[12,83739,83740,83741,83774],{},"Within this representation, a driven two-level system behaves like a spin-",[40,83742,83744,83762],{"className":83743},[1565],[40,83745,83747],{"className":83746},[1569],[1571,83748,83749],{"xmlns":1573},[1575,83750,83751,83759],{},[1578,83752,83753,83755,83757],{},[1735,83754,1410],{},[1584,83756,7610],{"mathvariant":1790},[1735,83758,1879],{},[1591,83760,83761],{"encoding":1593},"1\u002F2",[40,83763,83765],{"className":83764,"ariaHidden":1599},[1598],[40,83766,83768,83771],{"className":83767},[1603],[40,83769],{"className":83770,"style":1811},[1607],[40,83772,83761],{"className":83773},[1612]," particle precessing around an effective magnetic field. The drive strength supplies the transverse control component. Applied detuning supplies the longitudinal control component.",[55,83776,83778],{"id":83777},"rotating-frame-hamiltonian","Rotating-Frame Hamiltonian",[12,83780,83781],{},"In the laboratory frame, a minimal semiclassical Hamiltonian for the driven qubit is",[40,83783,83785],{"className":83784},[1827],[40,83786,83788,83856],{"className":83787},[1565],[40,83789,83791],{"className":83790},[1569],[1571,83792,83793],{"xmlns":1573,"display":1836},[1575,83794,83795,83853],{},[1578,83796,83797,83799,83801,83803,83805,83807,83821,83827,83829,83831,83833,83835,83837,83839,83841,83843,83845,83851],{},[1584,83798,2182],{},[1731,83800,241],{"stretchy":1797},[1584,83802,2342],{},[1731,83804,2693],{"stretchy":1797},[1731,83806,623],{},[8538,83808,83809,83819],{},[1578,83810,83811,83813],{},[1584,83812,81755],{"mathvariant":1790},[2212,83814,83815,83817],{},[1584,83816,80659],{},[1735,83818,791],{},[1735,83820,1879],{},[2212,83822,83823,83825],{},[1584,83824,83020],{},[1584,83826,19585],{},[1731,83828,1147],{},[1584,83830,81755],{"mathvariant":1790},[1584,83832,80975],{"mathvariant":1790},[1584,83834,36987],{},[1731,83836,35745],{},[1731,83838,241],{"stretchy":1797},[1584,83840,80659],{},[1584,83842,2342],{},[1731,83844,2693],{"stretchy":1797},[2212,83846,83847,83849],{},[1584,83848,83020],{},[1584,83850,8185],{},[1731,83852,1862],{"separator":1599},[1591,83854,83855],{"encoding":1593},"H(t)=\\frac{\\hbar\\omega_0}{2}\\sigma_z+\\hbar\\Omega\\cos(\\omega t)\\sigma_x,",[40,83857,83859,83886,84045],{"className":83858,"ariaHidden":1599},[1598],[40,83860,83862,83865,83868,83871,83874,83877,83880,83883],{"className":83861},[1603],[40,83863],{"className":83864,"style":1811},[1607],[40,83866,2182],{"className":83867,"style":2197},[1612,1653],[40,83869,241],{"className":83870},[1985],[40,83872,2342],{"className":83873},[1612,1653],[40,83875,2693],{"className":83876},[1822],[40,83878],{"className":83879,"style":1963},[1756],[40,83881,623],{"className":83882},[1967],[40,83884],{"className":83885,"style":1963},[1756],[40,83887,83889,83893,83996,84036,84039,84042],{"className":83888},[1603],[40,83890],{"className":83891,"style":83892},[1607],"height:2.0519em;vertical-align:-0.686em;",[40,83894,83896,83899,83993],{"className":83895},[1612],[40,83897],{"className":83898},[1985,8840],[40,83900,83902],{"className":83901},[8538],[40,83903,83905,83985],{"className":83904},[1625,2244],[40,83906,83908,83982],{"className":83907},[1629],[40,83909,83912,83923,83931],{"className":83910,"style":83911},[1633],"height:1.3659em;",[40,83913,83914,83917],{"style":10764},[40,83915],{"className":83916,"style":3354},[1641],[40,83918,83920],{"className":83919},[1612],[40,83921,1879],{"className":83922},[1612],[40,83924,83925,83928],{"style":8925},[40,83926],{"className":83927,"style":3354},[1641],[40,83929],{"className":83930,"style":8933},[8932],[40,83932,83933,83936],{"style":8936},[40,83934],{"className":83935,"style":3354},[1641],[40,83937,83939,83942],{"className":83938},[1612],[40,83940,81755],{"className":83941},[1612],[40,83943,83945,83948],{"className":83944},[1612],[40,83946,80659],{"className":83947,"style":1818},[1612,1653],[40,83949,83951],{"className":83950},[1621],[40,83952,83954,83974],{"className":83953},[1625,2244],[40,83955,83957,83971],{"className":83956},[1629],[40,83958,83960],{"className":83959,"style":13330},[1633],[40,83961,83962,83965],{"style":3616},[40,83963],{"className":83964,"style":1642},[1641],[40,83966,83968],{"className":83967},[1646,1647,1648,1649],[40,83969,791],{"className":83970},[1612,1649],[40,83972,2269],{"className":83973},[2268],[40,83975,83977],{"className":83976},[1629],[40,83978,83980],{"className":83979,"style":2276},[1633],[40,83981],{},[40,83983,2269],{"className":83984},[2268],[40,83986,83988],{"className":83987},[1629],[40,83989,83991],{"className":83990,"style":10802},[1633],[40,83992],{},[40,83994],{"className":83995},[1822,8840],[40,83997,83999,84002],{"className":83998},[1612],[40,84000,83020],{"className":84001,"style":1818},[1612,1653],[40,84003,84005],{"className":84004},[1621],[40,84006,84008,84028],{"className":84007},[1625,2244],[40,84009,84011,84025],{"className":84010},[1629],[40,84012,84014],{"className":84013,"style":3551},[1633],[40,84015,84016,84019],{"style":3616},[40,84017],{"className":84018,"style":1642},[1641],[40,84020,84022],{"className":84021},[1646,1647,1648,1649],[40,84023,19585],{"className":84024,"style":19632},[1612,1653,1649],[40,84026,2269],{"className":84027},[2268],[40,84029,84031],{"className":84030},[1629],[40,84032,84034],{"className":84033,"style":2276},[1633],[40,84035],{},[40,84037],{"className":84038,"style":1757},[1756],[40,84040,1147],{"className":84041},[1761],[40,84043],{"className":84044,"style":1757},[1756],[40,84046,84048,84051,84055,84058,84061,84064,84067,84070,84073,84113],{"className":84047},[1603],[40,84049],{"className":84050,"style":1811},[1607],[40,84052,84054],{"className":84053},[1612],"ℏΩ",[40,84056],{"className":84057,"style":2009},[1756],[40,84059,36987],{"className":84060},[6614],[40,84062,241],{"className":84063},[1985],[40,84065,80659],{"className":84066,"style":1818},[1612,1653],[40,84068,2342],{"className":84069},[1612,1653],[40,84071,2693],{"className":84072},[1822],[40,84074,84076,84079],{"className":84075},[1612],[40,84077,83020],{"className":84078,"style":1818},[1612,1653],[40,84080,84082],{"className":84081},[1621],[40,84083,84085,84105],{"className":84084},[1625,2244],[40,84086,84088,84102],{"className":84087},[1629],[40,84089,84091],{"className":84090,"style":3551},[1633],[40,84092,84093,84096],{"style":3616},[40,84094],{"className":84095,"style":1642},[1641],[40,84097,84099],{"className":84098},[1646,1647,1648,1649],[40,84100,8185],{"className":84101},[1612,1653,1649],[40,84103,2269],{"className":84104},[2268],[40,84106,84108],{"className":84107},[1629],[40,84109,84111],{"className":84110,"style":2276},[1633],[40,84112],{},[40,84114,1862],{"className":84115},[2005],[12,84117,27509,84118,84178,84179,84207,84208,84245],{},[40,84119,84121,84142],{"className":84120},[1565],[40,84122,84124],{"className":84123},[1569],[1571,84125,84126],{"xmlns":1573},[1575,84127,84128,84140],{},[1578,84129,84130,84132,84134,84136,84138],{},[1584,84131,80659],{},[1731,84133,623],{},[1735,84135,1879],{},[1584,84137,7607],{},[1584,84139,21331],{},[1591,84141,80920],{"encoding":1593},[40,84143,84145,84163],{"className":84144,"ariaHidden":1599},[1598],[40,84146,84148,84151,84154,84157,84160],{"className":84147},[1603],[40,84149],{"className":84150,"style":1681},[1607],[40,84152,80659],{"className":84153,"style":1818},[1612,1653],[40,84155],{"className":84156,"style":1963},[1756],[40,84158,623],{"className":84159},[1967],[40,84161],{"className":84162,"style":1963},[1756],[40,84164,84166,84169,84172,84175],{"className":84165},[1603],[40,84167],{"className":84168,"style":6079},[1607],[40,84170,1879],{"className":84171},[1612],[40,84173,7607],{"className":84174,"style":1818},[1612,1653],[40,84176,21331],{"className":84177,"style":26393},[1612,1653]," is the drive angular frequency and ",[40,84180,84182,84195],{"className":84181},[1565],[40,84183,84185],{"className":84184},[1569],[1571,84186,84187],{"xmlns":1573},[1575,84188,84189,84193],{},[1578,84190,84191],{},[1584,84192,80975],{"mathvariant":1790},[1591,84194,80978],{"encoding":1593},[40,84196,84198],{"className":84197,"ariaHidden":1599},[1598],[40,84199,84201,84204],{"className":84200},[1603],[40,84202],{"className":84203,"style":1608},[1607],[40,84205,80975],{"className":84206},[1612]," is the on-resonance Rabi angular frequency. The computational model specifies the drive strength using ",[40,84209,84211,84230],{"className":84210},[1565],[40,84212,84214],{"className":84213},[1569],[1571,84215,84216],{"xmlns":1573},[1575,84217,84218,84228],{},[1578,84219,84220,84222,84224,84226],{},[1584,84221,80975],{"mathvariant":1790},[1584,84223,7610],{"mathvariant":1790},[1735,84225,1879],{},[1584,84227,7607],{},[1591,84229,81020],{"encoding":1593},[40,84231,84233],{"className":84232,"ariaHidden":1599},[1598],[40,84234,84236,84239,84242],{"className":84235},[1603],[40,84237],{"className":84238,"style":1811},[1607],[40,84240,81033],{"className":84241},[1612],[40,84243,7607],{"className":84244,"style":1818},[1612,1653]," in MHz. Transforming into a frame rotating at the drive frequency and applying the rotating-wave approximation gives the effective time-independent Hamiltonian",[40,84247,84249],{"className":84248},[1827],[40,84250,84252,84311],{"className":84251},[1565],[40,84253,84255],{"className":84254},[1569],[1571,84256,84257],{"xmlns":1573,"display":1836},[1575,84258,84259,84308],{},[1578,84260,84261,84274,84276,84282,84306],{},[2212,84262,84263,84265],{},[1584,84264,2182],{},[1578,84266,84267,84269,84272],{},[1584,84268,19419],{"mathvariant":1790},[1584,84270,84271],{"mathvariant":1790},"W",[1584,84273,1700],{"mathvariant":1790},[1731,84275,623],{},[8538,84277,84278,84280],{},[1584,84279,81755],{"mathvariant":1790},[1735,84281,1879],{},[1578,84283,84284,84286,84288,84294,84296,84298,84304],{},[1731,84285,241],{"fence":1599},[1584,84287,81058],{"mathvariant":1790},[2212,84289,84290,84292],{},[1584,84291,83020],{},[1584,84293,19585],{},[1731,84295,1147],{},[1584,84297,80975],{"mathvariant":1790},[2212,84299,84300,84302],{},[1584,84301,83020],{},[1584,84303,8185],{},[1731,84305,2693],{"fence":1599},[1731,84307,1862],{"separator":1599},[1591,84309,84310],{"encoding":1593},"H_{\\mathrm{RWA}}=\\frac{\\hbar}{2}\\left(\\Delta\\sigma_z+\\Omega\\sigma_x\\right),",[40,84312,84314,84375],{"className":84313,"ariaHidden":1599},[1598],[40,84315,84317,84320,84366,84369,84372],{"className":84316},[1603],[40,84318],{"className":84319,"style":2231},[1607],[40,84321,84323,84326],{"className":84322},[1612],[40,84324,2182],{"className":84325,"style":2197},[1612,1653],[40,84327,84329],{"className":84328},[1621],[40,84330,84332,84358],{"className":84331},[1625,2244],[40,84333,84335,84355],{"className":84334},[1629],[40,84336,84338],{"className":84337,"style":2251},[1633],[40,84339,84340,84343],{"style":3554},[40,84341],{"className":84342,"style":1642},[1641],[40,84344,84346],{"className":84345},[1646,1647,1648,1649],[40,84347,84349],{"className":84348},[1612,1649],[40,84350,84352],{"className":84351},[1612,1649],[40,84353,81625],{"className":84354},[1612,82209,1649],[40,84356,2269],{"className":84357},[2268],[40,84359,84361],{"className":84360},[1629],[40,84362,84364],{"className":84363,"style":2276},[1633],[40,84365],{},[40,84367],{"className":84368,"style":1963},[1756],[40,84370,623],{"className":84371},[1967],[40,84373],{"className":84374,"style":1963},[1756],[40,84376,84378,84381,84443,84446,84550,84553],{"className":84377},[1603],[40,84379],{"className":84380,"style":83892},[1607],[40,84382,84384,84387,84440],{"className":84383},[1612],[40,84385],{"className":84386},[1985,8840],[40,84388,84390],{"className":84389},[8538],[40,84391,84393,84432],{"className":84392},[1625,2244],[40,84394,84396,84429],{"className":84395},[1629],[40,84397,84399,84410,84418],{"className":84398,"style":83911},[1633],[40,84400,84401,84404],{"style":10764},[40,84402],{"className":84403,"style":3354},[1641],[40,84405,84407],{"className":84406},[1612],[40,84408,1879],{"className":84409},[1612],[40,84411,84412,84415],{"style":8925},[40,84413],{"className":84414,"style":3354},[1641],[40,84416],{"className":84417,"style":8933},[8932],[40,84419,84420,84423],{"style":8936},[40,84421],{"className":84422,"style":3354},[1641],[40,84424,84426],{"className":84425},[1612],[40,84427,81755],{"className":84428},[1612],[40,84430,2269],{"className":84431},[2268],[40,84433,84435],{"className":84434},[1629],[40,84436,84438],{"className":84437,"style":10802},[1633],[40,84439],{},[40,84441],{"className":84442},[1822,8840],[40,84444],{"className":84445,"style":2009},[1756],[40,84447,84449,84452,84455,84495,84498,84501,84504,84507,84547],{"className":84448},[1981],[40,84450,241],{"className":84451,"style":1987},[1985,1986],[40,84453,81058],{"className":84454},[1612],[40,84456,84458,84461],{"className":84457},[1612],[40,84459,83020],{"className":84460,"style":1818},[1612,1653],[40,84462,84464],{"className":84463},[1621],[40,84465,84467,84487],{"className":84466},[1625,2244],[40,84468,84470,84484],{"className":84469},[1629],[40,84471,84473],{"className":84472,"style":3551},[1633],[40,84474,84475,84478],{"style":3616},[40,84476],{"className":84477,"style":1642},[1641],[40,84479,84481],{"className":84480},[1646,1647,1648,1649],[40,84482,19585],{"className":84483,"style":19632},[1612,1653,1649],[40,84485,2269],{"className":84486},[2268],[40,84488,84490],{"className":84489},[1629],[40,84491,84493],{"className":84492,"style":2276},[1633],[40,84494],{},[40,84496],{"className":84497,"style":1757},[1756],[40,84499,1147],{"className":84500},[1761],[40,84502],{"className":84503,"style":1757},[1756],[40,84505,80975],{"className":84506},[1612],[40,84508,84510,84513],{"className":84509},[1612],[40,84511,83020],{"className":84512,"style":1818},[1612,1653],[40,84514,84516],{"className":84515},[1621],[40,84517,84519,84539],{"className":84518},[1625,2244],[40,84520,84522,84536],{"className":84521},[1629],[40,84523,84525],{"className":84524,"style":3551},[1633],[40,84526,84527,84530],{"style":3616},[40,84528],{"className":84529,"style":1642},[1641],[40,84531,84533],{"className":84532},[1646,1647,1648,1649],[40,84534,8185],{"className":84535},[1612,1653,1649],[40,84537,2269],{"className":84538},[2268],[40,84540,84542],{"className":84541},[1629],[40,84543,84545],{"className":84544,"style":2276},[1633],[40,84546],{},[40,84548,2693],{"className":84549,"style":1987},[1822,1986],[40,84551],{"className":84552,"style":2009},[1756],[40,84554,1862],{"className":84555},[2005],[12,84557,84558],{},"with detuning",[40,84560,84562],{"className":84561},[1827],[40,84563,84565,84613],{"className":84564},[1565],[40,84566,84568],{"className":84567},[1569],[1571,84569,84570],{"xmlns":1573,"display":1836},[1575,84571,84572,84610],{},[1578,84573,84574,84576,84578,84580,84582,84588,84590,84592,84594,84596,84598,84600,84606,84608],{},[1584,84575,81058],{"mathvariant":1790},[1731,84577,623],{},[1584,84579,80659],{},[1731,84581,1733],{},[2212,84583,84584,84586],{},[1584,84585,80659],{},[1735,84587,791],{},[1731,84589,623],{},[1735,84591,1879],{},[1584,84593,7607],{},[1731,84595,241],{"stretchy":1797},[1584,84597,21331],{},[1731,84599,1733],{},[2212,84601,84602,84604],{},[1584,84603,21331],{},[1735,84605,791],{},[1731,84607,2693],{"stretchy":1797},[1584,84609,1823],{"mathvariant":1790},[1591,84611,84612],{"encoding":1593},"\\Delta=\\omega-\\omega_0=2\\pi(f-f_0).",[40,84614,84616,84634,84652,84707,84734],{"className":84615,"ariaHidden":1599},[1598],[40,84617,84619,84622,84625,84628,84631],{"className":84618},[1603],[40,84620],{"className":84621,"style":1608},[1607],[40,84623,81058],{"className":84624},[1612],[40,84626],{"className":84627,"style":1963},[1756],[40,84629,623],{"className":84630},[1967],[40,84632],{"className":84633,"style":1963},[1756],[40,84635,84637,84640,84643,84646,84649],{"className":84636},[1603],[40,84638],{"className":84639,"style":1749},[1607],[40,84641,80659],{"className":84642,"style":1818},[1612,1653],[40,84644],{"className":84645,"style":1757},[1756],[40,84647,1733],{"className":84648},[1761],[40,84650],{"className":84651,"style":1757},[1756],[40,84653,84655,84658,84698,84701,84704],{"className":84654},[1603],[40,84656],{"className":84657,"style":24251},[1607],[40,84659,84661,84664],{"className":84660},[1612],[40,84662,80659],{"className":84663,"style":1818},[1612,1653],[40,84665,84667],{"className":84666},[1621],[40,84668,84670,84690],{"className":84669},[1625,2244],[40,84671,84673,84687],{"className":84672},[1629],[40,84674,84676],{"className":84675,"style":13330},[1633],[40,84677,84678,84681],{"style":3616},[40,84679],{"className":84680,"style":1642},[1641],[40,84682,84684],{"className":84683},[1646,1647,1648,1649],[40,84685,791],{"className":84686},[1612,1649],[40,84688,2269],{"className":84689},[2268],[40,84691,84693],{"className":84692},[1629],[40,84694,84696],{"className":84695,"style":2276},[1633],[40,84697],{},[40,84699],{"className":84700,"style":1963},[1756],[40,84702,623],{"className":84703},[1967],[40,84705],{"className":84706,"style":1963},[1756],[40,84708,84710,84713,84716,84719,84722,84725,84728,84731],{"className":84709},[1603],[40,84711],{"className":84712,"style":1811},[1607],[40,84714,1879],{"className":84715},[1612],[40,84717,7607],{"className":84718,"style":1818},[1612,1653],[40,84720,241],{"className":84721},[1985],[40,84723,21331],{"className":84724,"style":26393},[1612,1653],[40,84726],{"className":84727,"style":1757},[1756],[40,84729,1733],{"className":84730},[1761],[40,84732],{"className":84733,"style":1757},[1756],[40,84735,84737,84740,84780,84783],{"className":84736},[1603],[40,84738],{"className":84739,"style":1811},[1607],[40,84741,84743,84746],{"className":84742},[1612],[40,84744,21331],{"className":84745,"style":26393},[1612,1653],[40,84747,84749],{"className":84748},[1621],[40,84750,84752,84772],{"className":84751},[1625,2244],[40,84753,84755,84769],{"className":84754},[1629],[40,84756,84758],{"className":84757,"style":13330},[1633],[40,84759,84760,84763],{"style":80580},[40,84761],{"className":84762,"style":1642},[1641],[40,84764,84766],{"className":84765},[1646,1647,1648,1649],[40,84767,791],{"className":84768},[1612,1649],[40,84770,2269],{"className":84771},[2268],[40,84773,84775],{"className":84774},[1629],[40,84776,84778],{"className":84777,"style":2276},[1633],[40,84779],{},[40,84781,2693],{"className":84782},[1822],[40,84784,1823],{"className":84785},[1612],[12,84787,84788,84789,9846,84817,84845,84846,84874,84875,84903],{},"This Hamiltonian describes precession around an effective control axis with components proportional to ",[40,84790,84792,84805],{"className":84791},[1565],[40,84793,84795],{"className":84794},[1569],[1571,84796,84797],{"xmlns":1573},[1575,84798,84799,84803],{},[1578,84800,84801],{},[1584,84802,80975],{"mathvariant":1790},[1591,84804,80978],{"encoding":1593},[40,84806,84808],{"className":84807,"ariaHidden":1599},[1598],[40,84809,84811,84814],{"className":84810},[1603],[40,84812],{"className":84813,"style":1608},[1607],[40,84815,80975],{"className":84816},[1612],[40,84818,84820,84833],{"className":84819},[1565],[40,84821,84823],{"className":84822},[1569],[1571,84824,84825],{"xmlns":1573},[1575,84826,84827,84831],{},[1578,84828,84829],{},[1584,84830,81058],{"mathvariant":1790},[1591,84832,81061],{"encoding":1593},[40,84834,84836],{"className":84835,"ariaHidden":1599},[1598],[40,84837,84839,84842],{"className":84838},[1603],[40,84840],{"className":84841,"style":1608},[1607],[40,84843,81058],{"className":84844},[1612],". The transverse component ",[40,84847,84849,84862],{"className":84848},[1565],[40,84850,84852],{"className":84851},[1569],[1571,84853,84854],{"xmlns":1573},[1575,84855,84856,84860],{},[1578,84857,84858],{},[1584,84859,80975],{"mathvariant":1790},[1591,84861,80978],{"encoding":1593},[40,84863,84865],{"className":84864,"ariaHidden":1599},[1598],[40,84866,84868,84871],{"className":84867},[1603],[40,84869],{"className":84870,"style":1608},[1607],[40,84872,80975],{"className":84873},[1612]," drives population transfer. A longitudinal component ",[40,84876,84878,84891],{"className":84877},[1565],[40,84879,84881],{"className":84880},[1569],[1571,84882,84883],{"xmlns":1573},[1575,84884,84885,84889],{},[1578,84886,84887],{},[1584,84888,81058],{"mathvariant":1790},[1591,84890,81061],{"encoding":1593},[40,84892,84894],{"className":84893,"ariaHidden":1599},[1598],[40,84895,84897,84900],{"className":84896},[1603],[40,84898],{"className":84899,"style":1608},[1607],[40,84901,81058],{"className":84902},[1612]," tilts the rotation axis away from the resonant direction.",[55,84905,84907],{"id":84906},"excited-state-probability","Excited-State Probability",[12,84909,84910,84911,84949,84950,84978],{},"The simulation assumes that the qubit begins in the ground state ",[40,84912,84914,84931],{"className":84913},[1565],[40,84915,84917],{"className":84916},[1569],[1571,84918,84919],{"xmlns":1573},[1575,84920,84921,84929],{},[1578,84922,84923,84925,84927],{},[1731,84924,1791],{"stretchy":1797},[1584,84926,36968],{},[1731,84928,1798],{"stretchy":1797},[1591,84930,80208],{"encoding":1593},[40,84932,84934],{"className":84933,"ariaHidden":1599},[1598],[40,84935,84937,84940,84943,84946],{"className":84936},[1603],[40,84938],{"className":84939,"style":1811},[1607],[40,84941,1791],{"className":84942},[1985],[40,84944,36968],{"className":84945,"style":1818},[1612,1653],[40,84947,1798],{"className":84948},[1822],". Evolving under the RWA Hamiltonian, the probability of occupying the excited state after pulse duration ",[40,84951,84953,84966],{"className":84952},[1565],[40,84954,84956],{"className":84955},[1569],[1571,84957,84958],{"xmlns":1573},[1575,84959,84960,84964],{},[1578,84961,84962],{},[1584,84963,80017],{},[1591,84965,80143],{"encoding":1593},[40,84967,84969],{"className":84968,"ariaHidden":1599},[1598],[40,84970,84972,84975],{"className":84971},[1603],[40,84973],{"className":84974,"style":1681},[1607],[40,84976,80017],{"className":84977,"style":80089},[1612,1653]," 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0 400000 1296",[5334,85669],{"d":85670},"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",[40,85672,2269],{"className":85673},[2268],[40,85675,85677],{"className":85676},[1629],[40,85678,85681],{"className":85679,"style":85680},[1633],"height:0.1777em;",[40,85682],{},[12,85684,85685],{},"is the generalized Rabi angular frequency. The amplitude coefficient",[40,85687,85689],{"className":85688},[1827],[40,85690,85692,85720],{"className":85691},[1565],[40,85693,85695],{"className":85694},[1569],[1571,85696,85697],{"xmlns":1573,"display":1836},[1575,85698,85699,85717],{},[1578,85700,85701],{},[8538,85702,85703,85709],{},[1581,85704,85705,85707],{},[1584,85706,80975],{"mathvariant":1790},[1735,85708,1879],{},[2819,85710,85711,85713,85715],{},[1584,85712,80975],{"mathvariant":1790},[1584,85714,19419],{},[1735,85716,1879],{},[1591,85718,85719],{"encoding":1593},"\\frac{\\Omega^2}{\\Omega_R^2}",[40,85721,85723],{"className":85722,"ariaHidden":1599},[1598],[40,85724,85726,85729],{"className":85725},[1603],[40,85727],{"className":85728,"style":85146},[1607],[40,85730,85732,85735,85862],{"className":85731},[1612],[40,85733],{"className":85734},[1985,8840],[40,85736,85738],{"className":85737},[8538],[40,85739,85741,85854],{"className":85740},[1625,2244],[40,85742,85744,85851],{"className":85743},[1629],[40,85745,85747,85806,85814],{"className":85746,"style":85165},[1633],[40,85748,85749,85752],{"style":10764},[40,85750],{"className":85751,"style":3354},[1641],[40,85753,85755],{"className":85754},[1612],[40,85756,85758,85761],{"className":85757},[1612],[40,85759,80975],{"className":85760},[1612],[40,85762,85764],{"className":85763},[1621],[40,85765,85767,85798],{"className":85766},[1625,2244],[40,85768,85770,85795],{"className":85769},[1629],[40,85771,85773,85784],{"className":85772,"style":85192},[1633],[40,85774,85775,85778],{"style":85195},[40,85776],{"className":85777,"style":1642},[1641],[40,85779,85781],{"className":85780},[1646,1647,1648,1649],[40,85782,19419],{"className":85783,"style":19444},[1612,1653,1649],[40,85785,85786,85789],{"style":85207},[40,85787],{"className":85788,"style":1642},[1641],[40,85790,85792],{"className":85791},[1646,1647,1648,1649],[40,85793,1879],{"className":85794},[1612,1649],[40,85796,2269],{"className":85797},[2268],[40,85799,85801],{"className":85800},[1629],[40,85802,85804],{"className":85803,"style":85226},[1633],[40,85805],{},[40,85807,85808,85811],{"style":8925},[40,85809],{"className":85810,"style":3354},[1641],[40,85812],{"className":85813,"style":8933},[8932],[40,85815,85816,85819],{"style":8936},[40,85817],{"className":85818,"style":3354},[1641],[40,85820,85822],{"className":85821},[1612],[40,85823,85825,85828],{"className":85824},[1612],[40,85826,80975],{"className":85827},[1612],[40,85829,85831],{"className":85830},[1621],[40,85832,85834],{"className":85833},[1625],[40,85835,85837],{"className":85836},[1629],[40,85838,85840],{"className":85839,"style":12271},[1633],[40,85841,85842,85845],{"style":1637},[40,85843],{"className":85844,"style":1642},[1641],[40,85846,85848],{"className":85847},[1646,1647,1648,1649],[40,85849,1879],{"className":85850},[1612,1649],[40,85852,2269],{"className":85853},[2268],[40,85855,85857],{"className":85856},[1629],[40,85858,85860],{"className":85859,"style":85283},[1633],[40,85861],{},[40,85863],{"className":85864},[1822,8840],[12,85866,85867,85868,85919],{},"sets the maximum possible population transfer. At exact resonance, ",[40,85869,85871,85889],{"className":85870},[1565],[40,85872,85874],{"className":85873},[1569],[1571,85875,85876],{"xmlns":1573},[1575,85877,85878,85886],{},[1578,85879,85880,85882,85884],{},[1584,85881,81058],{"mathvariant":1790},[1731,85883,623],{},[1735,85885,791],{},[1591,85887,85888],{"encoding":1593},"\\Delta=0",[40,85890,85892,85910],{"className":85891,"ariaHidden":1599},[1598],[40,85893,85895,85898,85901,85904,85907],{"className":85894},[1603],[40,85896],{"className":85897,"style":1608},[1607],[40,85899,81058],{"className":85900},[1612],[40,85902],{"className":85903,"style":1963},[1756],[40,85905,623],{"className":85906},[1967],[40,85908],{"className":85909,"style":1963},[1756],[40,85911,85913,85916],{"className":85912},[1603],[40,85914],{"className":85915,"style":1771},[1607],[40,85917,791],{"className":85918},[1612],", so",[40,85921,85923],{"className":85922},[1827],[40,85924,85926,85958],{"className":85925},[1565],[40,85927,85929],{"className":85928},[1569],[1571,85930,85931],{"xmlns":1573,"display":1836},[1575,85932,85933,85955],{},[1578,85934,85935,85951,85953],{},[8538,85936,85937,85943],{},[1581,85938,85939,85941],{},[1584,85940,80975],{"mathvariant":1790},[1735,85942,1879],{},[2819,85944,85945,85947,85949],{},[1584,85946,80975],{"mathvariant":1790},[1584,85948,19419],{},[1735,85950,1879],{},[1731,85952,623],{},[1735,85954,82656],{},[1591,85956,85957],{"encoding":1593},"\\frac{\\Omega^2}{\\Omega_R^2}=1.",[40,85959,85961,86112],{"className":85960,"ariaHidden":1599},[1598],[40,85962,85964,85967,86103,86106,86109],{"className":85963},[1603],[40,85965],{"className":85966,"style":85146},[1607],[40,85968,85970,85973,86100],{"className":85969},[1612],[40,85971],{"className":85972},[1985,8840],[40,85974,85976],{"className":85975},[8538],[40,85977,85979,86092],{"className":85978},[1625,2244],[40,85980,85982,86089],{"className":85981},[1629],[40,85983,85985,86044,86052],{"className":85984,"style":85165},[1633],[40,85986,85987,85990],{"style":10764},[40,85988],{"className":85989,"style":3354},[1641],[40,85991,85993],{"className":85992},[1612],[40,85994,85996,85999],{"className":85995},[1612],[40,85997,80975],{"className":85998},[1612],[40,86000,86002],{"className":86001},[1621],[40,86003,86005,86036],{"className":86004},[1625,2244],[40,86006,86008,86033],{"className":86007},[1629],[40,86009,86011,86022],{"className":86010,"style":85192},[1633],[40,86012,86013,86016],{"style":85195},[40,86014],{"className":86015,"style":1642},[1641],[40,86017,86019],{"className":86018},[1646,1647,1648,1649],[40,86020,19419],{"className":86021,"style":19444},[1612,1653,1649],[40,86023,86024,86027],{"style":85207},[40,86025],{"className":86026,"style":1642},[1641],[40,86028,86030],{"className":86029},[1646,1647,1648,1649],[40,86031,1879],{"className":86032},[1612,1649],[40,86034,2269],{"className":86035},[2268],[40,86037,86039],{"className":86038},[1629],[40,86040,86042],{"className":86041,"style":85226},[1633],[40,86043],{},[40,86045,86046,86049],{"style":8925},[40,86047],{"className":86048,"style":3354},[1641],[40,86050],{"className":86051,"style":8933},[8932],[40,86053,86054,86057],{"style":8936},[40,86055],{"className":86056,"style":3354},[1641],[40,86058,86060],{"className":86059},[1612],[40,86061,86063,86066],{"className":86062},[1612],[40,86064,80975],{"className":86065},[1612],[40,86067,86069],{"className":86068},[1621],[40,86070,86072],{"className":86071},[1625],[40,86073,86075],{"className":86074},[1629],[40,86076,86078],{"className":86077,"style":12271},[1633],[40,86079,86080,86083],{"style":1637},[40,86081],{"className":86082,"style":1642},[1641],[40,86084,86086],{"className":86085},[1646,1647,1648,1649],[40,86087,1879],{"className":86088},[1612,1649],[40,86090,2269],{"className":86091},[2268],[40,86093,86095],{"className":86094},[1629],[40,86096,86098],{"className":86097,"style":85283},[1633],[40,86099],{},[40,86101],{"className":86102},[1822,8840],[40,86104],{"className":86105,"style":1963},[1756],[40,86107,623],{"className":86108},[1967],[40,86110],{"className":86111,"style":1963},[1756],[40,86113,86115,86118],{"className":86114},[1603],[40,86116],{"className":86117,"style":1771},[1607],[40,86119,82656],{"className":86120},[1612],[12,86122,86123],{},"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.",[12,86125,86126],{},"At resonance, the excited-state probability simplifies to",[40,86128,86130],{"className":86129},[1827],[40,86131,86133,86187],{"className":86132},[1565],[40,86134,86136],{"className":86135},[1569],[1571,86137,86138],{"xmlns":1573,"display":1836},[1575,86139,86140,86184],{},[1578,86141,86142,86148,86150,86152,86154,86156,86166,86182],{},[2212,86143,86144,86146],{},[1584,86145,2218],{},[1584,86147,2330],{},[1731,86149,241],{"stretchy":1797},[1584,86151,80017],{},[1731,86153,2693],{"stretchy":1797},[1731,86155,623],{},[1581,86157,86158,86164],{},[1578,86159,86160,86162],{},[1584,86161,85034],{},[1731,86163,35745],{},[1735,86165,1879],{},[1578,86167,86168,86170,86180],{},[1731,86169,241],{"fence":1599},[8538,86171,86172,86178],{},[1578,86173,86174,86176],{},[1584,86175,80975],{"mathvariant":1790},[1584,86177,80017],{},[1735,86179,1879],{},[1731,86181,2693],{"fence":1599},[1584,86183,1823],{"mathvariant":1790},[1591,86185,86186],{"encoding":1593},"P_e(\\tau)=\\sin^2\\left(\\frac{\\Omega\\tau}{2}\\right).",[40,86188,86190,86254],{"className":86189,"ariaHidden":1599},[1598],[40,86191,86193,86196,86236,86239,86242,86245,86248,86251],{"className":86192},[1603],[40,86194],{"className":86195,"style":1811},[1607],[40,86197,86199,86202],{"className":86198},[1612],[40,86200,2218],{"className":86201,"style":2264},[1612,1653],[40,86203,86205],{"className":86204},[1621],[40,86206,86208,86228],{"className":86207},[1625,2244],[40,86209,86211,86225],{"className":86210},[1629],[40,86212,86214],{"className":86213,"style":3551},[1633],[40,86215,86216,86219],{"style":6681},[40,86217],{"className":86218,"style":1642},[1641],[40,86220,86222],{"className":86221},[1646,1647,1648,1649],[40,86223,2330],{"className":86224},[1612,1653,1649],[40,86226,2269],{"className":86227},[2268],[40,86229,86231],{"className":86230},[1629],[40,86232,86234],{"className":86233,"style":2276},[1633],[40,86235],{},[40,86237,241],{"className":86238},[1985],[40,86240,80017],{"className":86241,"style":80089},[1612,1653],[40,86243,2693],{"className":86244},[1822],[40,86246],{"className":86247,"style":1963},[1756],[40,86249,623],{"className":86250},[1967],[40,86252],{"className":86253,"style":1963},[1756],[40,86255,86257,86260,86289,86292,86372,86375],{"className":86256},[1603],[40,86258],{"className":86259,"style":36723},[1607],[40,86261,86263,86266],{"className":86262},[6614],[40,86264,85034],{"className":86265},[6614],[40,86267,86269],{"className":86268},[1621],[40,86270,86272],{"className":86271},[1625],[40,86273,86275],{"className":86274},[1629],[40,86276,86278],{"className":86277,"style":85310},[1633],[40,86279,86280,86283],{"style":85313},[40,86281],{"className":86282,"style":1642},[1641],[40,86284,86286],{"className":86285},[1646,1647,1648,1649],[40,86287,1879],{"className":86288},[1612,1649],[40,86290],{"className":86291,"style":2009},[1756],[40,86293,86295,86301,86366],{"className":86294},[1981],[40,86296,86298],{"className":86297,"style":1987},[1985,1986],[40,86299,241],{"className":86300},[1991,1648],[40,86302,86304,86307,86363],{"className":86303},[1612],[40,86305],{"className":86306},[1985,8840],[40,86308,86310],{"className":86309},[8538],[40,86311,86313,86355],{"className":86312},[1625,2244],[40,86314,86316,86352],{"className":86315},[1629],[40,86317,86319,86330,86338],{"className":86318,"style":85353},[1633],[40,86320,86321,86324],{"style":10764},[40,86322],{"className":86323,"style":3354},[1641],[40,86325,86327],{"className":86326},[1612],[40,86328,1879],{"className":86329},[1612],[40,86331,86332,86335],{"style":8925},[40,86333],{"className":86334,"style":3354},[1641],[40,86336],{"className":86337,"style":8933},[8932],[40,86339,86340,86343],{"style":8936},[40,86341],{"className":86342,"style":3354},[1641],[40,86344,86346,86349],{"className":86345},[1612],[40,86347,80975],{"className":86348},[1612],[40,86350,80017],{"className":86351,"style":80089},[1612,1653],[40,86353,2269],{"className":86354},[2268],[40,86356,86358],{"className":86357},[1629],[40,86359,86361],{"className":86360,"style":10802},[1633],[40,86362],{},[40,86364],{"className":86365},[1822,8840],[40,86367,86369],{"className":86368,"style":1987},[1822,1986],[40,86370,2693],{"className":86371},[1991,1648],[40,86373],{"className":86374,"style":2009},[1756],[40,86376,1823],{"className":86377},[1612],[12,86379,86380,86381,86410],{},"The pulse duration required for ideal population inversion defines the standard ",[40,86382,86384,86398],{"className":86383},[1565],[40,86385,86387],{"className":86386},[1569],[1571,86388,86389],{"xmlns":1573},[1575,86390,86391,86395],{},[1578,86392,86393],{},[1584,86394,7607],{},[1591,86396,86397],{"encoding":1593},"\\pi",[40,86399,86401],{"className":86400,"ariaHidden":1599},[1598],[40,86402,86404,86407],{"className":86403},[1603],[40,86405],{"className":86406,"style":1681},[1607],[40,86408,7607],{"className":86409,"style":1818},[1612,1653],"-pulse time,",[40,86412,86414],{"className":86413},[1827],[40,86415,86417,86467],{"className":86416},[1565],[40,86418,86420],{"className":86419},[1569],[1571,86421,86422],{"xmlns":1573,"display":1836},[1575,86423,86424,86464],{},[1578,86425,86426,86432,86434,86440,86442,86462],{},[2212,86427,86428,86430],{},[1584,86429,80017],{},[1584,86431,7607],{},[1731,86433,623],{},[8538,86435,86436,86438],{},[1584,86437,7607],{},[1584,86439,80975],{"mathvariant":1790},[1731,86441,623],{},[8538,86443,86444,86446],{},[1735,86445,1410],{},[1578,86447,86448,86450,86452,86454,86456,86458,86460],{},[1735,86449,1879],{},[1731,86451,241],{"stretchy":1797},[1584,86453,80975],{"mathvariant":1790},[1584,86455,7610],{"mathvariant":1790},[1735,86457,1879],{},[1584,86459,7607],{},[1731,86461,2693],{"stretchy":1797},[1584,86463,1823],{"mathvariant":1790},[1591,86465,86466],{"encoding":1593},"\\tau_\\pi=\\frac{\\pi}{\\Omega}=\\frac{1}{2(\\Omega\u002F2\\pi)}.",[40,86468,86470,86526,86605],{"className":86469,"ariaHidden":1599},[1598],[40,86471,86473,86476,86517,86520,86523],{"className":86472},[1603],[40,86474],{"className":86475,"style":24251},[1607],[40,86477,86479,86482],{"className":86478},[1612],[40,86480,80017],{"className":86481,"style":80089},[1612,1653],[40,86483,86485],{"className":86484},[1621],[40,86486,86488,86509],{"className":86487},[1625,2244],[40,86489,86491,86506],{"className":86490},[1629],[40,86492,86494],{"className":86493,"style":3551},[1633],[40,86495,86497,86500],{"style":86496},"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;",[40,86498],{"className":86499,"style":1642},[1641],[40,86501,86503],{"className":86502},[1646,1647,1648,1649],[40,86504,7607],{"className":86505,"style":1818},[1612,1653,1649],[40,86507,2269],{"className":86508},[2268],[40,86510,86512],{"className":86511},[1629],[40,86513,86515],{"className":86514,"style":2276},[1633],[40,86516],{},[40,86518],{"className":86519,"style":1963},[1756],[40,86521,623],{"className":86522},[1967],[40,86524],{"className":86525,"style":1963},[1756],[40,86527,86529,86533,86596,86599,86602],{"className":86528},[1603],[40,86530],{"className":86531,"style":86532},[1607],"height:1.7936em;vertical-align:-0.686em;",[40,86534,86536,86539,86593],{"className":86535},[1612],[40,86537],{"className":86538},[1985,8840],[40,86540,86542],{"className":86541},[8538],[40,86543,86545,86585],{"className":86544},[1625,2244],[40,86546,86548,86582],{"className":86547},[1629],[40,86549,86552,86563,86571],{"className":86550,"style":86551},[1633],"height:1.1076em;",[40,86553,86554,86557],{"style":10764},[40,86555],{"className":86556,"style":3354},[1641],[40,86558,86560],{"className":86559},[1612],[40,86561,80975],{"className":86562},[1612],[40,86564,86565,86568],{"style":8925},[40,86566],{"className":86567,"style":3354},[1641],[40,86569],{"className":86570,"style":8933},[8932],[40,86572,86573,86576],{"style":8936},[40,86574],{"className":86575,"style":3354},[1641],[40,86577,86579],{"className":86578},[1612],[40,86580,7607],{"className":86581,"style":1818},[1612,1653],[40,86583,2269],{"className":86584},[2268],[40,86586,86588],{"className":86587},[1629],[40,86589,86591],{"className":86590,"style":10802},[1633],[40,86592],{},[40,86594],{"className":86595},[1822,8840],[40,86597],{"className":86598,"style":1963},[1756],[40,86600,623],{"className":86601},[1967],[40,86603],{"className":86604,"style":1963},[1756],[40,86606,86608,86611,86685],{"className":86607},[1603],[40,86609],{"className":86610,"style":34893},[1607],[40,86612,86614,86617,86682],{"className":86613},[1612],[40,86615],{"className":86616},[1985,8840],[40,86618,86620],{"className":86619},[8538],[40,86621,86623,86674],{"className":86622},[1625,2244],[40,86624,86626,86671],{"className":86625},[1629],[40,86627,86629,86652,86660],{"className":86628,"style":8853},[1633],[40,86630,86631,86634],{"style":10764},[40,86632],{"className":86633,"style":3354},[1641],[40,86635,86637,86640,86643,86646,86649],{"className":86636},[1612],[40,86638,1879],{"className":86639},[1612],[40,86641,241],{"className":86642},[1985],[40,86644,81033],{"className":86645},[1612],[40,86647,7607],{"className":86648,"style":1818},[1612,1653],[40,86650,2693],{"className":86651},[1822],[40,86653,86654,86657],{"style":8925},[40,86655],{"className":86656,"style":3354},[1641],[40,86658],{"className":86659,"style":8933},[8932],[40,86661,86662,86665],{"style":8936},[40,86663],{"className":86664,"style":3354},[1641],[40,86666,86668],{"className":86667},[1612],[40,86669,1410],{"className":86670},[1612],[40,86672,2269],{"className":86673},[2268],[40,86675,86677],{"className":86676},[1629],[40,86678,86680],{"className":86679,"style":28611},[1633],[40,86681],{},[40,86683],{"className":86684},[1822,8840],[40,86686,1823],{"className":86687},[1612],[12,86689,86690],{},"Given the default drive strength",[40,86692,86694],{"className":86693},[1827],[40,86695,86697,86737],{"className":86696},[1565],[40,86698,86700],{"className":86699},[1569],[1571,86701,86702],{"xmlns":1573,"display":1836},[1575,86703,86704,86734],{},[1578,86705,86706,86716,86718,86721,86723,86732],{},[8538,86707,86708,86710],{},[1584,86709,80975],{"mathvariant":1790},[1578,86711,86712,86714],{},[1735,86713,1879],{},[1584,86715,7607],{},[1731,86717,623],{},[1735,86719,86720],{},"20",[5211,86722,5213],{},[1578,86724,86725,86728,86730],{},[1584,86726,86727],{"mathvariant":1790},"M",[1584,86729,2182],{"mathvariant":1790},[1584,86731,19585],{"mathvariant":1790},[1731,86733,1862],{"separator":1599},[1591,86735,86736],{"encoding":1593},"\\frac{\\Omega}{2\\pi}=20~\\mathrm{MHz},",[40,86738,86740,86821],{"className":86739,"ariaHidden":1599},[1598],[40,86741,86743,86747,86812,86815,86818],{"className":86742},[1603],[40,86744],{"className":86745,"style":86746},[1607],"height:2.0463em;vertical-align:-0.686em;",[40,86748,86750,86753,86809],{"className":86749},[1612],[40,86751],{"className":86752},[1985,8840],[40,86754,86756],{"className":86755},[8538],[40,86757,86759,86801],{"className":86758},[1625,2244],[40,86760,86762,86798],{"className":86761},[1629],[40,86763,86765,86779,86787],{"className":86764,"style":85353},[1633],[40,86766,86767,86770],{"style":10764},[40,86768],{"className":86769,"style":3354},[1641],[40,86771,86773,86776],{"className":86772},[1612],[40,86774,1879],{"className":86775},[1612],[40,86777,7607],{"className":86778,"style":1818},[1612,1653],[40,86780,86781,86784],{"style":8925},[40,86782],{"className":86783,"style":3354},[1641],[40,86785],{"className":86786,"style":8933},[8932],[40,86788,86789,86792],{"style":8936},[40,86790],{"className":86791,"style":3354},[1641],[40,86793,86795],{"className":86794},[1612],[40,86796,80975],{"className":86797},[1612],[40,86799,2269],{"className":86800},[2268],[40,86802,86804],{"className":86803},[1629],[40,86805,86807],{"className":86806,"style":10802},[1633],[40,86808],{},[40,86810],{"className":86811},[1822,8840],[40,86813],{"className":86814,"style":1963},[1756],[40,86816,623],{"className":86817},[1967],[40,86819],{"className":86820,"style":1963},[1756],[40,86822,86824,86828,86831,86834,86841],{"className":86823},[1603],[40,86825],{"className":86826,"style":86827},[1607],"height:0.8778em;vertical-align:-0.1944em;",[40,86829,86720],{"className":86830},[1612],[40,86832,5213],{"className":86833},[1756,82202],[40,86835,86837],{"className":86836},[1612],[40,86838,86840],{"className":86839},[1612,82209],"MHz",[40,86842,1862],{"className":86843},[2005],[12,86845,86846,86847,86875],{},"the ideal ",[40,86848,86850,86863],{"className":86849},[1565],[40,86851,86853],{"className":86852},[1569],[1571,86854,86855],{"xmlns":1573},[1575,86856,86857,86861],{},[1578,86858,86859],{},[1584,86860,7607],{},[1591,86862,86397],{"encoding":1593},[40,86864,86866],{"className":86865,"ariaHidden":1599},[1598],[40,86867,86869,86872],{"className":86868},[1603],[40,86870],{"className":86871,"style":1681},[1607],[40,86873,7607],{"className":86874,"style":1818},[1612,1653],"-pulse time is",[40,86877,86879],{"className":86878},[1827],[40,86880,86882,86915],{"className":86881},[1565],[40,86883,86885],{"className":86884},[1569],[1571,86886,86887],{"xmlns":1573,"display":1836},[1575,86888,86889,86912],{},[1578,86890,86891,86897,86899,86902,86904,86910],{},[2212,86892,86893,86895],{},[1584,86894,80017],{},[1584,86896,7607],{},[1731,86898,623],{},[1735,86900,86901],{},"25",[5211,86903,5213],{},[1578,86905,86906,86908],{},[1584,86907,3464],{"mathvariant":1790},[1584,86909,13582],{"mathvariant":1790},[1584,86911,1823],{"mathvariant":1790},[1591,86913,86914],{"encoding":1593},"\\tau_\\pi=25~\\mathrm{ns}.",[40,86916,86918,86973],{"className":86917,"ariaHidden":1599},[1598],[40,86919,86921,86924,86964,86967,86970],{"className":86920},[1603],[40,86922],{"className":86923,"style":24251},[1607],[40,86925,86927,86930],{"className":86926},[1612],[40,86928,80017],{"className":86929,"style":80089},[1612,1653],[40,86931,86933],{"className":86932},[1621],[40,86934,86936,86956],{"className":86935},[1625,2244],[40,86937,86939,86953],{"className":86938},[1629],[40,86940,86942],{"className":86941,"style":3551},[1633],[40,86943,86944,86947],{"style":86496},[40,86945],{"className":86946,"style":1642},[1641],[40,86948,86950],{"className":86949},[1646,1647,1648,1649],[40,86951,7607],{"className":86952,"style":1818},[1612,1653,1649],[40,86954,2269],{"className":86955},[2268],[40,86957,86959],{"className":86958},[1629],[40,86960,86962],{"className":86961,"style":2276},[1633],[40,86963],{},[40,86965],{"className":86966,"style":1963},[1756],[40,86968,623],{"className":86969},[1967],[40,86971],{"className":86972,"style":1963},[1756],[40,86974,86976,86979,86982,86985,86992],{"className":86975},[1603],[40,86977],{"className":86978,"style":1771},[1607],[40,86980,86901],{"className":86981},[1612],[40,86983,5213],{"className":86984},[1756,82202],[40,86986,86988],{"className":86987},[1612],[40,86989,86991],{"className":86990},[1612,82209],"ns",[40,86993,1823],{"className":86994},[1612],[12,86996,86997],{},"A full period of the resonant excited-state probability oscillation is",[40,86999,87001],{"className":87000},[1827],[40,87002,87004,87052],{"className":87003},[1565],[40,87005,87007],{"className":87006},[1569],[1571,87008,87009],{"xmlns":1573,"display":1836},[1575,87010,87011,87049],{},[1578,87012,87013,87019,87021,87031,87033,87047],{},[2212,87014,87015,87017],{},[1584,87016,36101],{},[1584,87018,19419],{},[1731,87020,623],{},[8538,87022,87023,87029],{},[1578,87024,87025,87027],{},[1735,87026,1879],{},[1584,87028,7607],{},[1584,87030,80975],{"mathvariant":1790},[1731,87032,623],{},[8538,87034,87035,87037],{},[1735,87036,1410],{},[1578,87038,87039,87041,87043,87045],{},[1584,87040,80975],{"mathvariant":1790},[1584,87042,7610],{"mathvariant":1790},[1735,87044,1879],{},[1584,87046,7607],{},[1584,87048,1823],{"mathvariant":1790},[1591,87050,87051],{"encoding":1593},"T_R=\\frac{2\\pi}{\\Omega}=\\frac{1}{\\Omega\u002F2\\pi}.",[40,87053,87055,87110,87191],{"className":87054,"ariaHidden":1599},[1598],[40,87056,87058,87061,87101,87104,87107],{"className":87057},[1603],[40,87059],{"className":87060,"style":2231},[1607],[40,87062,87064,87067],{"className":87063},[1612],[40,87065,36101],{"className":87066,"style":2264},[1612,1653],[40,87068,87070],{"className":87069},[1621],[40,87071,87073,87093],{"className":87072},[1625,2244],[40,87074,87076,87090],{"className":87075},[1629],[40,87077,87079],{"className":87078,"style":2251},[1633],[40,87080,87081,87084],{"style":6681},[40,87082],{"className":87083,"style":1642},[1641],[40,87085,87087],{"className":87086},[1646,1647,1648,1649],[40,87088,19419],{"className":87089,"style":19444},[1612,1653,1649],[40,87091,2269],{"className":87092},[2268],[40,87094,87096],{"className":87095},[1629],[40,87097,87099],{"className":87098,"style":2276},[1633],[40,87100],{},[40,87102],{"className":87103,"style":1963},[1756],[40,87105,623],{"className":87106},[1967],[40,87108],{"className":87109,"style":1963},[1756],[40,87111,87113,87117,87182,87185,87188],{"className":87112},[1603],[40,87114],{"className":87115,"style":87116},[1607],"height:2.0074em;vertical-align:-0.686em;",[40,87118,87120,87123,87179],{"className":87119},[1612],[40,87121],{"className":87122},[1985,8840],[40,87124,87126],{"className":87125},[8538],[40,87127,87129,87171],{"className":87128},[1625,2244],[40,87130,87132,87168],{"className":87131},[1629],[40,87133,87135,87146,87154],{"className":87134,"style":8853},[1633],[40,87136,87137,87140],{"style":10764},[40,87138],{"className":87139,"style":3354},[1641],[40,87141,87143],{"className":87142},[1612],[40,87144,80975],{"className":87145},[1612],[40,87147,87148,87151],{"style":8925},[40,87149],{"className":87150,"style":3354},[1641],[40,87152],{"className":87153,"style":8933},[8932],[40,87155,87156,87159],{"style":8936},[40,87157],{"className":87158,"style":3354},[1641],[40,87160,87162,87165],{"className":87161},[1612],[40,87163,1879],{"className":87164},[1612],[40,87166,7607],{"className":87167,"style":1818},[1612,1653],[40,87169,2269],{"className":87170},[2268],[40,87172,87174],{"className":87173},[1629],[40,87175,87177],{"className":87176,"style":10802},[1633],[40,87178],{},[40,87180],{"className":87181},[1822,8840],[40,87183],{"className":87184,"style":1963},[1756],[40,87186,623],{"className":87187},[1967],[40,87189],{"className":87190,"style":1963},[1756],[40,87192,87194,87197,87262],{"className":87193},[1603],[40,87195],{"className":87196,"style":34893},[1607],[40,87198,87200,87203,87259],{"className":87199},[1612],[40,87201],{"className":87202},[1985,8840],[40,87204,87206],{"className":87205},[8538],[40,87207,87209,87251],{"className":87208},[1625,2244],[40,87210,87212,87248],{"className":87211},[1629],[40,87213,87215,87229,87237],{"className":87214,"style":8853},[1633],[40,87216,87217,87220],{"style":10764},[40,87218],{"className":87219,"style":3354},[1641],[40,87221,87223,87226],{"className":87222},[1612],[40,87224,81033],{"className":87225},[1612],[40,87227,7607],{"className":87228,"style":1818},[1612,1653],[40,87230,87231,87234],{"style":8925},[40,87232],{"className":87233,"style":3354},[1641],[40,87235],{"className":87236,"style":8933},[8932],[40,87238,87239,87242],{"style":8936},[40,87240],{"className":87241,"style":3354},[1641],[40,87243,87245],{"className":87244},[1612],[40,87246,1410],{"className":87247},[1612],[40,87249,2269],{"className":87250},[2268],[40,87252,87254],{"className":87253},[1629],[40,87255,87257],{"className":87256,"style":28611},[1633],[40,87258],{},[40,87260],{"className":87261},[1822,8840],[40,87263,1823],{"className":87264},[1612],[12,87266,87267],{},"For the same default drive strength, the full temporal period is",[40,87269,87271],{"className":87270},[1827],[40,87272,87274,87307],{"className":87273},[1565],[40,87275,87277],{"className":87276},[1569],[1571,87278,87279],{"xmlns":1573,"display":1836},[1575,87280,87281,87304],{},[1578,87282,87283,87289,87291,87294,87296,87302],{},[2212,87284,87285,87287],{},[1584,87286,36101],{},[1584,87288,19419],{},[1731,87290,623],{},[1735,87292,87293],{},"50",[5211,87295,5213],{},[1578,87297,87298,87300],{},[1584,87299,3464],{"mathvariant":1790},[1584,87301,13582],{"mathvariant":1790},[1584,87303,1823],{"mathvariant":1790},[1591,87305,87306],{"encoding":1593},"T_R=50~\\mathrm{ns}.",[40,87308,87310,87365],{"className":87309,"ariaHidden":1599},[1598],[40,87311,87313,87316,87356,87359,87362],{"className":87312},[1603],[40,87314],{"className":87315,"style":2231},[1607],[40,87317,87319,87322],{"className":87318},[1612],[40,87320,36101],{"className":87321,"style":2264},[1612,1653],[40,87323,87325],{"className":87324},[1621],[40,87326,87328,87348],{"className":87327},[1625,2244],[40,87329,87331,87345],{"className":87330},[1629],[40,87332,87334],{"className":87333,"style":2251},[1633],[40,87335,87336,87339],{"style":6681},[40,87337],{"className":87338,"style":1642},[1641],[40,87340,87342],{"className":87341},[1646,1647,1648,1649],[40,87343,19419],{"className":87344,"style":19444},[1612,1653,1649],[40,87346,2269],{"className":87347},[2268],[40,87349,87351],{"className":87350},[1629],[40,87352,87354],{"className":87353,"style":2276},[1633],[40,87355],{},[40,87357],{"className":87358,"style":1963},[1756],[40,87360,623],{"className":87361},[1967],[40,87363],{"className":87364,"style":1963},[1756],[40,87366,87368,87371,87374,87377,87383],{"className":87367},[1603],[40,87369],{"className":87370,"style":1771},[1607],[40,87372,87293],{"className":87373},[1612],[40,87375,5213],{"className":87376},[1756,82202],[40,87378,87380],{"className":87379},[1612],[40,87381,86991],{"className":87382},[1612,82209],[40,87384,1823],{"className":87385},[1612],[55,87387,87389],{"id":87388},"phenomenological-dephasing","Phenomenological Dephasing",[12,87391,87392],{},"The computational model includes an optional phenomenological damping factor,",[40,87394,87396],{"className":87395},[1827],[40,87397,87399,87471],{"className":87398},[1565],[40,87400,87402],{"className":87401},[1569],[1571,87403,87404],{"xmlns":1573,"display":1836},[1575,87405,87406,87468],{},[1578,87407,87408,87414,87416,87418,87420,87422,87424,87426,87432,87434,87436,87438,87440,87442,87444,87446,87466],{},[2212,87409,87410,87412],{},[1584,87411,2218],{},[1584,87413,2330],{},[1731,87415,241],{"stretchy":1797},[1584,87417,21331],{},[1731,87419,1862],{"separator":1599},[1584,87421,80017],{},[1731,87423,2693],{"stretchy":1797},[1731,87425,13983],{},[2212,87427,87428,87430],{},[1584,87429,2218],{},[1584,87431,2330],{},[1731,87433,241],{"stretchy":1797},[1584,87435,21331],{},[1731,87437,1862],{"separator":1599},[1584,87439,80017],{},[1731,87441,2693],{"stretchy":1797},[1584,87443,23123],{},[1731,87445,35745],{},[1578,87447,87448,87450,87452,87464],{},[1731,87449,241],{"fence":1599},[1731,87451,1733],{},[8538,87453,87454,87456],{},[1584,87455,80017],{},[2819,87457,87458,87460,87462],{},[1584,87459,36101],{},[1735,87461,1879],{},[1731,87463,19819],{},[1731,87465,2693],{"fence":1599},[1584,87467,1823],{"mathvariant":1790},[1591,87469,87470],{"encoding":1593},"P_e(f,\\tau)\\rightarrow P_e(f,\\tau)\\exp\\left(-\\frac{\\tau}{T_2^\\ast}\\right).",[40,87472,87474,87547],{"className":87473,"ariaHidden":1599},[1598],[40,87475,87477,87480,87520,87523,87526,87529,87532,87535,87538,87541,87544],{"className":87476},[1603],[40,87478],{"className":87479,"style":1811},[1607],[40,87481,87483,87486],{"className":87482},[1612],[40,87484,2218],{"className":87485,"style":2264},[1612,1653],[40,87487,87489],{"className":87488},[1621],[40,87490,87492,87512],{"className":87491},[1625,2244],[40,87493,87495,87509],{"className":87494},[1629],[40,87496,87498],{"className":87497,"style":3551},[1633],[40,87499,87500,87503],{"style":6681},[40,87501],{"className":87502,"style":1642},[1641],[40,87504,87506],{"className":87505},[1646,1647,1648,1649],[40,87507,2330],{"className":87508},[1612,1653,1649],[40,87510,2269],{"className":87511},[2268],[40,87513,87515],{"className":87514},[1629],[40,87516,87518],{"className":87517,"style":2276},[1633],[40,87519],{},[40,87521,241],{"className":87522},[1985],[40,87524,21331],{"className":87525,"style":26393},[1612,1653],[40,87527,1862],{"className":87528},[2005],[40,87530],{"className":87531,"style":2009},[1756],[40,87533,80017],{"className":87534,"style":80089},[1612,1653],[40,87536,2693],{"className":87537},[1822],[40,87539],{"className":87540,"style":1963},[1756],[40,87542,13983],{"className":87543},[1967],[40,87545],{"className":87546,"style":1963},[1756],[40,87548,87550,87554,87594,87597,87600,87603,87606,87609,87612,87615,87618,87621,87753,87756],{"className":87549},[1603],[40,87551],{"className":87552,"style":87553},[1607],"height:2.4023em;vertical-align:-0.9523em;",[40,87555,87557,87560],{"className":87556},[1612],[40,87558,2218],{"className":87559,"style":2264},[1612,1653],[40,87561,87563],{"className":87562},[1621],[40,87564,87566,87586],{"className":87565},[1625,2244],[40,87567,87569,87583],{"className":87568},[1629],[40,87570,87572],{"className":87571,"style":3551},[1633],[40,87573,87574,87577],{"style":6681},[40,87575],{"className":87576,"style":1642},[1641],[40,87578,87580],{"className":87579},[1646,1647,1648,1649],[40,87581,2330],{"className":87582},[1612,1653,1649],[40,87584,2269],{"className":87585},[2268],[40,87587,87589],{"className":87588},[1629],[40,87590,87592],{"className":87591,"style":2276},[1633],[40,87593],{},[40,87595,241],{"className":87596},[1985],[40,87598,21331],{"className":87599,"style":26393},[1612,1653],[40,87601,1862],{"className":87602},[2005],[40,87604],{"className":87605,"style":2009},[1756],[40,87607,80017],{"className":87608,"style":80089},[1612,1653],[40,87610,2693],{"className":87611},[1822],[40,87613],{"className":87614,"style":2009},[1756],[40,87616,23123],{"className":87617},[6614],[40,87619],{"className":87620,"style":2009},[1756],[40,87622,87624,87630,87633,87747],{"className":87623},[1981],[40,87625,87627],{"className":87626,"style":1987},[1985,1986],[40,87628,241],{"className":87629},[1991,1648],[40,87631,1733],{"className":87632},[1612],[40,87634,87636,87639,87744],{"className":87635},[1612],[40,87637],{"className":87638},[1985,8840],[40,87640,87642],{"className":87641},[8538],[40,87643,87645,87735],{"className":87644},[1625,2244],[40,87646,87648,87732],{"className":87647},[1629],[40,87649,87651,87713,87721],{"className":87650,"style":86551},[1633],[40,87652,87653,87656],{"style":10764},[40,87654],{"className":87655,"style":3354},[1641],[40,87657,87659],{"className":87658},[1612],[40,87660,87662,87665],{"className":87661},[1612],[40,87663,36101],{"className":87664,"style":2264},[1612,1653],[40,87666,87668],{"className":87667},[1621],[40,87669,87671,87704],{"className":87670},[1625,2244],[40,87672,87674,87701],{"className":87673},[1629],[40,87675,87678,87690],{"className":87676,"style":87677},[1633],"height:0.6705em;",[40,87679,87681,87684],{"style":87680},"top:-2.4337em;margin-left:-0.1389em;margin-right:0.05em;",[40,87682],{"className":87683,"style":1642},[1641],[40,87685,87687],{"className":87686},[1646,1647,1648,1649],[40,87688,1879],{"className":87689},[1612,1649],[40,87691,87692,87695],{"style":85207},[40,87693],{"className":87694,"style":1642},[1641],[40,87696,87698],{"className":87697},[1646,1647,1648,1649],[40,87699,19819],{"className":87700},[1761,1649],[40,87702,2269],{"className":87703},[2268],[40,87705,87707],{"className":87706},[1629],[40,87708,87711],{"className":87709,"style":87710},[1633],"height:0.2663em;",[40,87712],{},[40,87714,87715,87718],{"style":8925},[40,87716],{"className":87717,"style":3354},[1641],[40,87719],{"className":87720,"style":8933},[8932],[40,87722,87723,87726],{"style":8936},[40,87724],{"className":87725,"style":3354},[1641],[40,87727,87729],{"className":87728},[1612],[40,87730,80017],{"className":87731,"style":80089},[1612,1653],[40,87733,2269],{"className":87734},[2268],[40,87736,87738],{"className":87737},[1629],[40,87739,87742],{"className":87740,"style":87741},[1633],"height:0.9523em;",[40,87743],{},[40,87745],{"className":87746},[1822,8840],[40,87748,87750],{"className":87749,"style":1987},[1822,1986],[40,87751,2693],{"className":87752},[1991,1648],[40,87754],{"className":87755,"style":2009},[1756],[40,87757,1823],{"className":87758},[1612],[12,87760,87761],{},"This exponential envelope attenuates the plotted Rabi fringes as pulse duration increases. The factor is a compact visualization device for contrast decay. A rigorous open-quantum-system treatment would require a density matrix model with relaxation channels, dephasing channels, finite temperature effects, shaped pulse envelopes, and stochastic dynamics. The default dephasing parameter is",[40,87763,87765],{"className":87764},[1827],[40,87766,87768,87803],{"className":87767},[1565],[40,87769,87771],{"className":87770},[1569],[1571,87772,87773],{"xmlns":1573,"display":1836},[1575,87774,87775,87800],{},[1578,87776,87777,87785,87787,87790,87792,87798],{},[2819,87778,87779,87781,87783],{},[1584,87780,36101],{},[1735,87782,1879],{},[1731,87784,19819],{},[1731,87786,623],{},[1735,87788,87789],{},"500",[5211,87791,5213],{},[1578,87793,87794,87796],{},[1584,87795,3464],{"mathvariant":1790},[1584,87797,13582],{"mathvariant":1790},[1584,87799,1823],{"mathvariant":1790},[1591,87801,87802],{"encoding":1593},"T_2^\\ast=500~\\mathrm{ns}.",[40,87804,87806,87874],{"className":87805,"ariaHidden":1599},[1598],[40,87807,87809,87813,87865,87868,87871],{"className":87808},[1603],[40,87810],{"className":87811,"style":87812},[1607],"height:0.9857em;vertical-align:-0.247em;",[40,87814,87816,87819],{"className":87815},[1612],[40,87817,36101],{"className":87818,"style":2264},[1612,1653],[40,87820,87822],{"className":87821},[1621],[40,87823,87825,87857],{"className":87824},[1625,2244],[40,87826,87828,87854],{"className":87827},[1629],[40,87829,87831,87843],{"className":87830,"style":36200},[1633],[40,87832,87834,87837],{"style":87833},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[40,87835],{"className":87836,"style":1642},[1641],[40,87838,87840],{"className":87839},[1646,1647,1648,1649],[40,87841,1879],{"className":87842},[1612,1649],[40,87844,87845,87848],{"style":1950},[40,87846],{"className":87847,"style":1642},[1641],[40,87849,87851],{"className":87850},[1646,1647,1648,1649],[40,87852,19819],{"className":87853},[1761,1649],[40,87855,2269],{"className":87856},[2268],[40,87858,87860],{"className":87859},[1629],[40,87861,87863],{"className":87862,"style":2957},[1633],[40,87864],{},[40,87866],{"className":87867,"style":1963},[1756],[40,87869,623],{"className":87870},[1967],[40,87872],{"className":87873,"style":1963},[1756],[40,87875,87877,87880,87883,87886,87892],{"className":87876},[1603],[40,87878],{"className":87879,"style":1771},[1607],[40,87881,87789],{"className":87882},[1612],[40,87884,5213],{"className":87885},[1756,82202],[40,87887,87889],{"className":87888},[1612],[40,87890,86991],{"className":87891},[1612,82209],[40,87893,1823],{"className":87894},[1612],[12,87896,87897,87898,87901],{},"Setting ",[37,87899,87900],{},"T2STAR_NS = None"," disables the phenomenological damping factor.",[55,87903,87905],{"id":87904},"detuning-and-generalized-rabi-frequency","Detuning and Generalized Rabi Frequency",[12,87907,87908],{},"Detuning quantifies the spectral mismatch between the applied drive and the natural qubit transition,",[40,87910,87912],{"className":87911},[1827],[40,87913,87915,87951],{"className":87914},[1565],[40,87916,87918],{"className":87917},[1569],[1571,87919,87920],{"xmlns":1573,"display":1836},[1575,87921,87922,87948],{},[1578,87923,87924,87926,87928,87930,87932,87934,87936,87938,87944,87946],{},[1584,87925,81058],{"mathvariant":1790},[1731,87927,623],{},[1735,87929,1879],{},[1584,87931,7607],{},[1731,87933,241],{"stretchy":1797},[1584,87935,21331],{},[1731,87937,1733],{},[2212,87939,87940,87942],{},[1584,87941,21331],{},[1735,87943,791],{},[1731,87945,2693],{"stretchy":1797},[1584,87947,1823],{"mathvariant":1790},[1591,87949,87950],{"encoding":1593},"\\Delta=2\\pi(f-f_0).",[40,87952,87954,87972,87999],{"className":87953,"ariaHidden":1599},[1598],[40,87955,87957,87960,87963,87966,87969],{"className":87956},[1603],[40,87958],{"className":87959,"style":1608},[1607],[40,87961,81058],{"className":87962},[1612],[40,87964],{"className":87965,"style":1963},[1756],[40,87967,623],{"className":87968},[1967],[40,87970],{"className":87971,"style":1963},[1756],[40,87973,87975,87978,87981,87984,87987,87990,87993,87996],{"className":87974},[1603],[40,87976],{"className":87977,"style":1811},[1607],[40,87979,1879],{"className":87980},[1612],[40,87982,7607],{"className":87983,"style":1818},[1612,1653],[40,87985,241],{"className":87986},[1985],[40,87988,21331],{"className":87989,"style":26393},[1612,1653],[40,87991],{"className":87992,"style":1757},[1756],[40,87994,1733],{"className":87995},[1761],[40,87997],{"className":87998,"style":1757},[1756],[40,88000,88002,88005,88045,88048],{"className":88001},[1603],[40,88003],{"className":88004,"style":1811},[1607],[40,88006,88008,88011],{"className":88007},[1612],[40,88009,21331],{"className":88010,"style":26393},[1612,1653],[40,88012,88014],{"className":88013},[1621],[40,88015,88017,88037],{"className":88016},[1625,2244],[40,88018,88020,88034],{"className":88019},[1629],[40,88021,88023],{"className":88022,"style":13330},[1633],[40,88024,88025,88028],{"style":80580},[40,88026],{"className":88027,"style":1642},[1641],[40,88029,88031],{"className":88030},[1646,1647,1648,1649],[40,88032,791],{"className":88033},[1612,1649],[40,88035,2269],{"className":88036},[2268],[40,88038,88040],{"className":88039},[1629],[40,88041,88043],{"className":88042,"style":2276},[1633],[40,88044],{},[40,88046,2693],{"className":88047},[1822],[40,88049,1823],{"className":88050},[1612],[12,88052,88053],{},"Expressed in ordinary frequency units, the generalized Rabi frequency is",[40,88055,88057],{"className":88056},[1827],[40,88058,88060,88148],{"className":88059},[1565],[40,88061,88063],{"className":88062},[1569],[1571,88064,88065],{"xmlns":1573,"display":1836},[1575,88066,88067,88145],{},[1578,88068,88069,88075,88077,88079,88081,88083,88097,88099,88143],{},[2212,88070,88071,88073],{},[1584,88072,81288],{},[1584,88074,19419],{},[1731,88076,241],{"stretchy":1797},[1584,88078,21331],{},[1731,88080,2693],{"stretchy":1797},[1731,88082,623],{},[8538,88084,88085,88091],{},[2212,88086,88087,88089],{},[1584,88088,80975],{"mathvariant":1790},[1584,88090,19419],{},[1578,88092,88093,88095],{},[1735,88094,1879],{},[1584,88096,7607],{},[1731,88098,623],{},[8543,88100,88101],{},[1578,88102,88103,88123,88125,88127,88129,88131,88137],{},[1581,88104,88105,88121],{},[1578,88106,88107,88109,88119],{},[1731,88108,241],{"fence":1599},[8538,88110,88111,88113],{},[1584,88112,80975],{"mathvariant":1790},[1578,88114,88115,88117],{},[1735,88116,1879],{},[1584,88118,7607],{},[1731,88120,2693],{"fence":1599},[1735,88122,1879],{},[1731,88124,1147],{},[1731,88126,241],{"stretchy":1797},[1584,88128,21331],{},[1731,88130,1733],{},[2212,88132,88133,88135],{},[1584,88134,21331],{},[1735,88136,791],{},[1581,88138,88139,88141],{},[1731,88140,2693],{"stretchy":1797},[1735,88142,1879],{},[1584,88144,1823],{"mathvariant":1790},[1591,88146,88147],{"encoding":1593},"\\nu_R(f)=\\frac{\\Omega_R}{2\\pi}=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[40,88149,88151,88215,88332],{"className":88150,"ariaHidden":1599},[1598],[40,88152,88154,88157,88197,88200,88203,88206,88209,88212],{"className":88153},[1603],[40,88155],{"className":88156,"style":1811},[1607],[40,88158,88160,88163],{"className":88159},[1612],[40,88161,81288],{"className":88162,"style":81309},[1612,1653],[40,88164,88166],{"className":88165},[1621],[40,88167,88169,88189],{"className":88168},[1625,2244],[40,88170,88172,88186],{"className":88171},[1629],[40,88173,88175],{"className":88174,"style":2251},[1633],[40,88176,88177,88180],{"style":81324},[40,88178],{"className":88179,"style":1642},[1641],[40,88181,88183],{"className":88182},[1646,1647,1648,1649],[40,88184,19419],{"className":88185,"style":19444},[1612,1653,1649],[40,88187,2269],{"className":88188},[2268],[40,88190,88192],{"className":88191},[1629],[40,88193,88195],{"className":88194,"style":2276},[1633],[40,88196],{},[40,88198,241],{"className":88199},[1985],[40,88201,21331],{"className":88202,"style":26393},[1612,1653],[40,88204,2693],{"className":88205},[1822],[40,88207],{"className":88208,"style":1963},[1756],[40,88210,623],{"className":88211},[1967],[40,88213],{"className":88214,"style":1963},[1756],[40,88216,88218,88221,88323,88326,88329],{"className":88217},[1603],[40,88219],{"className":88220,"style":86746},[1607],[40,88222,88224,88227,88320],{"className":88223},[1612],[40,88225],{"className":88226},[1985,8840],[40,88228,88230],{"className":88229},[8538],[40,88231,88233,88312],{"className":88232},[1625,2244],[40,88234,88236,88309],{"className":88235},[1629],[40,88237,88239,88253,88261],{"className":88238,"style":85353},[1633],[40,88240,88241,88244],{"style":10764},[40,88242],{"className":88243,"style":3354},[1641],[40,88245,88247,88250],{"className":88246},[1612],[40,88248,1879],{"className":88249},[1612],[40,88251,7607],{"className":88252,"style":1818},[1612,1653],[40,88254,88255,88258],{"style":8925},[40,88256],{"className":88257,"style":3354},[1641],[40,88259],{"className":88260,"style":8933},[8932],[40,88262,88263,88266],{"style":8936},[40,88264],{"className":88265,"style":3354},[1641],[40,88267,88269],{"className":88268},[1612],[40,88270,88272,88275],{"className":88271},[1612],[40,88273,80975],{"className":88274},[1612],[40,88276,88278],{"className":88277},[1621],[40,88279,88281,88301],{"className":88280},[1625,2244],[40,88282,88284,88298],{"className":88283},[1629],[40,88285,88287],{"className":88286,"style":2251},[1633],[40,88288,88289,88292],{"style":4442},[40,88290],{"className":88291,"style":1642},[1641],[40,88293,88295],{"className":88294},[1646,1647,1648,1649],[40,88296,19419],{"className":88297,"style":19444},[1612,1653,1649],[40,88299,2269],{"className":88300},[2268],[40,88302,88304],{"className":88303},[1629],[40,88305,88307],{"className":88306,"style":2276},[1633],[40,88308],{},[40,88310,2269],{"className":88311},[2268],[40,88313,88315],{"className":88314},[1629],[40,88316,88318],{"className":88317,"style":10802},[1633],[40,88319],{},[40,88321],{"className":88322},[1822,8840],[40,88324],{"className":88325,"style":1963},[1756],[40,88327,623],{"className":88328},[1967],[40,88330],{"className":88331,"style":1963},[1756],[40,88333,88335,88339,88593],{"className":88334},[1603],[40,88336],{"className":88337,"style":88338},[1607],"height:3.04em;vertical-align:-1.0541em;",[40,88340,88342],{"className":88341},[1612,8866],[40,88343,88345,88584],{"className":88344},[1625,2244],[40,88346,88348,88581],{"className":88347},[1629],[40,88349,88352,88564],{"className":88350,"style":88351},[1633],"height:1.9859em;",[40,88353,88356,88360],{"className":88354,"style":88355},[8880],"top:-5em;",[40,88357],{"className":88358,"style":88359},[1641],"height:5em;",[40,88361,88363,88471,88474,88477,88480,88483,88486,88489,88492,88495,88535],{"className":88362,"style":85586},[1612],[40,88364,88366,88446],{"className":88365},[1981],[40,88367,88369,88375,88440],{"className":88368},[1981],[40,88370,88372],{"className":88371,"style":1987},[1985,1986],[40,88373,241],{"className":88374},[1991,1648],[40,88376,88378,88381,88437],{"className":88377},[1612],[40,88379],{"className":88380},[1985,8840],[40,88382,88384],{"className":88383},[8538],[40,88385,88387,88429],{"className":88386},[1625,2244],[40,88388,88390,88426],{"className":88389},[1629],[40,88391,88393,88407,88415],{"className":88392,"style":85353},[1633],[40,88394,88395,88398],{"style":10764},[40,88396],{"className":88397,"style":3354},[1641],[40,88399,88401,88404],{"className":88400},[1612],[40,88402,1879],{"className":88403},[1612],[40,88405,7607],{"className":88406,"style":1818},[1612,1653],[40,88408,88409,88412],{"style":8925},[40,88410],{"className":88411,"style":3354},[1641],[40,88413],{"className":88414,"style":8933},[8932],[40,88416,88417,88420],{"style":8936},[40,88418],{"className":88419,"style":3354},[1641],[40,88421,88423],{"className":88422},[1612],[40,88424,80975],{"className":88425},[1612],[40,88427,2269],{"className":88428},[2268],[40,88430,88432],{"className":88431},[1629],[40,88433,88435],{"className":88434,"style":10802},[1633],[40,88436],{},[40,88438],{"className":88439},[1822,8840],[40,88441,88443],{"className":88442,"style":1987},[1822,1986],[40,88444,2693],{"className":88445},[1991,1648],[40,88447,88449],{"className":88448},[1621],[40,88450,88452],{"className":88451},[1625],[40,88453,88455],{"className":88454},[1629],[40,88456,88459],{"className":88457,"style":88458},[1633],"height:1.654em;",[40,88460,88462,88465],{"style":88461},"top:-3.9029em;margin-right:0.05em;",[40,88463],{"className":88464,"style":1642},[1641],[40,88466,88468],{"className":88467},[1646,1647,1648,1649],[40,88469,1879],{"className":88470},[1612,1649],[40,88472],{"className":88473,"style":1757},[1756],[40,88475,1147],{"className":88476},[1761],[40,88478],{"className":88479,"style":1757},[1756],[40,88481,241],{"className":88482},[1985],[40,88484,21331],{"className":88485,"style":26393},[1612,1653],[40,88487],{"className":88488,"style":1757},[1756],[40,88490,1733],{"className":88491},[1761],[40,88493],{"className":88494,"style":1757},[1756],[40,88496,88498,88501],{"className":88497},[1612],[40,88499,21331],{"className":88500,"style":26393},[1612,1653],[40,88502,88504],{"className":88503},[1621],[40,88505,88507,88527],{"className":88506},[1625,2244],[40,88508,88510,88524],{"className":88509},[1629],[40,88511,88513],{"className":88512,"style":13330},[1633],[40,88514,88515,88518],{"style":80580},[40,88516],{"className":88517,"style":1642},[1641],[40,88519,88521],{"className":88520},[1646,1647,1648,1649],[40,88522,791],{"className":88523},[1612,1649],[40,88525,2269],{"className":88526},[2268],[40,88528,88530],{"className":88529},[1629],[40,88531,88533],{"className":88532,"style":2276},[1633],[40,88534],{},[40,88536,88538,88541],{"className":88537},[1822],[40,88539,2693],{"className":88540},[1822],[40,88542,88544],{"className":88543},[1621],[40,88545,88547],{"className":88546},[1625],[40,88548,88550],{"className":88549},[1629],[40,88551,88553],{"className":88552,"style":28464},[1633],[40,88554,88555,88558],{"style":28467},[40,88556],{"className":88557,"style":1642},[1641],[40,88559,88561],{"className":88560},[1646,1647,1648,1649],[40,88562,1879],{"className":88563},[1612,1649],[40,88565,88567,88570],{"style":88566},"top:-3.9459em;",[40,88568],{"className":88569,"style":88359},[1641],[40,88571,88574],{"className":88572,"style":88573},[8900],"min-width:1.02em;height:3.08em;",[5325,88575,88578],{"xmlns":5327,"width":8904,"height":88576,"viewBox":88577,"preserveAspectRatio":8907},"3.08em","0 0 400000 3240",[5334,88579],{"d":88580},"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z",[40,88582,2269],{"className":88583},[2268],[40,88585,88587],{"className":88586},[1629],[40,88588,88591],{"className":88589,"style":88590},[1633],"height:1.0541em;",[40,88592],{},[40,88594,1823],{"className":88595},[1612],[12,88597,88598],{},"This expression explains the ridge visible in the Fourier-domain heatmap. At resonance, the generalized frequency equals the applied drive strength,",[40,88600,88602],{"className":88601},[1827],[40,88603,88605,88647],{"className":88604},[1565],[40,88606,88608],{"className":88607},[1569],[1571,88609,88610],{"xmlns":1573,"display":1836},[1575,88611,88612,88644],{},[1578,88613,88614,88620,88622,88628,88630,88632,88642],{},[2212,88615,88616,88618],{},[1584,88617,81288],{},[1584,88619,19419],{},[1731,88621,241],{"stretchy":1797},[2212,88623,88624,88626],{},[1584,88625,21331],{},[1735,88627,791],{},[1731,88629,2693],{"stretchy":1797},[1731,88631,623],{},[8538,88633,88634,88636],{},[1584,88635,80975],{"mathvariant":1790},[1578,88637,88638,88640],{},[1735,88639,1879],{},[1584,88641,7607],{},[1584,88643,1823],{"mathvariant":1790},[1591,88645,88646],{"encoding":1593},"\\nu_R(f_0)=\\frac{\\Omega}{2\\pi}.",[40,88648,88650,88751],{"className":88649,"ariaHidden":1599},[1598],[40,88651,88653,88656,88696,88699,88739,88742,88745,88748],{"className":88652},[1603],[40,88654],{"className":88655,"style":1811},[1607],[40,88657,88659,88662],{"className":88658},[1612],[40,88660,81288],{"className":88661,"style":81309},[1612,1653],[40,88663,88665],{"className":88664},[1621],[40,88666,88668,88688],{"className":88667},[1625,2244],[40,88669,88671,88685],{"className":88670},[1629],[40,88672,88674],{"className":88673,"style":2251},[1633],[40,88675,88676,88679],{"style":81324},[40,88677],{"className":88678,"style":1642},[1641],[40,88680,88682],{"className":88681},[1646,1647,1648,1649],[40,88683,19419],{"className":88684,"style":19444},[1612,1653,1649],[40,88686,2269],{"className":88687},[2268],[40,88689,88691],{"className":88690},[1629],[40,88692,88694],{"className":88693,"style":2276},[1633],[40,88695],{},[40,88697,241],{"className":88698},[1985],[40,88700,88702,88705],{"className":88701},[1612],[40,88703,21331],{"className":88704,"style":26393},[1612,1653],[40,88706,88708],{"className":88707},[1621],[40,88709,88711,88731],{"className":88710},[1625,2244],[40,88712,88714,88728],{"className":88713},[1629],[40,88715,88717],{"className":88716,"style":13330},[1633],[40,88718,88719,88722],{"style":80580},[40,88720],{"className":88721,"style":1642},[1641],[40,88723,88725],{"className":88724},[1646,1647,1648,1649],[40,88726,791],{"className":88727},[1612,1649],[40,88729,2269],{"className":88730},[2268],[40,88732,88734],{"className":88733},[1629],[40,88735,88737],{"className":88736,"style":2276},[1633],[40,88738],{},[40,88740,2693],{"className":88741},[1822],[40,88743],{"className":88744,"style":1963},[1756],[40,88746,623],{"className":88747},[1967],[40,88749],{"className":88750,"style":1963},[1756],[40,88752,88754,88757,88822],{"className":88753},[1603],[40,88755],{"className":88756,"style":86746},[1607],[40,88758,88760,88763,88819],{"className":88759},[1612],[40,88761],{"className":88762},[1985,8840],[40,88764,88766],{"className":88765},[8538],[40,88767,88769,88811],{"className":88768},[1625,2244],[40,88770,88772,88808],{"className":88771},[1629],[40,88773,88775,88789,88797],{"className":88774,"style":85353},[1633],[40,88776,88777,88780],{"style":10764},[40,88778],{"className":88779,"style":3354},[1641],[40,88781,88783,88786],{"className":88782},[1612],[40,88784,1879],{"className":88785},[1612],[40,88787,7607],{"className":88788,"style":1818},[1612,1653],[40,88790,88791,88794],{"style":8925},[40,88792],{"className":88793,"style":3354},[1641],[40,88795],{"className":88796,"style":8933},[8932],[40,88798,88799,88802],{"style":8936},[40,88800],{"className":88801,"style":3354},[1641],[40,88803,88805],{"className":88804},[1612],[40,88806,80975],{"className":88807},[1612],[40,88809,2269],{"className":88810},[2268],[40,88812,88814],{"className":88813},[1629],[40,88815,88817],{"className":88816,"style":10802},[1633],[40,88818],{},[40,88820],{"className":88821},[1822,8840],[40,88823,1823],{"className":88824},[1612],[12,88826,88827,88828,88897,88898,89000],{},"As the system moves away from resonance, ",[40,88829,88831,88848],{"className":88830},[1565],[40,88832,88834],{"className":88833},[1569],[1571,88835,88836],{"xmlns":1573},[1575,88837,88838,88846],{},[1578,88839,88840],{},[2212,88841,88842,88844],{},[1584,88843,81288],{},[1584,88845,19419],{},[1591,88847,81293],{"encoding":1593},[40,88849,88851],{"className":88850,"ariaHidden":1599},[1598],[40,88852,88854,88857],{"className":88853},[1603],[40,88855],{"className":88856,"style":24251},[1607],[40,88858,88860,88863],{"className":88859},[1612],[40,88861,81288],{"className":88862,"style":81309},[1612,1653],[40,88864,88866],{"className":88865},[1621],[40,88867,88869,88889],{"className":88868},[1625,2244],[40,88870,88872,88886],{"className":88871},[1629],[40,88873,88875],{"className":88874,"style":2251},[1633],[40,88876,88877,88880],{"style":81324},[40,88878],{"className":88879,"style":1642},[1641],[40,88881,88883],{"className":88882},[1646,1647,1648,1649],[40,88884,19419],{"className":88885,"style":19444},[1612,1653,1649],[40,88887,2269],{"className":88888},[2268],[40,88890,88892],{"className":88891},[1629],[40,88893,88895],{"className":88894,"style":2276},[1633],[40,88896],{}," increases symmetrically with ",[40,88899,88901,88927],{"className":88900},[1565],[40,88902,88904],{"className":88903},[1569],[1571,88905,88906],{"xmlns":1573},[1575,88907,88908,88924],{},[1578,88909,88910,88912,88914,88916,88922],{},[1731,88911,1791],{"stretchy":1797},[1584,88913,21331],{},[1731,88915,1733],{},[2212,88917,88918,88920],{},[1584,88919,21331],{},[1735,88921,791],{},[1731,88923,1791],{"stretchy":1797},[1591,88925,88926],{"encoding":1593},"\\lvert f-f_0\\rvert",[40,88928,88930,88951],{"className":88929,"ariaHidden":1599},[1598],[40,88931,88933,88936,88939,88942,88945,88948],{"className":88932},[1603],[40,88934],{"className":88935,"style":1811},[1607],[40,88937,1791],{"className":88938},[1985],[40,88940,21331],{"className":88941,"style":26393},[1612,1653],[40,88943],{"className":88944,"style":1757},[1756],[40,88946,1733],{"className":88947},[1761],[40,88949],{"className":88950,"style":1757},[1756],[40,88952,88954,88957,88997],{"className":88953},[1603],[40,88955],{"className":88956,"style":1811},[1607],[40,88958,88960,88963],{"className":88959},[1612],[40,88961,21331],{"className":88962,"style":26393},[1612,1653],[40,88964,88966],{"className":88965},[1621],[40,88967,88969,88989],{"className":88968},[1625,2244],[40,88970,88972,88986],{"className":88971},[1629],[40,88973,88975],{"className":88974,"style":13330},[1633],[40,88976,88977,88980],{"style":80580},[40,88978],{"className":88979,"style":1642},[1641],[40,88981,88983],{"className":88982},[1646,1647,1648,1649],[40,88984,791],{"className":88985},[1612,1649],[40,88987,2269],{"className":88988},[2268],[40,88990,88992],{"className":88991},[1629],[40,88993,88995],{"className":88994,"style":2276},[1633],[40,88996],{},[40,88998,1791],{"className":88999},[1822],". 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.",[55,89002,89004],{"id":89003},"quantum-information-interpretation","Quantum Information Interpretation",[12,89006,89007,89008,89078],{},"In quantum information language, a calibrated resonant Rabi pulse implements a single-qubit rotation. At exact resonance, the effective Hamiltonian is proportional to ",[40,89009,89011,89029],{"className":89010},[1565],[40,89012,89014],{"className":89013},[1569],[1571,89015,89016],{"xmlns":1573},[1575,89017,89018,89026],{},[1578,89019,89020],{},[2212,89021,89022,89024],{},[1584,89023,83020],{},[1584,89025,8185],{},[1591,89027,89028],{"encoding":1593},"\\sigma_x",[40,89030,89032],{"className":89031,"ariaHidden":1599},[1598],[40,89033,89035,89038],{"className":89034},[1603],[40,89036],{"className":89037,"style":24251},[1607],[40,89039,89041,89044],{"className":89040},[1612],[40,89042,83020],{"className":89043,"style":1818},[1612,1653],[40,89045,89047],{"className":89046},[1621],[40,89048,89050,89070],{"className":89049},[1625,2244],[40,89051,89053,89067],{"className":89052},[1629],[40,89054,89056],{"className":89055,"style":3551},[1633],[40,89057,89058,89061],{"style":3616},[40,89059],{"className":89060,"style":1642},[1641],[40,89062,89064],{"className":89063},[1646,1647,1648,1649],[40,89065,8185],{"className":89066},[1612,1653,1649],[40,89068,2269],{"className":89069},[2268],[40,89071,89073],{"className":89072},[1629],[40,89074,89076],{"className":89075,"style":2276},[1633],[40,89077],{},", so the microwave pulse implements",[40,89080,89082],{"className":89081},[1827],[40,89083,89085,89142],{"className":89084},[1565],[40,89086,89088],{"className":89087},[1569],[1571,89089,89090],{"xmlns":1573,"display":1836},[1575,89091,89092,89139],{},[1578,89093,89094,89100,89102,89105,89107,89109,89111,89113,89137],{},[2212,89095,89096,89098],{},[1584,89097,19419],{},[1584,89099,8185],{},[1731,89101,241],{"stretchy":1797},[1584,89103,89104],{},"θ",[1731,89106,2693],{"stretchy":1797},[1731,89108,623],{},[1584,89110,23123],{},[1731,89112,35745],{},[1578,89114,89115,89117,89119,89135],{},[1731,89116,241],{"fence":1599},[1731,89118,1733],{},[8538,89120,89121,89133],{},[1578,89122,89123,89125,89127],{},[1584,89124,2337],{},[1584,89126,89104],{},[2212,89128,89129,89131],{},[1584,89130,83020],{},[1584,89132,8185],{},[1735,89134,1879],{},[1731,89136,2693],{"fence":1599},[1731,89138,1862],{"separator":1599},[1591,89140,89141],{"encoding":1593},"R_x(\\theta)=\\exp\\left(-\\frac{i\\theta\\sigma_x}{2}\\right),",[40,89143,89145,89209],{"className":89144,"ariaHidden":1599},[1598],[40,89146,89148,89151,89191,89194,89197,89200,89203,89206],{"className":89147},[1603],[40,89149],{"className":89150,"style":1811},[1607],[40,89152,89154,89157],{"className":89153},[1612],[40,89155,19419],{"className":89156,"style":19444},[1612,1653],[40,89158,89160],{"className":89159},[1621],[40,89161,89163,89183],{"className":89162},[1625,2244],[40,89164,89166,89180],{"className":89165},[1629],[40,89167,89169],{"className":89168,"style":3551},[1633],[40,89170,89171,89174],{"style":19459},[40,89172],{"className":89173,"style":1642},[1641],[40,89175,89177],{"className":89176},[1646,1647,1648,1649],[40,89178,8185],{"className":89179},[1612,1653,1649],[40,89181,2269],{"className":89182},[2268],[40,89184,89186],{"className":89185},[1629],[40,89187,89189],{"className":89188,"style":2276},[1633],[40,89190],{},[40,89192,241],{"className":89193},[1985],[40,89195,89104],{"className":89196,"style":1654},[1612,1653],[40,89198,2693],{"className":89199},[1822],[40,89201],{"className":89202,"style":1963},[1756],[40,89204,623],{"className":89205},[1967],[40,89207],{"className":89208,"style":1963},[1756],[40,89210,89212,89215,89218,89221,89344,89347],{"className":89211},[1603],[40,89213],{"className":89214,"style":36723},[1607],[40,89216,23123],{"className":89217},[6614],[40,89219],{"className":89220,"style":2009},[1756],[40,89222,89224,89230,89233,89338],{"className":89223},[1981],[40,89225,89227],{"className":89226,"style":1987},[1985,1986],[40,89228,241],{"className":89229},[1991,1648],[40,89231,1733],{"className":89232},[1612],[40,89234,89236,89239,89335],{"className":89235},[1612],[40,89237],{"className":89238},[1985,8840],[40,89240,89242],{"className":89241},[8538],[40,89243,89245,89327],{"className":89244},[1625,2244],[40,89246,89248,89324],{"className":89247},[1629],[40,89249,89251,89262,89270],{"className":89250,"style":36813},[1633],[40,89252,89253,89256],{"style":10764},[40,89254],{"className":89255,"style":3354},[1641],[40,89257,89259],{"className":89258},[1612],[40,89260,1879],{"className":89261},[1612],[40,89263,89264,89267],{"style":8925},[40,89265],{"className":89266,"style":3354},[1641],[40,89268],{"className":89269,"style":8933},[8932],[40,89271,89272,89275],{"style":8936},[40,89273],{"className":89274,"style":3354},[1641],[40,89276,89278,89281,89284],{"className":89277},[1612],[40,89279,2337],{"className":89280},[1612,1653],[40,89282,89104],{"className":89283,"style":1654},[1612,1653],[40,89285,89287,89290],{"className":89286},[1612],[40,89288,83020],{"className":89289,"style":1818},[1612,1653],[40,89291,89293],{"className":89292},[1621],[40,89294,89296,89316],{"className":89295},[1625,2244],[40,89297,89299,89313],{"className":89298},[1629],[40,89300,89302],{"className":89301,"style":3551},[1633],[40,89303,89304,89307],{"style":3616},[40,89305],{"className":89306,"style":1642},[1641],[40,89308,89310],{"className":89309},[1646,1647,1648,1649],[40,89311,8185],{"className":89312},[1612,1653,1649],[40,89314,2269],{"className":89315},[2268],[40,89317,89319],{"className":89318},[1629],[40,89320,89322],{"className":89321,"style":2276},[1633],[40,89323],{},[40,89325,2269],{"className":89326},[2268],[40,89328,89330],{"className":89329},[1629],[40,89331,89333],{"className":89332,"style":10802},[1633],[40,89334],{},[40,89336],{"className":89337},[1822,8840],[40,89339,89341],{"className":89340,"style":1987},[1822,1986],[40,89342,2693],{"className":89343},[1991,1648],[40,89345],{"className":89346,"style":2009},[1756],[40,89348,1862],{"className":89349},[2005],[12,89351,85452],{},[40,89353,89355],{"className":89354},[1827],[40,89356,89358,89380],{"className":89357},[1565],[40,89359,89361],{"className":89360},[1569],[1571,89362,89363],{"xmlns":1573,"display":1836},[1575,89364,89365,89377],{},[1578,89366,89367,89369,89371,89373,89375],{},[1584,89368,89104],{},[1731,89370,623],{},[1584,89372,80975],{"mathvariant":1790},[1584,89374,80017],{},[1584,89376,1823],{"mathvariant":1790},[1591,89378,89379],{"encoding":1593},"\\theta=\\Omega\\tau.",[40,89381,89383,89401],{"className":89382,"ariaHidden":1599},[1598],[40,89384,89386,89389,89392,89395,89398],{"className":89385},[1603],[40,89387],{"className":89388,"style":4478},[1607],[40,89390,89104],{"className":89391,"style":1654},[1612,1653],[40,89393],{"className":89394,"style":1963},[1756],[40,89396,623],{"className":89397},[1967],[40,89399],{"className":89400,"style":1963},[1756],[40,89402,89404,89407,89410,89413],{"className":89403},[1603],[40,89405],{"className":89406,"style":1608},[1607],[40,89408,80975],{"className":89409},[1612],[40,89411,80017],{"className":89412,"style":80089},[1612,1653],[40,89414,1823],{"className":89415},[1612],[12,89417,89418,89419,89447],{},"A calibrated resonant ",[40,89420,89422,89435],{"className":89421},[1565],[40,89423,89425],{"className":89424},[1569],[1571,89426,89427],{"xmlns":1573},[1575,89428,89429,89433],{},[1578,89430,89431],{},[1584,89432,7607],{},[1591,89434,86397],{"encoding":1593},[40,89436,89438],{"className":89437,"ariaHidden":1599},[1598],[40,89439,89441,89444],{"className":89440},[1603],[40,89442],{"className":89443,"style":1681},[1607],[40,89445,7607],{"className":89446,"style":1818},[1612,1653],"-pulse enacts",[40,89449,89451],{"className":89450},[1827],[40,89452,89454,89492],{"className":89453},[1565],[40,89455,89457],{"className":89456},[1569],[1571,89458,89459],{"xmlns":1573,"display":1836},[1575,89460,89461,89489],{},[1578,89462,89463,89469,89471,89473,89475,89477,89479,89481,89487],{},[2212,89464,89465,89467],{},[1584,89466,19419],{},[1584,89468,8185],{},[1731,89470,241],{"stretchy":1797},[1584,89472,7607],{},[1731,89474,2693],{"stretchy":1797},[1731,89476,623],{},[1731,89478,1733],{},[1584,89480,2337],{},[2212,89482,89483,89485],{},[1584,89484,83020],{},[1584,89486,8185],{},[1731,89488,1862],{"separator":1599},[1591,89490,89491],{"encoding":1593},"R_x(\\pi)=-i\\sigma_x,",[40,89493,89495,89559],{"className":89494,"ariaHidden":1599},[1598],[40,89496,89498,89501,89541,89544,89547,89550,89553,89556],{"className":89497},[1603],[40,89499],{"className":89500,"style":1811},[1607],[40,89502,89504,89507],{"className":89503},[1612],[40,89505,19419],{"className":89506,"style":19444},[1612,1653],[40,89508,89510],{"className":89509},[1621],[40,89511,89513,89533],{"className":89512},[1625,2244],[40,89514,89516,89530],{"className":89515},[1629],[40,89517,89519],{"className":89518,"style":3551},[1633],[40,89520,89521,89524],{"style":19459},[40,89522],{"className":89523,"style":1642},[1641],[40,89525,89527],{"className":89526},[1646,1647,1648,1649],[40,89528,8185],{"className":89529},[1612,1653,1649],[40,89531,2269],{"className":89532},[2268],[40,89534,89536],{"className":89535},[1629],[40,89537,89539],{"className":89538,"style":2276},[1633],[40,89540],{},[40,89542,241],{"className":89543},[1985],[40,89545,7607],{"className":89546,"style":1818},[1612,1653],[40,89548,2693],{"className":89549},[1822],[40,89551],{"className":89552,"style":1963},[1756],[40,89554,623],{"className":89555},[1967],[40,89557],{"className":89558,"style":1963},[1756],[40,89560,89562,89565,89568,89571,89611],{"className":89561},[1603],[40,89563],{"className":89564,"style":27041},[1607],[40,89566,1733],{"className":89567},[1612],[40,89569,2337],{"className":89570},[1612,1653],[40,89572,89574,89577],{"className":89573},[1612],[40,89575,83020],{"className":89576,"style":1818},[1612,1653],[40,89578,89580],{"className":89579},[1621],[40,89581,89583,89603],{"className":89582},[1625,2244],[40,89584,89586,89600],{"className":89585},[1629],[40,89587,89589],{"className":89588,"style":3551},[1633],[40,89590,89591,89594],{"style":3616},[40,89592],{"className":89593,"style":1642},[1641],[40,89595,89597],{"className":89596},[1646,1647,1648,1649],[40,89598,8185],{"className":89599},[1612,1653,1649],[40,89601,2269],{"className":89602},[2268],[40,89604,89606],{"className":89605},[1629],[40,89607,89609],{"className":89608,"style":2276},[1633],[40,89610],{},[40,89612,1862],{"className":89613},[2005],[12,89615,89616,89617,89645,89646,89682],{},"which corresponds to a quantum ",[40,89618,89620,89633],{"className":89619},[1565],[40,89621,89623],{"className":89622},[1569],[1571,89624,89625],{"xmlns":1573},[1575,89626,89627,89631],{},[1578,89628,89629],{},[1584,89630,6775],{},[1591,89632,6775],{"encoding":1593},[40,89634,89636],{"className":89635,"ariaHidden":1599},[1598],[40,89637,89639,89642],{"className":89638},[1603],[40,89640],{"className":89641,"style":1608},[1607],[40,89643,6775],{"className":89644,"style":6920},[1612,1653]," gate up to a global phase. A resonant ",[40,89647,89649,89667],{"className":89648},[1565],[40,89650,89652],{"className":89651},[1569],[1571,89653,89654],{"xmlns":1573},[1575,89655,89656,89664],{},[1578,89657,89658,89660,89662],{},[1584,89659,7607],{},[1584,89661,7610],{"mathvariant":1790},[1735,89663,1879],{},[1591,89665,89666],{"encoding":1593},"\\pi\u002F2",[40,89668,89670],{"className":89669,"ariaHidden":1599},[1598],[40,89671,89673,89676,89679],{"className":89672},[1603],[40,89674],{"className":89675,"style":1811},[1607],[40,89677,7607],{"className":89678,"style":1818},[1612,1653],[40,89680,25175],{"className":89681},[1612],"-pulse prepares a balanced coherent superposition,",[40,89684,89686],{"className":89685},[1827],[40,89687,89689,89735],{"className":89688},[1565],[40,89690,89692],{"className":89691},[1569],[1571,89693,89694],{"xmlns":1573,"display":1836},[1575,89695,89696,89732],{},[1578,89697,89698,89700,89702,89704,89706,89730],{},[1731,89699,1791],{"stretchy":1797},[1584,89701,36968],{},[1731,89703,1798],{"stretchy":1797},[1731,89705,13983],{},[8538,89707,89708,89726],{},[1578,89709,89710,89712,89714,89716,89718,89720,89722,89724],{},[1731,89711,1791],{"stretchy":1797},[1584,89713,36968],{},[1731,89715,1798],{"stretchy":1797},[1731,89717,1733],{},[1584,89719,2337],{},[1731,89721,1791],{"stretchy":1797},[1584,89723,2330],{},[1731,89725,1798],{"stretchy":1797},[8543,89727,89728],{},[1735,89729,1879],{},[1731,89731,1862],{"separator":1599},[1591,89733,89734],{"encoding":1593},"\\lvert g\\rangle\\rightarrow\\frac{\\lvert g\\rangle-i\\lvert e\\rangle}{\\sqrt{2}},",[40,89736,89738,89762],{"className":89737,"ariaHidden":1599},[1598],[40,89739,89741,89744,89747,89750,89753,89756,89759],{"className":89740},[1603],[40,89742],{"className":89743,"style":1811},[1607],[40,89745,1791],{"className":89746},[1985],[40,89748,36968],{"className":89749,"style":1818},[1612,1653],[40,89751,1798],{"className":89752},[1822],[40,89754],{"className":89755,"style":1963},[1756],[40,89757,13983],{"className":89758},[1967],[40,89760],{"className":89761,"style":1963},[1756],[40,89763,89765,89769,89902],{"className":89764},[1603],[40,89766],{"className":89767,"style":89768},[1607],"height:2.357em;vertical-align:-0.93em;",[40,89770,89772,89775,89899],{"className":89771},[1612],[40,89773],{"className":89774},[1985,8840],[40,89776,89778],{"className":89777},[8538],[40,89779,89781,89891],{"className":89780},[1625,2244],[40,89782,89784,89888],{"className":89783},[1629],[40,89785,89787,89842,89850],{"className":89786,"style":28419},[1633],[40,89788,89789,89792],{"style":8856},[40,89790],{"className":89791,"style":3354},[1641],[40,89793,89795],{"className":89794},[1612],[40,89796,89798],{"className":89797},[1612,8866],[40,89799,89801,89834],{"className":89800},[1625,2244],[40,89802,89804,89831],{"className":89803},[1629],[40,89805,89807,89819],{"className":89806,"style":8876},[1633],[40,89808,89810,89813],{"className":89809,"style":3350},[8880],[40,89811],{"className":89812,"style":3354},[1641],[40,89814,89816],{"className":89815,"style":8887},[1612],[40,89817,1879],{"className":89818},[1612],[40,89820,89821,89824],{"style":8893},[40,89822],{"className":89823,"style":3354},[1641],[40,89825,89827],{"className":89826,"style":8901},[8900],[5325,89828,89829],{"xmlns":5327,"width":8904,"height":8905,"viewBox":8906,"preserveAspectRatio":8907},[5334,89830],{"d":8910},[40,89832,2269],{"className":89833},[2268],[40,89835,89837],{"className":89836},[1629],[40,89838,89840],{"className":89839,"style":8920},[1633],[40,89841],{},[40,89843,89844,89847],{"style":8925},[40,89845],{"className":89846,"style":3354},[1641],[40,89848],{"className":89849,"style":8933},[8932],[40,89851,89852,89855],{"style":8936},[40,89853],{"className":89854,"style":3354},[1641],[40,89856,89858,89861,89864,89867,89870,89873,89876,89879,89882,89885],{"className":89857},[1612],[40,89859,1791],{"className":89860},[1985],[40,89862,36968],{"className":89863,"style":1818},[1612,1653],[40,89865,1798],{"className":89866},[1822],[40,89868],{"className":89869,"style":1757},[1756],[40,89871,1733],{"className":89872},[1761],[40,89874],{"className":89875,"style":1757},[1756],[40,89877,2337],{"className":89878},[1612,1653],[40,89880,1791],{"className":89881},[1985],[40,89883,2330],{"className":89884},[1612,1653],[40,89886,1798],{"className":89887},[1822],[40,89889,2269],{"className":89890},[2268],[40,89892,89894],{"className":89893},[1629],[40,89895,89897],{"className":89896,"style":8955},[1633],[40,89898],{},[40,89900],{"className":89901},[1822,8840],[40,89903,1862],{"className":89904},[2005],[12,89906,89907],{},"with a phase convention set by the microwave drive phase.",[12,89909,89910],{},"Detuning changes the rotation axis so it is no longer purely transverse. The unitary can be written as",[40,89912,89914],{"className":89913},[1827],[40,89915,89917,89987],{"className":89916},[1565],[40,89918,89920],{"className":89919},[1569],[1571,89921,89922],{"xmlns":1573,"display":1836},[1575,89923,89924,89984],{},[1578,89925,89926,89937,89939,89941,89943,89945,89947,89949,89982],{},[2212,89927,89928,89930],{},[1584,89929,19419],{},[3315,89931,89932,89934],{"accent":1599},[1584,89933,3464],{},[1731,89935,89936],{},"^",[1731,89938,241],{"stretchy":1797},[1584,89940,89104],{},[1731,89942,2693],{"stretchy":1797},[1731,89944,623],{},[1584,89946,23123],{},[1731,89948,35745],{},[1578,89950,89951,89953,89955,89965,89971,89974,89980],{},[1731,89952,241],{"fence":1599},[1731,89954,1733],{},[8538,89956,89957,89963],{},[1578,89958,89959,89961],{},[1584,89960,2337],{},[1584,89962,89104],{},[1735,89964,1879],{},[3315,89966,89967,89969],{"accent":1599},[1584,89968,3464],{},[1731,89970,89936],{},[1731,89972,89973],{},"⋅",[3315,89975,89976,89978],{"accent":1599},[1584,89977,83020],{},[1731,89979,5221],{},[1731,89981,2693],{"fence":1599},[1731,89983,1862],{"separator":1599},[1591,89985,89986],{"encoding":1593},"R_{\\hat{n}}(\\theta)=\\exp\\left(-\\frac{i\\theta}{2}\\hat{n}\\cdot\\vec{\\sigma}\\right),",[40,89988,89990,90087],{"className":89989,"ariaHidden":1599},[1598],[40,89991,89993,89996,90069,90072,90075,90078,90081,90084],{"className":89992},[1603],[40,89994],{"className":89995,"style":1811},[1607],[40,89997,89999,90002],{"className":89998},[1612],[40,90000,19419],{"className":90001,"style":19444},[1612,1653],[40,90003,90005],{"className":90004},[1621],[40,90006,90008,90061],{"className":90007},[1625,2244],[40,90009,90011,90058],{"className":90010},[1629],[40,90012,90014],{"className":90013,"style":10290},[1633],[40,90015,90016,90019],{"style":19459},[40,90017],{"className":90018,"style":1642},[1641],[40,90020,90022],{"className":90021},[1646,1647,1648,1649],[40,90023,90025],{"className":90024},[1612,1649],[40,90026,90028],{"className":90027},[1612,3338,1649],[40,90029,90031],{"className":90030},[1625],[40,90032,90034],{"className":90033},[1629],[40,90035,90037,90046],{"className":90036,"style":4478},[1633],[40,90038,90040,90043],{"style":90039},"top:-2.7em;",[40,90041],{"className":90042,"style":1642},[1641],[40,90044,3464],{"className":90045},[1612,1653,1649],[40,90047,90048,90051],{"style":90039},[40,90049],{"className":90050,"style":1642},[1641],[40,90052,90055],{"className":90053,"style":90054},[3367],"left:-0.25em;",[40,90056,89936],{"className":90057},[1612,1649],[40,90059,2269],{"className":90060},[2268],[40,90062,90064],{"className":90063},[1629],[40,90065,90067],{"className":90066,"style":2276},[1633],[40,90068],{},[40,90070,241],{"className":90071},[1985],[40,90073,89104],{"className":90074,"style":1654},[1612,1653],[40,90076,2693],{"className":90077},[1822],[40,90079],{"className":90080,"style":1963},[1756],[40,90082,623],{"className":90083},[1967],[40,90085],{"className":90086,"style":1963},[1756],[40,90088,90090,90093,90096,90099,90258,90261],{"className":90089},[1603],[40,90091],{"className":90092,"style":36723},[1607],[40,90094,23123],{"className":90095},[6614],[40,90097],{"className":90098,"style":2009},[1756],[40,90100,90102,90108,90111,90176,90207,90210,90213,90216,90252],{"className":90101},[1981],[40,90103,90105],{"className":90104,"style":1987},[1985,1986],[40,90106,241],{"className":90107},[1991,1648],[40,90109,1733],{"className":90110},[1612],[40,90112,90114,90117,90173],{"className":90113},[1612],[40,90115],{"className":90116},[1985,8840],[40,90118,90120],{"className":90119},[8538],[40,90121,90123,90165],{"className":90122},[1625,2244],[40,90124,90126,90162],{"className":90125},[1629],[40,90127,90129,90140,90148],{"className":90128,"style":36813},[1633],[40,90130,90131,90134],{"style":10764},[40,90132],{"className":90133,"style":3354},[1641],[40,90135,90137],{"className":90136},[1612],[40,90138,1879],{"className":90139},[1612],[40,90141,90142,90145],{"style":8925},[40,90143],{"className":90144,"style":3354},[1641],[40,90146],{"className":90147,"style":8933},[8932],[40,90149,90150,90153],{"style":8936},[40,90151],{"className":90152,"style":3354},[1641],[40,90154,90156,90159],{"className":90155},[1612],[40,90157,2337],{"className":90158},[1612,1653],[40,90160,89104],{"className":90161,"style":1654},[1612,1653],[40,90163,2269],{"className":90164},[2268],[40,90166,90168],{"className":90167},[1629],[40,90169,90171],{"className":90170,"style":10802},[1633],[40,90172],{},[40,90174],{"className":90175},[1822,8840],[40,90177,90179],{"className":90178},[1612,3338],[40,90180,90182],{"className":90181},[1625],[40,90183,90185],{"className":90184},[1629],[40,90186,90188,90196],{"className":90187,"style":4478},[1633],[40,90189,90190,90193],{"style":3350},[40,90191],{"className":90192,"style":3354},[1641],[40,90194,3464],{"className":90195},[1612,1653],[40,90197,90198,90201],{"style":3350},[40,90199],{"className":90200,"style":3354},[1641],[40,90202,90204],{"className":90203,"style":90054},[3367],[40,90205,89936],{"className":90206},[1612],[40,90208],{"className":90209,"style":1757},[1756],[40,90211,89973],{"className":90212},[1761],[40,90214],{"className":90215,"style":1757},[1756],[40,90217,90219],{"className":90218},[1612,3338],[40,90220,90222],{"className":90221},[1625],[40,90223,90225],{"className":90224},[1629],[40,90226,90228,90236],{"className":90227,"style":5301},[1633],[40,90229,90230,90233],{"style":3350},[40,90231],{"className":90232,"style":3354},[1641],[40,90234,83020],{"className":90235,"style":1818},[1612,1653],[40,90237,90238,90241],{"style":3350},[40,90239],{"className":90240,"style":3354},[1641],[40,90242,90245],{"className":90243,"style":90244},[3367],"left:-0.2355em;",[40,90246,90248],{"className":90247,"style":5323},[5322],[5325,90249,90250],{"xmlns":5327,"width":5328,"height":5329,"style":5330,"viewBox":5331,"preserveAspectRatio":5332},[5334,90251],{"d":5336},[40,90253,90255],{"className":90254,"style":1987},[1822,1986],[40,90256,2693],{"className":90257},[1991,1648],[40,90259],{"className":90260,"style":2009},[1756],[40,90262,1862],{"className":90263},[2005],[12,90265,85452],{},[40,90267,90269],{"className":90268},[1827],[40,90270,90272,90334],{"className":90271},[1565],[40,90273,90275],{"className":90274},[1569],[1571,90276,90277],{"xmlns":1573,"display":1836},[1575,90278,90279,90331],{},[1578,90280,90281,90287,90289,90299,90301,90303,90305,90307,90309,90311,90313,90315,90317,90319,90321,90327,90329],{},[3315,90282,90283,90285],{"accent":1599},[1584,90284,3464],{},[1731,90286,89936],{},[1731,90288,623],{},[8538,90290,90291,90293],{},[1735,90292,1410],{},[2212,90294,90295,90297],{},[1584,90296,80975],{"mathvariant":1790},[1584,90298,19419],{},[1731,90300,241],{"stretchy":1797},[1584,90302,80975],{"mathvariant":1790},[1731,90304,1862],{"separator":1599},[1735,90306,791],{},[1731,90308,1862],{"separator":1599},[1584,90310,81058],{"mathvariant":1790},[1731,90312,2693],{"stretchy":1797},[1731,90314,1862],{"separator":1599},[1756,90316],{"width":17060},[1584,90318,89104],{},[1731,90320,623],{},[2212,90322,90323,90325],{},[1584,90324,80975],{"mathvariant":1790},[1584,90326,19419],{},[1584,90328,80017],{},[1584,90330,1823],{"mathvariant":1790},[1591,90332,90333],{"encoding":1593},"\\hat{n}=\\frac{1}{\\Omega_R}(\\Omega,0,\\Delta), \\qquad \\theta=\\Omega_R\\tau.",[40,90335,90337,90383,90538],{"className":90336,"ariaHidden":1599},[1598],[40,90338,90340,90343,90374,90377,90380],{"className":90339},[1603],[40,90341],{"className":90342,"style":4478},[1607],[40,90344,90346],{"className":90345},[1612,3338],[40,90347,90349],{"className":90348},[1625],[40,90350,90352],{"className":90351},[1629],[40,90353,90355,90363],{"className":90354,"style":4478},[1633],[40,90356,90357,90360],{"style":3350},[40,90358],{"className":90359,"style":3354},[1641],[40,90361,3464],{"className":90362},[1612,1653],[40,90364,90365,90368],{"style":3350},[40,90366],{"className":90367,"style":3354},[1641],[40,90369,90371],{"className":90370,"style":90054},[3367],[40,90372,89936],{"className":90373},[1612],[40,90375],{"className":90376,"style":1963},[1756],[40,90378,623],{"className":90379},[1967],[40,90381],{"className":90382,"style":1963},[1756],[40,90384,90386,90390,90490,90493,90496,90499,90502,90505,90508,90511,90514,90517,90520,90523,90526,90529,90532,90535],{"className":90385},[1603],[40,90387],{"className":90388,"style":90389},[1607],"height:2.1574em;vertical-align:-0.836em;",[40,90391,90393,90396,90487],{"className":90392},[1612],[40,90394],{"className":90395},[1985,8840],[40,90397,90399],{"className":90398},[8538],[40,90400,90402,90478],{"className":90401},[1625,2244],[40,90403,90405,90475],{"className":90404},[1629],[40,90406,90408,90456,90464],{"className":90407,"style":8853},[1633],[40,90409,90410,90413],{"style":10764},[40,90411],{"className":90412,"style":3354},[1641],[40,90414,90416],{"className":90415},[1612],[40,90417,90419,90422],{"className":90418},[1612],[40,90420,80975],{"className":90421},[1612],[40,90423,90425],{"className":90424},[1621],[40,90426,90428,90448],{"className":90427},[1625,2244],[40,90429,90431,90445],{"className":90430},[1629],[40,90432,90434],{"className":90433,"style":2251},[1633],[40,90435,90436,90439],{"style":4442},[40,90437],{"className":90438,"style":1642},[1641],[40,90440,90442],{"className":90441},[1646,1647,1648,1649],[40,90443,19419],{"className":90444,"style":19444},[1612,1653,1649],[40,90446,2269],{"className":90447},[2268],[40,90449,90451],{"className":90450},[1629],[40,90452,90454],{"className":90453,"style":2276},[1633],[40,90455],{},[40,90457,90458,90461],{"style":8925},[40,90459],{"className":90460,"style":3354},[1641],[40,90462],{"className":90463,"style":8933},[8932],[40,90465,90466,90469],{"style":8936},[40,90467],{"className":90468,"style":3354},[1641],[40,90470,90472],{"className":90471},[1612],[40,90473,1410],{"className":90474},[1612],[40,90476,2269],{"className":90477},[2268],[40,90479,90481],{"className":90480},[1629],[40,90482,90485],{"className":90483,"style":90484},[1633],"height:0.836em;",[40,90486],{},[40,90488],{"className":90489},[1822,8840],[40,90491,241],{"className":90492},[1985],[40,90494,80975],{"className":90495},[1612],[40,90497,1862],{"className":90498},[2005],[40,90500],{"className":90501,"style":2009},[1756],[40,90503,791],{"className":90504},[1612],[40,90506,1862],{"className":90507},[2005],[40,90509],{"className":90510,"style":2009},[1756],[40,90512,81058],{"className":90513},[1612],[40,90515,2693],{"className":90516},[1822],[40,90518,1862],{"className":90519},[2005],[40,90521],{"className":90522,"style":17678},[1756],[40,90524],{"className":90525,"style":2009},[1756],[40,90527,89104],{"className":90528,"style":1654},[1612,1653],[40,90530],{"className":90531,"style":1963},[1756],[40,90533,623],{"className":90534},[1967],[40,90536],{"className":90537,"style":1963},[1756],[40,90539,90541,90544,90584,90587],{"className":90540},[1603],[40,90542],{"className":90543,"style":2231},[1607],[40,90545,90547,90550],{"className":90546},[1612],[40,90548,80975],{"className":90549},[1612],[40,90551,90553],{"className":90552},[1621],[40,90554,90556,90576],{"className":90555},[1625,2244],[40,90557,90559,90573],{"className":90558},[1629],[40,90560,90562],{"className":90561,"style":2251},[1633],[40,90563,90564,90567],{"style":4442},[40,90565],{"className":90566,"style":1642},[1641],[40,90568,90570],{"className":90569},[1646,1647,1648,1649],[40,90571,19419],{"className":90572,"style":19444},[1612,1653,1649],[40,90574,2269],{"className":90575},[2268],[40,90577,90579],{"className":90578},[1629],[40,90580,90582],{"className":90581,"style":2276},[1633],[40,90583],{},[40,90585,80017],{"className":90586,"style":80089},[1612,1653],[40,90588,1823],{"className":90589},[1612],[12,90591,90592],{},"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.",[55,90594,90596],{"id":90595},"bloch-sphere-interpretation","Bloch-Sphere Interpretation",[12,90598,90599],{},"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.",[12,90601,90602,90603,90641,90642,90670,90671,90706],{},"A resonant microwave drive rotates the Bloch vector around a transverse axis. Starting from ",[40,90604,90606,90623],{"className":90605},[1565],[40,90607,90609],{"className":90608},[1569],[1571,90610,90611],{"xmlns":1573},[1575,90612,90613,90621],{},[1578,90614,90615,90617,90619],{},[1731,90616,1791],{"stretchy":1797},[1584,90618,36968],{},[1731,90620,1798],{"stretchy":1797},[1591,90622,80208],{"encoding":1593},[40,90624,90626],{"className":90625,"ariaHidden":1599},[1598],[40,90627,90629,90632,90635,90638],{"className":90628},[1603],[40,90630],{"className":90631,"style":1811},[1607],[40,90633,1791],{"className":90634},[1985],[40,90636,36968],{"className":90637,"style":1818},[1612,1653],[40,90639,1798],{"className":90640},[1822],", a properly timed ",[40,90643,90645,90658],{"className":90644},[1565],[40,90646,90648],{"className":90647},[1569],[1571,90649,90650],{"xmlns":1573},[1575,90651,90652,90656],{},[1578,90653,90654],{},[1584,90655,7607],{},[1591,90657,86397],{"encoding":1593},[40,90659,90661],{"className":90660,"ariaHidden":1599},[1598],[40,90662,90664,90667],{"className":90663},[1603],[40,90665],{"className":90666,"style":1681},[1607],[40,90668,7607],{"className":90669,"style":1818},[1612,1653],"-pulse moves the state to the excited-state pole. A ",[40,90672,90674,90691],{"className":90673},[1565],[40,90675,90677],{"className":90676},[1569],[1571,90678,90679],{"xmlns":1573},[1575,90680,90681,90689],{},[1578,90682,90683,90685,90687],{},[1584,90684,7607],{},[1584,90686,7610],{"mathvariant":1790},[1735,90688,1879],{},[1591,90690,89666],{"encoding":1593},[40,90692,90694],{"className":90693,"ariaHidden":1599},[1598],[40,90695,90697,90700,90703],{"className":90696},[1603],[40,90698],{"className":90699,"style":1811},[1607],[40,90701,7607],{"className":90702,"style":1818},[1612,1653],[40,90704,25175],{"className":90705},[1612],"-pulse moves the state to the equator, where the ground-state and excited-state measurement probabilities are equal.",[12,90708,90709,90710,90738],{},"Detuning tilts the rotation axis toward the ",[40,90711,90713,90726],{"className":90712},[1565],[40,90714,90716],{"className":90715},[1569],[1571,90717,90718],{"xmlns":1573},[1575,90719,90720,90724],{},[1578,90721,90722],{},[1584,90723,19585],{},[1591,90725,19585],{"encoding":1593},[40,90727,90729],{"className":90728,"ariaHidden":1599},[1598],[40,90730,90732,90735],{"className":90731},[1603],[40,90733],{"className":90734,"style":1681},[1607],[40,90736,19585],{"className":90737,"style":19632},[1612,1653],"-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",[40,90740,90742],{"className":90741},[1827],[40,90743,90745,90775],{"className":90744},[1565],[40,90746,90748],{"className":90747},[1569],[1571,90749,90750],{"xmlns":1573,"display":1836},[1575,90751,90752,90772],{},[1578,90753,90754,90770],{},[8538,90755,90756,90762],{},[1581,90757,90758,90760],{},[1584,90759,80975],{"mathvariant":1790},[1735,90761,1879],{},[2819,90763,90764,90766,90768],{},[1584,90765,80975],{"mathvariant":1790},[1584,90767,19419],{},[1735,90769,1879],{},[1584,90771,1823],{"mathvariant":1790},[1591,90773,90774],{"encoding":1593},"\\frac{\\Omega^2}{\\Omega_R^2}.",[40,90776,90778],{"className":90777,"ariaHidden":1599},[1598],[40,90779,90781,90784,90920],{"className":90780},[1603],[40,90782],{"className":90783,"style":85146},[1607],[40,90785,90787,90790,90917],{"className":90786},[1612],[40,90788],{"className":90789},[1985,8840],[40,90791,90793],{"className":90792},[8538],[40,90794,90796,90909],{"className":90795},[1625,2244],[40,90797,90799,90906],{"className":90798},[1629],[40,90800,90802,90861,90869],{"className":90801,"style":85165},[1633],[40,90803,90804,90807],{"style":10764},[40,90805],{"className":90806,"style":3354},[1641],[40,90808,90810],{"className":90809},[1612],[40,90811,90813,90816],{"className":90812},[1612],[40,90814,80975],{"className":90815},[1612],[40,90817,90819],{"className":90818},[1621],[40,90820,90822,90853],{"className":90821},[1625,2244],[40,90823,90825,90850],{"className":90824},[1629],[40,90826,90828,90839],{"className":90827,"style":85192},[1633],[40,90829,90830,90833],{"style":85195},[40,90831],{"className":90832,"style":1642},[1641],[40,90834,90836],{"className":90835},[1646,1647,1648,1649],[40,90837,19419],{"className":90838,"style":19444},[1612,1653,1649],[40,90840,90841,90844],{"style":85207},[40,90842],{"className":90843,"style":1642},[1641],[40,90845,90847],{"className":90846},[1646,1647,1648,1649],[40,90848,1879],{"className":90849},[1612,1649],[40,90851,2269],{"className":90852},[2268],[40,90854,90856],{"className":90855},[1629],[40,90857,90859],{"className":90858,"style":85226},[1633],[40,90860],{},[40,90862,90863,90866],{"style":8925},[40,90864],{"className":90865,"style":3354},[1641],[40,90867],{"className":90868,"style":8933},[8932],[40,90870,90871,90874],{"style":8936},[40,90872],{"className":90873,"style":3354},[1641],[40,90875,90877],{"className":90876},[1612],[40,90878,90880,90883],{"className":90879},[1612],[40,90881,80975],{"className":90882},[1612],[40,90884,90886],{"className":90885},[1621],[40,90887,90889],{"className":90888},[1625],[40,90890,90892],{"className":90891},[1629],[40,90893,90895],{"className":90894,"style":12271},[1633],[40,90896,90897,90900],{"style":1637},[40,90898],{"className":90899,"style":1642},[1641],[40,90901,90903],{"className":90902},[1646,1647,1648,1649],[40,90904,1879],{"className":90905},[1612,1649],[40,90907,2269],{"className":90908},[2268],[40,90910,90912],{"className":90911},[1629],[40,90913,90915],{"className":90914,"style":85283},[1633],[40,90916],{},[40,90918],{"className":90919},[1822,8840],[40,90921,1823],{"className":90922},[1612],[12,90924,90925,90926,90965],{},"As ",[40,90927,90929,90947],{"className":90928},[1565],[40,90930,90932],{"className":90931},[1569],[1571,90933,90934],{"xmlns":1573},[1575,90935,90936,90944],{},[1578,90937,90938,90940,90942],{},[1731,90939,1791],{"stretchy":1797},[1584,90941,81058],{"mathvariant":1790},[1731,90943,1791],{"stretchy":1797},[1591,90945,90946],{"encoding":1593},"\\lvert\\Delta\\rvert",[40,90948,90950],{"className":90949,"ariaHidden":1599},[1598],[40,90951,90953,90956,90959,90962],{"className":90952},[1603],[40,90954],{"className":90955,"style":1811},[1607],[40,90957,1791],{"className":90958},[1985],[40,90960,81058],{"className":90961},[1612],[40,90963,1791],{"className":90964},[1822]," grows, the rotation axis becomes more longitudinal, the oscillation frequency increases, and the maximum excited-state population decreases.",[55,90967,90969],{"id":90968},"fourier-analysis-dynamics","Fourier Analysis Dynamics",[12,90971,90972,90973,91001,91002,91100],{},"The spectral analysis computes a real Fast Fourier transform along the pulse-duration axis. For each sampled drive frequency ",[40,90974,90976,90989],{"className":90975},[1565],[40,90977,90979],{"className":90978},[1569],[1571,90980,90981],{"xmlns":1573},[1575,90982,90983,90987],{},[1578,90984,90985],{},[1584,90986,21331],{},[1591,90988,21331],{"encoding":1593},[40,90990,90992],{"className":90991,"ariaHidden":1599},[1598],[40,90993,90995,90998],{"className":90994},[1603],[40,90996],{"className":90997,"style":6079},[1607],[40,90999,21331],{"className":91000,"style":26393},[1612,1653],", the trace ",[40,91003,91005,91033],{"className":91004},[1565],[40,91006,91008],{"className":91007},[1569],[1571,91009,91010],{"xmlns":1573},[1575,91011,91012,91030],{},[1578,91013,91014,91020,91022,91024,91026,91028],{},[2212,91015,91016,91018],{},[1584,91017,2218],{},[1584,91019,2330],{},[1731,91021,241],{"stretchy":1797},[1584,91023,80017],{},[1731,91025,1862],{"separator":1599},[1584,91027,21331],{},[1731,91029,2693],{"stretchy":1797},[1591,91031,91032],{"encoding":1593},"P_e(\\tau,f)",[40,91034,91036],{"className":91035,"ariaHidden":1599},[1598],[40,91037,91039,91042,91082,91085,91088,91091,91094,91097],{"className":91038},[1603],[40,91040],{"className":91041,"style":1811},[1607],[40,91043,91045,91048],{"className":91044},[1612],[40,91046,2218],{"className":91047,"style":2264},[1612,1653],[40,91049,91051],{"className":91050},[1621],[40,91052,91054,91074],{"className":91053},[1625,2244],[40,91055,91057,91071],{"className":91056},[1629],[40,91058,91060],{"className":91059,"style":3551},[1633],[40,91061,91062,91065],{"style":6681},[40,91063],{"className":91064,"style":1642},[1641],[40,91066,91068],{"className":91067},[1646,1647,1648,1649],[40,91069,2330],{"className":91070},[1612,1653,1649],[40,91072,2269],{"className":91073},[2268],[40,91075,91077],{"className":91076},[1629],[40,91078,91080],{"className":91079,"style":2276},[1633],[40,91081],{},[40,91083,241],{"className":91084},[1985],[40,91086,80017],{"className":91087,"style":80089},[1612,1653],[40,91089,1862],{"className":91090},[2005],[40,91092],{"className":91093,"style":2009},[1756],[40,91095,21331],{"className":91096,"style":26393},[1612,1653],[40,91098,2693],{"className":91099},[1822]," is treated as a time-domain signal. The spectral amplitude is",[40,91102,91104],{"className":91103},[1827],[40,91105,91107,91170],{"className":91106},[1565],[40,91108,91110],{"className":91109},[1569],[1571,91111,91112],{"xmlns":1573,"display":1836},[1575,91113,91114,91167],{},[1578,91115,91116,91118,91120,91122,91124,91126,91128,91130,91165],{},[1584,91117,4901],{},[1731,91119,241],{"stretchy":1797},[1584,91121,81288],{},[1731,91123,1862],{"separator":1599},[1584,91125,21331],{},[1731,91127,2693],{"stretchy":1797},[1731,91129,623],{},[1578,91131,91132,91134,91141,91163],{},[1731,91133,1791],{"fence":1599},[2212,91135,91136,91139],{},[1584,91137,91138],{"mathvariant":1586},"F",[1584,91140,80017],{},[1578,91142,91143,91145,91151,91153,91155,91157,91159,91161],{},[1731,91144,1853],{"fence":1599},[2212,91146,91147,91149],{},[1584,91148,2218],{},[1584,91150,2330],{},[1731,91152,241],{"stretchy":1797},[1584,91154,80017],{},[1731,91156,1862],{"separator":1599},[1584,91158,21331],{},[1731,91160,2693],{"stretchy":1797},[1731,91162,1916],{"fence":1599},[1731,91164,1791],{"fence":1599},[1584,91166,1823],{"mathvariant":1790},[1591,91168,91169],{"encoding":1593},"S(\\nu,f)=\\left|\\mathcal{F}_{\\tau}\\left\\{P_e(\\tau,f)\\right\\}\\right|.",[40,91171,91173,91209],{"className":91172,"ariaHidden":1599},[1598],[40,91174,91176,91179,91182,91185,91188,91191,91194,91197,91200,91203,91206],{"className":91175},[1603],[40,91177],{"className":91178,"style":1811},[1607],[40,91180,4901],{"className":91181,"style":4936},[1612,1653],[40,91183,241],{"className":91184},[1985],[40,91186,81288],{"className":91187,"style":81309},[1612,1653],[40,91189,1862],{"className":91190},[2005],[40,91192],{"className":91193,"style":2009},[1756],[40,91195,21331],{"className":91196,"style":26393},[1612,1653],[40,91198,2693],{"className":91199},[1822],[40,91201],{"className":91202,"style":1963},[1756],[40,91204,623],{"className":91205},[1967],[40,91207],{"className":91208,"style":1963},[1756],[40,91210,91212,91215,91339,91342],{"className":91211},[1603],[40,91213],{"className":91214,"style":1811},[1607],[40,91216,91218,91221,91266,91269,91336],{"className":91217},[1981],[40,91219,1791],{"className":91220,"style":1987},[1985,1986],[40,91222,91224,91228],{"className":91223},[1612],[40,91225,91138],{"className":91226,"style":91227},[1612,1616],"margin-right:0.0993em;",[40,91229,91231],{"className":91230},[1621],[40,91232,91234,91258],{"className":91233},[1625,2244],[40,91235,91237,91255],{"className":91236},[1629],[40,91238,91240],{"className":91239,"style":3551},[1633],[40,91241,91243,91246],{"style":91242},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[40,91244],{"className":91245,"style":1642},[1641],[40,91247,91249],{"className":91248},[1646,1647,1648,1649],[40,91250,91252],{"className":91251},[1612,1649],[40,91253,80017],{"className":91254,"style":80089},[1612,1653,1649],[40,91256,2269],{"className":91257},[2268],[40,91259,91261],{"className":91260},[1629],[40,91262,91264],{"className":91263,"style":2276},[1633],[40,91265],{},[40,91267],{"className":91268,"style":2009},[1756],[40,91270,91272,91275,91315,91318,91321,91324,91327,91330,91333],{"className":91271},[1981],[40,91273,1853],{"className":91274,"style":1987},[1985,1986],[40,91276,91278,91281],{"className":91277},[1612],[40,91279,2218],{"className":91280,"style":2264},[1612,1653],[40,91282,91284],{"className":91283},[1621],[40,91285,91287,91307],{"className":91286},[1625,2244],[40,91288,91290,91304],{"className":91289},[1629],[40,91291,91293],{"className":91292,"style":3551},[1633],[40,91294,91295,91298],{"style":6681},[40,91296],{"className":91297,"style":1642},[1641],[40,91299,91301],{"className":91300},[1646,1647,1648,1649],[40,91302,2330],{"className":91303},[1612,1653,1649],[40,91305,2269],{"className":91306},[2268],[40,91308,91310],{"className":91309},[1629],[40,91311,91313],{"className":91312,"style":2276},[1633],[40,91314],{},[40,91316,241],{"className":91317},[1985],[40,91319,80017],{"className":91320,"style":80089},[1612,1653],[40,91322,1862],{"className":91323},[2005],[40,91325],{"className":91326,"style":2009},[1756],[40,91328,21331],{"className":91329,"style":26393},[1612,1653],[40,91331,2693],{"className":91332},[1822],[40,91334,1916],{"className":91335,"style":1987},[1822,1986],[40,91337,1791],{"className":91338,"style":1987},[1822,1986],[40,91340],{"className":91341,"style":2009},[1756],[40,91343,1823],{"className":91344},[1612],[12,91346,91347],{},"This probability expression contains a squared sinusoid. Using the identity",[40,91349,91351],{"className":91350},[1827],[40,91352,91354,91430],{"className":91353},[1565],[40,91355,91357],{"className":91356},[1569],[1571,91358,91359],{"xmlns":1573,"display":1836},[1575,91360,91361,91427],{},[1578,91362,91363,91373,91393,91395,91401],{},[1581,91364,91365,91371],{},[1578,91366,91367,91369],{},[1584,91368,85034],{},[1731,91370,35745],{},[1735,91372,1879],{},[1578,91374,91375,91377,91391],{},[1731,91376,241],{"fence":1599},[8538,91378,91379,91389],{},[1578,91380,91381,91387],{},[2212,91382,91383,91385],{},[1584,91384,80975],{"mathvariant":1790},[1584,91386,19419],{},[1584,91388,80017],{},[1735,91390,1879],{},[1731,91392,2693],{"fence":1599},[1731,91394,623],{},[8538,91396,91397,91399],{},[1735,91398,1410],{},[1735,91400,1879],{},[1578,91402,91403,91405,91407,91409,91411,91413,91415,91421,91423,91425],{},[1731,91404,11295],{"fence":1599},[1735,91406,1410],{},[1731,91408,1733],{},[1584,91410,36987],{},[1731,91412,35745],{},[1731,91414,241],{"stretchy":1797},[2212,91416,91417,91419],{},[1584,91418,80975],{"mathvariant":1790},[1584,91420,19419],{},[1584,91422,80017],{},[1731,91424,2693],{"stretchy":1797},[1731,91426,11320],{"fence":1599},[1591,91428,91429],{"encoding":1593},"\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right)=\\frac{1}{2}\\left[1-\\cos(\\Omega_R\\tau)\\right]",[40,91431,91433,91597],{"className":91432,"ariaHidden":1599},[1598],[40,91434,91436,91439,91468,91471,91588,91591,91594],{"className":91435},[1603],[40,91437],{"className":91438,"style":36723},[1607],[40,91440,91442,91445],{"className":91441},[6614],[40,91443,85034],{"className":91444},[6614],[40,91446,91448],{"className":91447},[1621],[40,91449,91451],{"className":91450},[1625],[40,91452,91454],{"className":91453},[1629],[40,91455,91457],{"className":91456,"style":85310},[1633],[40,91458,91459,91462],{"style":85313},[40,91460],{"className":91461,"style":1642},[1641],[40,91463,91465],{"className":91464},[1646,1647,1648,1649],[40,91466,1879],{"className":91467},[1612,1649],[40,91469],{"className":91470,"style":2009},[1756],[40,91472,91474,91480,91582],{"className":91473},[1981],[40,91475,91477],{"className":91476,"style":1987},[1985,1986],[40,91478,241],{"className":91479},[1991,1648],[40,91481,91483,91486,91579],{"className":91482},[1612],[40,91484],{"className":91485},[1985,8840],[40,91487,91489],{"className":91488},[8538],[40,91490,91492,91571],{"className":91491},[1625,2244],[40,91493,91495,91568],{"className":91494},[1629],[40,91496,91498,91509,91517],{"className":91497,"style":85353},[1633],[40,91499,91500,91503],{"style":10764},[40,91501],{"className":91502,"style":3354},[1641],[40,91504,91506],{"className":91505},[1612],[40,91507,1879],{"className":91508},[1612],[40,91510,91511,91514],{"style":8925},[40,91512],{"className":91513,"style":3354},[1641],[40,91515],{"className":91516,"style":8933},[8932],[40,91518,91519,91522],{"style":8936},[40,91520],{"className":91521,"style":3354},[1641],[40,91523,91525,91565],{"className":91524},[1612],[40,91526,91528,91531],{"className":91527},[1612],[40,91529,80975],{"className":91530},[1612],[40,91532,91534],{"className":91533},[1621],[40,91535,91537,91557],{"className":91536},[1625,2244],[40,91538,91540,91554],{"className":91539},[1629],[40,91541,91543],{"className":91542,"style":2251},[1633],[40,91544,91545,91548],{"style":4442},[40,91546],{"className":91547,"style":1642},[1641],[40,91549,91551],{"className":91550},[1646,1647,1648,1649],[40,91552,19419],{"className":91553,"style":19444},[1612,1653,1649],[40,91555,2269],{"className":91556},[2268],[40,91558,91560],{"className":91559},[1629],[40,91561,91563],{"className":91562,"style":2276},[1633],[40,91564],{},[40,91566,80017],{"className":91567,"style":80089},[1612,1653],[40,91569,2269],{"className":91570},[2268],[40,91572,91574],{"className":91573},[1629],[40,91575,91577],{"className":91576,"style":10802},[1633],[40,91578],{},[40,91580],{"className":91581},[1822,8840],[40,91583,91585],{"className":91584,"style":1987},[1822,1986],[40,91586,2693],{"className":91587},[1991,1648],[40,91589],{"className":91590,"style":1963},[1756],[40,91592,623],{"className":91593},[1967],[40,91595],{"className":91596,"style":1963},[1756],[40,91598,91600,91603,91665,91668],{"className":91599},[1603],[40,91601],{"className":91602,"style":87116},[1607],[40,91604,91606,91609,91662],{"className":91605},[1612],[40,91607],{"className":91608},[1985,8840],[40,91610,91612],{"className":91611},[8538],[40,91613,91615,91654],{"className":91614},[1625,2244],[40,91616,91618,91651],{"className":91617},[1629],[40,91619,91621,91632,91640],{"className":91620,"style":8853},[1633],[40,91622,91623,91626],{"style":10764},[40,91624],{"className":91625,"style":3354},[1641],[40,91627,91629],{"className":91628},[1612],[40,91630,1879],{"className":91631},[1612],[40,91633,91634,91637],{"style":8925},[40,91635],{"className":91636,"style":3354},[1641],[40,91638],{"className":91639,"style":8933},[8932],[40,91641,91642,91645],{"style":8936},[40,91643],{"className":91644,"style":3354},[1641],[40,91646,91648],{"className":91647},[1612],[40,91649,1410],{"className":91650},[1612],[40,91652,2269],{"className":91653},[2268],[40,91655,91657],{"className":91656},[1629],[40,91658,91660],{"className":91659,"style":10802},[1633],[40,91661],{},[40,91663],{"className":91664},[1822,8840],[40,91666],{"className":91667,"style":2009},[1756],[40,91669,91671,91674,91677,91680,91683,91686,91689,91692,91732,91735,91738],{"className":91670},[1981],[40,91672,11295],{"className":91673,"style":1987},[1985,1986],[40,91675,1410],{"className":91676},[1612],[40,91678],{"className":91679,"style":1757},[1756],[40,91681,1733],{"className":91682},[1761],[40,91684],{"className":91685,"style":1757},[1756],[40,91687,36987],{"className":91688},[6614],[40,91690,241],{"className":91691},[1985],[40,91693,91695,91698],{"className":91694},[1612],[40,91696,80975],{"className":91697},[1612],[40,91699,91701],{"className":91700},[1621],[40,91702,91704,91724],{"className":91703},[1625,2244],[40,91705,91707,91721],{"className":91706},[1629],[40,91708,91710],{"className":91709,"style":2251},[1633],[40,91711,91712,91715],{"style":4442},[40,91713],{"className":91714,"style":1642},[1641],[40,91716,91718],{"className":91717},[1646,1647,1648,1649],[40,91719,19419],{"className":91720,"style":19444},[1612,1653,1649],[40,91722,2269],{"className":91723},[2268],[40,91725,91727],{"className":91726},[1629],[40,91728,91730],{"className":91729,"style":2276},[1633],[40,91731],{},[40,91733,80017],{"className":91734,"style":80089},[1612,1653],[40,91736,2693],{"className":91737},[1822],[40,91739,11320],{"className":91740,"style":1987},[1822,1986],[12,91742,91743],{},"shows that the dominant oscillatory spectral component appears at the generalized frequency",[40,91745,91747],{"className":91746},[1827],[40,91748,91750,91786],{"className":91749},[1565],[40,91751,91753],{"className":91752},[1569],[1571,91754,91755],{"xmlns":1573,"display":1836},[1575,91756,91757,91783],{},[1578,91758,91759,91765,91767,91781],{},[2212,91760,91761,91763],{},[1584,91762,81288],{},[1584,91764,19419],{},[1731,91766,623],{},[8538,91768,91769,91775],{},[2212,91770,91771,91773],{},[1584,91772,80975],{"mathvariant":1790},[1584,91774,19419],{},[1578,91776,91777,91779],{},[1735,91778,1879],{},[1584,91780,7607],{},[1584,91782,1823],{"mathvariant":1790},[1591,91784,91785],{"encoding":1593},"\\nu_R=\\frac{\\Omega_R}{2\\pi}.",[40,91787,91789,91844],{"className":91788,"ariaHidden":1599},[1598],[40,91790,91792,91795,91835,91838,91841],{"className":91791},[1603],[40,91793],{"className":91794,"style":24251},[1607],[40,91796,91798,91801],{"className":91797},[1612],[40,91799,81288],{"className":91800,"style":81309},[1612,1653],[40,91802,91804],{"className":91803},[1621],[40,91805,91807,91827],{"className":91806},[1625,2244],[40,91808,91810,91824],{"className":91809},[1629],[40,91811,91813],{"className":91812,"style":2251},[1633],[40,91814,91815,91818],{"style":81324},[40,91816],{"className":91817,"style":1642},[1641],[40,91819,91821],{"className":91820},[1646,1647,1648,1649],[40,91822,19419],{"className":91823,"style":19444},[1612,1653,1649],[40,91825,2269],{"className":91826},[2268],[40,91828,91830],{"className":91829},[1629],[40,91831,91833],{"className":91832,"style":2276},[1633],[40,91834],{},[40,91836],{"className":91837,"style":1963},[1756],[40,91839,623],{"className":91840},[1967],[40,91842],{"className":91843,"style":1963},[1756],[40,91845,91847,91850,91952],{"className":91846},[1603],[40,91848],{"className":91849,"style":86746},[1607],[40,91851,91853,91856,91949],{"className":91852},[1612],[40,91854],{"className":91855},[1985,8840],[40,91857,91859],{"className":91858},[8538],[40,91860,91862,91941],{"className":91861},[1625,2244],[40,91863,91865,91938],{"className":91864},[1629],[40,91866,91868,91882,91890],{"className":91867,"style":85353},[1633],[40,91869,91870,91873],{"style":10764},[40,91871],{"className":91872,"style":3354},[1641],[40,91874,91876,91879],{"className":91875},[1612],[40,91877,1879],{"className":91878},[1612],[40,91880,7607],{"className":91881,"style":1818},[1612,1653],[40,91883,91884,91887],{"style":8925},[40,91885],{"className":91886,"style":3354},[1641],[40,91888],{"className":91889,"style":8933},[8932],[40,91891,91892,91895],{"style":8936},[40,91893],{"className":91894,"style":3354},[1641],[40,91896,91898],{"className":91897},[1612],[40,91899,91901,91904],{"className":91900},[1612],[40,91902,80975],{"className":91903},[1612],[40,91905,91907],{"className":91906},[1621],[40,91908,91910,91930],{"className":91909},[1625,2244],[40,91911,91913,91927],{"className":91912},[1629],[40,91914,91916],{"className":91915,"style":2251},[1633],[40,91917,91918,91921],{"style":4442},[40,91919],{"className":91920,"style":1642},[1641],[40,91922,91924],{"className":91923},[1646,1647,1648,1649],[40,91925,19419],{"className":91926,"style":19444},[1612,1653,1649],[40,91928,2269],{"className":91929},[2268],[40,91931,91933],{"className":91932},[1629],[40,91934,91936],{"className":91935,"style":2276},[1633],[40,91937],{},[40,91939,2269],{"className":91940},[2268],[40,91942,91944],{"className":91943},[1629],[40,91945,91947],{"className":91946,"style":10802},[1633],[40,91948],{},[40,91950],{"className":91951},[1822,8840],[40,91953,1823],{"className":91954},[1612],[12,91956,91957],{},"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.",[55,91959,91961],{"id":91960},"computational-methodology","Computational Methodology",[12,91963,91964,91965,91993,91994,92022,92023,9846,92051,92079,92080,92214,92215,92434],{},"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 ",[40,91966,91968,91981],{"className":91967},[1565],[40,91969,91971],{"className":91970},[1569],[1571,91972,91973],{"xmlns":1573},[1575,91974,91975,91979],{},[1578,91976,91977],{},[1584,91978,21331],{},[1591,91980,21331],{"encoding":1593},[40,91982,91984],{"className":91983,"ariaHidden":1599},[1598],[40,91985,91987,91990],{"className":91986},[1603],[40,91988],{"className":91989,"style":6079},[1607],[40,91991,21331],{"className":91992,"style":26393},[1612,1653]," in GHz and a pulse-duration grid ",[40,91995,91997,92010],{"className":91996},[1565],[40,91998,92000],{"className":91999},[1569],[1571,92001,92002],{"xmlns":1573},[1575,92003,92004,92008],{},[1578,92005,92006],{},[1584,92007,80017],{},[1591,92009,80143],{"encoding":1593},[40,92011,92013],{"className":92012,"ariaHidden":1599},[1598],[40,92014,92016,92019],{"className":92015},[1603],[40,92017],{"className":92018,"style":1681},[1607],[40,92020,80017],{"className":92021,"style":80089},[1612,1653]," in ns, then converts both to SI units. From a dense two-dimensional mesh over ",[40,92024,92026,92039],{"className":92025},[1565],[40,92027,92029],{"className":92028},[1569],[1571,92030,92031],{"xmlns":1573},[1575,92032,92033,92037],{},[1578,92034,92035],{},[1584,92036,21331],{},[1591,92038,21331],{"encoding":1593},[40,92040,92042],{"className":92041,"ariaHidden":1599},[1598],[40,92043,92045,92048],{"className":92044},[1603],[40,92046],{"className":92047,"style":6079},[1607],[40,92049,21331],{"className":92050,"style":26393},[1612,1653],[40,92052,92054,92067],{"className":92053},[1565],[40,92055,92057],{"className":92056},[1569],[1571,92058,92059],{"xmlns":1573},[1575,92060,92061,92065],{},[1578,92062,92063],{},[1584,92064,80017],{},[1591,92066,80143],{"encoding":1593},[40,92068,92070],{"className":92069,"ariaHidden":1599},[1598],[40,92071,92073,92076],{"className":92072},[1603],[40,92074],{"className":92075,"style":1681},[1607],[40,92077,80017],{"className":92078,"style":80089},[1612,1653],", the workflow computes the detuning 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evaluates the generalized Rabi angular frequency ",[40,92216,92218,92255],{"className":92217},[1565],[40,92219,92221],{"className":92220},[1569],[1571,92222,92223],{"xmlns":1573},[1575,92224,92225,92253],{},[1578,92226,92227,92233,92235],{},[2212,92228,92229,92231],{},[1584,92230,80975],{"mathvariant":1790},[1584,92232,19419],{},[1731,92234,623],{},[8543,92236,92237],{},[1578,92238,92239,92245,92247],{},[1581,92240,92241,92243],{},[1584,92242,80975],{"mathvariant":1790},[1735,92244,1879],{},[1731,92246,1147],{},[1581,92248,92249,92251],{},[1584,92250,81058],{"mathvariant":1790},[1735,92252,1879],{},[1591,92254,85496],{"encoding":1593},[40,92256,92258,92313],{"className":92257,"ariaHidden":1599},[1598],[40,92259,92261,92264,92304,92307,92310],{"className":92260},[1603],[40,92262],{"className":92263,"style":2231},[1607],[40,92265,92267,92270],{"className":92266},[1612],[40,92268,80975],{"className":92269},[1612],[40,92271,92273],{"className":92272},[1621],[40,92274,92276,92296],{"className":92275},[1625,2244],[40,92277,92279,92293],{"className":92278},[1629],[40,92280,92282],{"className":92281,"style":2251},[1633],[40,92283,92284,92287],{"style":4442},[40,92285],{"className":92286,"style":1642},[1641],[40,92288,92290],{"className":92289},[1646,1647,1648,1649],[40,92291,19419],{"className":92292,"style":19444},[1612,1653,1649],[40,92294,2269],{"className":92295},[2268],[40,92297,92299],{"className":92298},[1629],[40,92300,92302],{"className":92301,"style":2276},[1633],[40,92303],{},[40,92305],{"className":92306,"style":1963},[1756],[40,92308,623],{"className":92309},[1967],[40,92311],{"className":92312,"style":1963},[1756],[40,92314,92316,92320],{"className":92315},[1603],[40,92317],{"className":92318,"style":92319},[1607],"height:1.04em;vertical-align:-0.1266em;",[40,92321,92323],{"className":92322},[1612,8866],[40,92324,92326,92425],{"className":92325},[1625,2244],[40,92327,92329,92422],{"className":92328},[1629],[40,92330,92333,92409],{"className":92331,"style":92332},[1633],"height:0.9134em;",[40,92334,92336,92339],{"className":92335,"style":3350},[8880],[40,92337],{"className":92338,"style":3354},[1641],[40,92340,92342,92371,92374,92377,92380],{"className":92341,"style":8887},[1612],[40,92343,92345,92348],{"className":92344},[1612],[40,92346,80975],{"className":92347},[1612],[40,92349,92351],{"className":92350},[1621],[40,92352,92354],{"className":92353},[1625],[40,92355,92357],{"className":92356},[1629],[40,92358,92360],{"className":92359,"style":28464},[1633],[40,92361,92362,92365],{"style":28467},[40,92363],{"className":92364,"style":1642},[1641],[40,92366,92368],{"className":92367},[1646,1647,1648,1649],[40,92369,1879],{"className":92370},[1612,1649],[40,92372],{"className":92373,"style":1757},[1756],[40,92375,1147],{"className":92376},[1761],[40,92378],{"className":92379,"style":1757},[1756],[40,92381,92383,92386],{"className":92382},[1612],[40,92384,81058],{"className":92385},[1612],[40,92387,92389],{"className":92388},[1621],[40,92390,92392],{"className":92391},[1625],[40,92393,92395],{"className":92394},[1629],[40,92396,92398],{"className":92397,"style":28464},[1633],[40,92399,92400,92403],{"style":28467},[40,92401],{"className":92402,"style":1642},[1641],[40,92404,92406],{"className":92405},[1646,1647,1648,1649],[40,92407,1879],{"className":92408},[1612,1649],[40,92410,92412,92415],{"style":92411},"top:-2.8734em;",[40,92413],{"className":92414,"style":3354},[1641],[40,92416,92418],{"className":92417,"style":8901},[8900],[5325,92419,92420],{"xmlns":5327,"width":8904,"height":8905,"viewBox":8906,"preserveAspectRatio":8907},[5334,92421],{"d":8910},[40,92423,2269],{"className":92424},[2268],[40,92426,92428],{"className":92427},[1629],[40,92429,92432],{"className":92430,"style":92431},[1633],"height:0.1266em;",[40,92433],{},", and calculates the analytic excited-state probability",[40,92436,92438],{"className":92437},[1827],[40,92439,92441,92519],{"className":92440},[1565],[40,92442,92444],{"className":92443},[1569],[1571,92445,92446],{"xmlns":1573,"display":1836},[1575,92447,92448,92516],{},[1578,92449,92450,92456,92458,92460,92462,92464,92466,92468,92484,92494,92514],{},[2212,92451,92452,92454],{},[1584,92453,2218],{},[1584,92455,2330],{},[1731,92457,241],{"stretchy":1797},[1584,92459,21331],{},[1731,92461,1862],{"separator":1599},[1584,92463,80017],{},[1731,92465,2693],{"stretchy":1797},[1731,92467,623],{},[8538,92469,92470,92476],{},[1581,92471,92472,92474],{},[1584,92473,80975],{"mathvariant":1790},[1735,92475,1879],{},[2819,92477,92478,92480,92482],{},[1584,92479,80975],{"mathvariant":1790},[1584,92481,19419],{},[1735,92483,1879],{},[1581,92485,92486,92492],{},[1578,92487,92488,92490],{},[1584,92489,85034],{},[1731,92491,35745],{},[1735,92493,1879],{},[1578,92495,92496,92498,92512],{},[1731,92497,241],{"fence":1599},[8538,92499,92500,92510],{},[1578,92501,92502,92508],{},[2212,92503,92504,92506],{},[1584,92505,80975],{"mathvariant":1790},[1584,92507,19419],{},[1584,92509,80017],{},[1735,92511,1879],{},[1731,92513,2693],{"fence":1599},[1584,92515,1823],{"mathvariant":1790},[1591,92517,92518],{"encoding":1593},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right).",[40,92520,92522,92595],{"className":92521,"ariaHidden":1599},[1598],[40,92523,92525,92528,92568,92571,92574,92577,92580,92583,92586,92589,92592],{"className":92524},[1603],[40,92526],{"className":92527,"style":1811},[1607],[40,92529,92531,92534],{"className":92530},[1612],[40,92532,2218],{"className":92533,"style":2264},[1612,1653],[40,92535,92537],{"className":92536},[1621],[40,92538,92540,92560],{"className":92539},[1625,2244],[40,92541,92543,92557],{"className":92542},[1629],[40,92544,92546],{"className":92545,"style":3551},[1633],[40,92547,92548,92551],{"style":6681},[40,92549],{"className":92550,"style":1642},[1641],[40,92552,92554],{"className":92553},[1646,1647,1648,1649],[40,92555,2330],{"className":92556},[1612,1653,1649],[40,92558,2269],{"className":92559},[2268],[40,92561,92563],{"className":92562},[1629],[40,92564,92566],{"className":92565,"style":2276},[1633],[40,92567],{},[40,92569,241],{"className":92570},[1985],[40,92572,21331],{"className":92573,"style":26393},[1612,1653],[40,92575,1862],{"className":92576},[2005],[40,92578],{"className":92579,"style":2009},[1756],[40,92581,80017],{"className":92582,"style":80089},[1612,1653],[40,92584,2693],{"className":92585},[1822],[40,92587],{"className":92588,"style":1963},[1756],[40,92590,623],{"className":92591},[1967],[40,92593],{"className":92594,"style":1963},[1756],[40,92596,92598,92601,92737,92740,92769,92772,92889,92892],{"className":92597},[1603],[40,92599],{"className":92600,"style":85146},[1607],[40,92602,92604,92607,92734],{"className":92603},[1612],[40,92605],{"className":92606},[1985,8840],[40,92608,92610],{"className":92609},[8538],[40,92611,92613,92726],{"className":92612},[1625,2244],[40,92614,92616,92723],{"className":92615},[1629],[40,92617,92619,92678,92686],{"className":92618,"style":85165},[1633],[40,92620,92621,92624],{"style":10764},[40,92622],{"className":92623,"style":3354},[1641],[40,92625,92627],{"className":92626},[1612],[40,92628,92630,92633],{"className":92629},[1612],[40,92631,80975],{"className":92632},[1612],[40,92634,92636],{"className":92635},[1621],[40,92637,92639,92670],{"className":92638},[1625,2244],[40,92640,92642,92667],{"className":92641},[1629],[40,92643,92645,92656],{"className":92644,"style":85192},[1633],[40,92646,92647,92650],{"style":85195},[40,92648],{"className":92649,"style":1642},[1641],[40,92651,92653],{"className":92652},[1646,1647,1648,1649],[40,92654,19419],{"className":92655,"style":19444},[1612,1653,1649],[40,92657,92658,92661],{"style":85207},[40,92659],{"className":92660,"style":1642},[1641],[40,92662,92664],{"className":92663},[1646,1647,1648,1649],[40,92665,1879],{"className":92666},[1612,1649],[40,92668,2269],{"className":92669},[2268],[40,92671,92673],{"className":92672},[1629],[40,92674,92676],{"className":92675,"style":85226},[1633],[40,92677],{},[40,92679,92680,92683],{"style":8925},[40,92681],{"className":92682,"style":3354},[1641],[40,92684],{"className":92685,"style":8933},[8932],[40,92687,92688,92691],{"style":8936},[40,92689],{"className":92690,"style":3354},[1641],[40,92692,92694],{"className":92693},[1612],[40,92695,92697,92700],{"className":92696},[1612],[40,92698,80975],{"className":92699},[1612],[40,92701,92703],{"className":92702},[1621],[40,92704,92706],{"className":92705},[1625],[40,92707,92709],{"className":92708},[1629],[40,92710,92712],{"className":92711,"style":12271},[1633],[40,92713,92714,92717],{"style":1637},[40,92715],{"className":92716,"style":1642},[1641],[40,92718,92720],{"className":92719},[1646,1647,1648,1649],[40,92721,1879],{"className":92722},[1612,1649],[40,92724,2269],{"className":92725},[2268],[40,92727,92729],{"className":92728},[1629],[40,92730,92732],{"className":92731,"style":85283},[1633],[40,92733],{},[40,92735],{"className":92736},[1822,8840],[40,92738],{"className":92739,"style":2009},[1756],[40,92741,92743,92746],{"className":92742},[6614],[40,92744,85034],{"className":92745},[6614],[40,92747,92749],{"className":92748},[1621],[40,92750,92752],{"className":92751},[1625],[40,92753,92755],{"className":92754},[1629],[40,92756,92758],{"className":92757,"style":85310},[1633],[40,92759,92760,92763],{"style":85313},[40,92761],{"className":92762,"style":1642},[1641],[40,92764,92766],{"className":92765},[1646,1647,1648,1649],[40,92767,1879],{"className":92768},[1612,1649],[40,92770],{"className":92771,"style":2009},[1756],[40,92773,92775,92781,92883],{"className":92774},[1981],[40,92776,92778],{"className":92777,"style":1987},[1985,1986],[40,92779,241],{"className":92780},[1991,1648],[40,92782,92784,92787,92880],{"className":92783},[1612],[40,92785],{"className":92786},[1985,8840],[40,92788,92790],{"className":92789},[8538],[40,92791,92793,92872],{"className":92792},[1625,2244],[40,92794,92796,92869],{"className":92795},[1629],[40,92797,92799,92810,92818],{"className":92798,"style":85353},[1633],[40,92800,92801,92804],{"style":10764},[40,92802],{"className":92803,"style":3354},[1641],[40,92805,92807],{"className":92806},[1612],[40,92808,1879],{"className":92809},[1612],[40,92811,92812,92815],{"style":8925},[40,92813],{"className":92814,"style":3354},[1641],[40,92816],{"className":92817,"style":8933},[8932],[40,92819,92820,92823],{"style":8936},[40,92821],{"className":92822,"style":3354},[1641],[40,92824,92826,92866],{"className":92825},[1612],[40,92827,92829,92832],{"className":92828},[1612],[40,92830,80975],{"className":92831},[1612],[40,92833,92835],{"className":92834},[1621],[40,92836,92838,92858],{"className":92837},[1625,2244],[40,92839,92841,92855],{"className":92840},[1629],[40,92842,92844],{"className":92843,"style":2251},[1633],[40,92845,92846,92849],{"style":4442},[40,92847],{"className":92848,"style":1642},[1641],[40,92850,92852],{"className":92851},[1646,1647,1648,1649],[40,92853,19419],{"className":92854,"style":19444},[1612,1653,1649],[40,92856,2269],{"className":92857},[2268],[40,92859,92861],{"className":92860},[1629],[40,92862,92864],{"className":92863,"style":2276},[1633],[40,92865],{},[40,92867,80017],{"className":92868,"style":80089},[1612,1653],[40,92870,2269],{"className":92871},[2268],[40,92873,92875],{"className":92874},[1629],[40,92876,92878],{"className":92877,"style":10802},[1633],[40,92879],{},[40,92881],{"className":92882},[1822,8840],[40,92884,92886],{"className":92885,"style":1987},[1822,1986],[40,92887,2693],{"className":92888},[1991,1648],[40,92890],{"className":92891,"style":2009},[1756],[40,92893,1823],{"className":92894},[1612],[12,92896,92897,92898,93013],{},"The routine optionally applies the damping envelope ",[40,92899,92901,92935],{"className":92900},[1565],[40,92902,92904],{"className":92903},[1569],[1571,92905,92906],{"xmlns":1573},[1575,92907,92908,92932],{},[1578,92909,92910,92912,92914,92916,92918,92920,92922,92930],{},[1584,92911,23123],{},[1731,92913,35745],{},[1731,92915,241],{"stretchy":1797},[1731,92917,1733],{},[1584,92919,80017],{},[1584,92921,7610],{"mathvariant":1790},[2819,92923,92924,92926,92928],{},[1584,92925,36101],{},[1735,92927,1879],{},[1731,92929,19819],{},[1731,92931,2693],{"stretchy":1797},[1591,92933,92934],{"encoding":1593},"\\exp(-\\tau\u002FT_2^\\ast)",[40,92936,92938],{"className":92937,"ariaHidden":1599},[1598],[40,92939,92941,92944,92947,92950,92953,92956,92959,93010],{"className":92940},[1603],[40,92942],{"className":92943,"style":1811},[1607],[40,92945,23123],{"className":92946},[6614],[40,92948,241],{"className":92949},[1985],[40,92951,1733],{"className":92952},[1612],[40,92954,80017],{"className":92955,"style":80089},[1612,1653],[40,92957,7610],{"className":92958},[1612],[40,92960,92962,92965],{"className":92961},[1612],[40,92963,36101],{"className":92964,"style":2264},[1612,1653],[40,92966,92968],{"className":92967},[1621],[40,92969,92971,93002],{"className":92970},[1625,2244],[40,92972,92974,92999],{"className":92973},[1629],[40,92975,92977,92988],{"className":92976,"style":35637},[1633],[40,92978,92979,92982],{"style":81585},[40,92980],{"className":92981,"style":1642},[1641],[40,92983,92985],{"className":92984},[1646,1647,1648,1649],[40,92986,1879],{"className":92987},[1612,1649],[40,92989,92990,92993],{"style":1637},[40,92991],{"className":92992,"style":1642},[1641],[40,92994,92996],{"className":92995},[1646,1647,1648,1649],[40,92997,19819],{"className":92998},[1761,1649],[40,93000,2269],{"className":93001},[2268],[40,93003,93005],{"className":93004},[1629],[40,93006,93008],{"className":93007,"style":81615},[1633],[40,93009],{},[40,93011,2693],{"className":93012},[1822]," and clips the result to the physical interval",[40,93015,93017],{"className":93016},[1827],[40,93018,93020,93046],{"className":93019},[1565],[40,93021,93023],{"className":93022},[1569],[1571,93024,93025],{"xmlns":1573,"display":1836},[1575,93026,93027,93043],{},[1578,93028,93029,93031,93033,93039,93041],{},[1735,93030,791],{},[1731,93032,26564],{},[2212,93034,93035,93037],{},[1584,93036,2218],{},[1584,93038,2330],{},[1731,93040,26564],{},[1735,93042,82656],{},[1591,93044,93045],{"encoding":1593},"0\\le P_e\\le 1.",[40,93047,93049,93068,93123],{"className":93048,"ariaHidden":1599},[1598],[40,93050,93052,93056,93059,93062,93065],{"className":93051},[1603],[40,93053],{"className":93054,"style":93055},[1607],"height:0.7804em;vertical-align:-0.136em;",[40,93057,791],{"className":93058},[1612],[40,93060],{"className":93061,"style":1963},[1756],[40,93063,26564],{"className":93064},[1967],[40,93066],{"className":93067,"style":1963},[1756],[40,93069,93071,93074,93114,93117,93120],{"className":93070},[1603],[40,93072],{"className":93073,"style":2231},[1607],[40,93075,93077,93080],{"className":93076},[1612],[40,93078,2218],{"className":93079,"style":2264},[1612,1653],[40,93081,93083],{"className":93082},[1621],[40,93084,93086,93106],{"className":93085},[1625,2244],[40,93087,93089,93103],{"className":93088},[1629],[40,93090,93092],{"className":93091,"style":3551},[1633],[40,93093,93094,93097],{"style":6681},[40,93095],{"className":93096,"style":1642},[1641],[40,93098,93100],{"className":93099},[1646,1647,1648,1649],[40,93101,2330],{"className":93102},[1612,1653,1649],[40,93104,2269],{"className":93105},[2268],[40,93107,93109],{"className":93108},[1629],[40,93110,93112],{"className":93111,"style":2276},[1633],[40,93113],{},[40,93115],{"className":93116,"style":1963},[1756],[40,93118,26564],{"className":93119},[1967],[40,93121],{"className":93122,"style":1963},[1756],[40,93124,93126,93129],{"className":93125},[1603],[40,93127],{"className":93128,"style":1771},[1607],[40,93130,82656],{"className":93131},[1612],[12,93133,93134],{},"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.",[55,93136,93138],{"id":93137},"graphical-interpretation","Graphical Interpretation",[12,93140,93141,93142,93240],{},"The ",[40,93143,93145,93173],{"className":93144},[1565],[40,93146,93148],{"className":93147},[1569],[1571,93149,93150],{"xmlns":1573},[1575,93151,93152,93170],{},[1578,93153,93154,93160,93162,93164,93166,93168],{},[2212,93155,93156,93158],{},[1584,93157,2218],{},[1584,93159,2330],{},[1731,93161,241],{"stretchy":1797},[1584,93163,21331],{},[1731,93165,1862],{"separator":1599},[1584,93167,80017],{},[1731,93169,2693],{"stretchy":1797},[1591,93171,93172],{"encoding":1593},"P_e(f,\\tau)",[40,93174,93176],{"className":93175,"ariaHidden":1599},[1598],[40,93177,93179,93182,93222,93225,93228,93231,93234,93237],{"className":93178},[1603],[40,93180],{"className":93181,"style":1811},[1607],[40,93183,93185,93188],{"className":93184},[1612],[40,93186,2218],{"className":93187,"style":2264},[1612,1653],[40,93189,93191],{"className":93190},[1621],[40,93192,93194,93214],{"className":93193},[1625,2244],[40,93195,93197,93211],{"className":93196},[1629],[40,93198,93200],{"className":93199,"style":3551},[1633],[40,93201,93202,93205],{"style":6681},[40,93203],{"className":93204,"style":1642},[1641],[40,93206,93208],{"className":93207},[1646,1647,1648,1649],[40,93209,2330],{"className":93210},[1612,1653,1649],[40,93212,2269],{"className":93213},[2268],[40,93215,93217],{"className":93216},[1629],[40,93218,93220],{"className":93219,"style":2276},[1633],[40,93221],{},[40,93223,241],{"className":93224},[1985],[40,93226,21331],{"className":93227,"style":26393},[1612,1653],[40,93229,1862],{"className":93230},[2005],[40,93232],{"className":93233,"style":2009},[1756],[40,93235,80017],{"className":93236,"style":80089},[1612,1653],[40,93238,2693],{"className":93239},[1822]," 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.",[12,93242,93243],{},"The resonant interaction region appears near",[40,93245,93247],{"className":93246},[1827],[40,93248,93250,93273],{"className":93249},[1565],[40,93251,93253],{"className":93252},[1569],[1571,93254,93255],{"xmlns":1573,"display":1836},[1575,93256,93257,93271],{},[1578,93258,93259,93261,93263,93269],{},[1584,93260,21331],{},[1731,93262,623],{},[2212,93264,93265,93267],{},[1584,93266,21331],{},[1735,93268,791],{},[1584,93270,1823],{"mathvariant":1790},[1591,93272,82352],{"encoding":1593},[40,93274,93276,93294],{"className":93275,"ariaHidden":1599},[1598],[40,93277,93279,93282,93285,93288,93291],{"className":93278},[1603],[40,93280],{"className":93281,"style":6079},[1607],[40,93283,21331],{"className":93284,"style":26393},[1612,1653],[40,93286],{"className":93287,"style":1963},[1756],[40,93289,623],{"className":93290},[1967],[40,93292],{"className":93293,"style":1963},[1756],[40,93295,93297,93300,93340],{"className":93296},[1603],[40,93298],{"className":93299,"style":6079},[1607],[40,93301,93303,93306],{"className":93302},[1612],[40,93304,21331],{"className":93305,"style":26393},[1612,1653],[40,93307,93309],{"className":93308},[1621],[40,93310,93312,93332],{"className":93311},[1625,2244],[40,93313,93315,93329],{"className":93314},[1629],[40,93316,93318],{"className":93317,"style":13330},[1633],[40,93319,93320,93323],{"style":80580},[40,93321],{"className":93322,"style":1642},[1641],[40,93324,93326],{"className":93325},[1646,1647,1648,1649],[40,93327,791],{"className":93328},[1612,1649],[40,93330,2269],{"className":93331},[2268],[40,93333,93335],{"className":93334},[1629],[40,93336,93338],{"className":93337,"style":2276},[1633],[40,93339],{},[40,93341,1823],{"className":93342},[1612],[12,93344,93345,93346,93425],{},"At resonance, contrast is maximal and the pulse duration alone sets the rotation angle. With the default drive strength ",[40,93347,93349,93383],{"className":93348},[1565],[40,93350,93352],{"className":93351},[1569],[1571,93353,93354],{"xmlns":1573},[1575,93355,93356,93380],{},[1578,93357,93358,93360,93362,93364,93366,93368,93370,93372],{},[1584,93359,80975],{"mathvariant":1790},[1584,93361,7610],{"mathvariant":1790},[1735,93363,1879],{},[1584,93365,7607],{},[1731,93367,623],{},[1735,93369,86720],{},[5211,93371,5213],{},[1578,93373,93374,93376,93378],{},[1584,93375,86727],{"mathvariant":1790},[1584,93377,2182],{"mathvariant":1790},[1584,93379,19585],{"mathvariant":1790},[1591,93381,93382],{"encoding":1593},"\\Omega\u002F2\\pi=20~\\mathrm{MHz}",[40,93384,93386,93407],{"className":93385,"ariaHidden":1599},[1598],[40,93387,93389,93392,93395,93398,93401,93404],{"className":93388},[1603],[40,93390],{"className":93391,"style":1811},[1607],[40,93393,81033],{"className":93394},[1612],[40,93396,7607],{"className":93397,"style":1818},[1612,1653],[40,93399],{"className":93400,"style":1963},[1756],[40,93402,623],{"className":93403},[1967],[40,93405],{"className":93406,"style":1963},[1756],[40,93408,93410,93413,93416,93419],{"className":93409},[1603],[40,93411],{"className":93412,"style":1608},[1607],[40,93414,86720],{"className":93415},[1612],[40,93417,5213],{"className":93418},[1756,82202],[40,93420,93422],{"className":93421},[1612],[40,93423,86840],{"className":93424},[1612,82209],", the ideal resonant maxima appear near",[40,93427,93429],{"className":93428},[1827],[40,93430,93432,93505],{"className":93431},[1565],[40,93433,93435],{"className":93434},[1569],[1571,93436,93437],{"xmlns":1573,"display":1836},[1575,93438,93439,93502],{},[1578,93440,93441,93443,93445,93447,93449,93455,93457,93459,93462,93464,93470,93472,93474,93477,93479,93485,93487,93489,93492,93494,93500],{},[1584,93442,80017],{},[1731,93444,623],{},[1735,93446,86901],{},[5211,93448,5213],{},[1578,93450,93451,93453],{},[1584,93452,3464],{"mathvariant":1790},[1584,93454,13582],{"mathvariant":1790},[1731,93456,1862],{"separator":1599},[5211,93458,5213],{},[1735,93460,93461],{},"75",[5211,93463,5213],{},[1578,93465,93466,93468],{},[1584,93467,3464],{"mathvariant":1790},[1584,93469,13582],{"mathvariant":1790},[1731,93471,1862],{"separator":1599},[5211,93473,5213],{},[1735,93475,93476],{},"125",[5211,93478,5213],{},[1578,93480,93481,93483],{},[1584,93482,3464],{"mathvariant":1790},[1584,93484,13582],{"mathvariant":1790},[1731,93486,1862],{"separator":1599},[5211,93488,5213],{},[1735,93490,93491],{},"175",[5211,93493,5213],{},[1578,93495,93496,93498],{},[1584,93497,3464],{"mathvariant":1790},[1584,93499,13582],{"mathvariant":1790},[1731,93501,1862],{"separator":1599},[1591,93503,93504],{"encoding":1593},"\\tau=25~\\mathrm{ns},\\ 75~\\mathrm{ns},\\ 125~\\mathrm{ns},\\ 175~\\mathrm{ns},",[40,93506,93508,93526],{"className":93507,"ariaHidden":1599},[1598],[40,93509,93511,93514,93517,93520,93523],{"className":93510},[1603],[40,93512],{"className":93513,"style":1681},[1607],[40,93515,80017],{"className":93516,"style":80089},[1612,1653],[40,93518],{"className":93519,"style":1963},[1756],[40,93521,623],{"className":93522},[1967],[40,93524],{"className":93525,"style":1963},[1756],[40,93527,93529,93533,93536,93539,93545,93548,93551,93554,93557,93560,93566,93569,93572,93575,93578,93581,93587,93590,93593,93596,93599,93602,93608],{"className":93528},[1603],[40,93530],{"className":93531,"style":93532},[1607],"height:0.8389em;vertical-align:-0.1944em;",[40,93534,86901],{"className":93535},[1612],[40,93537,5213],{"className":93538},[1756,82202],[40,93540,93542],{"className":93541},[1612],[40,93543,86991],{"className":93544},[1612,82209],[40,93546,1862],{"className":93547},[2005],[40,93549,5213],{"className":93550},[1756],[40,93552],{"className":93553,"style":2009},[1756],[40,93555,93461],{"className":93556},[1612],[40,93558,5213],{"className":93559},[1756,82202],[40,93561,93563],{"className":93562},[1612],[40,93564,86991],{"className":93565},[1612,82209],[40,93567,1862],{"className":93568},[2005],[40,93570,5213],{"className":93571},[1756],[40,93573],{"className":93574,"style":2009},[1756],[40,93576,93476],{"className":93577},[1612],[40,93579,5213],{"className":93580},[1756,82202],[40,93582,93584],{"className":93583},[1612],[40,93585,86991],{"className":93586},[1612,82209],[40,93588,1862],{"className":93589},[2005],[40,93591,5213],{"className":93592},[1756],[40,93594],{"className":93595,"style":2009},[1756],[40,93597,93491],{"className":93598},[1612],[40,93600,5213],{"className":93601},[1756,82202],[40,93603,93605],{"className":93604},[1612],[40,93606,86991],{"className":93607},[1612,82209],[40,93609,1862],{"className":93610},[2005],[12,93612,93613],{},"before the optional phenomenological damping factor is applied.",[12,93615,93616,93617,93686,93687,93813],{},"Off resonance, the fringes show lower contrast and faster temporal oscillations. This behavior follows from the generalized Rabi frequency ",[40,93618,93620,93637],{"className":93619},[1565],[40,93621,93623],{"className":93622},[1569],[1571,93624,93625],{"xmlns":1573},[1575,93626,93627,93635],{},[1578,93628,93629],{},[2212,93630,93631,93633],{},[1584,93632,80975],{"mathvariant":1790},[1584,93634,19419],{},[1591,93636,81215],{"encoding":1593},[40,93638,93640],{"className":93639,"ariaHidden":1599},[1598],[40,93641,93643,93646],{"className":93642},[1603],[40,93644],{"className":93645,"style":2231},[1607],[40,93647,93649,93652],{"className":93648},[1612],[40,93650,80975],{"className":93651},[1612],[40,93653,93655],{"className":93654},[1621],[40,93656,93658,93678],{"className":93657},[1625,2244],[40,93659,93661,93675],{"className":93660},[1629],[40,93662,93664],{"className":93663,"style":2251},[1633],[40,93665,93666,93669],{"style":4442},[40,93667],{"className":93668,"style":1642},[1641],[40,93670,93672],{"className":93671},[1646,1647,1648,1649],[40,93673,19419],{"className":93674,"style":19444},[1612,1653,1649],[40,93676,2269],{"className":93677},[2268],[40,93679,93681],{"className":93680},[1629],[40,93682,93684],{"className":93683,"style":2276},[1633],[40,93685],{},", which increases with detuning, and from the amplitude coefficient ",[40,93688,93690,93718],{"className":93689},[1565],[40,93691,93693],{"className":93692},[1569],[1571,93694,93695],{"xmlns":1573},[1575,93696,93697,93715],{},[1578,93698,93699,93705,93707],{},[1581,93700,93701,93703],{},[1584,93702,80975],{"mathvariant":1790},[1735,93704,1879],{},[1584,93706,7610],{"mathvariant":1790},[2819,93708,93709,93711,93713],{},[1584,93710,80975],{"mathvariant":1790},[1584,93712,19419],{},[1735,93714,1879],{},[1591,93716,93717],{"encoding":1593},"\\Omega^2\u002F\\Omega_R^2",[40,93719,93721],{"className":93720,"ariaHidden":1599},[1598],[40,93722,93724,93728,93757,93760],{"className":93723},[1603],[40,93725],{"className":93726,"style":93727},[1607],"height:1.0894em;vertical-align:-0.2753em;",[40,93729,93731,93734],{"className":93730},[1612],[40,93732,80975],{"className":93733},[1612],[40,93735,93737],{"className":93736},[1621],[40,93738,93740],{"className":93739},[1625],[40,93741,93743],{"className":93742},[1629],[40,93744,93746],{"className":93745,"style":12271},[1633],[40,93747,93748,93751],{"style":1637},[40,93749],{"className":93750,"style":1642},[1641],[40,93752,93754],{"className":93753},[1646,1647,1648,1649],[40,93755,1879],{"className":93756},[1612,1649],[40,93758,7610],{"className":93759},[1612],[40,93761,93763,93766],{"className":93762},[1612],[40,93764,80975],{"className":93765},[1612],[40,93767,93769],{"className":93768},[1621],[40,93770,93772,93804],{"className":93771},[1625,2244],[40,93773,93775,93801],{"className":93774},[1629],[40,93776,93778,93790],{"className":93777,"style":12271},[1633],[40,93779,93781,93784],{"style":93780},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[40,93782],{"className":93783,"style":1642},[1641],[40,93785,93787],{"className":93786},[1646,1647,1648,1649],[40,93788,19419],{"className":93789,"style":19444},[1612,1653,1649],[40,93791,93792,93795],{"style":1637},[40,93793],{"className":93794,"style":1642},[1641],[40,93796,93798],{"className":93797},[1646,1647,1648,1649],[40,93799,1879],{"className":93800},[1612,1649],[40,93802,2269],{"className":93803},[2268],[40,93805,93807],{"className":93806},[1629],[40,93808,93811],{"className":93809,"style":93810},[1633],"height:0.2753em;",[40,93812],{},", which decreases with detuning.",[12,93815,93816],{},"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.",[12,93818,93819,93820,93917],{},"The FFT heatmap characterizes the oscillation-frequency content of ",[40,93821,93823,93850],{"className":93822},[1565],[40,93824,93826],{"className":93825},[1569],[1571,93827,93828],{"xmlns":1573},[1575,93829,93830,93848],{},[1578,93831,93832,93838,93840,93842,93844,93846],{},[2212,93833,93834,93836],{},[1584,93835,2218],{},[1584,93837,2330],{},[1731,93839,241],{"stretchy":1797},[1584,93841,80017],{},[1731,93843,1862],{"separator":1599},[1584,93845,21331],{},[1731,93847,2693],{"stretchy":1797},[1591,93849,91032],{"encoding":1593},[40,93851,93853],{"className":93852,"ariaHidden":1599},[1598],[40,93854,93856,93859,93899,93902,93905,93908,93911,93914],{"className":93855},[1603],[40,93857],{"className":93858,"style":1811},[1607],[40,93860,93862,93865],{"className":93861},[1612],[40,93863,2218],{"className":93864,"style":2264},[1612,1653],[40,93866,93868],{"className":93867},[1621],[40,93869,93871,93891],{"className":93870},[1625,2244],[40,93872,93874,93888],{"className":93873},[1629],[40,93875,93877],{"className":93876,"style":3551},[1633],[40,93878,93879,93882],{"style":6681},[40,93880],{"className":93881,"style":1642},[1641],[40,93883,93885],{"className":93884},[1646,1647,1648,1649],[40,93886,2330],{"className":93887},[1612,1653,1649],[40,93889,2269],{"className":93890},[2268],[40,93892,93894],{"className":93893},[1629],[40,93895,93897],{"className":93896,"style":2276},[1633],[40,93898],{},[40,93900,241],{"className":93901},[1985],[40,93903,80017],{"className":93904,"style":80089},[1612,1653],[40,93906,1862],{"className":93907},[2005],[40,93909],{"className":93910,"style":2009},[1756],[40,93912,21331],{"className":93913,"style":26393},[1612,1653],[40,93915,2693],{"className":93916},[1822],". An overlaid theoretical curve follows",[40,93919,93921],{"className":93920},[1827],[40,93922,93924,93996],{"className":93923},[1565],[40,93925,93927],{"className":93926},[1569],[1571,93928,93929],{"xmlns":1573,"display":1836},[1575,93930,93931,93993],{},[1578,93932,93933,93939,93941,93943,93945,93947,93991],{},[2212,93934,93935,93937],{},[1584,93936,81288],{},[1584,93938,19419],{},[1731,93940,241],{"stretchy":1797},[1584,93942,21331],{},[1731,93944,2693],{"stretchy":1797},[1731,93946,623],{},[8543,93948,93949],{},[1578,93950,93951,93971,93973,93975,93977,93979,93985],{},[1581,93952,93953,93969],{},[1578,93954,93955,93957,93967],{},[1731,93956,241],{"fence":1599},[8538,93958,93959,93961],{},[1584,93960,80975],{"mathvariant":1790},[1578,93962,93963,93965],{},[1735,93964,1879],{},[1584,93966,7607],{},[1731,93968,2693],{"fence":1599},[1735,93970,1879],{},[1731,93972,1147],{},[1731,93974,241],{"stretchy":1797},[1584,93976,21331],{},[1731,93978,1733],{},[2212,93980,93981,93983],{},[1584,93982,21331],{},[1735,93984,791],{},[1581,93986,93987,93989],{},[1731,93988,2693],{"stretchy":1797},[1735,93990,1879],{},[1584,93992,1823],{"mathvariant":1790},[1591,93994,93995],{"encoding":1593},"\\nu_R(f)=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[40,93997,93999,94063],{"className":93998,"ariaHidden":1599},[1598],[40,94000,94002,94005,94045,94048,94051,94054,94057,94060],{"className":94001},[1603],[40,94003],{"className":94004,"style":1811},[1607],[40,94006,94008,94011],{"className":94007},[1612],[40,94009,81288],{"className":94010,"style":81309},[1612,1653],[40,94012,94014],{"className":94013},[1621],[40,94015,94017,94037],{"className":94016},[1625,2244],[40,94018,94020,94034],{"className":94019},[1629],[40,94021,94023],{"className":94022,"style":2251},[1633],[40,94024,94025,94028],{"style":81324},[40,94026],{"className":94027,"style":1642},[1641],[40,94029,94031],{"className":94030},[1646,1647,1648,1649],[40,94032,19419],{"className":94033,"style":19444},[1612,1653,1649],[40,94035,2269],{"className":94036},[2268],[40,94038,94040],{"className":94039},[1629],[40,94041,94043],{"className":94042,"style":2276},[1633],[40,94044],{},[40,94046,241],{"className":94047},[1985],[40,94049,21331],{"className":94050,"style":26393},[1612,1653],[40,94052,2693],{"className":94053},[1822],[40,94055],{"className":94056,"style":1963},[1756],[40,94058,623],{"className":94059},[1967],[40,94061],{"className":94062,"style":1963},[1756],[40,94064,94066,94069,94312],{"className":94065},[1603],[40,94067],{"className":94068,"style":88338},[1607],[40,94070,94072],{"className":94071},[1612,8866],[40,94073,94075,94304],{"className":94074},[1625,2244],[40,94076,94078,94301],{"className":94077},[1629],[40,94079,94081,94289],{"className":94080,"style":88351},[1633],[40,94082,94084,94087],{"className":94083,"style":88355},[8880],[40,94085],{"className":94086,"style":88359},[1641],[40,94088,94090,94196,94199,94202,94205,94208,94211,94214,94217,94220,94260],{"className":94089,"style":85586},[1612],[40,94091,94093,94173],{"className":94092},[1981],[40,94094,94096,94102,94167],{"className":94095},[1981],[40,94097,94099],{"className":94098,"style":1987},[1985,1986],[40,94100,241],{"className":94101},[1991,1648],[40,94103,94105,94108,94164],{"className":94104},[1612],[40,94106],{"className":94107},[1985,8840],[40,94109,94111],{"className":94110},[8538],[40,94112,94114,94156],{"className":94113},[1625,2244],[40,94115,94117,94153],{"className":94116},[1629],[40,94118,94120,94134,94142],{"className":94119,"style":85353},[1633],[40,94121,94122,94125],{"style":10764},[40,94123],{"className":94124,"style":3354},[1641],[40,94126,94128,94131],{"className":94127},[1612],[40,94129,1879],{"className":94130},[1612],[40,94132,7607],{"className":94133,"style":1818},[1612,1653],[40,94135,94136,94139],{"style":8925},[40,94137],{"className":94138,"style":3354},[1641],[40,94140],{"className":94141,"style":8933},[8932],[40,94143,94144,94147],{"style":8936},[40,94145],{"className":94146,"style":3354},[1641],[40,94148,94150],{"className":94149},[1612],[40,94151,80975],{"className":94152},[1612],[40,94154,2269],{"className":94155},[2268],[40,94157,94159],{"className":94158},[1629],[40,94160,94162],{"className":94161,"style":10802},[1633],[40,94163],{},[40,94165],{"className":94166},[1822,8840],[40,94168,94170],{"className":94169,"style":1987},[1822,1986],[40,94171,2693],{"className":94172},[1991,1648],[40,94174,94176],{"className":94175},[1621],[40,94177,94179],{"className":94178},[1625],[40,94180,94182],{"className":94181},[1629],[40,94183,94185],{"className":94184,"style":88458},[1633],[40,94186,94187,94190],{"style":88461},[40,94188],{"className":94189,"style":1642},[1641],[40,94191,94193],{"className":94192},[1646,1647,1648,1649],[40,94194,1879],{"className":94195},[1612,1649],[40,94197],{"className":94198,"style":1757},[1756],[40,94200,1147],{"className":94201},[1761],[40,94203],{"className":94204,"style":1757},[1756],[40,94206,241],{"className":94207},[1985],[40,94209,21331],{"className":94210,"style":26393},[1612,1653],[40,94212],{"className":94213,"style":1757},[1756],[40,94215,1733],{"className":94216},[1761],[40,94218],{"className":94219,"style":1757},[1756],[40,94221,94223,94226],{"className":94222},[1612],[40,94224,21331],{"className":94225,"style":26393},[1612,1653],[40,94227,94229],{"className":94228},[1621],[40,94230,94232,94252],{"className":94231},[1625,2244],[40,94233,94235,94249],{"className":94234},[1629],[40,94236,94238],{"className":94237,"style":13330},[1633],[40,94239,94240,94243],{"style":80580},[40,94241],{"className":94242,"style":1642},[1641],[40,94244,94246],{"className":94245},[1646,1647,1648,1649],[40,94247,791],{"className":94248},[1612,1649],[40,94250,2269],{"className":94251},[2268],[40,94253,94255],{"className":94254},[1629],[40,94256,94258],{"className":94257,"style":2276},[1633],[40,94259],{},[40,94261,94263,94266],{"className":94262},[1822],[40,94264,2693],{"className":94265},[1822],[40,94267,94269],{"className":94268},[1621],[40,94270,94272],{"className":94271},[1625],[40,94273,94275],{"className":94274},[1629],[40,94276,94278],{"className":94277,"style":28464},[1633],[40,94279,94280,94283],{"style":28467},[40,94281],{"className":94282,"style":1642},[1641],[40,94284,94286],{"className":94285},[1646,1647,1648,1649],[40,94287,1879],{"className":94288},[1612,1649],[40,94290,94291,94294],{"style":88566},[40,94292],{"className":94293,"style":88359},[1641],[40,94295,94297],{"className":94296,"style":88573},[8900],[5325,94298,94299],{"xmlns":5327,"width":8904,"height":88576,"viewBox":88577,"preserveAspectRatio":8907},[5334,94300],{"d":88580},[40,94302,2269],{"className":94303},[2268],[40,94305,94307],{"className":94306},[1629],[40,94308,94310],{"className":94309,"style":88590},[1633],[40,94311],{},[40,94313,1823],{"className":94314},[1612],[55,94316,94318],{"id":94317},"primary-modeling-assumptions","Primary Modeling Assumptions",[12,94320,94321],{},"This notebook is an idealized mathematical physics visualization designed for conceptual clarity and calibration intuition.",[80165,94323,94324,94334],{},[80168,94325,94326],{},[80171,94327,94328,94331],{},[80174,94329,94330],{},"Assumption",[80174,94332,94333],{},"Implication",[80181,94335,94336,94420,94428,94435,94443,94480,94570,94727,94775,94783,94791,94799],{},[80171,94337,94338,94341],{},[80186,94339,94340],{},"Two-level truncation",[80186,94342,94343,94344,9846,94382,1823],{},"The simulated model includes only ",[40,94345,94347,94364],{"className":94346},[1565],[40,94348,94350],{"className":94349},[1569],[1571,94351,94352],{"xmlns":1573},[1575,94353,94354,94362],{},[1578,94355,94356,94358,94360],{},[1731,94357,1791],{"stretchy":1797},[1584,94359,36968],{},[1731,94361,1798],{"stretchy":1797},[1591,94363,80208],{"encoding":1593},[40,94365,94367],{"className":94366,"ariaHidden":1599},[1598],[40,94368,94370,94373,94376,94379],{"className":94369},[1603],[40,94371],{"className":94372,"style":1811},[1607],[40,94374,1791],{"className":94375},[1985],[40,94377,36968],{"className":94378,"style":1818},[1612,1653],[40,94380,1798],{"className":94381},[1822],[40,94383,94385,94402],{"className":94384},[1565],[40,94386,94388],{"className":94387},[1569],[1571,94389,94390],{"xmlns":1573},[1575,94391,94392,94400],{},[1578,94393,94394,94396,94398],{},[1731,94395,1791],{"stretchy":1797},[1584,94397,2330],{},[1731,94399,1798],{"stretchy":1797},[1591,94401,80254],{"encoding":1593},[40,94403,94405],{"className":94404,"ariaHidden":1599},[1598],[40,94406,94408,94411,94414,94417],{"className":94407},[1603],[40,94409],{"className":94410,"style":1811},[1607],[40,94412,1791],{"className":94413},[1985],[40,94415,2330],{"className":94416},[1612,1653],[40,94418,1798],{"className":94419},[1822],[80171,94421,94422,94425],{},[80186,94423,94424],{},"Classical drive",[80186,94426,94427],{},"The microwave field is prescribed externally instead of quantized as a field mode.",[80171,94429,94430,94432],{},[80186,94431,81628],{},[80186,94433,94434],{},"Fast counter-rotating terms are omitted.",[80171,94436,94437,94440],{},[80186,94438,94439],{},"Square pulse",[80186,94441,94442],{},"The drive is assumed to turn on and off instantaneously.",[80171,94444,94445,94448],{},[80186,94446,94447],{},"Uniform drive amplitude",[80186,94449,94450,94451,94479],{},"The parameter ",[40,94452,94454,94467],{"className":94453},[1565],[40,94455,94457],{"className":94456},[1569],[1571,94458,94459],{"xmlns":1573},[1575,94460,94461,94465],{},[1578,94462,94463],{},[1584,94464,80975],{"mathvariant":1790},[1591,94466,80978],{"encoding":1593},[40,94468,94470],{"className":94469,"ariaHidden":1599},[1598],[40,94471,94473,94476],{"className":94472},[1603],[40,94474],{"className":94475,"style":1608},[1607],[40,94477,80975],{"className":94478},[1612]," remains constant across the frequency sweep.",[80171,94481,94482,94485],{},[80186,94483,94484],{},"Optional damping envelope",[80186,94486,94450,94487,94569],{},[40,94488,94490,94509],{"className":94489},[1565],[40,94491,94493],{"className":94492},[1569],[1571,94494,94495],{"xmlns":1573},[1575,94496,94497,94507],{},[1578,94498,94499],{},[2819,94500,94501,94503,94505],{},[1584,94502,36101],{},[1735,94504,1879],{},[1731,94506,19819],{},[1591,94508,81554],{"encoding":1593},[40,94510,94512],{"className":94511,"ariaHidden":1599},[1598],[40,94513,94515,94518],{"className":94514},[1603],[40,94516],{"className":94517,"style":81564},[1607],[40,94519,94521,94524],{"className":94520},[1612],[40,94522,36101],{"className":94523,"style":2264},[1612,1653],[40,94525,94527],{"className":94526},[1621],[40,94528,94530,94561],{"className":94529},[1625,2244],[40,94531,94533,94558],{"className":94532},[1629],[40,94534,94536,94547],{"className":94535,"style":35637},[1633],[40,94537,94538,94541],{"style":81585},[40,94539],{"className":94540,"style":1642},[1641],[40,94542,94544],{"className":94543},[1646,1647,1648,1649],[40,94545,1879],{"className":94546},[1612,1649],[40,94548,94549,94552],{"style":1637},[40,94550],{"className":94551,"style":1642},[1641],[40,94553,94555],{"className":94554},[1646,1647,1648,1649],[40,94556,19819],{"className":94557},[1761,1649],[40,94559,2269],{"className":94560},[2268],[40,94562,94564],{"className":94563},[1629],[40,94565,94567],{"className":94566,"style":81615},[1633],[40,94568],{}," enters as a simple exponential contrast envelope.",[80171,94571,94572,94646],{},[80186,94573,94574,94575,94645],{},"No explicit ",[40,94576,94578,94596],{"className":94577},[1565],[40,94579,94581],{"className":94580},[1569],[1571,94582,94583],{"xmlns":1573},[1575,94584,94585,94593],{},[1578,94586,94587],{},[2212,94588,94589,94591],{},[1584,94590,36101],{},[1735,94592,1410],{},[1591,94594,94595],{"encoding":1593},"T_1",[40,94597,94599],{"className":94598,"ariaHidden":1599},[1598],[40,94600,94602,94605],{"className":94601},[1603],[40,94603],{"className":94604,"style":2231},[1607],[40,94606,94608,94611],{"className":94607},[1612],[40,94609,36101],{"className":94610,"style":2264},[1612,1653],[40,94612,94614],{"className":94613},[1621],[40,94615,94617,94637],{"className":94616},[1625,2244],[40,94618,94620,94634],{"className":94619},[1629],[40,94621,94623],{"className":94622,"style":13330},[1633],[40,94624,94625,94628],{"style":6681},[40,94626],{"className":94627,"style":1642},[1641],[40,94629,94631],{"className":94630},[1646,1647,1648,1649],[40,94632,1410],{"className":94633},[1612,1649],[40,94635,2269],{"className":94636},[2268],[40,94638,94640],{"className":94639},[1629],[40,94641,94643],{"className":94642,"style":2276},[1633],[40,94644],{}," relaxation",[80186,94647,94648,94649,94687,94688,94726],{},"Energy decay from ",[40,94650,94652,94669],{"className":94651},[1565],[40,94653,94655],{"className":94654},[1569],[1571,94656,94657],{"xmlns":1573},[1575,94658,94659,94667],{},[1578,94660,94661,94663,94665],{},[1731,94662,1791],{"stretchy":1797},[1584,94664,2330],{},[1731,94666,1798],{"stretchy":1797},[1591,94668,80254],{"encoding":1593},[40,94670,94672],{"className":94671,"ariaHidden":1599},[1598],[40,94673,94675,94678,94681,94684],{"className":94674},[1603],[40,94676],{"className":94677,"style":1811},[1607],[40,94679,1791],{"className":94680},[1985],[40,94682,2330],{"className":94683},[1612,1653],[40,94685,1798],{"className":94686},[1822]," to ",[40,94689,94691,94708],{"className":94690},[1565],[40,94692,94694],{"className":94693},[1569],[1571,94695,94696],{"xmlns":1573},[1575,94697,94698,94706],{},[1578,94699,94700,94702,94704],{},[1731,94701,1791],{"stretchy":1797},[1584,94703,36968],{},[1731,94705,1798],{"stretchy":1797},[1591,94707,80208],{"encoding":1593},[40,94709,94711],{"className":94710,"ariaHidden":1599},[1598],[40,94712,94714,94717,94720,94723],{"className":94713},[1603],[40,94715],{"className":94716,"style":1811},[1607],[40,94718,1791],{"className":94719},[1985],[40,94721,36968],{"className":94722,"style":1818},[1612,1653],[40,94724,1798],{"className":94725},[1822]," is omitted.",[80171,94728,94729,94732],{},[80186,94730,94731],{},"No higher-state leakage",[80186,94733,94734,94735,94774],{},"Higher levels, including a transmon ",[40,94736,94738,94756],{"className":94737},[1565],[40,94739,94741],{"className":94740},[1569],[1571,94742,94743],{"xmlns":1573},[1575,94744,94745,94753],{},[1578,94746,94747,94749,94751],{},[1731,94748,1791],{"stretchy":1797},[1584,94750,21331],{},[1731,94752,1798],{"stretchy":1797},[1591,94754,94755],{"encoding":1593},"\\lvert f\\rangle",[40,94757,94759],{"className":94758,"ariaHidden":1599},[1598],[40,94760,94762,94765,94768,94771],{"className":94761},[1603],[40,94763],{"className":94764,"style":1811},[1607],[40,94766,1791],{"className":94767},[1985],[40,94769,21331],{"className":94770,"style":26393},[1612,1653],[40,94772,1798],{"className":94773},[1822]," state, are excluded.",[80171,94776,94777,94780],{},[80186,94778,94779],{},"No AC Stark shift",[80186,94781,94782],{},"Drive-induced transition-frequency shifts are excluded.",[80171,94784,94785,94788],{},[80186,94786,94787],{},"No Bloch-Siegert shift",[80186,94789,94790],{},"Counter-rotating corrections are excluded.",[80171,94792,94793,94796],{},[80186,94794,94795],{},"No experimental readout model",[80186,94797,94798],{},"Assignment errors and finite signal-to-noise effects are omitted.",[80171,94800,94801,94804],{},[80186,94802,94803],{},"No pulse-envelope shaping",[80186,94805,94806],{},"Gaussian, DRAG, cosine, and hardware-specific envelopes are omitted.",[12,94808,94809],{},"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.",[55,94811,94813],{"id":94812},"notebook-control-knobs","Notebook Control Knobs",[12,94815,94816],{},"The first code section centralizes the main physical and numerical parameters for direct adjustment.",[80165,94818,94819,94832],{},[80168,94820,94821],{},[80171,94822,94823,94826,94830],{},[80174,94824,94825],{},"Parameter",[80174,94827,94829],{"align":94828},"right","Default",[80174,94831,80179],{},[80181,94833,94834,94918,94971,94986,95001,95016,95031,95046,95060],{},[80171,94835,94836,94841,94845],{},[80186,94837,94838],{},[37,94839,94840],{},"F0_GHZ",[80186,94842,94843],{"align":94828},[37,94844,82115],{},[80186,94846,94847,94848,94917],{},"Qubit transition frequency ",[40,94849,94851,94868],{"className":94850},[1565],[40,94852,94854],{"className":94853},[1569],[1571,94855,94856],{"xmlns":1573},[1575,94857,94858,94866],{},[1578,94859,94860],{},[2212,94861,94862,94864],{},[1584,94863,21331],{},[1735,94865,791],{},[1591,94867,80550],{"encoding":1593},[40,94869,94871],{"className":94870,"ariaHidden":1599},[1598],[40,94872,94874,94877],{"className":94873},[1603],[40,94875],{"className":94876,"style":6079},[1607],[40,94878,94880,94883],{"className":94879},[1612],[40,94881,21331],{"className":94882,"style":26393},[1612,1653],[40,94884,94886],{"className":94885},[1621],[40,94887,94889,94909],{"className":94888},[1625,2244],[40,94890,94892,94906],{"className":94891},[1629],[40,94893,94895],{"className":94894,"style":13330},[1633],[40,94896,94897,94900],{"style":80580},[40,94898],{"className":94899,"style":1642},[1641],[40,94901,94903],{"className":94902},[1646,1647,1648,1649],[40,94904,791],{"className":94905},[1612,1649],[40,94907,2269],{"className":94908},[2268],[40,94910,94912],{"className":94911},[1629],[40,94913,94915],{"className":94914,"style":2276},[1633],[40,94916],{}," in GHz",[80171,94919,94920,94925,94930],{},[80186,94921,94922],{},[37,94923,94924],{},"OMEGA_ONRESONANCE_MHZ",[80186,94926,94927],{"align":94828},[37,94928,94929],{},"20.0",[80186,94931,94932,94933,94970],{},"On-resonance Rabi rate ",[40,94934,94936,94955],{"className":94935},[1565],[40,94937,94939],{"className":94938},[1569],[1571,94940,94941],{"xmlns":1573},[1575,94942,94943,94953],{},[1578,94944,94945,94947,94949,94951],{},[1584,94946,80975],{"mathvariant":1790},[1584,94948,7610],{"mathvariant":1790},[1735,94950,1879],{},[1584,94952,7607],{},[1591,94954,81020],{"encoding":1593},[40,94956,94958],{"className":94957,"ariaHidden":1599},[1598],[40,94959,94961,94964,94967],{"className":94960},[1603],[40,94962],{"className":94963,"style":1811},[1607],[40,94965,81033],{"className":94966},[1612],[40,94968,7607],{"className":94969,"style":1818},[1612,1653]," in MHz",[80171,94972,94973,94978,94983],{},[80186,94974,94975],{},[37,94976,94977],{},"FREQ_MIN_GHZ",[80186,94979,94980],{"align":94828},[37,94981,94982],{},"4.90",[80186,94984,94985],{},"Minimum swept drive frequency in GHz",[80171,94987,94988,94993,94998],{},[80186,94989,94990],{},[37,94991,94992],{},"FREQ_MAX_GHZ",[80186,94994,94995],{"align":94828},[37,94996,94997],{},"5.10",[80186,94999,95000],{},"Maximum swept drive frequency in GHz",[80171,95002,95003,95008,95013],{},[80186,95004,95005],{},[37,95006,95007],{},"N_FREQ",[80186,95009,95010],{"align":94828},[37,95011,95012],{},"401",[80186,95014,95015],{},"Number of drive-frequency samples",[80171,95017,95018,95023,95028],{},[80186,95019,95020],{},[37,95021,95022],{},"DUR_MIN_NS",[80186,95024,95025],{"align":94828},[37,95026,95027],{},"0.0",[80186,95029,95030],{},"Minimum pulse duration in ns",[80171,95032,95033,95038,95043],{},[80186,95034,95035],{},[37,95036,95037],{},"DUR_MAX_NS",[80186,95039,95040],{"align":94828},[37,95041,95042],{},"200.0",[80186,95044,95045],{},"Maximum pulse duration in ns",[80171,95047,95048,95053,95057],{},[80186,95049,95050],{},[37,95051,95052],{},"N_DUR",[80186,95054,95055],{"align":94828},[37,95056,95012],{},[80186,95058,95059],{},"Number of pulse-duration samples",[80171,95061,95062,95067,95072],{},[80186,95063,95064],{},[37,95065,95066],{},"T2STAR_NS",[80186,95068,95069],{"align":94828},[37,95070,95071],{},"500.0",[80186,95073,95074,95075,95078],{},"Optional damping time in ns, with ",[37,95076,95077],{},"None"," disabling damping",[12,95080,95081],{},"Additional cross-section plotting cells expose localized slice controls.",[80165,95083,95084,95094],{},[80168,95085,95086],{},[80171,95087,95088,95090,95092],{},[80174,95089,94825],{},[80174,95091,94829],{"align":94828},[80174,95093,80179],{},[80181,95095,95096,95195],{},[80171,95097,95098,95103,95107],{},[80186,95099,95100],{},[37,95101,95102],{},"F_CROSS_GHZ",[80186,95104,95105],{"align":94828},[37,95106,82115],{},[80186,95108,95109,95110],{},"Drive-frequency slice used to plot ",[40,95111,95113,95137],{"className":95112},[1565],[40,95114,95116],{"className":95115},[1569],[1571,95117,95118],{"xmlns":1573},[1575,95119,95120,95134],{},[1578,95121,95122,95128,95130,95132],{},[2212,95123,95124,95126],{},[1584,95125,2218],{},[1584,95127,2330],{},[1731,95129,241],{"stretchy":1797},[1584,95131,80017],{},[1731,95133,2693],{"stretchy":1797},[1591,95135,95136],{"encoding":1593},"P_e(\\tau)",[40,95138,95140],{"className":95139,"ariaHidden":1599},[1598],[40,95141,95143,95146,95186,95189,95192],{"className":95142},[1603],[40,95144],{"className":95145,"style":1811},[1607],[40,95147,95149,95152],{"className":95148},[1612],[40,95150,2218],{"className":95151,"style":2264},[1612,1653],[40,95153,95155],{"className":95154},[1621],[40,95156,95158,95178],{"className":95157},[1625,2244],[40,95159,95161,95175],{"className":95160},[1629],[40,95162,95164],{"className":95163,"style":3551},[1633],[40,95165,95166,95169],{"style":6681},[40,95167],{"className":95168,"style":1642},[1641],[40,95170,95172],{"className":95171},[1646,1647,1648,1649],[40,95173,2330],{"className":95174},[1612,1653,1649],[40,95176,2269],{"className":95177},[2268],[40,95179,95181],{"className":95180},[1629],[40,95182,95184],{"className":95183,"style":2276},[1633],[40,95185],{},[40,95187,241],{"className":95188},[1985],[40,95190,80017],{"className":95191,"style":80089},[1612,1653],[40,95193,2693],{"className":95194},[1822],[80171,95196,95197,95202,95207],{},[80186,95198,95199],{},[37,95200,95201],{},"T_CROSS_NS",[80186,95203,95204],{"align":94828},[37,95205,95206],{},"100.0",[80186,95208,95209,95210],{},"Pulse-duration slice used to plot ",[40,95211,95213,95237],{"className":95212},[1565],[40,95214,95216],{"className":95215},[1569],[1571,95217,95218],{"xmlns":1573},[1575,95219,95220,95234],{},[1578,95221,95222,95228,95230,95232],{},[2212,95223,95224,95226],{},[1584,95225,2218],{},[1584,95227,2330],{},[1731,95229,241],{"stretchy":1797},[1584,95231,21331],{},[1731,95233,2693],{"stretchy":1797},[1591,95235,95236],{"encoding":1593},"P_e(f)",[40,95238,95240],{"className":95239,"ariaHidden":1599},[1598],[40,95241,95243,95246,95286,95289,95292],{"className":95242},[1603],[40,95244],{"className":95245,"style":1811},[1607],[40,95247,95249,95252],{"className":95248},[1612],[40,95250,2218],{"className":95251,"style":2264},[1612,1653],[40,95253,95255],{"className":95254},[1621],[40,95256,95258,95278],{"className":95257},[1625,2244],[40,95259,95261,95275],{"className":95260},[1629],[40,95262,95264],{"className":95263,"style":3551},[1633],[40,95265,95266,95269],{"style":6681},[40,95267],{"className":95268,"style":1642},[1641],[40,95270,95272],{"className":95271},[1646,1647,1648,1649],[40,95273,2330],{"className":95274},[1612,1653,1649],[40,95276,2269],{"className":95277},[2268],[40,95279,95281],{"className":95280},[1629],[40,95282,95284],{"className":95283,"style":2276},[1633],[40,95285],{},[40,95287,241],{"className":95288},[1985],[40,95290,21331],{"className":95291,"style":26393},[1612,1653],[40,95293,2693],{"className":95294},[1822],[12,95296,95297],{},"Separate FFT processing controls support spectral analysis.",[80165,95299,95300,95310],{},[80168,95301,95302],{},[80171,95303,95304,95306,95308],{},[80174,95305,94825],{},[80174,95307,94829],{"align":94828},[80174,95309,80179],{},[80181,95311,95312,95326,95341,95355,95370,95385],{},[80171,95313,95314,95319,95323],{},[80186,95315,95316],{},[37,95317,95318],{},"FFT_PAD",[80186,95320,95321],{"align":94828},[37,95322,34795],{},[80186,95324,95325],{},"Zero-padding factor used in the Fourier transform",[80171,95327,95328,95333,95338],{},[80186,95329,95330],{},[37,95331,95332],{},"WINDOW",[80186,95334,95335],{"align":94828},[37,95336,95337],{},"\"hann\"",[80186,95339,95340],{},"Window function applied along the pulse-duration axis",[80171,95342,95343,95348,95352],{},[80186,95344,95345],{},[37,95346,95347],{},"DETREND_MEAN",[80186,95349,95350],{"align":94828},[37,95351,12926],{},[80186,95353,95354],{},"Flag that subtracts the mean before the FFT",[80171,95356,95357,95362,95367],{},[80186,95358,95359],{},[37,95360,95361],{},"AMP_MODE",[80186,95363,95364],{"align":94828},[37,95365,95366],{},"\"magnitude\"",[80186,95368,95369],{},"Display mode for magnitude, power, or dB-scaled amplitude",[80171,95371,95372,95377,95382],{},[80186,95373,95374],{},[37,95375,95376],{},"NORM_MODE",[80186,95378,95379],{"align":94828},[37,95380,95381],{},"\"global\"",[80186,95383,95384],{},"Normalization mode applied globally or per drive-frequency column",[80171,95386,95387,95392,95397],{},[80186,95388,95389],{},[37,95390,95391],{},"FREQ_UNITS",[80186,95393,95394],{"align":94828},[37,95395,95396],{},"\"MHz\"",[80186,95398,95399],{},"Frequency units used on the Fourier-frequency axis",[12,95401,95402],{},"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.",[30,95404,95406],{"className":32,"code":95405,"language":34,"meta":35,"style":35},"# @title Install dependencies\nimport sys\n%pip -q install pyvista pythreejs trame\n",[37,95407,95408,95413,95420],{"__ignoreMap":35},[40,95409,95410],{"class":42,"line":43},[40,95411,95412],{"class":46},"# @title Install dependencies\n",[40,95414,95415,95417],{"class":42,"line":50},[40,95416,130],{"class":129},[40,95418,95419],{"class":133}," sys\n",[40,95421,95422,95424,95427,95429],{"class":42,"line":155},[40,95423,7814],{"class":441},[40,95425,95426],{"class":133},"pip ",[40,95428,1407],{"class":441},[40,95430,95431],{"class":133},"q install pyvista pythreejs trame\n",[30,95433,95435],{"className":32,"code":95434,"language":34,"meta":35,"style":35},"# @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",[37,95436,95437,95442,95449,95460,95470,95481,95493,95497,95504,95520,95530,95538,95547,95551,95566,95579,95583,95588,95604,95620,95636,95652,95668,95684,95700,95715,95734,95738],{"__ignoreMap":35},[40,95438,95439],{"class":42,"line":43},[40,95440,95441],{"class":46},"# @title Controls & imports\n",[40,95443,95444,95446],{"class":42,"line":50},[40,95445,130],{"class":129},[40,95447,95448],{"class":133}," math\n",[40,95450,95451,95453,95455,95457],{"class":42,"line":155},[40,95452,171],{"class":129},[40,95454,174],{"class":133},[40,95456,130],{"class":129},[40,95458,95459],{"class":133}," Optional, Tuple\n",[40,95461,95462,95464,95466,95468],{"class":42,"line":168},[40,95463,130],{"class":129},[40,95465,134],{"class":133},[40,95467,137],{"class":129},[40,95469,140],{"class":133},[40,95471,95472,95474,95477,95479],{"class":42,"line":182},[40,95473,130],{"class":129},[40,95475,95476],{"class":133}," matplotlib.pyplot ",[40,95478,137],{"class":129},[40,95480,165],{"class":133},[40,95482,95483,95485,95488,95490],{"class":42,"line":195},[40,95484,130],{"class":129},[40,95486,95487],{"class":133}," pandas ",[40,95489,137],{"class":129},[40,95491,95492],{"class":133}," pd\n",[40,95494,95495],{"class":42,"line":203},[40,95496,228],{"emptyLinePlaceholder":227},[40,95498,95499,95502],{"class":42,"line":216},[40,95500,95501],{"class":129},"try",[40,95503,766],{"class":133},[40,95505,95506,95509,95512,95514,95517],{"class":42,"line":224},[40,95507,95508],{"class":129},"    import",[40,95510,95511],{"class":133}," pyvista ",[40,95513,137],{"class":129},[40,95515,95516],{"class":133}," pv  ",[40,95518,95519],{"class":46},"# optional; only for 3D interactive\n",[40,95521,95522,95525,95527],{"class":42,"line":231},[40,95523,95524],{"class":462},"    PV_AVAILABLE",[40,95526,459],{"class":441},[40,95528,95529],{"class":462}," True\n",[40,95531,95532,95535],{"class":42,"line":251},[40,95533,95534],{"class":129},"except",[40,95536,95537],{"class":133}," Exception:\n",[40,95539,95540,95542,95544],{"class":42,"line":256},[40,95541,95524],{"class":462},[40,95543,459],{"class":441},[40,95545,95546],{"class":462}," False\n",[40,95548,95549],{"class":42,"line":269},[40,95550,228],{"emptyLinePlaceholder":227},[40,95552,95553,95556,95559,95561,95563],{"class":42,"line":282},[40,95554,95555],{"class":133},"plt.rcParams[",[40,95557,95558],{"class":244},"'figure.dpi'",[40,95560,7250],{"class":133},[40,95562,623],{"class":441},[40,95564,95565],{"class":462}," 250\n",[40,95567,95568,95570,95573,95575,95577],{"class":42,"line":295},[40,95569,95555],{"class":133},[40,95571,95572],{"class":244},"'axes.grid'",[40,95574,7250],{"class":133},[40,95576,623],{"class":441},[40,95578,95546],{"class":462},[40,95580,95581],{"class":42,"line":308},[40,95582,228],{"emptyLinePlaceholder":227},[40,95584,95585],{"class":42,"line":321},[40,95586,95587],{"class":46},"# ---------- Control knobs ----------\n",[40,95589,95590,95592,95594,95596,95598,95601],{"class":42,"line":334},[40,95591,94840],{"class":462},[40,95593,438],{"class":133},[40,95595,442],{"class":441},[40,95597,459],{"class":441},[40,95599,95600],{"class":462}," 5.000",[40,95602,95603],{"class":46},"        # Qubit transition freq f0 [GHz]\n",[40,95605,95606,95608,95610,95612,95614,95617],{"class":42,"line":347},[40,95607,94924],{"class":462},[40,95609,438],{"class":133},[40,95611,442],{"class":441},[40,95613,459],{"class":441},[40,95615,95616],{"class":462}," 20.0",[40,95618,95619],{"class":46},"  # Ω\u002F2π on-resonance [MHz]\n",[40,95621,95622,95624,95626,95628,95630,95633],{"class":42,"line":352},[40,95623,94977],{"class":462},[40,95625,438],{"class":133},[40,95627,442],{"class":441},[40,95629,459],{"class":441},[40,95631,95632],{"class":462}," 4.90",[40,95634,95635],{"class":46},"   # Sweep min f [GHz]\n",[40,95637,95638,95640,95642,95644,95646,95649],{"class":42,"line":357},[40,95639,94992],{"class":462},[40,95641,438],{"class":133},[40,95643,442],{"class":441},[40,95645,459],{"class":441},[40,95647,95648],{"class":462}," 5.10",[40,95650,95651],{"class":46},"   # Sweep max f [GHz]\n",[40,95653,95654,95656,95658,95660,95662,95665],{"class":42,"line":370},[40,95655,95007],{"class":462},[40,95657,438],{"class":133},[40,95659,456],{"class":441},[40,95661,459],{"class":441},[40,95663,95664],{"class":462}," 401",[40,95666,95667],{"class":46},"            # # of frequency points\n",[40,95669,95670,95672,95674,95676,95678,95681],{"class":42,"line":376},[40,95671,95022],{"class":462},[40,95673,438],{"class":133},[40,95675,442],{"class":441},[40,95677,459],{"class":441},[40,95679,95680],{"class":462}," 0.0",[40,95682,95683],{"class":46},"      # Sweep min duration τ [ns]\n",[40,95685,95686,95688,95690,95692,95694,95697],{"class":42,"line":387},[40,95687,95037],{"class":462},[40,95689,438],{"class":133},[40,95691,442],{"class":441},[40,95693,459],{"class":441},[40,95695,95696],{"class":462}," 200.0",[40,95698,95699],{"class":46},"    # Sweep max duration τ [ns]\n",[40,95701,95702,95704,95706,95708,95710,95712],{"class":42,"line":392},[40,95703,95052],{"class":462},[40,95705,438],{"class":133},[40,95707,456],{"class":441},[40,95709,459],{"class":441},[40,95711,95664],{"class":462},[40,95713,95714],{"class":46},"             # # of duration points\n",[40,95716,95717,95719,95722,95724,95726,95728,95731],{"class":42,"line":397},[40,95718,95066],{"class":462},[40,95720,95721],{"class":133},": Optional[",[40,95723,442],{"class":441},[40,95725,7250],{"class":133},[40,95727,623],{"class":441},[40,95729,95730],{"class":462}," 500.0",[40,95732,95733],{"class":46},"  # None to disable exp(-τ\u002FT2*)\n",[40,95735,95736],{"class":42,"line":402},[40,95737,228],{"emptyLinePlaceholder":227},[40,95739,95740,95742,95744,95747],{"class":42,"line":414},[40,95741,7569],{"class":441},[40,95743,241],{"class":133},[40,95745,95746],{"class":244},"'Controls loaded.'",[40,95748,248],{"class":133},[30,95750,95753],{"className":95751,"code":95752,"language":6083,"meta":35},[7575],"Controls loaded.\n",[37,95754,95752],{"__ignoreMap":35},[30,95756,95758],{"className":32,"code":95757,"language":34,"meta":35,"style":35},"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",[37,95759,95760,95769,95776,95783,95794,95805,95823,95828,95833,95837,95841,95846,95851,95856,95861,95866,95870,95874,95879,95883,95898,95913,95927,95942,95965,95969,95997,96019,96048,96052,96083,96103,96123,96146,96150,96174,96204,96218,96222],{"__ignoreMap":35},[40,95761,95762,95764,95767],{"class":42,"line":43},[40,95763,405],{"class":129},[40,95765,95766],{"class":237}," rabi_probability",[40,95768,411],{"class":133},[40,95770,95771,95774],{"class":42,"line":50},[40,95772,95773],{"class":417},"    f_ghz",[40,95775,421],{"class":133},[40,95777,95778,95781],{"class":42,"line":155},[40,95779,95780],{"class":417},"    tau_ns",[40,95782,421],{"class":133},[40,95784,95785,95788,95790,95792],{"class":42,"line":168},[40,95786,95787],{"class":417},"    f0_ghz",[40,95789,438],{"class":133},[40,95791,442],{"class":441},[40,95793,445],{"class":133},[40,95795,95796,95799,95801,95803],{"class":42,"line":182},[40,95797,95798],{"class":417},"    omega_onres_mhz",[40,95800,438],{"class":133},[40,95802,442],{"class":441},[40,95804,445],{"class":133},[40,95806,95807,95810,95812,95814,95816,95818,95821],{"class":42,"line":195},[40,95808,95809],{"class":417},"    t2star_ns",[40,95811,95721],{"class":133},[40,95813,442],{"class":441},[40,95815,7250],{"class":133},[40,95817,623],{"class":441},[40,95819,95820],{"class":462}," None",[40,95822,445],{"class":133},[40,95824,95825],{"class":42,"line":203},[40,95826,95827],{"class":133},") -> np.ndarray:\n",[40,95829,95830],{"class":42,"line":216},[40,95831,95832],{"class":244},"    \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n",[40,95834,95835],{"class":42,"line":224},[40,95836,228],{"emptyLinePlaceholder":227},[40,95838,95839],{"class":42,"line":231},[40,95840,522],{"class":244},[40,95842,95843],{"class":42,"line":251},[40,95844,95845],{"class":244},"        f_ghz: 1D array of drive frequencies [GHz].\n",[40,95847,95848],{"class":42,"line":256},[40,95849,95850],{"class":244},"        tau_ns: 1D array of pulse durations [ns].\n",[40,95852,95853],{"class":42,"line":269},[40,95854,95855],{"class":244},"        f0_ghz: Qubit frequency on resonance [GHz].\n",[40,95857,95858],{"class":42,"line":282},[40,95859,95860],{"class":244},"        omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n",[40,95862,95863],{"class":42,"line":295},[40,95864,95865],{"class":244},"        t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n",[40,95867,95868],{"class":42,"line":308},[40,95869,228],{"emptyLinePlaceholder":227},[40,95871,95872],{"class":42,"line":321},[40,95873,593],{"class":244},[40,95875,95876],{"class":42,"line":334},[40,95877,95878],{"class":244},"        2D array P_e with shape (len(tau_ns), len(f_ghz)).\n",[40,95880,95881],{"class":42,"line":347},[40,95882,505],{"class":244},[40,95884,95885,95888,95890,95893,95895],{"class":42,"line":352},[40,95886,95887],{"class":133},"    two_pi ",[40,95889,623],{"class":441},[40,95891,95892],{"class":462}," 2.0",[40,95894,7273],{"class":441},[40,95896,95897],{"class":133}," math.pi\n",[40,95899,95900,95903,95905,95908,95910],{"class":42,"line":357},[40,95901,95902],{"class":133},"    f ",[40,95904,623],{"class":441},[40,95906,95907],{"class":133}," f_ghz ",[40,95909,1433],{"class":441},[40,95911,95912],{"class":462}," 1e9\n",[40,95914,95915,95918,95920,95923,95925],{"class":42,"line":370},[40,95916,95917],{"class":133},"    f0 ",[40,95919,623],{"class":441},[40,95921,95922],{"class":133}," f0_ghz ",[40,95924,1433],{"class":441},[40,95926,95912],{"class":462},[40,95928,95929,95932,95934,95937,95939],{"class":42,"line":376},[40,95930,95931],{"class":133},"    tau ",[40,95933,623],{"class":441},[40,95935,95936],{"class":133}," tau_ns ",[40,95938,1433],{"class":441},[40,95940,95941],{"class":462}," 1e-9\n",[40,95943,95944,95947,95949,95952,95954,95957,95959,95962],{"class":42,"line":387},[40,95945,95946],{"class":133},"    omega ",[40,95948,623],{"class":441},[40,95950,95951],{"class":133}," two_pi ",[40,95953,1433],{"class":441},[40,95955,95956],{"class":133}," omega_onres_mhz ",[40,95958,1433],{"class":441},[40,95960,95961],{"class":462}," 1e6",[40,95963,95964],{"class":46},"  # rad\u002Fs\n",[40,95966,95967],{"class":42,"line":392},[40,95968,228],{"emptyLinePlaceholder":227},[40,95970,95971,95974,95977,95979,95981,95984,95987,95990,95992,95995],{"class":42,"line":397},[40,95972,95973],{"class":133},"    F, ",[40,95975,95976],{"class":462},"TAU",[40,95978,459],{"class":441},[40,95980,654],{"class":133},[40,95982,95983],{"class":237},"meshgrid",[40,95985,95986],{"class":133},"(f, tau, ",[40,95988,95989],{"class":1195},"indexing",[40,95991,623],{"class":441},[40,95993,95994],{"class":244},"'xy'",[40,95996,248],{"class":133},[40,95998,95999,96002,96004,96006,96008,96011,96013,96016],{"class":42,"line":402},[40,96000,96001],{"class":133},"    Delta ",[40,96003,623],{"class":441},[40,96005,95951],{"class":133},[40,96007,1433],{"class":441},[40,96009,96010],{"class":133}," (F ",[40,96012,1407],{"class":441},[40,96014,96015],{"class":133}," f0)               ",[40,96017,96018],{"class":46},"# rad\u002Fs\n",[40,96020,96021,96024,96026,96028,96030,96033,96035,96037,96039,96042,96044,96046],{"class":42,"line":414},[40,96022,96023],{"class":133},"    Omega_R ",[40,96025,623],{"class":441},[40,96027,654],{"class":133},[40,96029,8866],{"class":237},[40,96031,96032],{"class":133},"(omega",[40,96034,1419],{"class":441},[40,96036,1879],{"class":462},[40,96038,7357],{"class":441},[40,96040,96041],{"class":133}," Delta",[40,96043,1419],{"class":441},[40,96045,1879],{"class":462},[40,96047,248],{"class":133},[40,96049,96050],{"class":42,"line":424},[40,96051,228],{"emptyLinePlaceholder":227},[40,96053,96054,96057,96059,96062,96064,96067,96069,96072,96074,96077,96079,96081],{"class":42,"line":432},[40,96055,96056],{"class":129},"    with",[40,96058,654],{"class":133},[40,96060,96061],{"class":237},"errstate",[40,96063,241],{"class":133},[40,96065,96066],{"class":1195},"divide",[40,96068,623],{"class":441},[40,96070,96071],{"class":244},"'ignore'",[40,96073,917],{"class":133},[40,96075,96076],{"class":1195},"invalid",[40,96078,623],{"class":441},[40,96080,96071],{"class":244},[40,96082,923],{"class":133},[40,96084,96085,96088,96090,96093,96095,96098,96100],{"class":42,"line":448},[40,96086,96087],{"class":133},"        amp ",[40,96089,623],{"class":441},[40,96091,96092],{"class":133}," (omega ",[40,96094,7610],{"class":441},[40,96096,96097],{"class":133}," Omega_R) ",[40,96099,1419],{"class":441},[40,96101,96102],{"class":462}," 2\n",[40,96104,96105,96108,96110,96113,96115,96118,96120],{"class":42,"line":468},[40,96106,96107],{"class":133},"        phase ",[40,96109,623],{"class":441},[40,96111,96112],{"class":462}," 0.5",[40,96114,7273],{"class":441},[40,96116,96117],{"class":133}," Omega_R ",[40,96119,1433],{"class":441},[40,96121,96122],{"class":462}," TAU\n",[40,96124,96125,96128,96130,96133,96135,96137,96139,96142,96144],{"class":42,"line":486},[40,96126,96127],{"class":133},"        Pe ",[40,96129,623],{"class":441},[40,96131,96132],{"class":133}," amp ",[40,96134,1433],{"class":441},[40,96136,654],{"class":133},[40,96138,85034],{"class":237},[40,96140,96141],{"class":133},"(phase) ",[40,96143,1419],{"class":441},[40,96145,96102],{"class":462},[40,96147,96148],{"class":42,"line":502},[40,96149,228],{"emptyLinePlaceholder":227},[40,96151,96152,96154,96157,96159,96161,96163,96166,96168,96170,96172],{"class":42,"line":508},[40,96153,755],{"class":129},[40,96155,96156],{"class":133}," t2star_ns ",[40,96158,36063],{"class":129},[40,96160,21033],{"class":129},[40,96162,95820],{"class":462},[40,96164,96165],{"class":129}," and",[40,96167,96156],{"class":133},[40,96169,689],{"class":441},[40,96171,95680],{"class":462},[40,96173,766],{"class":133},[40,96175,96176,96179,96181,96183,96185,96187,96189,96191,96194,96197,96199,96202],{"class":42,"line":514},[40,96177,96178],{"class":133},"        decay ",[40,96180,623],{"class":441},[40,96182,654],{"class":133},[40,96184,23123],{"class":237},[40,96186,241],{"class":133},[40,96188,1407],{"class":441},[40,96190,95976],{"class":462},[40,96192,96193],{"class":441}," \u002F",[40,96195,96196],{"class":133}," (t2star_ns ",[40,96198,1433],{"class":441},[40,96200,96201],{"class":462}," 1e-9",[40,96203,21594],{"class":133},[40,96205,96206,96208,96210,96213,96215],{"class":42,"line":519},[40,96207,96127],{"class":133},[40,96209,623],{"class":441},[40,96211,96212],{"class":133}," Pe ",[40,96214,1433],{"class":441},[40,96216,96217],{"class":133}," decay\n",[40,96219,96220],{"class":42,"line":525},[40,96221,228],{"emptyLinePlaceholder":227},[40,96223,96224,96226,96228,96231,96234,96236,96238,96241],{"class":42,"line":531},[40,96225,1546],{"class":129},[40,96227,654],{"class":133},[40,96229,96230],{"class":237},"clip",[40,96232,96233],{"class":133},"(Pe, ",[40,96235,95027],{"class":462},[40,96237,917],{"class":133},[40,96239,96240],{"class":462},"1.0",[40,96242,248],{"class":133},[30,96244,96246],{"className":32,"code":96245,"language":34,"meta":35,"style":35},"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",[37,96247,96248,96274,96299,96323],{"__ignoreMap":35},[40,96249,96250,96253,96255,96257,96260,96262,96264,96266,96268,96270,96272],{"class":42,"line":43},[40,96251,96252],{"class":133},"freq_ghz ",[40,96254,623],{"class":441},[40,96256,654],{"class":133},[40,96258,96259],{"class":237},"linspace",[40,96261,241],{"class":133},[40,96263,94977],{"class":462},[40,96265,917],{"class":133},[40,96267,94992],{"class":462},[40,96269,917],{"class":133},[40,96271,95007],{"class":462},[40,96273,248],{"class":133},[40,96275,96276,96279,96281,96283,96285,96287,96289,96291,96293,96295,96297],{"class":42,"line":50},[40,96277,96278],{"class":133},"dur_ns ",[40,96280,623],{"class":441},[40,96282,654],{"class":133},[40,96284,96259],{"class":237},[40,96286,241],{"class":133},[40,96288,95022],{"class":462},[40,96290,917],{"class":133},[40,96292,95037],{"class":462},[40,96294,917],{"class":133},[40,96296,95052],{"class":462},[40,96298,248],{"class":133},[40,96300,96301,96304,96306,96308,96311,96313,96315,96317,96319,96321],{"class":42,"line":155},[40,96302,96303],{"class":133},"Pe ",[40,96305,623],{"class":441},[40,96307,95766],{"class":237},[40,96309,96310],{"class":133},"(freq_ghz, dur_ns, ",[40,96312,94840],{"class":462},[40,96314,917],{"class":133},[40,96316,94924],{"class":462},[40,96318,917],{"class":133},[40,96320,95066],{"class":462},[40,96322,248],{"class":133},[40,96324,96325,96327,96329,96332],{"class":42,"line":168},[40,96326,7569],{"class":441},[40,96328,241],{"class":133},[40,96330,96331],{"class":244},"'Grid shape (τ × f):'",[40,96333,96334],{"class":133},", Pe.shape)\n",[30,96336,96339],{"className":96337,"code":96338,"language":6083,"meta":35},[7575],"Grid shape (τ × f): (401, 401)\n",[37,96340,96338],{"__ignoreMap":35},[30,96342,96344],{"className":32,"code":96343,"language":34,"meta":35,"style":35},"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",[37,96345,96346,96361,96366,96381,96394,96407,96420,96433,96446,96459,96472,96485,96490,96509,96513],{"__ignoreMap":35},[40,96347,96348,96351,96353,96356,96359],{"class":42,"line":43},[40,96349,96350],{"class":133},"controls_table ",[40,96352,623],{"class":441},[40,96354,96355],{"class":133}," pd.",[40,96357,96358],{"class":237},"DataFrame",[40,96360,411],{"class":133},[40,96362,96363],{"class":42,"line":50},[40,96364,96365],{"class":133},"    [\n",[40,96367,96368,96371,96374,96376,96378],{"class":42,"line":155},[40,96369,96370],{"class":133},"        [",[40,96372,96373],{"class":244},"'f0 (GHz)'",[40,96375,917],{"class":133},[40,96377,94840],{"class":462},[40,96379,96380],{"class":133},"],\n",[40,96382,96383,96385,96388,96390,96392],{"class":42,"line":168},[40,96384,96370],{"class":133},[40,96386,96387],{"class":244},"'Ω\u002F2π on-res (MHz)'",[40,96389,917],{"class":133},[40,96391,94924],{"class":462},[40,96393,96380],{"class":133},[40,96395,96396,96398,96401,96403,96405],{"class":42,"line":182},[40,96397,96370],{"class":133},[40,96399,96400],{"class":244},"'f sweep min (GHz)'",[40,96402,917],{"class":133},[40,96404,94977],{"class":462},[40,96406,96380],{"class":133},[40,96408,96409,96411,96414,96416,96418],{"class":42,"line":195},[40,96410,96370],{"class":133},[40,96412,96413],{"class":244},"'f sweep max (GHz)'",[40,96415,917],{"class":133},[40,96417,94992],{"class":462},[40,96419,96380],{"class":133},[40,96421,96422,96424,96427,96429,96431],{"class":42,"line":203},[40,96423,96370],{"class":133},[40,96425,96426],{"class":244},"'N_f'",[40,96428,917],{"class":133},[40,96430,95007],{"class":462},[40,96432,96380],{"class":133},[40,96434,96435,96437,96440,96442,96444],{"class":42,"line":216},[40,96436,96370],{"class":133},[40,96438,96439],{"class":244},"'τ sweep min (ns)'",[40,96441,917],{"class":133},[40,96443,95022],{"class":462},[40,96445,96380],{"class":133},[40,96447,96448,96450,96453,96455,96457],{"class":42,"line":224},[40,96449,96370],{"class":133},[40,96451,96452],{"class":244},"'τ sweep max (ns)'",[40,96454,917],{"class":133},[40,96456,95037],{"class":462},[40,96458,96380],{"class":133},[40,96460,96461,96463,96466,96468,96470],{"class":42,"line":231},[40,96462,96370],{"class":133},[40,96464,96465],{"class":244},"'N_τ'",[40,96467,917],{"class":133},[40,96469,95052],{"class":462},[40,96471,96380],{"class":133},[40,96473,96474,96476,96479,96481,96483],{"class":42,"line":251},[40,96475,96370],{"class":133},[40,96477,96478],{"class":244},"'T2* (ns; None=off)'",[40,96480,917],{"class":133},[40,96482,95066],{"class":462},[40,96484,96380],{"class":133},[40,96486,96487],{"class":42,"line":256},[40,96488,96489],{"class":133},"    ],\n",[40,96491,96492,96495,96497,96499,96502,96504,96507],{"class":42,"line":269},[40,96493,96494],{"class":1195},"    columns",[40,96496,623],{"class":441},[40,96498,11295],{"class":133},[40,96500,96501],{"class":244},"'Parameter'",[40,96503,917],{"class":133},[40,96505,96506],{"class":244},"'Value'",[40,96508,833],{"class":133},[40,96510,96511],{"class":42,"line":282},[40,96512,248],{"class":133},[40,96514,96515],{"class":42,"line":295},[40,96516,96517],{"class":133},"controls_table\n",[30,96519,96522],{"className":96520,"code":96521,"language":6083,"meta":35},[7575],"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",[37,96523,96521],{"__ignoreMap":35},[30,96525,96527],{"className":32,"code":96526,"language":34,"meta":35,"style":35},"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",[37,96528,96529,96561,96587,96623,96638,96652,96676,96686],{"__ignoreMap":35},[40,96530,96531,96534,96536,96539,96542,96544,96547,96549,96551,96554,96556,96559],{"class":42,"line":43},[40,96532,96533],{"class":133},"fig, ax ",[40,96535,623],{"class":441},[40,96537,96538],{"class":133}," plt.",[40,96540,96541],{"class":237},"subplots",[40,96543,241],{"class":133},[40,96545,96546],{"class":1195},"figsize",[40,96548,623],{"class":441},[40,96550,241],{"class":133},[40,96552,96553],{"class":462},"7",[40,96555,917],{"class":133},[40,96557,96558],{"class":462},"5",[40,96560,21594],{"class":133},[40,96562,96563,96566,96568,96571,96574,96577,96580,96582,96585],{"class":42,"line":50},[40,96564,96565],{"class":133},"c ",[40,96567,623],{"class":441},[40,96569,96570],{"class":133}," ax.",[40,96572,96573],{"class":237},"pcolormesh",[40,96575,96576],{"class":133},"(freq_ghz, dur_ns, Pe, ",[40,96578,96579],{"class":1195},"shading",[40,96581,623],{"class":441},[40,96583,96584],{"class":244},"'auto'",[40,96586,248],{"class":133},[40,96588,96589,96592,96594,96597,96600,96603,96606,96608,96611,96613,96615,96617,96621],{"class":42,"line":155},[40,96590,96591],{"class":133},"cb ",[40,96593,623],{"class":441},[40,96595,96596],{"class":133}," fig.",[40,96598,96599],{"class":237},"colorbar",[40,96601,96602],{"class":133},"(c, ",[40,96604,96605],{"class":1195},"ax",[40,96607,623],{"class":441},[40,96609,96610],{"class":133},"ax, ",[40,96612,20062],{"class":1195},[40,96614,623],{"class":441},[40,96616,1256],{"class":129},[40,96618,96620],{"class":96619},"sVyAn","'$P_e$'",[40,96622,248],{"class":133},[40,96624,96625,96628,96631,96633,96636],{"class":42,"line":168},[40,96626,96627],{"class":133},"ax.",[40,96629,96630],{"class":237},"set_xlabel",[40,96632,241],{"class":133},[40,96634,96635],{"class":244},"'Drive frequency f (GHz)'",[40,96637,248],{"class":133},[40,96639,96640,96642,96645,96647,96650],{"class":42,"line":182},[40,96641,96627],{"class":133},[40,96643,96644],{"class":237},"set_ylabel",[40,96646,241],{"class":133},[40,96648,96649],{"class":244},"'Pulse duration τ (ns)'",[40,96651,248],{"class":133},[40,96653,96654,96656,96659,96661,96664,96667,96669,96671,96674],{"class":42,"line":195},[40,96655,96627],{"class":133},[40,96657,96658],{"class":237},"set_title",[40,96660,241],{"class":133},[40,96662,96663],{"class":244},"'Rabi excited-state probability $P_e(f,",[40,96665,96666],{"class":441},"\\\\",[40,96668,1862],{"class":244},[40,96670,96666],{"class":441},[40,96672,96673],{"class":244},"tau)$'",[40,96675,248],{"class":133},[40,96677,96678,96681,96684],{"class":42,"line":203},[40,96679,96680],{"class":133},"fig.",[40,96682,96683],{"class":237},"tight_layout",[40,96685,12875],{"class":133},[40,96687,96688,96690,96692],{"class":42,"line":216},[40,96689,24430],{"class":133},[40,96691,24667],{"class":237},[40,96693,12875],{"class":133},[8229,96695],{"alt":96696,"src":96697},"Output 1 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.webp",[30,96699,96701],{"className":32,"code":96700,"language":34,"meta":35,"style":35},"# @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",[37,96702,96703,96708,96713,96717,96724,96731,96737,96741,96746,96751,96755,96764,96768,96777,96789,96801,96813,96825,96829,96833,96859,96874,96881,96888,96892,96896,96906,96916,96920,96925,96935,96939,96948,96968,96977,96997,97001,97010,97019,97028,97037,97046,97050,97060,97070,97080,97090,97100,97104,97114,97124,97134,97144,97154,97158,97168,97186,97190,97194,97218,97223,97242,97246,97250,97259,97275,97283,97288,97299,97304,97308,97317,97322,97338,97349,97354,97358,97367,97371,97375,97382,97391,97421,97432,97437,97445,97450,97455,97462,97469,97476,97483,97490,97497,97505,97510,97520,97524,97532,97540,97544,97548,97554,97560,97566,97572,97578,97582,97592,97596,97600,97604,97615,97619,97629,97639,97651,97663,97668,97675,97689,97694,97707,97723,97743,97751,97757,97762,97773,97779,97791,97797,97802,97807,97812,97835,97841,97861,97866,97871,97881,97893,97905,97917,97937,97943,97967,97990,98006,98011,98038,98043,98067,98089,98110,98115,98129,98144,98162,98167,98172,98187,98193,98199,98205,98210,98215,98247,98274,98300,98305,98319,98334,98339,98344,98358,98363,98369,98375,98381,98386,98391,98396,98410,98415,98426,98434,98444,98449,98454,98459,98466,98482,98491,98500,98509,98518,98541,98547,98552,98557,98562,98572,98581,98590,98595,98601,98617,98628,98643,98653,98660,98669,98674,98696,98701,98718,98735,98740,98753,98767,98772,98780,98785,98790,98800,98808,98816,98827,98838,98847,98853,98863,98873,98882,98891,98902,98914,98925,98930,98935,98940,98950,98957,98965,98976,98987,98996,99002,99012,99022,99031,99040,99049,99061,99073,99082,99087,99092,99097,99107,99114,99121,99132,99143,99152,99158,99168,99176,99183,99190,99199,99208,99217,99226,99237,99249,99261,99270,99275,99280,99285,99295,99302,99309,99321,99332,99341,99347,99363,99368,99377,99387,99396,99405,99411,99420,99432,99445,99457,99462,99467,99472,99491,99497,99506,99513,99520,99528,99539,99550,99561,99566,99575,99582,99589,99597,99618,99629,99641,99652,99663,99668,99673,99678,99692,99704,99710,99715,99723,99733,99738,99750,99761,99771,99782,99787,99792,99804,99815,99824,99835,99840,99845,99855,99861,99875,99880,99890,99895,99907,99912,99917,99940,99945,99968,99979,100018,100028,100040,100060,100078,100083,100088,100100,100110,100120,100125,100130,100138,100148,100163,100174,100186,100191,100198,100207,100221,100232,100244,100249,100254,100264,100272,100282,100303,100314,100326,100331,100338,100347,100366,100377,100389,100394,100403,100408,100418,100426,100435,100449,100460,100486,100491,100498,100507,100520,100531,100554,100559,100564,100578,100586,100595,100608,100630,100641,100646,100653,100662,100675,100696,100707,100712,100717,100727,100737,100745,100755,100764,100776,100788,100800,100805,100812,100821,100830,100841,100852,100863,100868,100875,100884,100893,100904,100915,100936,100941,100948,100957,100966,100977,100988,101009,101014,101019,101024,101034,101045,101057,101069,101081,101093,101098,101103,101112,101117,101127,101142,101157,101162,101174,101199,101224,101249,101274,101300,101307,101313,101338,101359,101381,101386,101393,101399,101421,101442,101463,101468,101475,101481,101509,101514,101521,101527,101554,101559,101566,101572,101595],{"__ignoreMap":35},[40,96704,96705],{"class":42,"line":43},[40,96706,96707],{"class":46},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n",[40,96709,96710],{"class":42,"line":50},[40,96711,96712],{"class":244},"\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n",[40,96714,96715],{"class":42,"line":155},[40,96716,228],{"emptyLinePlaceholder":227},[40,96718,96719,96721],{"class":42,"line":168},[40,96720,130],{"class":129},[40,96722,96723],{"class":133}," importlib.util\n",[40,96725,96726,96728],{"class":42,"line":182},[40,96727,130],{"class":129},[40,96729,96730],{"class":133}," subprocess\n",[40,96732,96733,96735],{"class":42,"line":195},[40,96734,130],{"class":129},[40,96736,95419],{"class":133},[40,96738,96739],{"class":42,"line":203},[40,96740,228],{"emptyLinePlaceholder":227},[40,96742,96743],{"class":42,"line":216},[40,96744,96745],{"class":46},"# -------------------------------------------------------------------------\n",[40,96747,96748],{"class":42,"line":224},[40,96749,96750],{"class":46},"# Control knobs\n",[40,96752,96753],{"class":42,"line":231},[40,96754,96745],{"class":46},[40,96756,96757,96760,96762],{"class":42,"line":251},[40,96758,96759],{"class":462},"INSTALL_BLOCH_DEPENDENCIES",[40,96761,459],{"class":441},[40,96763,95529],{"class":462},[40,96765,96766],{"class":42,"line":256},[40,96767,228],{"emptyLinePlaceholder":227},[40,96769,96770,96773,96775],{"class":42,"line":269},[40,96771,96772],{"class":462},"BLOCH_REQUIRED_PACKAGES",[40,96774,459],{"class":441},[40,96776,22777],{"class":133},[40,96778,96779,96782,96784,96787],{"class":42,"line":282},[40,96780,96781],{"class":244},"    \"numpy\"",[40,96783,438],{"class":133},[40,96785,96786],{"class":244},"\"numpy\"",[40,96788,445],{"class":133},[40,96790,96791,96794,96796,96799],{"class":42,"line":295},[40,96792,96793],{"class":244},"    \"matplotlib\"",[40,96795,438],{"class":133},[40,96797,96798],{"class":244},"\"matplotlib\"",[40,96800,445],{"class":133},[40,96802,96803,96806,96808,96811],{"class":42,"line":308},[40,96804,96805],{"class":244},"    \"scipy\"",[40,96807,438],{"class":133},[40,96809,96810],{"class":244},"\"scipy\"",[40,96812,445],{"class":133},[40,96814,96815,96818,96820,96823],{"class":42,"line":321},[40,96816,96817],{"class":244},"    \"qutip\"",[40,96819,438],{"class":133},[40,96821,96822],{"class":244},"\"qutip\"",[40,96824,445],{"class":133},[40,96826,96827],{"class":42,"line":334},[40,96828,22929],{"class":133},[40,96830,96831],{"class":42,"line":347},[40,96832,228],{"emptyLinePlaceholder":227},[40,96834,96835,96838,96840,96843,96845,96848,96850,96853,96855,96857],{"class":42,"line":352},[40,96836,96837],{"class":462},"BLOCH_F0_GHZ",[40,96839,459],{"class":441},[40,96841,96842],{"class":441}," globals",[40,96844,13031],{"class":133},[40,96846,96847],{"class":237},"get",[40,96849,241],{"class":133},[40,96851,96852],{"class":244},"\"F0_GHZ\"",[40,96854,917],{"class":133},[40,96856,82115],{"class":462},[40,96858,248],{"class":133},[40,96860,96861,96864,96866,96868,96870,96872],{"class":42,"line":357},[40,96862,96863],{"class":462},"BLOCH_OMEGA_ONRESONANCE_MHZ",[40,96865,459],{"class":441},[40,96867,96842],{"class":441},[40,96869,13031],{"class":133},[40,96871,96847],{"class":237},[40,96873,411],{"class":133},[40,96875,96876,96879],{"class":42,"line":370},[40,96877,96878],{"class":244},"    \"OMEGA_ONRESONANCE_MHZ\"",[40,96880,445],{"class":133},[40,96882,96883,96886],{"class":42,"line":376},[40,96884,96885],{"class":462},"    20.0",[40,96887,445],{"class":133},[40,96889,96890],{"class":42,"line":387},[40,96891,248],{"class":133},[40,96893,96894],{"class":42,"line":392},[40,96895,228],{"emptyLinePlaceholder":227},[40,96897,96898,96901,96903],{"class":42,"line":397},[40,96899,96900],{"class":462},"BLOCH_RESONANT_DETUNING_MHZ",[40,96902,459],{"class":441},[40,96904,96905],{"class":462}," 0.0\n",[40,96907,96908,96911,96913],{"class":42,"line":402},[40,96909,96910],{"class":462},"BLOCH_DETUNED_DETUNING_MHZ",[40,96912,459],{"class":441},[40,96914,96915],{"class":462}," 40.0\n",[40,96917,96918],{"class":42,"line":414},[40,96919,228],{"emptyLinePlaceholder":227},[40,96921,96922],{"class":42,"line":424},[40,96923,96924],{"class":46},"# Use None to automatically plot one resonant pi pulse.\n",[40,96926,96927,96930,96932],{"class":42,"line":432},[40,96928,96929],{"class":462},"BLOCH_DURATION_NS",[40,96931,459],{"class":441},[40,96933,96934],{"class":462}," None\n",[40,96936,96937],{"class":42,"line":448},[40,96938,228],{"emptyLinePlaceholder":227},[40,96940,96941,96944,96946],{"class":42,"line":468},[40,96942,96943],{"class":462},"BLOCH_NUM_TIME_POINTS",[40,96945,459],{"class":441},[40,96947,22675],{"class":462},[40,96949,96950,96953,96955,96958,96961,96963,96966],{"class":42,"line":486},[40,96951,96952],{"class":462},"BLOCH_FIGSIZE",[40,96954,459],{"class":441},[40,96956,96957],{"class":133}," (",[40,96959,96960],{"class":462},"8.8",[40,96962,917],{"class":133},[40,96964,96965],{"class":462},"7.8",[40,96967,248],{"class":133},[40,96969,96970,96973,96975],{"class":42,"line":502},[40,96971,96972],{"class":462},"BLOCH_DPI",[40,96974,459],{"class":441},[40,96976,95565],{"class":462},[40,96978,96979,96982,96984,96986,96988,96991,96993,96995],{"class":42,"line":508},[40,96980,96981],{"class":462},"BLOCH_VIEW",[40,96983,459],{"class":441},[40,96985,7471],{"class":133},[40,96987,1407],{"class":441},[40,96989,96990],{"class":462},"60",[40,96992,917],{"class":133},[40,96994,86901],{"class":462},[40,96996,833],{"class":133},[40,96998,96999],{"class":42,"line":514},[40,97000,228],{"emptyLinePlaceholder":227},[40,97002,97003,97006,97008],{"class":42,"line":519},[40,97004,97005],{"class":462},"BLOCH_SHOW_CONTROL_AXIS_ARROWS",[40,97007,459],{"class":441},[40,97009,95529],{"class":462},[40,97011,97012,97015,97017],{"class":42,"line":525},[40,97013,97014],{"class":462},"BLOCH_SHOW_ARROW_TIP_LABELS",[40,97016,459],{"class":441},[40,97018,95529],{"class":462},[40,97020,97021,97024,97026],{"class":42,"line":531},[40,97022,97023],{"class":462},"BLOCH_SHOW_ENDPOINT_MARKERS",[40,97025,459],{"class":441},[40,97027,95529],{"class":462},[40,97029,97030,97033,97035],{"class":42,"line":537},[40,97031,97032],{"class":462},"BLOCH_SHOW_LEGEND",[40,97034,459],{"class":441},[40,97036,95529],{"class":462},[40,97038,97039,97042,97044],{"class":42,"line":543},[40,97040,97041],{"class":462},"BLOCH_PRINT_SUMMARY",[40,97043,459],{"class":441},[40,97045,95529],{"class":462},[40,97047,97048],{"class":42,"line":549},[40,97049,228],{"emptyLinePlaceholder":227},[40,97051,97052,97055,97057],{"class":42,"line":555},[40,97053,97054],{"class":462},"COLOR_RESONANT_TRAJECTORY",[40,97056,459],{"class":441},[40,97058,97059],{"class":244}," \"tab:blue\"\n",[40,97061,97062,97065,97067],{"class":42,"line":561},[40,97063,97064],{"class":462},"COLOR_DETUNED_TRAJECTORY",[40,97066,459],{"class":441},[40,97068,97069],{"class":244}," \"tab:orange\"\n",[40,97071,97072,97075,97077],{"class":42,"line":567},[40,97073,97074],{"class":462},"COLOR_RESONANT_AXIS",[40,97076,459],{"class":441},[40,97078,97079],{"class":244}," \"tab:green\"\n",[40,97081,97082,97085,97087],{"class":42,"line":573},[40,97083,97084],{"class":462},"COLOR_DETUNED_AXIS",[40,97086,459],{"class":441},[40,97088,97089],{"class":244}," \"tab:red\"\n",[40,97091,97092,97095,97097],{"class":42,"line":579},[40,97093,97094],{"class":462},"COLOR_INITIAL_STATE",[40,97096,459],{"class":441},[40,97098,97099],{"class":244}," 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1.16\n",[40,97155,97156],{"class":42,"line":617},[40,97157,228],{"emptyLinePlaceholder":227},[40,97159,97160,97163,97165],{"class":42,"line":635},[40,97161,97162],{"class":462},"LEGEND_LOCATION",[40,97164,459],{"class":441},[40,97166,97167],{"class":244}," \"upper left\"\n",[40,97169,97170,97173,97175,97177,97180,97182,97184],{"class":42,"line":646},[40,97171,97172],{"class":462},"LEGEND_BBOX_TO_ANCHOR",[40,97174,459],{"class":441},[40,97176,96957],{"class":133},[40,97178,97179],{"class":462},"1.02",[40,97181,917],{"class":133},[40,97183,97179],{"class":462},[40,97185,248],{"class":133},[40,97187,97188],{"class":42,"line":662},[40,97189,228],{"emptyLinePlaceholder":227},[40,97191,97192],{"class":42,"line":695},[40,97193,228],{"emptyLinePlaceholder":227},[40,97195,97196,97198,97201,97203,97206,97208,97211,97214,97216],{"class":42,"line":701},[40,97197,405],{"class":129},[40,97199,97200],{"class":237}," 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label",[40,100795,623],{"class":441},[40,100797,100183],{"class":244},[40,100799,445],{"class":133},[40,100801,100803],{"class":42,"line":100802},433,[40,100804,98546],{"class":133},[40,100806,100808,100810],{"class":42,"line":100807},434,[40,100809,100742],{"class":237},[40,100811,411],{"class":133},[40,100813,100815,100817,100819],{"class":42,"line":100814},435,[40,100816,100750],{"class":133},[40,100818,791],{"class":462},[40,100820,96380],{"class":133},[40,100822,100824,100826,100828],{"class":42,"line":100823},436,[40,100825,100750],{"class":133},[40,100827,791],{"class":462},[40,100829,96380],{"class":133},[40,100831,100833,100835,100837,100839],{"class":42,"line":100832},437,[40,100834,100769],{"class":1195},[40,100836,623],{"class":441},[40,100838,97064],{"class":462},[40,100840,445],{"class":133},[40,100842,100844,100846,100848,100850],{"class":42,"line":100843},438,[40,100845,100781],{"class":1195},[40,100847,623],{"class":441},[40,100849,97108],{"class":462},[40,100851,445],{"class":133},[40,100853,100855,100857,100859,100861],{"class":42,"line":100854},439,[40,100856,100793],{"class":1195},[40,100858,623],{"class":441},[40,100860,100241],{"class":244},[40,100862,445],{"class":133},[40,100864,100866],{"class":42,"line":100865},440,[40,100867,98546],{"class":133},[40,100869,100871,100873],{"class":42,"line":100870},441,[40,100872,100742],{"class":237},[40,100874,411],{"class":133},[40,100876,100878,100880,100882],{"class":42,"line":100877},442,[40,100879,100750],{"class":133},[40,100881,791],{"class":462},[40,100883,96380],{"class":133},[40,100885,100887,100889,100891],{"class":42,"line":100886},443,[40,100888,100750],{"class":133},[40,100890,791],{"class":462},[40,100892,96380],{"class":133},[40,100894,100896,100898,100900,100902],{"class":42,"line":100895},444,[40,100897,100769],{"class":1195},[40,100899,623],{"class":441},[40,100901,97074],{"class":462},[40,100903,445],{"class":133},[40,100905,100907,100909,100911,100913],{"class":42,"line":100906},445,[40,100908,100781],{"class":1195},[40,100910,623],{"class":441},[40,100912,97118],{"class":462},[40,100914,445],{"class":133},[40,100916,100918,100920,100922,100924,100926,100928,100930,100932,100934],{"class":42,"line":100917},446,[40,100919,100793],{"class":1195},[40,100921,623],{"class":441},[40,100923,1256],{"class":129},[40,100925,100471],{"class":96619},[40,100927,100474],{"class":441},[40,100929,100477],{"class":96619},[40,100931,100480],{"class":441},[40,100933,100483],{"class":96619},[40,100935,445],{"class":133},[40,100937,100939],{"class":42,"line":100938},447,[40,100940,98546],{"class":133},[40,100942,100944,100946],{"class":42,"line":100943},448,[40,100945,100742],{"class":237},[40,100947,411],{"class":133},[40,100949,100951,100953,100955],{"class":42,"line":100950},449,[40,100952,100750],{"class":133},[40,100954,791],{"class":462},[40,100956,96380],{"class":133},[40,100958,100960,100962,100964],{"class":42,"line":100959},450,[40,100961,100750],{"class":133},[40,100963,791],{"class":462},[40,100965,96380],{"class":133},[40,100967,100969,100971,100973,100975],{"class":42,"line":100968},451,[40,100970,100769],{"class":1195},[40,100972,623],{"class":441},[40,100974,97084],{"class":462},[40,100976,445],{"class":133},[40,100978,100980,100982,100984,100986],{"class":42,"line":100979},452,[40,100981,100781],{"class":1195},[40,100983,623],{"class":441},[40,100985,97118],{"class":462},[40,100987,445],{"class":133},[40,100989,100991,100993,100995,100997,100999,101001,101003,101005,101007],{"class":42,"line":100990},453,[40,100992,100793],{"class":1195},[40,100994,623],{"class":441},[40,100996,1256],{"class":129},[40,100998,100542],{"class":96619},[40,101000,100474],{"class":441},[40,101002,100477],{"class":96619},[40,101004,100480],{"class":441},[40,101006,100551],{"class":96619},[40,101008,445],{"class":133},[40,101010,101012],{"class":42,"line":101011},454,[40,101013,98546],{"class":133},[40,101015,101017],{"class":42,"line":101016},455,[40,101018,97353],{"class":133},[40,101020,101022],{"class":42,"line":101021},456,[40,101023,228],{"emptyLinePlaceholder":227},[40,101025,101027,101030,101032],{"class":42,"line":101026},457,[40,101028,101029],{"class":133},"    bloch_axis.",[40,101031,24612],{"class":237},[40,101033,411],{"class":133},[40,101035,101037,101040,101042],{"class":42,"line":101036},458,[40,101038,101039],{"class":1195},"        handles",[40,101041,623],{"class":441},[40,101043,101044],{"class":133},"legend_handles,\n",[40,101046,101048,101051,101053,101055],{"class":42,"line":101047},459,[40,101049,101050],{"class":1195},"        loc",[40,101052,623],{"class":441},[40,101054,97162],{"class":462},[40,101056,445],{"class":133},[40,101058,101060,101063,101065,101067],{"class":42,"line":101059},460,[40,101061,101062],{"class":1195},"        bbox_to_anchor",[40,101064,623],{"class":441},[40,101066,97172],{"class":462},[40,101068,445],{"class":133},[40,101070,101072,101075,101077,101079],{"class":42,"line":101071},461,[40,101073,101074],{"class":1195},"        frameon",[40,101076,623],{"class":441},[40,101078,12926],{"class":462},[40,101080,445],{"class":133},[40,101082,101084,101087,101089,101091],{"class":42,"line":101083},462,[40,101085,101086],{"class":1195},"        borderaxespad",[40,101088,623],{"class":441},[40,101090,95027],{"class":462},[40,101092,445],{"class":133},[40,101094,101096],{"class":42,"line":101095},463,[40,101097,698],{"class":133},[40,101099,101101],{"class":42,"line":101100},464,[40,101102,228],{"emptyLinePlaceholder":227},[40,101104,101106,101108,101110],{"class":42,"line":101105},465,[40,101107,24430],{"class":133},[40,101109,24667],{"class":237},[40,101111,12875],{"class":133},[40,101113,101115],{"class":42,"line":101114},466,[40,101116,228],{"emptyLinePlaceholder":227},[40,101118,101120,101122,101125],{"class":42,"line":101119},467,[40,101121,683],{"class":129},[40,101123,101124],{"class":462}," BLOCH_PRINT_SUMMARY",[40,101126,766],{"class":133},[40,101128,101130,101133,101135,101138,101140],{"class":42,"line":101129},468,[40,101131,101132],{"class":133},"    resonant_axis ",[40,101134,623],{"class":441},[40,101136,101137],{"class":133}," resonant_trajectory[",[40,101139,100446],{"class":244},[40,101141,833],{"class":133},[40,101143,101145,101148,101150,101153,101155],{"class":42,"line":101144},469,[40,101146,101147],{"class":133},"    detuned_axis ",[40,101149,623],{"class":441},[40,101151,101152],{"class":133}," detuned_trajectory[",[40,101154,100446],{"class":244},[40,101156,833],{"class":133},[40,101158,101160],{"class":42,"line":101159},470,[40,101161,228],{"emptyLinePlaceholder":227},[40,101163,101165,101167,101169,101172],{"class":42,"line":101164},471,[40,101166,1100],{"class":441},[40,101168,241],{"class":133},[40,101170,101171],{"class":244},"\"Bloch-sphere trajectory summary\"",[40,101173,248],{"class":133},[40,101175,101177,101179,101181,101183,101186,101189,101192,101194,101197],{"class":42,"line":101176},472,[40,101178,1100],{"class":441},[40,101180,241],{"class":133},[40,101182,21331],{"class":129},[40,101184,101185],{"class":244},"\"Natural transition frequency f0: ",[40,101187,101188],{"class":462},"{BLOCH_F0_GHZ",[40,101190,101191],{"class":129},":.6f",[40,101193,1916],{"class":462},[40,101195,101196],{"class":244}," GHz\"",[40,101198,248],{"class":133},[40,101200,101202,101204,101206,101208,101211,101213,101216,101218,101220,101222],{"class":42,"line":101201},473,[40,101203,1100],{"class":441},[40,101205,241],{"class":133},[40,101207,21331],{"class":129},[40,101209,101210],{"class":244},"\"Resonant drive frequency: ",[40,101212,1853],{"class":462},[40,101214,101215],{"class":133},"resonant_drive_ghz",[40,101217,101191],{"class":129},[40,101219,1916],{"class":462},[40,101221,101196],{"class":244},[40,101223,248],{"class":133},[40,101225,101227,101229,101231,101233,101236,101238,101241,101243,101245,101247],{"class":42,"line":101226},474,[40,101228,1100],{"class":441},[40,101230,241],{"class":133},[40,101232,21331],{"class":129},[40,101234,101235],{"class":244},"\"Detuned drive frequency: ",[40,101237,1853],{"class":462},[40,101239,101240],{"class":133},"detuned_drive_ghz",[40,101242,101191],{"class":129},[40,101244,1916],{"class":462},[40,101246,101196],{"class":244},[40,101248,248],{"class":133},[40,101250,101252,101254,101256,101258,101261,101264,101267,101269,101272],{"class":42,"line":101251},475,[40,101253,1100],{"class":441},[40,101255,241],{"class":133},[40,101257,21331],{"class":129},[40,101259,101260],{"class":244},"\"On-resonance Rabi rate: ",[40,101262,101263],{"class":462},"{BLOCH_OMEGA_ONRESONANCE_MHZ",[40,101265,101266],{"class":129},":.3f",[40,101268,1916],{"class":462},[40,101270,101271],{"class":244}," MHz\"",[40,101273,248],{"class":133},[40,101275,101277,101279,101281,101283,101286,101288,101291,101293,101295,101298],{"class":42,"line":101276},476,[40,101278,1100],{"class":441},[40,101280,241],{"class":133},[40,101282,21331],{"class":129},[40,101284,101285],{"class":244},"\"Plotted duration: ",[40,101287,1853],{"class":462},[40,101289,101290],{"class":133},"bloch_duration_ns",[40,101292,101266],{"class":129},[40,101294,1916],{"class":462},[40,101296,101297],{"class":244}," ns\"",[40,101299,248],{"class":133},[40,101301,101303,101305],{"class":42,"line":101302},477,[40,101304,1100],{"class":441},[40,101306,411],{"class":133},[40,101308,101310],{"class":42,"line":101309},478,[40,101311,101312],{"class":244},"        \"Resonant control axis n_res: \"\n",[40,101314,101316,101319,101322,101324,101327,101329,101331,101333,101335],{"class":42,"line":101315},479,[40,101317,101318],{"class":129},"        f",[40,101320,101321],{"class":244},"\"(",[40,101323,1853],{"class":462},[40,101325,101326],{"class":133},"resonant_axis[",[40,101328,791],{"class":462},[40,101330,11320],{"class":133},[40,101332,101191],{"class":129},[40,101334,1916],{"class":462},[40,101336,101337],{"class":244},", \"\n",[40,101339,101341,101343,101345,101347,101349,101351,101353,101355,101357],{"class":42,"line":101340},480,[40,101342,101318],{"class":129},[40,101344,21334],{"class":244},[40,101346,1853],{"class":462},[40,101348,101326],{"class":133},[40,101350,1410],{"class":462},[40,101352,11320],{"class":133},[40,101354,101191],{"class":129},[40,101356,1916],{"class":462},[40,101358,101337],{"class":244},[40,101360,101362,101364,101366,101368,101370,101372,101374,101376,101378],{"class":42,"line":101361},481,[40,101363,101318],{"class":129},[40,101365,21334],{"class":244},[40,101367,1853],{"class":462},[40,101369,101326],{"class":133},[40,101371,1879],{"class":462},[40,101373,11320],{"class":133},[40,101375,101191],{"class":129},[40,101377,1916],{"class":462},[40,101379,101380],{"class":244},")\"\n",[40,101382,101384],{"class":42,"line":101383},482,[40,101385,698],{"class":133},[40,101387,101389,101391],{"class":42,"line":101388},483,[40,101390,1100],{"class":441},[40,101392,411],{"class":133},[40,101394,101396],{"class":42,"line":101395},484,[40,101397,101398],{"class":244},"        \"Detuned control axis n_det: \"\n",[40,101400,101402,101404,101406,101408,101411,101413,101415,101417,101419],{"class":42,"line":101401},485,[40,101403,101318],{"class":129},[40,101405,101321],{"class":244},[40,101407,1853],{"class":462},[40,101409,101410],{"class":133},"detuned_axis[",[40,101412,791],{"class":462},[40,101414,11320],{"class":133},[40,101416,101191],{"class":129},[40,101418,1916],{"class":462},[40,101420,101337],{"class":244},[40,101422,101424,101426,101428,101430,101432,101434,101436,101438,101440],{"class":42,"line":101423},486,[40,101425,101318],{"class":129},[40,101427,21334],{"class":244},[40,101429,1853],{"class":462},[40,101431,101410],{"class":133},[40,101433,1410],{"class":462},[40,101435,11320],{"class":133},[40,101437,101191],{"class":129},[40,101439,1916],{"class":462},[40,101441,101337],{"class":244},[40,101443,101445,101447,101449,101451,101453,101455,101457,101459,101461],{"class":42,"line":101444},487,[40,101446,101318],{"class":129},[40,101448,21334],{"class":244},[40,101450,1853],{"class":462},[40,101452,101410],{"class":133},[40,101454,1879],{"class":462},[40,101456,11320],{"class":133},[40,101458,101191],{"class":129},[40,101460,1916],{"class":462},[40,101462,101380],{"class":244},[40,101464,101466],{"class":42,"line":101465},488,[40,101467,698],{"class":133},[40,101469,101471,101473],{"class":42,"line":101470},489,[40,101472,1100],{"class":441},[40,101474,411],{"class":133},[40,101476,101478],{"class":42,"line":101477},490,[40,101479,101480],{"class":244},"        \"Resonant final excited-state probability: \"\n",[40,101482,101484,101486,101488,101490,101492,101495,101497,101499,101501,101503,101505,101507],{"class":42,"line":101483},491,[40,101485,101318],{"class":129},[40,101487,21334],{"class":244},[40,101489,1853],{"class":462},[40,101491,100157],{"class":133},[40,101493,101494],{"class":244},"'excited_probability'",[40,101496,794],{"class":133},[40,101498,1407],{"class":441},[40,101500,1410],{"class":462},[40,101502,11320],{"class":133},[40,101504,101191],{"class":129},[40,101506,1916],{"class":462},[40,101508,100039],{"class":244},[40,101510,101512],{"class":42,"line":101511},492,[40,101513,698],{"class":133},[40,101515,101517,101519],{"class":42,"line":101516},493,[40,101518,1100],{"class":441},[40,101520,411],{"class":133},[40,101522,101524],{"class":42,"line":101523},494,[40,101525,101526],{"class":244},"        \"Detuned final excited-state probability: \"\n",[40,101528,101530,101532,101534,101536,101538,101540,101542,101544,101546,101548,101550,101552],{"class":42,"line":101529},495,[40,101531,101318],{"class":129},[40,101533,21334],{"class":244},[40,101535,1853],{"class":462},[40,101537,100216],{"class":133},[40,101539,101494],{"class":244},[40,101541,794],{"class":133},[40,101543,1407],{"class":441},[40,101545,1410],{"class":462},[40,101547,11320],{"class":133},[40,101549,101191],{"class":129},[40,101551,1916],{"class":462},[40,101553,100039],{"class":244},[40,101555,101557],{"class":42,"line":101556},496,[40,101558,698],{"class":133},[40,101560,101562,101564],{"class":42,"line":101561},497,[40,101563,1100],{"class":441},[40,101565,411],{"class":133},[40,101567,101569],{"class":42,"line":101568},498,[40,101570,101571],{"class":244},"        \"Detuned generalized Rabi frequency: \"\n",[40,101573,101575,101577,101579,101581,101583,101586,101588,101590,101592],{"class":42,"line":101574},499,[40,101576,101318],{"class":129},[40,101578,21334],{"class":244},[40,101580,1853],{"class":462},[40,101582,100216],{"class":133},[40,101584,101585],{"class":244},"'generalized_rabi_mhz'",[40,101587,11320],{"class":133},[40,101589,101191],{"class":129},[40,101591,1916],{"class":462},[40,101593,101594],{"class":244}," MHz\"\n",[40,101596,101598],{"class":42,"line":101597},500,[40,101599,698],{"class":133},[8229,101601],{"alt":101602,"src":101603},"Output 2 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-02.webp",[30,101605,101608],{"className":101606,"code":101607,"language":6083,"meta":35},[7575],"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",[37,101609,101607],{"__ignoreMap":35},[30,101611,101613],{"className":32,"code":101612,"language":34,"meta":35,"style":35},"# --- 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",[37,101614,101615,101620,101632,101642,101652,101656,101661,101677,101694,101715,101719,101724,101729,101753,101779,101811,101828,101843,101847,101852,101875,101901,101929,101946,101960,101964,101969,101991,101995,102000,102038,102067,102071,102085,102090,102114,102134,102145,102165,102169,102173,102186,102216,102231,102244,102248,102253,102292,102308,102312,102320],{"__ignoreMap":35},[40,101616,101617],{"class":42,"line":43},[40,101618,101619],{"class":46},"# --- High-resolution 3D surface with colormap & legend  ---\n",[40,101621,101622,101624,101627,101629],{"class":42,"line":50},[40,101623,171],{"class":129},[40,101625,101626],{"class":133}," mpl_toolkits.mplot3d ",[40,101628,130],{"class":129},[40,101630,101631],{"class":133}," Axes3D\n",[40,101633,101634,101636,101638,101640],{"class":42,"line":155},[40,101635,130],{"class":129},[40,101637,134],{"class":133},[40,101639,137],{"class":129},[40,101641,140],{"class":133},[40,101643,101644,101646,101648,101650],{"class":42,"line":168},[40,101645,130],{"class":129},[40,101647,95476],{"class":133},[40,101649,137],{"class":129},[40,101651,165],{"class":133},[40,101653,101654],{"class":42,"line":182},[40,101655,228],{"emptyLinePlaceholder":227},[40,101657,101658],{"class":42,"line":195},[40,101659,101660],{"class":46},"# Controls for surface quality and appearance\n",[40,101662,101663,101666,101668,101670,101672,101674],{"class":42,"line":203},[40,101664,101665],{"class":462},"UPSAMPLE_FACTOR",[40,101667,438],{"class":133},[40,101669,456],{"class":441},[40,101671,459],{"class":441},[40,101673,1356],{"class":462},[40,101675,101676],{"class":46},"           # 2=~4× more faces; try 3 for even smoother\n",[40,101678,101679,101682,101684,101686,101688,101691],{"class":42,"line":216},[40,101680,101681],{"class":462},"SURFACE_CMAP",[40,101683,438],{"class":133},[40,101685,97210],{"class":441},[40,101687,459],{"class":441},[40,101689,101690],{"class":244}," 'inferno'",[40,101692,101693],{"class":46},"      # e.g., 'viridis', 'plasma', 'magma', 'inferno'\n",[40,101695,101696,101699,101701,101704,101706,101708,101710,101712],{"class":42,"line":224},[40,101697,101698],{"class":462},"Z_MIN",[40,101700,917],{"class":133},[40,101702,101703],{"class":462},"Z_MAX",[40,101705,459],{"class":441},[40,101707,95680],{"class":462},[40,101709,917],{"class":133},[40,101711,96240],{"class":462},[40,101713,101714],{"class":46},"            # clamp\u002Fnormalize expected Pe range\n",[40,101716,101717],{"class":42,"line":231},[40,101718,228],{"emptyLinePlaceholder":227},[40,101720,101721],{"class":42,"line":251},[40,101722,101723],{"class":46},"# --- Bilinear upsampling (SciPy-free) ---\n",[40,101725,101726],{"class":42,"line":256},[40,101727,101728],{"class":46},"# 1) Upsample frequency axis for each fixed τ\n",[40,101730,101731,101734,101736,101738,101740,101743,101745,101748,101750],{"class":42,"line":269},[40,101732,101733],{"class":133},"freq_hi ",[40,101735,623],{"class":441},[40,101737,654],{"class":133},[40,101739,96259],{"class":237},[40,101741,101742],{"class":133},"(freq_ghz.",[40,101744,24284],{"class":237},[40,101746,101747],{"class":133},"(), freq_ghz.",[40,101749,35767],{"class":237},[40,101751,101752],{"class":133},"(),\n",[40,101754,101755,101758,101760,101762,101764,101767,101769,101771,101773,101775,101777],{"class":42,"line":282},[40,101756,101757],{"class":462},"                      UPSAMPLE_FACTOR",[40,101759,7273],{"class":441},[40,101761,96957],{"class":133},[40,101763,802],{"class":441},[40,101765,101766],{"class":133},"(freq_ghz) ",[40,101768,1407],{"class":441},[40,101770,463],{"class":462},[40,101772,1416],{"class":133},[40,101774,1147],{"class":441},[40,101776,463],{"class":462},[40,101778,248],{"class":133},[40,101780,101781,101784,101786,101788,101791,101794,101796,101799,101801,101804,101806,101809],{"class":42,"line":295},[40,101782,101783],{"class":133},"Pe_tau_f ",[40,101785,623],{"class":441},[40,101787,654],{"class":133},[40,101789,101790],{"class":237},"empty",[40,101792,101793],{"class":133},"((",[40,101795,802],{"class":441},[40,101797,101798],{"class":133},"(dur_ns), ",[40,101800,802],{"class":441},[40,101802,101803],{"class":133},"(freq_hi)), ",[40,101805,1196],{"class":1195},[40,101807,101808],{"class":441},"=float",[40,101810,248],{"class":133},[40,101812,101813,101815,101817,101819,101821,101823,101825],{"class":42,"line":308},[40,101814,668],{"class":129},[40,101816,7290],{"class":133},[40,101818,674],{"class":129},[40,101820,7265],{"class":441},[40,101822,241],{"class":133},[40,101824,802],{"class":441},[40,101826,101827],{"class":133},"(dur_ns)):\n",[40,101829,101830,101833,101835,101837,101840],{"class":42,"line":321},[40,101831,101832],{"class":133},"    Pe_tau_f[i, :] ",[40,101834,623],{"class":441},[40,101836,654],{"class":133},[40,101838,101839],{"class":237},"interp",[40,101841,101842],{"class":133},"(freq_hi, freq_ghz, Pe[i, :])\n",[40,101844,101845],{"class":42,"line":334},[40,101846,228],{"emptyLinePlaceholder":227},[40,101848,101849],{"class":42,"line":347},[40,101850,101851],{"class":46},"# 2) Upsample τ axis for each fixed frequency\n",[40,101853,101854,101857,101859,101861,101863,101866,101868,101871,101873],{"class":42,"line":352},[40,101855,101856],{"class":133},"dur_hi ",[40,101858,623],{"class":441},[40,101860,654],{"class":133},[40,101862,96259],{"class":237},[40,101864,101865],{"class":133},"(dur_ns.",[40,101867,24284],{"class":237},[40,101869,101870],{"class":133},"(), dur_ns.",[40,101872,35767],{"class":237},[40,101874,101752],{"class":133},[40,101876,101877,101880,101882,101884,101886,101889,101891,101893,101895,101897,101899],{"class":42,"line":357},[40,101878,101879],{"class":462},"                     UPSAMPLE_FACTOR",[40,101881,7273],{"class":441},[40,101883,96957],{"class":133},[40,101885,802],{"class":441},[40,101887,101888],{"class":133},"(dur_ns) ",[40,101890,1407],{"class":441},[40,101892,463],{"class":462},[40,101894,1416],{"class":133},[40,101896,1147],{"class":441},[40,101898,463],{"class":462},[40,101900,248],{"class":133},[40,101902,101903,101906,101908,101910,101912,101914,101916,101919,101921,101923,101925,101927],{"class":42,"line":370},[40,101904,101905],{"class":133},"Pe_hi ",[40,101907,623],{"class":441},[40,101909,654],{"class":133},[40,101911,101790],{"class":237},[40,101913,101793],{"class":133},[40,101915,802],{"class":441},[40,101917,101918],{"class":133},"(dur_hi), ",[40,101920,802],{"class":441},[40,101922,101803],{"class":133},[40,101924,1196],{"class":1195},[40,101926,101808],{"class":441},[40,101928,248],{"class":133},[40,101930,101931,101933,101935,101937,101939,101941,101943],{"class":42,"line":376},[40,101932,668],{"class":129},[40,101934,7725],{"class":133},[40,101936,674],{"class":129},[40,101938,7265],{"class":441},[40,101940,241],{"class":133},[40,101942,802],{"class":441},[40,101944,101945],{"class":133},"(freq_hi)):\n",[40,101947,101948,101951,101953,101955,101957],{"class":42,"line":387},[40,101949,101950],{"class":133},"    Pe_hi[:, j] ",[40,101952,623],{"class":441},[40,101954,654],{"class":133},[40,101956,101839],{"class":237},[40,101958,101959],{"class":133},"(dur_hi, dur_ns, Pe_tau_f[:, j])\n",[40,101961,101962],{"class":42,"line":392},[40,101963,228],{"emptyLinePlaceholder":227},[40,101965,101966],{"class":42,"line":397},[40,101967,101968],{"class":46},"# Mesh for plotting\n",[40,101970,101971,101974,101976,101978,101980,101983,101985,101987,101989],{"class":42,"line":402},[40,101972,101973],{"class":133},"F_hi, T_hi ",[40,101975,623],{"class":441},[40,101977,654],{"class":133},[40,101979,95983],{"class":237},[40,101981,101982],{"class":133},"(freq_hi, dur_hi, ",[40,101984,95989],{"class":1195},[40,101986,623],{"class":441},[40,101988,95994],{"class":244},[40,101990,248],{"class":133},[40,101992,101993],{"class":42,"line":414},[40,101994,228],{"emptyLinePlaceholder":227},[40,101996,101997],{"class":42,"line":424},[40,101998,101999],{"class":46},"# --- Plot ---\n",[40,102001,102002,102005,102007,102009,102011,102013,102015,102017,102019,102021,102023,102026,102028,102031,102033,102036],{"class":42,"line":432},[40,102003,102004],{"class":133},"fig ",[40,102006,623],{"class":441},[40,102008,96538],{"class":133},[40,102010,99929],{"class":237},[40,102012,241],{"class":133},[40,102014,96546],{"class":1195},[40,102016,623],{"class":441},[40,102018,241],{"class":133},[40,102020,34795],{"class":462},[40,102022,917],{"class":133},[40,102024,102025],{"class":462},"6",[40,102027,1253],{"class":133},[40,102029,102030],{"class":1195},"dpi",[40,102032,623],{"class":441},[40,102034,102035],{"class":462},"200",[40,102037,248],{"class":133},[40,102039,102040,102043,102045,102047,102050,102052,102055,102057,102060,102062,102065],{"class":42,"line":448},[40,102041,102042],{"class":133},"ax ",[40,102044,623],{"class":441},[40,102046,96596],{"class":133},[40,102048,102049],{"class":237},"add_subplot",[40,102051,241],{"class":133},[40,102053,102054],{"class":462},"111",[40,102056,917],{"class":133},[40,102058,102059],{"class":1195},"projection",[40,102061,623],{"class":441},[40,102063,102064],{"class":244},"'3d'",[40,102066,248],{"class":133},[40,102068,102069],{"class":42,"line":468},[40,102070,228],{"emptyLinePlaceholder":227},[40,102072,102073,102076,102078,102080,102083],{"class":42,"line":486},[40,102074,102075],{"class":133},"surf ",[40,102077,623],{"class":441},[40,102079,96570],{"class":133},[40,102081,102082],{"class":237},"plot_surface",[40,102084,411],{"class":133},[40,102086,102087],{"class":42,"line":502},[40,102088,102089],{"class":133},"    F_hi, T_hi, Pe_hi,\n",[40,102091,102092,102095,102097,102099,102101,102104,102106,102108,102111],{"class":42,"line":508},[40,102093,102094],{"class":1195},"    rstride",[40,102096,623],{"class":441},[40,102098,1410],{"class":462},[40,102100,917],{"class":133},[40,102102,102103],{"class":1195},"cstride",[40,102105,623],{"class":441},[40,102107,1410],{"class":462},[40,102109,102110],{"class":133},",             ",[40,102112,102113],{"class":46},"# use every row\u002Fcolumn (after upsampling)\n",[40,102115,102116,102119,102121,102123,102125,102128,102130,102132],{"class":42,"line":514},[40,102117,102118],{"class":1195},"    linewidth",[40,102120,623],{"class":441},[40,102122,791],{"class":462},[40,102124,917],{"class":133},[40,102126,102127],{"class":1195},"antialiased",[40,102129,623],{"class":441},[40,102131,12926],{"class":462},[40,102133,445],{"class":133},[40,102135,102136,102139,102141,102143],{"class":42,"line":519},[40,102137,102138],{"class":1195},"    cmap",[40,102140,623],{"class":441},[40,102142,101681],{"class":462},[40,102144,445],{"class":133},[40,102146,102147,102150,102152,102154,102156,102159,102161,102163],{"class":42,"line":525},[40,102148,102149],{"class":1195},"    vmin",[40,102151,623],{"class":441},[40,102153,101698],{"class":462},[40,102155,917],{"class":133},[40,102157,102158],{"class":1195},"vmax",[40,102160,623],{"class":441},[40,102162,101703],{"class":462},[40,102164,445],{"class":133},[40,102166,102167],{"class":42,"line":531},[40,102168,248],{"class":133},[40,102170,102171],{"class":42,"line":537},[40,102172,228],{"emptyLinePlaceholder":227},[40,102174,102175,102177,102179,102181,102184],{"class":42,"line":543},[40,102176,96627],{"class":133},[40,102178,96630],{"class":237},[40,102180,241],{"class":133},[40,102182,102183],{"class":244},"'f (GHz)'",[40,102185,248],{"class":133},[40,102187,102188,102190,102192,102194,102196,102199,102202,102205,102207,102209,102211,102214],{"class":42,"line":549},[40,102189,96627],{"class":133},[40,102191,96644],{"class":237},[40,102193,241],{"class":133},[40,102195,1256],{"class":129},[40,102197,102198],{"class":96619},"'$",[40,102200,102201],{"class":441},"\\t",[40,102203,102204],{"class":96619},"au$ ",[40,102206,241],{"class":462},[40,102208,86991],{"class":96619},[40,102210,2693],{"class":462},[40,102212,102213],{"class":96619},"'",[40,102215,248],{"class":133},[40,102217,102218,102220,102223,102225,102227,102229],{"class":42,"line":555},[40,102219,96627],{"class":133},[40,102221,102222],{"class":237},"set_zlabel",[40,102224,241],{"class":133},[40,102226,1256],{"class":129},[40,102228,96620],{"class":96619},[40,102230,248],{"class":133},[40,102232,102233,102235,102237,102239,102242],{"class":42,"line":561},[40,102234,96627],{"class":133},[40,102236,96658],{"class":237},[40,102238,241],{"class":133},[40,102240,102241],{"class":244},"'Rabi $P_e$ surface'",[40,102243,248],{"class":133},[40,102245,102246],{"class":42,"line":567},[40,102247,228],{"emptyLinePlaceholder":227},[40,102249,102250],{"class":42,"line":573},[40,102251,102252],{"class":46},"# Colorbar (legend for the colormap)\n",[40,102254,102255,102257,102259,102261,102263,102266,102268,102270,102272,102275,102277,102280,102282,102285,102287,102290],{"class":42,"line":579},[40,102256,96591],{"class":133},[40,102258,623],{"class":441},[40,102260,96596],{"class":133},[40,102262,96599],{"class":237},[40,102264,102265],{"class":133},"(surf, ",[40,102267,96605],{"class":1195},[40,102269,623],{"class":441},[40,102271,96610],{"class":133},[40,102273,102274],{"class":1195},"shrink",[40,102276,623],{"class":441},[40,102278,102279],{"class":462},"0.7",[40,102281,917],{"class":133},[40,102283,102284],{"class":1195},"pad",[40,102286,623],{"class":441},[40,102288,102289],{"class":462},"0.08",[40,102291,248],{"class":133},[40,102293,102294,102297,102300,102302,102304,102306],{"class":42,"line":585},[40,102295,102296],{"class":133},"cb.",[40,102298,102299],{"class":237},"set_label",[40,102301,241],{"class":133},[40,102303,1256],{"class":129},[40,102305,96620],{"class":96619},[40,102307,248],{"class":133},[40,102309,102310],{"class":42,"line":590},[40,102311,228],{"emptyLinePlaceholder":227},[40,102313,102314,102316,102318],{"class":42,"line":596},[40,102315,96680],{"class":133},[40,102317,96683],{"class":237},[40,102319,12875],{"class":133},[40,102321,102322,102324,102326],{"class":42,"line":602},[40,102323,24430],{"class":133},[40,102325,24667],{"class":237},[40,102327,12875],{"class":133},[8229,102329],{"alt":102330,"src":102331},"Output 3 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-03.webp",[30,102333,102335],{"className":32,"code":102334,"language":34,"meta":35,"style":35},"# @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",[37,102336,102337,102342,102347,102362,102378,102382,102392,102402,102406,102442,102447,102472,102494,102543,102557,102581,102585,102590,102611,102621,102625,102652,102671,102697,102711,102716,102762,102779,102787,102795,102799,102804,102825,102835,102839,102865,102882,102894,102908,102954,102970,102978],{"__ignoreMap":35},[40,102338,102339],{"class":42,"line":43},[40,102340,102341],{"class":46},"# @title Cross-sections\n",[40,102343,102344],{"class":42,"line":50},[40,102345,102346],{"class":46},"# Slicing controls\n",[40,102348,102349,102351,102353,102355,102357,102359],{"class":42,"line":155},[40,102350,95102],{"class":462},[40,102352,438],{"class":133},[40,102354,442],{"class":441},[40,102356,459],{"class":441},[40,102358,95600],{"class":462},[40,102360,102361],{"class":46},"  # drive frequency slice [GHz]\n",[40,102363,102364,102366,102368,102370,102372,102375],{"class":42,"line":168},[40,102365,95201],{"class":462},[40,102367,438],{"class":133},[40,102369,442],{"class":441},[40,102371,459],{"class":441},[40,102373,102374],{"class":462}," 100.0",[40,102376,102377],{"class":46},"   # duration slice [ns]\n",[40,102379,102380],{"class":42,"line":182},[40,102381,228],{"emptyLinePlaceholder":227},[40,102383,102384,102386,102388,102390],{"class":42,"line":195},[40,102385,130],{"class":129},[40,102387,134],{"class":133},[40,102389,137],{"class":129},[40,102391,140],{"class":133},[40,102393,102394,102396,102398,102400],{"class":42,"line":203},[40,102395,130],{"class":129},[40,102397,95476],{"class":133},[40,102399,137],{"class":129},[40,102401,165],{"class":133},[40,102403,102404],{"class":42,"line":216},[40,102405,228],{"emptyLinePlaceholder":227},[40,102407,102408,102410,102413,102415,102418,102421,102424,102426,102428,102430,102432,102434,102436,102438,102440],{"class":42,"line":224},[40,102409,405],{"class":129},[40,102411,102412],{"class":237}," _nearest_index",[40,102414,241],{"class":133},[40,102416,102417],{"class":417},"vec",[40,102419,102420],{"class":133},": np.ndarray, ",[40,102422,102423],{"class":417},"value",[40,102425,438],{"class":133},[40,102427,442],{"class":441},[40,102429,917],{"class":133},[40,102431,21061],{"class":417},[40,102433,438],{"class":133},[40,102435,97210],{"class":441},[40,102437,97213],{"class":133},[40,102439,456],{"class":441},[40,102441,766],{"class":133},[40,102443,102444],{"class":42,"line":231},[40,102445,102446],{"class":244},"    \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n",[40,102448,102449,102452,102454,102456,102459,102461,102464,102466,102468,102470],{"class":42,"line":251},[40,102450,102451],{"class":133},"    vmin, vmax ",[40,102453,623],{"class":441},[40,102455,97933],{"class":441},[40,102457,102458],{"class":133},"(vec.",[40,102460,24284],{"class":237},[40,102462,102463],{"class":133},"()), ",[40,102465,442],{"class":441},[40,102467,102458],{"class":133},[40,102469,35767],{"class":237},[40,102471,21140],{"class":133},[40,102473,102474,102476,102479,102481,102484,102487,102489,102491],{"class":42,"line":256},[40,102475,755],{"class":129},[40,102477,102478],{"class":133}," value ",[40,102480,4721],{"class":441},[40,102482,102483],{"class":133}," vmin ",[40,102485,102486],{"class":129},"or",[40,102488,102478],{"class":133},[40,102490,689],{"class":441},[40,102492,102493],{"class":133}," vmax:\n",[40,102495,102496,102499,102501,102503,102506,102508,102510,102512,102514,102516,102518,102520,102523,102525,102528,102530,102532,102534,102536,102538,102541],{"class":42,"line":269},[40,102497,102498],{"class":441},"        print",[40,102500,241],{"class":133},[40,102502,21331],{"class":129},[40,102504,102505],{"class":244},"\"[Note] ",[40,102507,1853],{"class":462},[40,102509,21061],{"class":133},[40,102511,1916],{"class":462},[40,102513,623],{"class":244},[40,102515,1853],{"class":462},[40,102517,102423],{"class":133},[40,102519,1916],{"class":462},[40,102521,102522],{"class":244}," is outside sweep; clamping to [",[40,102524,1853],{"class":462},[40,102526,102527],{"class":133},"vmin",[40,102529,1916],{"class":462},[40,102531,917],{"class":244},[40,102533,1853],{"class":462},[40,102535,102158],{"class":133},[40,102537,1916],{"class":462},[40,102539,102540],{"class":244},"].\"",[40,102542,248],{"class":133},[40,102544,102545,102548,102550,102552,102554],{"class":42,"line":282},[40,102546,102547],{"class":133},"        value ",[40,102549,623],{"class":441},[40,102551,654],{"class":133},[40,102553,96230],{"class":237},[40,102555,102556],{"class":133},"(value, vmin, vmax)\n",[40,102558,102559,102561,102563,102565,102568,102570,102573,102576,102578],{"class":42,"line":295},[40,102560,1546],{"class":129},[40,102562,1135],{"class":441},[40,102564,98230],{"class":133},[40,102566,102567],{"class":237},"argmin",[40,102569,98230],{"class":133},[40,102571,102572],{"class":237},"abs",[40,102574,102575],{"class":133},"(vec ",[40,102577,1407],{"class":441},[40,102579,102580],{"class":133}," value)))\n",[40,102582,102583],{"class":42,"line":308},[40,102584,228],{"emptyLinePlaceholder":227},[40,102586,102587],{"class":42,"line":321},[40,102588,102589],{"class":46},"# --- Slice at f = F_CROSS_GHZ ---\n",[40,102591,102592,102595,102597,102599,102602,102604,102606,102609],{"class":42,"line":334},[40,102593,102594],{"class":133},"j ",[40,102596,623],{"class":441},[40,102598,102412],{"class":237},[40,102600,102601],{"class":133},"(freq_ghz, ",[40,102603,95102],{"class":462},[40,102605,917],{"class":133},[40,102607,102608],{"class":244},"\"F_CROSS_GHZ\"",[40,102610,248],{"class":133},[40,102612,102613,102616,102618],{"class":42,"line":347},[40,102614,102615],{"class":133},"Pe_vs_tau ",[40,102617,623],{"class":441},[40,102619,102620],{"class":133}," Pe[:, j]\n",[40,102622,102623],{"class":42,"line":352},[40,102624,228],{"emptyLinePlaceholder":227},[40,102626,102627,102629,102631,102633,102635,102637,102639,102641,102643,102645,102647,102650],{"class":42,"line":357},[40,102628,96533],{"class":133},[40,102630,623],{"class":441},[40,102632,96538],{"class":133},[40,102634,96541],{"class":237},[40,102636,241],{"class":133},[40,102638,96546],{"class":1195},[40,102640,623],{"class":441},[40,102642,241],{"class":133},[40,102644,102025],{"class":462},[40,102646,917],{"class":133},[40,102648,102649],{"class":462},"3.5",[40,102651,21594],{"class":133},[40,102653,102654,102656,102658,102661,102664,102666,102669],{"class":42,"line":370},[40,102655,96627],{"class":133},[40,102657,24433],{"class":237},[40,102659,102660],{"class":133},"(dur_ns, Pe_vs_tau, ",[40,102662,102663],{"class":1195},"color",[40,102665,623],{"class":441},[40,102667,102668],{"class":244},"'blue'",[40,102670,248],{"class":133},[40,102672,102673,102675,102677,102679,102681,102683,102685,102687,102689,102691,102693,102695],{"class":42,"line":376},[40,102674,96627],{"class":133},[40,102676,96630],{"class":237},[40,102678,241],{"class":133},[40,102680,1256],{"class":129},[40,102682,102198],{"class":96619},[40,102684,102201],{"class":441},[40,102686,102204],{"class":96619},[40,102688,241],{"class":462},[40,102690,86991],{"class":96619},[40,102692,2693],{"class":462},[40,102694,102213],{"class":96619},[40,102696,248],{"class":133},[40,102698,102699,102701,102703,102705,102707,102709],{"class":42,"line":387},[40,102700,96627],{"class":133},[40,102702,96644],{"class":237},[40,102704,241],{"class":133},[40,102706,1256],{"class":129},[40,102708,96620],{"class":96619},[40,102710,248],{"class":133},[40,102712,102713],{"class":42,"line":392},[40,102714,102715],{"class":46},"# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\n",[40,102717,102718,102720,102722,102724,102726,102729,102731,102733,102736,102738,102741,102743,102746,102748,102750,102753,102755,102757,102760],{"class":42,"line":397},[40,102719,96627],{"class":133},[40,102721,96658],{"class":237},[40,102723,241],{"class":133},[40,102725,1256],{"class":129},[40,102727,102728],{"class":96619},"'$P_e",[40,102730,241],{"class":462},[40,102732,102201],{"class":441},[40,102734,102735],{"class":96619},"au",[40,102737,2693],{"class":462},[40,102739,102740],{"class":96619},"$ at f = '",[40,102742,7357],{"class":441},[40,102744,102745],{"class":129}," f",[40,102747,102213],{"class":244},[40,102749,1853],{"class":462},[40,102751,102752],{"class":133},"freq_ghz[j]",[40,102754,101191],{"class":129},[40,102756,1916],{"class":462},[40,102758,102759],{"class":244}," GHz'",[40,102761,248],{"class":133},[40,102763,102764,102766,102769,102771,102773,102775,102777],{"class":42,"line":402},[40,102765,96627],{"class":133},[40,102767,102768],{"class":237},"set_ylim",[40,102770,241],{"class":133},[40,102772,95027],{"class":462},[40,102774,917],{"class":133},[40,102776,96240],{"class":462},[40,102778,248],{"class":133},[40,102780,102781,102783,102785],{"class":42,"line":414},[40,102782,96680],{"class":133},[40,102784,96683],{"class":237},[40,102786,12875],{"class":133},[40,102788,102789,102791,102793],{"class":42,"line":424},[40,102790,24430],{"class":133},[40,102792,24667],{"class":237},[40,102794,12875],{"class":133},[40,102796,102797],{"class":42,"line":432},[40,102798,228],{"emptyLinePlaceholder":227},[40,102800,102801],{"class":42,"line":448},[40,102802,102803],{"class":46},"# --- Slice at τ = T_CROSS_NS ---\n",[40,102805,102806,102809,102811,102813,102816,102818,102820,102823],{"class":42,"line":468},[40,102807,102808],{"class":133},"i ",[40,102810,623],{"class":441},[40,102812,102412],{"class":237},[40,102814,102815],{"class":133},"(dur_ns, ",[40,102817,95201],{"class":462},[40,102819,917],{"class":133},[40,102821,102822],{"class":244},"\"T_CROSS_NS\"",[40,102824,248],{"class":133},[40,102826,102827,102830,102832],{"class":42,"line":486},[40,102828,102829],{"class":133},"Pe_vs_f ",[40,102831,623],{"class":441},[40,102833,102834],{"class":133}," Pe[i, :]\n",[40,102836,102837],{"class":42,"line":502},[40,102838,228],{"emptyLinePlaceholder":227},[40,102840,102841,102843,102845,102847,102849,102851,102853,102855,102857,102859,102861,102863],{"class":42,"line":508},[40,102842,96533],{"class":133},[40,102844,623],{"class":441},[40,102846,96538],{"class":133},[40,102848,96541],{"class":237},[40,102850,241],{"class":133},[40,102852,96546],{"class":1195},[40,102854,623],{"class":441},[40,102856,241],{"class":133},[40,102858,102025],{"class":462},[40,102860,917],{"class":133},[40,102862,102649],{"class":462},[40,102864,21594],{"class":133},[40,102866,102867,102869,102871,102874,102876,102878,102880],{"class":42,"line":514},[40,102868,96627],{"class":133},[40,102870,24433],{"class":237},[40,102872,102873],{"class":133},"(freq_ghz, Pe_vs_f, ",[40,102875,102663],{"class":1195},[40,102877,623],{"class":441},[40,102879,102668],{"class":244},[40,102881,248],{"class":133},[40,102883,102884,102886,102888,102890,102892],{"class":42,"line":519},[40,102885,96627],{"class":133},[40,102887,96630],{"class":237},[40,102889,241],{"class":133},[40,102891,102183],{"class":244},[40,102893,248],{"class":133},[40,102895,102896,102898,102900,102902,102904,102906],{"class":42,"line":525},[40,102897,96627],{"class":133},[40,102899,96644],{"class":237},[40,102901,241],{"class":133},[40,102903,1256],{"class":129},[40,102905,96620],{"class":96619},[40,102907,248],{"class":133},[40,102909,102910,102912,102914,102916,102918,102920,102922,102924,102926,102929,102931,102934,102936,102938,102940,102942,102945,102947,102949,102952],{"class":42,"line":531},[40,102911,96627],{"class":133},[40,102913,96658],{"class":237},[40,102915,241],{"class":133},[40,102917,1256],{"class":129},[40,102919,102728],{"class":96619},[40,102921,241],{"class":462},[40,102923,21331],{"class":96619},[40,102925,2693],{"class":462},[40,102927,102928],{"class":96619},"$ at $",[40,102930,102201],{"class":441},[40,102932,102933],{"class":96619},"au$ = '",[40,102935,7357],{"class":441},[40,102937,102745],{"class":129},[40,102939,102213],{"class":244},[40,102941,1853],{"class":462},[40,102943,102944],{"class":133},"dur_ns[i]",[40,102946,101266],{"class":129},[40,102948,1916],{"class":462},[40,102950,102951],{"class":244}," ns'",[40,102953,248],{"class":133},[40,102955,102956,102958,102960,102962,102964,102966,102968],{"class":42,"line":537},[40,102957,96627],{"class":133},[40,102959,102768],{"class":237},[40,102961,241],{"class":133},[40,102963,95027],{"class":462},[40,102965,917],{"class":133},[40,102967,96240],{"class":462},[40,102969,248],{"class":133},[40,102971,102972,102974,102976],{"class":42,"line":543},[40,102973,96680],{"class":133},[40,102975,96683],{"class":237},[40,102977,12875],{"class":133},[40,102979,102980,102982,102984],{"class":42,"line":549},[40,102981,24430],{"class":133},[40,102983,24667],{"class":237},[40,102985,12875],{"class":133},[8229,102987],{"alt":102988,"src":102989},"Output 4 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-04.webp",[8229,102991],{"alt":102992,"src":102993},"Output 5 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-05.webp",[30,102995,102997],{"className":32,"code":102996,"language":34,"meta":35,"style":35},"# @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",[37,102998,102999,103004,103014,103024,103028,103033,103049,103065,103081,103097,103113,103129,103133,103138,103164,103182,103195,103216,103220,103229,103236,103243,103258,103273,103288,103303,103318,103333,103338,103343,103347,103351,103356,103361,103366,103371,103376,103381,103386,103391,103395,103399,103404,103409,103413,103430,103447,103467,103471,103476,103509,103519,103562,103574,103578,103583,103592,103607,103614,103651,103665,103669,103674,103709,103734,103738,103743,103761,103779,103785,103798,103802,103807,103825,103865,103879,103885,103907,103911,103928,103959,103963,103968,103987,104001,104007,104020,104024,104031,104035,104040,104051,104056,104066,104076,104086,104096,104106,104116,104120],{"__ignoreMap":35},[40,103000,103001],{"class":42,"line":43},[40,103002,103003],{"class":46},"# @title FFT of Rabi oscillations along tau (controls + computation)\n",[40,103005,103006,103008,103010,103012],{"class":42,"line":50},[40,103007,130],{"class":129},[40,103009,134],{"class":133},[40,103011,137],{"class":129},[40,103013,140],{"class":133},[40,103015,103016,103018,103020,103022],{"class":42,"line":155},[40,103017,130],{"class":129},[40,103019,95476],{"class":133},[40,103021,137],{"class":129},[40,103023,165],{"class":133},[40,103025,103026],{"class":42,"line":168},[40,103027,228],{"emptyLinePlaceholder":227},[40,103029,103030],{"class":42,"line":182},[40,103031,103032],{"class":46},"# -------- Control knobs (FFT & display) --------\n",[40,103034,103035,103037,103039,103041,103043,103046],{"class":42,"line":195},[40,103036,95318],{"class":462},[40,103038,438],{"class":133},[40,103040,456],{"class":441},[40,103042,459],{"class":441},[40,103044,103045],{"class":462}," 8",[40,103047,103048],{"class":46},"                 # zero-padding factor (2, 4, 8, ...)\n",[40,103050,103051,103053,103055,103057,103059,103062],{"class":42,"line":203},[40,103052,95332],{"class":462},[40,103054,438],{"class":133},[40,103056,97210],{"class":441},[40,103058,459],{"class":441},[40,103060,103061],{"class":244}," \"hann\"",[40,103063,103064],{"class":46},"             # \"hann\", \"boxcar\"\n",[40,103066,103067,103069,103071,103073,103075,103078],{"class":42,"line":216},[40,103068,95347],{"class":462},[40,103070,438],{"class":133},[40,103072,476],{"class":441},[40,103074,459],{"class":441},[40,103076,103077],{"class":462}," True",[40,103079,103080],{"class":46},"        # subtract mean along τ before FFT\n",[40,103082,103083,103085,103087,103089,103091,103094],{"class":42,"line":224},[40,103084,95361],{"class":462},[40,103086,438],{"class":133},[40,103088,97210],{"class":441},[40,103090,459],{"class":441},[40,103092,103093],{"class":244}," \"magnitude\"",[40,103095,103096],{"class":46},"      # \"magnitude\" | \"power\" | \"db\"\n",[40,103098,103099,103101,103103,103105,103107,103110],{"class":42,"line":231},[40,103100,95376],{"class":462},[40,103102,438],{"class":133},[40,103104,97210],{"class":441},[40,103106,459],{"class":441},[40,103108,103109],{"class":244}," \"global\"",[40,103111,103112],{"class":46},"        # \"global\" | \"per_f\" (column-wise)\n",[40,103114,103115,103117,103119,103121,103123,103126],{"class":42,"line":251},[40,103116,95391],{"class":462},[40,103118,438],{"class":133},[40,103120,97210],{"class":441},[40,103122,459],{"class":441},[40,103124,103125],{"class":244}," \"MHz\"",[40,103127,103128],{"class":46},"          # \"MHz\" or \"GHz\" for Fourier axis\n",[40,103130,103131],{"class":42,"line":256},[40,103132,228],{"emptyLinePlaceholder":227},[40,103134,103135],{"class":42,"line":269},[40,103136,103137],{"class":46},"# -------- Helpers --------\n",[40,103139,103140,103142,103145,103147,103149,103151,103153,103155,103158,103160,103162],{"class":42,"line":282},[40,103141,405],{"class":129},[40,103143,103144],{"class":237}," _make_window",[40,103146,241],{"class":133},[40,103148,3464],{"class":417},[40,103150,438],{"class":133},[40,103152,456],{"class":441},[40,103154,917],{"class":133},[40,103156,103157],{"class":417},"kind",[40,103159,438],{"class":133},[40,103161,97210],{"class":441},[40,103163,95827],{"class":133},[40,103165,103166,103168,103171,103174,103176,103178,103180],{"class":42,"line":295},[40,103167,755],{"class":129},[40,103169,103170],{"class":133}," kind.",[40,103172,103173],{"class":237},"lower",[40,103175,715],{"class":133},[40,103177,761],{"class":441},[40,103179,103061],{"class":244},[40,103181,766],{"class":133},[40,103183,103184,103187,103189,103192],{"class":42,"line":308},[40,103185,103186],{"class":129},"        return",[40,103188,654],{"class":133},[40,103190,103191],{"class":237},"hanning",[40,103193,103194],{"class":133},"(n)\n",[40,103196,103197,103199,103201,103204,103207,103209,103211,103213],{"class":42,"line":321},[40,103198,1546],{"class":129},[40,103200,654],{"class":133},[40,103202,103203],{"class":237},"ones",[40,103205,103206],{"class":133},"(n, ",[40,103208,1196],{"class":1195},[40,103210,101808],{"class":441},[40,103212,21772],{"class":133},[40,103214,103215],{"class":46},"# boxcar\n",[40,103217,103218],{"class":42,"line":334},[40,103219,228],{"emptyLinePlaceholder":227},[40,103221,103222,103224,103227],{"class":42,"line":347},[40,103223,405],{"class":129},[40,103225,103226],{"class":237}," rfft_spectrogram",[40,103228,411],{"class":133},[40,103230,103231,103234],{"class":42,"line":352},[40,103232,103233],{"class":417},"    Pe",[40,103235,421],{"class":133},[40,103237,103238,103241],{"class":42,"line":357},[40,103239,103240],{"class":417},"    dur_ns",[40,103242,421],{"class":133},[40,103244,103245,103248,103250,103252,103254,103256],{"class":42,"line":370},[40,103246,103247],{"class":417},"    pad",[40,103249,438],{"class":133},[40,103251,456],{"class":441},[40,103253,459],{"class":441},[40,103255,103045],{"class":462},[40,103257,445],{"class":133},[40,103259,103260,103263,103265,103267,103269,103271],{"class":42,"line":376},[40,103261,103262],{"class":417},"    window",[40,103264,438],{"class":133},[40,103266,97210],{"class":441},[40,103268,459],{"class":441},[40,103270,103061],{"class":244},[40,103272,445],{"class":133},[40,103274,103275,103278,103280,103282,103284,103286],{"class":42,"line":387},[40,103276,103277],{"class":417},"    detrend_mean",[40,103279,438],{"class":133},[40,103281,476],{"class":441},[40,103283,459],{"class":441},[40,103285,103077],{"class":462},[40,103287,445],{"class":133},[40,103289,103290,103293,103295,103297,103299,103301],{"class":42,"line":392},[40,103291,103292],{"class":417},"    amp_mode",[40,103294,438],{"class":133},[40,103296,97210],{"class":441},[40,103298,459],{"class":441},[40,103300,103093],{"class":244},[40,103302,445],{"class":133},[40,103304,103305,103308,103310,103312,103314,103316],{"class":42,"line":397},[40,103306,103307],{"class":417},"    norm_mode",[40,103309,438],{"class":133},[40,103311,97210],{"class":441},[40,103313,459],{"class":441},[40,103315,103109],{"class":244},[40,103317,445],{"class":133},[40,103319,103320,103323,103325,103327,103329,103331],{"class":42,"line":402},[40,103321,103322],{"class":417},"    freq_units",[40,103324,438],{"class":133},[40,103326,97210],{"class":441},[40,103328,459],{"class":441},[40,103330,103125],{"class":244},[40,103332,445],{"class":133},[40,103334,103335],{"class":42,"line":414},[40,103336,103337],{"class":133},") -> tuple[np.ndarray, np.ndarray]:\n",[40,103339,103340],{"class":42,"line":424},[40,103341,103342],{"class":244},"    \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n",[40,103344,103345],{"class":42,"line":432},[40,103346,228],{"emptyLinePlaceholder":227},[40,103348,103349],{"class":42,"line":448},[40,103350,522],{"class":244},[40,103352,103353],{"class":42,"line":468},[40,103354,103355],{"class":244},"        Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n",[40,103357,103358],{"class":42,"line":486},[40,103359,103360],{"class":244},"        dur_ns: τ grid [ns], length N_tau, evenly spaced.\n",[40,103362,103363],{"class":42,"line":502},[40,103364,103365],{"class":244},"        pad: zero-padding factor (≥1).\n",[40,103367,103368],{"class":42,"line":508},[40,103369,103370],{"class":244},"        window: \"hann\" or \"boxcar\".\n",[40,103372,103373],{"class":42,"line":514},[40,103374,103375],{"class":244},"        detrend_mean: subtract column-wise mean before FFT.\n",[40,103377,103378],{"class":42,"line":519},[40,103379,103380],{"class":244},"        amp_mode: \"magnitude\", \"power\", or \"db\".\n",[40,103382,103383],{"class":42,"line":525},[40,103384,103385],{"class":244},"        norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n",[40,103387,103388],{"class":42,"line":531},[40,103389,103390],{"class":244},"        freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n",[40,103392,103393],{"class":42,"line":537},[40,103394,228],{"emptyLinePlaceholder":227},[40,103396,103397],{"class":42,"line":543},[40,103398,593],{"class":244},[40,103400,103401],{"class":42,"line":549},[40,103402,103403],{"class":244},"        f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n",[40,103405,103406],{"class":42,"line":555},[40,103407,103408],{"class":244},"        S:     2D nonnegative spectrogram (N_w, N_f).\n",[40,103410,103411],{"class":42,"line":561},[40,103412,505],{"class":244},[40,103414,103415,103418,103420,103422,103424,103426,103428],{"class":42,"line":567},[40,103416,103417],{"class":133},"    Pe ",[40,103419,623],{"class":441},[40,103421,654],{"class":133},[40,103423,98233],{"class":237},[40,103425,96233],{"class":133},[40,103427,442],{"class":441},[40,103429,248],{"class":133},[40,103431,103432,103435,103437,103439,103441,103443,103445],{"class":42,"line":573},[40,103433,103434],{"class":133},"    dur_ns ",[40,103436,623],{"class":441},[40,103438,654],{"class":133},[40,103440,98233],{"class":237},[40,103442,102815],{"class":133},[40,103444,442],{"class":441},[40,103446,248],{"class":133},[40,103448,103449,103452,103455,103457,103459,103461,103464],{"class":42,"line":579},[40,103450,103451],{"class":129},"    assert",[40,103453,103454],{"class":133}," Pe.shape[",[40,103456,791],{"class":462},[40,103458,7250],{"class":133},[40,103460,761],{"class":441},[40,103462,103463],{"class":133}," dur_ns.size, ",[40,103465,103466],{"class":244},"\"Pe must be shaped (N_tau, N_f).\"\n",[40,103468,103469],{"class":42,"line":585},[40,103470,228],{"emptyLinePlaceholder":227},[40,103472,103473],{"class":42,"line":590},[40,103474,103475],{"class":46},"    # Uniform sampling assumption (true for np.linspace)\n",[40,103477,103478,103481,103483,103485,103488,103490,103492,103494,103497,103499,103502,103504,103506],{"class":42,"line":596},[40,103479,103480],{"class":133},"    dt ",[40,103482,623],{"class":441},[40,103484,97933],{"class":441},[40,103486,103487],{"class":133},"(dur_ns[",[40,103489,1410],{"class":462},[40,103491,7250],{"class":133},[40,103493,1407],{"class":441},[40,103495,103496],{"class":133}," dur_ns[",[40,103498,791],{"class":462},[40,103500,103501],{"class":133},"]) ",[40,103503,1433],{"class":441},[40,103505,96201],{"class":462},[40,103507,103508],{"class":46},"               # [s]\n",[40,103510,103511,103514,103516],{"class":42,"line":602},[40,103512,103513],{"class":133},"    n_tau ",[40,103515,623],{"class":441},[40,103517,103518],{"class":133}," dur_ns.size\n",[40,103520,103521,103524,103526,103528,103530,103532,103535,103537,103539,103542,103544,103547,103550,103552,103555,103557,103559],{"class":42,"line":607},[40,103522,103523],{"class":133},"    nfft ",[40,103525,623],{"class":441},[40,103527,1135],{"class":441},[40,103529,241],{"class":133},[40,103531,1879],{"class":462},[40,103533,103534],{"class":441}," **",[40,103536,1135],{"class":441},[40,103538,98230],{"class":133},[40,103540,103541],{"class":237},"ceil",[40,103543,98230],{"class":133},[40,103545,103546],{"class":237},"log2",[40,103548,103549],{"class":133},"(n_tau))) ",[40,103551,1433],{"class":441},[40,103553,103554],{"class":441}," max",[40,103556,241],{"class":133},[40,103558,1410],{"class":462},[40,103560,103561],{"class":133},", pad))\n",[40,103563,103564,103567,103569,103571],{"class":42,"line":612},[40,103565,103566],{"class":133},"    w ",[40,103568,623],{"class":441},[40,103570,103144],{"class":237},[40,103572,103573],{"class":133},"(n_tau, window)\n",[40,103575,103576],{"class":42,"line":617},[40,103577,228],{"emptyLinePlaceholder":227},[40,103579,103580],{"class":42,"line":635},[40,103581,103582],{"class":46},"    # Prepare output\n",[40,103584,103585,103588,103590],{"class":42,"line":646},[40,103586,103587],{"class":133},"    S_list ",[40,103589,623],{"class":441},[40,103591,950],{"class":133},[40,103593,103594,103597,103599,103602,103605],{"class":42,"line":662},[40,103595,103596],{"class":133},"    X ",[40,103598,623],{"class":441},[40,103600,103601],{"class":133}," Pe.",[40,103603,103604],{"class":237},"copy",[40,103606,12875],{"class":133},[40,103608,103609,103611],{"class":42,"line":695},[40,103610,755],{"class":129},[40,103612,103613],{"class":133}," detrend_mean:\n",[40,103615,103616,103619,103621,103624,103626,103629,103632,103634,103636,103638,103640,103642,103645,103647,103649],{"class":42,"line":701},[40,103617,103618],{"class":133},"        X ",[40,103620,623],{"class":441},[40,103622,103623],{"class":133}," X ",[40,103625,1407],{"class":441},[40,103627,103628],{"class":133}," X.",[40,103630,103631],{"class":237},"mean",[40,103633,241],{"class":133},[40,103635,99484],{"class":1195},[40,103637,623],{"class":441},[40,103639,791],{"class":462},[40,103641,917],{"class":133},[40,103643,103644],{"class":1195},"keepdims",[40,103646,623],{"class":441},[40,103648,12926],{"class":462},[40,103650,248],{"class":133},[40,103652,103653,103655,103658,103661,103663],{"class":42,"line":729},[40,103654,103596],{"class":133},[40,103656,103657],{"class":441},"*=",[40,103659,103660],{"class":133}," w[:, ",[40,103662,95077],{"class":462},[40,103664,833],{"class":133},[40,103666,103667],{"class":42,"line":752},[40,103668,228],{"emptyLinePlaceholder":227},[40,103670,103671],{"class":42,"line":769},[40,103672,103673],{"class":46},"    # Real FFT along τ (axis=0)\n",[40,103675,103676,103679,103681,103684,103687,103690,103692,103694,103697,103699,103701,103703,103706],{"class":42,"line":778},[40,103677,103678],{"class":133},"    Xf ",[40,103680,623],{"class":441},[40,103682,103683],{"class":133}," np.fft.",[40,103685,103686],{"class":237},"rfft",[40,103688,103689],{"class":133},"(X, ",[40,103691,3464],{"class":1195},[40,103693,623],{"class":441},[40,103695,103696],{"class":133},"nfft, ",[40,103698,99484],{"class":1195},[40,103700,623],{"class":441},[40,103702,791],{"class":462},[40,103704,103705],{"class":133},")                     ",[40,103707,103708],{"class":46},"# shape (N_w, N_f)\n",[40,103710,103711,103714,103716,103718,103721,103724,103726,103728,103731],{"class":42,"line":808},[40,103712,103713],{"class":133},"    f_fft_hz ",[40,103715,623],{"class":441},[40,103717,103683],{"class":133},[40,103719,103720],{"class":237},"rfftfreq",[40,103722,103723],{"class":133},"(nfft, ",[40,103725,4003],{"class":1195},[40,103727,623],{"class":441},[40,103729,103730],{"class":133},"dt)                  ",[40,103732,103733],{"class":46},"# [Hz]\n",[40,103735,103736],{"class":42,"line":816},[40,103737,228],{"emptyLinePlaceholder":227},[40,103739,103740],{"class":42,"line":836},[40,103741,103742],{"class":46},"    # Amplitude selection\n",[40,103744,103745,103747,103750,103752,103754,103756,103759],{"class":42,"line":844},[40,103746,755],{"class":129},[40,103748,103749],{"class":133}," amp_mode.",[40,103751,103173],{"class":237},[40,103753,715],{"class":133},[40,103755,761],{"class":441},[40,103757,103758],{"class":244}," \"power\"",[40,103760,766],{"class":133},[40,103762,103763,103766,103768,103770,103772,103775,103777],{"class":42,"line":851},[40,103764,103765],{"class":133},"        S ",[40,103767,623],{"class":441},[40,103769,654],{"class":133},[40,103771,102572],{"class":237},[40,103773,103774],{"class":133},"(Xf) ",[40,103776,1419],{"class":441},[40,103778,96102],{"class":462},[40,103780,103781,103783],{"class":42,"line":871},[40,103782,839],{"class":129},[40,103784,766],{"class":133},[40,103786,103787,103789,103791,103793,103795],{"class":42,"line":878},[40,103788,103765],{"class":133},[40,103790,623],{"class":441},[40,103792,654],{"class":133},[40,103794,102572],{"class":237},[40,103796,103797],{"class":133},"(Xf)\n",[40,103799,103800],{"class":42,"line":894},[40,103801,228],{"emptyLinePlaceholder":227},[40,103803,103804],{"class":42,"line":899},[40,103805,103806],{"class":46},"    # Normalization\n",[40,103808,103809,103811,103814,103816,103818,103820,103823],{"class":42,"line":904},[40,103810,755],{"class":129},[40,103812,103813],{"class":133}," norm_mode.",[40,103815,103173],{"class":237},[40,103817,715],{"class":133},[40,103819,761],{"class":441},[40,103821,103822],{"class":244}," \"per_f\"",[40,103824,766],{"class":133},[40,103826,103827,103830,103832,103834,103837,103840,103842,103844,103846,103848,103850,103852,103854,103856,103858,103860,103863],{"class":42,"line":926},[40,103828,103829],{"class":133},"        col_max ",[40,103831,623],{"class":441},[40,103833,654],{"class":133},[40,103835,103836],{"class":237},"maximum",[40,103838,103839],{"class":133},"(S.",[40,103841,35767],{"class":237},[40,103843,241],{"class":133},[40,103845,99484],{"class":1195},[40,103847,623],{"class":441},[40,103849,791],{"class":462},[40,103851,917],{"class":133},[40,103853,103644],{"class":1195},[40,103855,623],{"class":441},[40,103857,12926],{"class":462},[40,103859,1253],{"class":133},[40,103861,103862],{"class":462},"1e-15",[40,103864,248],{"class":133},[40,103866,103867,103869,103871,103874,103876],{"class":42,"line":931},[40,103868,103765],{"class":133},[40,103870,623],{"class":441},[40,103872,103873],{"class":133}," S ",[40,103875,7610],{"class":441},[40,103877,103878],{"class":133}," col_max\n",[40,103880,103881,103883],{"class":42,"line":937},[40,103882,839],{"class":129},[40,103884,766],{"class":133},[40,103886,103887,103889,103891,103893,103895,103897,103899,103901,103903,103905],{"class":42,"line":942},[40,103888,103765],{"class":133},[40,103890,623],{"class":441},[40,103892,103873],{"class":133},[40,103894,7610],{"class":441},[40,103896,103554],{"class":441},[40,103898,103839],{"class":133},[40,103900,35767],{"class":237},[40,103902,12998],{"class":133},[40,103904,103862],{"class":462},[40,103906,248],{"class":133},[40,103908,103909],{"class":42,"line":953},[40,103910,228],{"emptyLinePlaceholder":227},[40,103912,103913,103915,103917,103919,103921,103923,103926],{"class":42,"line":973},[40,103914,755],{"class":129},[40,103916,103749],{"class":133},[40,103918,103173],{"class":237},[40,103920,715],{"class":133},[40,103922,761],{"class":441},[40,103924,103925],{"class":244}," \"db\"",[40,103927,766],{"class":133},[40,103929,103930,103932,103934,103936,103938,103940,103943,103945,103947,103950,103953,103956],{"class":42,"line":1013},[40,103931,103765],{"class":133},[40,103933,623],{"class":441},[40,103935,95616],{"class":462},[40,103937,7273],{"class":441},[40,103939,654],{"class":133},[40,103941,103942],{"class":237},"log10",[40,103944,98230],{"class":133},[40,103946,103836],{"class":237},[40,103948,103949],{"class":133},"(S, ",[40,103951,103952],{"class":462},"1e-12",[40,103954,103955],{"class":133},"))  ",[40,103957,103958],{"class":46},"# dB, clipped floor\n",[40,103960,103961],{"class":42,"line":1018},[40,103962,228],{"emptyLinePlaceholder":227},[40,103964,103965],{"class":42,"line":1028},[40,103966,103967],{"class":46},"    # Units\n",[40,103969,103970,103972,103975,103978,103980,103982,103985],{"class":42,"line":1043},[40,103971,755],{"class":129},[40,103973,103974],{"class":133}," freq_units.",[40,103976,103977],{"class":237},"upper",[40,103979,715],{"class":133},[40,103981,761],{"class":441},[40,103983,103984],{"class":244}," \"GHz\"",[40,103986,766],{"class":133},[40,103988,103989,103992,103994,103997,103999],{"class":42,"line":1076},[40,103990,103991],{"class":133},"        f_fft ",[40,103993,623],{"class":441},[40,103995,103996],{"class":133}," f_fft_hz ",[40,103998,1433],{"class":441},[40,104000,95941],{"class":462},[40,104002,104003,104005],{"class":42,"line":1081},[40,104004,839],{"class":129},[40,104006,766],{"class":133},[40,104008,104009,104011,104013,104015,104017],{"class":42,"line":1092},[40,104010,103991],{"class":133},[40,104012,623],{"class":441},[40,104014,103996],{"class":133},[40,104016,1433],{"class":441},[40,104018,104019],{"class":462}," 1e-6\n",[40,104021,104022],{"class":42,"line":1097},[40,104023,228],{"emptyLinePlaceholder":227},[40,104025,104026,104028],{"class":42,"line":1111},[40,104027,1546],{"class":129},[40,104029,104030],{"class":133}," f_fft, S\n",[40,104032,104033],{"class":42,"line":1116},[40,104034,228],{"emptyLinePlaceholder":227},[40,104036,104037],{"class":42,"line":1127},[40,104038,104039],{"class":46},"# Compute spectrogram with current grid\n",[40,104041,104042,104045,104047,104049],{"class":42,"line":1155},[40,104043,104044],{"class":133},"f_fft_axis, S_fft ",[40,104046,623],{"class":441},[40,104048,103226],{"class":237},[40,104050,411],{"class":133},[40,104052,104053],{"class":42,"line":1174},[40,104054,104055],{"class":133},"    Pe, dur_ns,\n",[40,104057,104058,104060,104062,104064],{"class":42,"line":1179},[40,104059,103247],{"class":1195},[40,104061,623],{"class":441},[40,104063,95318],{"class":462},[40,104065,445],{"class":133},[40,104067,104068,104070,104072,104074],{"class":42,"line":1204},[40,104069,103262],{"class":1195},[40,104071,623],{"class":441},[40,104073,95332],{"class":462},[40,104075,445],{"class":133},[40,104077,104078,104080,104082,104084],{"class":42,"line":1209},[40,104079,103277],{"class":1195},[40,104081,623],{"class":441},[40,104083,95347],{"class":462},[40,104085,445],{"class":133},[40,104087,104088,104090,104092,104094],{"class":42,"line":1220},[40,104089,103292],{"class":1195},[40,104091,623],{"class":441},[40,104093,95361],{"class":462},[40,104095,445],{"class":133},[40,104097,104098,104100,104102,104104],{"class":42,"line":1264},[40,104099,103307],{"class":1195},[40,104101,623],{"class":441},[40,104103,95376],{"class":462},[40,104105,445],{"class":133},[40,104107,104108,104110,104112,104114],{"class":42,"line":1273},[40,104109,103322],{"class":1195},[40,104111,623],{"class":441},[40,104113,95391],{"class":462},[40,104115,445],{"class":133},[40,104117,104118],{"class":42,"line":1278},[40,104119,248],{"class":133},[40,104121,104122,104124,104126,104129,104132,104135,104138,104140,104142,104145,104147,104149,104151,104153,104155],{"class":42,"line":1289},[40,104123,7569],{"class":441},[40,104125,241],{"class":133},[40,104127,104128],{"class":244},"\"FFT grid:\"",[40,104130,104131],{"class":133},", S_fft.shape, ",[40,104133,104134],{"class":244},"\"| Fourier freq range:\"",[40,104136,104137],{"class":133},", f_fft_axis[",[40,104139,791],{"class":462},[40,104141,799],{"class":133},[40,104143,104144],{"class":244},"\"→\"",[40,104146,104137],{"class":133},[40,104148,1407],{"class":441},[40,104150,1410],{"class":462},[40,104152,799],{"class":133},[40,104154,95391],{"class":462},[40,104156,248],{"class":133},[30,104158,104161],{"className":104159,"code":104160,"language":6083,"meta":35},[7575],"FFT grid: (2049, 401) | Fourier freq range: 0.0 → 999.9999999999999 MHz\n",[37,104162,104160],{"__ignoreMap":35},[30,104164,104166],{"className":32,"code":104165,"language":34,"meta":35,"style":35},"# @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",[37,104167,104168,104173,104179,104192,104206,104234,104254,104286,104309,104328,104342,104348],{"__ignoreMap":35},[40,104169,104170],{"class":42,"line":43},[40,104171,104172],{"class":46},"# @title Theoretical Rabi ridge\n",[40,104174,104175,104177],{"class":42,"line":50},[40,104176,130],{"class":129},[40,104178,95448],{"class":133},[40,104180,104181,104184,104186,104188,104190],{"class":42,"line":155},[40,104182,104183],{"class":133},"two_pi ",[40,104185,623],{"class":441},[40,104187,95892],{"class":462},[40,104189,7273],{"class":441},[40,104191,95897],{"class":133},[40,104193,104194,104197,104199,104202,104204],{"class":42,"line":168},[40,104195,104196],{"class":133},"f0 ",[40,104198,623],{"class":441},[40,104200,104201],{"class":462}," F0_GHZ",[40,104203,7273],{"class":441},[40,104205,95912],{"class":462},[40,104207,104208,104211,104213,104215,104217,104220,104222,104225,104228,104231],{"class":42,"line":182},[40,104209,104210],{"class":133},"delta_hz ",[40,104212,623],{"class":441},[40,104214,95951],{"class":133},[40,104216,1433],{"class":441},[40,104218,104219],{"class":133}," (freq_ghz",[40,104221,1433],{"class":441},[40,104223,104224],{"class":462},"1e9",[40,104226,104227],{"class":441}," -",[40,104229,104230],{"class":133}," f0)   ",[40,104232,104233],{"class":46},"# rad\u002Fs detuning\n",[40,104235,104236,104239,104241,104243,104245,104248,104250,104252],{"class":42,"line":195},[40,104237,104238],{"class":133},"omega_onres ",[40,104240,623],{"class":441},[40,104242,95951],{"class":133},[40,104244,1433],{"class":441},[40,104246,104247],{"class":462}," OMEGA_ONRESONANCE_MHZ",[40,104249,7273],{"class":441},[40,104251,95961],{"class":462},[40,104253,95964],{"class":46},[40,104255,104256,104259,104261,104263,104265,104268,104270,104272,104274,104277,104279,104281,104284],{"class":42,"line":203},[40,104257,104258],{"class":133},"Omega_R ",[40,104260,623],{"class":441},[40,104262,654],{"class":133},[40,104264,8866],{"class":237},[40,104266,104267],{"class":133},"(omega_onres",[40,104269,1419],{"class":441},[40,104271,1879],{"class":462},[40,104273,7357],{"class":441},[40,104275,104276],{"class":133}," delta_hz",[40,104278,1419],{"class":441},[40,104280,1879],{"class":462},[40,104282,104283],{"class":133},")     ",[40,104285,96018],{"class":46},[40,104287,104288,104291,104293,104295,104297,104299,104301,104303,104306],{"class":42,"line":216},[40,104289,104290],{"class":133},"ridge_hz ",[40,104292,623],{"class":441},[40,104294,96117],{"class":133},[40,104296,7610],{"class":441},[40,104298,96957],{"class":133},[40,104300,97855],{"class":462},[40,104302,7273],{"class":441},[40,104304,104305],{"class":133}," math.pi)                ",[40,104307,104308],{"class":46},"# Hz\n",[40,104310,104311,104313,104316,104318,104320,104322,104324,104326],{"class":42,"line":224},[40,104312,683],{"class":129},[40,104314,104315],{"class":462}," FREQ_UNITS",[40,104317,1823],{"class":133},[40,104319,103977],{"class":237},[40,104321,715],{"class":133},[40,104323,761],{"class":441},[40,104325,103984],{"class":244},[40,104327,766],{"class":133},[40,104329,104330,104333,104335,104338,104340],{"class":42,"line":231},[40,104331,104332],{"class":133},"    ridge_axis ",[40,104334,623],{"class":441},[40,104336,104337],{"class":133}," ridge_hz ",[40,104339,1433],{"class":441},[40,104341,95941],{"class":462},[40,104343,104344,104346],{"class":42,"line":251},[40,104345,99720],{"class":129},[40,104347,766],{"class":133},[40,104349,104350,104352,104354,104356,104358,104361],{"class":42,"line":256},[40,104351,104332],{"class":133},[40,104353,623],{"class":441},[40,104355,104337],{"class":133},[40,104357,1433],{"class":441},[40,104359,104360],{"class":462}," 1e-6",[40,104362,104363],{"class":46},"  # MHz\n",[30,104365,104367],{"className":32,"code":104366,"language":34,"meta":35,"style":35},"# @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",[37,104368,104369,104374,104402,104435,104455,104498,104502,104548,104561,104582,104620,104638,104646],{"__ignoreMap":35},[40,104370,104371],{"class":42,"line":43},[40,104372,104373],{"class":46},"# @title 2D heatmap\n",[40,104375,104376,104378,104380,104382,104384,104386,104388,104390,104392,104395,104397,104400],{"class":42,"line":50},[40,104377,96533],{"class":133},[40,104379,623],{"class":441},[40,104381,96538],{"class":133},[40,104383,96541],{"class":237},[40,104385,241],{"class":133},[40,104387,96546],{"class":1195},[40,104389,623],{"class":441},[40,104391,241],{"class":133},[40,104393,104394],{"class":462},"7.5",[40,104396,917],{"class":133},[40,104398,104399],{"class":462},"5.5",[40,104401,21594],{"class":133},[40,104403,104404,104407,104409,104411,104413,104416,104418,104420,104423,104425,104428,104430,104433],{"class":42,"line":155},[40,104405,104406],{"class":133},"im ",[40,104408,623],{"class":441},[40,104410,96570],{"class":133},[40,104412,96573],{"class":237},[40,104414,104415],{"class":133},"(freq_ghz, f_fft_axis, S_fft, ",[40,104417,96579],{"class":1195},[40,104419,623],{"class":441},[40,104421,104422],{"class":244},"\"auto\"",[40,104424,917],{"class":133},[40,104426,104427],{"class":1195},"cmap",[40,104429,623],{"class":441},[40,104431,104432],{"class":244},"\"inferno\"",[40,104434,248],{"class":133},[40,104436,104437,104439,104441,104443,104445,104448,104450,104452],{"class":42,"line":168},[40,104438,96591],{"class":133},[40,104440,623],{"class":441},[40,104442,96596],{"class":133},[40,104444,96599],{"class":237},[40,104446,104447],{"class":133},"(im, ",[40,104449,96605],{"class":1195},[40,104451,623],{"class":441},[40,104453,104454],{"class":133},"ax)\n",[40,104456,104457,104459,104461,104463,104466,104468,104470,104473,104476,104479,104481,104483,104486,104488,104491,104493,104496],{"class":42,"line":182},[40,104458,102296],{"class":133},[40,104460,102299],{"class":237},[40,104462,241],{"class":133},[40,104464,104465],{"class":244},"\"FFT amplitude\"",[40,104467,7357],{"class":441},[40,104469,96957],{"class":133},[40,104471,104472],{"class":244},"\" (dB)\"",[40,104474,104475],{"class":129}," if",[40,104477,104478],{"class":462}," AMP_MODE",[40,104480,1823],{"class":133},[40,104482,103173],{"class":237},[40,104484,104485],{"class":133},"()",[40,104487,761],{"class":441},[40,104489,104490],{"class":244},"\"db\"",[40,104492,7524],{"class":129},[40,104494,104495],{"class":244}," \" (arb.)\"",[40,104497,21594],{"class":133},[40,104499,104500],{"class":42,"line":195},[40,104501,228],{"emptyLinePlaceholder":227},[40,104503,104504,104506,104508,104511,104514,104516,104519,104521,104524,104526,104528,104530,104532,104534,104537,104540,104543,104546],{"class":42,"line":203},[40,104505,96627],{"class":133},[40,104507,24433],{"class":237},[40,104509,104510],{"class":133},"(freq_ghz, ridge_axis, ",[40,104512,104513],{"class":244},"\"--\"",[40,104515,917],{"class":133},[40,104517,104518],{"class":1195},"lw",[40,104520,623],{"class":441},[40,104522,104523],{"class":462},"1.5",[40,104525,917],{"class":133},[40,104527,20062],{"class":1195},[40,104529,623],{"class":441},[40,104531,1256],{"class":129},[40,104533,99993],{"class":96619},[40,104535,104536],{"class":441},"\\O",[40,104538,104539],{"class":96619},"mega_R\u002F2",[40,104541,104542],{"class":441},"\\p",[40,104544,104545],{"class":96619},"i$\"",[40,104547,248],{"class":133},[40,104549,104550,104552,104554,104556,104559],{"class":42,"line":216},[40,104551,96627],{"class":133},[40,104553,96630],{"class":237},[40,104555,241],{"class":133},[40,104557,104558],{"class":244},"\"Drive frequency f (GHz)\"",[40,104560,248],{"class":133},[40,104562,104563,104565,104567,104569,104571,104574,104577,104580],{"class":42,"line":224},[40,104564,96627],{"class":133},[40,104566,96644],{"class":237},[40,104568,241],{"class":133},[40,104570,21331],{"class":129},[40,104572,104573],{"class":244},"\"Fourier frequency (",[40,104575,104576],{"class":462},"{FREQ_UNITS}",[40,104578,104579],{"class":244},")\"",[40,104581,248],{"class":133},[40,104583,104584,104586,104588,104590,104593,104595,104597,104600,104603,104605,104608,104610,104613,104615,104618],{"class":42,"line":231},[40,104585,96627],{"class":133},[40,104587,96658],{"class":237},[40,104589,241],{"class":133},[40,104591,104592],{"class":244},"\"Fourier transform along pulse length: $|",[40,104594,96666],{"class":441},[40,104596,1616],{"class":244},[40,104598,104599],{"class":462},"{F}",[40,104601,104602],{"class":244},"_",[40,104604,96666],{"class":441},[40,104606,104607],{"class":244},"tau",[40,104609,96666],{"class":441},[40,104611,104612],{"class":244},"{P_e",[40,104614,96666],{"class":441},[40,104616,104617],{"class":244},"}|$\"",[40,104619,248],{"class":133},[40,104621,104622,104624,104626,104628,104631,104633,104636],{"class":42,"line":251},[40,104623,96627],{"class":133},[40,104625,24612],{"class":237},[40,104627,241],{"class":133},[40,104629,104630],{"class":1195},"loc",[40,104632,623],{"class":441},[40,104634,104635],{"class":244},"\"upper right\"",[40,104637,248],{"class":133},[40,104639,104640,104642,104644],{"class":42,"line":256},[40,104641,96680],{"class":133},[40,104643,96683],{"class":237},[40,104645,12875],{"class":133},[40,104647,104648,104650,104652],{"class":42,"line":269},[40,104649,24430],{"class":133},[40,104651,24667],{"class":237},[40,104653,12875],{"class":133},[8229,104655],{"alt":104656,"src":104657},"Output 6 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-06.webp",[30,104659,104661],{"className":32,"code":104660,"language":34,"meta":35,"style":35},"# --- 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",[37,104662,104663,104668,104678,104688,104698,104702,104707,104724,104741,104758,104775,104779,104784,104801,104817,104821,104826,104839,104843,104848,104886,104907,104911,104931,104944,104958,104962,104967,104972,105007,105036,105040,105045,105079,105089,105099,105103,105107,105133,105137,105142,105176,105201,105213,105223,105241,105259,105269,105285,105289,105293,105306,105325,105361,105374,105413,105417,105422,105449,105453,105461],{"__ignoreMap":35},[40,104664,104665],{"class":42,"line":43},[40,104666,104667],{"class":46},"# --- Adaptive high-res 3D surface ---\n",[40,104669,104670,104672,104674,104676],{"class":42,"line":50},[40,104671,171],{"class":129},[40,104673,101626],{"class":133},[40,104675,130],{"class":129},[40,104677,101631],{"class":133},[40,104679,104680,104682,104684,104686],{"class":42,"line":155},[40,104681,130],{"class":129},[40,104683,134],{"class":133},[40,104685,137],{"class":129},[40,104687,140],{"class":133},[40,104689,104690,104692,104694,104696],{"class":42,"line":168},[40,104691,130],{"class":129},[40,104693,95476],{"class":133},[40,104695,137],{"class":129},[40,104697,165],{"class":133},[40,104699,104700],{"class":42,"line":182},[40,104701,228],{"emptyLinePlaceholder":227},[40,104703,104704],{"class":42,"line":195},[40,104705,104706],{"class":46},"# ------ Control knobs ------\n",[40,104708,104709,104712,104714,104716,104718,104721],{"class":42,"line":203},[40,104710,104711],{"class":462},"UPSAMPLE_F_FREQ",[40,104713,438],{"class":133},[40,104715,456],{"class":441},[40,104717,459],{"class":441},[40,104719,104720],{"class":462}," 3",[40,104722,104723],{"class":46},"       # try 2–3; upsample the drive-frequency axis only\n",[40,104725,104726,104729,104731,104733,104735,104738],{"class":42,"line":216},[40,104727,104728],{"class":462},"MAX_FACES",[40,104730,438],{"class":133},[40,104732,456],{"class":441},[40,104734,459],{"class":441},[40,104736,104737],{"class":462}," 300_000",[40,104739,104740],{"class":46},"       # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\n",[40,104742,104743,104746,104748,104750,104752,104755],{"class":42,"line":224},[40,104744,104745],{"class":462},"MIN_ROWS",[40,104747,438],{"class":133},[40,104749,456],{"class":441},[40,104751,459],{"class":441},[40,104753,104754],{"class":462}," 128",[40,104756,104757],{"class":46},"            # keep at least this many Fourier-frequency rows\n",[40,104759,104760,104763,104765,104767,104769,104772],{"class":42,"line":231},[40,104761,104762],{"class":462},"MAX_ROWS",[40,104764,438],{"class":133},[40,104766,456],{"class":441},[40,104768,459],{"class":441},[40,104770,104771],{"class":462}," 1024",[40,104773,104774],{"class":46},"           # hard cap for rows to keep plotting fast\n",[40,104776,104777],{"class":42,"line":251},[40,104778,228],{"emptyLinePlaceholder":227},[40,104780,104781],{"class":42,"line":256},[40,104782,104783],{"class":46},"# Preserve global color limits so the legend stays consistent\n",[40,104785,104786,104789,104791,104793,104795,104798],{"class":42,"line":269},[40,104787,104788],{"class":133},"vmin ",[40,104790,623],{"class":441},[40,104792,97933],{"class":441},[40,104794,98230],{"class":133},[40,104796,104797],{"class":237},"nanmin",[40,104799,104800],{"class":133},"(S_fft))\n",[40,104802,104803,104806,104808,104810,104812,104815],{"class":42,"line":282},[40,104804,104805],{"class":133},"vmax ",[40,104807,623],{"class":441},[40,104809,97933],{"class":441},[40,104811,98230],{"class":133},[40,104813,104814],{"class":237},"nanmax",[40,104816,104800],{"class":133},[40,104818,104819],{"class":42,"line":295},[40,104820,228],{"emptyLinePlaceholder":227},[40,104822,104823],{"class":42,"line":308},[40,104824,104825],{"class":46},"# Shapes\n",[40,104827,104828,104831,104833,104836],{"class":42,"line":321},[40,104829,104830],{"class":133},"n_rows, n_cols ",[40,104832,623],{"class":441},[40,104834,104835],{"class":133}," S_fft.shape        ",[40,104837,104838],{"class":46},"# rows=Fourier freq (ν), cols=drive freq (f)\n",[40,104840,104841],{"class":42,"line":334},[40,104842,228],{"emptyLinePlaceholder":227},[40,104844,104845],{"class":42,"line":347},[40,104846,104847],{"class":46},"# 1) Upsample along the typically smaller axis (drive frequency) only\n",[40,104849,104850,104853,104855,104857,104859,104861,104863,104866,104868,104870,104872,104874,104876,104878,104881,104883],{"class":42,"line":352},[40,104851,104852],{"class":133},"n_cols_up ",[40,104854,623],{"class":441},[40,104856,1518],{"class":441},[40,104858,241],{"class":133},[40,104860,104711],{"class":462},[40,104862,7273],{"class":441},[40,104864,104865],{"class":133}," (n_cols ",[40,104867,1407],{"class":441},[40,104869,463],{"class":462},[40,104871,1416],{"class":133},[40,104873,1147],{"class":441},[40,104875,463],{"class":462},[40,104877,917],{"class":133},[40,104879,104880],{"class":462},"1200",[40,104882,21772],{"class":133},[40,104884,104885],{"class":46},"# sensible upper bound\n",[40,104887,104888,104890,104892,104894,104896,104898,104900,104902,104904],{"class":42,"line":357},[40,104889,101733],{"class":133},[40,104891,623],{"class":441},[40,104893,654],{"class":133},[40,104895,96259],{"class":237},[40,104897,101742],{"class":133},[40,104899,24284],{"class":237},[40,104901,101747],{"class":133},[40,104903,35767],{"class":237},[40,104905,104906],{"class":133},"(), n_cols_up)\n",[40,104908,104909],{"class":42,"line":370},[40,104910,228],{"emptyLinePlaceholder":227},[40,104912,104913,104916,104918,104920,104922,104925,104927,104929],{"class":42,"line":376},[40,104914,104915],{"class":133},"S_up_f ",[40,104917,623],{"class":441},[40,104919,654],{"class":133},[40,104921,101790],{"class":237},[40,104923,104924],{"class":133},"((n_rows, n_cols_up), ",[40,104926,1196],{"class":1195},[40,104928,101808],{"class":441},[40,104930,248],{"class":133},[40,104932,104933,104935,104937,104939,104941],{"class":42,"line":387},[40,104934,668],{"class":129},[40,104936,7290],{"class":133},[40,104938,674],{"class":129},[40,104940,7265],{"class":441},[40,104942,104943],{"class":133},"(n_rows):\n",[40,104945,104946,104949,104951,104953,104955],{"class":42,"line":392},[40,104947,104948],{"class":133},"    S_up_f[i, :] ",[40,104950,623],{"class":441},[40,104952,654],{"class":133},[40,104954,101839],{"class":237},[40,104956,104957],{"class":133},"(freq_hi, freq_ghz, S_fft[i, :])\n",[40,104959,104960],{"class":42,"line":397},[40,104961,228],{"emptyLinePlaceholder":227},[40,104963,104964],{"class":42,"line":402},[40,104965,104966],{"class":46},"# 2) Determine how many Fourier rows we can afford for MAX_FACES\n",[40,104968,104969],{"class":42,"line":414},[40,104970,104971],{"class":46},"# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\n",[40,104973,104974,104977,104979,104981,104983,104985,104987,104989,104992,104994,104996,104998,105000,105003,105005],{"class":42,"line":424},[40,104975,104976],{"class":133},"n_rows_cap ",[40,104978,623],{"class":441},[40,104980,1135],{"class":441},[40,104982,241],{"class":133},[40,104984,104728],{"class":462},[40,104986,96193],{"class":441},[40,104988,103554],{"class":441},[40,104990,104991],{"class":133},"(n_cols_up ",[40,104993,1407],{"class":441},[40,104995,463],{"class":462},[40,104997,917],{"class":133},[40,104999,1410],{"class":462},[40,105001,105002],{"class":133},")) ",[40,105004,1147],{"class":441},[40,105006,1124],{"class":462},[40,105008,105009,105012,105014,105016,105018,105020,105023,105025,105027,105029,105031,105033],{"class":42,"line":432},[40,105010,105011],{"class":133},"n_rows_plot ",[40,105013,623],{"class":441},[40,105015,1135],{"class":441},[40,105017,98230],{"class":133},[40,105019,96230],{"class":237},[40,105021,105022],{"class":133},"(n_rows_cap, ",[40,105024,104745],{"class":462},[40,105026,917],{"class":133},[40,105028,24284],{"class":441},[40,105030,241],{"class":133},[40,105032,104762],{"class":462},[40,105034,105035],{"class":133},", n_rows)))\n",[40,105037,105038],{"class":42,"line":448},[40,105039,228],{"emptyLinePlaceholder":227},[40,105041,105042],{"class":42,"line":468},[40,105043,105044],{"class":46},"# Down-sample ν-axis evenly to n_rows_plot\n",[40,105046,105047,105050,105052,105054,105056,105058,105060,105063,105065,105067,105070,105073,105075,105077],{"class":42,"line":486},[40,105048,105049],{"class":133},"row_idx ",[40,105051,623],{"class":441},[40,105053,654],{"class":133},[40,105055,96259],{"class":237},[40,105057,241],{"class":133},[40,105059,791],{"class":462},[40,105061,105062],{"class":133},", n_rows ",[40,105064,1407],{"class":441},[40,105066,463],{"class":462},[40,105068,105069],{"class":133},", n_rows_plot).",[40,105071,105072],{"class":237},"astype",[40,105074,241],{"class":133},[40,105076,456],{"class":441},[40,105078,248],{"class":133},[40,105080,105081,105084,105086],{"class":42,"line":502},[40,105082,105083],{"class":133},"f_fft_plot ",[40,105085,623],{"class":441},[40,105087,105088],{"class":133}," f_fft_axis[row_idx]\n",[40,105090,105091,105094,105096],{"class":42,"line":508},[40,105092,105093],{"class":133},"S_final ",[40,105095,623],{"class":441},[40,105097,105098],{"class":133}," S_up_f[row_idx, :]\n",[40,105100,105101],{"class":42,"line":514},[40,105102,228],{"emptyLinePlaceholder":227},[40,105104,105105],{"class":42,"line":519},[40,105106,101968],{"class":46},[40,105108,105109,105112,105115,105117,105119,105121,105124,105126,105128,105131],{"class":42,"line":525},[40,105110,105111],{"class":133},"F2_hi, ",[40,105113,105114],{"class":462},"FFOUR_hi",[40,105116,459],{"class":441},[40,105118,654],{"class":133},[40,105120,95983],{"class":237},[40,105122,105123],{"class":133},"(freq_hi, f_fft_plot, ",[40,105125,95989],{"class":1195},[40,105127,623],{"class":441},[40,105129,105130],{"class":244},"\"xy\"",[40,105132,248],{"class":133},[40,105134,105135],{"class":42,"line":531},[40,105136,228],{"emptyLinePlaceholder":227},[40,105138,105139],{"class":42,"line":537},[40,105140,105141],{"class":46},"# 3) Plot (keeps 'inferno')\n",[40,105143,105144,105146,105148,105150,105152,105154,105156,105158,105160,105162,105164,105166,105168,105170,105172,105174],{"class":42,"line":543},[40,105145,102004],{"class":133},[40,105147,623],{"class":441},[40,105149,96538],{"class":133},[40,105151,99929],{"class":237},[40,105153,241],{"class":133},[40,105155,96546],{"class":1195},[40,105157,623],{"class":441},[40,105159,241],{"class":133},[40,105161,34795],{"class":462},[40,105163,917],{"class":133},[40,105165,102025],{"class":462},[40,105167,1253],{"class":133},[40,105169,102030],{"class":1195},[40,105171,623],{"class":441},[40,105173,102035],{"class":462},[40,105175,248],{"class":133},[40,105177,105178,105180,105182,105184,105186,105188,105190,105192,105194,105196,105199],{"class":42,"line":549},[40,105179,102042],{"class":133},[40,105181,623],{"class":441},[40,105183,96596],{"class":133},[40,105185,102049],{"class":237},[40,105187,241],{"class":133},[40,105189,102054],{"class":462},[40,105191,917],{"class":133},[40,105193,102059],{"class":1195},[40,105195,623],{"class":441},[40,105197,105198],{"class":244},"\"3d\"",[40,105200,248],{"class":133},[40,105202,105203,105205,105207,105209,105211],{"class":42,"line":555},[40,105204,102075],{"class":133},[40,105206,623],{"class":441},[40,105208,96570],{"class":133},[40,105210,102082],{"class":237},[40,105212,411],{"class":133},[40,105214,105215,105218,105220],{"class":42,"line":561},[40,105216,105217],{"class":133},"    F2_hi, ",[40,105219,105114],{"class":462},[40,105221,105222],{"class":133},", S_final,\n",[40,105224,105225,105227,105229,105231,105233,105235,105237,105239],{"class":42,"line":567},[40,105226,102094],{"class":1195},[40,105228,623],{"class":441},[40,105230,1410],{"class":462},[40,105232,917],{"class":133},[40,105234,102103],{"class":1195},[40,105236,623],{"class":441},[40,105238,1410],{"class":462},[40,105240,445],{"class":133},[40,105242,105243,105245,105247,105249,105251,105253,105255,105257],{"class":42,"line":573},[40,105244,102118],{"class":1195},[40,105246,623],{"class":441},[40,105248,791],{"class":462},[40,105250,917],{"class":133},[40,105252,102127],{"class":1195},[40,105254,623],{"class":441},[40,105256,12926],{"class":462},[40,105258,445],{"class":133},[40,105260,105261,105263,105265,105267],{"class":42,"line":579},[40,105262,102138],{"class":1195},[40,105264,623],{"class":441},[40,105266,104432],{"class":244},[40,105268,445],{"class":133},[40,105270,105271,105273,105275,105278,105280,105282],{"class":42,"line":585},[40,105272,102149],{"class":1195},[40,105274,623],{"class":441},[40,105276,105277],{"class":133},"vmin, ",[40,105279,102158],{"class":1195},[40,105281,623],{"class":441},[40,105283,105284],{"class":133},"vmax\n",[40,105286,105287],{"class":42,"line":590},[40,105288,248],{"class":133},[40,105290,105291],{"class":42,"line":596},[40,105292,228],{"emptyLinePlaceholder":227},[40,105294,105295,105297,105299,105301,105304],{"class":42,"line":602},[40,105296,96627],{"class":133},[40,105298,96630],{"class":237},[40,105300,241],{"class":133},[40,105302,105303],{"class":244},"\"f (GHz)\"",[40,105305,248],{"class":133},[40,105307,105308,105310,105312,105314,105316,105319,105321,105323],{"class":42,"line":607},[40,105309,96627],{"class":133},[40,105311,96644],{"class":237},[40,105313,241],{"class":133},[40,105315,21331],{"class":129},[40,105317,105318],{"class":244},"\"Fourier freq (",[40,105320,104576],{"class":462},[40,105322,104579],{"class":244},[40,105324,248],{"class":133},[40,105326,105327,105329,105331,105333,105335,105337,105339,105341,105343,105345,105347,105349,105351,105353,105355,105357,105359],{"class":42,"line":612},[40,105328,96627],{"class":133},[40,105330,102222],{"class":237},[40,105332,241],{"class":133},[40,105334,104465],{"class":244},[40,105336,7357],{"class":441},[40,105338,96957],{"class":133},[40,105340,104472],{"class":244},[40,105342,104475],{"class":129},[40,105344,104478],{"class":462},[40,105346,1823],{"class":133},[40,105348,103173],{"class":237},[40,105350,715],{"class":133},[40,105352,761],{"class":441},[40,105354,103925],{"class":244},[40,105356,7524],{"class":129},[40,105358,104495],{"class":244},[40,105360,21594],{"class":133},[40,105362,105363,105365,105367,105369,105372],{"class":42,"line":617},[40,105364,96627],{"class":133},[40,105366,96658],{"class":237},[40,105368,241],{"class":133},[40,105370,105371],{"class":244},"\"Rabi FFT surface (adaptive resolution)\"",[40,105373,248],{"class":133},[40,105375,105376,105378,105380,105382,105384,105386,105388,105390,105392,105394,105396,105398,105400,105402,105404,105406,105408,105411],{"class":42,"line":635},[40,105377,96680],{"class":133},[40,105379,96599],{"class":237},[40,105381,102265],{"class":133},[40,105383,96605],{"class":1195},[40,105385,623],{"class":441},[40,105387,96610],{"class":133},[40,105389,102274],{"class":1195},[40,105391,623],{"class":441},[40,105393,102279],{"class":462},[40,105395,917],{"class":133},[40,105397,102284],{"class":1195},[40,105399,623],{"class":441},[40,105401,102289],{"class":462},[40,105403,917],{"class":133},[40,105405,20062],{"class":1195},[40,105407,623],{"class":441},[40,105409,105410],{"class":244},"\"Amplitude\"",[40,105412,248],{"class":133},[40,105414,105415],{"class":42,"line":646},[40,105416,228],{"emptyLinePlaceholder":227},[40,105418,105419],{"class":42,"line":662},[40,105420,105421],{"class":46},"# A clearer default view\n",[40,105423,105424,105426,105429,105431,105434,105436,105438,105440,105443,105445,105447],{"class":42,"line":695},[40,105425,96627],{"class":133},[40,105427,105428],{"class":237},"view_init",[40,105430,241],{"class":133},[40,105432,105433],{"class":1195},"elev",[40,105435,623],{"class":441},[40,105437,86901],{"class":462},[40,105439,917],{"class":133},[40,105441,105442],{"class":1195},"azim",[40,105444,13164],{"class":441},[40,105446,96990],{"class":462},[40,105448,248],{"class":133},[40,105450,105451],{"class":42,"line":701},[40,105452,228],{"emptyLinePlaceholder":227},[40,105454,105455,105457,105459],{"class":42,"line":729},[40,105456,96680],{"class":133},[40,105458,96683],{"class":237},[40,105460,12875],{"class":133},[40,105462,105463,105465,105467],{"class":42,"line":752},[40,105464,24430],{"class":133},[40,105466,24667],{"class":237},[40,105468,12875],{"class":133},[8229,105470],{"alt":105471,"src":105472},"Output 7 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-07.webp",[30,105474,105476],{"className":32,"code":105475,"language":34,"meta":35,"style":35},"# @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",[37,105477,105478,105483,105506,105538,105542,105547,105571,105599,105616,105661,105679,105715,105742,105750,105763,105767,105772,105798,105824,105841,105853,105889,105918],{"__ignoreMap":35},[40,105479,105480],{"class":42,"line":43},[40,105481,105482],{"class":46},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\n",[40,105484,105485,105488,105490,105492,105494,105496,105498,105500,105503],{"class":42,"line":50},[40,105486,105487],{"class":462},"F_STAR_GHZ",[40,105489,438],{"class":133},[40,105491,442],{"class":441},[40,105493,459],{"class":441},[40,105495,97933],{"class":441},[40,105497,241],{"class":133},[40,105499,94840],{"class":462},[40,105501,105502],{"class":133},")   ",[40,105504,105505],{"class":46},"# slice near resonance\n",[40,105507,105508,105511,105513,105515,105518,105520,105522,105524,105527,105529,105532,105535],{"class":42,"line":155},[40,105509,105510],{"class":462},"NU_STAR",[40,105512,459],{"class":441},[40,105514,97933],{"class":441},[40,105516,105517],{"class":133},"(ridge_axis[np.",[40,105519,102567],{"class":237},[40,105521,98230],{"class":133},[40,105523,102572],{"class":237},[40,105525,105526],{"class":133},"(freq_ghz ",[40,105528,1407],{"class":441},[40,105530,105531],{"class":462}," F_STAR_GHZ",[40,105533,105534],{"class":133},"))])  ",[40,105536,105537],{"class":46},"# near Ω_R\u002F2π\n",[40,105539,105540],{"class":42,"line":168},[40,105541,228],{"emptyLinePlaceholder":227},[40,105543,105544],{"class":42,"line":182},[40,105545,105546],{"class":46},"# Slice at f = F_STAR_GHZ\n",[40,105548,105549,105551,105553,105555,105557,105559,105561,105563,105565,105567,105569],{"class":42,"line":195},[40,105550,102594],{"class":133},[40,105552,623],{"class":441},[40,105554,1135],{"class":441},[40,105556,98230],{"class":133},[40,105558,102567],{"class":237},[40,105560,98230],{"class":133},[40,105562,102572],{"class":237},[40,105564,105526],{"class":133},[40,105566,1407],{"class":441},[40,105568,105531],{"class":462},[40,105570,13115],{"class":133},[40,105572,105573,105575,105577,105579,105581,105583,105585,105587,105589,105592,105594,105597],{"class":42,"line":203},[40,105574,96533],{"class":133},[40,105576,623],{"class":441},[40,105578,96538],{"class":133},[40,105580,96541],{"class":237},[40,105582,241],{"class":133},[40,105584,96546],{"class":1195},[40,105586,623],{"class":441},[40,105588,241],{"class":133},[40,105590,105591],{"class":462},"6.0",[40,105593,917],{"class":133},[40,105595,105596],{"class":462},"3.8",[40,105598,21594],{"class":133},[40,105600,105601,105603,105605,105608,105610,105612,105614],{"class":42,"line":216},[40,105602,96627],{"class":133},[40,105604,24433],{"class":237},[40,105606,105607],{"class":133},"(f_fft_axis, S_fft[:, j], ",[40,105609,102663],{"class":1195},[40,105611,623],{"class":441},[40,105613,102668],{"class":244},[40,105615,248],{"class":133},[40,105617,105618,105620,105623,105626,105629,105631,105633,105635,105637,105639,105641,105643,105645,105647,105649,105651,105653,105655,105657,105659],{"class":42,"line":224},[40,105619,96627],{"class":133},[40,105621,105622],{"class":237},"axvline",[40,105624,105625],{"class":133},"(ridge_axis[j], ",[40,105627,105628],{"class":1195},"ls",[40,105630,623],{"class":441},[40,105632,104513],{"class":244},[40,105634,917],{"class":133},[40,105636,104518],{"class":1195},[40,105638,623],{"class":441},[40,105640,96240],{"class":462},[40,105642,917],{"class":133},[40,105644,20062],{"class":1195},[40,105646,623],{"class":441},[40,105648,1256],{"class":129},[40,105650,99993],{"class":96619},[40,105652,104536],{"class":441},[40,105654,104539],{"class":96619},[40,105656,104542],{"class":441},[40,105658,104545],{"class":96619},[40,105660,248],{"class":133},[40,105662,105663,105665,105667,105669,105671,105673,105675,105677],{"class":42,"line":231},[40,105664,96627],{"class":133},[40,105666,96630],{"class":237},[40,105668,241],{"class":133},[40,105670,21331],{"class":129},[40,105672,104573],{"class":244},[40,105674,104576],{"class":462},[40,105676,104579],{"class":244},[40,105678,248],{"class":133},[40,105680,105681,105683,105685,105687,105689,105691,105693,105695,105697,105699,105701,105703,105705,105707,105709,105711,105713],{"class":42,"line":251},[40,105682,96627],{"class":133},[40,105684,96644],{"class":237},[40,105686,241],{"class":133},[40,105688,104465],{"class":244},[40,105690,7357],{"class":441},[40,105692,96957],{"class":133},[40,105694,104472],{"class":244},[40,105696,104475],{"class":129},[40,105698,104478],{"class":462},[40,105700,1823],{"class":133},[40,105702,103173],{"class":237},[40,105704,104485],{"class":133},[40,105706,761],{"class":441},[40,105708,104490],{"class":244},[40,105710,7524],{"class":129},[40,105712,104495],{"class":244},[40,105714,21594],{"class":133},[40,105716,105717,105719,105721,105723,105726,105729,105731,105733,105735,105737,105740],{"class":42,"line":256},[40,105718,96627],{"class":133},[40,105720,96658],{"class":237},[40,105722,241],{"class":133},[40,105724,105725],{"class":129},"fr",[40,105727,105728],{"class":244},"\"Slice at $f=",[40,105730,1853],{"class":462},[40,105732,102752],{"class":133},[40,105734,101191],{"class":129},[40,105736,1916],{"class":462},[40,105738,105739],{"class":244},"$ GHz\"",[40,105741,248],{"class":133},[40,105743,105744,105746,105748],{"class":42,"line":269},[40,105745,96627],{"class":133},[40,105747,24612],{"class":237},[40,105749,12875],{"class":133},[40,105751,105752,105754,105756,105759,105761],{"class":42,"line":282},[40,105753,96680],{"class":133},[40,105755,96683],{"class":237},[40,105757,105758],{"class":133},"(); plt.",[40,105760,24667],{"class":237},[40,105762,12875],{"class":133},[40,105764,105765],{"class":42,"line":295},[40,105766,228],{"emptyLinePlaceholder":227},[40,105768,105769],{"class":42,"line":308},[40,105770,105771],{"class":46},"# Slice at ν = NU_STAR\n",[40,105773,105774,105776,105778,105780,105782,105784,105786,105788,105791,105793,105796],{"class":42,"line":321},[40,105775,102808],{"class":133},[40,105777,623],{"class":441},[40,105779,1135],{"class":441},[40,105781,98230],{"class":133},[40,105783,102567],{"class":237},[40,105785,98230],{"class":133},[40,105787,102572],{"class":237},[40,105789,105790],{"class":133},"(f_fft_axis ",[40,105792,1407],{"class":441},[40,105794,105795],{"class":462}," NU_STAR",[40,105797,13115],{"class":133},[40,105799,105800,105802,105804,105806,105808,105810,105812,105814,105816,105818,105820,105822],{"class":42,"line":334},[40,105801,96533],{"class":133},[40,105803,623],{"class":441},[40,105805,96538],{"class":133},[40,105807,96541],{"class":237},[40,105809,241],{"class":133},[40,105811,96546],{"class":1195},[40,105813,623],{"class":441},[40,105815,241],{"class":133},[40,105817,105591],{"class":462},[40,105819,917],{"class":133},[40,105821,105596],{"class":462},[40,105823,21594],{"class":133},[40,105825,105826,105828,105830,105833,105835,105837,105839],{"class":42,"line":347},[40,105827,96627],{"class":133},[40,105829,24433],{"class":237},[40,105831,105832],{"class":133},"(freq_ghz, S_fft[i, :], ",[40,105834,102663],{"class":1195},[40,105836,623],{"class":441},[40,105838,102668],{"class":244},[40,105840,248],{"class":133},[40,105842,105843,105845,105847,105849,105851],{"class":42,"line":352},[40,105844,96627],{"class":133},[40,105846,96630],{"class":237},[40,105848,241],{"class":133},[40,105850,104558],{"class":244},[40,105852,248],{"class":133},[40,105854,105855,105857,105859,105861,105863,105865,105867,105869,105871,105873,105875,105877,105879,105881,105883,105885,105887],{"class":42,"line":357},[40,105856,96627],{"class":133},[40,105858,96644],{"class":237},[40,105860,241],{"class":133},[40,105862,104465],{"class":244},[40,105864,7357],{"class":441},[40,105866,96957],{"class":133},[40,105868,104472],{"class":244},[40,105870,104475],{"class":129},[40,105872,104478],{"class":462},[40,105874,1823],{"class":133},[40,105876,103173],{"class":237},[40,105878,104485],{"class":133},[40,105880,761],{"class":441},[40,105882,104490],{"class":244},[40,105884,7524],{"class":129},[40,105886,104495],{"class":244},[40,105888,21594],{"class":133},[40,105890,105891,105893,105895,105897,105899,105902,105904,105907,105909,105911,105914,105916],{"class":42,"line":370},[40,105892,96627],{"class":133},[40,105894,96658],{"class":237},[40,105896,241],{"class":133},[40,105898,105725],{"class":129},[40,105900,105901],{"class":244},"\"Slice 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Sampling-Budget & Bit-String Analogy for Qubit Classifications",[40728],{"type":9,"value":105992,"toc":110574},[105993,106000,110415,110421,110424,110427,110433,110439,110442,110444,110562,110564,110571],[12,105994,105995],{},[15,105996,17,105997,79984],{},[19,105998,79983],{"href":105999},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[30,106001,106003],{"className":32,"code":106002,"language":34,"meta":35,"style":35},"# 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",[37,106004,106005,106010,106015,106020,106025,106030,106035,106040,106045,106050,106055,106059,106066,106072,106084,106095,106099,106109,106119,106129,106141,106153,106157,106170,106174,106179,106183,106230,106234,106239,106244,106248,106261,106275,106289,106314,106350,106416,106449,106453,106457,106504,106508,106513,106518,106522,106546,106567,106580,106586,106616,106638,106642,106646,106651,106655,106671,106682,106690,106700,106707,106711,106715,106727,106734,106741,106748,106753,106757,106763,106770,106777,106782,106786,106792,106799,106806,106811,106815,106821,106828,106835,106840,106844,106850,106857,106864,106869,106873,106877,106881,106886,106890,106902,106914,106926,106952,106959,106971,106981,106985,106992,107002,107006,107010,107014,107039,107043,107047,107052,107056,107088,107092,107097,107101,107134,107138,107142,107173,107177,107182,107186,107197,107214,107231,107264,107276,107280,107289,107297,107305,107313,107321,107329,107337,107345,107359,107372,107385,107389,107401,107438,107445,107449,107453,107492,107496,107501,107505,107522,107526,107530,107535,107539,107553,107562,107566,107577,107591,107613,107623,107648,107652,107666,107670,107675,107697,107717,107739,107753,107771,107788,107792,107796,107801,107805,107810,107826,107863,107894,107898,107908,107926,107930,107935,107960,107984,107988,107999,108003,108008,108012,108017,108032,108054,108073,108093,108097,108102,108107,108112,108139,108165,108189,108194,108198,108227,108281,108318,108331,108354,108362,108372,108400,108408,108412,108417,108439,108449,108469,108493,108497,108519,108544,108601,108605,108644,108657,108678,108692,108709,108717,108727,108752,108760,108764,108775,108779,108784,108788,108805,108818,108838,108858,108881,108903,108922,108939,108944,108948,108952,108957,108961,108974,108979,108983,109014,109018,109099,109103,109107,109118,109123,109128,109133,109138,109142,109150,109157,109168,109175,109182,109186,109197,109202,109207,109212,109217,109221,109235,109240,109245,109250,109255,109260,109264,109268,109280,109284,109288,109299,109304,109309,109314,109319,109323,109328,109333,109337,109357,109369,109379,109389,109409,109469,109500,109504,109519,109535,109586,109590,109605,109616,109627,109639,109644,109659,109674,109679,109683,109688,109706,109711,109715,109720,109728,109740,109745,109769,109773,109788,109815,109819,109824,109835,109862,109866,109871,109884,109892,109903,109911,109915,109933,109955,109967,109971,109976,109981,109997,110005,110019,110027,110053,110057,110074,110095,110106,110118,110129,110141,110145,110152,110159,110163,110167,110182,110196,110207,110217,110226,110230,110234,110244,110271,110282,110286,110291,110314,110325,110340,110362,110384],{"__ignoreMap":35},[40,106006,106007],{"class":42,"line":43},[40,106008,106009],{"class":46},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n",[40,106011,106012],{"class":42,"line":50},[40,106013,106014],{"class":46},"# generates tables and figures, so you can iterate further.\n",[40,106016,106017],{"class":42,"line":155},[40,106018,106019],{"class":46},"#\n",[40,106021,106022],{"class":42,"line":168},[40,106023,106024],{"class":46},"# Contents created in the current directory:\n",[40,106026,106027],{"class":42,"line":182},[40,106028,106029],{"class":46},"#  - Sampling_Budget_Qubit_Bitstrings.ipynb      (Colab notebook)\n",[40,106031,106032],{"class":42,"line":195},[40,106033,106034],{"class":46},"#  - summary.csv                                 (summary table)\n",[40,106036,106037],{"class":42,"line":203},[40,106038,106039],{"class":46},"#  - bitstrings_\u003CMODEL>.csv                      (raw bit-strings per model)\n",[40,106041,106042],{"class":42,"line":216},[40,106043,106044],{"class":46},"#  - fig_ones_fraction.png                       (bar chart with Wilson CI)\n",[40,106046,106047],{"class":42,"line":224},[40,106048,106049],{"class":46},"#  - fig_required_shots.png                      (required shots for ±ε)\n",[40,106051,106052],{"class":42,"line":231},[40,106053,106054],{"class":46},"#  - simulation_config.json                      (run parameters)\n",[40,106056,106057],{"class":42,"line":251},[40,106058,228],{"emptyLinePlaceholder":227},[40,106060,106061,106063],{"class":42,"line":256},[40,106062,130],{"class":129},[40,106064,106065],{"class":133}," json\n",[40,106067,106068,106070],{"class":42,"line":269},[40,106069,130],{"class":129},[40,106071,95448],{"class":133},[40,106073,106074,106076,106079,106081],{"class":42,"line":282},[40,106075,171],{"class":129},[40,106077,106078],{"class":133}," dataclasses ",[40,106080,130],{"class":129},[40,106082,106083],{"class":133}," dataclass\n",[40,106085,106086,106088,106090,106092],{"class":42,"line":295},[40,106087,171],{"class":129},[40,106089,174],{"class":133},[40,106091,130],{"class":129},[40,106093,106094],{"class":133}," List, Dict, Tuple\n",[40,106096,106097],{"class":42,"line":308},[40,106098,228],{"emptyLinePlaceholder":227},[40,106100,106101,106103,106105,106107],{"class":42,"line":321},[40,106102,130],{"class":129},[40,106104,134],{"class":133},[40,106106,137],{"class":129},[40,106108,140],{"class":133},[40,106110,106111,106113,106115,106117],{"class":42,"line":334},[40,106112,130],{"class":129},[40,106114,95487],{"class":133},[40,106116,137],{"class":129},[40,106118,95492],{"class":133},[40,106120,106121,106123,106125,106127],{"class":42,"line":347},[40,106122,130],{"class":129},[40,106124,95476],{"class":133},[40,106126,137],{"class":129},[40,106128,165],{"class":133},[40,106130,106131,106133,106136,106138],{"class":42,"line":352},[40,106132,171],{"class":129},[40,106134,106135],{"class":133}," IPython.display ",[40,106137,130],{"class":129},[40,106139,106140],{"class":133}," display, Markdown\n",[40,106142,106143,106145,106148,106150],{"class":42,"line":357},[40,106144,130],{"class":129},[40,106146,106147],{"class":133}," nbformat ",[40,106149,137],{"class":129},[40,106151,106152],{"class":133}," nbf\n",[40,106154,106155],{"class":42,"line":370},[40,106156,228],{"emptyLinePlaceholder":227},[40,106158,106159,106161,106163,106165,106167],{"class":42,"line":376},[40,106160,95555],{"class":133},[40,106162,95558],{"class":244},[40,106164,7250],{"class":133},[40,106166,623],{"class":441},[40,106168,106169],{"class":462}," 200\n",[40,106171,106172],{"class":42,"line":387},[40,106173,228],{"emptyLinePlaceholder":227},[40,106175,106176],{"class":42,"line":392},[40,106177,106178],{"class":46},"# ---------- Helper math ----------\n",[40,106180,106181],{"class":42,"line":397},[40,106182,228],{"emptyLinePlaceholder":227},[40,106184,106185,106187,106190,106192,106194,106196,106198,106200,106202,106204,106206,106208,106210,106212,106214,106216,106219,106222,106224,106226,106228],{"class":42,"line":402},[40,106186,405],{"class":129},[40,106188,106189],{"class":237}," wilson_ci",[40,106191,241],{"class":133},[40,106193,2670],{"class":417},[40,106195,438],{"class":133},[40,106197,456],{"class":441},[40,106199,917],{"class":133},[40,106201,3464],{"class":417},[40,106203,438],{"class":133},[40,106205,456],{"class":441},[40,106207,917],{"class":133},[40,106209,19585],{"class":417},[40,106211,438],{"class":133},[40,106213,442],{"class":441},[40,106215,459],{"class":441},[40,106217,106218],{"class":462}," 1.96",[40,106220,106221],{"class":133},") -> Tuple[",[40,106223,442],{"class":441},[40,106225,917],{"class":133},[40,106227,442],{"class":441},[40,106229,97936],{"class":133},[40,106231,106232],{"class":42,"line":414},[40,106233,505],{"class":244},[40,106235,106236],{"class":42,"line":424},[40,106237,106238],{"class":244},"    Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n",[40,106240,106241],{"class":42,"line":432},[40,106242,106243],{"class":244},"    Returns (low, high). Works for edge cases k=0 or k=n.\n",[40,106245,106246],{"class":42,"line":448},[40,106247,505],{"class":244},[40,106249,106250,106252,106255,106257,106259],{"class":42,"line":468},[40,106251,755],{"class":129},[40,106253,106254],{"class":133}," n ",[40,106256,761],{"class":441},[40,106258,20308],{"class":462},[40,106260,766],{"class":133},[40,106262,106263,106265,106267,106269,106271,106273],{"class":42,"line":486},[40,106264,103186],{"class":129},[40,106266,96957],{"class":133},[40,106268,95027],{"class":462},[40,106270,917],{"class":133},[40,106272,96240],{"class":462},[40,106274,248],{"class":133},[40,106276,106277,106280,106282,106284,106286],{"class":42,"line":502},[40,106278,106279],{"class":133},"    p_hat ",[40,106281,623],{"class":441},[40,106283,758],{"class":133},[40,106285,7610],{"class":441},[40,106287,106288],{"class":133}," n\n",[40,106290,106291,106294,106296,106299,106301,106304,106306,106308,106310,106312],{"class":42,"line":508},[40,106292,106293],{"class":133},"    denom ",[40,106295,623],{"class":441},[40,106297,106298],{"class":462}," 1.0",[40,106300,7357],{"class":441},[40,106302,106303],{"class":133}," (z ",[40,106305,1419],{"class":441},[40,106307,1356],{"class":462},[40,106309,1416],{"class":133},[40,106311,7610],{"class":441},[40,106313,106288],{"class":133},[40,106315,106316,106319,106321,106324,106326,106328,106330,106332,106334,106336,106338,106340,106342,106345,106347],{"class":42,"line":514},[40,106317,106318],{"class":133},"    center ",[40,106320,623],{"class":441},[40,106322,106323],{"class":133}," (p_hat ",[40,106325,1147],{"class":441},[40,106327,106303],{"class":133},[40,106329,1419],{"class":441},[40,106331,1356],{"class":462},[40,106333,1416],{"class":133},[40,106335,7610],{"class":441},[40,106337,96957],{"class":133},[40,106339,1879],{"class":462},[40,106341,7273],{"class":441},[40,106343,106344],{"class":133}," n)) ",[40,106346,7610],{"class":441},[40,106348,106349],{"class":133}," denom\n",[40,106351,106352,106355,106357,106359,106361,106364,106366,106369,106371,106373,106375,106377,106380,106382,106385,106387,106389,106391,106393,106395,106397,106399,106401,106403,106405,106407,106409,106412,106414],{"class":42,"line":519},[40,106353,106354],{"class":133},"    span ",[40,106356,623],{"class":441},[40,106358,106303],{"class":133},[40,106360,1433],{"class":441},[40,106362,106363],{"class":133}," math.",[40,106365,8866],{"class":237},[40,106367,106368],{"class":133},"((p_hat ",[40,106370,1433],{"class":441},[40,106372,96957],{"class":133},[40,106374,1410],{"class":462},[40,106376,104227],{"class":441},[40,106378,106379],{"class":133}," p_hat) ",[40,106381,7610],{"class":441},[40,106383,106384],{"class":133}," n) ",[40,106386,1147],{"class":441},[40,106388,106303],{"class":133},[40,106390,1419],{"class":441},[40,106392,1356],{"class":462},[40,106394,1416],{"class":133},[40,106396,7610],{"class":441},[40,106398,96957],{"class":133},[40,106400,11199],{"class":462},[40,106402,7273],{"class":441},[40,106404,106254],{"class":133},[40,106406,1419],{"class":441},[40,106408,1356],{"class":462},[40,106410,106411],{"class":133},"))) ",[40,106413,7610],{"class":441},[40,106415,106349],{"class":133},[40,106417,106418,106420,106422,106424,106426,106428,106431,106433,106436,106438,106440,106442,106444,106446],{"class":42,"line":525},[40,106419,1546],{"class":129},[40,106421,96957],{"class":133},[40,106423,35767],{"class":441},[40,106425,241],{"class":133},[40,106427,95027],{"class":462},[40,106429,106430],{"class":133},", center ",[40,106432,1407],{"class":441},[40,106434,106435],{"class":133}," span), ",[40,106437,24284],{"class":441},[40,106439,241],{"class":133},[40,106441,96240],{"class":462},[40,106443,106430],{"class":133},[40,106445,1147],{"class":441},[40,106447,106448],{"class":133}," span))\n",[40,106450,106451],{"class":42,"line":531},[40,106452,228],{"emptyLinePlaceholder":227},[40,106454,106455],{"class":42,"line":537},[40,106456,228],{"emptyLinePlaceholder":227},[40,106458,106459,106461,106464,106466,106468,106470,106472,106474,106477,106479,106481,106483,106486,106488,106490,106492,106494,106496,106498,106500,106502],{"class":42,"line":543},[40,106460,405],{"class":129},[40,106462,106463],{"class":237}," shots_for_epsilon",[40,106465,241],{"class":133},[40,106467,12],{"class":417},[40,106469,438],{"class":133},[40,106471,442],{"class":441},[40,106473,917],{"class":133},[40,106475,106476],{"class":417},"epsilon",[40,106478,438],{"class":133},[40,106480,442],{"class":441},[40,106482,459],{"class":441},[40,106484,106485],{"class":462}," 0.01",[40,106487,917],{"class":133},[40,106489,19585],{"class":417},[40,106491,438],{"class":133},[40,106493,442],{"class":441},[40,106495,459],{"class":441},[40,106497,106218],{"class":462},[40,106499,97213],{"class":133},[40,106501,456],{"class":441},[40,106503,766],{"class":133},[40,106505,106506],{"class":42,"line":549},[40,106507,505],{"class":244},[40,106509,106510],{"class":42,"line":555},[40,106511,106512],{"class":244},"    Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n",[40,106514,106515],{"class":42,"line":561},[40,106516,106517],{"class":244},"    at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n",[40,106519,106520],{"class":42,"line":567},[40,106521,505],{"class":244},[40,106523,106524,106527,106529,106531,106533,106535,106538,106540,106542,106544],{"class":42,"line":573},[40,106525,106526],{"class":133},"    p ",[40,106528,623],{"class":441},[40,106530,1518],{"class":441},[40,106532,241],{"class":133},[40,106534,35767],{"class":441},[40,106536,106537],{"class":133},"(p, ",[40,106539,95027],{"class":462},[40,106541,1253],{"class":133},[40,106543,96240],{"class":462},[40,106545,248],{"class":133},[40,106547,106548,106551,106553,106556,106558,106560,106562,106564],{"class":42,"line":579},[40,106549,106550],{"class":133},"    var ",[40,106552,623],{"class":441},[40,106554,106555],{"class":133}," p ",[40,106557,1433],{"class":441},[40,106559,96957],{"class":133},[40,106561,96240],{"class":462},[40,106563,104227],{"class":441},[40,106565,106566],{"class":133}," p)\n",[40,106568,106569,106571,106574,106576,106578],{"class":42,"line":585},[40,106570,755],{"class":129},[40,106572,106573],{"class":133}," var ",[40,106575,761],{"class":441},[40,106577,95680],{"class":462},[40,106579,766],{"class":133},[40,106581,106582,106584],{"class":42,"line":590},[40,106583,103186],{"class":129},[40,106585,1124],{"class":462},[40,106587,106588,106591,106593,106595,106597,106599,106601,106603,106605,106607,106610,106612,106614],{"class":42,"line":596},[40,106589,106590],{"class":133},"    n ",[40,106592,623],{"class":441},[40,106594,106303],{"class":133},[40,106596,1419],{"class":441},[40,106598,1356],{"class":462},[40,106600,1416],{"class":133},[40,106602,1433],{"class":441},[40,106604,106573],{"class":133},[40,106606,7610],{"class":441},[40,106608,106609],{"class":133}," (epsilon ",[40,106611,1419],{"class":441},[40,106613,1356],{"class":462},[40,106615,248],{"class":133},[40,106617,106618,106620,106622,106624,106626,106628,106630,106633,106635],{"class":42,"line":602},[40,106619,1546],{"class":129},[40,106621,103554],{"class":441},[40,106623,241],{"class":133},[40,106625,1410],{"class":462},[40,106627,917],{"class":133},[40,106629,456],{"class":441},[40,106631,106632],{"class":133},"(math.",[40,106634,103541],{"class":237},[40,106636,106637],{"class":133},"(n)))\n",[40,106639,106640],{"class":42,"line":607},[40,106641,228],{"emptyLinePlaceholder":227},[40,106643,106644],{"class":42,"line":612},[40,106645,228],{"emptyLinePlaceholder":227},[40,106647,106648],{"class":42,"line":617},[40,106649,106650],{"class":46},"# ---------- Domain model ----------\n",[40,106652,106653],{"class":42,"line":635},[40,106654,228],{"emptyLinePlaceholder":227},[40,106656,106657,106660,106662,106665,106667,106669],{"class":42,"line":646},[40,106658,106659],{"class":237},"@dataclass",[40,106661,241],{"class":133},[40,106663,106664],{"class":1195},"frozen",[40,106666,623],{"class":441},[40,106668,12926],{"class":462},[40,106670,248],{"class":133},[40,106672,106673,106676,106680],{"class":42,"line":662},[40,106674,106675],{"class":129},"class",[40,106677,106679],{"class":106678},"sU0A5"," QubitModel",[40,106681,766],{"class":133},[40,106683,106684,106687],{"class":42,"line":695},[40,106685,106686],{"class":133},"    name: ",[40,106688,106689],{"class":441},"str\n",[40,106691,106692,106695,106697],{"class":42,"line":701},[40,106693,106694],{"class":133},"    satisfaction_prob: ",[40,106696,442],{"class":441},[40,106698,106699],{"class":46},"  # Probability a single shot meets the analytical criterion (bit = 1)\n",[40,106701,106702,106705],{"class":42,"line":729},[40,106703,106704],{"class":133},"    description: ",[40,106706,106689],{"class":441},[40,106708,106709],{"class":42,"line":752},[40,106710,228],{"emptyLinePlaceholder":227},[40,106712,106713],{"class":42,"line":769},[40,106714,228],{"emptyLinePlaceholder":227},[40,106716,106717,106720,106723,106725],{"class":42,"line":778},[40,106718,106719],{"class":462},"DEFAULT_MODELS",[40,106721,106722],{"class":133},": List[QubitModel] ",[40,106724,623],{"class":441},[40,106726,1217],{"class":133},[40,106728,106729,106732],{"class":42,"line":808},[40,106730,106731],{"class":237},"    QubitModel",[40,106733,411],{"class":133},[40,106735,106736,106739],{"class":42,"line":816},[40,106737,106738],{"class":244},"        \"Toy working qubit\"",[40,106740,445],{"class":133},[40,106742,106743,106746],{"class":42,"line":836},[40,106744,106745],{"class":462},"        0.50",[40,106747,445],{"class":133},[40,106749,106750],{"class":42,"line":844},[40,106751,106752],{"class":244},"        \"Simple demonstrator; about half shots meet the criterion.\"\n",[40,106754,106755],{"class":42,"line":851},[40,106756,21686],{"class":133},[40,106758,106759,106761],{"class":42,"line":871},[40,106760,106731],{"class":237},[40,106762,411],{"class":133},[40,106764,106765,106768],{"class":42,"line":878},[40,106766,106767],{"class":244},"        \"NISQ qubit\"",[40,106769,445],{"class":133},[40,106771,106772,106775],{"class":42,"line":894},[40,106773,106774],{"class":462},"        0.60",[40,106776,445],{"class":133},[40,106778,106779],{"class":42,"line":899},[40,106780,106781],{"class":244},"        \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n",[40,106783,106784],{"class":42,"line":904},[40,106785,21686],{"class":133},[40,106787,106788,106790],{"class":42,"line":926},[40,106789,106731],{"class":237},[40,106791,411],{"class":133},[40,106793,106794,106797],{"class":42,"line":931},[40,106795,106796],{"class":244},"        \"Fault-tolerant (logical)\"",[40,106798,445],{"class":133},[40,106800,106801,106804],{"class":42,"line":937},[40,106802,106803],{"class":462},"        0.90",[40,106805,445],{"class":133},[40,106807,106808],{"class":42,"line":942},[40,106809,106810],{"class":244},"        \"Error-corrected logical qubit; high single-shot success probability.\"\n",[40,106812,106813],{"class":42,"line":953},[40,106814,21686],{"class":133},[40,106816,106817,106819],{"class":42,"line":973},[40,106818,106731],{"class":237},[40,106820,411],{"class":133},[40,106822,106823,106826],{"class":42,"line":1013},[40,106824,106825],{"class":244},"        \"Topologically protected\"",[40,106827,445],{"class":133},[40,106829,106830,106833],{"class":42,"line":1018},[40,106831,106832],{"class":462},"        0.98",[40,106834,445],{"class":133},[40,106836,106837],{"class":42,"line":1028},[40,106838,106839],{"class":244},"        \"Intrinsic protection; very high single-shot success probability.\"\n",[40,106841,106842],{"class":42,"line":1043},[40,106843,21686],{"class":133},[40,106845,106846,106848],{"class":42,"line":1076},[40,106847,106731],{"class":237},[40,106849,411],{"class":133},[40,106851,106852,106855],{"class":42,"line":1081},[40,106853,106854],{"class":244},"        \"Ideal qubit\"",[40,106856,445],{"class":133},[40,106858,106859,106862],{"class":42,"line":1092},[40,106860,106861],{"class":462},"        1.00",[40,106863,445],{"class":133},[40,106865,106866],{"class":42,"line":1097},[40,106867,106868],{"class":244},"        \"Theoretical perfect qubit; always successful in a single shot.\"\n",[40,106870,106871],{"class":42,"line":1111},[40,106872,21686],{"class":133},[40,106874,106875],{"class":42,"line":1116},[40,106876,833],{"class":133},[40,106878,106879],{"class":42,"line":1127},[40,106880,228],{"emptyLinePlaceholder":227},[40,106882,106883],{"class":42,"line":1155},[40,106884,106885],{"class":46},"# ---------- Simulation configuration ----------\n",[40,106887,106888],{"class":42,"line":1174},[40,106889,228],{"emptyLinePlaceholder":227},[40,106891,106892,106895,106898,106900],{"class":42,"line":1179},[40,106893,106894],{"class":462},"CONFIG",[40,106896,106897],{"class":133},": Dict ",[40,106899,623],{"class":441},[40,106901,22777],{"class":133},[40,106903,106904,106907,106909,106912],{"class":42,"line":1204},[40,106905,106906],{"class":244},"    \"random_seed\"",[40,106908,438],{"class":133},[40,106910,106911],{"class":462},"123456",[40,106913,445],{"class":133},[40,106915,106916,106919,106921,106924],{"class":42,"line":1209},[40,106917,106918],{"class":244},"    \"shots_per_model\"",[40,106920,438],{"class":133},[40,106922,106923],{"class":462},"10000",[40,106925,445],{"class":133},[40,106927,106928,106931,106933,106936,106938,106941,106943,106946,106949],{"class":42,"line":1220},[40,106929,106930],{"class":244},"    \"epsilons\"",[40,106932,22911],{"class":133},[40,106934,106935],{"class":462},"0.02",[40,106937,917],{"class":133},[40,106939,106940],{"class":462},"0.01",[40,106942,917],{"class":133},[40,106944,106945],{"class":462},"0.005",[40,106947,106948],{"class":133},"],  ",[40,106950,106951],{"class":46},"# ±2%, ±1%, ±0.5% targets\n",[40,106953,106954,106957],{"class":42,"line":1264},[40,106955,106956],{"class":244},"    \"cost_model\"",[40,106958,22796],{"class":133},[40,106960,106961,106964,106966,106969],{"class":42,"line":1273},[40,106962,106963],{"class":244},"        \"default_cost_per_shot_usd\"",[40,106965,438],{"class":133},[40,106967,106968],{"class":462},"0.00005",[40,106970,445],{"class":133},[40,106972,106973,106976,106978],{"class":42,"line":1278},[40,106974,106975],{"class":244},"        \"fixed_overhead_usd\"",[40,106977,438],{"class":133},[40,106979,106980],{"class":462},"0.10\n",[40,106982,106983],{"class":42,"line":1289},[40,106984,22817],{"class":133},[40,106986,106987,106990],{"class":42,"line":1304},[40,106988,106989],{"class":244},"    \"timing\"",[40,106991,22796],{"class":133},[40,106993,106994,106997,106999],{"class":42,"line":1319},[40,106995,106996],{"class":244},"        \"time_per_shot_seconds\"",[40,106998,438],{"class":133},[40,107000,107001],{"class":462},"2e-6\n",[40,107003,107004],{"class":42,"line":1330},[40,107005,97796],{"class":133},[40,107007,107008],{"class":42,"line":1335},[40,107009,22929],{"class":133},[40,107011,107012],{"class":42,"line":1361},[40,107013,228],{"emptyLinePlaceholder":227},[40,107015,107016,107019,107021,107024,107027,107029,107031,107033,107036],{"class":42,"line":1377},[40,107017,107018],{"class":133},"rng ",[40,107020,623],{"class":441},[40,107022,107023],{"class":133}," np.random.",[40,107025,107026],{"class":237},"default_rng",[40,107028,241],{"class":133},[40,107030,106894],{"class":462},[40,107032,11295],{"class":133},[40,107034,107035],{"class":244},"\"random_seed\"",[40,107037,107038],{"class":133},"])\n",[40,107040,107041],{"class":42,"line":1396},[40,107042,228],{"emptyLinePlaceholder":227},[40,107044,107045],{"class":42,"line":1438},[40,107046,228],{"emptyLinePlaceholder":227},[40,107048,107049],{"class":42,"line":1461},[40,107050,107051],{"class":46},"# ---------- Core simulation ----------\n",[40,107053,107054],{"class":42,"line":1467},[40,107055,228],{"emptyLinePlaceholder":227},[40,107057,107058,107060,107063,107065,107067,107069,107071,107073,107076,107078,107080,107082,107085],{"class":42,"line":1475},[40,107059,405],{"class":129},[40,107061,107062],{"class":237}," simulate_bitstring",[40,107064,241],{"class":133},[40,107066,12],{"class":417},[40,107068,438],{"class":133},[40,107070,442],{"class":441},[40,107072,917],{"class":133},[40,107074,107075],{"class":417},"shots",[40,107077,438],{"class":133},[40,107079,456],{"class":441},[40,107081,917],{"class":133},[40,107083,107084],{"class":417},"rng_",[40,107086,107087],{"class":133},": np.random.Generator) -> np.ndarray:\n",[40,107089,107090],{"class":42,"line":1484},[40,107091,505],{"class":244},[40,107093,107094],{"class":42,"line":1489},[40,107095,107096],{"class":244},"    Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n",[40,107098,107099],{"class":42,"line":1505},[40,107100,505],{"class":244},[40,107102,107103,107105,107108,107111,107113,107115,107118,107121,107123,107126,107128,107130,107132],{"class":42,"line":1510},[40,107104,1546],{"class":129},[40,107106,107107],{"class":133}," rng_.",[40,107109,107110],{"class":237},"binomial",[40,107112,241],{"class":133},[40,107114,1410],{"class":462},[40,107116,107117],{"class":133},", p, ",[40,107119,107120],{"class":1195},"size",[40,107122,623],{"class":441},[40,107124,107125],{"class":133},"shots).",[40,107127,105072],{"class":237},[40,107129,241],{"class":133},[40,107131,456],{"class":441},[40,107133,248],{"class":133},[40,107135,107136],{"class":42,"line":1530},[40,107137,228],{"emptyLinePlaceholder":227},[40,107139,107140],{"class":42,"line":1543},[40,107141,228],{"emptyLinePlaceholder":227},[40,107143,107144,107146,107149,107151,107154,107157,107160,107162,107165,107168,107170],{"class":42,"line":97665},[40,107145,405],{"class":129},[40,107147,107148],{"class":237}," summarize_model",[40,107150,241],{"class":133},[40,107152,107153],{"class":417},"model",[40,107155,107156],{"class":133},": QubitModel, ",[40,107158,107159],{"class":417},"bits",[40,107161,102420],{"class":133},[40,107163,107164],{"class":417},"epsilons",[40,107166,107167],{"class":133},": List[",[40,107169,442],{"class":441},[40,107171,107172],{"class":133},"]) -> Dict:\n",[40,107174,107175],{"class":42,"line":97670},[40,107176,505],{"class":244},[40,107178,107179],{"class":42,"line":97677},[40,107180,107181],{"class":244},"    Calculate summary statistics for a given model and its simulated bitstring.\n",[40,107183,107184],{"class":42,"line":97691},[40,107185,505],{"class":244},[40,107187,107188,107190,107192,107194],{"class":42,"line":97696},[40,107189,106590],{"class":133},[40,107191,623],{"class":441},[40,107193,1135],{"class":441},[40,107195,107196],{"class":133},"(bits.size)\n",[40,107198,107199,107202,107204,107206,107209,107212],{"class":42,"line":97709},[40,107200,107201],{"class":133},"    ones ",[40,107203,623],{"class":441},[40,107205,1135],{"class":441},[40,107207,107208],{"class":133},"(bits.",[40,107210,107211],{"class":237},"sum",[40,107213,21140],{"class":133},[40,107215,107216,107219,107221,107223,107226,107228],{"class":42,"line":97725},[40,107217,107218],{"class":133},"    zeros ",[40,107220,623],{"class":441},[40,107222,1135],{"class":441},[40,107224,107225],{"class":133},"(n ",[40,107227,1407],{"class":441},[40,107229,107230],{"class":133}," ones)\n",[40,107232,107233,107236,107238,107241,107243,107245,107247,107249,107251,107253,107255,107257,107259,107262],{"class":42,"line":97745},[40,107234,107235],{"class":133},"    frac ",[40,107237,623],{"class":441},[40,107239,107240],{"class":133}," ones ",[40,107242,7610],{"class":441},[40,107244,106254],{"class":133},[40,107246,683],{"class":129},[40,107248,106254],{"class":133},[40,107250,689],{"class":441},[40,107252,20308],{"class":462},[40,107254,7524],{"class":129},[40,107256,97933],{"class":441},[40,107258,241],{"class":133},[40,107260,107261],{"class":244},"\"nan\"",[40,107263,248],{"class":133},[40,107265,107266,107269,107271,107273],{"class":42,"line":97753},[40,107267,107268],{"class":133},"    lo, hi ",[40,107270,623],{"class":441},[40,107272,106189],{"class":237},[40,107274,107275],{"class":133},"(ones, n)\n",[40,107277,107278],{"class":42,"line":97759},[40,107279,228],{"emptyLinePlaceholder":227},[40,107281,107282,107285,107287],{"class":42,"line":97764},[40,107283,107284],{"class":133},"    row ",[40,107286,623],{"class":441},[40,107288,22777],{"class":133},[40,107290,107291,107294],{"class":42,"line":97775},[40,107292,107293],{"class":244},"        \"Model\"",[40,107295,107296],{"class":133},": model.name,\n",[40,107298,107299,107302],{"class":42,"line":97781},[40,107300,107301],{"class":244},"        \"Description\"",[40,107303,107304],{"class":133},": model.description,\n",[40,107306,107307,107310],{"class":42,"line":97793},[40,107308,107309],{"class":244},"        \"Configured p (target)\"",[40,107311,107312],{"class":133},": model.satisfaction_prob,\n",[40,107314,107315,107318],{"class":42,"line":97799},[40,107316,107317],{"class":244},"        \"Shots\"",[40,107319,107320],{"class":133},": n,\n",[40,107322,107323,107326],{"class":42,"line":97804},[40,107324,107325],{"class":244},"        \"Ones (successes)\"",[40,107327,107328],{"class":133},": ones,\n",[40,107330,107331,107334],{"class":42,"line":97809},[40,107332,107333],{"class":244},"        \"Zeros (failures)\"",[40,107335,107336],{"class":133},": zeros,\n",[40,107338,107339,107342],{"class":42,"line":97814},[40,107340,107341],{"class":244},"        \"Ones fraction (observed)\"",[40,107343,107344],{"class":133},": frac,\n",[40,107346,107347,107350,107353,107356],{"class":42,"line":97837},[40,107348,107349],{"class":244},"        \"Wilson CI 95",[40,107351,107352],{"class":462},"% lo",[40,107354,107355],{"class":244},"w\"",[40,107357,107358],{"class":133},": lo,\n",[40,107360,107361,107363,107366,107369],{"class":42,"line":97843},[40,107362,107349],{"class":244},[40,107364,107365],{"class":462},"% hi",[40,107367,107368],{"class":244},"gh\"",[40,107370,107371],{"class":133},": hi,\n",[40,107373,107374,107377,107380,107382],{"class":42,"line":97863},[40,107375,107376],{"class":244},"        \"CI width (95%)\"",[40,107378,107379],{"class":133},": hi ",[40,107381,1407],{"class":441},[40,107383,107384],{"class":133}," lo,\n",[40,107386,107387],{"class":42,"line":97868},[40,107388,97796],{"class":133},[40,107390,107391,107393,107396,107398],{"class":42,"line":97873},[40,107392,956],{"class":129},[40,107394,107395],{"class":133}," eps ",[40,107397,674],{"class":129},[40,107399,107400],{"class":133}," epsilons:\n",[40,107402,107403,107406,107408,107411,107413,107415,107418,107420,107422,107424,107426,107429,107431,107433,107435],{"class":42,"line":97883},[40,107404,107405],{"class":133},"        row[",[40,107407,21331],{"class":129},[40,107409,107410],{"class":244},"\"Required shots for ±",[40,107412,1853],{"class":462},[40,107414,456],{"class":441},[40,107416,107417],{"class":133},"(eps",[40,107419,1433],{"class":441},[40,107421,4726],{"class":462},[40,107423,2693],{"class":133},[40,107425,1916],{"class":462},[40,107427,107428],{"class":244},"%\"",[40,107430,7250],{"class":133},[40,107432,623],{"class":441},[40,107434,106463],{"class":237},[40,107436,107437],{"class":133},"(model.satisfaction_prob, eps)\n",[40,107439,107440,107442],{"class":42,"line":97895},[40,107441,1546],{"class":129},[40,107443,107444],{"class":133}," row\n",[40,107446,107447],{"class":42,"line":97907},[40,107448,228],{"emptyLinePlaceholder":227},[40,107450,107451],{"class":42,"line":97919},[40,107452,228],{"emptyLinePlaceholder":227},[40,107454,107455,107457,107460,107462,107464,107466,107468,107470,107473,107475,107477,107479,107482,107484,107486,107488,107490],{"class":42,"line":97939},[40,107456,405],{"class":129},[40,107458,107459],{"class":237}," apply_costs",[40,107461,241],{"class":133},[40,107463,107075],{"class":417},[40,107465,438],{"class":133},[40,107467,456],{"class":441},[40,107469,917],{"class":133},[40,107471,107472],{"class":417},"cost_per_shot",[40,107474,438],{"class":133},[40,107476,442],{"class":441},[40,107478,917],{"class":133},[40,107480,107481],{"class":417},"fixed_overhead",[40,107483,438],{"class":133},[40,107485,442],{"class":441},[40,107487,97213],{"class":133},[40,107489,442],{"class":441},[40,107491,766],{"class":133},[40,107493,107494],{"class":42,"line":97945},[40,107495,505],{"class":244},[40,107497,107498],{"class":42,"line":97969},[40,107499,107500],{"class":244},"    Calculate illustrative cost based on shots and cost model.\n",[40,107502,107503],{"class":42,"line":97992},[40,107504,505],{"class":244},[40,107506,107507,107509,107512,107514,107517,107519],{"class":42,"line":98008},[40,107508,1546],{"class":129},[40,107510,107511],{"class":133}," fixed_overhead ",[40,107513,1147],{"class":441},[40,107515,107516],{"class":133}," shots ",[40,107518,1433],{"class":441},[40,107520,107521],{"class":133}," cost_per_shot\n",[40,107523,107524],{"class":42,"line":98013},[40,107525,228],{"emptyLinePlaceholder":227},[40,107527,107528],{"class":42,"line":98040},[40,107529,228],{"emptyLinePlaceholder":227},[40,107531,107532],{"class":42,"line":98045},[40,107533,107534],{"class":46},"# ---------- Run the simulation ----------\n",[40,107536,107537],{"class":42,"line":98069},[40,107538,228],{"emptyLinePlaceholder":227},[40,107540,107541,107544,107546,107549,107551],{"class":42,"line":98091},[40,107542,107543],{"class":133},"all_bitstrings: Dict[",[40,107545,97210],{"class":441},[40,107547,107548],{"class":133},", np.ndarray] ",[40,107550,623],{"class":441},[40,107552,22762],{"class":133},[40,107554,107555,107558,107560],{"class":42,"line":98112},[40,107556,107557],{"class":133},"summary_rows: List[Dict] ",[40,107559,623],{"class":441},[40,107561,950],{"class":133},[40,107563,107564],{"class":42,"line":98117},[40,107565,228],{"emptyLinePlaceholder":227},[40,107567,107568,107570,107572,107575],{"class":42,"line":98131},[40,107569,7569],{"class":441},[40,107571,241],{"class":133},[40,107573,107574],{"class":244},"\"Running simulation...\"",[40,107576,248],{"class":133},[40,107578,107579,107581,107584,107586,107589],{"class":42,"line":98146},[40,107580,668],{"class":129},[40,107582,107583],{"class":133}," m ",[40,107585,674],{"class":129},[40,107587,107588],{"class":462}," DEFAULT_MODELS",[40,107590,766],{"class":133},[40,107592,107593,107596,107598,107600,107603,107605,107607,107610],{"class":42,"line":98164},[40,107594,107595],{"class":133},"    bits ",[40,107597,623],{"class":441},[40,107599,107062],{"class":237},[40,107601,107602],{"class":133},"(m.satisfaction_prob, ",[40,107604,106894],{"class":462},[40,107606,11295],{"class":133},[40,107608,107609],{"class":244},"\"shots_per_model\"",[40,107611,107612],{"class":133},"], rng)\n",[40,107614,107615,107618,107620],{"class":42,"line":98169},[40,107616,107617],{"class":133},"    all_bitstrings[m.name] ",[40,107619,623],{"class":441},[40,107621,107622],{"class":133}," bits\n",[40,107624,107625,107628,107630,107632,107635,107638,107640,107642,107645],{"class":42,"line":98174},[40,107626,107627],{"class":133},"    summary_rows.",[40,107629,979],{"class":237},[40,107631,241],{"class":133},[40,107633,107634],{"class":237},"summarize_model",[40,107636,107637],{"class":133},"(m, bits, ",[40,107639,106894],{"class":462},[40,107641,11295],{"class":133},[40,107643,107644],{"class":244},"\"epsilons\"",[40,107646,107647],{"class":133},"]))\n",[40,107649,107650],{"class":42,"line":98189},[40,107651,228],{"emptyLinePlaceholder":227},[40,107653,107654,107657,107659,107661,107663],{"class":42,"line":98195},[40,107655,107656],{"class":133},"summary_df ",[40,107658,623],{"class":441},[40,107660,96355],{"class":133},[40,107662,96358],{"class":237},[40,107664,107665],{"class":133},"(summary_rows)\n",[40,107667,107668],{"class":42,"line":98201},[40,107669,228],{"emptyLinePlaceholder":227},[40,107671,107672],{"class":42,"line":98207},[40,107673,107674],{"class":46},"# Add cost estimates\n",[40,107676,107677,107680,107682,107685,107687,107690,107692,107695],{"class":42,"line":98212},[40,107678,107679],{"class":133},"cps ",[40,107681,623],{"class":441},[40,107683,107684],{"class":462}," CONFIG",[40,107686,11295],{"class":133},[40,107688,107689],{"class":244},"\"cost_model\"",[40,107691,794],{"class":133},[40,107693,107694],{"class":244},"\"default_cost_per_shot_usd\"",[40,107696,833],{"class":133},[40,107698,107699,107702,107704,107706,107708,107710,107712,107715],{"class":42,"line":98217},[40,107700,107701],{"class":133},"over ",[40,107703,623],{"class":441},[40,107705,107684],{"class":462},[40,107707,11295],{"class":133},[40,107709,107689],{"class":244},[40,107711,794],{"class":133},[40,107713,107714],{"class":244},"\"fixed_overhead_usd\"",[40,107716,833],{"class":133},[40,107718,107719,107721,107724,107726,107728,107730,107733,107735,107737],{"class":42,"line":98249},[40,107720,683],{"class":129},[40,107722,107723],{"class":133}," cps ",[40,107725,689],{"class":441},[40,107727,20308],{"class":462},[40,107729,21220],{"class":129},[40,107731,107732],{"class":133}," over ",[40,107734,689],{"class":441},[40,107736,20308],{"class":462},[40,107738,766],{"class":133},[40,107740,107741,107744,107747,107749,107751],{"class":42,"line":98276},[40,107742,107743],{"class":133},"    summary_df[",[40,107745,107746],{"class":244},"\"Illustrative cost (USD)\"",[40,107748,7250],{"class":133},[40,107750,623],{"class":441},[40,107752,1217],{"class":133},[40,107754,107755,107758,107760,107762,107765,107768],{"class":42,"line":98302},[40,107756,107757],{"class":237},"        apply_costs",[40,107759,241],{"class":133},[40,107761,456],{"class":441},[40,107763,107764],{"class":133},"(row[",[40,107766,107767],{"class":244},"\"Shots\"",[40,107769,107770],{"class":133},"]), cps, over)\n",[40,107772,107773,107775,107778,107780,107783,107786],{"class":42,"line":98307},[40,107774,1307],{"class":129},[40,107776,107777],{"class":133}," _, row ",[40,107779,674],{"class":129},[40,107781,107782],{"class":133}," summary_df.",[40,107784,107785],{"class":237},"iterrows",[40,107787,12875],{"class":133},[40,107789,107790],{"class":42,"line":98321},[40,107791,97353],{"class":133},[40,107793,107794],{"class":42,"line":98336},[40,107795,228],{"emptyLinePlaceholder":227},[40,107797,107798],{"class":42,"line":98341},[40,107799,107800],{"class":46},"# ---------- Save artifacts ----------\n",[40,107802,107803],{"class":42,"line":98346},[40,107804,228],{"emptyLinePlaceholder":227},[40,107806,107807],{"class":42,"line":98360},[40,107808,107809],{"class":46},"# Save raw bitstrings and summary CSV\n",[40,107811,107812,107814,107817,107819,107822,107824],{"class":42,"line":98365},[40,107813,668],{"class":129},[40,107815,107816],{"class":133}," name, arr ",[40,107818,674],{"class":129},[40,107820,107821],{"class":133}," all_bitstrings.",[40,107823,21661],{"class":237},[40,107825,970],{"class":133},[40,107827,107828,107831,107833,107835,107838,107840,107843,107846,107848,107851,107853,107856,107858,107860],{"class":42,"line":98371},[40,107829,107830],{"class":133},"    out_path ",[40,107832,623],{"class":441},[40,107834,102745],{"class":129},[40,107836,107837],{"class":244},"\"bitstrings_",[40,107839,1853],{"class":462},[40,107841,107842],{"class":133},"name.",[40,107844,107845],{"class":237},"replace",[40,107847,241],{"class":133},[40,107849,107850],{"class":244},"' '",[40,107852,917],{"class":133},[40,107854,107855],{"class":244},"'_'",[40,107857,2693],{"class":133},[40,107859,1916],{"class":462},[40,107861,107862],{"class":244},".csv\"\n",[40,107864,107865,107868,107870,107873,107876,107879,107882,107885,107888,107890,107892],{"class":42,"line":98377},[40,107866,107867],{"class":133},"    pd.",[40,107869,96358],{"class":237},[40,107871,107872],{"class":133},"({",[40,107874,107875],{"class":244},"\"bit\"",[40,107877,107878],{"class":133},": arr}).",[40,107880,107881],{"class":237},"to_csv",[40,107883,107884],{"class":133},"(out_path, ",[40,107886,107887],{"class":1195},"index",[40,107889,623],{"class":441},[40,107891,21017],{"class":462},[40,107893,248],{"class":133},[40,107895,107896],{"class":42,"line":98383},[40,107897,228],{"emptyLinePlaceholder":227},[40,107899,107900,107903,107905],{"class":42,"line":98388},[40,107901,107902],{"class":133},"summary_csv_path ",[40,107904,623],{"class":441},[40,107906,107907],{"class":244}," \"summary.csv\"\n",[40,107909,107910,107913,107915,107918,107920,107922,107924],{"class":42,"line":98393},[40,107911,107912],{"class":133},"summary_df.",[40,107914,107881],{"class":237},[40,107916,107917],{"class":133},"(summary_csv_path, ",[40,107919,107887],{"class":1195},[40,107921,623],{"class":441},[40,107923,21017],{"class":462},[40,107925,248],{"class":133},[40,107927,107928],{"class":42,"line":98398},[40,107929,228],{"emptyLinePlaceholder":227},[40,107931,107932],{"class":42,"line":98412},[40,107933,107934],{"class":46},"# Save config JSON\n",[40,107936,107937,107940,107943,107945,107948,107950,107953,107955,107957],{"class":42,"line":98417},[40,107938,107939],{"class":129},"with",[40,107941,107942],{"class":441}," open",[40,107944,241],{"class":133},[40,107946,107947],{"class":244},"\"simulation_config.json\"",[40,107949,917],{"class":133},[40,107951,107952],{"class":244},"\"w\"",[40,107954,1416],{"class":133},[40,107956,137],{"class":129},[40,107958,107959],{"class":133}," f:\n",[40,107961,107962,107965,107968,107970,107972,107975,107978,107980,107982],{"class":42,"line":98428},[40,107963,107964],{"class":133},"    json.",[40,107966,107967],{"class":237},"dump",[40,107969,241],{"class":133},[40,107971,106894],{"class":462},[40,107973,107974],{"class":133},", f, ",[40,107976,107977],{"class":1195},"indent",[40,107979,623],{"class":441},[40,107981,1879],{"class":462},[40,107983,248],{"class":133},[40,107985,107986],{"class":42,"line":98436},[40,107987,228],{"emptyLinePlaceholder":227},[40,107989,107990,107992,107994,107997],{"class":42,"line":98446},[40,107991,7569],{"class":441},[40,107993,241],{"class":133},[40,107995,107996],{"class":244},"\"Saved data to CSV and JSON files.\"",[40,107998,248],{"class":133},[40,108000,108001],{"class":42,"line":98451},[40,108002,228],{"emptyLinePlaceholder":227},[40,108004,108005],{"class":42,"line":98456},[40,108006,108007],{"class":46},"# ---------- Visualizations ----------\n",[40,108009,108010],{"class":42,"line":98461},[40,108011,228],{"emptyLinePlaceholder":227},[40,108013,108014],{"class":42,"line":98468},[40,108015,108016],{"class":46},"# Figure 1: Observed ones fraction with Wilson CI\n",[40,108018,108019,108022,108025,108027,108030],{"class":42,"line":98484},[40,108020,108021],{"class":133},"plt.style.",[40,108023,108024],{"class":237},"use",[40,108026,241],{"class":133},[40,108028,108029],{"class":244},"'seaborn-v0_8-whitegrid'",[40,108031,248],{"class":133},[40,108033,108034,108036,108038,108040,108042,108044,108046,108048,108050,108052],{"class":42,"line":98493},[40,108035,24430],{"class":133},[40,108037,99929],{"class":237},[40,108039,241],{"class":133},[40,108041,96546],{"class":1195},[40,108043,623],{"class":441},[40,108045,241],{"class":133},[40,108047,22859],{"class":462},[40,108049,917],{"class":133},[40,108051,102025],{"class":462},[40,108053,21594],{"class":133},[40,108055,108056,108059,108061,108063,108066,108068,108070],{"class":42,"line":98502},[40,108057,108058],{"class":133},"x ",[40,108060,623],{"class":441},[40,108062,654],{"class":133},[40,108064,108065],{"class":237},"arange",[40,108067,241],{"class":133},[40,108069,802],{"class":441},[40,108071,108072],{"class":133},"(summary_df))\n",[40,108074,108075,108078,108080,108083,108086,108088,108091],{"class":42,"line":98511},[40,108076,108077],{"class":133},"y ",[40,108079,623],{"class":441},[40,108081,108082],{"class":133}," summary_df[",[40,108084,108085],{"class":244},"\"Ones fraction (observed)\"",[40,108087,21959],{"class":133},[40,108089,108090],{"class":237},"to_numpy",[40,108092,12875],{"class":133},[40,108094,108095],{"class":42,"line":98520},[40,108096,228],{"emptyLinePlaceholder":227},[40,108098,108099],{"class":42,"line":98543},[40,108100,108101],{"class":46},"# --- FIX APPLIED HERE ---\n",[40,108103,108104],{"class":42,"line":98549},[40,108105,108106],{"class":46},"# Take the absolute value to prevent tiny negative numbers from floating-point\n",[40,108108,108109],{"class":42,"line":98554},[40,108110,108111],{"class":46},"# inaccuracies from causing a ValueError in the errorbar plot.\n",[40,108113,108114,108117,108119,108122,108124,108126,108129,108131,108133,108135,108137],{"class":42,"line":98559},[40,108115,108116],{"class":133},"lower_error ",[40,108118,623],{"class":441},[40,108120,108121],{"class":133}," y ",[40,108123,1407],{"class":441},[40,108125,108082],{"class":133},[40,108127,108128],{"class":244},"\"Wilson CI 95",[40,108130,107352],{"class":462},[40,108132,107355],{"class":244},[40,108134,21959],{"class":133},[40,108136,108090],{"class":237},[40,108138,12875],{"class":133},[40,108140,108141,108144,108146,108148,108150,108152,108154,108156,108158,108160,108162],{"class":42,"line":98564},[40,108142,108143],{"class":133},"upper_error ",[40,108145,623],{"class":441},[40,108147,108082],{"class":133},[40,108149,108128],{"class":244},[40,108151,107365],{"class":462},[40,108153,107368],{"class":244},[40,108155,21959],{"class":133},[40,108157,108090],{"class":237},[40,108159,715],{"class":133},[40,108161,1407],{"class":441},[40,108163,108164],{"class":133}," y\n",[40,108166,108167,108170,108172,108174,108176,108179,108181,108184,108186],{"class":42,"line":98574},[40,108168,108169],{"class":133},"yerr ",[40,108171,623],{"class":441},[40,108173,654],{"class":133},[40,108175,22127],{"class":237},[40,108177,108178],{"class":133},"([np.",[40,108180,102572],{"class":237},[40,108182,108183],{"class":133},"(lower_error), np.",[40,108185,102572],{"class":237},[40,108187,108188],{"class":133},"(upper_error)])\n",[40,108190,108191],{"class":42,"line":98583},[40,108192,108193],{"class":46},"# --- END OF FIX ---\n",[40,108195,108196],{"class":42,"line":98592},[40,108197,228],{"emptyLinePlaceholder":227},[40,108199,108200,108202,108205,108208,108210,108212,108215,108217,108220,108222,108225],{"class":42,"line":98597},[40,108201,24430],{"class":133},[40,108203,108204],{"class":237},"bar",[40,108206,108207],{"class":133},"(x, y, ",[40,108209,102663],{"class":1195},[40,108211,623],{"class":441},[40,108213,108214],{"class":244},"'skyblue'",[40,108216,917],{"class":133},[40,108218,108219],{"class":1195},"edgecolor",[40,108221,623],{"class":441},[40,108223,108224],{"class":244},"'black'",[40,108226,248],{"class":133},[40,108228,108229,108231,108234,108236,108239,108241,108244,108247,108249,108252,108254,108257,108259,108261,108263,108265,108267,108270,108272,108275,108277,108279],{"class":42,"line":98603},[40,108230,24430],{"class":133},[40,108232,108233],{"class":237},"errorbar",[40,108235,108207],{"class":133},[40,108237,108238],{"class":1195},"yerr",[40,108240,623],{"class":441},[40,108242,108243],{"class":133},"yerr, ",[40,108245,108246],{"class":1195},"fmt",[40,108248,623],{"class":441},[40,108250,108251],{"class":244},"\"none\"",[40,108253,917],{"class":133},[40,108255,108256],{"class":1195},"capsize",[40,108258,623],{"class":441},[40,108260,96558],{"class":462},[40,108262,917],{"class":133},[40,108264,102663],{"class":1195},[40,108266,623],{"class":441},[40,108268,108269],{"class":244},"\"black\"",[40,108271,917],{"class":133},[40,108273,108274],{"class":1195},"elinewidth",[40,108276,623],{"class":441},[40,108278,104523],{"class":462},[40,108280,248],{"class":133},[40,108282,108283,108285,108287,108290,108293,108295,108297,108299,108302,108304,108306,108308,108311,108313,108316],{"class":42,"line":98619},[40,108284,24430],{"class":133},[40,108286,24575],{"class":237},[40,108288,108289],{"class":133},"(x, summary_df[",[40,108291,108292],{"class":244},"\"Model\"",[40,108294,21959],{"class":133},[40,108296,23424],{"class":237},[40,108298,12998],{"class":133},[40,108300,108301],{"class":1195},"rotation",[40,108303,623],{"class":441},[40,108305,86901],{"class":462},[40,108307,917],{"class":133},[40,108309,108310],{"class":1195},"ha",[40,108312,623],{"class":441},[40,108314,108315],{"class":244},"\"right\"",[40,108317,248],{"class":133},[40,108319,108320,108322,108324,108326,108329],{"class":42,"line":98630},[40,108321,24430],{"class":133},[40,108323,24635],{"class":237},[40,108325,241],{"class":133},[40,108327,108328],{"class":244},"\"Observed Ones Fraction\"",[40,108330,248],{"class":133},[40,108332,108333,108335,108337,108339,108342,108344,108347,108349,108352],{"class":42,"line":98645},[40,108334,24430],{"class":133},[40,108336,24649],{"class":237},[40,108338,241],{"class":133},[40,108340,108341],{"class":244},"\"Observed Single-Shot Success Rates with 95% Wilson CI\"",[40,108343,917],{"class":133},[40,108345,108346],{"class":1195},"fontsize",[40,108348,623],{"class":441},[40,108350,108351],{"class":462},"14",[40,108353,248],{"class":133},[40,108355,108356,108358,108360],{"class":42,"line":98655},[40,108357,24430],{"class":133},[40,108359,96683],{"class":237},[40,108361,12875],{"class":133},[40,108363,108364,108367,108369],{"class":42,"line":98662},[40,108365,108366],{"class":133},"fig1_path ",[40,108368,623],{"class":441},[40,108370,108371],{"class":244}," \"fig_ones_fraction.png\"\n",[40,108373,108374,108376,108379,108382,108384,108386,108388,108390,108393,108395,108398],{"class":42,"line":98671},[40,108375,24430],{"class":133},[40,108377,108378],{"class":237},"savefig",[40,108380,108381],{"class":133},"(fig1_path, ",[40,108383,102030],{"class":1195},[40,108385,623],{"class":441},[40,108387,102035],{"class":462},[40,108389,917],{"class":133},[40,108391,108392],{"class":1195},"bbox_inches",[40,108394,623],{"class":441},[40,108396,108397],{"class":244},"'tight'",[40,108399,248],{"class":133},[40,108401,108402,108404,108406],{"class":42,"line":98676},[40,108403,24430],{"class":133},[40,108405,24667],{"class":237},[40,108407,12875],{"class":133},[40,108409,108410],{"class":42,"line":98698},[40,108411,228],{"emptyLinePlaceholder":227},[40,108413,108414],{"class":42,"line":98703},[40,108415,108416],{"class":46},"# Figure 2: Required shots at several ±epsilon targets\n",[40,108418,108419,108421,108423,108425,108427,108429,108431,108433,108435,108437],{"class":42,"line":98720},[40,108420,24430],{"class":133},[40,108422,99929],{"class":237},[40,108424,241],{"class":133},[40,108426,96546],{"class":1195},[40,108428,623],{"class":441},[40,108430,241],{"class":133},[40,108432,22859],{"class":462},[40,108434,917],{"class":133},[40,108436,102025],{"class":462},[40,108438,21594],{"class":133},[40,108440,108441,108444,108446],{"class":42,"line":98737},[40,108442,108443],{"class":133},"width ",[40,108445,623],{"class":441},[40,108447,108448],{"class":462}," 0.2\n",[40,108450,108451,108453,108455,108457,108459,108461,108463,108465,108467],{"class":42,"line":98742},[40,108452,108058],{"class":133},[40,108454,623],{"class":441},[40,108456,654],{"class":133},[40,108458,108065],{"class":237},[40,108460,241],{"class":133},[40,108462,802],{"class":441},[40,108464,241],{"class":133},[40,108466,106719],{"class":462},[40,108468,21594],{"class":133},[40,108470,108471,108474,108476,108478,108481,108483,108486,108488,108491],{"class":42,"line":98755},[40,108472,108473],{"class":133},"colors ",[40,108475,623],{"class":441},[40,108477,7471],{"class":133},[40,108479,108480],{"class":244},"'#ff9999'",[40,108482,1862],{"class":133},[40,108484,108485],{"class":244},"'#66b3ff'",[40,108487,1862],{"class":133},[40,108489,108490],{"class":244},"'#99ff99'",[40,108492,833],{"class":133},[40,108494,108495],{"class":42,"line":98769},[40,108496,228],{"emptyLinePlaceholder":227},[40,108498,108499,108501,108504,108506,108508,108510,108512,108514,108516],{"class":42,"line":98774},[40,108500,668],{"class":129},[40,108502,108503],{"class":133}," i, eps ",[40,108505,674],{"class":129},[40,108507,997],{"class":441},[40,108509,241],{"class":133},[40,108511,106894],{"class":462},[40,108513,11295],{"class":133},[40,108515,107644],{"class":244},[40,108517,108518],{"class":133},"]):\n",[40,108520,108521,108524,108526,108528,108531,108534,108536,108538,108540,108542],{"class":42,"line":98782},[40,108522,108523],{"class":133},"    req ",[40,108525,623],{"class":441},[40,108527,7471],{"class":133},[40,108529,108530],{"class":237},"shots_for_epsilon",[40,108532,108533],{"class":133},"(m.satisfaction_prob, eps) ",[40,108535,668],{"class":129},[40,108537,107583],{"class":133},[40,108539,674],{"class":129},[40,108541,107588],{"class":462},[40,108543,833],{"class":133},[40,108545,108546,108549,108551,108554,108556,108558,108560,108563,108565,108567,108569,108572,108574,108577,108580,108582,108584,108586,108588,108590,108593,108595,108597,108599],{"class":42,"line":98787},[40,108547,108548],{"class":133},"    plt.",[40,108550,108204],{"class":237},[40,108552,108553],{"class":133},"(x ",[40,108555,1147],{"class":441},[40,108557,7290],{"class":133},[40,108559,1433],{"class":441},[40,108561,108562],{"class":133}," width, req, width, ",[40,108564,20062],{"class":1195},[40,108566,623],{"class":441},[40,108568,21331],{"class":129},[40,108570,108571],{"class":244},"\"±",[40,108573,1853],{"class":462},[40,108575,108576],{"class":133},"eps",[40,108578,108579],{"class":129},":.1%",[40,108581,1916],{"class":462},[40,108583,21334],{"class":244},[40,108585,917],{"class":133},[40,108587,102663],{"class":1195},[40,108589,623],{"class":441},[40,108591,108592],{"class":133},"colors[i], ",[40,108594,108219],{"class":1195},[40,108596,623],{"class":441},[40,108598,108224],{"class":244},[40,108600,248],{"class":133},[40,108602,108603],{"class":42,"line":98792},[40,108604,228],{"emptyLinePlaceholder":227},[40,108606,108607,108609,108611,108613,108615,108618,108620,108622,108624,108626,108628,108630,108632,108634,108636,108638,108640,108642],{"class":42,"line":98802},[40,108608,24430],{"class":133},[40,108610,24575],{"class":237},[40,108612,108553],{"class":133},[40,108614,1147],{"class":441},[40,108616,108617],{"class":133}," width, [m.name ",[40,108619,668],{"class":129},[40,108621,107583],{"class":133},[40,108623,674],{"class":129},[40,108625,107588],{"class":462},[40,108627,799],{"class":133},[40,108629,108301],{"class":1195},[40,108631,623],{"class":441},[40,108633,86901],{"class":462},[40,108635,917],{"class":133},[40,108637,108310],{"class":1195},[40,108639,623],{"class":441},[40,108641,108315],{"class":244},[40,108643,248],{"class":133},[40,108645,108646,108648,108650,108652,108655],{"class":42,"line":98810},[40,108647,24430],{"class":133},[40,108649,24635],{"class":237},[40,108651,241],{"class":133},[40,108653,108654],{"class":244},"\"Required Shots (Log Scale)\"",[40,108656,248],{"class":133},[40,108658,108659,108661,108663,108665,108668,108670,108672,108674,108676],{"class":42,"line":98818},[40,108660,24430],{"class":133},[40,108662,24649],{"class":237},[40,108664,241],{"class":133},[40,108666,108667],{"class":244},"\"Approximate Shots to Estimate Success Rate within ±ε\"",[40,108669,917],{"class":133},[40,108671,108346],{"class":1195},[40,108673,623],{"class":441},[40,108675,108351],{"class":462},[40,108677,248],{"class":133},[40,108679,108680,108682,108685,108687,108690],{"class":42,"line":98829},[40,108681,24430],{"class":133},[40,108683,108684],{"class":237},"yscale",[40,108686,241],{"class":133},[40,108688,108689],{"class":244},"'log'",[40,108691,248],{"class":133},[40,108693,108694,108696,108698,108700,108702,108704,108707],{"class":42,"line":98840},[40,108695,24430],{"class":133},[40,108697,24612],{"class":237},[40,108699,241],{"class":133},[40,108701,24649],{"class":1195},[40,108703,623],{"class":441},[40,108705,108706],{"class":244},"\"Error Tolerance\"",[40,108708,248],{"class":133},[40,108710,108711,108713,108715],{"class":42,"line":98849},[40,108712,24430],{"class":133},[40,108714,96683],{"class":237},[40,108716,12875],{"class":133},[40,108718,108719,108722,108724],{"class":42,"line":98855},[40,108720,108721],{"class":133},"fig2_path ",[40,108723,623],{"class":441},[40,108725,108726],{"class":244}," \"fig_required_shots.png\"\n",[40,108728,108729,108731,108733,108736,108738,108740,108742,108744,108746,108748,108750],{"class":42,"line":98865},[40,108730,24430],{"class":133},[40,108732,108378],{"class":237},[40,108734,108735],{"class":133},"(fig2_path, ",[40,108737,102030],{"class":1195},[40,108739,623],{"class":441},[40,108741,102035],{"class":462},[40,108743,917],{"class":133},[40,108745,108392],{"class":1195},[40,108747,623],{"class":441},[40,108749,108397],{"class":244},[40,108751,248],{"class":133},[40,108753,108754,108756,108758],{"class":42,"line":98875},[40,108755,24430],{"class":133},[40,108757,24667],{"class":237},[40,108759,12875],{"class":133},[40,108761,108762],{"class":42,"line":98884},[40,108763,228],{"emptyLinePlaceholder":227},[40,108765,108766,108768,108770,108773],{"class":42,"line":98893},[40,108767,7569],{"class":441},[40,108769,241],{"class":133},[40,108771,108772],{"class":244},"\"Generated and saved plots.\"",[40,108774,248],{"class":133},[40,108776,108777],{"class":42,"line":98904},[40,108778,228],{"emptyLinePlaceholder":227},[40,108780,108781],{"class":42,"line":98916},[40,108782,108783],{"class":46},"# ---------- Display interactive table ----------\n",[40,108785,108786],{"class":42,"line":98927},[40,108787,228],{"emptyLinePlaceholder":227},[40,108789,108790,108793,108795,108798,108800,108803],{"class":42,"line":98932},[40,108791,108792],{"class":237},"display",[40,108794,241],{"class":133},[40,108796,108797],{"class":237},"Markdown",[40,108799,241],{"class":133},[40,108801,108802],{"class":244},"\"##  Qubit Sampling Budget - Summary\"",[40,108804,21594],{"class":133},[40,108806,108807,108809,108812,108815],{"class":42,"line":98937},[40,108808,108792],{"class":237},[40,108810,108811],{"class":133},"(summary_df.style.",[40,108813,108814],{"class":237},"format",[40,108816,108817],{"class":133},"({\n",[40,108819,108820,108823,108825,108827,108829,108832,108834,108836],{"class":42,"line":98942},[40,108821,108822],{"class":244},"    \"Configured p (target)\"",[40,108824,438],{"class":133},[40,108826,21334],{"class":244},[40,108828,1853],{"class":462},[40,108830,108831],{"class":129},":.2f",[40,108833,1916],{"class":462},[40,108835,21334],{"class":244},[40,108837,445],{"class":133},[40,108839,108840,108843,108845,108847,108849,108852,108854,108856],{"class":42,"line":98952},[40,108841,108842],{"class":244},"    \"Ones fraction (observed)\"",[40,108844,438],{"class":133},[40,108846,21334],{"class":244},[40,108848,1853],{"class":462},[40,108850,108851],{"class":129},":.4f",[40,108853,1916],{"class":462},[40,108855,21334],{"class":244},[40,108857,445],{"class":133},[40,108859,108860,108863,108865,108867,108869,108871,108873,108875,108877,108879],{"class":42,"line":98959},[40,108861,108862],{"class":244},"    \"Wilson CI 95",[40,108864,107352],{"class":462},[40,108866,107355],{"class":244},[40,108868,438],{"class":133},[40,108870,21334],{"class":244},[40,108872,1853],{"class":462},[40,108874,108851],{"class":129},[40,108876,1916],{"class":462},[40,108878,21334],{"class":244},[40,108880,445],{"class":133},[40,108882,108883,108885,108887,108889,108891,108893,108895,108897,108899,108901],{"class":42,"line":98967},[40,108884,108862],{"class":244},[40,108886,107365],{"class":462},[40,108888,107368],{"class":244},[40,108890,438],{"class":133},[40,108892,21334],{"class":244},[40,108894,1853],{"class":462},[40,108896,108851],{"class":129},[40,108898,1916],{"class":462},[40,108900,21334],{"class":244},[40,108902,445],{"class":133},[40,108904,108905,108908,108910,108912,108914,108916,108918,108920],{"class":42,"line":98978},[40,108906,108907],{"class":244},"    \"CI width (95%)\"",[40,108909,438],{"class":133},[40,108911,21334],{"class":244},[40,108913,1853],{"class":462},[40,108915,108851],{"class":129},[40,108917,1916],{"class":462},[40,108919,21334],{"class":244},[40,108921,445],{"class":133},[40,108923,108924,108927,108929,108931,108933,108935,108937],{"class":42,"line":98989},[40,108925,108926],{"class":244},"    \"Illustrative cost (USD)\"",[40,108928,438],{"class":133},[40,108930,99993],{"class":244},[40,108932,1853],{"class":462},[40,108934,108831],{"class":129},[40,108936,1916],{"class":462},[40,108938,100039],{"class":244},[40,108940,108941],{"class":42,"line":98998},[40,108942,108943],{"class":133},"}))\n",[40,108945,108946],{"class":42,"line":99004},[40,108947,228],{"emptyLinePlaceholder":227},[40,108949,108950],{"class":42,"line":99014},[40,108951,228],{"emptyLinePlaceholder":227},[40,108953,108954],{"class":42,"line":99024},[40,108955,108956],{"class":46},"# ---------- Build a ready-to-run Colab notebook ----------\n",[40,108958,108959],{"class":42,"line":99033},[40,108960,228],{"emptyLinePlaceholder":227},[40,108962,108963,108966,108968,108971],{"class":42,"line":99042},[40,108964,108965],{"class":133},"md_intro ",[40,108967,623],{"class":441},[40,108969,108970],{"class":129}," r",[40,108972,108973],{"class":96619},"\"\"\"\n",[40,108975,108976],{"class":42,"line":99051},[40,108977,108978],{"class":46},"# Sampling Budget & Bit-String Analogy for Qubit Classes\n",[40,108980,108981],{"class":42,"line":99063},[40,108982,228],{"emptyLinePlaceholder":227},[40,108984,108985,108987,108990,108992,108995,108997,108999,109001,109004,109006,109009,109011],{"class":42,"line":99075},[40,108986,1419],{"class":462},[40,108988,108989],{"class":96619},"High school level ",[40,108991,241],{"class":462},[40,108993,108994],{"class":96619},"concise",[40,108996,2693],{"class":462},[40,108998,24039],{"class":96619},[40,109000,1419],{"class":462},[40,109002,109003],{"class":96619}," We treat each measurement ",[40,109005,241],{"class":462},[40,109007,109008],{"class":96619},"“single shot”",[40,109010,2693],{"class":462},[40,109012,109013],{"class":96619}," 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",[40,109015,109016],{"class":42,"line":99084},[40,109017,228],{"emptyLinePlaceholder":227},[40,109019,109020,109022,109025,109027,109029,109031,109033,109035,109038,109040,109043,109045,109048,109050,109053,109055,109058,109060,109063,109065,109068,109071,109074,109077,109079,109081,109084,109086,109088,109091,109094,109096],{"class":42,"line":99089},[40,109021,1419],{"class":462},[40,109023,109024],{"class":96619},"Graduate level ",[40,109026,241],{"class":462},[40,109028,108994],{"class":96619},[40,109030,2693],{"class":462},[40,109032,24039],{"class":96619},[40,109034,1419],{"class":462},[40,109036,109037],{"class":96619}," We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ ",[40,109039,241],{"class":462},[40,109041,109042],{"class":96619},"noisy intermediate-scale quantum",[40,109044,2693],{"class":462},[40,109046,109047],{"class":96619},", fault-tolerant ",[40,109049,241],{"class":462},[40,109051,109052],{"class":96619},"logical",[40,109054,2693],{"class":462},[40,109056,109057],{"class":96619},", topologically protected, and ideal. We report Wilson score intervals ",[40,109059,241],{"class":462},[40,109061,109062],{"class":96619},"95%",[40,109064,2693],{"class":462},[40,109066,109067],{"class":96619}," for observed proportions and estimate the required shot budget $n ",[40,109069,109070],{"class":441},"\\a",[40,109072,109073],{"class":96619},"pprox z^2",[40,109075,109076],{"class":441},"\\,",[40,109078,12],{"class":96619},[40,109080,241],{"class":462},[40,109082,109083],{"class":96619},"1-p",[40,109085,2693],{"class":462},[40,109087,7610],{"class":96619},[40,109089,109090],{"class":441},"\\e",[40,109092,109093],{"class":96619},"psilon^2$ to bound absolute error by ±$",[40,109095,109090],{"class":441},[40,109097,109098],{"class":96619},"psilon$ at ~95% confidence.\n",[40,109100,109101],{"class":42,"line":99094},[40,109102,108973],{"class":96619},[40,109104,109105],{"class":42,"line":99099},[40,109106,228],{"emptyLinePlaceholder":227},[40,109108,109109,109112,109114,109116],{"class":42,"line":99109},[40,109110,109111],{"class":133},"code_config ",[40,109113,623],{"class":441},[40,109115,108970],{"class":129},[40,109117,108973],{"class":96619},[40,109119,109120],{"class":42,"line":99116},[40,109121,109122],{"class":46},"# @title Configuration\n",[40,109124,109125],{"class":42,"line":99123},[40,109126,109127],{"class":96619},"from dataclasses import dataclass\n",[40,109129,109130],{"class":42,"line":99134},[40,109131,109132],{"class":96619},"from typing import List, Dict\n",[40,109134,109135],{"class":42,"line":99145},[40,109136,109137],{"class":96619},"import numpy as np\n",[40,109139,109140],{"class":42,"line":99154},[40,109141,228],{"emptyLinePlaceholder":227},[40,109143,109144,109147],{"class":42,"line":99160},[40,109145,109146],{"class":96619},"RANDOM_SEED = 579345  ",[40,109148,109149],{"class":46},"# @param {type:\"number\"}\n",[40,109151,109152,109155],{"class":42,"line":99170},[40,109153,109154],{"class":96619},"SHOTS_PER_MODEL = 10000  ",[40,109156,109149],{"class":46},[40,109158,109159,109162,109165],{"class":42,"line":99178},[40,109160,109161],{"class":96619},"EPSILONS = ",[40,109163,109164],{"class":462},"[0.02, 0.01, 0.005]",[40,109166,109167],{"class":46},"  # @param\n",[40,109169,109170,109173],{"class":42,"line":99185},[40,109171,109172],{"class":96619},"COST_PER_SHOT_USD = 0.00005  ",[40,109174,109149],{"class":46},[40,109176,109177,109180],{"class":42,"line":99192},[40,109178,109179],{"class":96619},"FIXED_OVERHEAD_USD = 0.10    ",[40,109181,109149],{"class":46},[40,109183,109184],{"class":42,"line":99201},[40,109185,228],{"emptyLinePlaceholder":227},[40,109187,109188,109190,109192,109195],{"class":42,"line":99210},[40,109189,106659],{"class":96619},[40,109191,241],{"class":462},[40,109193,109194],{"class":96619},"frozen=True",[40,109196,248],{"class":462},[40,109198,109199],{"class":42,"line":99219},[40,109200,109201],{"class":96619},"class QubitModel:\n",[40,109203,109204],{"class":42,"line":99228},[40,109205,109206],{"class":96619},"    name: str\n",[40,109208,109209],{"class":42,"line":99239},[40,109210,109211],{"class":96619},"    satisfaction_prob: float\n",[40,109213,109214],{"class":42,"line":99251},[40,109215,109216],{"class":96619},"    description: str\n",[40,109218,109219],{"class":42,"line":99263},[40,109220,228],{"emptyLinePlaceholder":227},[40,109222,109223,109226,109229,109232],{"class":42,"line":99272},[40,109224,109225],{"class":96619},"MODELS: List",[40,109227,109228],{"class":462},"[QubitModel]",[40,109230,109231],{"class":96619}," = ",[40,109233,109234],{"class":462},"[\n",[40,109236,109237],{"class":42,"line":99277},[40,109238,109239],{"class":462},"    QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n",[40,109241,109242],{"class":42,"line":99282},[40,109243,109244],{"class":462},"    QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n",[40,109246,109247],{"class":42,"line":99287},[40,109248,109249],{"class":462},"    QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n",[40,109251,109252],{"class":42,"line":99297},[40,109253,109254],{"class":462},"    QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n",[40,109256,109257],{"class":42,"line":99304},[40,109258,109259],{"class":462},"    QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n",[40,109261,109262],{"class":42,"line":99311},[40,109263,833],{"class":462},[40,109265,109266],{"class":42,"line":99323},[40,109267,228],{"emptyLinePlaceholder":227},[40,109269,109270,109273,109275,109278],{"class":42,"line":99334},[40,109271,109272],{"class":96619},"rng = np.random.default_rng",[40,109274,241],{"class":462},[40,109276,109277],{"class":96619},"RANDOM_SEED",[40,109279,248],{"class":462},[40,109281,109282],{"class":42,"line":99343},[40,109283,108973],{"class":96619},[40,109285,109286],{"class":42,"line":99349},[40,109287,228],{"emptyLinePlaceholder":227},[40,109289,109290,109293,109295,109297],{"class":42,"line":99365},[40,109291,109292],{"class":133},"code_lib_and_run ",[40,109294,623],{"class":441},[40,109296,108970],{"class":129},[40,109298,108973],{"class":96619},[40,109300,109301],{"class":42,"line":99370},[40,109302,109303],{"class":46},"# @title Run Simulation & Display Results\n",[40,109305,109306],{"class":42,"line":99379},[40,109307,109308],{"class":96619},"import math\n",[40,109310,109311],{"class":42,"line":99389},[40,109312,109313],{"class":96619},"from typing import Tuple, Dict\n",[40,109315,109316],{"class":42,"line":99398},[40,109317,109318],{"class":96619},"import pandas as pd\n",[40,109320,109321],{"class":42,"line":99407},[40,109322,109137],{"class":96619},[40,109324,109325],{"class":42,"line":99413},[40,109326,109327],{"class":96619},"import matplotlib.pyplot as plt\n",[40,109329,109330],{"class":42,"line":99422},[40,109331,109332],{"class":96619},"from IPython.display import display, Markdown\n",[40,109334,109335],{"class":42,"line":99434},[40,109336,228],{"emptyLinePlaceholder":227},[40,109338,109339,109342,109344,109347,109349,109352,109355],{"class":42,"line":99447},[40,109340,109341],{"class":96619},"def wilson_ci",[40,109343,241],{"class":462},[40,109345,109346],{"class":96619},"k: int, n: int, z: float = 1.96",[40,109348,2693],{"class":462},[40,109350,109351],{"class":96619}," -> Tuple",[40,109353,109354],{"class":462},"[float, float]",[40,109356,766],{"class":96619},[40,109358,109359,109362,109364,109367],{"class":42,"line":99459},[40,109360,109361],{"class":96619},"    if n == 0: return ",[40,109363,241],{"class":462},[40,109365,109366],{"class":96619},"0.0, 1.0",[40,109368,248],{"class":462},[40,109370,109371,109374,109376],{"class":42,"line":99464},[40,109372,109373],{"class":96619},"    p_hat, z2_n = k \u002F n, z",[40,109375,1419],{"class":462},[40,109377,109378],{"class":96619},"2 \u002F n\n",[40,109380,109381,109384,109386],{"class":42,"line":99469},[40,109382,109383],{"class":96619},"    denom = 1.0 ",[40,109385,1147],{"class":462},[40,109387,109388],{"class":96619}," z2_n\n",[40,109390,109391,109394,109396,109399,109401,109404,109406],{"class":42,"line":99474},[40,109392,109393],{"class":96619},"    center = ",[40,109395,241],{"class":462},[40,109397,109398],{"class":96619},"p_hat ",[40,109400,1147],{"class":462},[40,109402,109403],{"class":96619}," z2_n \u002F 2",[40,109405,2693],{"class":462},[40,109407,109408],{"class":96619}," \u002F denom\n",[40,109410,109411,109414,109416,109419,109421,109424,109426,109428,109430,109432,109435,109437,109440,109442,109445,109447,109450,109452,109455,109457,109460,109462,109464,109467],{"class":42,"line":99493},[40,109412,109413],{"class":96619},"    span = ",[40,109415,241],{"class":462},[40,109417,109418],{"class":96619},"z ",[40,109420,1433],{"class":462},[40,109422,109423],{"class":96619}," math.sqrt",[40,109425,241],{"class":462},[40,109427,109398],{"class":96619},[40,109429,1433],{"class":462},[40,109431,96957],{"class":462},[40,109433,109434],{"class":96619},"1 - p_hat",[40,109436,2693],{"class":462},[40,109438,109439],{"class":96619}," \u002F n ",[40,109441,1147],{"class":462},[40,109443,109444],{"class":96619}," z",[40,109446,1419],{"class":462},[40,109448,109449],{"class":96619},"2 \u002F ",[40,109451,241],{"class":462},[40,109453,109454],{"class":96619},"4 ",[40,109456,1433],{"class":462},[40,109458,109459],{"class":96619}," n",[40,109461,1419],{"class":462},[40,109463,1879],{"class":96619},[40,109465,109466],{"class":462},")))",[40,109468,109408],{"class":96619},[40,109470,109471,109474,109476,109478,109480,109483,109485,109488,109490,109493,109495,109498],{"class":42,"line":99499},[40,109472,109473],{"class":96619},"    return ",[40,109475,241],{"class":462},[40,109477,35767],{"class":96619},[40,109479,241],{"class":462},[40,109481,109482],{"class":96619},"0.0, center - span",[40,109484,2693],{"class":462},[40,109486,109487],{"class":96619},", min",[40,109489,241],{"class":462},[40,109491,109492],{"class":96619},"1.0, center ",[40,109494,1147],{"class":462},[40,109496,109497],{"class":96619}," span",[40,109499,21594],{"class":462},[40,109501,109502],{"class":42,"line":99508},[40,109503,228],{"emptyLinePlaceholder":227},[40,109505,109506,109509,109511,109514,109516],{"class":42,"line":99515},[40,109507,109508],{"class":96619},"def shots_for_epsilon",[40,109510,241],{"class":462},[40,109512,109513],{"class":96619},"p: float, epsilon: float = 0.01, z: float = 1.96",[40,109515,2693],{"class":462},[40,109517,109518],{"class":96619}," -> int:\n",[40,109520,109521,109524,109526,109528,109530,109532],{"class":42,"line":99522},[40,109522,109523],{"class":96619},"    if p ",[40,109525,1433],{"class":462},[40,109527,96957],{"class":462},[40,109529,109083],{"class":96619},[40,109531,2693],{"class":462},[40,109533,109534],{"class":96619}," == 0: return 1\n",[40,109536,109537,109540,109542,109545,109547,109550,109552,109554,109556,109559,109561,109563,109565,109567,109569,109572,109575,109577,109579,109581,109583],{"class":42,"line":99530},[40,109538,109539],{"class":96619},"    return max",[40,109541,241],{"class":462},[40,109543,109544],{"class":96619},"1, int",[40,109546,241],{"class":462},[40,109548,109549],{"class":96619},"math.ceil",[40,109551,101793],{"class":462},[40,109553,19585],{"class":96619},[40,109555,1419],{"class":462},[40,109557,109558],{"class":96619},"2 ",[40,109560,1433],{"class":462},[40,109562,106555],{"class":96619},[40,109564,1433],{"class":462},[40,109566,96957],{"class":462},[40,109568,109083],{"class":96619},[40,109570,109571],{"class":462},"))",[40,109573,109574],{"class":96619}," \u002F ",[40,109576,241],{"class":462},[40,109578,106476],{"class":96619},[40,109580,1419],{"class":462},[40,109582,1879],{"class":96619},[40,109584,109585],{"class":462},"))))\n",[40,109587,109588],{"class":42,"line":99541},[40,109589,228],{"emptyLinePlaceholder":227},[40,109591,109592,109595,109597,109600,109602],{"class":42,"line":99552},[40,109593,109594],{"class":96619},"def summarize_model",[40,109596,241],{"class":462},[40,109598,109599],{"class":96619},"model, bits: np.ndarray, epsilons",[40,109601,2693],{"class":462},[40,109603,109604],{"class":96619}," -> Dict:\n",[40,109606,109607,109610,109612,109614],{"class":42,"line":99563},[40,109608,109609],{"class":96619},"    n = len",[40,109611,241],{"class":462},[40,109613,107159],{"class":96619},[40,109615,248],{"class":462},[40,109617,109618,109621,109623,109625],{"class":42,"line":99568},[40,109619,109620],{"class":96619},"    ones = sum",[40,109622,241],{"class":462},[40,109624,107159],{"class":96619},[40,109626,248],{"class":462},[40,109628,109629,109632,109634,109637],{"class":42,"line":99577},[40,109630,109631],{"class":96619},"    lo, hi = wilson_ci",[40,109633,241],{"class":462},[40,109635,109636],{"class":96619},"ones, n",[40,109638,248],{"class":462},[40,109640,109641],{"class":42,"line":99584},[40,109642,109643],{"class":96619},"    row = {\n",[40,109645,109646,109649,109651,109654,109656],{"class":42,"line":99591},[40,109647,109648],{"class":96619},"        \"Model\": model.name, \"p ",[40,109650,241],{"class":462},[40,109652,109653],{"class":96619},"target",[40,109655,2693],{"class":462},[40,109657,109658],{"class":96619},"\": model.satisfaction_prob, \"Shots\": n,\n",[40,109660,109661,109664,109666,109669,109671],{"class":42,"line":99599},[40,109662,109663],{"class":96619},"        \"Ones\": ones, \"p ",[40,109665,241],{"class":462},[40,109667,109668],{"class":96619},"observed",[40,109670,2693],{"class":462},[40,109672,109673],{"class":96619},"\": ones \u002F n if n > 0 else 0,\n",[40,109675,109676],{"class":42,"line":99620},[40,109677,109678],{"class":96619},"        \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n",[40,109680,109681],{"class":42,"line":99631},[40,109682,97796],{"class":96619},[40,109684,109685],{"class":42,"line":99643},[40,109686,109687],{"class":96619},"    for eps in epsilons:\n",[40,109689,109690,109693,109696,109699,109701,109704],{"class":42,"line":99654},[40,109691,109692],{"class":96619},"        row",[40,109694,109695],{"class":462},"[f\"Shots for ±{eps:.1%}\"]",[40,109697,109698],{"class":96619}," = shots_for_epsilon",[40,109700,241],{"class":462},[40,109702,109703],{"class":96619},"model.satisfaction_prob, eps",[40,109705,248],{"class":462},[40,109707,109708],{"class":42,"line":99665},[40,109709,109710],{"class":96619},"    return row\n",[40,109712,109713],{"class":42,"line":99670},[40,109714,228],{"emptyLinePlaceholder":227},[40,109716,109717],{"class":42,"line":99675},[40,109718,109719],{"class":46},"# --- Run Simulation ---\n",[40,109721,109722,109725],{"class":42,"line":99680},[40,109723,109724],{"class":96619},"summary_rows = ",[40,109726,109727],{"class":462},"[summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\n",[40,109729,109730,109733,109735,109738],{"class":42,"line":99694},[40,109731,109732],{"class":96619},"summary_df = pd.DataFrame",[40,109734,241],{"class":462},[40,109736,109737],{"class":96619},"summary_rows",[40,109739,248],{"class":462},[40,109741,109742],{"class":42,"line":99706},[40,109743,109744],{"class":96619},"if COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n",[40,109746,109747,109750,109753,109756,109758,109761,109764,109766],{"class":42,"line":99712},[40,109748,109749],{"class":96619},"    summary_df",[40,109751,109752],{"class":462},"[\"Cost (USD)\"]",[40,109754,109755],{"class":96619}," = FIXED_OVERHEAD_USD ",[40,109757,1147],{"class":462},[40,109759,109760],{"class":96619}," summary_df",[40,109762,109763],{"class":462},"[\"Shots\"]",[40,109765,7273],{"class":462},[40,109767,109768],{"class":96619}," COST_PER_SHOT_USD\n",[40,109770,109771],{"class":42,"line":99717},[40,109772,228],{"emptyLinePlaceholder":227},[40,109774,109775,109777,109779,109781,109783,109785],{"class":42,"line":99725},[40,109776,108792],{"class":96619},[40,109778,241],{"class":462},[40,109780,108797],{"class":96619},[40,109782,241],{"class":462},[40,109784,21334],{"class":96619},[40,109786,109787],{"class":46},"### Simulation Summary\"))\n",[40,109789,109790,109792,109794,109797,109799,109802,109804,109807,109809,109812],{"class":42,"line":99735},[40,109791,108792],{"class":96619},[40,109793,241],{"class":462},[40,109795,109796],{"class":96619},"summary_df.style.format",[40,109798,241],{"class":462},[40,109800,109801],{"class":96619},"precision=4",[40,109803,2693],{"class":462},[40,109805,109806],{"class":96619},".background_gradient",[40,109808,241],{"class":462},[40,109810,109811],{"class":96619},"cmap='viridis', subset=",[40,109813,109814],{"class":462},"['p (observed)', 'CI width']))\n",[40,109816,109817],{"class":42,"line":99740},[40,109818,228],{"emptyLinePlaceholder":227},[40,109820,109821],{"class":42,"line":99752},[40,109822,109823],{"class":46},"# --- Plotting ---\n",[40,109825,109826,109829,109831,109833],{"class":42,"line":99763},[40,109827,109828],{"class":96619},"plt.style.use",[40,109830,241],{"class":462},[40,109832,108029],{"class":96619},[40,109834,248],{"class":462},[40,109836,109837,109840,109842,109845,109847,109850,109852,109855,109857,109860],{"class":42,"line":99773},[40,109838,109839],{"class":96619},"fig, ",[40,109841,241],{"class":462},[40,109843,109844],{"class":96619},"ax1, ax2",[40,109846,2693],{"class":462},[40,109848,109849],{"class":96619}," = plt.subplots",[40,109851,241],{"class":462},[40,109853,109854],{"class":96619},"1, 2, figsize=",[40,109856,241],{"class":462},[40,109858,109859],{"class":96619},"16, 6",[40,109861,21594],{"class":462},[40,109863,109864],{"class":42,"line":99784},[40,109865,228],{"emptyLinePlaceholder":227},[40,109867,109868],{"class":42,"line":99789},[40,109869,109870],{"class":46},"# Plot 1\n",[40,109872,109873,109876,109879,109882],{"class":42,"line":99794},[40,109874,109875],{"class":96619},"y = summary_df",[40,109877,109878],{"class":462},"[\"p (observed)\"]",[40,109880,109881],{"class":96619},".to_numpy",[40,109883,12875],{"class":462},[40,109885,109886,109889],{"class":42,"line":99806},[40,109887,109888],{"class":96619},"lower_error = y - summary_df",[40,109890,109891],{"class":462},"[\"CI 95% low\"]\n",[40,109893,109894,109897,109900],{"class":42,"line":99817},[40,109895,109896],{"class":96619},"upper_error = summary_df",[40,109898,109899],{"class":462},"[\"CI 95% high\"]",[40,109901,109902],{"class":96619}," - y\n",[40,109904,109905,109908],{"class":42,"line":99826},[40,109906,109907],{"class":96619},"yerr = np.vstack",[40,109909,109910],{"class":462},"([np.abs(lower_error), np.abs(upper_error)])\n",[40,109912,109913],{"class":42,"line":99837},[40,109914,228],{"emptyLinePlaceholder":227},[40,109916,109917,109920,109922,109925,109928,109931],{"class":42,"line":99842},[40,109918,109919],{"class":96619},"ax1.bar",[40,109921,241],{"class":462},[40,109923,109924],{"class":96619},"summary_df",[40,109926,109927],{"class":462},"[\"Model\"]",[40,109929,109930],{"class":96619},", y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black'",[40,109932,248],{"class":462},[40,109934,109935,109938,109940,109943,109945,109948,109950,109953],{"class":42,"line":99847},[40,109936,109937],{"class":96619},"ax1.set_title",[40,109939,241],{"class":462},[40,109941,109942],{"class":96619},"\"Observed Success Rate ",[40,109944,241],{"class":462},[40,109946,109947],{"class":96619},"95% Wilson CI",[40,109949,2693],{"class":462},[40,109951,109952],{"class":96619},"\", fontsize=14",[40,109954,248],{"class":462},[40,109956,109957,109960,109962,109965],{"class":42,"line":99857},[40,109958,109959],{"class":96619},"ax1.tick_params",[40,109961,241],{"class":462},[40,109963,109964],{"class":96619},"axis='x', rotation=30, labelsize=10",[40,109966,248],{"class":462},[40,109968,109969],{"class":42,"line":99863},[40,109970,228],{"emptyLinePlaceholder":227},[40,109972,109973],{"class":42,"line":99877},[40,109974,109975],{"class":46},"# Plot 2\n",[40,109977,109978],{"class":42,"line":99882},[40,109979,109980],{"class":96619},"width = 0.2\n",[40,109982,109983,109986,109988,109990,109992,109995],{"class":42,"line":99892},[40,109984,109985],{"class":96619},"x = np.arange",[40,109987,241],{"class":462},[40,109989,802],{"class":96619},[40,109991,241],{"class":462},[40,109993,109994],{"class":96619},"MODELS",[40,109996,21594],{"class":462},[40,109998,109999,110002],{"class":42,"line":99897},[40,110000,110001],{"class":96619},"colors = ",[40,110003,110004],{"class":462},"['#ff9999','#66b3ff','#99ff99']\n",[40,110006,110007,110010,110012,110015,110017],{"class":42,"line":99909},[40,110008,110009],{"class":96619},"for i, eps in enumerate",[40,110011,241],{"class":462},[40,110013,110014],{"class":96619},"EPSILONS",[40,110016,2693],{"class":462},[40,110018,766],{"class":96619},[40,110020,110021,110024],{"class":42,"line":99914},[40,110022,110023],{"class":96619},"    req = summary_df",[40,110025,110026],{"class":462},"[f\"Shots for ±{eps:.1%}\"]\n",[40,110028,110029,110032,110034,110036,110038,110040,110042,110045,110048,110051],{"class":42,"line":99919},[40,110030,110031],{"class":96619},"    ax2.bar",[40,110033,241],{"class":462},[40,110035,108058],{"class":96619},[40,110037,1147],{"class":462},[40,110039,7290],{"class":96619},[40,110041,1433],{"class":462},[40,110043,110044],{"class":96619}," width, req, width, label=f\"±{eps:.1%}\", color=colors",[40,110046,110047],{"class":462},"[i]",[40,110049,110050],{"class":96619},", edgecolor='black'",[40,110052,248],{"class":462},[40,110054,110055],{"class":42,"line":99942},[40,110056,228],{"emptyLinePlaceholder":227},[40,110058,110059,110062,110064,110066,110068,110071],{"class":42,"line":99947},[40,110060,110061],{"class":96619},"ax2.set_xticks",[40,110063,241],{"class":462},[40,110065,108058],{"class":96619},[40,110067,1147],{"class":462},[40,110069,110070],{"class":96619}," width, summary_df",[40,110072,110073],{"class":462},"[\"Model\"])\n",[40,110075,110076,110079,110081,110084,110086,110089,110091,110093],{"class":42,"line":99970},[40,110077,110078],{"class":96619},"ax2.set_ylabel",[40,110080,241],{"class":462},[40,110082,110083],{"class":96619},"\"Required Shots ",[40,110085,241],{"class":462},[40,110087,110088],{"class":96619},"Log Scale",[40,110090,2693],{"class":462},[40,110092,21334],{"class":96619},[40,110094,248],{"class":462},[40,110096,110097,110100,110102,110104],{"class":42,"line":99981},[40,110098,110099],{"class":96619},"ax2.set_yscale",[40,110101,241],{"class":462},[40,110103,108689],{"class":96619},[40,110105,248],{"class":462},[40,110107,110108,110111,110113,110116],{"class":42,"line":100020},[40,110109,110110],{"class":96619},"ax2.set_title",[40,110112,241],{"class":462},[40,110114,110115],{"class":96619},"\"Shots Needed for ±ε Tolerance\", fontsize=14",[40,110117,248],{"class":462},[40,110119,110120,110123,110125,110127],{"class":42,"line":100030},[40,110121,110122],{"class":96619},"ax2.tick_params",[40,110124,241],{"class":462},[40,110126,109964],{"class":96619},[40,110128,248],{"class":462},[40,110130,110131,110134,110136,110139],{"class":42,"line":100042},[40,110132,110133],{"class":96619},"ax2.legend",[40,110135,241],{"class":462},[40,110137,110138],{"class":96619},"title=\"Tolerance\"",[40,110140,248],{"class":462},[40,110142,110143],{"class":42,"line":100062},[40,110144,228],{"emptyLinePlaceholder":227},[40,110146,110147,110150],{"class":42,"line":100080},[40,110148,110149],{"class":96619},"plt.tight_layout",[40,110151,12875],{"class":462},[40,110153,110154,110157],{"class":42,"line":100085},[40,110155,110156],{"class":96619},"plt.show",[40,110158,12875],{"class":462},[40,110160,110161],{"class":42,"line":100090},[40,110162,108973],{"class":96619},[40,110164,110165],{"class":42,"line":100102},[40,110166,228],{"emptyLinePlaceholder":227},[40,110168,110169,110172,110174,110177,110180],{"class":42,"line":100112},[40,110170,110171],{"class":133},"nb ",[40,110173,623],{"class":441},[40,110175,110176],{"class":133}," nbf.v4.",[40,110178,110179],{"class":237},"new_notebook",[40,110181,12875],{"class":133},[40,110183,110184,110187,110190,110192,110194],{"class":42,"line":100122},[40,110185,110186],{"class":133},"nb[",[40,110188,110189],{"class":244},"\"cells\"",[40,110191,7250],{"class":133},[40,110193,623],{"class":441},[40,110195,1217],{"class":133},[40,110197,110198,110201,110204],{"class":42,"line":100127},[40,110199,110200],{"class":133},"    nbf.v4.",[40,110202,110203],{"class":237},"new_markdown_cell",[40,110205,110206],{"class":133},"(md_intro),\n",[40,110208,110209,110211,110214],{"class":42,"line":100132},[40,110210,110200],{"class":133},[40,110212,110213],{"class":237},"new_code_cell",[40,110215,110216],{"class":133},"(code_config),\n",[40,110218,110219,110221,110223],{"class":42,"line":100140},[40,110220,110200],{"class":133},[40,110222,110213],{"class":237},[40,110224,110225],{"class":133},"(code_lib_and_run),\n",[40,110227,110228],{"class":42,"line":100150},[40,110229,833],{"class":133},[40,110231,110232],{"class":42,"line":100165},[40,110233,228],{"emptyLinePlaceholder":227},[40,110235,110236,110239,110241],{"class":42,"line":100176},[40,110237,110238],{"class":133},"nb_path ",[40,110240,623],{"class":441},[40,110242,110243],{"class":244}," \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\n",[40,110245,110246,110248,110250,110253,110255,110257,110260,110262,110265,110267,110269],{"class":42,"line":100188},[40,110247,107939],{"class":129},[40,110249,107942],{"class":441},[40,110251,110252],{"class":133},"(nb_path, ",[40,110254,107952],{"class":244},[40,110256,917],{"class":133},[40,110258,110259],{"class":1195},"encoding",[40,110261,623],{"class":441},[40,110263,110264],{"class":244},"\"utf-8\"",[40,110266,1416],{"class":133},[40,110268,137],{"class":129},[40,110270,107959],{"class":133},[40,110272,110273,110276,110279],{"class":42,"line":100193},[40,110274,110275],{"class":133},"    nbf.",[40,110277,110278],{"class":237},"write",[40,110280,110281],{"class":133},"(nb, f)\n",[40,110283,110284],{"class":42,"line":100200},[40,110285,228],{"emptyLinePlaceholder":227},[40,110287,110288],{"class":42,"line":100209},[40,110289,110290],{"class":46},"# ---------- Final confirmation ----------\n",[40,110292,110293,110295,110297,110299,110301,110303,110305,110308,110310,110312],{"class":42,"line":100223},[40,110294,7569],{"class":441},[40,110296,241],{"class":133},[40,110298,21334],{"class":244},[40,110300,100036],{"class":441},[40,110302,21334],{"class":244},[40,110304,7357],{"class":441},[40,110306,110307],{"class":244}," \"=\"",[40,110309,1433],{"class":441},[40,110311,87293],{"class":462},[40,110313,248],{"class":133},[40,110315,110316,110318,110320,110323],{"class":42,"line":100234},[40,110317,7569],{"class":441},[40,110319,241],{"class":133},[40,110321,110322],{"class":244},"\"SCRIPT COMPLETE\"",[40,110324,248],{"class":133},[40,110326,110327,110329,110331,110334,110336,110338],{"class":42,"line":100246},[40,110328,7569],{"class":441},[40,110330,241],{"class":133},[40,110332,110333],{"class":244},"\"=\"",[40,110335,1433],{"class":441},[40,110337,87293],{"class":462},[40,110339,248],{"class":133},[40,110341,110342,110344,110346,110348,110351,110353,110356,110358,110360],{"class":42,"line":100251},[40,110343,7569],{"class":441},[40,110345,241],{"class":133},[40,110347,21331],{"class":129},[40,110349,110350],{"class":244},"\"Google Colab notebook saved to: ",[40,110352,1853],{"class":462},[40,110354,110355],{"class":133},"nb_path",[40,110357,1916],{"class":462},[40,110359,21334],{"class":244},[40,110361,248],{"class":133},[40,110363,110364,110366,110368,110370,110373,110375,110378,110380,110382],{"class":42,"line":100256},[40,110365,7569],{"class":441},[40,110367,241],{"class":133},[40,110369,21331],{"class":129},[40,110371,110372],{"class":244},"\"Summary data saved to: ",[40,110374,1853],{"class":462},[40,110376,110377],{"class":133},"summary_csv_path",[40,110379,1916],{"class":462},[40,110381,21334],{"class":244},[40,110383,248],{"class":133},[40,110385,110386,110388,110390,110392,110395,110397,110400,110402,110404,110406,110409,110411,110413],{"class":42,"line":100266},[40,110387,7569],{"class":441},[40,110389,241],{"class":133},[40,110391,21331],{"class":129},[40,110393,110394],{"class":244},"\"Plots saved to: ",[40,110396,1853],{"class":462},[40,110398,110399],{"class":133},"fig1_path",[40,110401,1916],{"class":462},[40,110403,917],{"class":244},[40,110405,1853],{"class":462},[40,110407,110408],{"class":133},"fig2_path",[40,110410,1916],{"class":462},[40,110412,21334],{"class":244},[40,110414,248],{"class":133},[30,110416,110419],{"className":110417,"code":110418,"language":6083,"meta":35},[7575],"Running simulation...\nSaved data to CSV and JSON files.\n",[37,110420,110418],{"__ignoreMap":35},[8229,110422],{"alt":96696,"src":110423},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.webp",[8229,110425],{"alt":101602,"src":110426},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-02.webp",[30,110428,110431],{"className":110429,"code":110430,"language":6083,"meta":35},[7575],"Generated and saved plots.\n",[37,110432,110430],{"__ignoreMap":35},[30,110434,110437],{"className":110435,"code":110436,"language":6083,"meta":35},[7575],"==================================================\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",[37,110438,110436],{"__ignoreMap":35},[110440,110441],"hr",{},[55,110443,40611],{"id":40610},[26425,110445,110446,110453,110460,110467,110474,110481,110488,110495,110502,110509,110516,110523,110530,110541,110548,110555],{},[101,110447,110448,110449,110452],{},"Agresti, A. and Coull, B.A. (1998) 'Approximate is better than “exact” for interval estimation of binomial proportions', ",[15,110450,110451],{},"The American Statistician",", 52(2), pp. 119-126.",[101,110454,110455,110456,110459],{},"Brown, L.D., Cai, T.T. and DasGupta, A. (2001) 'Interval Estimation for a Binomial Proportion', ",[15,110457,110458],{},"Statistical Science",", 16(2), pp. 101-133.",[101,110461,110462,110463,110466],{},"Cochran, W.G. (1977) ",[15,110464,110465],{},"Sampling Techniques",". 3rd edn. New York: John Wiley & Sons.",[101,110468,110469,110470,110473],{},"Gottesman, D. (2009) 'An introduction to quantum error correction and fault-tolerant quantum computation', in ",[15,110471,110472],{},"Quantum Information Science and Its Contributions to Mathematics, Proceedings of Symposia in Applied Mathematics",", 72, pp. 13-58.",[101,110475,110476,110477,110480],{},"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', ",[15,110478,110479],{},"Nature",", 585, pp. 357–362.",[101,110482,110483,110484,110487],{},"Hunter, J.D. (2007) 'Matplotlib: A 2D Graphics Environment', ",[15,110485,110486],{},"Computing in Science & Engineering",", 9(3), pp. 90-95.",[101,110489,110490,110491,110494],{},"Kitaev, A.Y. (2003) 'Fault-tolerant quantum computation by anyons', ",[15,110492,110493],{},"Annals of Physics",", 303(1), pp. 2-30.",[101,110496,110497,110498,110501],{},"McKinney, W. (2010) 'Data Structures for Statistical Computing in Python', in ",[15,110499,110500],{},"Proceedings of the 9th Python in Science Conference",", pp. 56-61.",[101,110503,110504,110505,110508],{},"Nayak, C., Simon, S.H., Stern, A., Freedman, M. and Das Sarma, S. (2008) 'Non-Abelian anyons and topological quantum computation', ",[15,110506,110507],{},"Reviews of Modern Physics",", 80(3), pp. 1083-1159.",[101,110510,110511,110512,110515],{},"Newman, M. (2013) ",[15,110513,110514],{},"Computational Physics",". CreateSpace Independent Publishing Platform.",[101,110517,110518,110519,110522],{},"Nielsen, M.A. and Chuang, I.L. (2010) ",[15,110520,110521],{},"Quantum Computation and Quantum Information: 10th Anniversary Edition",". Cambridge: Cambridge University Press.",[101,110524,110525,110526,110529],{},"Preskill, J. (2018) 'Quantum Computing in the NISQ era and beyond', ",[15,110527,110528],{},"Quantum",", 2, p. 79.",[101,110531,110532,110533,110536,110537,1823],{},"The pandas development team (2020) ",[15,110534,110535],{},"pandas-dev\u002Fpandas: Pandas",". Zenodo. Available at: ",[19,110538,110540],{"href":110539},"https:\u002F\u002Fwww.google.com\u002Fsearch?q=https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134","https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134",[101,110542,110543,110544,110547],{},"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', ",[15,110545,110546],{},"Nature Methods",", 17, pp. 261–272.",[101,110549,110550,110551,110554],{},"Wilde, M.M. (2017) ",[15,110552,110553],{},"Quantum Information Theory",". 2nd edn. Cambridge: Cambridge University Press.",[101,110556,110557,110558,110561],{},"Wilson, E.B. 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