[{"data":1,"prerenderedAt":95977},["ShallowReactive",2],{"blog-post-expert-notes\u002Frabi-oscillation-visualization":3,"sibling-dives-expert-notes\u002Frabi-oscillation-visualization":26492,"learn-track-expert-notes\u002Frabi-oscillation-visualization":95976},{"id":4,"title":5,"authors":6,"body":8,"breadcrumb":26456,"builders":26461,"byline":26472,"challenge":26474,"courseAuthor":26474,"courseLead":26474,"dek":26474,"description":26475,"draft":26476,"extension":26477,"eyebrow":26478,"finish":26474,"fork":26474,"hero":26474,"heroAlt":26474,"heroCta":26474,"heroImage":26474,"kind":26479,"lessonCount":26474,"meta":26480,"navigation":15768,"newsItems":26474,"next":26474,"ogImage":26481,"order":26474,"outcomes":26474,"path":26482,"publishDate":26483,"readingTime":26484,"related":26485,"relatedProjects":26474,"seo":26486,"stem":26487,"tags":26488,"track":26474,"trackName":26474,"__hash__":26491},"blog\u002Fblog\u002Fexpert-notes\u002Frabi-oscillation-visualization.md","Rabi Oscillations: Visualizing Excited-State Probability",[7],"onri-jay-benally",{"type":9,"value":10,"toc":26439},"minimark",[11,24,27,186,250,255,258,1767,1771,1852,2068,2071,2223,2226,2354,2463,2563,2566,2570,2573,2695,2756,2918,2921,3144,3147,3906,3943,3947,3950,4297,4427,4739,4742,4969,5087,5091,5162,5634,5637,5878,5881,6060,6115,6316,6319,6322,6573,6606,6886,6889,7042,7074,7195,7198,7465,7468,7586,7590,7593,7961,7964,8099,8107,8111,8114,8256,8259,8801,8804,9030,9206,9210,9284,9558,9560,9625,9657,9824,9895,10129,10132,10135,10502,10504,10828,10831,10835,10838,10945,10977,11161,11204,11208,11339,11588,11591,11984,11987,12198,12201,12205,12678,13138,13257,13376,13379,13383,13485,13488,13587,13670,13855,13858,14058,14061,14162,14559,14563,14566,15051,15054,15058,15061,15323,15326,15539,15542,15646,15649,15693,16053,16061,16583,16675,16681,16859,16865,17040,17045,22055,22059,22065,22795,22799,23457,23461,23465,24635,24641,24842,25133,25137,25951,25955,26413,26417,26421,26425,26435],[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\u002Fgithub.com\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FRabi_Oscillation_Visualization_for_Excited_State_Probability.ipynb","the original notebook"," and shared under CC-BY-4.0. The text, code, and figures below are Onri's, as published.",[12,25,26],{},"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",[28,29,32],"span",{"className":30},[31],"katex-display",[28,33,36,86],{"className":34},[35],"katex",[28,37,40],{"className":38},[39],"katex-mathml",[41,42,45],"math",{"xmlns":43,"display":44},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[46,47,48,81],"semantics",{},[49,50,51,61,66,69,73,76,79],"mrow",{},[52,53,54,58],"msub",{},[55,56,57],"mi",{},"P",[55,59,60],{},"e",[62,63,65],"mo",{"stretchy":64},"false","(",[55,67,68],{},"f",[62,70,72],{"separator":71},"true",",",[55,74,75],{},"τ",[62,77,78],{"stretchy":64},")",[62,80,72],{"separator":71},[82,83,85],"annotation",{"encoding":84},"application\u002Fx-tex","P_e(f,\\tau),",[28,87,90],{"className":88,"ariaHidden":71},[89],"katex-html",[28,91,94,99,158,162,166,170,175,179,183],{"className":92},[93],"base",[28,95],{"className":96,"style":98},[97],"strut","height:1em;vertical-align:-0.25em;",[28,100,103,108],{"className":101},[102],"mord",[28,104,57],{"className":105,"style":107},[102,106],"mathnormal","margin-right:0.1389em;",[28,109,112],{"className":110},[111],"msupsub",[28,113,117,149],{"className":114},[115,116],"vlist-t","vlist-t2",[28,118,121,144],{"className":119},[120],"vlist-r",[28,122,126],{"className":123,"style":125},[124],"vlist","height:0.1514em;",[28,127,129,134],{"style":128},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[28,130],{"className":131,"style":133},[132],"pstrut","height:2.7em;",[28,135,141],{"className":136},[137,138,139,140],"sizing","reset-size6","size3","mtight",[28,142,60],{"className":143},[102,106,140],[28,145,148],{"className":146},[147],"vlist-s","​",[28,150,152],{"className":151},[120],[28,153,156],{"className":154,"style":155},[124],"height:0.15em;",[28,157],{},[28,159,65],{"className":160},[161],"mopen",[28,163,68],{"className":164,"style":165},[102,106],"margin-right:0.1076em;",[28,167,72],{"className":168},[169],"mpunct",[28,171],{"className":172,"style":174},[173],"mspace","margin-right:0.1667em;",[28,176,75],{"className":177,"style":178},[102,106],"margin-right:0.1132em;",[28,180,78],{"className":181},[182],"mclose",[28,184,72],{"className":185},[169],[12,187,188,189,218,219,249],{},"where ",[28,190,192,205],{"className":191},[35],[28,193,195],{"className":194},[39],[41,196,197],{"xmlns":43},[46,198,199,203],{},[49,200,201],{},[55,202,68],{},[82,204,68],{"encoding":84},[28,206,208],{"className":207,"ariaHidden":71},[89],[28,209,211,215],{"className":210},[93],[28,212],{"className":213,"style":214},[97],"height:0.8889em;vertical-align:-0.1944em;",[28,216,68],{"className":217,"style":165},[102,106]," denotes the applied drive frequency and ",[28,220,222,236],{"className":221},[35],[28,223,225],{"className":224},[39],[41,226,227],{"xmlns":43},[46,228,229,233],{},[49,230,231],{},[55,232,75],{},[82,234,235],{"encoding":84},"\\tau",[28,237,239],{"className":238,"ariaHidden":71},[89],[28,240,242,246],{"className":241},[93],[28,243],{"className":244,"style":245},[97],"height:0.4306em;",[28,247,75],{"className":248,"style":178},[102,106]," 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.",[251,252,254],"h2",{"id":253},"nomenclature-and-definitions","Nomenclature and Definitions",[12,256,257],{},"The following table defines the physical and mathematical quantities that recur throughout the visualization workflow.",[259,260,261,274],"table",{},[262,263,264],"thead",{},[265,266,267,271],"tr",{},[268,269,270],"th",{},"Symbol or Term",[268,272,273],{},"Meaning",[275,276,277,328,374,421,551,629,710,745,973,1070,1109,1156,1311,1392,1618,1653,1751,1759],"tbody",{},[265,278,279,325],{},[280,281,282],"td",{},[28,283,285,306],{"className":284},[35],[28,286,288],{"className":287},[39],[41,289,290],{"xmlns":43},[46,291,292,303],{},[49,293,294,297,300],{},[62,295,296],{"stretchy":64},"∣",[55,298,299],{},"g",[62,301,302],{"stretchy":64},"⟩",[82,304,305],{"encoding":84},"\\lvert g\\rangle",[28,307,309],{"className":308,"ariaHidden":71},[89],[28,310,312,315,318,322],{"className":311},[93],[28,313],{"className":314,"style":98},[97],[28,316,296],{"className":317},[161],[28,319,299],{"className":320,"style":321},[102,106],"margin-right:0.0359em;",[28,323,302],{"className":324},[182],[280,326,327],{},"Ground state of the qubit",[265,329,330,371],{},[280,331,332],{},[28,333,335,353],{"className":334},[35],[28,336,338],{"className":337},[39],[41,339,340],{"xmlns":43},[46,341,342,350],{},[49,343,344,346,348],{},[62,345,296],{"stretchy":64},[55,347,60],{},[62,349,302],{"stretchy":64},[82,351,352],{"encoding":84},"\\lvert e\\rangle",[28,354,356],{"className":355,"ariaHidden":71},[89],[28,357,359,362,365,368],{"className":358},[93],[28,360],{"className":361,"style":98},[97],[28,363,296],{"className":364},[161],[28,366,60],{"className":367},[102,106],[28,369,302],{"className":370},[182],[280,372,373],{},"Excited state of the 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",[28,475,477,494],{"className":476},[35],[28,478,480],{"className":479},[39],[41,481,482],{"xmlns":43},[46,483,484,492],{},[49,485,486,488,490],{},[62,487,296],{"stretchy":64},[55,489,299],{},[62,491,302],{"stretchy":64},[82,493,305],{"encoding":84},[28,495,497],{"className":496,"ariaHidden":71},[89],[28,498,500,503,506,509],{"className":499},[93],[28,501],{"className":502,"style":98},[97],[28,504,296],{"className":505},[161],[28,507,299],{"className":508,"style":321},[102,106],[28,510,302],{"className":511},[182]," and 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state",[265,630,631,707],{},[280,632,633],{},[28,634,636,656],{"className":635},[35],[28,637,639],{"className":638},[39],[41,640,641],{"xmlns":43},[46,642,643,653],{},[49,644,645],{},[52,646,647,649],{},[55,648,68],{},[650,651,652],"mn",{},"0",[82,654,655],{"encoding":84},"f_0",[28,657,659],{"className":658,"ariaHidden":71},[89],[28,660,662,665],{"className":661},[93],[28,663],{"className":664,"style":214},[97],[28,666,668,671],{"className":667},[102],[28,669,68],{"className":670,"style":165},[102,106],[28,672,674],{"className":673},[111],[28,675,677,699],{"className":676},[115,116],[28,678,680,696],{"className":679},[120],[28,681,684],{"className":682,"style":683},[124],"height:0.3011em;",[28,685,687,690],{"style":686},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[28,688],{"className":689,"style":133},[132],[28,691,693],{"className":692},[137,138,139,140],[28,694,652],{"className":695},[102,140],[28,697,148],{"className":698},[147],[28,700,702],{"className":701},[120],[28,703,705],{"className":704,"style":155},[124],[28,706],{},[280,708,709],{},"Natural transition frequency of the qubit",[265,711,712,742],{},[280,713,714],{},[28,715,717,730],{"className":716},[35],[28,718,720],{"className":719},[39],[41,721,722],{"xmlns":43},[46,723,724,728],{},[49,725,726],{},[55,727,68],{},[82,729,68],{"encoding":84},[28,731,733],{"className":732,"ariaHidden":71},[89],[28,734,736,739],{"className":735},[93],[28,737],{"className":738,"style":214},[97],[28,740,68],{"className":741,"style":165},[102,106],[280,743,744],{},"Applied drive 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Rabi angular frequency",[265,1110,1111,1153],{},[280,1112,1113],{},[28,1114,1116,1137],{"className":1115},[35],[28,1117,1119],{"className":1118},[39],[41,1120,1121],{"xmlns":43},[46,1122,1123,1134],{},[49,1124,1125,1127,1130,1132],{},[55,1126,1089],{"mathvariant":1088},[55,1128,1129],{"mathvariant":1088},"\u002F",[650,1131,848],{},[55,1133,851],{},[82,1135,1136],{"encoding":84},"\\Omega\u002F2\\pi",[28,1138,1140],{"className":1139,"ariaHidden":71},[89],[28,1141,1143,1146,1150],{"className":1142},[93],[28,1144],{"className":1145,"style":98},[97],[28,1147,1149],{"className":1148},[102],"Ω\u002F2",[28,1151,851],{"className":1152,"style":321},[102,106],[280,1154,1155],{},"On-resonance Rabi frequency in ordinary frequency units",[265,1157,1158,1190],{},[280,1159,1160],{},[28,1161,1163,1178],{"className":1162},[35],[28,1164,1166],{"className":1165},[39],[41,1167,1168],{"xmlns":43},[46,1169,1170,1175],{},[49,1171,1172],{},[55,1173,1174],{"mathvariant":1088},"Δ",[82,1176,1177],{"encoding":84},"\\Delta",[28,1179,1181],{"className":1180,"ariaHidden":71},[89],[28,1182,1184,1187],{"className":1183},[93],[28,1185],{"className":1186,"style":1102},[97],[28,1188,1174],{"className":1189},[102],[280,1191,1192,1193],{},"Drive detuning, ",[28,1194,1196,1223],{"className":1195},[35],[28,1197,1199],{"className":1198},[39],[41,1200,1201],{"xmlns":43},[46,1202,1203,1220],{},[49,1204,1205,1207,1209,1211,1214],{},[55,1206,1174],{"mathvariant":1088},[62,1208,845],{},[55,1210,765],{},[62,1212,1213],{},"−",[52,1215,1216,1218],{},[55,1217,765],{},[650,1219,652],{},[82,1221,1222],{"encoding":84},"\\Delta=\\omega-\\omega_0",[28,1224,1226,1244,1265],{"className":1225,"ariaHidden":71},[89],[28,1227,1229,1232,1235,1238,1241],{"className":1228},[93],[28,1230],{"className":1231,"style":1102},[97],[28,1233,1174],{"className":1234},[102],[28,1236],{"className":1237,"style":913},[173],[28,1239,845],{"className":1240},[917],[28,1242],{"className":1243,"style":913},[173],[28,1245,1247,1251,1254,1258,1262],{"className":1246},[93],[28,1248],{"className":1249,"style":1250},[97],"height:0.6667em;vertical-align:-0.0833em;",[28,1252,765],{"className":1253,"style":321},[102,106],[28,1255],{"className":1256,"style":1257},[173],"margin-right:0.2222em;",[28,1259,1213],{"className":1260},[1261],"mbin",[28,1263],{"className":1264,"style":1257},[173],[28,1266,1268,1271],{"className":1267},[93],[28,1269],{"className":1270,"style":780},[97],[28,1272,1274,1277],{"className":1273},[102],[28,1275,765],{"className":1276,"style":321},[102,106],[28,1278,1280],{"className":1279},[111],[28,1281,1283,1303],{"className":1282},[115,116],[28,1284,1286,1300],{"className":1285},[120],[28,1287,1289],{"className":1288,"style":683},[124],[28,1290,1291,1294],{"style":801},[28,1292],{"className":1293,"style":133},[132],[28,1295,1297],{"className":1296},[137,138,139,140],[28,1298,652],{"className":1299},[102,140],[28,1301,148],{"className":1302},[147],[28,1304,1306],{"className":1305},[120],[28,1307,1309],{"className":1308,"style":155},[124],[28,1310],{},[265,1312,1313,1389],{},[280,1314,1315],{},[28,1316,1318,1337],{"className":1317},[35],[28,1319,1321],{"className":1320},[39],[41,1322,1323],{"xmlns":43},[46,1324,1325,1334],{},[49,1326,1327],{},[52,1328,1329,1331],{},[55,1330,1089],{"mathvariant":1088},[55,1332,1333],{},"R",[82,1335,1336],{"encoding":84},"\\Omega_R",[28,1338,1340],{"className":1339,"ariaHidden":71},[89],[28,1341,1343,1346],{"className":1342},[93],[28,1344],{"className":1345,"style":585},[97],[28,1347,1349,1352],{"className":1348},[102],[28,1350,1089],{"className":1351},[102],[28,1353,1355],{"className":1354},[111],[28,1356,1358,1381],{"className":1357},[115,116],[28,1359,1361,1378],{"className":1360},[120],[28,1362,1365],{"className":1363,"style":1364},[124],"height:0.3283em;",[28,1366,1368,1371],{"style":1367},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[28,1369],{"className":1370,"style":133},[132],[28,1372,1374],{"className":1373},[137,138,139,140],[28,1375,1333],{"className":1376,"style":1377},[102,106,140],"margin-right:0.0077em;",[28,1379,148],{"className":1380},[147],[28,1382,1384],{"className":1383},[120],[28,1385,1387],{"className":1386,"style":155},[124],[28,1388],{},[280,1390,1391],{},"Generalized Rabi angular frequency",[265,1393,1394,1469],{},[280,1395,1396],{},[28,1397,1399,1418],{"className":1398},[35],[28,1400,1402],{"className":1401},[39],[41,1403,1404],{"xmlns":43},[46,1405,1406,1415],{},[49,1407,1408],{},[52,1409,1410,1413],{},[55,1411,1412],{},"ν",[55,1414,1333],{},[82,1416,1417],{"encoding":84},"\\nu_R",[28,1419,1421],{"className":1420,"ariaHidden":71},[89],[28,1422,1424,1427],{"className":1423},[93],[28,1425],{"className":1426,"style":780},[97],[28,1428,1430,1434],{"className":1429},[102],[28,1431,1412],{"className":1432,"style":1433},[102,106],"margin-right:0.0637em;",[28,1435,1437],{"className":1436},[111],[28,1438,1440,1461],{"className":1439},[115,116],[28,1441,1443,1458],{"className":1442},[120],[28,1444,1446],{"className":1445,"style":1364},[124],[28,1447,1449,1452],{"style":1448},"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;",[28,1450],{"className":1451,"style":133},[132],[28,1453,1455],{"className":1454},[137,138,139,140],[28,1456,1333],{"className":1457,"style":1377},[102,106,140],[28,1459,148],{"className":1460},[147],[28,1462,1464],{"className":1463},[120],[28,1465,1467],{"className":1466,"style":155},[124],[28,1468],{},[280,1470,1471,1472],{},"Generalized Rabi frequency, ",[28,1473,1475,1507],{"className":1474},[35],[28,1476,1478],{"className":1477},[39],[41,1479,1480],{"xmlns":43},[46,1481,1482,1504],{},[49,1483,1484,1490,1492,1498,1500,1502],{},[52,1485,1486,1488],{},[55,1487,1412],{},[55,1489,1333],{},[62,1491,845],{},[52,1493,1494,1496],{},[55,1495,1089],{"mathvariant":1088},[55,1497,1333],{},[55,1499,1129],{"mathvariant":1088},[650,1501,848],{},[55,1503,851],{},[82,1505,1506],{"encoding":84},"\\nu_R=\\Omega_R\u002F2\\pi",[28,1508,1510,1565],{"className":1509,"ariaHidden":71},[89],[28,1511,1513,1516,1556,1559,1562],{"className":1512},[93],[28,1514],{"className":1515,"style":780},[97],[28,1517,1519,1522],{"className":1518},[102],[28,1520,1412],{"className":1521,"style":1433},[102,106],[28,1523,1525],{"className":1524},[111],[28,1526,1528,1548],{"className":1527},[115,116],[28,1529,1531,1545],{"className":1530},[120],[28,1532,1534],{"className":1533,"style":1364},[124],[28,1535,1536,1539],{"style":1448},[28,1537],{"className":1538,"style":133},[132],[28,1540,1542],{"className":1541},[137,138,139,140],[28,1543,1333],{"className":1544,"style":1377},[102,106,140],[28,1546,148],{"className":1547},[147],[28,1549,1551],{"className":1550},[120],[28,1552,1554],{"className":1553,"style":155},[124],[28,1555],{},[28,1557],{"className":1558,"style":913},[173],[28,1560,845],{"className":1561},[917],[28,1563],{"className":1564,"style":913},[173],[28,1566,1568,1571,1611,1615],{"className":1567},[93],[28,1569],{"className":1570,"style":98},[97],[28,1572,1574,1577],{"className":1573},[102],[28,1575,1089],{"className":1576},[102],[28,1578,1580],{"className":1579},[111],[28,1581,1583,1603],{"className":1582},[115,116],[28,1584,1586,1600],{"className":1585},[120],[28,1587,1589],{"className":1588,"style":1364},[124],[28,1590,1591,1594],{"style":1367},[28,1592],{"className":1593,"style":133},[132],[28,1595,1597],{"className":1596},[137,138,139,140],[28,1598,1333],{"className":1599,"style":1377},[102,106,140],[28,1601,148],{"className":1602},[147],[28,1604,1606],{"className":1605},[120],[28,1607,1609],{"className":1608,"style":155},[124],[28,1610],{},[28,1612,1614],{"className":1613},[102],"\u002F2",[28,1616,851],{"className":1617,"style":321},[102,106],[265,1619,1620,1650],{},[280,1621,1622],{},[28,1623,1625,1638],{"className":1624},[35],[28,1626,1628],{"className":1627},[39],[41,1629,1630],{"xmlns":43},[46,1631,1632,1636],{},[49,1633,1634],{},[55,1635,75],{},[82,1637,235],{"encoding":84},[28,1639,1641],{"className":1640,"ariaHidden":71},[89],[28,1642,1644,1647],{"className":1643},[93],[28,1645],{"className":1646,"style":245},[97],[28,1648,75],{"className":1649,"style":178},[102,106],[280,1651,1652],{},"Pulse duration",[265,1654,1655,1748],{},[280,1656,1657],{},[28,1658,1660,1683],{"className":1659},[35],[28,1661,1663],{"className":1662},[39],[41,1664,1665],{"xmlns":43},[46,1666,1667,1680],{},[49,1668,1669],{},[1670,1671,1672,1675,1677],"msubsup",{},[55,1673,1674],{},"T",[650,1676,848],{},[62,1678,1679],{},"∗",[82,1681,1682],{"encoding":84},"T_2^\\ast",[28,1684,1686],{"className":1685,"ariaHidden":71},[89],[28,1687,1689,1693],{"className":1688},[93],[28,1690],{"className":1691,"style":1692},[97],"height:0.9368em;vertical-align:-0.2481em;",[28,1694,1696,1699],{"className":1695},[102],[28,1697,1674],{"className":1698,"style":107},[102,106],[28,1700,1702],{"className":1701},[111],[28,1703,1705,1739],{"className":1704},[115,116],[28,1706,1708,1736],{"className":1707},[120],[28,1709,1712,1724],{"className":1710,"style":1711},[124],"height:0.6887em;",[28,1713,1715,1718],{"style":1714},"top:-2.4519em;margin-left:-0.1389em;margin-right:0.05em;",[28,1716],{"className":1717,"style":133},[132],[28,1719,1721],{"className":1720},[137,138,139,140],[28,1722,848],{"className":1723},[102,140],[28,1725,1727,1730],{"style":1726},"top:-3.063em;margin-right:0.05em;",[28,1728],{"className":1729,"style":133},[132],[28,1731,1733],{"className":1732},[137,138,139,140],[28,1734,1679],{"className":1735},[1261,140],[28,1737,148],{"className":1738},[147],[28,1740,1742],{"className":1741},[120],[28,1743,1746],{"className":1744,"style":1745},[124],"height:0.2481em;",[28,1747],{},[280,1749,1750],{},"Phenomenological dephasing time used as an optional damping parameter",[265,1752,1753,1756],{},[280,1754,1755],{},"RWA",[280,1757,1758],{},"Rotating-wave approximation",[265,1760,1761,1764],{},[280,1762,1763],{},"FFT",[280,1765,1766],{},"Fast Fourier transform",[251,1768,1770],{"id":1769},"physical-model","Physical Model",[12,1772,1773,1774,1812,1813,1851],{},"A qubit can be modeled as a two-level quantum system with a ground state ",[28,1775,1777,1794],{"className":1776},[35],[28,1778,1780],{"className":1779},[39],[41,1781,1782],{"xmlns":43},[46,1783,1784,1792],{},[49,1785,1786,1788,1790],{},[62,1787,296],{"stretchy":64},[55,1789,299],{},[62,1791,302],{"stretchy":64},[82,1793,305],{"encoding":84},[28,1795,1797],{"className":1796,"ariaHidden":71},[89],[28,1798,1800,1803,1806,1809],{"className":1799},[93],[28,1801],{"className":1802,"style":98},[97],[28,1804,296],{"className":1805},[161],[28,1807,299],{"className":1808,"style":321},[102,106],[28,1810,302],{"className":1811},[182]," and an excited state ",[28,1814,1816,1833],{"className":1815},[35],[28,1817,1819],{"className":1818},[39],[41,1820,1821],{"xmlns":43},[46,1822,1823,1831],{},[49,1824,1825,1827,1829],{},[62,1826,296],{"stretchy":64},[55,1828,60],{},[62,1830,302],{"stretchy":64},[82,1832,352],{"encoding":84},[28,1834,1836],{"className":1835,"ariaHidden":71},[89],[28,1837,1839,1842,1845,1848],{"className":1838},[93],[28,1840],{"className":1841,"style":98},[97],[28,1843,296],{"className":1844},[161],[28,1846,60],{"className":1847},[102,106],[28,1849,302],{"className":1850},[182],". In the absence of a microwave drive, the energy splitting is",[28,1853,1855],{"className":1854},[31],[28,1856,1858,1898],{"className":1857},[35],[28,1859,1861],{"className":1860},[39],[41,1862,1863],{"xmlns":43,"display":44},[46,1864,1865,1895],{},[49,1866,1867,1874,1876,1882,1884,1887,1893],{},[52,1868,1869,1872],{},[55,1870,1871],{},"E",[55,1873,60],{},[62,1875,1213],{},[52,1877,1878,1880],{},[55,1879,1871],{},[55,1881,299],{},[62,1883,845],{},[55,1885,1886],{"mathvariant":1088},"ℏ",[52,1888,1889,1891],{},[55,1890,765],{},[650,1892,652],{},[62,1894,72],{"separator":71},[82,1896,1897],{"encoding":84},"E_e-E_g=\\hbar\\omega_0,",[28,1899,1901,1958,2015],{"className":1900,"ariaHidden":71},[89],[28,1902,1904,1907,1949,1952,1955],{"className":1903},[93],[28,1905],{"className":1906,"style":585},[97],[28,1908,1910,1914],{"className":1909},[102],[28,1911,1871],{"className":1912,"style":1913},[102,106],"margin-right:0.0576em;",[28,1915,1917],{"className":1916},[111],[28,1918,1920,1941],{"className":1919},[115,116],[28,1921,1923,1938],{"className":1922},[120],[28,1924,1926],{"className":1925,"style":125},[124],[28,1927,1929,1932],{"style":1928},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[28,1930],{"className":1931,"style":133},[132],[28,1933,1935],{"className":1934},[137,138,139,140],[28,1936,60],{"className":1937},[102,106,140],[28,1939,148],{"className":1940},[147],[28,1942,1944],{"className":1943},[120],[28,1945,1947],{"className":1946,"style":155},[124],[28,1948],{},[28,1950],{"className":1951,"style":1257},[173],[28,1953,1213],{"className":1954},[1261],[28,1956],{"className":1957,"style":1257},[173],[28,1959,1961,1965,2006,2009,2012],{"className":1960},[93],[28,1962],{"className":1963,"style":1964},[97],"height:0.9694em;vertical-align:-0.2861em;",[28,1966,1968,1971],{"className":1967},[102],[28,1969,1871],{"className":1970,"style":1913},[102,106],[28,1972,1974],{"className":1973},[111],[28,1975,1977,1997],{"className":1976},[115,116],[28,1978,1980,1994],{"className":1979},[120],[28,1981,1983],{"className":1982,"style":125},[124],[28,1984,1985,1988],{"style":1928},[28,1986],{"className":1987,"style":133},[132],[28,1989,1991],{"className":1990},[137,138,139,140],[28,1992,299],{"className":1993,"style":321},[102,106,140],[28,1995,148],{"className":1996},[147],[28,1998,2000],{"className":1999},[120],[28,2001,2004],{"className":2002,"style":2003},[124],"height:0.2861em;",[28,2005],{},[28,2007],{"className":2008,"style":913},[173],[28,2010,845],{"className":2011},[917],[28,2013],{"className":2014,"style":913},[173],[28,2016,2018,2022,2025,2065],{"className":2017},[93],[28,2019],{"className":2020,"style":2021},[97],"height:0.8833em;vertical-align:-0.1944em;",[28,2023,1886],{"className":2024},[102],[28,2026,2028,2031],{"className":2027},[102],[28,2029,765],{"className":2030,"style":321},[102,106],[28,2032,2034],{"className":2033},[111],[28,2035,2037,2057],{"className":2036},[115,116],[28,2038,2040,2054],{"className":2039},[120],[28,2041,2043],{"className":2042,"style":683},[124],[28,2044,2045,2048],{"style":801},[28,2046],{"className":2047,"style":133},[132],[28,2049,2051],{"className":2050},[137,138,139,140],[28,2052,652],{"className":2053},[102,140],[28,2055,148],{"className":2056},[147],[28,2058,2060],{"className":2059},[120],[28,2061,2063],{"className":2062,"style":155},[124],[28,2064],{},[28,2066,72],{"className":2067},[169],[12,2069,2070],{},"where the angular transition frequency is",[28,2072,2074],{"className":2073},[31],[28,2075,2077,2110],{"className":2076},[35],[28,2078,2080],{"className":2079},[39],[41,2081,2082],{"xmlns":43,"display":44},[46,2083,2084,2107],{},[49,2085,2086,2092,2094,2096,2098,2104],{},[52,2087,2088,2090],{},[55,2089,765],{},[650,2091,652],{},[62,2093,845],{},[650,2095,848],{},[55,2097,851],{},[52,2099,2100,2102],{},[55,2101,68],{},[650,2103,652],{},[55,2105,2106],{"mathvariant":1088},".",[82,2108,2109],{"encoding":84},"\\omega_0=2\\pi f_0.",[28,2111,2113,2168],{"className":2112,"ariaHidden":71},[89],[28,2114,2116,2119,2159,2162,2165],{"className":2115},[93],[28,2117],{"className":2118,"style":780},[97],[28,2120,2122,2125],{"className":2121},[102],[28,2123,765],{"className":2124,"style":321},[102,106],[28,2126,2128],{"className":2127},[111],[28,2129,2131,2151],{"className":2130},[115,116],[28,2132,2134,2148],{"className":2133},[120],[28,2135,2137],{"className":2136,"style":683},[124],[28,2138,2139,2142],{"style":801},[28,2140],{"className":2141,"style":133},[132],[28,2143,2145],{"className":2144},[137,138,139,140],[28,2146,652],{"className":2147},[102,140],[28,2149,148],{"className":2150},[147],[28,2152,2154],{"className":2153},[120],[28,2155,2157],{"className":2156,"style":155},[124],[28,2158],{},[28,2160],{"className":2161,"style":913},[173],[28,2163,845],{"className":2164},[917],[28,2166],{"className":2167,"style":913},[173],[28,2169,2171,2174,2177,2180,2220],{"className":2170},[93],[28,2172],{"className":2173,"style":214},[97],[28,2175,848],{"className":2176},[102],[28,2178,851],{"className":2179,"style":321},[102,106],[28,2181,2183,2186],{"className":2182},[102],[28,2184,68],{"className":2185,"style":165},[102,106],[28,2187,2189],{"className":2188},[111],[28,2190,2192,2212],{"className":2191},[115,116],[28,2193,2195,2209],{"className":2194},[120],[28,2196,2198],{"className":2197,"style":683},[124],[28,2199,2200,2203],{"style":686},[28,2201],{"className":2202,"style":133},[132],[28,2204,2206],{"className":2205},[137,138,139,140],[28,2207,652],{"className":2208},[102,140],[28,2210,148],{"className":2211},[147],[28,2213,2215],{"className":2214},[120],[28,2216,2218],{"className":2217,"style":155},[124],[28,2219],{},[28,2221,2106],{"className":2222},[102],[12,2224,2225],{},"The default configuration uses a transition frequency of",[28,2227,2229],{"className":2228},[31],[28,2230,2232,2272],{"className":2231},[35],[28,2233,2235],{"className":2234},[39],[41,2236,2237],{"xmlns":43,"display":44},[46,2238,2239,2269],{},[49,2240,2241,2247,2249,2252,2256,2267],{},[52,2242,2243,2245],{},[55,2244,68],{},[650,2246,652],{},[62,2248,845],{},[650,2250,2251],{},"5.000",[2253,2254,2255],"mtext",{}," ",[49,2257,2258,2261,2264],{},[55,2259,2260],{"mathvariant":1088},"G",[55,2262,2263],{"mathvariant":1088},"H",[55,2265,2266],{"mathvariant":1088},"z",[55,2268,2106],{"mathvariant":1088},[82,2270,2271],{"encoding":84},"f_0=5.000~\\mathrm{GHz}.",[28,2273,2275,2330],{"className":2274,"ariaHidden":71},[89],[28,2276,2278,2281,2321,2324,2327],{"className":2277},[93],[28,2279],{"className":2280,"style":214},[97],[28,2282,2284,2287],{"className":2283},[102],[28,2285,68],{"className":2286,"style":165},[102,106],[28,2288,2290],{"className":2289},[111],[28,2291,2293,2313],{"className":2292},[115,116],[28,2294,2296,2310],{"className":2295},[120],[28,2297,2299],{"className":2298,"style":683},[124],[28,2300,2301,2304],{"style":686},[28,2302],{"className":2303,"style":133},[132],[28,2305,2307],{"className":2306},[137,138,139,140],[28,2308,652],{"className":2309},[102,140],[28,2311,148],{"className":2312},[147],[28,2314,2316],{"className":2315},[120],[28,2317,2319],{"className":2318,"style":155},[124],[28,2320],{},[28,2322],{"className":2323,"style":913},[173],[28,2325,845],{"className":2326},[917],[28,2328],{"className":2329,"style":913},[173],[28,2331,2333,2336,2339,2343,2351],{"className":2332},[93],[28,2334],{"className":2335,"style":1102},[97],[28,2337,2251],{"className":2338},[102],[28,2340,2255],{"className":2341},[173,2342],"nobreak",[28,2344,2346],{"className":2345},[102],[28,2347,2350],{"className":2348},[102,2349],"mathrm","GHz",[28,2352,2106],{"className":2353},[102],[12,2355,2356,2357,2385,2386,512,2424,2462],{},"Applying a microwave drive with frequency ",[28,2358,2360,2373],{"className":2359},[35],[28,2361,2363],{"className":2362},[39],[41,2364,2365],{"xmlns":43},[46,2366,2367,2371],{},[49,2368,2369],{},[55,2370,68],{},[82,2372,68],{"encoding":84},[28,2374,2376],{"className":2375,"ariaHidden":71},[89],[28,2377,2379,2382],{"className":2378},[93],[28,2380],{"className":2381,"style":214},[97],[28,2383,68],{"className":2384,"style":165},[102,106]," induces coherent rotations between ",[28,2387,2389,2406],{"className":2388},[35],[28,2390,2392],{"className":2391},[39],[41,2393,2394],{"xmlns":43},[46,2395,2396,2404],{},[49,2397,2398,2400,2402],{},[62,2399,296],{"stretchy":64},[55,2401,299],{},[62,2403,302],{"stretchy":64},[82,2405,305],{"encoding":84},[28,2407,2409],{"className":2408,"ariaHidden":71},[89],[28,2410,2412,2415,2418,2421],{"className":2411},[93],[28,2413],{"className":2414,"style":98},[97],[28,2416,296],{"className":2417},[161],[28,2419,299],{"className":2420,"style":321},[102,106],[28,2422,302],{"className":2423},[182],[28,2425,2427,2444],{"className":2426},[35],[28,2428,2430],{"className":2429},[39],[41,2431,2432],{"xmlns":43},[46,2433,2434,2442],{},[49,2435,2436,2438,2440],{},[62,2437,296],{"stretchy":64},[55,2439,60],{},[62,2441,302],{"stretchy":64},[82,2443,352],{"encoding":84},[28,2445,2447],{"className":2446,"ariaHidden":71},[89],[28,2448,2450,2453,2456,2459],{"className":2449},[93],[28,2451],{"className":2452,"style":98},[97],[28,2454,296],{"className":2455},[161],[28,2457,60],{"className":2458},[102,106],[28,2460,302],{"className":2461},[182],". Exact resonance occurs when",[28,2464,2466],{"className":2465},[31],[28,2467,2469,2493],{"className":2468},[35],[28,2470,2472],{"className":2471},[39],[41,2473,2474],{"xmlns":43,"display":44},[46,2475,2476,2490],{},[49,2477,2478,2480,2482,2488],{},[55,2479,68],{},[62,2481,845],{},[52,2483,2484,2486],{},[55,2485,68],{},[650,2487,652],{},[55,2489,2106],{"mathvariant":1088},[82,2491,2492],{"encoding":84},"f=f_0.",[28,2494,2496,2514],{"className":2495,"ariaHidden":71},[89],[28,2497,2499,2502,2505,2508,2511],{"className":2498},[93],[28,2500],{"className":2501,"style":214},[97],[28,2503,68],{"className":2504,"style":165},[102,106],[28,2506],{"className":2507,"style":913},[173],[28,2509,845],{"className":2510},[917],[28,2512],{"className":2513,"style":913},[173],[28,2515,2517,2520,2560],{"className":2516},[93],[28,2518],{"className":2519,"style":214},[97],[28,2521,2523,2526],{"className":2522},[102],[28,2524,68],{"className":2525,"style":165},[102,106],[28,2527,2529],{"className":2528},[111],[28,2530,2532,2552],{"className":2531},[115,116],[28,2533,2535,2549],{"className":2534},[120],[28,2536,2538],{"className":2537,"style":683},[124],[28,2539,2540,2543],{"style":686},[28,2541],{"className":2542,"style":133},[132],[28,2544,2546],{"className":2545},[137,138,139,140],[28,2547,652],{"className":2548},[102,140],[28,2550,148],{"className":2551},[147],[28,2553,2555],{"className":2554},[120],[28,2556,2558],{"className":2557,"style":155},[124],[28,2559],{},[28,2561,2106],{"className":2562},[102],[12,2564,2565],{},"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.",[251,2567,2569],{"id":2568},"mathematical-foundation","Mathematical Foundation",[12,2571,2572],{},"The quantum state is a normalized vector in a two-dimensional Hilbert space,",[28,2574,2576],{"className":2575},[31],[28,2577,2579,2620],{"className":2578},[35],[28,2580,2582],{"className":2581},[39],[41,2583,2584],{"xmlns":43,"display":44},[46,2585,2586,2617],{},[49,2587,2588,2590,2592,2594,2596,2598,2600,2602,2604,2607,2609,2611,2613,2615],{},[62,2589,296],{"stretchy":64},[55,2591,394],{},[62,2593,302],{"stretchy":64},[62,2595,845],{},[55,2597,439],{},[62,2599,296],{"stretchy":64},[55,2601,299],{},[62,2603,302],{"stretchy":64},[62,2605,2606],{},"+",[55,2608,444],{},[62,2610,296],{"stretchy":64},[55,2612,60],{},[62,2614,302],{"stretchy":64},[62,2616,72],{"separator":71},[82,2618,2619],{"encoding":84},"\\lvert \\psi\\rangle=\\alpha\\lvert g\\rangle+\\beta\\lvert e\\rangle,",[28,2621,2623,2647,2674],{"className":2622,"ariaHidden":71},[89],[28,2624,2626,2629,2632,2635,2638,2641,2644],{"className":2625},[93],[28,2627],{"className":2628,"style":98},[97],[28,2630,296],{"className":2631},[161],[28,2633,394],{"className":2634,"style":321},[102,106],[28,2636,302],{"className":2637},[182],[28,2639],{"className":2640,"style":913},[173],[28,2642,845],{"className":2643},[917],[28,2645],{"className":2646,"style":913},[173],[28,2648,2650,2653,2656,2659,2662,2665,2668,2671],{"className":2649},[93],[28,2651],{"className":2652,"style":98},[97],[28,2654,439],{"className":2655,"style":460},[102,106],[28,2657,296],{"className":2658},[161],[28,2660,299],{"className":2661,"style":321},[102,106],[28,2663,302],{"className":2664},[182],[28,2666],{"className":2667,"style":1257},[173],[28,2669,2606],{"className":2670},[1261],[28,2672],{"className":2673,"style":1257},[173],[28,2675,2677,2680,2683,2686,2689,2692],{"className":2676},[93],[28,2678],{"className":2679,"style":98},[97],[28,2681,444],{"className":2682,"style":470},[102,106],[28,2684,296],{"className":2685},[161],[28,2687,60],{"className":2688},[102,106],[28,2690,302],{"className":2691},[182],[28,2693,72],{"className":2694},[169],[12,2696,188,2697,512,2726,2755],{},[28,2698,2700,2714],{"className":2699},[35],[28,2701,2703],{"className":2702},[39],[41,2704,2705],{"xmlns":43},[46,2706,2707,2711],{},[49,2708,2709],{},[55,2710,439],{},[82,2712,2713],{"encoding":84},"\\alpha",[28,2715,2717],{"className":2716,"ariaHidden":71},[89],[28,2718,2720,2723],{"className":2719},[93],[28,2721],{"className":2722,"style":245},[97],[28,2724,439],{"className":2725,"style":460},[102,106],[28,2727,2729,2743],{"className":2728},[35],[28,2730,2732],{"className":2731},[39],[41,2733,2734],{"xmlns":43},[46,2735,2736,2740],{},[49,2737,2738],{},[55,2739,444],{},[82,2741,2742],{"encoding":84},"\\beta",[28,2744,2746],{"className":2745,"ariaHidden":71},[89],[28,2747,2749,2752],{"className":2748},[93],[28,2750],{"className":2751,"style":214},[97],[28,2753,444],{"className":2754,"style":470},[102,106]," are complex probability amplitudes. Normalization requires",[28,2757,2759],{"className":2758},[31],[28,2760,2762,2802],{"className":2761},[35],[28,2763,2765],{"className":2764},[39],[41,2766,2767],{"xmlns":43,"display":44},[46,2768,2769,2799],{},[49,2770,2771,2773,2775,2782,2784,2786,2788,2794,2796],{},[55,2772,296],{"mathvariant":1088},[55,2774,439],{},[2776,2777,2778,2780],"msup",{},[55,2779,296],{"mathvariant":1088},[650,2781,848],{},[62,2783,2606],{},[55,2785,296],{"mathvariant":1088},[55,2787,444],{},[2776,2789,2790,2792],{},[55,2791,296],{"mathvariant":1088},[650,2793,848],{},[62,2795,845],{},[650,2797,2798],{},"1.",[82,2800,2801],{"encoding":84},"|\\alpha|^2+|\\beta|^2=1.",[28,2803,2805,2858,2908],{"className":2804,"ariaHidden":71},[89],[28,2806,2808,2812,2815,2818,2849,2852,2855],{"className":2807},[93],[28,2809],{"className":2810,"style":2811},[97],"height:1.1141em;vertical-align:-0.25em;",[28,2813,296],{"className":2814},[102],[28,2816,439],{"className":2817,"style":460},[102,106],[28,2819,2821,2824],{"className":2820},[102],[28,2822,296],{"className":2823},[102],[28,2825,2827],{"className":2826},[111],[28,2828,2830],{"className":2829},[115],[28,2831,2833],{"className":2832},[120],[28,2834,2837],{"className":2835,"style":2836},[124],"height:0.8641em;",[28,2838,2840,2843],{"style":2839},"top:-3.113em;margin-right:0.05em;",[28,2841],{"className":2842,"style":133},[132],[28,2844,2846],{"className":2845},[137,138,139,140],[28,2847,848],{"className":2848},[102,140],[28,2850],{"className":2851,"style":1257},[173],[28,2853,2606],{"className":2854},[1261],[28,2856],{"className":2857,"style":1257},[173],[28,2859,2861,2864,2867,2870,2899,2902,2905],{"className":2860},[93],[28,2862],{"className":2863,"style":2811},[97],[28,2865,296],{"className":2866},[102],[28,2868,444],{"className":2869,"style":470},[102,106],[28,2871,2873,2876],{"className":2872},[102],[28,2874,296],{"className":2875},[102],[28,2877,2879],{"className":2878},[111],[28,2880,2882],{"className":2881},[115],[28,2883,2885],{"className":2884},[120],[28,2886,2888],{"className":2887,"style":2836},[124],[28,2889,2890,2893],{"style":2839},[28,2891],{"className":2892,"style":133},[132],[28,2894,2896],{"className":2895},[137,138,139,140],[28,2897,848],{"className":2898},[102,140],[28,2900],{"className":2901,"style":913},[173],[28,2903,845],{"className":2904},[917],[28,2906],{"className":2907,"style":913},[173],[28,2909,2911,2915],{"className":2910},[93],[28,2912],{"className":2913,"style":2914},[97],"height:0.6444em;",[28,2916,2798],{"className":2917},[102],[12,2919,2920],{},"The Born rule gives the probability of measuring the excited state,",[28,2922,2924],{"className":2923},[31],[28,2925,2927,2980],{"className":2926},[35],[28,2928,2930],{"className":2929},[39],[41,2931,2932],{"xmlns":43,"display":44},[46,2933,2934,2977],{},[49,2935,2936,2942,2944,2946,2949,2951,2953,2955,2957,2963,2965,2967,2969,2975],{},[52,2937,2938,2940],{},[55,2939,57],{},[55,2941,60],{},[62,2943,845],{},[55,2945,296],{"mathvariant":1088},[62,2947,2948],{"stretchy":64},"⟨",[55,2950,60],{},[62,2952,296],{"stretchy":64},[55,2954,394],{},[62,2956,302],{"stretchy":64},[2776,2958,2959,2961],{},[55,2960,296],{"mathvariant":1088},[650,2962,848],{},[62,2964,845],{},[55,2966,296],{"mathvariant":1088},[55,2968,444],{},[2776,2970,2971,2973],{},[55,2972,296],{"mathvariant":1088},[650,2974,848],{},[55,2976,2106],{"mathvariant":1088},[82,2978,2979],{"encoding":84},"P_e=|\\langle e\\lvert\\psi\\rangle|^2=|\\beta|^2.",[28,2981,2983,3038,3100],{"className":2982,"ariaHidden":71},[89],[28,2984,2986,2989,3029,3032,3035],{"className":2985},[93],[28,2987],{"className":2988,"style":585},[97],[28,2990,2992,2995],{"className":2991},[102],[28,2993,57],{"className":2994,"style":107},[102,106],[28,2996,2998],{"className":2997},[111],[28,2999,3001,3021],{"className":3000},[115,116],[28,3002,3004,3018],{"className":3003},[120],[28,3005,3007],{"className":3006,"style":125},[124],[28,3008,3009,3012],{"style":128},[28,3010],{"className":3011,"style":133},[132],[28,3013,3015],{"className":3014},[137,138,139,140],[28,3016,60],{"className":3017},[102,106,140],[28,3019,148],{"className":3020},[147],[28,3022,3024],{"className":3023},[120],[28,3025,3027],{"className":3026,"style":155},[124],[28,3028],{},[28,3030],{"className":3031,"style":913},[173],[28,3033,845],{"className":3034},[917],[28,3036],{"className":3037,"style":913},[173],[28,3039,3041,3044,3047,3050,3053,3056,3059,3062,3091,3094,3097],{"className":3040},[93],[28,3042],{"className":3043,"style":2811},[97],[28,3045,296],{"className":3046},[102],[28,3048,2948],{"className":3049},[161],[28,3051,60],{"className":3052},[102,106],[28,3054,296],{"className":3055},[161],[28,3057,394],{"className":3058,"style":321},[102,106],[28,3060,302],{"className":3061},[182],[28,3063,3065,3068],{"className":3064},[102],[28,3066,296],{"className":3067},[102],[28,3069,3071],{"className":3070},[111],[28,3072,3074],{"className":3073},[115],[28,3075,3077],{"className":3076},[120],[28,3078,3080],{"className":3079,"style":2836},[124],[28,3081,3082,3085],{"style":2839},[28,3083],{"className":3084,"style":133},[132],[28,3086,3088],{"className":3087},[137,138,139,140],[28,3089,848],{"className":3090},[102,140],[28,3092],{"className":3093,"style":913},[173],[28,3095,845],{"className":3096},[917],[28,3098],{"className":3099,"style":913},[173],[28,3101,3103,3106,3109,3112,3141],{"className":3102},[93],[28,3104],{"className":3105,"style":2811},[97],[28,3107,296],{"className":3108},[102],[28,3110,444],{"className":3111,"style":470},[102,106],[28,3113,3115,3118],{"className":3114},[102],[28,3116,296],{"className":3117},[102],[28,3119,3121],{"className":3120},[111],[28,3122,3124],{"className":3123},[115],[28,3125,3127],{"className":3126},[120],[28,3128,3130],{"className":3129,"style":2836},[124],[28,3131,3132,3135],{"style":2839},[28,3133],{"className":3134,"style":133},[132],[28,3136,3138],{"className":3137},[137,138,139,140],[28,3139,848],{"className":3140},[102,140],[28,3142,2106],{"className":3143},[102],[12,3145,3146],{},"Pauli matrices then provide a natural operator basis for the two-level system,",[28,3148,3150],{"className":3149},[31],[28,3151,3153,3330],{"className":3152},[35],[28,3154,3156],{"className":3155},[39],[41,3157,3158],{"xmlns":43,"display":44},[46,3159,3160,3327],{},[49,3161,3162,3170,3172,3218,3220,3223,3230,3232,3273,3275,3277,3283,3285,3325],{},[52,3163,3164,3167],{},[55,3165,3166],{},"σ",[55,3168,3169],{},"x",[62,3171,845],{},[49,3173,3174,3177,3215],{},[62,3175,3176],{"fence":71},"[",[3178,3179,3183,3201],"mtable",{"rowspacing":3180,"columnalign":3181,"columnspacing":3182},"0.16em","center 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Applied detuning supplies the longitudinal control component.",[251,3944,3946],{"id":3945},"rotating-frame-hamiltonian","Rotating-Frame Hamiltonian",[12,3948,3949],{},"In the laboratory frame, a minimal semiclassical Hamiltonian for the driven qubit 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t)\\sigma_x,",[28,4029,4031,4059,4225],{"className":4030,"ariaHidden":71},[89],[28,4032,4034,4037,4041,4044,4047,4050,4053,4056],{"className":4033},[93],[28,4035],{"className":4036,"style":98},[97],[28,4038,2263],{"className":4039,"style":4040},[102,106],"margin-right:0.0813em;",[28,4042,65],{"className":4043},[161],[28,4045,3971],{"className":4046},[102,106],[28,4048,78],{"className":4049},[182],[28,4051],{"className":4052,"style":913},[173],[28,4054,845],{"className":4055},[917],[28,4057],{"className":4058,"style":913},[173],[28,4060,4062,4066,4176,4216,4219,4222],{"className":4061},[93],[28,4063],{"className":4064,"style":4065},[97],"height:2.0519em;vertical-align:-0.686em;",[28,4067,4069,4073,4173],{"className":4068},[102],[28,4070],{"className":4071},[161,4072],"nulldelimiter",[28,4074,4076],{"className":4075},[3977],[28,4077,4079,4164],{"className":4078},[115,116],[28,4080,4082,4161],{"className":4081},[120],[28,4083,4086,4098,4109],{"className":4084,"style":4085},[124],"height:1.3659em;",[28,4087,4089,4092],{"style":4088},"top:-2.314em;",[28,4090],{"className":4091,"style":3434},[132],[28,4093,4095],{"className":4094},[102],[28,4096,848],{"className":4097},[102],[28,4099,4101,4104],{"style":4100},"top:-3.23em;",[28,4102],{"className":4103,"style":3434},[132],[28,4105],{"className":4106,"style":4108},[4107],"frac-line","border-bottom-width:0.04em;",[28,4110,4112,4115],{"style":4111},"top:-3.677em;",[28,4113],{"className":4114,"style":3434},[132],[28,4116,4118,4121],{"className":4117},[102],[28,4119,1886],{"className":4120},[102],[28,4122,4124,4127],{"className":4123},[102],[28,4125,765],{"className":4126,"style":321},[102,106],[28,4128,4130],{"className":4129},[111],[28,4131,4133,4153],{"className":4132},[115,116],[28,4134,4136,4150],{"className":4135},[120],[28,4137,4139],{"className":4138,"style":683},[124],[28,4140,4141,4144],{"style":801},[28,4142],{"className":4143,"style":133},[132],[28,4145,4147],{"className":4146},[137,138,139,140],[28,4148,652],{"className":4149},[102,140],[28,4151,148],{"className":4152},[147],[28,4154,4156],{"className":4155},[120],[28,4157,4159],{"className":4158,"style":155},[124],[28,4160],{},[28,4162,148],{"className":4163},[147],[28,4165,4167],{"className":4166},[120],[28,4168,4171],{"className":4169,"style":4170},[124],"height:0.686em;",[28,4172],{},[28,4174],{"className":4175},[182,4072],[28,4177,4179,4182],{"className":4178},[102],[28,4180,3166],{"className":4181,"style":321},[102,106],[28,4183,4185],{"className":4184},[111],[28,4186,4188,4208],{"className":4187},[115,116],[28,4189,4191,4205],{"className":4190},[120],[28,4192,4194],{"className":4193,"style":125},[124],[28,4195,4196,4199],{"style":801},[28,4197],{"className":4198,"style":133},[132],[28,4200,4202],{"className":4201},[137,138,139,140],[28,4203,2266],{"className":4204,"style":3753},[102,106,140],[28,4206,148],{"className":4207},[147],[28,4209,4211],{"className":4210},[120],[28,4212,4214],{"className":4213,"style":155},[124],[28,4215],{},[28,4217],{"className":4218,"style":1257},[173],[28,4220,2606],{"className":4221},[1261],[28,4223],{"className":4224,"style":1257},[173],[28,4226,4228,4231,4235,4238,4242,4245,4248,4251,4254,4294],{"className":4227},[93],[28,4229],{"className":4230,"style":98},[97],[28,4232,4234],{"className":4233},[102],"ℏΩ",[28,4236],{"className":4237,"style":174},[173],[28,4239,4005],{"className":4240},[4241],"mop",[28,4243,65],{"className":4244},[161],[28,4246,765],{"className":4247,"style":321},[102,106],[28,4249,3971],{"className":4250},[102,106],[28,4252,78],{"className":4253},[182],[28,4255,4257,4260],{"className":4256},[102],[28,4258,3166],{"className":4259,"style":321},[102,106],[28,4261,4263],{"className":4262},[111],[28,4264,4266,4286],{"className":4265},[115,116],[28,4267,4269,4283],{"className":4268},[120],[28,4270,4272],{"className":4271,"style":125},[124],[28,4273,4274,4277],{"style":801},[28,4275],{"className":4276,"style":133},[132],[28,4278,4280],{"className":4279},[137,138,139,140],[28,4281,3169],{"className":4282},[102,106,140],[28,4284,148],{"className":4285},[147],[28,4287,4289],{"className":4288},[120],[28,4290,4292],{"className":4291,"style":155},[124],[28,4293],{},[28,4295,72],{"className":4296},[169],[12,4298,188,4299,4359,4360,4388,4389,4426],{},[28,4300,4302,4323],{"className":4301},[35],[28,4303,4305],{"className":4304},[39],[41,4306,4307],{"xmlns":43},[46,4308,4309,4321],{},[49,4310,4311,4313,4315,4317,4319],{},[55,4312,765],{},[62,4314,845],{},[650,4316,848],{},[55,4318,851],{},[55,4320,68],{},[82,4322,1033],{"encoding":84},[28,4324,4326,4344],{"className":4325,"ariaHidden":71},[89],[28,4327,4329,4332,4335,4338,4341],{"className":4328},[93],[28,4330],{"className":4331,"style":245},[97],[28,4333,765],{"className":4334,"style":321},[102,106],[28,4336],{"className":4337,"style":913},[173],[28,4339,845],{"className":4340},[917],[28,4342],{"className":4343,"style":913},[173],[28,4345,4347,4350,4353,4356],{"className":4346},[93],[28,4348],{"className":4349,"style":214},[97],[28,4351,848],{"className":4352},[102],[28,4354,851],{"className":4355,"style":321},[102,106],[28,4357,68],{"className":4358,"style":165},[102,106]," is the drive angular frequency and ",[28,4361,4363,4376],{"className":4362},[35],[28,4364,4366],{"className":4365},[39],[41,4367,4368],{"xmlns":43},[46,4369,4370,4374],{},[49,4371,4372],{},[55,4373,1089],{"mathvariant":1088},[82,4375,1092],{"encoding":84},[28,4377,4379],{"className":4378,"ariaHidden":71},[89],[28,4380,4382,4385],{"className":4381},[93],[28,4383],{"className":4384,"style":1102},[97],[28,4386,1089],{"className":4387},[102]," is the on-resonance Rabi angular frequency. The computational model specifies the drive strength using ",[28,4390,4392,4411],{"className":4391},[35],[28,4393,4395],{"className":4394},[39],[41,4396,4397],{"xmlns":43},[46,4398,4399,4409],{},[49,4400,4401,4403,4405,4407],{},[55,4402,1089],{"mathvariant":1088},[55,4404,1129],{"mathvariant":1088},[650,4406,848],{},[55,4408,851],{},[82,4410,1136],{"encoding":84},[28,4412,4414],{"className":4413,"ariaHidden":71},[89],[28,4415,4417,4420,4423],{"className":4416},[93],[28,4418],{"className":4419,"style":98},[97],[28,4421,1149],{"className":4422},[102],[28,4424,851],{"className":4425,"style":321},[102,106]," in MHz. Transforming into a frame rotating at the drive frequency and applying the rotating-wave approximation gives the effective time-independent Hamiltonian",[28,4428,4430],{"className":4429},[31],[28,4431,4433,4493],{"className":4432},[35],[28,4434,4436],{"className":4435},[39],[41,4437,4438],{"xmlns":43,"display":44},[46,4439,4440,4490],{},[49,4441,4442,4456,4458,4464,4488],{},[52,4443,4444,4446],{},[55,4445,2263],{},[49,4447,4448,4450,4453],{},[55,4449,1333],{"mathvariant":1088},[55,4451,4452],{"mathvariant":1088},"W",[55,4454,4455],{"mathvariant":1088},"A",[62,4457,845],{},[3977,4459,4460,4462],{},[55,4461,1886],{"mathvariant":1088},[650,4463,848],{},[49,4465,4466,4468,4470,4476,4478,4480,4486],{},[62,4467,65],{"fence":71},[55,4469,1174],{"mathvariant":1088},[52,4471,4472,4474],{},[55,4473,3166],{},[55,4475,2266],{},[62,4477,2606],{},[55,4479,1089],{"mathvariant":1088},[52,4481,4482,4484],{},[55,4483,3166],{},[55,4485,3169],{},[62,4487,78],{"fence":71},[62,4489,72],{"separator":71},[82,4491,4492],{"encoding":84},"H_{\\mathrm{RWA}}=\\frac{\\hbar}{2}\\left(\\Delta\\sigma_z+\\Omega\\sigma_x\\right),",[28,4494,4496,4558],{"className":4495,"ariaHidden":71},[89],[28,4497,4499,4502,4549,4552,4555],{"className":4498},[93],[28,4500],{"className":4501,"style":585},[97],[28,4503,4505,4508],{"className":4504},[102],[28,4506,2263],{"className":4507,"style":4040},[102,106],[28,4509,4511],{"className":4510},[111],[28,4512,4514,4541],{"className":4513},[115,116],[28,4515,4517,4538],{"className":4516},[120],[28,4518,4520],{"className":4519,"style":1364},[124],[28,4521,4523,4526],{"style":4522},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[28,4524],{"className":4525,"style":133},[132],[28,4527,4529],{"className":4528},[137,138,139,140],[28,4530,4532],{"className":4531},[102,140],[28,4533,4535],{"className":4534},[102,140],[28,4536,1755],{"className":4537},[102,2349,140],[28,4539,148],{"className":4540},[147],[28,4542,4544],{"className":4543},[120],[28,4545,4547],{"className":4546,"style":155},[124],[28,4548],{},[28,4550],{"className":4551,"style":913},[173],[28,4553,845],{"className":4554},[917],[28,4556],{"className":4557,"style":913},[173],[28,4559,4561,4564,4626,4629,4733,4736],{"className":4560},[93],[28,4562],{"className":4563,"style":4065},[97],[28,4565,4567,4570,4623],{"className":4566},[102],[28,4568],{"className":4569},[161,4072],[28,4571,4573],{"className":4572},[3977],[28,4574,4576,4615],{"className":4575},[115,116],[28,4577,4579,4612],{"className":4578},[120],[28,4580,4582,4593,4601],{"className":4581,"style":4085},[124],[28,4583,4584,4587],{"style":4088},[28,4585],{"className":4586,"style":3434},[132],[28,4588,4590],{"className":4589},[102],[28,4591,848],{"className":4592},[102],[28,4594,4595,4598],{"style":4100},[28,4596],{"className":4597,"style":3434},[132],[28,4599],{"className":4600,"style":4108},[4107],[28,4602,4603,4606],{"style":4111},[28,4604],{"className":4605,"style":3434},[132],[28,4607,4609],{"className":4608},[102],[28,4610,1886],{"className":4611},[102],[28,4613,148],{"className":4614},[147],[28,4616,4618],{"className":4617},[120],[28,4619,4621],{"className":4620,"style":4170},[124],[28,4622],{},[28,4624],{"className":4625},[182,4072],[28,4627],{"className":4628,"style":174},[173],[28,4630,4632,4635,4638,4678,4681,4684,4687,4690,4730],{"className":4631},[3398],[28,4633,65],{"className":4634,"style":3403},[161,3402],[28,4636,1174],{"className":4637},[102],[28,4639,4641,4644],{"className":4640},[102],[28,4642,3166],{"className":4643,"style":321},[102,106],[28,4645,4647],{"className":4646},[111],[28,4648,4650,4670],{"className":4649},[115,116],[28,4651,4653,4667],{"className":4652},[120],[28,4654,4656],{"className":4655,"style":125},[124],[28,4657,4658,4661],{"style":801},[28,4659],{"className":4660,"style":133},[132],[28,4662,4664],{"className":4663},[137,138,139,140],[28,4665,2266],{"className":4666,"style":3753},[102,106,140],[28,4668,148],{"className":4669},[147],[28,4671,4673],{"className":4672},[120],[28,4674,4676],{"className":4675,"style":155},[124],[28,4677],{},[28,4679],{"className":4680,"style":1257},[173],[28,4682,2606],{"className":4683},[1261],[28,4685],{"className":4686,"style":1257},[173],[28,4688,1089],{"className":4689},[102],[28,4691,4693,4696],{"className":4692},[102],[28,4694,3166],{"className":4695,"style":321},[102,106],[28,4697,4699],{"className":4698},[111],[28,4700,4702,4722],{"className":4701},[115,116],[28,4703,4705,4719],{"className":4704},[120],[28,4706,4708],{"className":4707,"style":125},[124],[28,4709,4710,4713],{"style":801},[28,4711],{"className":4712,"style":133},[132],[28,4714,4716],{"className":4715},[137,138,139,140],[28,4717,3169],{"className":4718},[102,106,140],[28,4720,148],{"className":4721},[147],[28,4723,4725],{"className":4724},[120],[28,4726,4728],{"className":4727,"style":155},[124],[28,4729],{},[28,4731,78],{"className":4732,"style":3403},[182,3402],[28,4734],{"className":4735,"style":174},[173],[28,4737,72],{"className":4738},[169],[12,4740,4741],{},"with detuning",[28,4743,4745],{"className":4744},[31],[28,4746,4748,4796],{"className":4747},[35],[28,4749,4751],{"className":4750},[39],[41,4752,4753],{"xmlns":43,"display":44},[46,4754,4755,4793],{},[49,4756,4757,4759,4761,4763,4765,4771,4773,4775,4777,4779,4781,4783,4789,4791],{},[55,4758,1174],{"mathvariant":1088},[62,4760,845],{},[55,4762,765],{},[62,4764,1213],{},[52,4766,4767,4769],{},[55,4768,765],{},[650,4770,652],{},[62,4772,845],{},[650,4774,848],{},[55,4776,851],{},[62,4778,65],{"stretchy":64},[55,4780,68],{},[62,4782,1213],{},[52,4784,4785,4787],{},[55,4786,68],{},[650,4788,652],{},[62,4790,78],{"stretchy":64},[55,4792,2106],{"mathvariant":1088},[82,4794,4795],{"encoding":84},"\\Delta=\\omega-\\omega_0=2\\pi(f-f_0).",[28,4797,4799,4817,4835,4890,4917],{"className":4798,"ariaHidden":71},[89],[28,4800,4802,4805,4808,4811,4814],{"className":4801},[93],[28,4803],{"className":4804,"style":1102},[97],[28,4806,1174],{"className":4807},[102],[28,4809],{"className":4810,"style":913},[173],[28,4812,845],{"className":4813},[917],[28,4815],{"className":4816,"style":913},[173],[28,4818,4820,4823,4826,4829,4832],{"className":4819},[93],[28,4821],{"className":4822,"style":1250},[97],[28,4824,765],{"className":4825,"style":321},[102,106],[28,4827],{"className":4828,"style":1257},[173],[28,4830,1213],{"className":4831},[1261],[28,4833],{"className":4834,"style":1257},[173],[28,4836,4838,4841,4881,4884,4887],{"className":4837},[93],[28,4839],{"className":4840,"style":780},[97],[28,4842,4844,4847],{"className":4843},[102],[28,4845,765],{"className":4846,"style":321},[102,106],[28,4848,4850],{"className":4849},[111],[28,4851,4853,4873],{"className":4852},[115,116],[28,4854,4856,4870],{"className":4855},[120],[28,4857,4859],{"className":4858,"style":683},[124],[28,4860,4861,4864],{"style":801},[28,4862],{"className":4863,"style":133},[132],[28,4865,4867],{"className":4866},[137,138,139,140],[28,4868,652],{"className":4869},[102,140],[28,4871,148],{"className":4872},[147],[28,4874,4876],{"className":4875},[120],[28,4877,4879],{"className":4878,"style":155},[124],[28,4880],{},[28,4882],{"className":4883,"style":913},[173],[28,4885,845],{"className":4886},[917],[28,4888],{"className":4889,"style":913},[173],[28,4891,4893,4896,4899,4902,4905,4908,4911,4914],{"className":4892},[93],[28,4894],{"className":4895,"style":98},[97],[28,4897,848],{"className":4898},[102],[28,4900,851],{"className":4901,"style":321},[102,106],[28,4903,65],{"className":4904},[161],[28,4906,68],{"className":4907,"style":165},[102,106],[28,4909],{"className":4910,"style":1257},[173],[28,4912,1213],{"className":4913},[1261],[28,4915],{"className":4916,"style":1257},[173],[28,4918,4920,4923,4963,4966],{"className":4919},[93],[28,4921],{"className":4922,"style":98},[97],[28,4924,4926,4929],{"className":4925},[102],[28,4927,68],{"className":4928,"style":165},[102,106],[28,4930,4932],{"className":4931},[111],[28,4933,4935,4955],{"className":4934},[115,116],[28,4936,4938,4952],{"className":4937},[120],[28,4939,4941],{"className":4940,"style":683},[124],[28,4942,4943,4946],{"style":686},[28,4944],{"className":4945,"style":133},[132],[28,4947,4949],{"className":4948},[137,138,139,140],[28,4950,652],{"className":4951},[102,140],[28,4953,148],{"className":4954},[147],[28,4956,4958],{"className":4957},[120],[28,4959,4961],{"className":4960,"style":155},[124],[28,4962],{},[28,4964,78],{"className":4965},[182],[28,4967,2106],{"className":4968},[102],[12,4970,4971,4972,512,5000,5028,5029,5057,5058,5086],{},"This Hamiltonian describes precession around an effective control axis with components proportional to ",[28,4973,4975,4988],{"className":4974},[35],[28,4976,4978],{"className":4977},[39],[41,4979,4980],{"xmlns":43},[46,4981,4982,4986],{},[49,4983,4984],{},[55,4985,1089],{"mathvariant":1088},[82,4987,1092],{"encoding":84},[28,4989,4991],{"className":4990,"ariaHidden":71},[89],[28,4992,4994,4997],{"className":4993},[93],[28,4995],{"className":4996,"style":1102},[97],[28,4998,1089],{"className":4999},[102],[28,5001,5003,5016],{"className":5002},[35],[28,5004,5006],{"className":5005},[39],[41,5007,5008],{"xmlns":43},[46,5009,5010,5014],{},[49,5011,5012],{},[55,5013,1174],{"mathvariant":1088},[82,5015,1177],{"encoding":84},[28,5017,5019],{"className":5018,"ariaHidden":71},[89],[28,5020,5022,5025],{"className":5021},[93],[28,5023],{"className":5024,"style":1102},[97],[28,5026,1174],{"className":5027},[102],". The transverse component ",[28,5030,5032,5045],{"className":5031},[35],[28,5033,5035],{"className":5034},[39],[41,5036,5037],{"xmlns":43},[46,5038,5039,5043],{},[49,5040,5041],{},[55,5042,1089],{"mathvariant":1088},[82,5044,1092],{"encoding":84},[28,5046,5048],{"className":5047,"ariaHidden":71},[89],[28,5049,5051,5054],{"className":5050},[93],[28,5052],{"className":5053,"style":1102},[97],[28,5055,1089],{"className":5056},[102]," drives population transfer. A longitudinal component ",[28,5059,5061,5074],{"className":5060},[35],[28,5062,5064],{"className":5063},[39],[41,5065,5066],{"xmlns":43},[46,5067,5068,5072],{},[49,5069,5070],{},[55,5071,1174],{"mathvariant":1088},[82,5073,1177],{"encoding":84},[28,5075,5077],{"className":5076,"ariaHidden":71},[89],[28,5078,5080,5083],{"className":5079},[93],[28,5081],{"className":5082,"style":1102},[97],[28,5084,1174],{"className":5085},[102]," tilts the rotation axis away from the resonant direction.",[251,5088,5090],{"id":5089},"excited-state-probability","Excited-State Probability",[12,5092,5093,5094,5132,5133,5161],{},"The simulation assumes that the qubit begins in the ground state ",[28,5095,5097,5114],{"className":5096},[35],[28,5098,5100],{"className":5099},[39],[41,5101,5102],{"xmlns":43},[46,5103,5104,5112],{},[49,5105,5106,5108,5110],{},[62,5107,296],{"stretchy":64},[55,5109,299],{},[62,5111,302],{"stretchy":64},[82,5113,305],{"encoding":84},[28,5115,5117],{"className":5116,"ariaHidden":71},[89],[28,5118,5120,5123,5126,5129],{"className":5119},[93],[28,5121],{"className":5122,"style":98},[97],[28,5124,296],{"className":5125},[161],[28,5127,299],{"className":5128,"style":321},[102,106],[28,5130,302],{"className":5131},[182],". Evolving under the RWA Hamiltonian, the probability of occupying the excited state after pulse duration ",[28,5134,5136,5149],{"className":5135},[35],[28,5137,5139],{"className":5138},[39],[41,5140,5141],{"xmlns":43},[46,5142,5143,5147],{},[49,5144,5145],{},[55,5146,75],{},[82,5148,235],{"encoding":84},[28,5150,5152],{"className":5151,"ariaHidden":71},[89],[28,5153,5155,5158],{"className":5154},[93],[28,5156],{"className":5157,"style":245},[97],[28,5159,75],{"className":5160,"style":178},[102,106]," 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The amplitude coefficient",[28,5882,5884],{"className":5883},[31],[28,5885,5887,5915],{"className":5886},[35],[28,5888,5890],{"className":5889},[39],[41,5891,5892],{"xmlns":43,"display":44},[46,5893,5894,5912],{},[49,5895,5896],{},[3977,5897,5898,5904],{},[2776,5899,5900,5902],{},[55,5901,1089],{"mathvariant":1088},[650,5903,848],{},[1670,5905,5906,5908,5910],{},[55,5907,1089],{"mathvariant":1088},[55,5909,1333],{},[650,5911,848],{},[82,5913,5914],{"encoding":84},"\\frac{\\Omega^2}{\\Omega_R^2}",[28,5916,5918],{"className":5917,"ariaHidden":71},[89],[28,5919,5921,5924],{"className":5920},[93],[28,5922],{"className":5923,"style":5329},[97],[28,5925,5927,5930,6057],{"className":5926},[102],[28,5928],{"className":5929},[161,4072],[28,5931,5933],{"className":5932},[3977],[28,5934,5936,6049],{"className":5935},[115,116],[28,5937,5939,6046],{"className":5938},[120],[28,5940,5942,6001,6009],{"className":5941,"style":5348},[124],[28,5943,5944,5947],{"style":4088},[28,5945],{"className":5946,"style":3434},[132],[28,5948,5950],{"className":5949},[102],[28,5951,5953,5956],{"className":5952},[102],[28,5954,1089],{"className":5955},[102],[28,5957,5959],{"className":5958},[111],[28,5960,5962,5993],{"className":5961},[115,116],[28,5963,5965,5990],{"className":5964},[120],[28,5966,5968,5979],{"className":5967,"style":5375},[124],[28,5969,5970,5973],{"style":5378},[28,5971],{"className":5972,"style":133},[132],[28,5974,5976],{"className":5975},[137,138,139,140],[28,5977,1333],{"className":5978,"style":1377},[102,106,140],[28,5980,5981,5984],{"style":5390},[28,5982],{"className":5983,"style":133},[132],[28,5985,5987],{"className":5986},[137,138,139,140],[28,5988,848],{"className":5989},[102,140],[28,5991,148],{"className":5992},[147],[28,5994,5996],{"className":5995},[120],[28,5997,5999],{"className":5998,"style":5409},[124],[28,6000],{},[28,6002,6003,6006],{"style":4100},[28,6004],{"className":6005,"style":3434},[132],[28,6007],{"className":6008,"style":4108},[4107],[28,6010,6011,6014],{"style":4111},[28,6012],{"className":6013,"style":3434},[132],[28,6015,6017],{"className":6016},[102],[28,6018,6020,6023],{"className":6019},[102],[28,6021,1089],{"className":6022},[102],[28,6024,6026],{"className":6025},[111],[28,6027,6029],{"className":6028},[115],[28,6030,6032],{"className":6031},[120],[28,6033,6035],{"className":6034,"style":5446},[124],[28,6036,6037,6040],{"style":1726},[28,6038],{"className":6039,"style":133},[132],[28,6041,6043],{"className":6042},[137,138,139,140],[28,6044,848],{"className":6045},[102,140],[28,6047,148],{"className":6048},[147],[28,6050,6052],{"className":6051},[120],[28,6053,6055],{"className":6054,"style":5467},[124],[28,6056],{},[28,6058],{"className":6059},[182,4072],[12,6061,6062,6063,6114],{},"sets the maximum possible population transfer. At exact resonance, ",[28,6064,6066,6084],{"className":6065},[35],[28,6067,6069],{"className":6068},[39],[41,6070,6071],{"xmlns":43},[46,6072,6073,6081],{},[49,6074,6075,6077,6079],{},[55,6076,1174],{"mathvariant":1088},[62,6078,845],{},[650,6080,652],{},[82,6082,6083],{"encoding":84},"\\Delta=0",[28,6085,6087,6105],{"className":6086,"ariaHidden":71},[89],[28,6088,6090,6093,6096,6099,6102],{"className":6089},[93],[28,6091],{"className":6092,"style":1102},[97],[28,6094,1174],{"className":6095},[102],[28,6097],{"className":6098,"style":913},[173],[28,6100,845],{"className":6101},[917],[28,6103],{"className":6104,"style":913},[173],[28,6106,6108,6111],{"className":6107},[93],[28,6109],{"className":6110,"style":2914},[97],[28,6112,652],{"className":6113},[102],", so",[28,6116,6118],{"className":6117},[31],[28,6119,6121,6153],{"className":6120},[35],[28,6122,6124],{"className":6123},[39],[41,6125,6126],{"xmlns":43,"display":44},[46,6127,6128,6150],{},[49,6129,6130,6146,6148],{},[3977,6131,6132,6138],{},[2776,6133,6134,6136],{},[55,6135,1089],{"mathvariant":1088},[650,6137,848],{},[1670,6139,6140,6142,6144],{},[55,6141,1089],{"mathvariant":1088},[55,6143,1333],{},[650,6145,848],{},[62,6147,845],{},[650,6149,2798],{},[82,6151,6152],{"encoding":84},"\\frac{\\Omega^2}{\\Omega_R^2}=1.",[28,6154,6156,6307],{"className":6155,"ariaHidden":71},[89],[28,6157,6159,6162,6298,6301,6304],{"className":6158},[93],[28,6160],{"className":6161,"style":5329},[97],[28,6163,6165,6168,6295],{"className":6164},[102],[28,6166],{"className":6167},[161,4072],[28,6169,6171],{"className":6170},[3977],[28,6172,6174,6287],{"className":6173},[115,116],[28,6175,6177,6284],{"className":6176},[120],[28,6178,6180,6239,6247],{"className":6179,"style":5348},[124],[28,6181,6182,6185],{"style":4088},[28,6183],{"className":6184,"style":3434},[132],[28,6186,6188],{"className":6187},[102],[28,6189,6191,6194],{"className":6190},[102],[28,6192,1089],{"className":6193},[102],[28,6195,6197],{"className":6196},[111],[28,6198,6200,6231],{"className":6199},[115,116],[28,6201,6203,6228],{"className":6202},[120],[28,6204,6206,6217],{"className":6205,"style":5375},[124],[28,6207,6208,6211],{"style":5378},[28,6209],{"className":6210,"style":133},[132],[28,6212,6214],{"className":6213},[137,138,139,140],[28,6215,1333],{"className":6216,"style":1377},[102,106,140],[28,6218,6219,6222],{"style":5390},[28,6220],{"className":6221,"style":133},[132],[28,6223,6225],{"className":6224},[137,138,139,140],[28,6226,848],{"className":6227},[102,140],[28,6229,148],{"className":6230},[147],[28,6232,6234],{"className":6233},[120],[28,6235,6237],{"className":6236,"style":5409},[124],[28,6238],{},[28,6240,6241,6244],{"style":4100},[28,6242],{"className":6243,"style":3434},[132],[28,6245],{"className":6246,"style":4108},[4107],[28,6248,6249,6252],{"style":4111},[28,6250],{"className":6251,"style":3434},[132],[28,6253,6255],{"className":6254},[102],[28,6256,6258,6261],{"className":6257},[102],[28,6259,1089],{"className":6260},[102],[28,6262,6264],{"className":6263},[111],[28,6265,6267],{"className":6266},[115],[28,6268,6270],{"className":6269},[120],[28,6271,6273],{"className":6272,"style":5446},[124],[28,6274,6275,6278],{"style":1726},[28,6276],{"className":6277,"style":133},[132],[28,6279,6281],{"className":6280},[137,138,139,140],[28,6282,848],{"className":6283},[102,140],[28,6285,148],{"className":6286},[147],[28,6288,6290],{"className":6289},[120],[28,6291,6293],{"className":6292,"style":5467},[124],[28,6294],{},[28,6296],{"className":6297},[182,4072],[28,6299],{"className":6300,"style":913},[173],[28,6302,845],{"className":6303},[917],[28,6305],{"className":6306,"style":913},[173],[28,6308,6310,6313],{"className":6309},[93],[28,6311],{"className":6312,"style":2914},[97],[28,6314,2798],{"className":6315},[102],[12,6317,6318],{},"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.",[12,6320,6321],{},"At resonance, the excited-state probability simplifies to",[28,6323,6325],{"className":6324},[31],[28,6326,6328,6382],{"className":6327},[35],[28,6329,6331],{"className":6330},[39],[41,6332,6333],{"xmlns":43,"display":44},[46,6334,6335,6379],{},[49,6336,6337,6343,6345,6347,6349,6351,6361,6377],{},[52,6338,6339,6341],{},[55,6340,57],{},[55,6342,60],{},[62,6344,65],{"stretchy":64},[55,6346,75],{},[62,6348,78],{"stretchy":64},[62,6350,845],{},[2776,6352,6353,6359],{},[49,6354,6355,6357],{},[55,6356,5217],{},[62,6358,4008],{},[650,6360,848],{},[49,6362,6363,6365,6375],{},[62,6364,65],{"fence":71},[3977,6366,6367,6373],{},[49,6368,6369,6371],{},[55,6370,1089],{"mathvariant":1088},[55,6372,75],{},[650,6374,848],{},[62,6376,78],{"fence":71},[55,6378,2106],{"mathvariant":1088},[82,6380,6381],{"encoding":84},"P_e(\\tau)=\\sin^2\\left(\\frac{\\Omega\\tau}{2}\\right).",[28,6383,6385,6449],{"className":6384,"ariaHidden":71},[89],[28,6386,6388,6391,6431,6434,6437,6440,6443,6446],{"className":6387},[93],[28,6389],{"className":6390,"style":98},[97],[28,6392,6394,6397],{"className":6393},[102],[28,6395,57],{"className":6396,"style":107},[102,106],[28,6398,6400],{"className":6399},[111],[28,6401,6403,6423],{"className":6402},[115,116],[28,6404,6406,6420],{"className":6405},[120],[28,6407,6409],{"className":6408,"style":125},[124],[28,6410,6411,6414],{"style":128},[28,6412],{"className":6413,"style":133},[132],[28,6415,6417],{"className":6416},[137,138,139,140],[28,6418,60],{"className":6419},[102,106,140],[28,6421,148],{"className":6422},[147],[28,6424,6426],{"className":6425},[120],[28,6427,6429],{"className":6428,"style":155},[124],[28,6430],{},[28,6432,65],{"className":6433},[161],[28,6435,75],{"className":6436,"style":178},[102,106],[28,6438,78],{"className":6439},[182],[28,6441],{"className":6442,"style":913},[173],[28,6444,845],{"className":6445},[917],[28,6447],{"className":6448,"style":913},[173],[28,6450,6452,6455,6484,6487,6567,6570],{"className":6451},[93],[28,6453],{"className":6454,"style":3394},[97],[28,6456,6458,6461],{"className":6457},[4241],[28,6459,5217],{"className":6460},[4241],[28,6462,6464],{"className":6463},[111],[28,6465,6467],{"className":6466},[115],[28,6468,6470],{"className":6469},[120],[28,6471,6473],{"className":6472,"style":5494},[124],[28,6474,6475,6478],{"style":5497},[28,6476],{"className":6477,"style":133},[132],[28,6479,6481],{"className":6480},[137,138,139,140],[28,6482,848],{"className":6483},[102,140],[28,6485],{"className":6486,"style":174},[173],[28,6488,6490,6496,6561],{"className":6489},[3398],[28,6491,6493],{"className":6492,"style":3403},[161,3402],[28,6494,65],{"className":6495},[3407,139],[28,6497,6499,6502,6558],{"className":6498},[102],[28,6500],{"className":6501},[161,4072],[28,6503,6505],{"className":6504},[3977],[28,6506,6508,6550],{"className":6507},[115,116],[28,6509,6511,6547],{"className":6510},[120],[28,6512,6514,6525,6533],{"className":6513,"style":5537},[124],[28,6515,6516,6519],{"style":4088},[28,6517],{"className":6518,"style":3434},[132],[28,6520,6522],{"className":6521},[102],[28,6523,848],{"className":6524},[102],[28,6526,6527,6530],{"style":4100},[28,6528],{"className":6529,"style":3434},[132],[28,6531],{"className":6532,"style":4108},[4107],[28,6534,6535,6538],{"style":4111},[28,6536],{"className":6537,"style":3434},[132],[28,6539,6541,6544],{"className":6540},[102],[28,6542,1089],{"className":6543},[102],[28,6545,75],{"className":6546,"style":178},[102,106],[28,6548,148],{"className":6549},[147],[28,6551,6553],{"className":6552},[120],[28,6554,6556],{"className":6555,"style":4170},[124],[28,6557],{},[28,6559],{"className":6560},[182,4072],[28,6562,6564],{"className":6563,"style":3403},[182,3402],[28,6565,78],{"className":6566},[3407,139],[28,6568],{"className":6569,"style":174},[173],[28,6571,2106],{"className":6572},[102],[12,6574,6575,6576,6605],{},"The pulse duration required for ideal population inversion defines the standard ",[28,6577,6579,6593],{"className":6578},[35],[28,6580,6582],{"className":6581},[39],[41,6583,6584],{"xmlns":43},[46,6585,6586,6590],{},[49,6587,6588],{},[55,6589,851],{},[82,6591,6592],{"encoding":84},"\\pi",[28,6594,6596],{"className":6595,"ariaHidden":71},[89],[28,6597,6599,6602],{"className":6598},[93],[28,6600],{"className":6601,"style":245},[97],[28,6603,851],{"className":6604,"style":321},[102,106],"-pulse time,",[28,6607,6609],{"className":6608},[31],[28,6610,6612,6662],{"className":6611},[35],[28,6613,6615],{"className":6614},[39],[41,6616,6617],{"xmlns":43,"display":44},[46,6618,6619,6659],{},[49,6620,6621,6627,6629,6635,6637,6657],{},[52,6622,6623,6625],{},[55,6624,75],{},[55,6626,851],{},[62,6628,845],{},[3977,6630,6631,6633],{},[55,6632,851],{},[55,6634,1089],{"mathvariant":1088},[62,6636,845],{},[3977,6638,6639,6641],{},[650,6640,3200],{},[49,6642,6643,6645,6647,6649,6651,6653,6655],{},[650,6644,848],{},[62,6646,65],{"stretchy":64},[55,6648,1089],{"mathvariant":1088},[55,6650,1129],{"mathvariant":1088},[650,6652,848],{},[55,6654,851],{},[62,6656,78],{"stretchy":64},[55,6658,2106],{"mathvariant":1088},[82,6660,6661],{"encoding":84},"\\tau_\\pi=\\frac{\\pi}{\\Omega}=\\frac{1}{2(\\Omega\u002F2\\pi)}.",[28,6663,6665,6721,6800],{"className":6664,"ariaHidden":71},[89],[28,6666,6668,6671,6712,6715,6718],{"className":6667},[93],[28,6669],{"className":6670,"style":780},[97],[28,6672,6674,6677],{"className":6673},[102],[28,6675,75],{"className":6676,"style":178},[102,106],[28,6678,6680],{"className":6679},[111],[28,6681,6683,6704],{"className":6682},[115,116],[28,6684,6686,6701],{"className":6685},[120],[28,6687,6689],{"className":6688,"style":125},[124],[28,6690,6692,6695],{"style":6691},"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;",[28,6693],{"className":6694,"style":133},[132],[28,6696,6698],{"className":6697},[137,138,139,140],[28,6699,851],{"className":6700,"style":321},[102,106,140],[28,6702,148],{"className":6703},[147],[28,6705,6707],{"className":6706},[120],[28,6708,6710],{"className":6709,"style":155},[124],[28,6711],{},[28,6713],{"className":6714,"style":913},[173],[28,6716,845],{"className":6717},[917],[28,6719],{"className":6720,"style":913},[173],[28,6722,6724,6728,6791,6794,6797],{"className":6723},[93],[28,6725],{"className":6726,"style":6727},[97],"height:1.7936em;vertical-align:-0.686em;",[28,6729,6731,6734,6788],{"className":6730},[102],[28,6732],{"className":6733},[161,4072],[28,6735,6737],{"className":6736},[3977],[28,6738,6740,6780],{"className":6739},[115,116],[28,6741,6743,6777],{"className":6742},[120],[28,6744,6747,6758,6766],{"className":6745,"style":6746},[124],"height:1.1076em;",[28,6748,6749,6752],{"style":4088},[28,6750],{"className":6751,"style":3434},[132],[28,6753,6755],{"className":6754},[102],[28,6756,1089],{"className":6757},[102],[28,6759,6760,6763],{"style":4100},[28,6761],{"className":6762,"style":3434},[132],[28,6764],{"className":6765,"style":4108},[4107],[28,6767,6768,6771],{"style":4111},[28,6769],{"className":6770,"style":3434},[132],[28,6772,6774],{"className":6773},[102],[28,6775,851],{"className":6776,"style":321},[102,106],[28,6778,148],{"className":6779},[147],[28,6781,6783],{"className":6782},[120],[28,6784,6786],{"className":6785,"style":4170},[124],[28,6787],{},[28,6789],{"className":6790},[182,4072],[28,6792],{"className":6793,"style":913},[173],[28,6795,845],{"className":6796},[917],[28,6798],{"className":6799,"style":913},[173],[28,6801,6803,6807,6883],{"className":6802},[93],[28,6804],{"className":6805,"style":6806},[97],"height:2.2574em;vertical-align:-0.936em;",[28,6808,6810,6813,6880],{"className":6809},[102],[28,6811],{"className":6812},[161,4072],[28,6814,6816],{"className":6815},[3977],[28,6817,6819,6871],{"className":6818},[115,116],[28,6820,6822,6868],{"className":6821},[120],[28,6823,6826,6849,6857],{"className":6824,"style":6825},[124],"height:1.3214em;",[28,6827,6828,6831],{"style":4088},[28,6829],{"className":6830,"style":3434},[132],[28,6832,6834,6837,6840,6843,6846],{"className":6833},[102],[28,6835,848],{"className":6836},[102],[28,6838,65],{"className":6839},[161],[28,6841,1149],{"className":6842},[102],[28,6844,851],{"className":6845,"style":321},[102,106],[28,6847,78],{"className":6848},[182],[28,6850,6851,6854],{"style":4100},[28,6852],{"className":6853,"style":3434},[132],[28,6855],{"className":6856,"style":4108},[4107],[28,6858,6859,6862],{"style":4111},[28,6860],{"className":6861,"style":3434},[132],[28,6863,6865],{"className":6864},[102],[28,6866,3200],{"className":6867},[102],[28,6869,148],{"className":6870},[147],[28,6872,6874],{"className":6873},[120],[28,6875,6878],{"className":6876,"style":6877},[124],"height:0.936em;",[28,6879],{},[28,6881],{"className":6882},[182,4072],[28,6884,2106],{"className":6885},[102],[12,6887,6888],{},"Given the default drive strength",[28,6890,6892],{"className":6891},[31],[28,6893,6895,6935],{"className":6894},[35],[28,6896,6898],{"className":6897},[39],[41,6899,6900],{"xmlns":43,"display":44},[46,6901,6902,6932],{},[49,6903,6904,6914,6916,6919,6921,6930],{},[3977,6905,6906,6908],{},[55,6907,1089],{"mathvariant":1088},[49,6909,6910,6912],{},[650,6911,848],{},[55,6913,851],{},[62,6915,845],{},[650,6917,6918],{},"20",[2253,6920,2255],{},[49,6922,6923,6926,6928],{},[55,6924,6925],{"mathvariant":1088},"M",[55,6927,2263],{"mathvariant":1088},[55,6929,2266],{"mathvariant":1088},[62,6931,72],{"separator":71},[82,6933,6934],{"encoding":84},"\\frac{\\Omega}{2\\pi}=20~\\mathrm{MHz},",[28,6936,6938,7019],{"className":6937,"ariaHidden":71},[89],[28,6939,6941,6945,7010,7013,7016],{"className":6940},[93],[28,6942],{"className":6943,"style":6944},[97],"height:2.0463em;vertical-align:-0.686em;",[28,6946,6948,6951,7007],{"className":6947},[102],[28,6949],{"className":6950},[161,4072],[28,6952,6954],{"className":6953},[3977],[28,6955,6957,6999],{"className":6956},[115,116],[28,6958,6960,6996],{"className":6959},[120],[28,6961,6963,6977,6985],{"className":6962,"style":5537},[124],[28,6964,6965,6968],{"style":4088},[28,6966],{"className":6967,"style":3434},[132],[28,6969,6971,6974],{"className":6970},[102],[28,6972,848],{"className":6973},[102],[28,6975,851],{"className":6976,"style":321},[102,106],[28,6978,6979,6982],{"style":4100},[28,6980],{"className":6981,"style":3434},[132],[28,6983],{"className":6984,"style":4108},[4107],[28,6986,6987,6990],{"style":4111},[28,6988],{"className":6989,"style":3434},[132],[28,6991,6993],{"className":6992},[102],[28,6994,1089],{"className":6995},[102],[28,6997,148],{"className":6998},[147],[28,7000,7002],{"className":7001},[120],[28,7003,7005],{"className":7004,"style":4170},[124],[28,7006],{},[28,7008],{"className":7009},[182,4072],[28,7011],{"className":7012,"style":913},[173],[28,7014,845],{"className":7015},[917],[28,7017],{"className":7018,"style":913},[173],[28,7020,7022,7026,7029,7032,7039],{"className":7021},[93],[28,7023],{"className":7024,"style":7025},[97],"height:0.8778em;vertical-align:-0.1944em;",[28,7027,6918],{"className":7028},[102],[28,7030,2255],{"className":7031},[173,2342],[28,7033,7035],{"className":7034},[102],[28,7036,7038],{"className":7037},[102,2349],"MHz",[28,7040,72],{"className":7041},[169],[12,7043,7044,7045,7073],{},"the ideal ",[28,7046,7048,7061],{"className":7047},[35],[28,7049,7051],{"className":7050},[39],[41,7052,7053],{"xmlns":43},[46,7054,7055,7059],{},[49,7056,7057],{},[55,7058,851],{},[82,7060,6592],{"encoding":84},[28,7062,7064],{"className":7063,"ariaHidden":71},[89],[28,7065,7067,7070],{"className":7066},[93],[28,7068],{"className":7069,"style":245},[97],[28,7071,851],{"className":7072,"style":321},[102,106],"-pulse time is",[28,7075,7077],{"className":7076},[31],[28,7078,7080,7115],{"className":7079},[35],[28,7081,7083],{"className":7082},[39],[41,7084,7085],{"xmlns":43,"display":44},[46,7086,7087,7112],{},[49,7088,7089,7095,7097,7100,7102,7110],{},[52,7090,7091,7093],{},[55,7092,75],{},[55,7094,851],{},[62,7096,845],{},[650,7098,7099],{},"25",[2253,7101,2255],{},[49,7103,7104,7107],{},[55,7105,7106],{"mathvariant":1088},"n",[55,7108,7109],{"mathvariant":1088},"s",[55,7111,2106],{"mathvariant":1088},[82,7113,7114],{"encoding":84},"\\tau_\\pi=25~\\mathrm{ns}.",[28,7116,7118,7173],{"className":7117,"ariaHidden":71},[89],[28,7119,7121,7124,7164,7167,7170],{"className":7120},[93],[28,7122],{"className":7123,"style":780},[97],[28,7125,7127,7130],{"className":7126},[102],[28,7128,75],{"className":7129,"style":178},[102,106],[28,7131,7133],{"className":7132},[111],[28,7134,7136,7156],{"className":7135},[115,116],[28,7137,7139,7153],{"className":7138},[120],[28,7140,7142],{"className":7141,"style":125},[124],[28,7143,7144,7147],{"style":6691},[28,7145],{"className":7146,"style":133},[132],[28,7148,7150],{"className":7149},[137,138,139,140],[28,7151,851],{"className":7152,"style":321},[102,106,140],[28,7154,148],{"className":7155},[147],[28,7157,7159],{"className":7158},[120],[28,7160,7162],{"className":7161,"style":155},[124],[28,7163],{},[28,7165],{"className":7166,"style":913},[173],[28,7168,845],{"className":7169},[917],[28,7171],{"className":7172,"style":913},[173],[28,7174,7176,7179,7182,7185,7192],{"className":7175},[93],[28,7177],{"className":7178,"style":2914},[97],[28,7180,7099],{"className":7181},[102],[28,7183,2255],{"className":7184},[173,2342],[28,7186,7188],{"className":7187},[102],[28,7189,7191],{"className":7190},[102,2349],"ns",[28,7193,2106],{"className":7194},[102],[12,7196,7197],{},"A full period of the resonant excited-state probability oscillation is",[28,7199,7201],{"className":7200},[31],[28,7202,7204,7252],{"className":7203},[35],[28,7205,7207],{"className":7206},[39],[41,7208,7209],{"xmlns":43,"display":44},[46,7210,7211,7249],{},[49,7212,7213,7219,7221,7231,7233,7247],{},[52,7214,7215,7217],{},[55,7216,1674],{},[55,7218,1333],{},[62,7220,845],{},[3977,7222,7223,7229],{},[49,7224,7225,7227],{},[650,7226,848],{},[55,7228,851],{},[55,7230,1089],{"mathvariant":1088},[62,7232,845],{},[3977,7234,7235,7237],{},[650,7236,3200],{},[49,7238,7239,7241,7243,7245],{},[55,7240,1089],{"mathvariant":1088},[55,7242,1129],{"mathvariant":1088},[650,7244,848],{},[55,7246,851],{},[55,7248,2106],{"mathvariant":1088},[82,7250,7251],{"encoding":84},"T_R=\\frac{2\\pi}{\\Omega}=\\frac{1}{\\Omega\u002F2\\pi}.",[28,7253,7255,7310,7391],{"className":7254,"ariaHidden":71},[89],[28,7256,7258,7261,7301,7304,7307],{"className":7257},[93],[28,7259],{"className":7260,"style":585},[97],[28,7262,7264,7267],{"className":7263},[102],[28,7265,1674],{"className":7266,"style":107},[102,106],[28,7268,7270],{"className":7269},[111],[28,7271,7273,7293],{"className":7272},[115,116],[28,7274,7276,7290],{"className":7275},[120],[28,7277,7279],{"className":7278,"style":1364},[124],[28,7280,7281,7284],{"style":128},[28,7282],{"className":7283,"style":133},[132],[28,7285,7287],{"className":7286},[137,138,139,140],[28,7288,1333],{"className":7289,"style":1377},[102,106,140],[28,7291,148],{"className":7292},[147],[28,7294,7296],{"className":7295},[120],[28,7297,7299],{"className":7298,"style":155},[124],[28,7300],{},[28,7302],{"className":7303,"style":913},[173],[28,7305,845],{"className":7306},[917],[28,7308],{"className":7309,"style":913},[173],[28,7311,7313,7317,7382,7385,7388],{"className":7312},[93],[28,7314],{"className":7315,"style":7316},[97],"height:2.0074em;vertical-align:-0.686em;",[28,7318,7320,7323,7379],{"className":7319},[102],[28,7321],{"className":7322},[161,4072],[28,7324,7326],{"className":7325},[3977],[28,7327,7329,7371],{"className":7328},[115,116],[28,7330,7332,7368],{"className":7331},[120],[28,7333,7335,7346,7354],{"className":7334,"style":6825},[124],[28,7336,7337,7340],{"style":4088},[28,7338],{"className":7339,"style":3434},[132],[28,7341,7343],{"className":7342},[102],[28,7344,1089],{"className":7345},[102],[28,7347,7348,7351],{"style":4100},[28,7349],{"className":7350,"style":3434},[132],[28,7352],{"className":7353,"style":4108},[4107],[28,7355,7356,7359],{"style":4111},[28,7357],{"className":7358,"style":3434},[132],[28,7360,7362,7365],{"className":7361},[102],[28,7363,848],{"className":7364},[102],[28,7366,851],{"className":7367,"style":321},[102,106],[28,7369,148],{"className":7370},[147],[28,7372,7374],{"className":7373},[120],[28,7375,7377],{"className":7376,"style":4170},[124],[28,7378],{},[28,7380],{"className":7381},[182,4072],[28,7383],{"className":7384,"style":913},[173],[28,7386,845],{"className":7387},[917],[28,7389],{"className":7390,"style":913},[173],[28,7392,7394,7397,7462],{"className":7393},[93],[28,7395],{"className":7396,"style":6806},[97],[28,7398,7400,7403,7459],{"className":7399},[102],[28,7401],{"className":7402},[161,4072],[28,7404,7406],{"className":7405},[3977],[28,7407,7409,7451],{"className":7408},[115,116],[28,7410,7412,7448],{"className":7411},[120],[28,7413,7415,7429,7437],{"className":7414,"style":6825},[124],[28,7416,7417,7420],{"style":4088},[28,7418],{"className":7419,"style":3434},[132],[28,7421,7423,7426],{"className":7422},[102],[28,7424,1149],{"className":7425},[102],[28,7427,851],{"className":7428,"style":321},[102,106],[28,7430,7431,7434],{"style":4100},[28,7432],{"className":7433,"style":3434},[132],[28,7435],{"className":7436,"style":4108},[4107],[28,7438,7439,7442],{"style":4111},[28,7440],{"className":7441,"style":3434},[132],[28,7443,7445],{"className":7444},[102],[28,7446,3200],{"className":7447},[102],[28,7449,148],{"className":7450},[147],[28,7452,7454],{"className":7453},[120],[28,7455,7457],{"className":7456,"style":6877},[124],[28,7458],{},[28,7460],{"className":7461},[182,4072],[28,7463,2106],{"className":7464},[102],[12,7466,7467],{},"For the same default drive strength, the full temporal period is",[28,7469,7471],{"className":7470},[31],[28,7472,7474,7507],{"className":7473},[35],[28,7475,7477],{"className":7476},[39],[41,7478,7479],{"xmlns":43,"display":44},[46,7480,7481,7504],{},[49,7482,7483,7489,7491,7494,7496,7502],{},[52,7484,7485,7487],{},[55,7486,1674],{},[55,7488,1333],{},[62,7490,845],{},[650,7492,7493],{},"50",[2253,7495,2255],{},[49,7497,7498,7500],{},[55,7499,7106],{"mathvariant":1088},[55,7501,7109],{"mathvariant":1088},[55,7503,2106],{"mathvariant":1088},[82,7505,7506],{"encoding":84},"T_R=50~\\mathrm{ns}.",[28,7508,7510,7565],{"className":7509,"ariaHidden":71},[89],[28,7511,7513,7516,7556,7559,7562],{"className":7512},[93],[28,7514],{"className":7515,"style":585},[97],[28,7517,7519,7522],{"className":7518},[102],[28,7520,1674],{"className":7521,"style":107},[102,106],[28,7523,7525],{"className":7524},[111],[28,7526,7528,7548],{"className":7527},[115,116],[28,7529,7531,7545],{"className":7530},[120],[28,7532,7534],{"className":7533,"style":1364},[124],[28,7535,7536,7539],{"style":128},[28,7537],{"className":7538,"style":133},[132],[28,7540,7542],{"className":7541},[137,138,139,140],[28,7543,1333],{"className":7544,"style":1377},[102,106,140],[28,7546,148],{"className":7547},[147],[28,7549,7551],{"className":7550},[120],[28,7552,7554],{"className":7553,"style":155},[124],[28,7555],{},[28,7557],{"className":7558,"style":913},[173],[28,7560,845],{"className":7561},[917],[28,7563],{"className":7564,"style":913},[173],[28,7566,7568,7571,7574,7577,7583],{"className":7567},[93],[28,7569],{"className":7570,"style":2914},[97],[28,7572,7493],{"className":7573},[102],[28,7575,2255],{"className":7576},[173,2342],[28,7578,7580],{"className":7579},[102],[28,7581,7191],{"className":7582},[102,2349],[28,7584,2106],{"className":7585},[102],[251,7587,7589],{"id":7588},"phenomenological-dephasing","Phenomenological Dephasing",[12,7591,7592],{},"The computational model includes an optional phenomenological damping factor,",[28,7594,7596],{"className":7595},[31],[28,7597,7599,7673],{"className":7598},[35],[28,7600,7602],{"className":7601},[39],[41,7603,7604],{"xmlns":43,"display":44},[46,7605,7606,7670],{},[49,7607,7608,7614,7616,7618,7620,7622,7624,7627,7633,7635,7637,7639,7641,7643,7646,7648,7668],{},[52,7609,7610,7612],{},[55,7611,57],{},[55,7613,60],{},[62,7615,65],{"stretchy":64},[55,7617,68],{},[62,7619,72],{"separator":71},[55,7621,75],{},[62,7623,78],{"stretchy":64},[62,7625,7626],{},"→",[52,7628,7629,7631],{},[55,7630,57],{},[55,7632,60],{},[62,7634,65],{"stretchy":64},[55,7636,68],{},[62,7638,72],{"separator":71},[55,7640,75],{},[62,7642,78],{"stretchy":64},[55,7644,7645],{},"exp",[62,7647,4008],{},[49,7649,7650,7652,7654,7666],{},[62,7651,65],{"fence":71},[62,7653,1213],{},[3977,7655,7656,7658],{},[55,7657,75],{},[1670,7659,7660,7662,7664],{},[55,7661,1674],{},[650,7663,848],{},[62,7665,1679],{},[62,7667,78],{"fence":71},[55,7669,2106],{"mathvariant":1088},[82,7671,7672],{"encoding":84},"P_e(f,\\tau)\\rightarrow P_e(f,\\tau)\\exp\\left(-\\frac{\\tau}{T_2^\\ast}\\right).",[28,7674,7676,7749],{"className":7675,"ariaHidden":71},[89],[28,7677,7679,7682,7722,7725,7728,7731,7734,7737,7740,7743,7746],{"className":7678},[93],[28,7680],{"className":7681,"style":98},[97],[28,7683,7685,7688],{"className":7684},[102],[28,7686,57],{"className":7687,"style":107},[102,106],[28,7689,7691],{"className":7690},[111],[28,7692,7694,7714],{"className":7693},[115,116],[28,7695,7697,7711],{"className":7696},[120],[28,7698,7700],{"className":7699,"style":125},[124],[28,7701,7702,7705],{"style":128},[28,7703],{"className":7704,"style":133},[132],[28,7706,7708],{"className":7707},[137,138,139,140],[28,7709,60],{"className":7710},[102,106,140],[28,7712,148],{"className":7713},[147],[28,7715,7717],{"className":7716},[120],[28,7718,7720],{"className":7719,"style":155},[124],[28,7721],{},[28,7723,65],{"className":7724},[161],[28,7726,68],{"className":7727,"style":165},[102,106],[28,7729,72],{"className":7730},[169],[28,7732],{"className":7733,"style":174},[173],[28,7735,75],{"className":7736,"style":178},[102,106],[28,7738,78],{"className":7739},[182],[28,7741],{"className":7742,"style":913},[173],[28,7744,7626],{"className":7745},[917],[28,7747],{"className":7748,"style":913},[173],[28,7750,7752,7756,7796,7799,7802,7805,7808,7811,7814,7817,7820,7823,7955,7958],{"className":7751},[93],[28,7753],{"className":7754,"style":7755},[97],"height:2.4023em;vertical-align:-0.9523em;",[28,7757,7759,7762],{"className":7758},[102],[28,7760,57],{"className":7761,"style":107},[102,106],[28,7763,7765],{"className":7764},[111],[28,7766,7768,7788],{"className":7767},[115,116],[28,7769,7771,7785],{"className":7770},[120],[28,7772,7774],{"className":7773,"style":125},[124],[28,7775,7776,7779],{"style":128},[28,7777],{"className":7778,"style":133},[132],[28,7780,7782],{"className":7781},[137,138,139,140],[28,7783,60],{"className":7784},[102,106,140],[28,7786,148],{"className":7787},[147],[28,7789,7791],{"className":7790},[120],[28,7792,7794],{"className":7793,"style":155},[124],[28,7795],{},[28,7797,65],{"className":7798},[161],[28,7800,68],{"className":7801,"style":165},[102,106],[28,7803,72],{"className":7804},[169],[28,7806],{"className":7807,"style":174},[173],[28,7809,75],{"className":7810,"style":178},[102,106],[28,7812,78],{"className":7813},[182],[28,7815],{"className":7816,"style":174},[173],[28,7818,7645],{"className":7819},[4241],[28,7821],{"className":7822,"style":174},[173],[28,7824,7826,7832,7835,7949],{"className":7825},[3398],[28,7827,7829],{"className":7828,"style":3403},[161,3402],[28,7830,65],{"className":7831},[3407,139],[28,7833,1213],{"className":7834},[102],[28,7836,7838,7841,7946],{"className":7837},[102],[28,7839],{"className":7840},[161,4072],[28,7842,7844],{"className":7843},[3977],[28,7845,7847,7937],{"className":7846},[115,116],[28,7848,7850,7934],{"className":7849},[120],[28,7851,7853,7915,7923],{"className":7852,"style":6746},[124],[28,7854,7855,7858],{"style":4088},[28,7856],{"className":7857,"style":3434},[132],[28,7859,7861],{"className":7860},[102],[28,7862,7864,7867],{"className":7863},[102],[28,7865,1674],{"className":7866,"style":107},[102,106],[28,7868,7870],{"className":7869},[111],[28,7871,7873,7906],{"className":7872},[115,116],[28,7874,7876,7903],{"className":7875},[120],[28,7877,7880,7892],{"className":7878,"style":7879},[124],"height:0.6705em;",[28,7881,7883,7886],{"style":7882},"top:-2.4337em;margin-left:-0.1389em;margin-right:0.05em;",[28,7884],{"className":7885,"style":133},[132],[28,7887,7889],{"className":7888},[137,138,139,140],[28,7890,848],{"className":7891},[102,140],[28,7893,7894,7897],{"style":5390},[28,7895],{"className":7896,"style":133},[132],[28,7898,7900],{"className":7899},[137,138,139,140],[28,7901,1679],{"className":7902},[1261,140],[28,7904,148],{"className":7905},[147],[28,7907,7909],{"className":7908},[120],[28,7910,7913],{"className":7911,"style":7912},[124],"height:0.2663em;",[28,7914],{},[28,7916,7917,7920],{"style":4100},[28,7918],{"className":7919,"style":3434},[132],[28,7921],{"className":7922,"style":4108},[4107],[28,7924,7925,7928],{"style":4111},[28,7926],{"className":7927,"style":3434},[132],[28,7929,7931],{"className":7930},[102],[28,7932,75],{"className":7933,"style":178},[102,106],[28,7935,148],{"className":7936},[147],[28,7938,7940],{"className":7939},[120],[28,7941,7944],{"className":7942,"style":7943},[124],"height:0.9523em;",[28,7945],{},[28,7947],{"className":7948},[182,4072],[28,7950,7952],{"className":7951,"style":3403},[182,3402],[28,7953,78],{"className":7954},[3407,139],[28,7956],{"className":7957,"style":174},[173],[28,7959,2106],{"className":7960},[102],[12,7962,7963],{},"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 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in ordinary frequency units, the generalized Rabi frequency 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[161,4072],[28,8587,8589],{"className":8588},[3977],[28,8590,8592,8634],{"className":8591},[115,116],[28,8593,8595,8631],{"className":8594},[120],[28,8596,8598,8612,8620],{"className":8597,"style":5537},[124],[28,8599,8600,8603],{"style":4088},[28,8601],{"className":8602,"style":3434},[132],[28,8604,8606,8609],{"className":8605},[102],[28,8607,848],{"className":8608},[102],[28,8610,851],{"className":8611,"style":321},[102,106],[28,8613,8614,8617],{"style":4100},[28,8615],{"className":8616,"style":3434},[132],[28,8618],{"className":8619,"style":4108},[4107],[28,8621,8622,8625],{"style":4111},[28,8623],{"className":8624,"style":3434},[132],[28,8626,8628],{"className":8627},[102],[28,8629,1089],{"className":8630},[102],[28,8632,148],{"className":8633},[147],[28,8635,8637],{"className":8636},[120],[28,8638,8640],{"className":8639,"style":4170},[124],[28,8641],{},[28,8643],{"className":8644},[182,4072],[28,8646,8648],{"className":8647,"style":3403},[182,3402],[28,8649,78],{"className":8650},[3407,139],[28,8652,8654],{"className":8653},[111],[28,8655,8657],{"className":8656},[115],[28,8658,8660],{"className":8659},[120],[28,8661,8664],{"className":8662,"style":8663},[124],"height:1.654em;",[28,8665,8667,8670],{"style":8666},"top:-3.9029em;margin-right:0.05em;",[28,8668],{"className":8669,"style":133},[132],[28,8671,8673],{"className":8672},[137,138,139,140],[28,8674,848],{"className":8675},[102,140],[28,8677],{"className":8678,"style":1257},[173],[28,8680,2606],{"className":8681},[1261],[28,8683],{"className":8684,"style":1257},[173],[28,8686,65],{"className":8687},[161],[28,8689,68],{"className":8690,"style":165},[102,106],[28,8692],{"className":8693,"style":1257},[173],[28,8695,1213],{"className":8696},[1261],[28,8698],{"className":8699,"style":1257},[173],[28,8701,8703,8706],{"className":8702},[102],[28,8704,68],{"className":8705,"style":165},[102,106],[28,8707,8709],{"className":8708},[111],[28,8710,8712,8732],{"className":8711},[115,116],[28,8713,8715,8729],{"className":8714},[120],[28,8716,8718],{"className":8717,"style":683},[124],[28,8719,8720,8723],{"style":686},[28,8721],{"className":8722,"style":133},[132],[28,8724,8726],{"className":8725},[137,138,139,140],[28,8727,652],{"className":8728},[102,140],[28,8730,148],{"className":8731},[147],[28,8733,8735],{"className":8734},[120],[28,8736,8738],{"className":8737,"style":155},[124],[28,8739],{},[28,8741,8743,8746],{"className":8742},[182],[28,8744,78],{"className":8745},[182],[28,8747,8749],{"className":8748},[111],[28,8750,8752],{"className":8751},[115],[28,8753,8755],{"className":8754},[120],[28,8756,8758],{"className":8757,"style":5792},[124],[28,8759,8760,8763],{"style":5795},[28,8761],{"className":8762,"style":133},[132],[28,8764,8766],{"className":8765},[137,138,139,140],[28,8767,848],{"className":8768},[102,140],[28,8770,8772,8775],{"style":8771},"top:-3.9459em;",[28,8773],{"className":8774,"style":8564},[132],[28,8776,8779],{"className":8777,"style":8778},[5852],"min-width:1.02em;height:3.08em;",[5855,8780,8783],{"xmlns":5857,"width":5858,"height":8781,"viewBox":8782,"preserveAspectRatio":5861},"3.08em","0 0 400000 3240",[5863,8784],{"d":8785},"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",[28,8787,148],{"className":8788},[147],[28,8790,8792],{"className":8791},[120],[28,8793,8796],{"className":8794,"style":8795},[124],"height:1.0541em;",[28,8797],{},[28,8799,2106],{"className":8800},[102],[12,8802,8803],{},"This expression explains the ridge visible in the Fourier-domain heatmap. At resonance, the generalized frequency equals the applied drive strength,",[28,8805,8807],{"className":8806},[31],[28,8808,8810,8852],{"className":8809},[35],[28,8811,8813],{"className":8812},[39],[41,8814,8815],{"xmlns":43,"display":44},[46,8816,8817,8849],{},[49,8818,8819,8825,8827,8833,8835,8837,8847],{},[52,8820,8821,8823],{},[55,8822,1412],{},[55,8824,1333],{},[62,8826,65],{"stretchy":64},[52,8828,8829,8831],{},[55,8830,68],{},[650,8832,652],{},[62,8834,78],{"stretchy":64},[62,8836,845],{},[3977,8838,8839,8841],{},[55,8840,1089],{"mathvariant":1088},[49,8842,8843,8845],{},[650,8844,848],{},[55,8846,851],{},[55,8848,2106],{"mathvariant":1088},[82,8850,8851],{"encoding":84},"\\nu_R(f_0)=\\frac{\\Omega}{2\\pi}.",[28,8853,8855,8956],{"className":8854,"ariaHidden":71},[89],[28,8856,8858,8861,8901,8904,8944,8947,8950,8953],{"className":8857},[93],[28,8859],{"className":8860,"style":98},[97],[28,8862,8864,8867],{"className":8863},[102],[28,8865,1412],{"className":8866,"style":1433},[102,106],[28,8868,8870],{"className":8869},[111],[28,8871,8873,8893],{"className":8872},[115,116],[28,8874,8876,8890],{"className":8875},[120],[28,8877,8879],{"className":8878,"style":1364},[124],[28,8880,8881,8884],{"style":1448},[28,8882],{"className":8883,"style":133},[132],[28,8885,8887],{"className":8886},[137,138,139,140],[28,8888,1333],{"className":8889,"style":1377},[102,106,140],[28,8891,148],{"className":8892},[147],[28,8894,8896],{"className":8895},[120],[28,8897,8899],{"className":8898,"style":155},[124],[28,8900],{},[28,8902,65],{"className":8903},[161],[28,8905,8907,8910],{"className":8906},[102],[28,8908,68],{"className":8909,"style":165},[102,106],[28,8911,8913],{"className":8912},[111],[28,8914,8916,8936],{"className":8915},[115,116],[28,8917,8919,8933],{"className":8918},[120],[28,8920,8922],{"className":8921,"style":683},[124],[28,8923,8924,8927],{"style":686},[28,8925],{"className":8926,"style":133},[132],[28,8928,8930],{"className":8929},[137,138,139,140],[28,8931,652],{"className":8932},[102,140],[28,8934,148],{"className":8935},[147],[28,8937,8939],{"className":8938},[120],[28,8940,8942],{"className":8941,"style":155},[124],[28,8943],{},[28,8945,78],{"className":8946},[182],[28,8948],{"className":8949,"style":913},[173],[28,8951,845],{"className":8952},[917],[28,8954],{"className":8955,"style":913},[173],[28,8957,8959,8962,9027],{"className":8958},[93],[28,8960],{"className":8961,"style":6944},[97],[28,8963,8965,8968,9024],{"className":8964},[102],[28,8966],{"className":8967},[161,4072],[28,8969,8971],{"className":8970},[3977],[28,8972,8974,9016],{"className":8973},[115,116],[28,8975,8977,9013],{"className":8976},[120],[28,8978,8980,8994,9002],{"className":8979,"style":5537},[124],[28,8981,8982,8985],{"style":4088},[28,8983],{"className":8984,"style":3434},[132],[28,8986,8988,8991],{"className":8987},[102],[28,8989,848],{"className":8990},[102],[28,8992,851],{"className":8993,"style":321},[102,106],[28,8995,8996,8999],{"style":4100},[28,8997],{"className":8998,"style":3434},[132],[28,9000],{"className":9001,"style":4108},[4107],[28,9003,9004,9007],{"style":4111},[28,9005],{"className":9006,"style":3434},[132],[28,9008,9010],{"className":9009},[102],[28,9011,1089],{"className":9012},[102],[28,9014,148],{"className":9015},[147],[28,9017,9019],{"className":9018},[120],[28,9020,9022],{"className":9021,"style":4170},[124],[28,9023],{},[28,9025],{"className":9026},[182,4072],[28,9028,2106],{"className":9029},[102],[12,9031,9032,9033,9102,9103,9205],{},"As the system moves away from resonance, ",[28,9034,9036,9053],{"className":9035},[35],[28,9037,9039],{"className":9038},[39],[41,9040,9041],{"xmlns":43},[46,9042,9043,9051],{},[49,9044,9045],{},[52,9046,9047,9049],{},[55,9048,1412],{},[55,9050,1333],{},[82,9052,1417],{"encoding":84},[28,9054,9056],{"className":9055,"ariaHidden":71},[89],[28,9057,9059,9062],{"className":9058},[93],[28,9060],{"className":9061,"style":780},[97],[28,9063,9065,9068],{"className":9064},[102],[28,9066,1412],{"className":9067,"style":1433},[102,106],[28,9069,9071],{"className":9070},[111],[28,9072,9074,9094],{"className":9073},[115,116],[28,9075,9077,9091],{"className":9076},[120],[28,9078,9080],{"className":9079,"style":1364},[124],[28,9081,9082,9085],{"style":1448},[28,9083],{"className":9084,"style":133},[132],[28,9086,9088],{"className":9087},[137,138,139,140],[28,9089,1333],{"className":9090,"style":1377},[102,106,140],[28,9092,148],{"className":9093},[147],[28,9095,9097],{"className":9096},[120],[28,9098,9100],{"className":9099,"style":155},[124],[28,9101],{}," increases symmetrically with ",[28,9104,9106,9132],{"className":9105},[35],[28,9107,9109],{"className":9108},[39],[41,9110,9111],{"xmlns":43},[46,9112,9113,9129],{},[49,9114,9115,9117,9119,9121,9127],{},[62,9116,296],{"stretchy":64},[55,9118,68],{},[62,9120,1213],{},[52,9122,9123,9125],{},[55,9124,68],{},[650,9126,652],{},[62,9128,296],{"stretchy":64},[82,9130,9131],{"encoding":84},"\\lvert f-f_0\\rvert",[28,9133,9135,9156],{"className":9134,"ariaHidden":71},[89],[28,9136,9138,9141,9144,9147,9150,9153],{"className":9137},[93],[28,9139],{"className":9140,"style":98},[97],[28,9142,296],{"className":9143},[161],[28,9145,68],{"className":9146,"style":165},[102,106],[28,9148],{"className":9149,"style":1257},[173],[28,9151,1213],{"className":9152},[1261],[28,9154],{"className":9155,"style":1257},[173],[28,9157,9159,9162,9202],{"className":9158},[93],[28,9160],{"className":9161,"style":98},[97],[28,9163,9165,9168],{"className":9164},[102],[28,9166,68],{"className":9167,"style":165},[102,106],[28,9169,9171],{"className":9170},[111],[28,9172,9174,9194],{"className":9173},[115,116],[28,9175,9177,9191],{"className":9176},[120],[28,9178,9180],{"className":9179,"style":683},[124],[28,9181,9182,9185],{"style":686},[28,9183],{"className":9184,"style":133},[132],[28,9186,9188],{"className":9187},[137,138,139,140],[28,9189,652],{"className":9190},[102,140],[28,9192,148],{"className":9193},[147],[28,9195,9197],{"className":9196},[120],[28,9198,9200],{"className":9199,"style":155},[124],[28,9201],{},[28,9203,296],{"className":9204},[182],". 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.",[251,9207,9209],{"id":9208},"quantum-information-interpretation","Quantum Information Interpretation",[12,9211,9212,9213,9283],{},"In quantum information language, a calibrated resonant Rabi pulse implements a single-qubit rotation. At exact resonance, the effective Hamiltonian is proportional to 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so the microwave pulse 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a phase convention set by the microwave drive phase.",[12,10133,10134],{},"Detuning changes the rotation axis so it is no longer purely transverse. The unitary can be written as",[28,10136,10138],{"className":10137},[31],[28,10139,10141,10213],{"className":10140},[35],[28,10142,10144],{"className":10143},[39],[41,10145,10146],{"xmlns":43,"display":44},[46,10147,10148,10210],{},[49,10149,10150,10162,10164,10166,10168,10170,10172,10174,10208],{},[52,10151,10152,10154],{},[55,10153,1333],{},[10155,10156,10157,10159],"mover",{"accent":71},[55,10158,7106],{},[62,10160,10161],{},"^",[62,10163,65],{"stretchy":64},[55,10165,9309],{},[62,10167,78],{"stretchy":64},[62,10169,845],{},[55,10171,7645],{},[62,10173,4008],{},[49,10175,10176,10178,10180,10190,10196,10199,10206],{},[62,10177,65],{"fence":71},[62,10179,1213],{},[3977,10181,10182,10188],{},[49,10183,10184,10186],{},[55,10185,3256],{},[55,10187,9309],{},[650,10189,848],{},[10155,10191,10192,10194],{"accent":71},[55,10193,7106],{},[62,10195,10161],{},[62,10197,10198],{},"⋅",[10155,10200,10201,10203],{"accent":71},[55,10202,3166],{},[62,10204,10205],{},"⃗",[62,10207,78],{"fence":71},[62,10209,72],{"separator":71},[82,10211,10212],{"encoding":84},"R_{\\hat{n}}(\\theta)=\\exp\\left(-\\frac{i\\theta}{2}\\hat{n}\\cdot\\vec{\\sigma}\\right),",[28,10214,10216,10316],{"className":10215,"ariaHidden":71},[89],[28,10217,10219,10222,10298,10301,10304,10307,10310,10313],{"className":10218},[93],[28,10220],{"className":10221,"style":98},[97],[28,10223,10225,10228],{"className":10224},[102],[28,10226,1333],{"className":10227,"style":1377},[102,106],[28,10229,10231],{"className":10230},[111],[28,10232,10234,10290],{"className":10233},[115,116],[28,10235,10237,10287],{"className":10236},[120],[28,10238,10241],{"className":10239,"style":10240},[124],"height:0.3361em;",[28,10242,10243,10246],{"style":9376},[28,10244],{"className":10245,"style":133},[132],[28,10247,10249],{"className":10248},[137,138,139,140],[28,10250,10252],{"className":10251},[102,140],[28,10253,10256],{"className":10254},[102,10255,140],"accent",[28,10257,10259],{"className":10258},[115],[28,10260,10262],{"className":10261},[120],[28,10263,10265,10274],{"className":10264,"style":9597},[124],[28,10266,10268,10271],{"style":10267},"top:-2.7em;",[28,10269],{"className":10270,"style":133},[132],[28,10272,7106],{"className":10273},[102,106,140],[28,10275,10276,10279],{"style":10267},[28,10277],{"className":10278,"style":133},[132],[28,10280,10284],{"className":10281,"style":10283},[10282],"accent-body","left:-0.25em;",[28,10285,10161],{"className":10286},[102,140],[28,10288,148],{"className":10289},[147],[28,10291,10293],{"className":10292},[120],[28,10294,10296],{"className":10295,"style":155},[124],[28,10297],{},[28,10299,65],{"className":10300},[161],[28,10302,9309],{"className":10303,"style":9403},[102,106],[28,10305,78],{"className":10306},[182],[28,10308],{"className":10309,"style":913},[173],[28,10311,845],{"className":10312},[917],[28,10314],{"className":10315,"style":913},[173],[28,10317,10319,10322,10325,10328,10496,10499],{"className":10318},[93],[28,10320],{"className":10321,"style":3394},[97],[28,10323,7645],{"className":10324},[4241],[28,10326],{"className":10327,"style":174},[173],[28,10329,10331,10337,10340,10405,10436,10439,10442,10445,10490],{"className":10330},[3398],[28,10332,10334],{"className":10333,"style":3403},[161,3402],[28,10335,65],{"className":10336},[3407,139],[28,10338,1213],{"className":10339},[102],[28,10341,10343,10346,10402],{"className":10342},[102],[28,10344],{"className":10345},[161,4072],[28,10347,10349],{"className":10348},[3977],[28,10350,10352,10394],{"className":10351},[115,116],[28,10353,10355,10391],{"className":10354},[120],[28,10356,10358,10369,10377],{"className":10357,"style":9458},[124],[28,10359,10360,10363],{"style":4088},[28,10361],{"className":10362,"style":3434},[132],[28,10364,10366],{"className":10365},[102],[28,10367,848],{"className":10368},[102],[28,10370,10371,10374],{"style":4100},[28,10372],{"className":10373,"style":3434},[132],[28,10375],{"className":10376,"style":4108},[4107],[28,10378,10379,10382],{"style":4111},[28,10380],{"className":10381,"style":3434},[132],[28,10383,10385,10388],{"className":10384},[102],[28,10386,3256],{"className":10387},[102,106],[28,10389,9309],{"className":10390,"style":9403},[102,106],[28,10392,148],{"className":10393},[147],[28,10395,10397],{"className":10396},[120],[28,10398,10400],{"className":10399,"style":4170},[124],[28,10401],{},[28,10403],{"className":10404},[182,4072],[28,10406,10408],{"className":10407},[102,10255],[28,10409,10411],{"className":10410},[115],[28,10412,10414],{"className":10413},[120],[28,10415,10417,10425],{"className":10416,"style":9597},[124],[28,10418,10419,10422],{"style":10025},[28,10420],{"className":10421,"style":3434},[132],[28,10423,7106],{"className":10424},[102,106],[28,10426,10427,10430],{"style":10025},[28,10428],{"className":10429,"style":3434},[132],[28,10431,10433],{"className":10432,"style":10283},[10282],[28,10434,10161],{"className":10435},[102],[28,10437],{"className":10438,"style":1257},[173],[28,10440,10198],{"className":10441},[1261],[28,10443],{"className":10444,"style":1257},[173],[28,10446,10448],{"className":10447},[102,10255],[28,10449,10451],{"className":10450},[115],[28,10452,10454],{"className":10453},[120],[28,10455,10458,10466],{"className":10456,"style":10457},[124],"height:0.714em;",[28,10459,10460,10463],{"style":10025},[28,10461],{"className":10462,"style":3434},[132],[28,10464,3166],{"className":10465,"style":321},[102,106],[28,10467,10468,10471],{"style":10025},[28,10469],{"className":10470,"style":3434},[132],[28,10472,10475],{"className":10473,"style":10474},[10282],"left:-0.2355em;",[28,10476,10480],{"className":10477,"style":10479},[10478],"overlay","height:0.714em;width:0.471em;",[5855,10481,10487],{"xmlns":5857,"width":10482,"height":10483,"style":10484,"viewBox":10485,"preserveAspectRatio":10486},"0.471em","0.714em","width:0.471em","0 0 471 714","xMinYMin",[5863,10488],{"d":10489},"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z",[28,10491,10493],{"className":10492,"style":3403},[182,3402],[28,10494,78],{"className":10495},[3407,139],[28,10497],{"className":10498,"style":174},[173],[28,10500,72],{"className":10501},[169],[12,10503,5636],{},[28,10505,10507],{"className":10506},[31],[28,10508,10510,10572],{"className":10509},[35],[28,10511,10513],{"className":10512},[39],[41,10514,10515],{"xmlns":43,"display":44},[46,10516,10517,10569],{},[49,10518,10519,10525,10527,10537,10539,10541,10543,10545,10547,10549,10551,10553,10555,10557,10559,10565,10567],{},[10155,10520,10521,10523],{"accent":71},[55,10522,7106],{},[62,10524,10161],{},[62,10526,845],{},[3977,10528,10529,10531],{},[650,10530,3200],{},[52,10532,10533,10535],{},[55,10534,1089],{"mathvariant":1088},[55,10536,1333],{},[62,10538,65],{"stretchy":64},[55,10540,1089],{"mathvariant":1088},[62,10542,72],{"separator":71},[650,10544,652],{},[62,10546,72],{"separator":71},[55,10548,1174],{"mathvariant":1088},[62,10550,78],{"stretchy":64},[62,10552,72],{"separator":71},[173,10554],{"width":3222},[55,10556,9309],{},[62,10558,845],{},[52,10560,10561,10563],{},[55,10562,1089],{"mathvariant":1088},[55,10564,1333],{},[55,10566,75],{},[55,10568,2106],{"mathvariant":1088},[82,10570,10571],{"encoding":84},"\\hat{n}=\\frac{1}{\\Omega_R}(\\Omega,0,\\Delta), \\qquad \\theta=\\Omega_R\\tau.",[28,10573,10575,10621,10776],{"className":10574,"ariaHidden":71},[89],[28,10576,10578,10581,10612,10615,10618],{"className":10577},[93],[28,10579],{"className":10580,"style":9597},[97],[28,10582,10584],{"className":10583},[102,10255],[28,10585,10587],{"className":10586},[115],[28,10588,10590],{"className":10589},[120],[28,10591,10593,10601],{"className":10592,"style":9597},[124],[28,10594,10595,10598],{"style":10025},[28,10596],{"className":10597,"style":3434},[132],[28,10599,7106],{"className":10600},[102,106],[28,10602,10603,10606],{"style":10025},[28,10604],{"className":10605,"style":3434},[132],[28,10607,10609],{"className":10608,"style":10283},[10282],[28,10610,10161],{"className":10611},[102],[28,10613],{"className":10614,"style":913},[173],[28,10616,845],{"className":10617},[917],[28,10619],{"className":10620,"style":913},[173],[28,10622,10624,10628,10728,10731,10734,10737,10740,10743,10746,10749,10752,10755,10758,10761,10764,10767,10770,10773],{"className":10623},[93],[28,10625],{"className":10626,"style":10627},[97],"height:2.1574em;vertical-align:-0.836em;",[28,10629,10631,10634,10725],{"className":10630},[102],[28,10632],{"className":10633},[161,4072],[28,10635,10637],{"className":10636},[3977],[28,10638,10640,10716],{"className":10639},[115,116],[28,10641,10643,10713],{"className":10642},[120],[28,10644,10646,10694,10702],{"className":10645,"style":6825},[124],[28,10647,10648,10651],{"style":4088},[28,10649],{"className":10650,"style":3434},[132],[28,10652,10654],{"className":10653},[102],[28,10655,10657,10660],{"className":10656},[102],[28,10658,1089],{"className":10659},[102],[28,10661,10663],{"className":10662},[111],[28,10664,10666,10686],{"className":10665},[115,116],[28,10667,10669,10683],{"className":10668},[120],[28,10670,10672],{"className":10671,"style":1364},[124],[28,10673,10674,10677],{"style":1367},[28,10675],{"className":10676,"style":133},[132],[28,10678,10680],{"className":10679},[137,138,139,140],[28,10681,1333],{"className":10682,"style":1377},[102,106,140],[28,10684,148],{"className":10685},[147],[28,10687,10689],{"className":10688},[120],[28,10690,10692],{"className":10691,"style":155},[124],[28,10693],{},[28,10695,10696,10699],{"style":4100},[28,10697],{"className":10698,"style":3434},[132],[28,10700],{"className":10701,"style":4108},[4107],[28,10703,10704,10707],{"style":4111},[28,10705],{"className":10706,"style":3434},[132],[28,10708,10710],{"className":10709},[102],[28,10711,3200],{"className":10712},[102],[28,10714,148],{"className":10715},[147],[28,10717,10719],{"className":10718},[120],[28,10720,10723],{"className":10721,"style":10722},[124],"height:0.836em;",[28,10724],{},[28,10726],{"className":10727},[182,4072],[28,10729,65],{"className":10730},[161],[28,10732,1089],{"className":10733},[102],[28,10735,72],{"className":10736},[169],[28,10738],{"className":10739,"style":174},[173],[28,10741,652],{"className":10742},[102],[28,10744,72],{"className":10745},[169],[28,10747],{"className":10748,"style":174},[173],[28,10750,1174],{"className":10751},[102],[28,10753,78],{"className":10754},[182],[28,10756,72],{"className":10757},[169],[28,10759],{"className":10760,"style":3533},[173],[28,10762],{"className":10763,"style":174},[173],[28,10765,9309],{"className":10766,"style":9403},[102,106],[28,10768],{"className":10769,"style":913},[173],[28,10771,845],{"className":10772},[917],[28,10774],{"className":10775,"style":913},[173],[28,10777,10779,10782,10822,10825],{"className":10778},[93],[28,10780],{"className":10781,"style":585},[97],[28,10783,10785,10788],{"className":10784},[102],[28,10786,1089],{"className":10787},[102],[28,10789,10791],{"className":10790},[111],[28,10792,10794,10814],{"className":10793},[115,116],[28,10795,10797,10811],{"className":10796},[120],[28,10798,10800],{"className":10799,"style":1364},[124],[28,10801,10802,10805],{"style":1367},[28,10803],{"className":10804,"style":133},[132],[28,10806,10808],{"className":10807},[137,138,139,140],[28,10809,1333],{"className":10810,"style":1377},[102,106,140],[28,10812,148],{"className":10813},[147],[28,10815,10817],{"className":10816},[120],[28,10818,10820],{"className":10819,"style":155},[124],[28,10821],{},[28,10823,75],{"className":10824,"style":178},[102,106],[28,10826,2106],{"className":10827},[102],[12,10829,10830],{},"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.",[251,10832,10834],{"id":10833},"bloch-sphere-interpretation","Bloch-Sphere Interpretation",[12,10836,10837],{},"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.",[12,10839,10840,10841,10879,10880,10908,10909,10944],{},"A resonant microwave drive rotates the Bloch vector around a transverse axis. Starting from ",[28,10842,10844,10861],{"className":10843},[35],[28,10845,10847],{"className":10846},[39],[41,10848,10849],{"xmlns":43},[46,10850,10851,10859],{},[49,10852,10853,10855,10857],{},[62,10854,296],{"stretchy":64},[55,10856,299],{},[62,10858,302],{"stretchy":64},[82,10860,305],{"encoding":84},[28,10862,10864],{"className":10863,"ariaHidden":71},[89],[28,10865,10867,10870,10873,10876],{"className":10866},[93],[28,10868],{"className":10869,"style":98},[97],[28,10871,296],{"className":10872},[161],[28,10874,299],{"className":10875,"style":321},[102,106],[28,10877,302],{"className":10878},[182],", a properly timed ",[28,10881,10883,10896],{"className":10882},[35],[28,10884,10886],{"className":10885},[39],[41,10887,10888],{"xmlns":43},[46,10889,10890,10894],{},[49,10891,10892],{},[55,10893,851],{},[82,10895,6592],{"encoding":84},[28,10897,10899],{"className":10898,"ariaHidden":71},[89],[28,10900,10902,10905],{"className":10901},[93],[28,10903],{"className":10904,"style":245},[97],[28,10906,851],{"className":10907,"style":321},[102,106],"-pulse moves the state to the excited-state pole. A ",[28,10910,10912,10929],{"className":10911},[35],[28,10913,10915],{"className":10914},[39],[41,10916,10917],{"xmlns":43},[46,10918,10919,10927],{},[49,10920,10921,10923,10925],{},[55,10922,851],{},[55,10924,1129],{"mathvariant":1088},[650,10926,848],{},[82,10928,9878],{"encoding":84},[28,10930,10932],{"className":10931,"ariaHidden":71},[89],[28,10933,10935,10938,10941],{"className":10934},[93],[28,10936],{"className":10937,"style":98},[97],[28,10939,851],{"className":10940,"style":321},[102,106],[28,10942,1614],{"className":10943},[102],"-pulse moves the state to the equator, where the ground-state and excited-state measurement probabilities are equal.",[12,10946,10947,10948,10976],{},"Detuning tilts the rotation axis toward the ",[28,10949,10951,10964],{"className":10950},[35],[28,10952,10954],{"className":10953},[39],[41,10955,10956],{"xmlns":43},[46,10957,10958,10962],{},[49,10959,10960],{},[55,10961,2266],{},[82,10963,2266],{"encoding":84},[28,10965,10967],{"className":10966,"ariaHidden":71},[89],[28,10968,10970,10973],{"className":10969},[93],[28,10971],{"className":10972,"style":245},[97],[28,10974,2266],{"className":10975,"style":3753},[102,106],"-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",[28,10978,10980],{"className":10979},[31],[28,10981,10983,11013],{"className":10982},[35],[28,10984,10986],{"className":10985},[39],[41,10987,10988],{"xmlns":43,"display":44},[46,10989,10990,11010],{},[49,10991,10992,11008],{},[3977,10993,10994,11000],{},[2776,10995,10996,10998],{},[55,10997,1089],{"mathvariant":1088},[650,10999,848],{},[1670,11001,11002,11004,11006],{},[55,11003,1089],{"mathvariant":1088},[55,11005,1333],{},[650,11007,848],{},[55,11009,2106],{"mathvariant":1088},[82,11011,11012],{"encoding":84},"\\frac{\\Omega^2}{\\Omega_R^2}.",[28,11014,11016],{"className":11015,"ariaHidden":71},[89],[28,11017,11019,11022,11158],{"className":11018},[93],[28,11020],{"className":11021,"style":5329},[97],[28,11023,11025,11028,11155],{"className":11024},[102],[28,11026],{"className":11027},[161,4072],[28,11029,11031],{"className":11030},[3977],[28,11032,11034,11147],{"className":11033},[115,116],[28,11035,11037,11144],{"className":11036},[120],[28,11038,11040,11099,11107],{"className":11039,"style":5348},[124],[28,11041,11042,11045],{"style":4088},[28,11043],{"className":11044,"style":3434},[132],[28,11046,11048],{"className":11047},[102],[28,11049,11051,11054],{"className":11050},[102],[28,11052,1089],{"className":11053},[102],[28,11055,11057],{"className":11056},[111],[28,11058,11060,11091],{"className":11059},[115,116],[28,11061,11063,11088],{"className":11062},[120],[28,11064,11066,11077],{"className":11065,"style":5375},[124],[28,11067,11068,11071],{"style":5378},[28,11069],{"className":11070,"style":133},[132],[28,11072,11074],{"className":11073},[137,138,139,140],[28,11075,1333],{"className":11076,"style":1377},[102,106,140],[28,11078,11079,11082],{"style":5390},[28,11080],{"className":11081,"style":133},[132],[28,11083,11085],{"className":11084},[137,138,139,140],[28,11086,848],{"className":11087},[102,140],[28,11089,148],{"className":11090},[147],[28,11092,11094],{"className":11093},[120],[28,11095,11097],{"className":11096,"style":5409},[124],[28,11098],{},[28,11100,11101,11104],{"style":4100},[28,11102],{"className":11103,"style":3434},[132],[28,11105],{"className":11106,"style":4108},[4107],[28,11108,11109,11112],{"style":4111},[28,11110],{"className":11111,"style":3434},[132],[28,11113,11115],{"className":11114},[102],[28,11116,11118,11121],{"className":11117},[102],[28,11119,1089],{"className":11120},[102],[28,11122,11124],{"className":11123},[111],[28,11125,11127],{"className":11126},[115],[28,11128,11130],{"className":11129},[120],[28,11131,11133],{"className":11132,"style":5446},[124],[28,11134,11135,11138],{"style":1726},[28,11136],{"className":11137,"style":133},[132],[28,11139,11141],{"className":11140},[137,138,139,140],[28,11142,848],{"className":11143},[102,140],[28,11145,148],{"className":11146},[147],[28,11148,11150],{"className":11149},[120],[28,11151,11153],{"className":11152,"style":5467},[124],[28,11154],{},[28,11156],{"className":11157},[182,4072],[28,11159,2106],{"className":11160},[102],[12,11162,11163,11164,11203],{},"As ",[28,11165,11167,11185],{"className":11166},[35],[28,11168,11170],{"className":11169},[39],[41,11171,11172],{"xmlns":43},[46,11173,11174,11182],{},[49,11175,11176,11178,11180],{},[62,11177,296],{"stretchy":64},[55,11179,1174],{"mathvariant":1088},[62,11181,296],{"stretchy":64},[82,11183,11184],{"encoding":84},"\\lvert\\Delta\\rvert",[28,11186,11188],{"className":11187,"ariaHidden":71},[89],[28,11189,11191,11194,11197,11200],{"className":11190},[93],[28,11192],{"className":11193,"style":98},[97],[28,11195,296],{"className":11196},[161],[28,11198,1174],{"className":11199},[102],[28,11201,296],{"className":11202},[182]," grows, the rotation axis becomes more longitudinal, the oscillation frequency increases, and the maximum excited-state population decreases.",[251,11205,11207],{"id":11206},"fourier-analysis-dynamics","Fourier Analysis Dynamics",[12,11209,11210,11211,11239,11240,11338],{},"The spectral analysis computes a real Fast Fourier transform along the pulse-duration axis. For each sampled drive frequency ",[28,11212,11214,11227],{"className":11213},[35],[28,11215,11217],{"className":11216},[39],[41,11218,11219],{"xmlns":43},[46,11220,11221,11225],{},[49,11222,11223],{},[55,11224,68],{},[82,11226,68],{"encoding":84},[28,11228,11230],{"className":11229,"ariaHidden":71},[89],[28,11231,11233,11236],{"className":11232},[93],[28,11234],{"className":11235,"style":214},[97],[28,11237,68],{"className":11238,"style":165},[102,106],", the trace ",[28,11241,11243,11271],{"className":11242},[35],[28,11244,11246],{"className":11245},[39],[41,11247,11248],{"xmlns":43},[46,11249,11250,11268],{},[49,11251,11252,11258,11260,11262,11264,11266],{},[52,11253,11254,11256],{},[55,11255,57],{},[55,11257,60],{},[62,11259,65],{"stretchy":64},[55,11261,75],{},[62,11263,72],{"separator":71},[55,11265,68],{},[62,11267,78],{"stretchy":64},[82,11269,11270],{"encoding":84},"P_e(\\tau,f)",[28,11272,11274],{"className":11273,"ariaHidden":71},[89],[28,11275,11277,11280,11320,11323,11326,11329,11332,11335],{"className":11276},[93],[28,11278],{"className":11279,"style":98},[97],[28,11281,11283,11286],{"className":11282},[102],[28,11284,57],{"className":11285,"style":107},[102,106],[28,11287,11289],{"className":11288},[111],[28,11290,11292,11312],{"className":11291},[115,116],[28,11293,11295,11309],{"className":11294},[120],[28,11296,11298],{"className":11297,"style":125},[124],[28,11299,11300,11303],{"style":128},[28,11301],{"className":11302,"style":133},[132],[28,11304,11306],{"className":11305},[137,138,139,140],[28,11307,60],{"className":11308},[102,106,140],[28,11310,148],{"className":11311},[147],[28,11313,11315],{"className":11314},[120],[28,11316,11318],{"className":11317,"style":155},[124],[28,11319],{},[28,11321,65],{"className":11322},[161],[28,11324,75],{"className":11325,"style":178},[102,106],[28,11327,72],{"className":11328},[169],[28,11330],{"className":11331,"style":174},[173],[28,11333,68],{"className":11334,"style":165},[102,106],[28,11336,78],{"className":11337},[182]," is treated as a time-domain signal. The spectral amplitude is",[28,11340,11342],{"className":11341},[31],[28,11343,11345,11412],{"className":11344},[35],[28,11346,11348],{"className":11347},[39],[41,11349,11350],{"xmlns":43,"display":44},[46,11351,11352,11409],{},[49,11353,11354,11357,11359,11361,11363,11365,11367,11369,11407],{},[55,11355,11356],{},"S",[62,11358,65],{"stretchy":64},[55,11360,1412],{},[62,11362,72],{"separator":71},[55,11364,68],{},[62,11366,78],{"stretchy":64},[62,11368,845],{},[49,11370,11371,11373,11381,11405],{},[62,11372,296],{"fence":71},[52,11374,11375,11379],{},[55,11376,11378],{"mathvariant":11377},"script","F",[55,11380,75],{},[49,11382,11383,11386,11392,11394,11396,11398,11400,11402],{},[62,11384,11385],{"fence":71},"{",[52,11387,11388,11390],{},[55,11389,57],{},[55,11391,60],{},[62,11393,65],{"stretchy":64},[55,11395,75],{},[62,11397,72],{"separator":71},[55,11399,68],{},[62,11401,78],{"stretchy":64},[62,11403,11404],{"fence":71},"}",[62,11406,296],{"fence":71},[55,11408,2106],{"mathvariant":1088},[82,11410,11411],{"encoding":84},"S(\\nu,f)=\\left|\\mathcal{F}_{\\tau}\\left\\{P_e(\\tau,f)\\right\\}\\right|.",[28,11413,11415,11451],{"className":11414,"ariaHidden":71},[89],[28,11416,11418,11421,11424,11427,11430,11433,11436,11439,11442,11445,11448],{"className":11417},[93],[28,11419],{"className":11420,"style":98},[97],[28,11422,11356],{"className":11423,"style":1913},[102,106],[28,11425,65],{"className":11426},[161],[28,11428,1412],{"className":11429,"style":1433},[102,106],[28,11431,72],{"className":11432},[169],[28,11434],{"className":11435,"style":174},[173],[28,11437,68],{"className":11438,"style":165},[102,106],[28,11440,78],{"className":11441},[182],[28,11443],{"className":11444,"style":913},[173],[28,11446,845],{"className":11447},[917],[28,11449],{"className":11450,"style":913},[173],[28,11452,11454,11457,11582,11585],{"className":11453},[93],[28,11455],{"className":11456,"style":98},[97],[28,11458,11460,11463,11509,11512,11579],{"className":11459},[3398],[28,11461,296],{"className":11462,"style":3403},[161,3402],[28,11464,11466,11471],{"className":11465},[102],[28,11467,11378],{"className":11468,"style":11470},[102,11469],"mathcal","margin-right:0.0993em;",[28,11472,11474],{"className":11473},[111],[28,11475,11477,11501],{"className":11476},[115,116],[28,11478,11480,11498],{"className":11479},[120],[28,11481,11483],{"className":11482,"style":125},[124],[28,11484,11486,11489],{"style":11485},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[28,11487],{"className":11488,"style":133},[132],[28,11490,11492],{"className":11491},[137,138,139,140],[28,11493,11495],{"className":11494},[102,140],[28,11496,75],{"className":11497,"style":178},[102,106,140],[28,11499,148],{"className":11500},[147],[28,11502,11504],{"className":11503},[120],[28,11505,11507],{"className":11506,"style":155},[124],[28,11508],{},[28,11510],{"className":11511,"style":174},[173],[28,11513,11515,11518,11558,11561,11564,11567,11570,11573,11576],{"className":11514},[3398],[28,11516,11385],{"className":11517,"style":3403},[161,3402],[28,11519,11521,11524],{"className":11520},[102],[28,11522,57],{"className":11523,"style":107},[102,106],[28,11525,11527],{"className":11526},[111],[28,11528,11530,11550],{"className":11529},[115,116],[28,11531,11533,11547],{"className":11532},[120],[28,11534,11536],{"className":11535,"style":125},[124],[28,11537,11538,11541],{"style":128},[28,11539],{"className":11540,"style":133},[132],[28,11542,11544],{"className":11543},[137,138,139,140],[28,11545,60],{"className":11546},[102,106,140],[28,11548,148],{"className":11549},[147],[28,11551,11553],{"className":11552},[120],[28,11554,11556],{"className":11555,"style":155},[124],[28,11557],{},[28,11559,65],{"className":11560},[161],[28,11562,75],{"className":11563,"style":178},[102,106],[28,11565,72],{"className":11566},[169],[28,11568],{"className":11569,"style":174},[173],[28,11571,68],{"className":11572,"style":165},[102,106],[28,11574,78],{"className":11575},[182],[28,11577,11404],{"className":11578,"style":3403},[182,3402],[28,11580,296],{"className":11581,"style":3403},[182,3402],[28,11583],{"className":11584,"style":174},[173],[28,11586,2106],{"className":11587},[102],[12,11589,11590],{},"This probability expression contains a squared sinusoid. Using the identity",[28,11592,11594],{"className":11593},[31],[28,11595,11597,11673],{"className":11596},[35],[28,11598,11600],{"className":11599},[39],[41,11601,11602],{"xmlns":43,"display":44},[46,11603,11604,11670],{},[49,11605,11606,11616,11636,11638,11644],{},[2776,11607,11608,11614],{},[49,11609,11610,11612],{},[55,11611,5217],{},[62,11613,4008],{},[650,11615,848],{},[49,11617,11618,11620,11634],{},[62,11619,65],{"fence":71},[3977,11621,11622,11632],{},[49,11623,11624,11630],{},[52,11625,11626,11628],{},[55,11627,1089],{"mathvariant":1088},[55,11629,1333],{},[55,11631,75],{},[650,11633,848],{},[62,11635,78],{"fence":71},[62,11637,845],{},[3977,11639,11640,11642],{},[650,11641,3200],{},[650,11643,848],{},[49,11645,11646,11648,11650,11652,11654,11656,11658,11664,11666,11668],{},[62,11647,3176],{"fence":71},[650,11649,3200],{},[62,11651,1213],{},[55,11653,4005],{},[62,11655,4008],{},[62,11657,65],{"stretchy":64},[52,11659,11660,11662],{},[55,11661,1089],{"mathvariant":1088},[55,11663,1333],{},[55,11665,75],{},[62,11667,78],{"stretchy":64},[62,11669,3217],{"fence":71},[82,11671,11672],{"encoding":84},"\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right)=\\frac{1}{2}\\left[1-\\cos(\\Omega_R\\tau)\\right]",[28,11674,11676,11840],{"className":11675,"ariaHidden":71},[89],[28,11677,11679,11682,11711,11714,11831,11834,11837],{"className":11678},[93],[28,11680],{"className":11681,"style":3394},[97],[28,11683,11685,11688],{"className":11684},[4241],[28,11686,5217],{"className":11687},[4241],[28,11689,11691],{"className":11690},[111],[28,11692,11694],{"className":11693},[115],[28,11695,11697],{"className":11696},[120],[28,11698,11700],{"className":11699,"style":5494},[124],[28,11701,11702,11705],{"style":5497},[28,11703],{"className":11704,"style":133},[132],[28,11706,11708],{"className":11707},[137,138,139,140],[28,11709,848],{"className":11710},[102,140],[28,11712],{"className":11713,"style":174},[173],[28,11715,11717,11723,11825],{"className":11716},[3398],[28,11718,11720],{"className":11719,"style":3403},[161,3402],[28,11721,65],{"className":11722},[3407,139],[28,11724,11726,11729,11822],{"className":11725},[102],[28,11727],{"className":11728},[161,4072],[28,11730,11732],{"className":11731},[3977],[28,11733,11735,11814],{"className":11734},[115,116],[28,11736,11738,11811],{"className":11737},[120],[28,11739,11741,11752,11760],{"className":11740,"style":5537},[124],[28,11742,11743,11746],{"style":4088},[28,11744],{"className":11745,"style":3434},[132],[28,11747,11749],{"className":11748},[102],[28,11750,848],{"className":11751},[102],[28,11753,11754,11757],{"style":4100},[28,11755],{"className":11756,"style":3434},[132],[28,11758],{"className":11759,"style":4108},[4107],[28,11761,11762,11765],{"style":4111},[28,11763],{"className":11764,"style":3434},[132],[28,11766,11768,11808],{"className":11767},[102],[28,11769,11771,11774],{"className":11770},[102],[28,11772,1089],{"className":11773},[102],[28,11775,11777],{"className":11776},[111],[28,11778,11780,11800],{"className":11779},[115,116],[28,11781,11783,11797],{"className":11782},[120],[28,11784,11786],{"className":11785,"style":1364},[124],[28,11787,11788,11791],{"style":1367},[28,11789],{"className":11790,"style":133},[132],[28,11792,11794],{"className":11793},[137,138,139,140],[28,11795,1333],{"className":11796,"style":1377},[102,106,140],[28,11798,148],{"className":11799},[147],[28,11801,11803],{"className":11802},[120],[28,11804,11806],{"className":11805,"style":155},[124],[28,11807],{},[28,11809,75],{"className":11810,"style":178},[102,106],[28,11812,148],{"className":11813},[147],[28,11815,11817],{"className":11816},[120],[28,11818,11820],{"className":11819,"style":4170},[124],[28,11821],{},[28,11823],{"className":11824},[182,4072],[28,11826,11828],{"className":11827,"style":3403},[182,3402],[28,11829,78],{"className":11830},[3407,139],[28,11832],{"className":11833,"style":913},[173],[28,11835,845],{"className":11836},[917],[28,11838],{"className":11839,"style":913},[173],[28,11841,11843,11846,11908,11911],{"className":11842},[93],[28,11844],{"className":11845,"style":7316},[97],[28,11847,11849,11852,11905],{"className":11848},[102],[28,11850],{"className":11851},[161,4072],[28,11853,11855],{"className":11854},[3977],[28,11856,11858,11897],{"className":11857},[115,116],[28,11859,11861,11894],{"className":11860},[120],[28,11862,11864,11875,11883],{"className":11863,"style":6825},[124],[28,11865,11866,11869],{"style":4088},[28,11867],{"className":11868,"style":3434},[132],[28,11870,11872],{"className":11871},[102],[28,11873,848],{"className":11874},[102],[28,11876,11877,11880],{"style":4100},[28,11878],{"className":11879,"style":3434},[132],[28,11881],{"className":11882,"style":4108},[4107],[28,11884,11885,11888],{"style":4111},[28,11886],{"className":11887,"style":3434},[132],[28,11889,11891],{"className":11890},[102],[28,11892,3200],{"className":11893},[102],[28,11895,148],{"className":11896},[147],[28,11898,11900],{"className":11899},[120],[28,11901,11903],{"className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that the dominant oscillatory spectral component appears at the generalized frequency",[28,11988,11990],{"className":11989},[31],[28,11991,11993,12029],{"className":11992},[35],[28,11994,11996],{"className":11995},[39],[41,11997,11998],{"xmlns":43,"display":44},[46,11999,12000,12026],{},[49,12001,12002,12008,12010,12024],{},[52,12003,12004,12006],{},[55,12005,1412],{},[55,12007,1333],{},[62,12009,845],{},[3977,12011,12012,12018],{},[52,12013,12014,12016],{},[55,12015,1089],{"mathvariant":1088},[55,12017,1333],{},[49,12019,12020,12022],{},[650,12021,848],{},[55,12023,851],{},[55,12025,2106],{"mathvariant":1088},[82,12027,12028],{"encoding":84},"\\nu_R=\\frac{\\Omega_R}{2\\pi}.",[28,12030,12032,12087],{"className":12031,"ariaHidden":71},[89],[28,12033,12035,12038,12078,12081,12084],{"className":12034},[93],[28,12036],{"className":12037,"style":780},[97],[28,12039,12041,12044],{"className":12040},[102],[28,12042,1412],{"className":12043,"style":1433},[102,106],[28,12045,12047],{"className":12046},[111],[28,12048,12050,12070],{"className":12049},[115,116],[28,12051,12053,12067],{"className":12052},[120],[28,12054,12056],{"className":12055,"style":1364},[124],[28,12057,12058,12061],{"style":1448},[28,12059],{"className":12060,"style":133},[132],[28,12062,12064],{"className":12063},[137,138,139,140],[28,12065,1333],{"className":12066,"style":1377},[102,106,140],[28,12068,148],{"className":12069},[147],[28,12071,12073],{"className":12072},[120],[28,12074,12076],{"className":12075,"style":155},[124],[28,12077],{},[28,12079],{"className":12080,"style":913},[173],[28,12082,845],{"className":12083},[917],[28,12085],{"className":12086,"style":913},[173],[28,12088,12090,12093,12195],{"className":12089},[93],[28,12091],{"className":12092,"style":6944},[97],[28,12094,12096,12099,12192],{"className":12095},[102],[28,12097],{"className":12098},[161,4072],[28,12100,12102],{"className":12101},[3977],[28,12103,12105,12184],{"className":12104},[115,116],[28,12106,12108,12181],{"className":12107},[120],[28,12109,12111,12125,12133],{"className":12110,"style":5537},[124],[28,12112,12113,12116],{"style":4088},[28,12114],{"className":12115,"style":3434},[132],[28,12117,12119,12122],{"className":12118},[102],[28,12120,848],{"className":12121},[102],[28,12123,851],{"className":12124,"style":321},[102,106],[28,12126,12127,12130],{"style":4100},[28,12128],{"className":12129,"style":3434},[132],[28,12131],{"className":12132,"style":4108},[4107],[28,12134,12135,12138],{"style":4111},[28,12136],{"className":12137,"style":3434},[132],[28,12139,12141],{"className":12140},[102],[28,12142,12144,12147],{"className":12143},[102],[28,12145,1089],{"className":12146},[102],[28,12148,12150],{"className":12149},[111],[28,12151,12153,12173],{"className":12152},[115,116],[28,12154,12156,12170],{"className":12155},[120],[28,12157,12159],{"className":12158,"style":1364},[124],[28,12160,12161,12164],{"style":1367},[28,12162],{"className":12163,"style":133},[132],[28,12165,12167],{"className":12166},[137,138,139,140],[28,12168,1333],{"className":12169,"style":1377},[102,106,140],[28,12171,148],{"className":12172},[147],[28,12174,12176],{"className":12175},[120],[28,12177,12179],{"className":12178,"style":155},[124],[28,12180],{},[28,12182,148],{"className":12183},[147],[28,12185,12187],{"className":12186},[120],[28,12188,12190],{"className":12189,"style":4170},[124],[28,12191],{},[28,12193],{"className":12194},[182,4072],[28,12196,2106],{"className":12197},[102],[12,12199,12200],{},"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.",[251,12202,12204],{"id":12203},"computational-methodology","Computational Methodology",[12,12206,12207,12208,12236,12237,12265,12266,512,12294,12322,12323,12457,12458,12677],{},"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 ",[28,12209,12211,12224],{"className":12210},[35],[28,12212,12214],{"className":12213},[39],[41,12215,12216],{"xmlns":43},[46,12217,12218,12222],{},[49,12219,12220],{},[55,12221,68],{},[82,12223,68],{"encoding":84},[28,12225,12227],{"className":12226,"ariaHidden":71},[89],[28,12228,12230,12233],{"className":12229},[93],[28,12231],{"className":12232,"style":214},[97],[28,12234,68],{"className":12235,"style":165},[102,106]," in GHz and a pulse-duration grid ",[28,12238,12240,12253],{"className":12239},[35],[28,12241,12243],{"className":12242},[39],[41,12244,12245],{"xmlns":43},[46,12246,12247,12251],{},[49,12248,12249],{},[55,12250,75],{},[82,12252,235],{"encoding":84},[28,12254,12256],{"className":12255,"ariaHidden":71},[89],[28,12257,12259,12262],{"className":12258},[93],[28,12260],{"className":12261,"style":245},[97],[28,12263,75],{"className":12264,"style":178},[102,106]," in ns, then converts both to SI units. From a dense two-dimensional mesh over ",[28,12267,12269,12282],{"className":12268},[35],[28,12270,12272],{"className":12271},[39],[41,12273,12274],{"xmlns":43},[46,12275,12276,12280],{},[49,12277,12278],{},[55,12279,68],{},[82,12281,68],{"encoding":84},[28,12283,12285],{"className":12284,"ariaHidden":71},[89],[28,12286,12288,12291],{"className":12287},[93],[28,12289],{"className":12290,"style":214},[97],[28,12292,68],{"className":12293,"style":165},[102,106],[28,12295,12297,12310],{"className":12296},[35],[28,12298,12300],{"className":12299},[39],[41,12301,12302],{"xmlns":43},[46,12303,12304,12308],{},[49,12305,12306],{},[55,12307,75],{},[82,12309,235],{"encoding":84},[28,12311,12313],{"className":12312,"ariaHidden":71},[89],[28,12314,12316,12319],{"className":12315},[93],[28,12317],{"className":12318,"style":245},[97],[28,12320,75],{"className":12321,"style":178},[102,106],", the workflow computes the detuning ",[28,12324,12326,12360],{"className":12325},[35],[28,12327,12329],{"className":12328},[39],[41,12330,12331],{"xmlns":43},[46,12332,12333,12357],{},[49,12334,12335,12337,12339,12341,12343,12345,12347,12349,12355],{},[55,12336,1174],{"mathvariant":1088},[62,12338,845],{},[650,12340,848],{},[55,12342,851],{},[62,12344,65],{"stretchy":64},[55,12346,68],{},[62,12348,1213],{},[52,12350,12351,12353],{},[55,12352,68],{},[650,12354,652],{},[62,12356,78],{"stretchy":64},[82,12358,12359],{"encoding":84},"\\Delta=2\\pi(f-f_0)",[28,12361,12363,12381,12408],{"className":12362,"ariaHidden":71},[89],[28,12364,12366,12369,12372,12375,12378],{"className":12365},[93],[28,12367],{"className":12368,"style":1102},[97],[28,12370,1174],{"className":12371},[102],[28,12373],{"className":12374,"style":913},[173],[28,12376,845],{"className":12377},[917],[28,12379],{"className":12380,"style":913},[173],[28,12382,12384,12387,12390,12393,12396,12399,12402,12405],{"className":12383},[93],[28,12385],{"className":12386,"style":98},[97],[28,12388,848],{"className":12389},[102],[28,12391,851],{"className":12392,"style":321},[102,106],[28,12394,65],{"className":12395},[161],[28,12397,68],{"className":12398,"style":165},[102,106],[28,12400],{"className":12401,"style":1257},[173],[28,12403,1213],{"className":12404},[1261],[28,12406],{"className":12407,"style":1257},[173],[28,12409,12411,12414,12454],{"className":12410},[93],[28,12412],{"className":12413,"style":98},[97],[28,12415,12417,12420],{"className":12416},[102],[28,12418,68],{"className":12419,"style":165},[102,106],[28,12421,12423],{"className":12422},[111],[28,12424,12426,12446],{"className":12425},[115,116],[28,12427,12429,12443],{"className":12428},[120],[28,12430,12432],{"className":12431,"style":683},[124],[28,12433,12434,12437],{"style":686},[28,12435],{"className":12436,"style":133},[132],[28,12438,12440],{"className":12439},[137,138,139,140],[28,12441,652],{"className":12442},[102,140],[28,12444,148],{"className":12445},[147],[28,12447,12449],{"className":12448},[120],[28,12450,12452],{"className":12451,"style":155},[124],[28,12453],{},[28,12455,78],{"className":12456},[182],", evaluates the generalized Rabi angular frequency ",[28,12459,12461,12498],{"className":12460},[35],[28,12462,12464],{"className":12463},[39],[41,12465,12466],{"xmlns":43},[46,12467,12468,12496],{},[49,12469,12470,12476,12478],{},[52,12471,12472,12474],{},[55,12473,1089],{"mathvariant":1088},[55,12475,1333],{},[62,12477,845],{},[5661,12479,12480],{},[49,12481,12482,12488,12490],{},[2776,12483,12484,12486],{},[55,12485,1089],{"mathvariant":1088},[650,12487,848],{},[62,12489,2606],{},[2776,12491,12492,12494],{},[55,12493,1174],{"mathvariant":1088},[650,12495,848],{},[82,12497,5681],{"encoding":84},[28,12499,12501,12556],{"className":12500,"ariaHidden":71},[89],[28,12502,12504,12507,12547,12550,12553],{"className":12503},[93],[28,12505],{"className":12506,"style":585},[97],[28,12508,12510,12513],{"className":12509},[102],[28,12511,1089],{"className":12512},[102],[28,12514,12516],{"className":12515},[111],[28,12517,12519,12539],{"className":12518},[115,116],[28,12520,12522,12536],{"className":12521},[120],[28,12523,12525],{"className":12524,"style":1364},[124],[28,12526,12527,12530],{"style":1367},[28,12528],{"className":12529,"style":133},[132],[28,12531,12533],{"className":12532},[137,138,139,140],[28,12534,1333],{"className":12535,"style":1377},[102,106,140],[28,12537,148],{"className":12538},[147],[28,12540,12542],{"className":12541},[120],[28,12543,12545],{"className":12544,"style":155},[124],[28,12546],{},[28,12548],{"className":12549,"style":913},[173],[28,12551,845],{"className":12552},[917],[28,12554],{"className":12555,"style":913},[173],[28,12557,12559,12563],{"className":12558},[93],[28,12560],{"className":12561,"style":12562},[97],"height:1.04em;vertical-align:-0.1266em;",[28,12564,12566],{"className":12565},[102,5750],[28,12567,12569,12668],{"className":12568},[115,116],[28,12570,12572,12665],{"className":12571},[120],[28,12573,12576,12652],{"className":12574,"style":12575},[124],"height:0.9134em;",[28,12577,12579,12582],{"className":12578,"style":10025},[5764],[28,12580],{"className":12581,"style":3434},[132],[28,12583,12585,12614,12617,12620,12623],{"className":12584,"style":10032},[102],[28,12586,12588,12591],{"className":12587},[102],[28,12589,1089],{"className":12590},[102],[28,12592,12594],{"className":12593},[111],[28,12595,12597],{"className":12596},[115],[28,12598,12600],{"className":12599},[120],[28,12601,12603],{"className":12602,"style":5792},[124],[28,12604,12605,12608],{"style":5795},[28,12606],{"className":12607,"style":133},[132],[28,12609,12611],{"className":12610},[137,138,139,140],[28,12612,848],{"className":12613},[102,140],[28,12615],{"className":12616,"style":1257},[173],[28,12618,2606],{"className":12619},[1261],[28,12621],{"className":12622,"style":1257},[173],[28,12624,12626,12629],{"className":12625},[102],[28,12627,1174],{"className":12628},[102],[28,12630,12632],{"className":12631},[111],[28,12633,12635],{"className":12634},[115],[28,12636,12638],{"className":12637},[120],[28,12639,12641],{"className":12640,"style":5792},[124],[28,12642,12643,12646],{"style":5795},[28,12644],{"className":12645,"style":133},[132],[28,12647,12649],{"className":12648},[137,138,139,140],[28,12650,848],{"className":12651},[102,140],[28,12653,12655,12658],{"style":12654},"top:-2.8734em;",[28,12656],{"className":12657,"style":3434},[132],[28,12659,12661],{"className":12660,"style":10045},[5852],[5855,12662,12663],{"xmlns":5857,"width":5858,"height":10048,"viewBox":10049,"preserveAspectRatio":5861},[5863,12664],{"d":10052},[28,12666,148],{"className":12667},[147],[28,12669,12671],{"className":12670},[120],[28,12672,12675],{"className":12673,"style":12674},[124],"height:0.1266em;",[28,12676],{},", and calculates the analytic excited-state probability",[28,12679,12681],{"className":12680},[31],[28,12682,12684,12762],{"className":12683},[35],[28,12685,12687],{"className":12686},[39],[41,12688,12689],{"xmlns":43,"display":44},[46,12690,12691,12759],{},[49,12692,12693,12699,12701,12703,12705,12707,12709,12711,12727,12737,12757],{},[52,12694,12695,12697],{},[55,12696,57],{},[55,12698,60],{},[62,12700,65],{"stretchy":64},[55,12702,68],{},[62,12704,72],{"separator":71},[55,12706,75],{},[62,12708,78],{"stretchy":64},[62,12710,845],{},[3977,12712,12713,12719],{},[2776,12714,12715,12717],{},[55,12716,1089],{"mathvariant":1088},[650,12718,848],{},[1670,12720,12721,12723,12725],{},[55,12722,1089],{"mathvariant":1088},[55,12724,1333],{},[650,12726,848],{},[2776,12728,12729,12735],{},[49,12730,12731,12733],{},[55,12732,5217],{},[62,12734,4008],{},[650,12736,848],{},[49,12738,12739,12741,12755],{},[62,12740,65],{"fence":71},[3977,12742,12743,12753],{},[49,12744,12745,12751],{},[52,12746,12747,12749],{},[55,12748,1089],{"mathvariant":1088},[55,12750,1333],{},[55,12752,75],{},[650,12754,848],{},[62,12756,78],{"fence":71},[55,12758,2106],{"mathvariant":1088},[82,12760,12761],{"encoding":84},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right).",[28,12763,12765,12838],{"className":12764,"ariaHidden":71},[89],[28,12766,12768,12771,12811,12814,12817,12820,12823,12826,12829,12832,12835],{"className":12767},[93],[28,12769],{"className":12770,"style":98},[97],[28,12772,12774,12777],{"className":12773},[102],[28,12775,57],{"className":12776,"style":107},[102,106],[28,12778,12780],{"className":12779},[111],[28,12781,12783,12803],{"className":12782},[115,116],[28,12784,12786,12800],{"className":12785},[120],[28,12787,12789],{"className":12788,"style":125},[124],[28,12790,12791,12794],{"style":128},[28,12792],{"className":12793,"style":133},[132],[28,12795,12797],{"className":12796},[137,138,139,140],[28,12798,60],{"className":12799},[102,106,140],[28,12801,148],{"className":12802},[147],[28,12804,12806],{"className":12805},[120],[28,12807,12809],{"className":12808,"style":155},[124],[28,12810],{},[28,12812,65],{"className":12813},[161],[28,12815,68],{"className":12816,"style":165},[102,106],[28,12818,72],{"className":12819},[169],[28,12821],{"className":12822,"style":174},[173],[28,12824,75],{"className":12825,"style":178},[102,106],[28,12827,78],{"className":12828},[182],[28,12830],{"className":12831,"style":913},[173],[28,12833,845],{"className":12834},[917],[28,12836],{"className":12837,"style":913},[173],[28,12839,12841,12844,12980,12983,13012,13015,13132,13135],{"className":12840},[93],[28,12842],{"className":12843,"style":5329},[97],[28,12845,12847,12850,12977],{"className":12846},[102],[28,12848],{"className":12849},[161,4072],[28,12851,12853],{"className":12852},[3977],[28,12854,12856,12969],{"className":12855},[115,116],[28,12857,12859,12966],{"className":12858},[120],[28,12860,12862,12921,12929],{"className":12861,"style":5348},[124],[28,12863,12864,12867],{"style":4088},[28,12865],{"className":12866,"style":3434},[132],[28,12868,12870],{"className":12869},[102],[28,12871,12873,12876],{"className":12872},[102],[28,12874,1089],{"className":12875},[102],[28,12877,12879],{"className":12878},[111],[28,12880,12882,12913],{"className":12881},[115,116],[28,12883,12885,12910],{"className":12884},[120],[28,12886,12888,12899],{"className":12887,"style":5375},[124],[28,12889,12890,12893],{"style":5378},[28,12891],{"className":12892,"style":133},[132],[28,12894,12896],{"className":12895},[137,138,139,140],[28,12897,1333],{"className":12898,"style":1377},[102,106,140],[28,12900,12901,12904],{"style":5390},[28,12902],{"className":12903,"style":133},[132],[28,12905,12907],{"className":12906},[137,138,139,140],[28,12908,848],{"className":12909},[102,140],[28,12911,148],{"className":12912},[147],[28,12914,12916],{"className":12915},[120],[28,12917,12919],{"className":12918,"style":5409},[124],[28,12920],{},[28,12922,12923,12926],{"style":4100},[28,12924],{"className":12925,"style":3434},[132],[28,12927],{"className":12928,"style":4108},[4107],[28,12930,12931,12934],{"style":4111},[28,12932],{"className":12933,"style":3434},[132],[28,12935,12937],{"className":12936},[102],[28,12938,12940,12943],{"className":12939},[102],[28,12941,1089],{"className":12942},[102],[28,12944,12946],{"className":12945},[111],[28,12947,12949],{"className":12948},[115],[28,12950,12952],{"className":12951},[120],[28,12953,12955],{"className":12954,"style":5446},[124],[28,12956,12957,12960],{"style":1726},[28,12958],{"className":12959,"style":133},[132],[28,12961,12963],{"className":12962},[137,138,139,140],[28,12964,848],{"className":12965},[102,140],[28,12967,148],{"className":12968},[147],[28,12970,12972],{"className":12971},[120],[28,12973,12975],{"className":12974,"style":5467},[124],[28,12976],{},[28,12978],{"className":12979},[182,4072],[28,12981],{"className":12982,"style":174},[173],[28,12984,12986,12989],{"className":12985},[4241],[28,12987,5217],{"className":12988},[4241],[28,12990,12992],{"className":12991},[111],[28,12993,12995],{"className":12994},[115],[28,12996,12998],{"className":12997},[120],[28,12999,13001],{"className":13000,"style":5494},[124],[28,13002,13003,13006],{"style":5497},[28,13004],{"className":13005,"style":133},[132],[28,13007,13009],{"className":13008},[137,138,139,140],[28,13010,848],{"className":13011},[102,140],[28,13013],{"className":13014,"style":174},[173],[28,13016,13018,13024,13126],{"className":13017},[3398],[28,13019,13021],{"className":13020,"style":3403},[161,3402],[28,13022,65],{"className":13023},[3407,139],[28,13025,13027,13030,13123],{"className":13026},[102],[28,13028],{"className":13029},[161,4072],[28,13031,13033],{"className":13032},[3977],[28,13034,13036,13115],{"className":13035},[115,116],[28,13037,13039,13112],{"className":13038},[120],[28,13040,13042,13053,13061],{"className":13041,"style":5537},[124],[28,13043,13044,13047],{"style":4088},[28,13045],{"className":13046,"style":3434},[132],[28,13048,13050],{"className":13049},[102],[28,13051,848],{"className":13052},[102],[28,13054,13055,13058],{"style":4100},[28,13056],{"className":13057,"style":3434},[132],[28,13059],{"className":13060,"style":4108},[4107],[28,13062,13063,13066],{"style":4111},[28,13064],{"className":13065,"style":3434},[132],[28,13067,13069,13109],{"className":13068},[102],[28,13070,13072,13075],{"className":13071},[102],[28,13073,1089],{"className":13074},[102],[28,13076,13078],{"className":13077},[111],[28,13079,13081,13101],{"className":13080},[115,116],[28,13082,13084,13098],{"className":13083},[120],[28,13085,13087],{"className":13086,"style":1364},[124],[28,13088,13089,13092],{"style":1367},[28,13090],{"className":13091,"style":133},[132],[28,13093,13095],{"className":13094},[137,138,139,140],[28,13096,1333],{"className":13097,"style":1377},[102,106,140],[28,13099,148],{"className":13100},[147],[28,13102,13104],{"className":13103},[120],[28,13105,13107],{"className":13106,"style":155},[124],[28,13108],{},[28,13110,75],{"className":13111,"style":178},[102,106],[28,13113,148],{"className":13114},[147],[28,13116,13118],{"className":13117},[120],[28,13119,13121],{"className":13120,"style":4170},[124],[28,13122],{},[28,13124],{"className":13125},[182,4072],[28,13127,13129],{"className":13128,"style":3403},[182,3402],[28,13130,78],{"className":13131},[3407,139],[28,13133],{"className":13134,"style":174},[173],[28,13136,2106],{"className":13137},[102],[12,13139,13140,13141,13256],{},"The routine optionally applies the damping envelope ",[28,13142,13144,13178],{"className":13143},[35],[28,13145,13147],{"className":13146},[39],[41,13148,13149],{"xmlns":43},[46,13150,13151,13175],{},[49,13152,13153,13155,13157,13159,13161,13163,13165,13173],{},[55,13154,7645],{},[62,13156,4008],{},[62,13158,65],{"stretchy":64},[62,13160,1213],{},[55,13162,75],{},[55,13164,1129],{"mathvariant":1088},[1670,13166,13167,13169,13171],{},[55,13168,1674],{},[650,13170,848],{},[62,13172,1679],{},[62,13174,78],{"stretchy":64},[82,13176,13177],{"encoding":84},"\\exp(-\\tau\u002FT_2^\\ast)",[28,13179,13181],{"className":13180,"ariaHidden":71},[89],[28,13182,13184,13187,13190,13193,13196,13199,13202,13253],{"className":13183},[93],[28,13185],{"className":13186,"style":98},[97],[28,13188,7645],{"className":13189},[4241],[28,13191,65],{"className":13192},[161],[28,13194,1213],{"className":13195},[102],[28,13197,75],{"className":13198,"style":178},[102,106],[28,13200,1129],{"className":13201},[102],[28,13203,13205,13208],{"className":13204},[102],[28,13206,1674],{"className":13207,"style":107},[102,106],[28,13209,13211],{"className":13210},[111],[28,13212,13214,13245],{"className":13213},[115,116],[28,13215,13217,13242],{"className":13216},[120],[28,13218,13220,13231],{"className":13219,"style":1711},[124],[28,13221,13222,13225],{"style":1714},[28,13223],{"className":13224,"style":133},[132],[28,13226,13228],{"className":13227},[137,138,139,140],[28,13229,848],{"className":13230},[102,140],[28,13232,13233,13236],{"style":1726},[28,13234],{"className":13235,"style":133},[132],[28,13237,13239],{"className":13238},[137,138,139,140],[28,13240,1679],{"className":13241},[1261,140],[28,13243,148],{"className":13244},[147],[28,13246,13248],{"className":13247},[120],[28,13249,13251],{"className":13250,"style":1745},[124],[28,13252],{},[28,13254,78],{"className":13255},[182]," and clips the result to the physical interval",[28,13258,13260],{"className":13259},[31],[28,13261,13263,13290],{"className":13262},[35],[28,13264,13266],{"className":13265},[39],[41,13267,13268],{"xmlns":43,"display":44},[46,13269,13270,13287],{},[49,13271,13272,13274,13277,13283,13285],{},[650,13273,652],{},[62,13275,13276],{},"≤",[52,13278,13279,13281],{},[55,13280,57],{},[55,13282,60],{},[62,13284,13276],{},[650,13286,2798],{},[82,13288,13289],{"encoding":84},"0\\le P_e\\le 1.",[28,13291,13293,13312,13367],{"className":13292,"ariaHidden":71},[89],[28,13294,13296,13300,13303,13306,13309],{"className":13295},[93],[28,13297],{"className":13298,"style":13299},[97],"height:0.7804em;vertical-align:-0.136em;",[28,13301,652],{"className":13302},[102],[28,13304],{"className":13305,"style":913},[173],[28,13307,13276],{"className":13308},[917],[28,13310],{"className":13311,"style":913},[173],[28,13313,13315,13318,13358,13361,13364],{"className":13314},[93],[28,13316],{"className":13317,"style":585},[97],[28,13319,13321,13324],{"className":13320},[102],[28,13322,57],{"className":13323,"style":107},[102,106],[28,13325,13327],{"className":13326},[111],[28,13328,13330,13350],{"className":13329},[115,116],[28,13331,13333,13347],{"className":13332},[120],[28,13334,13336],{"className":13335,"style":125},[124],[28,13337,13338,13341],{"style":128},[28,13339],{"className":13340,"style":133},[132],[28,13342,13344],{"className":13343},[137,138,139,140],[28,13345,60],{"className":13346},[102,106,140],[28,13348,148],{"className":13349},[147],[28,13351,13353],{"className":13352},[120],[28,13354,13356],{"className":13355,"style":155},[124],[28,13357],{},[28,13359],{"className":13360,"style":913},[173],[28,13362,13276],{"className":13363},[917],[28,13365],{"className":13366,"style":913},[173],[28,13368,13370,13373],{"className":13369},[93],[28,13371],{"className":13372,"style":2914},[97],[28,13374,2798],{"className":13375},[102],[12,13377,13378],{},"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.",[251,13380,13382],{"id":13381},"graphical-interpretation","Graphical Interpretation",[12,13384,13385,13386,13484],{},"The ",[28,13387,13389,13417],{"className":13388},[35],[28,13390,13392],{"className":13391},[39],[41,13393,13394],{"xmlns":43},[46,13395,13396,13414],{},[49,13397,13398,13404,13406,13408,13410,13412],{},[52,13399,13400,13402],{},[55,13401,57],{},[55,13403,60],{},[62,13405,65],{"stretchy":64},[55,13407,68],{},[62,13409,72],{"separator":71},[55,13411,75],{},[62,13413,78],{"stretchy":64},[82,13415,13416],{"encoding":84},"P_e(f,\\tau)",[28,13418,13420],{"className":13419,"ariaHidden":71},[89],[28,13421,13423,13426,13466,13469,13472,13475,13478,13481],{"className":13422},[93],[28,13424],{"className":13425,"style":98},[97],[28,13427,13429,13432],{"className":13428},[102],[28,13430,57],{"className":13431,"style":107},[102,106],[28,13433,13435],{"className":13434},[111],[28,13436,13438,13458],{"className":13437},[115,116],[28,13439,13441,13455],{"className":13440},[120],[28,13442,13444],{"className":13443,"style":125},[124],[28,13445,13446,13449],{"style":128},[28,13447],{"className":13448,"style":133},[132],[28,13450,13452],{"className":13451},[137,138,139,140],[28,13453,60],{"className":13454},[102,106,140],[28,13456,148],{"className":13457},[147],[28,13459,13461],{"className":13460},[120],[28,13462,13464],{"className":13463,"style":155},[124],[28,13465],{},[28,13467,65],{"className":13468},[161],[28,13470,68],{"className":13471,"style":165},[102,106],[28,13473,72],{"className":13474},[169],[28,13476],{"className":13477,"style":174},[173],[28,13479,75],{"className":13480,"style":178},[102,106],[28,13482,78],{"className":13483},[182]," 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.",[12,13486,13487],{},"The resonant interaction region appears near",[28,13489,13491],{"className":13490},[31],[28,13492,13494,13517],{"className":13493},[35],[28,13495,13497],{"className":13496},[39],[41,13498,13499],{"xmlns":43,"display":44},[46,13500,13501,13515],{},[49,13502,13503,13505,13507,13513],{},[55,13504,68],{},[62,13506,845],{},[52,13508,13509,13511],{},[55,13510,68],{},[650,13512,652],{},[55,13514,2106],{"mathvariant":1088},[82,13516,2492],{"encoding":84},[28,13518,13520,13538],{"className":13519,"ariaHidden":71},[89],[28,13521,13523,13526,13529,13532,13535],{"className":13522},[93],[28,13524],{"className":13525,"style":214},[97],[28,13527,68],{"className":13528,"style":165},[102,106],[28,13530],{"className":13531,"style":913},[173],[28,13533,845],{"className":13534},[917],[28,13536],{"className":13537,"style":913},[173],[28,13539,13541,13544,13584],{"className":13540},[93],[28,13542],{"className":13543,"style":214},[97],[28,13545,13547,13550],{"className":13546},[102],[28,13548,68],{"className":13549,"style":165},[102,106],[28,13551,13553],{"className":13552},[111],[28,13554,13556,13576],{"className":13555},[115,116],[28,13557,13559,13573],{"className":13558},[120],[28,13560,13562],{"className":13561,"style":683},[124],[28,13563,13564,13567],{"style":686},[28,13565],{"className":13566,"style":133},[132],[28,13568,13570],{"className":13569},[137,138,139,140],[28,13571,652],{"className":13572},[102,140],[28,13574,148],{"className":13575},[147],[28,13577,13579],{"className":13578},[120],[28,13580,13582],{"className":13581,"style":155},[124],[28,13583],{},[28,13585,2106],{"className":13586},[102],[12,13588,13589,13590,13669],{},"At resonance, contrast is maximal and the pulse duration alone sets the rotation angle. With the default drive strength ",[28,13591,13593,13627],{"className":13592},[35],[28,13594,13596],{"className":13595},[39],[41,13597,13598],{"xmlns":43},[46,13599,13600,13624],{},[49,13601,13602,13604,13606,13608,13610,13612,13614,13616],{},[55,13603,1089],{"mathvariant":1088},[55,13605,1129],{"mathvariant":1088},[650,13607,848],{},[55,13609,851],{},[62,13611,845],{},[650,13613,6918],{},[2253,13615,2255],{},[49,13617,13618,13620,13622],{},[55,13619,6925],{"mathvariant":1088},[55,13621,2263],{"mathvariant":1088},[55,13623,2266],{"mathvariant":1088},[82,13625,13626],{"encoding":84},"\\Omega\u002F2\\pi=20~\\mathrm{MHz}",[28,13628,13630,13651],{"className":13629,"ariaHidden":71},[89],[28,13631,13633,13636,13639,13642,13645,13648],{"className":13632},[93],[28,13634],{"className":13635,"style":98},[97],[28,13637,1149],{"className":13638},[102],[28,13640,851],{"className":13641,"style":321},[102,106],[28,13643],{"className":13644,"style":913},[173],[28,13646,845],{"className":13647},[917],[28,13649],{"className":13650,"style":913},[173],[28,13652,13654,13657,13660,13663],{"className":13653},[93],[28,13655],{"className":13656,"style":1102},[97],[28,13658,6918],{"className":13659},[102],[28,13661,2255],{"className":13662},[173,2342],[28,13664,13666],{"className":13665},[102],[28,13667,7038],{"className":13668},[102,2349],", the ideal resonant maxima appear near",[28,13671,13673],{"className":13672},[31],[28,13674,13676,13749],{"className":13675},[35],[28,13677,13679],{"className":13678},[39],[41,13680,13681],{"xmlns":43,"display":44},[46,13682,13683,13746],{},[49,13684,13685,13687,13689,13691,13693,13699,13701,13703,13706,13708,13714,13716,13718,13721,13723,13729,13731,13733,13736,13738,13744],{},[55,13686,75],{},[62,13688,845],{},[650,13690,7099],{},[2253,13692,2255],{},[49,13694,13695,13697],{},[55,13696,7106],{"mathvariant":1088},[55,13698,7109],{"mathvariant":1088},[62,13700,72],{"separator":71},[2253,13702,2255],{},[650,13704,13705],{},"75",[2253,13707,2255],{},[49,13709,13710,13712],{},[55,13711,7106],{"mathvariant":1088},[55,13713,7109],{"mathvariant":1088},[62,13715,72],{"separator":71},[2253,13717,2255],{},[650,13719,13720],{},"125",[2253,13722,2255],{},[49,13724,13725,13727],{},[55,13726,7106],{"mathvariant":1088},[55,13728,7109],{"mathvariant":1088},[62,13730,72],{"separator":71},[2253,13732,2255],{},[650,13734,13735],{},"175",[2253,13737,2255],{},[49,13739,13740,13742],{},[55,13741,7106],{"mathvariant":1088},[55,13743,7109],{"mathvariant":1088},[62,13745,72],{"separator":71},[82,13747,13748],{"encoding":84},"\\tau=25~\\mathrm{ns},\\ 75~\\mathrm{ns},\\ 125~\\mathrm{ns},\\ 175~\\mathrm{ns},",[28,13750,13752,13770],{"className":13751,"ariaHidden":71},[89],[28,13753,13755,13758,13761,13764,13767],{"className":13754},[93],[28,13756],{"className":13757,"style":245},[97],[28,13759,75],{"className":13760,"style":178},[102,106],[28,13762],{"className":13763,"style":913},[173],[28,13765,845],{"className":13766},[917],[28,13768],{"className":13769,"style":913},[173],[28,13771,13773,13777,13780,13783,13789,13792,13795,13798,13801,13804,13810,13813,13816,13819,13822,13825,13831,13834,13837,13840,13843,13846,13852],{"className":13772},[93],[28,13774],{"className":13775,"style":13776},[97],"height:0.8389em;vertical-align:-0.1944em;",[28,13778,7099],{"className":13779},[102],[28,13781,2255],{"className":13782},[173,2342],[28,13784,13786],{"className":13785},[102],[28,13787,7191],{"className":13788},[102,2349],[28,13790,72],{"className":13791},[169],[28,13793,2255],{"className":13794},[173],[28,13796],{"className":13797,"style":174},[173],[28,13799,13705],{"className":13800},[102],[28,13802,2255],{"className":13803},[173,2342],[28,13805,13807],{"className":13806},[102],[28,13808,7191],{"className":13809},[102,2349],[28,13811,72],{"className":13812},[169],[28,13814,2255],{"className":13815},[173],[28,13817],{"className":13818,"style":174},[173],[28,13820,13720],{"className":13821},[102],[28,13823,2255],{"className":13824},[173,2342],[28,13826,13828],{"className":13827},[102],[28,13829,7191],{"className":13830},[102,2349],[28,13832,72],{"className":13833},[169],[28,13835,2255],{"className":13836},[173],[28,13838],{"className":13839,"style":174},[173],[28,13841,13735],{"className":13842},[102],[28,13844,2255],{"className":13845},[173,2342],[28,13847,13849],{"className":13848},[102],[28,13850,7191],{"className":13851},[102,2349],[28,13853,72],{"className":13854},[169],[12,13856,13857],{},"before the optional phenomenological damping factor is applied.",[12,13859,13860,13861,13930,13931,14057],{},"Off resonance, the fringes show lower contrast and faster temporal oscillations. This behavior follows from the generalized Rabi frequency ",[28,13862,13864,13881],{"className":13863},[35],[28,13865,13867],{"className":13866},[39],[41,13868,13869],{"xmlns":43},[46,13870,13871,13879],{},[49,13872,13873],{},[52,13874,13875,13877],{},[55,13876,1089],{"mathvariant":1088},[55,13878,1333],{},[82,13880,1336],{"encoding":84},[28,13882,13884],{"className":13883,"ariaHidden":71},[89],[28,13885,13887,13890],{"className":13886},[93],[28,13888],{"className":13889,"style":585},[97],[28,13891,13893,13896],{"className":13892},[102],[28,13894,1089],{"className":13895},[102],[28,13897,13899],{"className":13898},[111],[28,13900,13902,13922],{"className":13901},[115,116],[28,13903,13905,13919],{"className":13904},[120],[28,13906,13908],{"className":13907,"style":1364},[124],[28,13909,13910,13913],{"style":1367},[28,13911],{"className":13912,"style":133},[132],[28,13914,13916],{"className":13915},[137,138,139,140],[28,13917,1333],{"className":13918,"style":1377},[102,106,140],[28,13920,148],{"className":13921},[147],[28,13923,13925],{"className":13924},[120],[28,13926,13928],{"className":13927,"style":155},[124],[28,13929],{},", which increases with detuning, and from the amplitude coefficient ",[28,13932,13934,13962],{"className":13933},[35],[28,13935,13937],{"className":13936},[39],[41,13938,13939],{"xmlns":43},[46,13940,13941,13959],{},[49,13942,13943,13949,13951],{},[2776,13944,13945,13947],{},[55,13946,1089],{"mathvariant":1088},[650,13948,848],{},[55,13950,1129],{"mathvariant":1088},[1670,13952,13953,13955,13957],{},[55,13954,1089],{"mathvariant":1088},[55,13956,1333],{},[650,13958,848],{},[82,13960,13961],{"encoding":84},"\\Omega^2\u002F\\Omega_R^2",[28,13963,13965],{"className":13964,"ariaHidden":71},[89],[28,13966,13968,13972,14001,14004],{"className":13967},[93],[28,13969],{"className":13970,"style":13971},[97],"height:1.0894em;vertical-align:-0.2753em;",[28,13973,13975,13978],{"className":13974},[102],[28,13976,1089],{"className":13977},[102],[28,13979,13981],{"className":13980},[111],[28,13982,13984],{"className":13983},[115],[28,13985,13987],{"className":13986},[120],[28,13988,13990],{"className":13989,"style":5446},[124],[28,13991,13992,13995],{"style":1726},[28,13993],{"className":13994,"style":133},[132],[28,13996,13998],{"className":13997},[137,138,139,140],[28,13999,848],{"className":14000},[102,140],[28,14002,1129],{"className":14003},[102],[28,14005,14007,14010],{"className":14006},[102],[28,14008,1089],{"className":14009},[102],[28,14011,14013],{"className":14012},[111],[28,14014,14016,14048],{"className":14015},[115,116],[28,14017,14019,14045],{"className":14018},[120],[28,14020,14022,14034],{"className":14021,"style":5446},[124],[28,14023,14025,14028],{"style":14024},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[28,14026],{"className":14027,"style":133},[132],[28,14029,14031],{"className":14030},[137,138,139,140],[28,14032,1333],{"className":14033,"style":1377},[102,106,140],[28,14035,14036,14039],{"style":1726},[28,14037],{"className":14038,"style":133},[132],[28,14040,14042],{"className":14041},[137,138,139,140],[28,14043,848],{"className":14044},[102,140],[28,14046,148],{"className":14047},[147],[28,14049,14051],{"className":14050},[120],[28,14052,14055],{"className":14053,"style":14054},[124],"height:0.2753em;",[28,14056],{},", which decreases with detuning.",[12,14059,14060],{},"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.",[12,14062,14063,14064,14161],{},"The FFT heatmap characterizes the oscillation-frequency content of ",[28,14065,14067,14094],{"className":14066},[35],[28,14068,14070],{"className":14069},[39],[41,14071,14072],{"xmlns":43},[46,14073,14074,14092],{},[49,14075,14076,14082,14084,14086,14088,14090],{},[52,14077,14078,14080],{},[55,14079,57],{},[55,14081,60],{},[62,14083,65],{"stretchy":64},[55,14085,75],{},[62,14087,72],{"separator":71},[55,14089,68],{},[62,14091,78],{"stretchy":64},[82,14093,11270],{"encoding":84},[28,14095,14097],{"className":14096,"ariaHidden":71},[89],[28,14098,14100,14103,14143,14146,14149,14152,14155,14158],{"className":14099},[93],[28,14101],{"className":14102,"style":98},[97],[28,14104,14106,14109],{"className":14105},[102],[28,14107,57],{"className":14108,"style":107},[102,106],[28,14110,14112],{"className":14111},[111],[28,14113,14115,14135],{"className":14114},[115,116],[28,14116,14118,14132],{"className":14117},[120],[28,14119,14121],{"className":14120,"style":125},[124],[28,14122,14123,14126],{"style":128},[28,14124],{"className":14125,"style":133},[132],[28,14127,14129],{"className":14128},[137,138,139,140],[28,14130,60],{"className":14131},[102,106,140],[28,14133,148],{"className":14134},[147],[28,14136,14138],{"className":14137},[120],[28,14139,14141],{"className":14140,"style":155},[124],[28,14142],{},[28,14144,65],{"className":14145},[161],[28,14147,75],{"className":14148,"style":178},[102,106],[28,14150,72],{"className":14151},[169],[28,14153],{"className":14154,"style":174},[173],[28,14156,68],{"className":14157,"style":165},[102,106],[28,14159,78],{"className":14160},[182],". An overlaid theoretical curve follows",[28,14163,14165],{"className":14164},[31],[28,14166,14168,14240],{"className":14167},[35],[28,14169,14171],{"className":14170},[39],[41,14172,14173],{"xmlns":43,"display":44},[46,14174,14175,14237],{},[49,14176,14177,14183,14185,14187,14189,14191,14235],{},[52,14178,14179,14181],{},[55,14180,1412],{},[55,14182,1333],{},[62,14184,65],{"stretchy":64},[55,14186,68],{},[62,14188,78],{"stretchy":64},[62,14190,845],{},[5661,14192,14193],{},[49,14194,14195,14215,14217,14219,14221,14223,14229],{},[2776,14196,14197,14213],{},[49,14198,14199,14201,14211],{},[62,14200,65],{"fence":71},[3977,14202,14203,14205],{},[55,14204,1089],{"mathvariant":1088},[49,14206,14207,14209],{},[650,14208,848],{},[55,14210,851],{},[62,14212,78],{"fence":71},[650,14214,848],{},[62,14216,2606],{},[62,14218,65],{"stretchy":64},[55,14220,68],{},[62,14222,1213],{},[52,14224,14225,14227],{},[55,14226,68],{},[650,14228,652],{},[2776,14230,14231,14233],{},[62,14232,78],{"stretchy":64},[650,14234,848],{},[55,14236,2106],{"mathvariant":1088},[82,14238,14239],{"encoding":84},"\\nu_R(f)=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[28,14241,14243,14307],{"className":14242,"ariaHidden":71},[89],[28,14244,14246,14249,14289,14292,14295,14298,14301,14304],{"className":14245},[93],[28,14247],{"className":14248,"style":98},[97],[28,14250,14252,14255],{"className":14251},[102],[28,14253,1412],{"className":14254,"style":1433},[102,106],[28,14256,14258],{"className":14257},[111],[28,14259,14261,14281],{"className":14260},[115,116],[28,14262,14264,14278],{"className":14263},[120],[28,14265,14267],{"className":14266,"style":1364},[124],[28,14268,14269,14272],{"style":1448},[28,14270],{"className":14271,"style":133},[132],[28,14273,14275],{"className":14274},[137,138,139,140],[28,14276,1333],{"className":14277,"style":1377},[102,106,140],[28,14279,148],{"className":14280},[147],[28,14282,14284],{"className":14283},[120],[28,14285,14287],{"className":14286,"style":155},[124],[28,14288],{},[28,14290,65],{"className":14291},[161],[28,14293,68],{"className":14294,"style":165},[102,106],[28,14296,78],{"className":14297},[182],[28,14299],{"className":14300,"style":913},[173],[28,14302,845],{"className":14303},[917],[28,14305],{"className":14306,"style":913},[173],[28,14308,14310,14313,14556],{"className":14309},[93],[28,14311],{"className":14312,"style":8543},[97],[28,14314,14316],{"className":14315},[102,5750],[28,14317,14319,14548],{"className":14318},[115,116],[28,14320,14322,14545],{"className":14321},[120],[28,14323,14325,14533],{"className":14324,"style":8556},[124],[28,14326,14328,14331],{"className":14327,"style":8560},[5764],[28,14329],{"className":14330,"style":8564},[132],[28,14332,14334,14440,14443,14446,14449,14452,14455,14458,14461,14464,14504],{"className":14333,"style":5773},[102],[28,14335,14337,14417],{"className":14336},[3398],[28,14338,14340,14346,14411],{"className":14339},[3398],[28,14341,14343],{"className":14342,"style":3403},[161,3402],[28,14344,65],{"className":14345},[3407,139],[28,14347,14349,14352,14408],{"className":14348},[102],[28,14350],{"className":14351},[161,4072],[28,14353,14355],{"className":14354},[3977],[28,14356,14358,14400],{"className":14357},[115,116],[28,14359,14361,14397],{"className":14360},[120],[28,14362,14364,14378,14386],{"className":14363,"style":5537},[124],[28,14365,14366,14369],{"style":4088},[28,14367],{"className":14368,"style":3434},[132],[28,14370,14372,14375],{"className":14371},[102],[28,14373,848],{"className":14374},[102],[28,14376,851],{"className":14377,"style":321},[102,106],[28,14379,14380,14383],{"style":4100},[28,14381],{"className":14382,"style":3434},[132],[28,14384],{"className":14385,"style":4108},[4107],[28,14387,14388,14391],{"style":4111},[28,14389],{"className":14390,"style":3434},[132],[28,14392,14394],{"className":14393},[102],[28,14395,1089],{"className":14396},[102],[28,14398,148],{"className":14399},[147],[28,14401,14403],{"className":14402},[120],[28,14404,14406],{"className":14405,"style":4170},[124],[28,14407],{},[28,14409],{"className":14410},[182,4072],[28,14412,14414],{"className":14413,"style":3403},[182,3402],[28,14415,78],{"className":14416},[3407,139],[28,14418,14420],{"className":14419},[111],[28,14421,14423],{"className":14422},[115],[28,14424,14426],{"className":14425},[120],[28,14427,14429],{"className":14428,"style":8663},[124],[28,14430,14431,14434],{"style":8666},[28,14432],{"className":14433,"style":133},[132],[28,14435,14437],{"className":14436},[137,138,139,140],[28,14438,848],{"className":14439},[102,140],[28,14441],{"className":14442,"style":1257},[173],[28,14444,2606],{"className":14445},[1261],[28,14447],{"className":14448,"style":1257},[173],[28,14450,65],{"className":14451},[161],[28,14453,68],{"className":14454,"style":165},[102,106],[28,14456],{"className":14457,"style":1257},[173],[28,14459,1213],{"className":14460},[1261],[28,14462],{"className":14463,"style":1257},[173],[28,14465,14467,14470],{"className":14466},[102],[28,14468,68],{"className":14469,"style":165},[102,106],[28,14471,14473],{"className":14472},[111],[28,14474,14476,14496],{"className":14475},[115,116],[28,14477,14479,14493],{"className":14478},[120],[28,14480,14482],{"className":14481,"style":683},[124],[28,14483,14484,14487],{"style":686},[28,14485],{"className":14486,"style":133},[132],[28,14488,14490],{"className":14489},[137,138,139,140],[28,14491,652],{"className":14492},[102,140],[28,14494,148],{"className":14495},[147],[28,14497,14499],{"className":14498},[120],[28,14500,14502],{"className":14501,"style":155},[124],[28,14503],{},[28,14505,14507,14510],{"className":14506},[182],[28,14508,78],{"className":14509},[182],[28,14511,14513],{"className":14512},[111],[28,14514,14516],{"className":14515},[115],[28,14517,14519],{"className":14518},[120],[28,14520,14522],{"className":14521,"style":5792},[124],[28,14523,14524,14527],{"style":5795},[28,14525],{"className":14526,"style":133},[132],[28,14528,14530],{"className":14529},[137,138,139,140],[28,14531,848],{"className":14532},[102,140],[28,14534,14535,14538],{"style":8771},[28,14536],{"className":14537,"style":8564},[132],[28,14539,14541],{"className":14540,"style":8778},[5852],[5855,14542,14543],{"xmlns":5857,"width":5858,"height":8781,"viewBox":8782,"preserveAspectRatio":5861},[5863,14544],{"d":8785},[28,14546,148],{"className":14547},[147],[28,14549,14551],{"className":14550},[120],[28,14552,14554],{"className":14553,"style":8795},[124],[28,14555],{},[28,14557,2106],{"className":14558},[102],[251,14560,14562],{"id":14561},"primary-modeling-assumptions","Primary Modeling Assumptions",[12,14564,14565],{},"This notebook is an idealized mathematical physics visualization designed for conceptual clarity and calibration intuition.",[259,14567,14568,14578],{},[262,14569,14570],{},[265,14571,14572,14575],{},[268,14573,14574],{},"Assumption",[268,14576,14577],{},"Implication",[275,14579,14580,14664,14672,14679,14687,14724,14814,14971,15019,15027,15035,15043],{},[265,14581,14582,14585],{},[280,14583,14584],{},"Two-level truncation",[280,14586,14587,14588,512,14626,2106],{},"The simulated model includes only ",[28,14589,14591,14608],{"className":14590},[35],[28,14592,14594],{"className":14593},[39],[41,14595,14596],{"xmlns":43},[46,14597,14598,14606],{},[49,14599,14600,14602,14604],{},[62,14601,296],{"stretchy":64},[55,14603,299],{},[62,14605,302],{"stretchy":64},[82,14607,305],{"encoding":84},[28,14609,14611],{"className":14610,"ariaHidden":71},[89],[28,14612,14614,14617,14620,14623],{"className":14613},[93],[28,14615],{"className":14616,"style":98},[97],[28,14618,296],{"className":14619},[161],[28,14621,299],{"className":14622,"style":321},[102,106],[28,14624,302],{"className":14625},[182],[28,14627,14629,14646],{"className":14628},[35],[28,14630,14632],{"className":14631},[39],[41,14633,14634],{"xmlns":43},[46,14635,14636,14644],{},[49,14637,14638,14640,14642],{},[62,14639,296],{"stretchy":64},[55,14641,60],{},[62,14643,302],{"stretchy":64},[82,14645,352],{"encoding":84},[28,14647,14649],{"className":14648,"ariaHidden":71},[89],[28,14650,14652,14655,14658,14661],{"className":14651},[93],[28,14653],{"className":14654,"style":98},[97],[28,14656,296],{"className":14657},[161],[28,14659,60],{"className":14660},[102,106],[28,14662,302],{"className":14663},[182],[265,14665,14666,14669],{},[280,14667,14668],{},"Classical drive",[280,14670,14671],{},"The microwave field is prescribed externally instead of quantized as a field mode.",[265,14673,14674,14676],{},[280,14675,1758],{},[280,14677,14678],{},"Fast counter-rotating terms are omitted.",[265,14680,14681,14684],{},[280,14682,14683],{},"Square pulse",[280,14685,14686],{},"The drive is assumed to turn on and off instantaneously.",[265,14688,14689,14692],{},[280,14690,14691],{},"Uniform drive amplitude",[280,14693,14694,14695,14723],{},"The parameter ",[28,14696,14698,14711],{"className":14697},[35],[28,14699,14701],{"className":14700},[39],[41,14702,14703],{"xmlns":43},[46,14704,14705,14709],{},[49,14706,14707],{},[55,14708,1089],{"mathvariant":1088},[82,14710,1092],{"encoding":84},[28,14712,14714],{"className":14713,"ariaHidden":71},[89],[28,14715,14717,14720],{"className":14716},[93],[28,14718],{"className":14719,"style":1102},[97],[28,14721,1089],{"className":14722},[102]," remains constant across the frequency sweep.",[265,14725,14726,14729],{},[280,14727,14728],{},"Optional damping envelope",[280,14730,14694,14731,14813],{},[28,14732,14734,14753],{"className":14733},[35],[28,14735,14737],{"className":14736},[39],[41,14738,14739],{"xmlns":43},[46,14740,14741,14751],{},[49,14742,14743],{},[1670,14744,14745,14747,14749],{},[55,14746,1674],{},[650,14748,848],{},[62,14750,1679],{},[82,14752,1682],{"encoding":84},[28,14754,14756],{"className":14755,"ariaHidden":71},[89],[28,14757,14759,14762],{"className":14758},[93],[28,14760],{"className":14761,"style":1692},[97],[28,14763,14765,14768],{"className":14764},[102],[28,14766,1674],{"className":14767,"style":107},[102,106],[28,14769,14771],{"className":14770},[111],[28,14772,14774,14805],{"className":14773},[115,116],[28,14775,14777,14802],{"className":14776},[120],[28,14778,14780,14791],{"className":14779,"style":1711},[124],[28,14781,14782,14785],{"style":1714},[28,14783],{"className":14784,"style":133},[132],[28,14786,14788],{"className":14787},[137,138,139,140],[28,14789,848],{"className":14790},[102,140],[28,14792,14793,14796],{"style":1726},[28,14794],{"className":14795,"style":133},[132],[28,14797,14799],{"className":14798},[137,138,139,140],[28,14800,1679],{"className":14801},[1261,140],[28,14803,148],{"className":14804},[147],[28,14806,14808],{"className":14807},[120],[28,14809,14811],{"className":14810,"style":1745},[124],[28,14812],{}," enters as a simple exponential contrast envelope.",[265,14815,14816,14890],{},[280,14817,14818,14819,14889],{},"No explicit ",[28,14820,14822,14840],{"className":14821},[35],[28,14823,14825],{"className":14824},[39],[41,14826,14827],{"xmlns":43},[46,14828,14829,14837],{},[49,14830,14831],{},[52,14832,14833,14835],{},[55,14834,1674],{},[650,14836,3200],{},[82,14838,14839],{"encoding":84},"T_1",[28,14841,14843],{"className":14842,"ariaHidden":71},[89],[28,14844,14846,14849],{"className":14845},[93],[28,14847],{"className":14848,"style":585},[97],[28,14850,14852,14855],{"className":14851},[102],[28,14853,1674],{"className":14854,"style":107},[102,106],[28,14856,14858],{"className":14857},[111],[28,14859,14861,14881],{"className":14860},[115,116],[28,14862,14864,14878],{"className":14863},[120],[28,14865,14867],{"className":14866,"style":683},[124],[28,14868,14869,14872],{"style":128},[28,14870],{"className":14871,"style":133},[132],[28,14873,14875],{"className":14874},[137,138,139,140],[28,14876,3200],{"className":14877},[102,140],[28,14879,148],{"className":14880},[147],[28,14882,14884],{"className":14883},[120],[28,14885,14887],{"className":14886,"style":155},[124],[28,14888],{}," relaxation",[280,14891,14892,14893,14931,14932,14970],{},"Energy decay from ",[28,14894,14896,14913],{"className":14895},[35],[28,14897,14899],{"className":14898},[39],[41,14900,14901],{"xmlns":43},[46,14902,14903,14911],{},[49,14904,14905,14907,14909],{},[62,14906,296],{"stretchy":64},[55,14908,60],{},[62,14910,302],{"stretchy":64},[82,14912,352],{"encoding":84},[28,14914,14916],{"className":14915,"ariaHidden":71},[89],[28,14917,14919,14922,14925,14928],{"className":14918},[93],[28,14920],{"className":14921,"style":98},[97],[28,14923,296],{"className":14924},[161],[28,14926,60],{"className":14927},[102,106],[28,14929,302],{"className":14930},[182]," to ",[28,14933,14935,14952],{"className":14934},[35],[28,14936,14938],{"className":14937},[39],[41,14939,14940],{"xmlns":43},[46,14941,14942,14950],{},[49,14943,14944,14946,14948],{},[62,14945,296],{"stretchy":64},[55,14947,299],{},[62,14949,302],{"stretchy":64},[82,14951,305],{"encoding":84},[28,14953,14955],{"className":14954,"ariaHidden":71},[89],[28,14956,14958,14961,14964,14967],{"className":14957},[93],[28,14959],{"className":14960,"style":98},[97],[28,14962,296],{"className":14963},[161],[28,14965,299],{"className":14966,"style":321},[102,106],[28,14968,302],{"className":14969},[182]," is omitted.",[265,14972,14973,14976],{},[280,14974,14975],{},"No higher-state leakage",[280,14977,14978,14979,15018],{},"Higher levels, including a transmon ",[28,14980,14982,15000],{"className":14981},[35],[28,14983,14985],{"className":14984},[39],[41,14986,14987],{"xmlns":43},[46,14988,14989,14997],{},[49,14990,14991,14993,14995],{},[62,14992,296],{"stretchy":64},[55,14994,68],{},[62,14996,302],{"stretchy":64},[82,14998,14999],{"encoding":84},"\\lvert f\\rangle",[28,15001,15003],{"className":15002,"ariaHidden":71},[89],[28,15004,15006,15009,15012,15015],{"className":15005},[93],[28,15007],{"className":15008,"style":98},[97],[28,15010,296],{"className":15011},[161],[28,15013,68],{"className":15014,"style":165},[102,106],[28,15016,302],{"className":15017},[182]," state, are excluded.",[265,15020,15021,15024],{},[280,15022,15023],{},"No AC Stark shift",[280,15025,15026],{},"Drive-induced transition-frequency shifts are excluded.",[265,15028,15029,15032],{},[280,15030,15031],{},"No Bloch-Siegert shift",[280,15033,15034],{},"Counter-rotating corrections are excluded.",[265,15036,15037,15040],{},[280,15038,15039],{},"No experimental readout model",[280,15041,15042],{},"Assignment errors and finite signal-to-noise effects are omitted.",[265,15044,15045,15048],{},[280,15046,15047],{},"No pulse-envelope shaping",[280,15049,15050],{},"Gaussian, DRAG, cosine, and hardware-specific envelopes are omitted.",[12,15052,15053],{},"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.",[251,15055,15057],{"id":15056},"notebook-control-knobs","Notebook Control Knobs",[12,15059,15060],{},"The first code section centralizes the main physical and numerical parameters for direct adjustment.",[259,15062,15063,15076],{},[262,15064,15065],{},[265,15066,15067,15070,15074],{},[268,15068,15069],{},"Parameter",[268,15071,15073],{"align":15072},"right","Default",[268,15075,273],{},[275,15077,15078,15162,15215,15230,15245,15260,15275,15290,15304],{},[265,15079,15080,15085,15089],{},[280,15081,15082],{},[8103,15083,15084],{},"F0_GHZ",[280,15086,15087],{"align":15072},[8103,15088,2251],{},[280,15090,15091,15092,15161],{},"Qubit transition frequency ",[28,15093,15095,15112],{"className":15094},[35],[28,15096,15098],{"className":15097},[39],[41,15099,15100],{"xmlns":43},[46,15101,15102,15110],{},[49,15103,15104],{},[52,15105,15106,15108],{},[55,15107,68],{},[650,15109,652],{},[82,15111,655],{"encoding":84},[28,15113,15115],{"className":15114,"ariaHidden":71},[89],[28,15116,15118,15121],{"className":15117},[93],[28,15119],{"className":15120,"style":214},[97],[28,15122,15124,15127],{"className":15123},[102],[28,15125,68],{"className":15126,"style":165},[102,106],[28,15128,15130],{"className":15129},[111],[28,15131,15133,15153],{"className":15132},[115,116],[28,15134,15136,15150],{"className":15135},[120],[28,15137,15139],{"className":15138,"style":683},[124],[28,15140,15141,15144],{"style":686},[28,15142],{"className":15143,"style":133},[132],[28,15145,15147],{"className":15146},[137,138,139,140],[28,15148,652],{"className":15149},[102,140],[28,15151,148],{"className":15152},[147],[28,15154,15156],{"className":15155},[120],[28,15157,15159],{"className":15158,"style":155},[124],[28,15160],{}," in GHz",[265,15163,15164,15169,15174],{},[280,15165,15166],{},[8103,15167,15168],{},"OMEGA_ONRESONANCE_MHZ",[280,15170,15171],{"align":15072},[8103,15172,15173],{},"20.0",[280,15175,15176,15177,15214],{},"On-resonance Rabi rate ",[28,15178,15180,15199],{"className":15179},[35],[28,15181,15183],{"className":15182},[39],[41,15184,15185],{"xmlns":43},[46,15186,15187,15197],{},[49,15188,15189,15191,15193,15195],{},[55,15190,1089],{"mathvariant":1088},[55,15192,1129],{"mathvariant":1088},[650,15194,848],{},[55,15196,851],{},[82,15198,1136],{"encoding":84},[28,15200,15202],{"className":15201,"ariaHidden":71},[89],[28,15203,15205,15208,15211],{"className":15204},[93],[28,15206],{"className":15207,"style":98},[97],[28,15209,1149],{"className":15210},[102],[28,15212,851],{"className":15213,"style":321},[102,106]," in MHz",[265,15216,15217,15222,15227],{},[280,15218,15219],{},[8103,15220,15221],{},"FREQ_MIN_GHZ",[280,15223,15224],{"align":15072},[8103,15225,15226],{},"4.90",[280,15228,15229],{},"Minimum swept drive frequency in GHz",[265,15231,15232,15237,15242],{},[280,15233,15234],{},[8103,15235,15236],{},"FREQ_MAX_GHZ",[280,15238,15239],{"align":15072},[8103,15240,15241],{},"5.10",[280,15243,15244],{},"Maximum swept drive frequency in GHz",[265,15246,15247,15252,15257],{},[280,15248,15249],{},[8103,15250,15251],{},"N_FREQ",[280,15253,15254],{"align":15072},[8103,15255,15256],{},"401",[280,15258,15259],{},"Number of drive-frequency samples",[265,15261,15262,15267,15272],{},[280,15263,15264],{},[8103,15265,15266],{},"DUR_MIN_NS",[280,15268,15269],{"align":15072},[8103,15270,15271],{},"0.0",[280,15273,15274],{},"Minimum pulse duration in ns",[265,15276,15277,15282,15287],{},[280,15278,15279],{},[8103,15280,15281],{},"DUR_MAX_NS",[280,15283,15284],{"align":15072},[8103,15285,15286],{},"200.0",[280,15288,15289],{},"Maximum pulse duration in ns",[265,15291,15292,15297,15301],{},[280,15293,15294],{},[8103,15295,15296],{},"N_DUR",[280,15298,15299],{"align":15072},[8103,15300,15256],{},[280,15302,15303],{},"Number of pulse-duration samples",[265,15305,15306,15311,15316],{},[280,15307,15308],{},[8103,15309,15310],{},"T2STAR_NS",[280,15312,15313],{"align":15072},[8103,15314,15315],{},"500.0",[280,15317,15318,15319,15322],{},"Optional damping time in ns, with ",[8103,15320,15321],{},"None"," disabling damping",[12,15324,15325],{},"Additional cross-section plotting cells expose localized slice controls.",[259,15327,15328,15338],{},[262,15329,15330],{},[265,15331,15332,15334,15336],{},[268,15333,15069],{},[268,15335,15073],{"align":15072},[268,15337,273],{},[275,15339,15340,15439],{},[265,15341,15342,15347,15351],{},[280,15343,15344],{},[8103,15345,15346],{},"F_CROSS_GHZ",[280,15348,15349],{"align":15072},[8103,15350,2251],{},[280,15352,15353,15354],{},"Drive-frequency slice used to plot ",[28,15355,15357,15381],{"className":15356},[35],[28,15358,15360],{"className":15359},[39],[41,15361,15362],{"xmlns":43},[46,15363,15364,15378],{},[49,15365,15366,15372,15374,15376],{},[52,15367,15368,15370],{},[55,15369,57],{},[55,15371,60],{},[62,15373,65],{"stretchy":64},[55,15375,75],{},[62,15377,78],{"stretchy":64},[82,15379,15380],{"encoding":84},"P_e(\\tau)",[28,15382,15384],{"className":15383,"ariaHidden":71},[89],[28,15385,15387,15390,15430,15433,15436],{"className":15386},[93],[28,15388],{"className":15389,"style":98},[97],[28,15391,15393,15396],{"className":15392},[102],[28,15394,57],{"className":15395,"style":107},[102,106],[28,15397,15399],{"className":15398},[111],[28,15400,15402,15422],{"className":15401},[115,116],[28,15403,15405,15419],{"className":15404},[120],[28,15406,15408],{"className":15407,"style":125},[124],[28,15409,15410,15413],{"style":128},[28,15411],{"className":15412,"style":133},[132],[28,15414,15416],{"className":15415},[137,138,139,140],[28,15417,60],{"className":15418},[102,106,140],[28,15420,148],{"className":15421},[147],[28,15423,15425],{"className":15424},[120],[28,15426,15428],{"className":15427,"style":155},[124],[28,15429],{},[28,15431,65],{"className":15432},[161],[28,15434,75],{"className":15435,"style":178},[102,106],[28,15437,78],{"className":15438},[182],[265,15440,15441,15446,15451],{},[280,15442,15443],{},[8103,15444,15445],{},"T_CROSS_NS",[280,15447,15448],{"align":15072},[8103,15449,15450],{},"100.0",[280,15452,15453,15454],{},"Pulse-duration slice used to plot ",[28,15455,15457,15481],{"className":15456},[35],[28,15458,15460],{"className":15459},[39],[41,15461,15462],{"xmlns":43},[46,15463,15464,15478],{},[49,15465,15466,15472,15474,15476],{},[52,15467,15468,15470],{},[55,15469,57],{},[55,15471,60],{},[62,15473,65],{"stretchy":64},[55,15475,68],{},[62,15477,78],{"stretchy":64},[82,15479,15480],{"encoding":84},"P_e(f)",[28,15482,15484],{"className":15483,"ariaHidden":71},[89],[28,15485,15487,15490,15530,15533,15536],{"className":15486},[93],[28,15488],{"className":15489,"style":98},[97],[28,15491,15493,15496],{"className":15492},[102],[28,15494,57],{"className":15495,"style":107},[102,106],[28,15497,15499],{"className":15498},[111],[28,15500,15502,15522],{"className":15501},[115,116],[28,15503,15505,15519],{"className":15504},[120],[28,15506,15508],{"className":15507,"style":125},[124],[28,15509,15510,15513],{"style":128},[28,15511],{"className":15512,"style":133},[132],[28,15514,15516],{"className":15515},[137,138,139,140],[28,15517,60],{"className":15518},[102,106,140],[28,15520,148],{"className":15521},[147],[28,15523,15525],{"className":15524},[120],[28,15526,15528],{"className":15527,"style":155},[124],[28,15529],{},[28,15531,65],{"className":15532},[161],[28,15534,68],{"className":15535,"style":165},[102,106],[28,15537,78],{"className":15538},[182],[12,15540,15541],{},"Separate FFT processing controls support spectral analysis.",[259,15543,15544,15554],{},[262,15545,15546],{},[265,15547,15548,15550,15552],{},[268,15549,15069],{},[268,15551,15073],{"align":15072},[268,15553,273],{},[275,15555,15556,15571,15586,15601,15616,15631],{},[265,15557,15558,15563,15568],{},[280,15559,15560],{},[8103,15561,15562],{},"FFT_PAD",[280,15564,15565],{"align":15072},[8103,15566,15567],{},"8",[280,15569,15570],{},"Zero-padding factor used in the Fourier transform",[265,15572,15573,15578,15583],{},[280,15574,15575],{},[8103,15576,15577],{},"WINDOW",[280,15579,15580],{"align":15072},[8103,15581,15582],{},"\"hann\"",[280,15584,15585],{},"Window function applied along the pulse-duration axis",[265,15587,15588,15593,15598],{},[280,15589,15590],{},[8103,15591,15592],{},"DETREND_MEAN",[280,15594,15595],{"align":15072},[8103,15596,15597],{},"True",[280,15599,15600],{},"Flag that subtracts the mean before the FFT",[265,15602,15603,15608,15613],{},[280,15604,15605],{},[8103,15606,15607],{},"AMP_MODE",[280,15609,15610],{"align":15072},[8103,15611,15612],{},"\"magnitude\"",[280,15614,15615],{},"Display mode for magnitude, power, or dB-scaled amplitude",[265,15617,15618,15623,15628],{},[280,15619,15620],{},[8103,15621,15622],{},"NORM_MODE",[280,15624,15625],{"align":15072},[8103,15626,15627],{},"\"global\"",[280,15629,15630],{},"Normalization mode applied globally or per drive-frequency column",[265,15632,15633,15638,15643],{},[280,15634,15635],{},[8103,15636,15637],{},"FREQ_UNITS",[280,15639,15640],{"align":15072},[8103,15641,15642],{},"\"MHz\"",[280,15644,15645],{},"Frequency units used on the Fourier-frequency axis",[12,15647,15648],{},"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.",[15650,15651,15656],"pre",{"className":15652,"code":15653,"language":15654,"meta":15655,"style":15655},"language-python shiki shiki-themes one-dark-pro","# @title Install dependencies\nimport sys\n%pip -q install pyvista pythreejs trame\n","python","",[8103,15657,15658,15666,15677],{"__ignoreMap":15655},[28,15659,15662],{"class":15660,"line":15661},"line",1,[28,15663,15665],{"class":15664},"sV9Aq","# @title Install dependencies\n",[28,15667,15669,15673],{"class":15660,"line":15668},2,[28,15670,15672],{"class":15671},"seHd6","import",[28,15674,15676],{"class":15675},"sn6KH"," sys\n",[28,15678,15680,15684,15687,15690],{"class":15660,"line":15679},3,[28,15681,15683],{"class":15682},"sjrmR","%",[28,15685,15686],{"class":15675},"pip ",[28,15688,15689],{"class":15682},"-",[28,15691,15692],{"class":15675},"q install pyvista pythreejs trame\n",[15650,15694,15696],{"className":15652,"code":15695,"language":15654,"meta":15655,"style":15655},"# @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",[8103,15697,15698,15703,15710,15723,15737,15750,15763,15770,15779,15796,15809,15818,15828,15833,15851,15865,15870,15876,15895,15912,15929,15946,15964,15981,15998,16014,16034,16039],{"__ignoreMap":15655},[28,15699,15700],{"class":15660,"line":15661},[28,15701,15702],{"class":15664},"# @title Controls & imports\n",[28,15704,15705,15707],{"class":15660,"line":15668},[28,15706,15672],{"class":15671},[28,15708,15709],{"class":15675}," math\n",[28,15711,15712,15715,15718,15720],{"class":15660,"line":15679},[28,15713,15714],{"class":15671},"from",[28,15716,15717],{"class":15675}," typing ",[28,15719,15672],{"class":15671},[28,15721,15722],{"class":15675}," Optional, Tuple\n",[28,15724,15726,15728,15731,15734],{"class":15660,"line":15725},4,[28,15727,15672],{"class":15671},[28,15729,15730],{"class":15675}," numpy ",[28,15732,15733],{"class":15671},"as",[28,15735,15736],{"class":15675}," np\n",[28,15738,15740,15742,15745,15747],{"class":15660,"line":15739},5,[28,15741,15672],{"class":15671},[28,15743,15744],{"class":15675}," matplotlib.pyplot ",[28,15746,15733],{"class":15671},[28,15748,15749],{"class":15675}," plt\n",[28,15751,15753,15755,15758,15760],{"class":15660,"line":15752},6,[28,15754,15672],{"class":15671},[28,15756,15757],{"class":15675}," pandas ",[28,15759,15733],{"class":15671},[28,15761,15762],{"class":15675}," pd\n",[28,15764,15766],{"class":15660,"line":15765},7,[28,15767,15769],{"emptyLinePlaceholder":15768},true,"\n",[28,15771,15773,15776],{"class":15660,"line":15772},8,[28,15774,15775],{"class":15671},"try",[28,15777,15778],{"class":15675},":\n",[28,15780,15782,15785,15788,15790,15793],{"class":15660,"line":15781},9,[28,15783,15784],{"class":15671},"    import",[28,15786,15787],{"class":15675}," pyvista ",[28,15789,15733],{"class":15671},[28,15791,15792],{"class":15675}," pv  ",[28,15794,15795],{"class":15664},"# optional; only for 3D interactive\n",[28,15797,15799,15803,15806],{"class":15660,"line":15798},10,[28,15800,15802],{"class":15801},"sVC51","    PV_AVAILABLE",[28,15804,15805],{"class":15682}," =",[28,15807,15808],{"class":15801}," True\n",[28,15810,15812,15815],{"class":15660,"line":15811},11,[28,15813,15814],{"class":15671},"except",[28,15816,15817],{"class":15675}," Exception:\n",[28,15819,15821,15823,15825],{"class":15660,"line":15820},12,[28,15822,15802],{"class":15801},[28,15824,15805],{"class":15682},[28,15826,15827],{"class":15801}," False\n",[28,15829,15831],{"class":15660,"line":15830},13,[28,15832,15769],{"emptyLinePlaceholder":15768},[28,15834,15836,15839,15843,15846,15848],{"class":15660,"line":15835},14,[28,15837,15838],{"class":15675},"plt.rcParams[",[28,15840,15842],{"class":15841},"subq3","'figure.dpi'",[28,15844,15845],{"class":15675},"] ",[28,15847,845],{"class":15682},[28,15849,15850],{"class":15801}," 250\n",[28,15852,15854,15856,15859,15861,15863],{"class":15660,"line":15853},15,[28,15855,15838],{"class":15675},[28,15857,15858],{"class":15841},"'axes.grid'",[28,15860,15845],{"class":15675},[28,15862,845],{"class":15682},[28,15864,15827],{"class":15801},[28,15866,15868],{"class":15660,"line":15867},16,[28,15869,15769],{"emptyLinePlaceholder":15768},[28,15871,15873],{"class":15660,"line":15872},17,[28,15874,15875],{"class":15664},"# ---------- Control knobs ----------\n",[28,15877,15879,15881,15884,15887,15889,15892],{"class":15660,"line":15878},18,[28,15880,15084],{"class":15801},[28,15882,15883],{"class":15675},": ",[28,15885,15886],{"class":15682},"float",[28,15888,15805],{"class":15682},[28,15890,15891],{"class":15801}," 5.000",[28,15893,15894],{"class":15664},"        # Qubit transition freq f0 [GHz]\n",[28,15896,15898,15900,15902,15904,15906,15909],{"class":15660,"line":15897},19,[28,15899,15168],{"class":15801},[28,15901,15883],{"class":15675},[28,15903,15886],{"class":15682},[28,15905,15805],{"class":15682},[28,15907,15908],{"class":15801}," 20.0",[28,15910,15911],{"class":15664},"  # Ω\u002F2π on-resonance [MHz]\n",[28,15913,15915,15917,15919,15921,15923,15926],{"class":15660,"line":15914},20,[28,15916,15221],{"class":15801},[28,15918,15883],{"class":15675},[28,15920,15886],{"class":15682},[28,15922,15805],{"class":15682},[28,15924,15925],{"class":15801}," 4.90",[28,15927,15928],{"class":15664},"   # Sweep min f [GHz]\n",[28,15930,15932,15934,15936,15938,15940,15943],{"class":15660,"line":15931},21,[28,15933,15236],{"class":15801},[28,15935,15883],{"class":15675},[28,15937,15886],{"class":15682},[28,15939,15805],{"class":15682},[28,15941,15942],{"class":15801}," 5.10",[28,15944,15945],{"class":15664},"   # Sweep max f [GHz]\n",[28,15947,15949,15951,15953,15956,15958,15961],{"class":15660,"line":15948},22,[28,15950,15251],{"class":15801},[28,15952,15883],{"class":15675},[28,15954,15955],{"class":15682},"int",[28,15957,15805],{"class":15682},[28,15959,15960],{"class":15801}," 401",[28,15962,15963],{"class":15664},"            # # of frequency points\n",[28,15965,15967,15969,15971,15973,15975,15978],{"class":15660,"line":15966},23,[28,15968,15266],{"class":15801},[28,15970,15883],{"class":15675},[28,15972,15886],{"class":15682},[28,15974,15805],{"class":15682},[28,15976,15977],{"class":15801}," 0.0",[28,15979,15980],{"class":15664},"      # Sweep min duration τ [ns]\n",[28,15982,15984,15986,15988,15990,15992,15995],{"class":15660,"line":15983},24,[28,15985,15281],{"class":15801},[28,15987,15883],{"class":15675},[28,15989,15886],{"class":15682},[28,15991,15805],{"class":15682},[28,15993,15994],{"class":15801}," 200.0",[28,15996,15997],{"class":15664},"    # Sweep max duration τ [ns]\n",[28,15999,16001,16003,16005,16007,16009,16011],{"class":15660,"line":16000},25,[28,16002,15296],{"class":15801},[28,16004,15883],{"class":15675},[28,16006,15955],{"class":15682},[28,16008,15805],{"class":15682},[28,16010,15960],{"class":15801},[28,16012,16013],{"class":15664},"             # # of duration points\n",[28,16015,16017,16019,16022,16024,16026,16028,16031],{"class":15660,"line":16016},26,[28,16018,15310],{"class":15801},[28,16020,16021],{"class":15675},": Optional[",[28,16023,15886],{"class":15682},[28,16025,15845],{"class":15675},[28,16027,845],{"class":15682},[28,16029,16030],{"class":15801}," 500.0",[28,16032,16033],{"class":15664},"  # None to disable exp(-τ\u002FT2*)\n",[28,16035,16037],{"class":15660,"line":16036},27,[28,16038,15769],{"emptyLinePlaceholder":15768},[28,16040,16042,16045,16047,16050],{"class":15660,"line":16041},28,[28,16043,16044],{"class":15682},"print",[28,16046,65],{"class":15675},[28,16048,16049],{"class":15841},"'Controls loaded.'",[28,16051,16052],{"class":15675},")\n",[15650,16054,16059],{"className":16055,"code":16057,"language":16058,"meta":15655},[16056],"language-text","Controls loaded.\n","text",[8103,16060,16057],{"__ignoreMap":15655},[15650,16062,16064],{"className":15652,"code":16063,"language":15654,"meta":15655,"style":15655},"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",[8103,16065,16066,16078,16087,16094,16106,16117,16135,16140,16145,16149,16154,16159,16164,16169,16174,16179,16183,16188,16193,16198,16214,16230,16244,16259,16282,16286,16316,16338,16369,16374,16408,16429,16450,16474,16479,16508,16540,16555,16560],{"__ignoreMap":15655},[28,16067,16068,16071,16075],{"class":15660,"line":15661},[28,16069,16070],{"class":15671},"def",[28,16072,16074],{"class":16073},"sVbv2"," rabi_probability",[28,16076,16077],{"class":15675},"(\n",[28,16079,16080,16084],{"class":15660,"line":15668},[28,16081,16083],{"class":16082},"sb9H8","    f_ghz",[28,16085,16086],{"class":15675},": np.ndarray,\n",[28,16088,16089,16092],{"class":15660,"line":15679},[28,16090,16091],{"class":16082},"    tau_ns",[28,16093,16086],{"class":15675},[28,16095,16096,16099,16101,16103],{"class":15660,"line":15725},[28,16097,16098],{"class":16082},"    f0_ghz",[28,16100,15883],{"class":15675},[28,16102,15886],{"class":15682},[28,16104,16105],{"class":15675},",\n",[28,16107,16108,16111,16113,16115],{"class":15660,"line":15739},[28,16109,16110],{"class":16082},"    omega_onres_mhz",[28,16112,15883],{"class":15675},[28,16114,15886],{"class":15682},[28,16116,16105],{"class":15675},[28,16118,16119,16122,16124,16126,16128,16130,16133],{"class":15660,"line":15752},[28,16120,16121],{"class":16082},"    t2star_ns",[28,16123,16021],{"class":15675},[28,16125,15886],{"class":15682},[28,16127,15845],{"class":15675},[28,16129,845],{"class":15682},[28,16131,16132],{"class":15801}," None",[28,16134,16105],{"class":15675},[28,16136,16137],{"class":15660,"line":15765},[28,16138,16139],{"class":15675},") -> np.ndarray:\n",[28,16141,16142],{"class":15660,"line":15772},[28,16143,16144],{"class":15841},"    \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n",[28,16146,16147],{"class":15660,"line":15781},[28,16148,15769],{"emptyLinePlaceholder":15768},[28,16150,16151],{"class":15660,"line":15798},[28,16152,16153],{"class":15841},"    Args:\n",[28,16155,16156],{"class":15660,"line":15811},[28,16157,16158],{"class":15841},"        f_ghz: 1D array of drive frequencies [GHz].\n",[28,16160,16161],{"class":15660,"line":15820},[28,16162,16163],{"class":15841},"        tau_ns: 1D array of pulse durations [ns].\n",[28,16165,16166],{"class":15660,"line":15830},[28,16167,16168],{"class":15841},"        f0_ghz: Qubit frequency on resonance [GHz].\n",[28,16170,16171],{"class":15660,"line":15835},[28,16172,16173],{"class":15841},"        omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n",[28,16175,16176],{"class":15660,"line":15853},[28,16177,16178],{"class":15841},"        t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n",[28,16180,16181],{"class":15660,"line":15867},[28,16182,15769],{"emptyLinePlaceholder":15768},[28,16184,16185],{"class":15660,"line":15872},[28,16186,16187],{"class":15841},"    Returns:\n",[28,16189,16190],{"class":15660,"line":15878},[28,16191,16192],{"class":15841},"        2D array P_e with shape (len(tau_ns), len(f_ghz)).\n",[28,16194,16195],{"class":15660,"line":15897},[28,16196,16197],{"class":15841},"    \"\"\"\n",[28,16199,16200,16203,16205,16208,16211],{"class":15660,"line":15914},[28,16201,16202],{"class":15675},"    two_pi ",[28,16204,845],{"class":15682},[28,16206,16207],{"class":15801}," 2.0",[28,16209,16210],{"class":15682}," *",[28,16212,16213],{"class":15675}," math.pi\n",[28,16215,16216,16219,16221,16224,16227],{"class":15660,"line":15931},[28,16217,16218],{"class":15675},"    f ",[28,16220,845],{"class":15682},[28,16222,16223],{"class":15675}," f_ghz ",[28,16225,16226],{"class":15682},"*",[28,16228,16229],{"class":15801}," 1e9\n",[28,16231,16232,16235,16237,16240,16242],{"class":15660,"line":15948},[28,16233,16234],{"class":15675},"    f0 ",[28,16236,845],{"class":15682},[28,16238,16239],{"class":15675}," f0_ghz ",[28,16241,16226],{"class":15682},[28,16243,16229],{"class":15801},[28,16245,16246,16249,16251,16254,16256],{"class":15660,"line":15966},[28,16247,16248],{"class":15675},"    tau ",[28,16250,845],{"class":15682},[28,16252,16253],{"class":15675}," tau_ns ",[28,16255,16226],{"class":15682},[28,16257,16258],{"class":15801}," 1e-9\n",[28,16260,16261,16264,16266,16269,16271,16274,16276,16279],{"class":15660,"line":15983},[28,16262,16263],{"class":15675},"    omega ",[28,16265,845],{"class":15682},[28,16267,16268],{"class":15675}," two_pi ",[28,16270,16226],{"class":15682},[28,16272,16273],{"class":15675}," omega_onres_mhz ",[28,16275,16226],{"class":15682},[28,16277,16278],{"class":15801}," 1e6",[28,16280,16281],{"class":15664},"  # rad\u002Fs\n",[28,16283,16284],{"class":15660,"line":16000},[28,16285,15769],{"emptyLinePlaceholder":15768},[28,16287,16288,16291,16294,16296,16299,16302,16305,16309,16311,16314],{"class":15660,"line":16016},[28,16289,16290],{"class":15675},"    F, ",[28,16292,16293],{"class":15801},"TAU",[28,16295,15805],{"class":15682},[28,16297,16298],{"class":15675}," np.",[28,16300,16301],{"class":16073},"meshgrid",[28,16303,16304],{"class":15675},"(f, tau, ",[28,16306,16308],{"class":16307},"s_ZVi","indexing",[28,16310,845],{"class":15682},[28,16312,16313],{"class":15841},"'xy'",[28,16315,16052],{"class":15675},[28,16317,16318,16321,16323,16325,16327,16330,16332,16335],{"class":15660,"line":16036},[28,16319,16320],{"class":15675},"    Delta ",[28,16322,845],{"class":15682},[28,16324,16268],{"class":15675},[28,16326,16226],{"class":15682},[28,16328,16329],{"class":15675}," (F ",[28,16331,15689],{"class":15682},[28,16333,16334],{"class":15675}," f0)               ",[28,16336,16337],{"class":15664},"# rad\u002Fs\n",[28,16339,16340,16343,16345,16347,16349,16352,16355,16357,16360,16363,16365,16367],{"class":15660,"line":16041},[28,16341,16342],{"class":15675},"    Omega_R ",[28,16344,845],{"class":15682},[28,16346,16298],{"class":15675},[28,16348,5750],{"class":16073},[28,16350,16351],{"class":15675},"(omega",[28,16353,16354],{"class":15682},"**",[28,16356,848],{"class":15801},[28,16358,16359],{"class":15682}," +",[28,16361,16362],{"class":15675}," Delta",[28,16364,16354],{"class":15682},[28,16366,848],{"class":15801},[28,16368,16052],{"class":15675},[28,16370,16372],{"class":15660,"line":16371},29,[28,16373,15769],{"emptyLinePlaceholder":15768},[28,16375,16377,16380,16382,16385,16387,16390,16392,16395,16398,16401,16403,16405],{"class":15660,"line":16376},30,[28,16378,16379],{"class":15671},"    with",[28,16381,16298],{"class":15675},[28,16383,16384],{"class":16073},"errstate",[28,16386,65],{"class":15675},[28,16388,16389],{"class":16307},"divide",[28,16391,845],{"class":15682},[28,16393,16394],{"class":15841},"'ignore'",[28,16396,16397],{"class":15675},", ",[28,16399,16400],{"class":16307},"invalid",[28,16402,845],{"class":15682},[28,16404,16394],{"class":15841},[28,16406,16407],{"class":15675},"):\n",[28,16409,16411,16414,16416,16419,16421,16424,16426],{"class":15660,"line":16410},31,[28,16412,16413],{"class":15675},"        amp ",[28,16415,845],{"class":15682},[28,16417,16418],{"class":15675}," (omega ",[28,16420,1129],{"class":15682},[28,16422,16423],{"class":15675}," Omega_R) ",[28,16425,16354],{"class":15682},[28,16427,16428],{"class":15801}," 2\n",[28,16430,16432,16435,16437,16440,16442,16445,16447],{"class":15660,"line":16431},32,[28,16433,16434],{"class":15675},"        phase ",[28,16436,845],{"class":15682},[28,16438,16439],{"class":15801}," 0.5",[28,16441,16210],{"class":15682},[28,16443,16444],{"class":15675}," Omega_R ",[28,16446,16226],{"class":15682},[28,16448,16449],{"class":15801}," TAU\n",[28,16451,16453,16456,16458,16461,16463,16465,16467,16470,16472],{"class":15660,"line":16452},33,[28,16454,16455],{"class":15675},"        Pe ",[28,16457,845],{"class":15682},[28,16459,16460],{"class":15675}," amp ",[28,16462,16226],{"class":15682},[28,16464,16298],{"class":15675},[28,16466,5217],{"class":16073},[28,16468,16469],{"class":15675},"(phase) ",[28,16471,16354],{"class":15682},[28,16473,16428],{"class":15801},[28,16475,16477],{"class":15660,"line":16476},34,[28,16478,15769],{"emptyLinePlaceholder":15768},[28,16480,16482,16485,16488,16491,16494,16496,16499,16501,16504,16506],{"class":15660,"line":16481},35,[28,16483,16484],{"class":15671},"    if",[28,16486,16487],{"class":15675}," t2star_ns ",[28,16489,16490],{"class":15671},"is",[28,16492,16493],{"class":15671}," not",[28,16495,16132],{"class":15801},[28,16497,16498],{"class":15671}," and",[28,16500,16487],{"class":15675},[28,16502,16503],{"class":15682},">",[28,16505,15977],{"class":15801},[28,16507,15778],{"class":15675},[28,16509,16511,16514,16516,16518,16520,16522,16524,16526,16529,16532,16534,16537],{"class":15660,"line":16510},36,[28,16512,16513],{"class":15675},"        decay ",[28,16515,845],{"class":15682},[28,16517,16298],{"class":15675},[28,16519,7645],{"class":16073},[28,16521,65],{"class":15675},[28,16523,15689],{"class":15682},[28,16525,16293],{"class":15801},[28,16527,16528],{"class":15682}," \u002F",[28,16530,16531],{"class":15675}," (t2star_ns ",[28,16533,16226],{"class":15682},[28,16535,16536],{"class":15801}," 1e-9",[28,16538,16539],{"class":15675},"))\n",[28,16541,16543,16545,16547,16550,16552],{"class":15660,"line":16542},37,[28,16544,16455],{"class":15675},[28,16546,845],{"class":15682},[28,16548,16549],{"class":15675}," Pe ",[28,16551,16226],{"class":15682},[28,16553,16554],{"class":15675}," decay\n",[28,16556,16558],{"class":15660,"line":16557},38,[28,16559,15769],{"emptyLinePlaceholder":15768},[28,16561,16563,16566,16568,16571,16574,16576,16578,16581],{"class":15660,"line":16562},39,[28,16564,16565],{"class":15671},"    return",[28,16567,16298],{"class":15675},[28,16569,16570],{"class":16073},"clip",[28,16572,16573],{"class":15675},"(Pe, ",[28,16575,15271],{"class":15801},[28,16577,16397],{"class":15675},[28,16579,16580],{"class":15801},"1.0",[28,16582,16052],{"class":15675},[15650,16584,16586],{"className":15652,"code":16585,"language":15654,"meta":15655,"style":15655},"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",[8103,16587,16588,16614,16639,16663],{"__ignoreMap":15655},[28,16589,16590,16593,16595,16597,16600,16602,16604,16606,16608,16610,16612],{"class":15660,"line":15661},[28,16591,16592],{"class":15675},"freq_ghz ",[28,16594,845],{"class":15682},[28,16596,16298],{"class":15675},[28,16598,16599],{"class":16073},"linspace",[28,16601,65],{"class":15675},[28,16603,15221],{"class":15801},[28,16605,16397],{"class":15675},[28,16607,15236],{"class":15801},[28,16609,16397],{"class":15675},[28,16611,15251],{"class":15801},[28,16613,16052],{"class":15675},[28,16615,16616,16619,16621,16623,16625,16627,16629,16631,16633,16635,16637],{"class":15660,"line":15668},[28,16617,16618],{"class":15675},"dur_ns ",[28,16620,845],{"class":15682},[28,16622,16298],{"class":15675},[28,16624,16599],{"class":16073},[28,16626,65],{"class":15675},[28,16628,15266],{"class":15801},[28,16630,16397],{"class":15675},[28,16632,15281],{"class":15801},[28,16634,16397],{"class":15675},[28,16636,15296],{"class":15801},[28,16638,16052],{"class":15675},[28,16640,16641,16644,16646,16648,16651,16653,16655,16657,16659,16661],{"class":15660,"line":15679},[28,16642,16643],{"class":15675},"Pe ",[28,16645,845],{"class":15682},[28,16647,16074],{"class":16073},[28,16649,16650],{"class":15675},"(freq_ghz, dur_ns, ",[28,16652,15084],{"class":15801},[28,16654,16397],{"class":15675},[28,16656,15168],{"class":15801},[28,16658,16397],{"class":15675},[28,16660,15310],{"class":15801},[28,16662,16052],{"class":15675},[28,16664,16665,16667,16669,16672],{"class":15660,"line":15725},[28,16666,16044],{"class":15682},[28,16668,65],{"class":15675},[28,16670,16671],{"class":15841},"'Grid shape (τ × f):'",[28,16673,16674],{"class":15675},", Pe.shape)\n",[15650,16676,16679],{"className":16677,"code":16678,"language":16058,"meta":15655},[16056],"Grid shape (τ × f): (401, 401)\n",[8103,16680,16678],{"__ignoreMap":15655},[15650,16682,16684],{"className":15652,"code":16683,"language":15654,"meta":15655,"style":15655},"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",[8103,16685,16686,16701,16706,16721,16734,16747,16760,16773,16786,16799,16812,16825,16830,16850,16854],{"__ignoreMap":15655},[28,16687,16688,16691,16693,16696,16699],{"class":15660,"line":15661},[28,16689,16690],{"class":15675},"controls_table ",[28,16692,845],{"class":15682},[28,16694,16695],{"class":15675}," pd.",[28,16697,16698],{"class":16073},"DataFrame",[28,16700,16077],{"class":15675},[28,16702,16703],{"class":15660,"line":15668},[28,16704,16705],{"class":15675},"    [\n",[28,16707,16708,16711,16714,16716,16718],{"class":15660,"line":15679},[28,16709,16710],{"class":15675},"        [",[28,16712,16713],{"class":15841},"'f0 (GHz)'",[28,16715,16397],{"class":15675},[28,16717,15084],{"class":15801},[28,16719,16720],{"class":15675},"],\n",[28,16722,16723,16725,16728,16730,16732],{"class":15660,"line":15725},[28,16724,16710],{"class":15675},[28,16726,16727],{"class":15841},"'Ω\u002F2π on-res (MHz)'",[28,16729,16397],{"class":15675},[28,16731,15168],{"class":15801},[28,16733,16720],{"class":15675},[28,16735,16736,16738,16741,16743,16745],{"class":15660,"line":15739},[28,16737,16710],{"class":15675},[28,16739,16740],{"class":15841},"'f sweep min (GHz)'",[28,16742,16397],{"class":15675},[28,16744,15221],{"class":15801},[28,16746,16720],{"class":15675},[28,16748,16749,16751,16754,16756,16758],{"class":15660,"line":15752},[28,16750,16710],{"class":15675},[28,16752,16753],{"class":15841},"'f sweep max (GHz)'",[28,16755,16397],{"class":15675},[28,16757,15236],{"class":15801},[28,16759,16720],{"class":15675},[28,16761,16762,16764,16767,16769,16771],{"class":15660,"line":15765},[28,16763,16710],{"class":15675},[28,16765,16766],{"class":15841},"'N_f'",[28,16768,16397],{"class":15675},[28,16770,15251],{"class":15801},[28,16772,16720],{"class":15675},[28,16774,16775,16777,16780,16782,16784],{"class":15660,"line":15772},[28,16776,16710],{"class":15675},[28,16778,16779],{"class":15841},"'τ sweep min (ns)'",[28,16781,16397],{"class":15675},[28,16783,15266],{"class":15801},[28,16785,16720],{"class":15675},[28,16787,16788,16790,16793,16795,16797],{"class":15660,"line":15781},[28,16789,16710],{"class":15675},[28,16791,16792],{"class":15841},"'τ sweep max (ns)'",[28,16794,16397],{"class":15675},[28,16796,15281],{"class":15801},[28,16798,16720],{"class":15675},[28,16800,16801,16803,16806,16808,16810],{"class":15660,"line":15798},[28,16802,16710],{"class":15675},[28,16804,16805],{"class":15841},"'N_τ'",[28,16807,16397],{"class":15675},[28,16809,15296],{"class":15801},[28,16811,16720],{"class":15675},[28,16813,16814,16816,16819,16821,16823],{"class":15660,"line":15811},[28,16815,16710],{"class":15675},[28,16817,16818],{"class":15841},"'T2* (ns; None=off)'",[28,16820,16397],{"class":15675},[28,16822,15310],{"class":15801},[28,16824,16720],{"class":15675},[28,16826,16827],{"class":15660,"line":15820},[28,16828,16829],{"class":15675},"    ],\n",[28,16831,16832,16835,16837,16839,16842,16844,16847],{"class":15660,"line":15830},[28,16833,16834],{"class":16307},"    columns",[28,16836,845],{"class":15682},[28,16838,3176],{"class":15675},[28,16840,16841],{"class":15841},"'Parameter'",[28,16843,16397],{"class":15675},[28,16845,16846],{"class":15841},"'Value'",[28,16848,16849],{"class":15675},"]\n",[28,16851,16852],{"class":15660,"line":15835},[28,16853,16052],{"class":15675},[28,16855,16856],{"class":15660,"line":15853},[28,16857,16858],{"class":15675},"controls_table\n",[15650,16860,16863],{"className":16861,"code":16862,"language":16058,"meta":15655},[16056],"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",[8103,16864,16862],{"__ignoreMap":15655},[15650,16866,16868],{"className":15652,"code":16867,"language":15654,"meta":15655,"style":15655},"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",[8103,16869,16870,16902,16928,16966,16981,16995,17019,17030],{"__ignoreMap":15655},[28,16871,16872,16875,16877,16880,16883,16885,16888,16890,16892,16895,16897,16900],{"class":15660,"line":15661},[28,16873,16874],{"class":15675},"fig, ax ",[28,16876,845],{"class":15682},[28,16878,16879],{"class":15675}," plt.",[28,16881,16882],{"class":16073},"subplots",[28,16884,65],{"class":15675},[28,16886,16887],{"class":16307},"figsize",[28,16889,845],{"class":15682},[28,16891,65],{"class":15675},[28,16893,16894],{"class":15801},"7",[28,16896,16397],{"class":15675},[28,16898,16899],{"class":15801},"5",[28,16901,16539],{"class":15675},[28,16903,16904,16907,16909,16912,16915,16918,16921,16923,16926],{"class":15660,"line":15668},[28,16905,16906],{"class":15675},"c ",[28,16908,845],{"class":15682},[28,16910,16911],{"class":15675}," ax.",[28,16913,16914],{"class":16073},"pcolormesh",[28,16916,16917],{"class":15675},"(freq_ghz, dur_ns, Pe, ",[28,16919,16920],{"class":16307},"shading",[28,16922,845],{"class":15682},[28,16924,16925],{"class":15841},"'auto'",[28,16927,16052],{"class":15675},[28,16929,16930,16933,16935,16938,16941,16944,16947,16949,16952,16955,16957,16960,16964],{"class":15660,"line":15679},[28,16931,16932],{"class":15675},"cb ",[28,16934,845],{"class":15682},[28,16936,16937],{"class":15675}," fig.",[28,16939,16940],{"class":16073},"colorbar",[28,16942,16943],{"class":15675},"(c, ",[28,16945,16946],{"class":16307},"ax",[28,16948,845],{"class":15682},[28,16950,16951],{"class":15675},"ax, ",[28,16953,16954],{"class":16307},"label",[28,16956,845],{"class":15682},[28,16958,16959],{"class":15671},"r",[28,16961,16963],{"class":16962},"sVyAn","'$P_e$'",[28,16965,16052],{"class":15675},[28,16967,16968,16971,16974,16976,16979],{"class":15660,"line":15725},[28,16969,16970],{"class":15675},"ax.",[28,16972,16973],{"class":16073},"set_xlabel",[28,16975,65],{"class":15675},[28,16977,16978],{"class":15841},"'Drive frequency f (GHz)'",[28,16980,16052],{"class":15675},[28,16982,16983,16985,16988,16990,16993],{"class":15660,"line":15739},[28,16984,16970],{"class":15675},[28,16986,16987],{"class":16073},"set_ylabel",[28,16989,65],{"class":15675},[28,16991,16992],{"class":15841},"'Pulse duration τ (ns)'",[28,16994,16052],{"class":15675},[28,16996,16997,16999,17002,17004,17007,17010,17012,17014,17017],{"class":15660,"line":15752},[28,16998,16970],{"class":15675},[28,17000,17001],{"class":16073},"set_title",[28,17003,65],{"class":15675},[28,17005,17006],{"class":15841},"'Rabi excited-state probability $P_e(f,",[28,17008,17009],{"class":15682},"\\\\",[28,17011,72],{"class":15841},[28,17013,17009],{"class":15682},[28,17015,17016],{"class":15841},"tau)$'",[28,17018,16052],{"class":15675},[28,17020,17021,17024,17027],{"class":15660,"line":15765},[28,17022,17023],{"class":15675},"fig.",[28,17025,17026],{"class":16073},"tight_layout",[28,17028,17029],{"class":15675},"()\n",[28,17031,17032,17035,17038],{"class":15660,"line":15772},[28,17033,17034],{"class":15675},"plt.",[28,17036,17037],{"class":16073},"show",[28,17039,17029],{"class":15675},[17041,17042],"blog-figure",{"alt":17043,"src":17044},"Output 1 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.webp",[15650,17046,17048],{"className":15652,"code":17047,"language":15654,"meta":15655,"style":15655},"# @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",[8103,17049,17050,17055,17060,17064,17071,17078,17084,17088,17093,17098,17102,17111,17115,17125,17137,17149,17161,17173,17178,17182,17209,17224,17231,17238,17242,17246,17256,17266,17270,17275,17285,17289,17299,17319,17328,17349,17353,17362,17371,17380,17390,17400,17405,17416,17427,17438,17449,17460,17465,17476,17487,17498,17509,17520,17525,17536,17555,17560,17565,17591,17597,17617,17622,17627,17637,17654,17663,17669,17681,17687,17692,17703,17709,17729,17742,17748,17753,17763,17768,17773,17781,17792,17823,17835,17841,17850,17856,17862,17870,17878,17886,17894,17902,17910,17919,17925,17936,17941,17950,17959,17964,17969,17976,17983,17990,17997,18004,18009,18020,18025,18030,18035,18047,18052,18063,18074,18087,18100,18105,18112,18126,18131,18144,18160,18180,18188,18194,18199,18210,18216,18228,18234,18239,18244,18249,18272,18278,18298,18303,18308,18318,18330,18342,18354,18374,18380,18405,18428,18444,18449,18476,18481,18505,18527,18548,18553,18568,18583,18601,18607,18612,18627,18633,18639,18645,18650,18655,18687,18714,18740,18745,18759,18774,18779,18784,18799,18805,18811,18817,18823,18829,18834,18839,18854,18859,18870,18878,18888,18893,18898,18903,18910,18926,18935,18944,18953,18962,18985,18991,18996,19001,19006,19016,19025,19034,19039,19045,19061,19072,19087,19097,19105,19114,19119,19141,19146,19163,19180,19185,19198,19212,19217,19225,19230,19235,19245,19253,19261,19273,19285,19294,19300,19311,19321,19330,19339,19350,19362,19373,19378,19383,19388,19398,19405,19413,19424,19435,19444,19450,19460,19470,19479,19488,19497,19509,19521,19530,19535,19540,19545,19555,19562,19569,19580,19591,19600,19606,19616,19624,19631,19638,19647,19656,19665,19674,19685,19697,19710,19719,19724,19729,19734,19744,19751,19758,19770,19781,19790,19796,19812,19817,19826,19836,19845,19854,19860,19869,19882,19895,19907,19912,19917,19922,19941,19947,19956,19963,19970,19978,19989,20000,20011,20016,20025,20032,20039,20047,20068,20079,20091,20102,20113,20118,20123,20128,20143,20155,20161,20166,20174,20184,20189,20201,20212,20222,20233,20238,20243,20255,20266,20275,20286,20291,20296,20306,20312,20326,20331,20341,20346,20358,20363,20368,20391,20396,20419,20430,20469,20479,20491,20511,20529,20534,20539,20551,20561,20571,20576,20581,20589,20599,20614,20625,20637,20642,20649,20658,20672,20683,20695,20700,20705,20715,20723,20733,20754,20765,20777,20782,20789,20798,20817,20828,20840,20845,20854,20859,20869,20877,20886,20900,20911,20937,20942,20949,20958,20971,20982,21005,21010,21015,21029,21037,21046,21059,21081,21092,21097,21104,21113,21126,21147,21158,21163,21168,21178,21188,21196,21206,21215,21227,21239,21251,21256,21263,21272,21281,21292,21303,21314,21319,21326,21335,21344,21355,21366,21387,21392,21399,21408,21417,21428,21439,21460,21465,21470,21475,21486,21497,21509,21521,21533,21545,21550,21555,21564,21569,21579,21594,21609,21614,21627,21652,21677,21702,21727,21753,21760,21766,21791,21813,21835,21840,21847,21853,21875,21896,21917,21922,21929,21935,21964,21969,21976,21982,22009,22014,22021,22027,22050],{"__ignoreMap":15655},[28,17051,17052],{"class":15660,"line":15661},[28,17053,17054],{"class":15664},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n",[28,17056,17057],{"class":15660,"line":15668},[28,17058,17059],{"class":15841},"\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n",[28,17061,17062],{"class":15660,"line":15679},[28,17063,15769],{"emptyLinePlaceholder":15768},[28,17065,17066,17068],{"class":15660,"line":15725},[28,17067,15672],{"class":15671},[28,17069,17070],{"class":15675}," importlib.util\n",[28,17072,17073,17075],{"class":15660,"line":15739},[28,17074,15672],{"class":15671},[28,17076,17077],{"class":15675}," subprocess\n",[28,17079,17080,17082],{"class":15660,"line":15752},[28,17081,15672],{"class":15671},[28,17083,15676],{"class":15675},[28,17085,17086],{"class":15660,"line":15765},[28,17087,15769],{"emptyLinePlaceholder":15768},[28,17089,17090],{"class":15660,"line":15772},[28,17091,17092],{"class":15664},"# -------------------------------------------------------------------------\n",[28,17094,17095],{"class":15660,"line":15781},[28,17096,17097],{"class":15664},"# Control knobs\n",[28,17099,17100],{"class":15660,"line":15798},[28,17101,17092],{"class":15664},[28,17103,17104,17107,17109],{"class":15660,"line":15811},[28,17105,17106],{"class":15801},"INSTALL_BLOCH_DEPENDENCIES",[28,17108,15805],{"class":15682},[28,17110,15808],{"class":15801},[28,17112,17113],{"class":15660,"line":15820},[28,17114,15769],{"emptyLinePlaceholder":15768},[28,17116,17117,17120,17122],{"class":15660,"line":15830},[28,17118,17119],{"class":15801},"BLOCH_REQUIRED_PACKAGES",[28,17121,15805],{"class":15682},[28,17123,17124],{"class":15675}," {\n",[28,17126,17127,17130,17132,17135],{"class":15660,"line":15835},[28,17128,17129],{"class":15841},"    \"numpy\"",[28,17131,15883],{"class":15675},[28,17133,17134],{"class":15841},"\"numpy\"",[28,17136,16105],{"class":15675},[28,17138,17139,17142,17144,17147],{"class":15660,"line":15853},[28,17140,17141],{"class":15841},"    \"matplotlib\"",[28,17143,15883],{"class":15675},[28,17145,17146],{"class":15841},"\"matplotlib\"",[28,17148,16105],{"class":15675},[28,17150,17151,17154,17156,17159],{"class":15660,"line":15867},[28,17152,17153],{"class":15841},"    \"scipy\"",[28,17155,15883],{"class":15675},[28,17157,17158],{"class":15841},"\"scipy\"",[28,17160,16105],{"class":15675},[28,17162,17163,17166,17168,17171],{"class":15660,"line":15872},[28,17164,17165],{"class":15841},"    \"qutip\"",[28,17167,15883],{"class":15675},[28,17169,17170],{"class":15841},"\"qutip\"",[28,17172,16105],{"class":15675},[28,17174,17175],{"class":15660,"line":15878},[28,17176,17177],{"class":15675},"}\n",[28,17179,17180],{"class":15660,"line":15897},[28,17181,15769],{"emptyLinePlaceholder":15768},[28,17183,17184,17187,17189,17192,17195,17198,17200,17203,17205,17207],{"class":15660,"line":15914},[28,17185,17186],{"class":15801},"BLOCH_F0_GHZ",[28,17188,15805],{"class":15682},[28,17190,17191],{"class":15682}," globals",[28,17193,17194],{"class":15675},"().",[28,17196,17197],{"class":16073},"get",[28,17199,65],{"class":15675},[28,17201,17202],{"class":15841},"\"F0_GHZ\"",[28,17204,16397],{"class":15675},[28,17206,2251],{"class":15801},[28,17208,16052],{"class":15675},[28,17210,17211,17214,17216,17218,17220,17222],{"class":15660,"line":15931},[28,17212,17213],{"class":15801},"BLOCH_OMEGA_ONRESONANCE_MHZ",[28,17215,15805],{"class":15682},[28,17217,17191],{"class":15682},[28,17219,17194],{"class":15675},[28,17221,17197],{"class":16073},[28,17223,16077],{"class":15675},[28,17225,17226,17229],{"class":15660,"line":15948},[28,17227,17228],{"class":15841},"    \"OMEGA_ONRESONANCE_MHZ\"",[28,17230,16105],{"class":15675},[28,17232,17233,17236],{"class":15660,"line":15966},[28,17234,17235],{"class":15801},"    20.0",[28,17237,16105],{"class":15675},[28,17239,17240],{"class":15660,"line":15983},[28,17241,16052],{"class":15675},[28,17243,17244],{"class":15660,"line":16000},[28,17245,15769],{"emptyLinePlaceholder":15768},[28,17247,17248,17251,17253],{"class":15660,"line":16016},[28,17249,17250],{"class":15801},"BLOCH_RESONANT_DETUNING_MHZ",[28,17252,15805],{"class":15682},[28,17254,17255],{"class":15801}," 0.0\n",[28,17257,17258,17261,17263],{"class":15660,"line":16036},[28,17259,17260],{"class":15801},"BLOCH_DETUNED_DETUNING_MHZ",[28,17262,15805],{"class":15682},[28,17264,17265],{"class":15801}," 40.0\n",[28,17267,17268],{"class":15660,"line":16041},[28,17269,15769],{"emptyLinePlaceholder":15768},[28,17271,17272],{"class":15660,"line":16371},[28,17273,17274],{"class":15664},"# Use None to automatically plot one resonant pi pulse.\n",[28,17276,17277,17280,17282],{"class":15660,"line":16376},[28,17278,17279],{"class":15801},"BLOCH_DURATION_NS",[28,17281,15805],{"class":15682},[28,17283,17284],{"class":15801}," None\n",[28,17286,17287],{"class":15660,"line":16410},[28,17288,15769],{"emptyLinePlaceholder":15768},[28,17290,17291,17294,17296],{"class":15660,"line":16431},[28,17292,17293],{"class":15801},"BLOCH_NUM_TIME_POINTS",[28,17295,15805],{"class":15682},[28,17297,17298],{"class":15801}," 300\n",[28,17300,17301,17304,17306,17309,17312,17314,17317],{"class":15660,"line":16452},[28,17302,17303],{"class":15801},"BLOCH_FIGSIZE",[28,17305,15805],{"class":15682},[28,17307,17308],{"class":15675}," 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0.12\n",[28,17510,17512,17515,17517],{"class":15660,"line":17511},53,[28,17513,17514],{"class":15801},"ARROW_LABEL_SCALE",[28,17516,15805],{"class":15682},[28,17518,17519],{"class":15801}," 1.16\n",[28,17521,17523],{"class":15660,"line":17522},54,[28,17524,15769],{"emptyLinePlaceholder":15768},[28,17526,17528,17531,17533],{"class":15660,"line":17527},55,[28,17529,17530],{"class":15801},"LEGEND_LOCATION",[28,17532,15805],{"class":15682},[28,17534,17535],{"class":15841}," \"upper left\"\n",[28,17537,17539,17542,17544,17546,17549,17551,17553],{"class":15660,"line":17538},56,[28,17540,17541],{"class":15801},"LEGEND_BBOX_TO_ANCHOR",[28,17543,15805],{"class":15682},[28,17545,17308],{"class":15675},[28,17547,17548],{"class":15801},"1.02",[28,17550,16397],{"class":15675},[28,17552,17548],{"class":15801},[28,17554,16052],{"class":15675},[28,17556,17558],{"class":15660,"line":17557},57,[28,17559,15769],{"emptyLinePlaceholder":15768},[28,17561,17563],{"class":15660,"line":17562},58,[28,17564,15769],{"emptyLinePlaceholder":15768},[28,17566,17568,17570,17573,17575,17578,17580,17583,17586,17589],{"class":15660,"line":17567},59,[28,17569,16070],{"class":15671},[28,17571,17572],{"class":16073}," package_is_available",[28,17574,65],{"class":15675},[28,17576,17577],{"class":16082},"package_name",[28,17579,15883],{"class":15675},[28,17581,17582],{"class":15682},"str",[28,17584,17585],{"class":15675},") -> 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try",[28,17780,15778],{"class":15675},[28,17782,17784,17787,17790],{"class":15660,"line":17783},81,[28,17785,17786],{"class":15675},"        subprocess.",[28,17788,17789],{"class":16073},"run",[28,17791,16077],{"class":15675},[28,17793,17795,17798,17801,17803,17806,17808,17811,17813,17816,17818,17821],{"class":15660,"line":17794},82,[28,17796,17797],{"class":15675},"            [sys.executable, ",[28,17799,17800],{"class":15841},"\"-m\"",[28,17802,16397],{"class":15675},[28,17804,17805],{"class":15841},"\"pip\"",[28,17807,16397],{"class":15675},[28,17809,17810],{"class":15841},"\"install\"",[28,17812,16397],{"class":15675},[28,17814,17815],{"class":15841},"\"-q\"",[28,17817,16397],{"class":15675},[28,17819,17820],{"class":15841},"\"uv\"",[28,17822,16720],{"class":15675},[28,17824,17826,17829,17831,17833],{"class":15660,"line":17825},83,[28,17827,17828],{"class":16307},"            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],\n",[28,17926,17928,17930,17932,17934],{"class":15660,"line":17927},96,[28,17929,17828],{"class":16307},[28,17931,845],{"class":15682},[28,17933,15597],{"class":15801},[28,17935,16105],{"class":15675},[28,17937,17939],{"class":15660,"line":17938},97,[28,17940,17840],{"class":15675},[28,17942,17944,17947],{"class":15660,"line":17943},98,[28,17945,17946],{"class":15671},"    except",[28,17948,17949],{"class":15675}," (subprocess.CalledProcessError, 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vector",[28,20740,845],{"class":15682},[28,20742,20608],{"class":15675},[28,20744,20611],{"class":15841},[28,20746,20747],{"class":15675},"][:, ",[28,20749,15689],{"class":15682},[28,20751,3200],{"class":15801},[28,20753,16720],{"class":15675},[28,20755,20757,20759,20761,20763],{"class":15660,"line":20756},386,[28,20758,19344],{"class":16307},[28,20760,845],{"class":15682},[28,20762,17410],{"class":15801},[28,20764,16105],{"class":15675},[28,20766,20768,20770,20772,20775],{"class":15660,"line":20767},387,[28,20769,19367],{"class":16307},[28,20771,845],{"class":15682},[28,20773,20774],{"class":15841},"\"Resonant final state\"",[28,20776,16105],{"class":15675},[28,20778,20780],{"class":15660,"line":20779},388,[28,20781,18606],{"class":15675},[28,20783,20785,20787],{"class":15660,"line":20784},389,[28,20786,20720],{"class":16073},[28,20788,16077],{"class":15675},[28,20790,20792,20794,20796],{"class":15660,"line":20791},390,[28,20793,20728],{"class":16307},[28,20795,845],{"class":15682},[28,20797,20598],{"class":15675},[28,20799,20801,20803,20805,20807,20809,20811,20813,20815],{"class":15660,"line":20800},391,[28,20802,20738],{"class":16307},[28,20804,845],{"class":15682},[28,20806,20667],{"class":15675},[28,20808,20611],{"class":15841},[28,20810,20747],{"class":15675},[28,20812,15689],{"class":15682},[28,20814,3200],{"class":15801},[28,20816,16720],{"class":15675},[28,20818,20820,20822,20824,20826],{"class":15660,"line":20819},392,[28,20821,19344],{"class":16307},[28,20823,845],{"class":15682},[28,20825,17421],{"class":15801},[28,20827,16105],{"class":15675},[28,20829,20831,20833,20835,20838],{"class":15660,"line":20830},393,[28,20832,19367],{"class":16307},[28,20834,845],{"class":15682},[28,20836,20837],{"class":15841},"\"Detuned final state\"",[28,20839,16105],{"class":15675},[28,20841,20843],{"class":15660,"line":20842},394,[28,20844,18606],{"class":15675},[28,20846,20848,20851],{"class":15660,"line":20847},395,[28,20849,20850],{"class":16073},"    add_initial_state_label",[28,20852,20853],{"class":15675},"(bloch_axis)\n",[28,20855,20857],{"class":15660,"line":20856},396,[28,20858,15769],{"emptyLinePlaceholder":15768},[28,20860,20862,20864,20867],{"class":15660,"line":20861},397,[28,20863,20133],{"class":15671},[28,20865,20866],{"class":15801}," BLOCH_SHOW_CONTROL_AXIS_ARROWS",[28,20868,15778],{"class":15675},[28,20870,20872,20875],{"class":15660,"line":20871},398,[28,20873,20874],{"class":16073},"    draw_control_axis_arrow",[28,20876,16077],{"class":15675},[28,20878,20880,20882,20884],{"class":15660,"line":20879},399,[28,20881,20728],{"class":16307},[28,20883,845],{"class":15682},[28,20885,20598],{"class":15675},[28,20887,20889,20891,20893,20895,20898],{"class":15660,"line":20888},400,[28,20890,20738],{"class":16307},[28,20892,845],{"class":15682},[28,20894,20608],{"class":15675},[28,20896,20897],{"class":15841},"\"control_axis\"",[28,20899,16720],{"class":15675},[28,20901,20903,20905,20907,20909],{"class":15660,"line":20902},401,[28,20904,19344],{"class":16307},[28,20906,845],{"class":15682},[28,20908,17432],{"class":15801},[28,20910,16105],{"class":15675},[28,20912,20914,20916,20918,20920,20923,20926,20929,20932,20935],{"class":15660,"line":20913},402,[28,20915,19367],{"class":16307},[28,20917,845],{"class":15682},[28,20919,16959],{"class":15671},[28,20921,20922],{"class":16962},"\"Resonant control axis $",[28,20924,20925],{"class":15682},"\\h",[28,20927,20928],{"class":16962},"at{n}_{",[28,20930,20931],{"class":15682},"\\m",[28,20933,20934],{"class":16962},"athrm{res}}$\"",[28,20936,16105],{"class":15675},[28,20938,20940],{"class":15660,"line":20939},403,[28,20941,18606],{"class":15675},[28,20943,20945,20947],{"class":15660,"line":20944},404,[28,20946,20874],{"class":16073},[28,20948,16077],{"class":15675},[28,20950,20952,20954,20956],{"class":15660,"line":20951},405,[28,20953,20728],{"class":16307},[28,20955,845],{"class":15682},[28,20957,20598],{"class":15675},[28,20959,20961,20963,20965,20967,20969],{"class":15660,"line":20960},406,[28,20962,20738],{"class":16307},[28,20964,845],{"class":15682},[28,20966,20667],{"class":15675},[28,20968,20897],{"class":15841},[28,20970,16720],{"class":15675},[28,20972,20974,20976,20978,20980],{"class":15660,"line":20973},407,[28,20975,19344],{"class":16307},[28,20977,845],{"class":15682},[28,20979,17443],{"class":15801},[28,20981,16105],{"class":15675},[28,20983,20985,20987,20989,20991,20994,20996,20998,21000,21003],{"class":15660,"line":20984},408,[28,20986,19367],{"class":16307},[28,20988,845],{"class":15682},[28,20990,16959],{"class":15671},[28,20992,20993],{"class":16962},"\"Detuned control axis $",[28,20995,20925],{"class":15682},[28,20997,20928],{"class":16962},[28,20999,20931],{"class":15682},[28,21001,21002],{"class":16962},"athrm{det}}$\"",[28,21004,16105],{"class":15675},[28,21006,21008],{"class":15660,"line":21007},409,[28,21009,18606],{"class":15675},[28,21011,21013],{"class":15660,"line":21012},410,[28,21014,15769],{"emptyLinePlaceholder":15768},[28,21016,21018,21020,21023,21025,21027],{"class":15660,"line":21017},411,[28,21019,20133],{"class":15671},[28,21021,21022],{"class":15801}," BLOCH_SHOW_ARROW_TIP_LABELS",[28,21024,16498],{"class":15671},[28,21026,20866],{"class":15801},[28,21028,15778],{"class":15675},[28,21030,21032,21035],{"class":15660,"line":21031},412,[28,21033,21034],{"class":16073},"    add_arrow_tip_label",[28,21036,16077],{"class":15675},[28,21038,21040,21042,21044],{"class":15660,"line":21039},413,[28,21041,20728],{"class":16307},[28,21043,845],{"class":15682},[28,21045,20598],{"class":15675},[28,21047,21049,21051,21053,21055,21057],{"class":15660,"line":21048},414,[28,21050,20738],{"class":16307},[28,21052,845],{"class":15682},[28,21054,20608],{"class":15675},[28,21056,20897],{"class":15841},[28,21058,16720],{"class":15675},[28,21060,21062,21065,21067,21069,21071,21073,21075,21077,21079],{"class":15660,"line":21061},415,[28,21063,21064],{"class":16307},"        text",[28,21066,845],{"class":15682},[28,21068,16959],{"class":15671},[28,21070,20444],{"class":16962},[28,21072,20925],{"class":15682},[28,21074,20928],{"class":16962},[28,21076,20931],{"class":15682},[28,21078,20934],{"class":16962},[28,21080,16105],{"class":15675},[28,21082,21084,21086,21088,21090],{"class":15660,"line":21083},416,[28,21085,19344],{"class":16307},[28,21087,845],{"class":15682},[28,21089,17432],{"class":15801},[28,21091,16105],{"class":15675},[28,21093,21095],{"class":15660,"line":21094},417,[28,21096,18606],{"class":15675},[28,21098,21100,21102],{"class":15660,"line":21099},418,[28,21101,21034],{"class":16073},[28,21103,16077],{"class":15675},[28,21105,21107,21109,21111],{"class":15660,"line":21106},419,[28,21108,20728],{"class":16307},[28,21110,845],{"class":15682},[28,21112,20598],{"class":15675},[28,21114,21116,21118,21120,21122,21124],{"class":15660,"line":21115},420,[28,21117,20738],{"class":16307},[28,21119,845],{"class":15682},[28,21121,20667],{"class":15675},[28,21123,20897],{"class":15841},[28,21125,16720],{"class":15675},[28,21127,21129,21131,21133,21135,21137,21139,21141,21143,21145],{"class":15660,"line":21128},421,[28,21130,21064],{"class":16307},[28,21132,845],{"class":15682},[28,21134,16959],{"class":15671},[28,21136,20444],{"class":16962},[28,21138,20925],{"class":15682},[28,21140,20928],{"class":16962},[28,21142,20931],{"class":15682},[28,21144,21002],{"class":16962},[28,21146,16105],{"class":15675},[28,21148,21150,21152,21154,21156],{"class":15660,"line":21149},422,[28,21151,19344],{"class":16307},[28,21153,845],{"class":15682},[28,21155,17443],{"class":15801},[28,21157,16105],{"class":15675},[28,21159,21161],{"class":15660,"line":21160},423,[28,21162,18606],{"class":15675},[28,21164,21166],{"class":15660,"line":21165},424,[28,21167,15769],{"emptyLinePlaceholder":15768},[28,21169,21171,21173,21176],{"class":15660,"line":21170},425,[28,21172,20133],{"class":15671},[28,21174,21175],{"class":15801}," BLOCH_SHOW_LEGEND",[28,21177,15778],{"class":15675},[28,21179,21181,21184,21186],{"class":15660,"line":21180},426,[28,21182,21183],{"class":15675},"    legend_handles ",[28,21185,845],{"class":15682},[28,21187,17702],{"class":15675},[28,21189,21191,21194],{"class":15660,"line":21190},427,[28,21192,21193],{"class":16073},"        Line2D",[28,21195,16077],{"class":15675},[28,21197,21199,21202,21204],{"class":15660,"line":21198},428,[28,21200,21201],{"class":15675},"            [",[28,21203,652],{"class":15801},[28,21205,16720],{"class":15675},[28,21207,21209,21211,21213],{"class":15660,"line":21208},429,[28,21210,21201],{"class":15675},[28,21212,652],{"class":15801},[28,21214,16720],{"class":15675},[28,21216,21218,21221,21223,21225],{"class":15660,"line":21217},430,[28,21219,21220],{"class":16307},"            color",[28,21222,845],{"class":15682},[28,21224,17410],{"class":15801},[28,21226,16105],{"class":15675},[28,21228,21230,21233,21235,21237],{"class":15660,"line":21229},431,[28,21231,21232],{"class":16307},"            linewidth",[28,21234,845],{"class":15682},[28,21236,17470],{"class":15801},[28,21238,16105],{"class":15675},[28,21240,21242,21245,21247,21249],{"class":15660,"line":21241},432,[28,21243,21244],{"class":16307},"            label",[28,21246,845],{"class":15682},[28,21248,20634],{"class":15841},[28,21250,16105],{"class":15675},[28,21252,21254],{"class":15660,"line":21253},433,[28,21255,18990],{"class":15675},[28,21257,21259,21261],{"class":15660,"line":21258},434,[28,21260,21193],{"class":16073},[28,21262,16077],{"class":15675},[28,21264,21266,21268,21270],{"class":15660,"line":21265},435,[28,21267,21201],{"class":15675},[28,21269,652],{"class":15801},[28,21271,16720],{"class":15675},[28,21273,21275,21277,21279],{"class":15660,"line":21274},436,[28,21276,21201],{"class":15675},[28,21278,652],{"class":15801},[28,21280,16720],{"class":15675},[28,21282,21284,21286,21288,21290],{"class":15660,"line":21283},437,[28,21285,21220],{"class":16307},[28,21287,845],{"class":15682},[28,21289,17421],{"class":15801},[28,21291,16105],{"class":15675},[28,21293,21295,21297,21299,21301],{"class":15660,"line":21294},438,[28,21296,21232],{"class":16307},[28,21298,845],{"class":15682},[28,21300,17470],{"class":15801},[28,21302,16105],{"class":15675},[28,21304,21306,21308,21310,21312],{"class":15660,"line":21305},439,[28,21307,21244],{"class":16307},[28,21309,845],{"class":15682},[28,21311,20692],{"class":15841},[28,21313,16105],{"class":15675},[28,21315,21317],{"class":15660,"line":21316},440,[28,21318,18990],{"class":15675},[28,21320,21322,21324],{"class":15660,"line":21321},441,[28,21323,21193],{"class":16073},[28,21325,16077],{"class":15675},[28,21327,21329,21331,21333],{"class":15660,"line":21328},442,[28,21330,21201],{"class":15675},[28,21332,652],{"class":15801},[28,21334,16720],{"class":15675},[28,21336,21338,21340,21342],{"class":15660,"line":21337},443,[28,21339,21201],{"class":15675},[28,21341,652],{"class":15801},[28,21343,16720],{"class":15675},[28,21345,21347,21349,21351,21353],{"class":15660,"line":21346},444,[28,21348,21220],{"class":16307},[28,21350,845],{"class":15682},[28,21352,17432],{"class":15801},[28,21354,16105],{"class":15675},[28,21356,21358,21360,21362,21364],{"class":15660,"line":21357},445,[28,21359,21232],{"class":16307},[28,21361,845],{"class":15682},[28,21363,17481],{"class":15801},[28,21365,16105],{"class":15675},[28,21367,21369,21371,21373,21375,21377,21379,21381,21383,21385],{"class":15660,"line":21368},446,[28,21370,21244],{"class":16307},[28,21372,845],{"class":15682},[28,21374,16959],{"class":15671},[28,21376,20922],{"class":16962},[28,21378,20925],{"class":15682},[28,21380,20928],{"class":16962},[28,21382,20931],{"class":15682},[28,21384,20934],{"class":16962},[28,21386,16105],{"class":15675},[28,21388,21390],{"class":15660,"line":21389},447,[28,21391,18990],{"class":15675},[28,21393,21395,21397],{"class":15660,"line":21394},448,[28,21396,21193],{"class":16073},[28,21398,16077],{"class":15675},[28,21400,21402,21404,21406],{"class":15660,"line":21401},449,[28,21403,21201],{"class":15675},[28,21405,652],{"class":15801},[28,21407,16720],{"class":15675},[28,21409,21411,21413,21415],{"class":15660,"line":21410},450,[28,21412,21201],{"class":15675},[28,21414,652],{"class":15801},[28,21416,16720],{"class":15675},[28,21418,21420,21422,21424,21426],{"class":15660,"line":21419},451,[28,21421,21220],{"class":16307},[28,21423,845],{"class":15682},[28,21425,17443],{"class":15801},[28,21427,16105],{"class":15675},[28,21429,21431,21433,21435,21437],{"class":15660,"line":21430},452,[28,21432,21232],{"class":16307},[28,21434,845],{"class":15682},[28,21436,17481],{"class":15801},[28,21438,16105],{"class":15675},[28,21440,21442,21444,21446,21448,21450,21452,21454,21456,21458],{"class":15660,"line":21441},453,[28,21443,21244],{"class":16307},[28,21445,845],{"class":15682},[28,21447,16959],{"class":15671},[28,21449,20993],{"class":16962},[28,21451,20925],{"class":15682},[28,21453,20928],{"class":16962},[28,21455,20931],{"class":15682},[28,21457,21002],{"class":16962},[28,21459,16105],{"class":15675},[28,21461,21463],{"class":15660,"line":21462},454,[28,21464,18990],{"class":15675},[28,21466,21468],{"class":15660,"line":21467},455,[28,21469,17747],{"class":15675},[28,21471,21473],{"class":15660,"line":21472},456,[28,21474,15769],{"emptyLinePlaceholder":15768},[28,21476,21478,21481,21484],{"class":15660,"line":21477},457,[28,21479,21480],{"class":15675},"    bloch_axis.",[28,21482,21483],{"class":16073},"legend",[28,21485,16077],{"class":15675},[28,21487,21489,21492,21494],{"class":15660,"line":21488},458,[28,21490,21491],{"class":16307},"        handles",[28,21493,845],{"class":15682},[28,21495,21496],{"class":15675},"legend_handles,\n",[28,21498,21500,21503,21505,21507],{"class":15660,"line":21499},459,[28,21501,21502],{"class":16307},"        loc",[28,21504,845],{"class":15682},[28,21506,17530],{"class":15801},[28,21508,16105],{"class":15675},[28,21510,21512,21515,21517,21519],{"class":15660,"line":21511},460,[28,21513,21514],{"class":16307},"        bbox_to_anchor",[28,21516,845],{"class":15682},[28,21518,17541],{"class":15801},[28,21520,16105],{"class":15675},[28,21522,21524,21527,21529,21531],{"class":15660,"line":21523},461,[28,21525,21526],{"class":16307},"        frameon",[28,21528,845],{"class":15682},[28,21530,15597],{"class":15801},[28,21532,16105],{"class":15675},[28,21534,21536,21539,21541,21543],{"class":15660,"line":21535},462,[28,21537,21538],{"class":16307},"        borderaxespad",[28,21540,845],{"class":15682},[28,21542,15271],{"class":15801},[28,21544,16105],{"class":15675},[28,21546,21548],{"class":15660,"line":21547},463,[28,21549,18606],{"class":15675},[28,21551,21553],{"class":15660,"line":21552},464,[28,21554,15769],{"emptyLinePlaceholder":15768},[28,21556,21558,21560,21562],{"class":15660,"line":21557},465,[28,21559,17034],{"class":15675},[28,21561,17037],{"class":16073},[28,21563,17029],{"class":15675},[28,21565,21567],{"class":15660,"line":21566},466,[28,21568,15769],{"emptyLinePlaceholder":15768},[28,21570,21572,21574,21577],{"class":15660,"line":21571},467,[28,21573,20133],{"class":15671},[28,21575,21576],{"class":15801}," BLOCH_PRINT_SUMMARY",[28,21578,15778],{"class":15675},[28,21580,21582,21585,21587,21590,21592],{"class":15660,"line":21581},468,[28,21583,21584],{"class":15675},"    resonant_axis ",[28,21586,845],{"class":15682},[28,21588,21589],{"class":15675}," resonant_trajectory[",[28,21591,20897],{"class":15841},[28,21593,16849],{"class":15675},[28,21595,21597,21600,21602,21605,21607],{"class":15660,"line":21596},469,[28,21598,21599],{"class":15675},"    detuned_axis ",[28,21601,845],{"class":15682},[28,21603,21604],{"class":15675}," detuned_trajectory[",[28,21606,20897],{"class":15841},[28,21608,16849],{"class":15675},[28,21610,21612],{"class":15660,"line":21611},470,[28,21613,15769],{"emptyLinePlaceholder":15768},[28,21615,21617,21620,21622,21625],{"class":15660,"line":21616},471,[28,21618,21619],{"class":15682},"    print",[28,21621,65],{"class":15675},[28,21623,21624],{"class":15841},"\"Bloch-sphere trajectory summary\"",[28,21626,16052],{"class":15675},[28,21628,21630,21632,21634,21636,21639,21642,21645,21647,21650],{"class":15660,"line":21629},472,[28,21631,21619],{"class":15682},[28,21633,65],{"class":15675},[28,21635,68],{"class":15671},[28,21637,21638],{"class":15841},"\"Natural transition frequency f0: ",[28,21640,21641],{"class":15801},"{BLOCH_F0_GHZ",[28,21643,21644],{"class":15671},":.6f",[28,21646,11404],{"class":15801},[28,21648,21649],{"class":15841}," GHz\"",[28,21651,16052],{"class":15675},[28,21653,21655,21657,21659,21661,21664,21666,21669,21671,21673,21675],{"class":15660,"line":21654},473,[28,21656,21619],{"class":15682},[28,21658,65],{"class":15675},[28,21660,68],{"class":15671},[28,21662,21663],{"class":15841},"\"Resonant drive frequency: ",[28,21665,11385],{"class":15801},[28,21667,21668],{"class":15675},"resonant_drive_ghz",[28,21670,21644],{"class":15671},[28,21672,11404],{"class":15801},[28,21674,21649],{"class":15841},[28,21676,16052],{"class":15675},[28,21678,21680,21682,21684,21686,21689,21691,21694,21696,21698,21700],{"class":15660,"line":21679},474,[28,21681,21619],{"class":15682},[28,21683,65],{"class":15675},[28,21685,68],{"class":15671},[28,21687,21688],{"class":15841},"\"Detuned drive frequency: ",[28,21690,11385],{"class":15801},[28,21692,21693],{"class":15675},"detuned_drive_ghz",[28,21695,21644],{"class":15671},[28,21697,11404],{"class":15801},[28,21699,21649],{"class":15841},[28,21701,16052],{"class":15675},[28,21703,21705,21707,21709,21711,21714,21717,21720,21722,21725],{"class":15660,"line":21704},475,[28,21706,21619],{"class":15682},[28,21708,65],{"class":15675},[28,21710,68],{"class":15671},[28,21712,21713],{"class":15841},"\"On-resonance Rabi rate: ",[28,21715,21716],{"class":15801},"{BLOCH_OMEGA_ONRESONANCE_MHZ",[28,21718,21719],{"class":15671},":.3f",[28,21721,11404],{"class":15801},[28,21723,21724],{"class":15841}," MHz\"",[28,21726,16052],{"class":15675},[28,21728,21730,21732,21734,21736,21739,21741,21744,21746,21748,21751],{"class":15660,"line":21729},476,[28,21731,21619],{"class":15682},[28,21733,65],{"class":15675},[28,21735,68],{"class":15671},[28,21737,21738],{"class":15841},"\"Plotted duration: ",[28,21740,11385],{"class":15801},[28,21742,21743],{"class":15675},"bloch_duration_ns",[28,21745,21719],{"class":15671},[28,21747,11404],{"class":15801},[28,21749,21750],{"class":15841}," ns\"",[28,21752,16052],{"class":15675},[28,21754,21756,21758],{"class":15660,"line":21755},477,[28,21757,21619],{"class":15682},[28,21759,16077],{"class":15675},[28,21761,21763],{"class":15660,"line":21762},478,[28,21764,21765],{"class":15841},"        \"Resonant control axis n_res: \"\n",[28,21767,21769,21772,21775,21777,21780,21782,21784,21786,21788],{"class":15660,"line":21768},479,[28,21770,21771],{"class":15671},"        f",[28,21773,21774],{"class":15841},"\"(",[28,21776,11385],{"class":15801},[28,21778,21779],{"class":15675},"resonant_axis[",[28,21781,652],{"class":15801},[28,21783,3217],{"class":15675},[28,21785,21644],{"class":15671},[28,21787,11404],{"class":15801},[28,21789,21790],{"class":15841},", \"\n",[28,21792,21794,21796,21799,21801,21803,21805,21807,21809,21811],{"class":15660,"line":21793},480,[28,21795,21771],{"class":15671},[28,21797,21798],{"class":15841},"\"",[28,21800,11385],{"class":15801},[28,21802,21779],{"class":15675},[28,21804,3200],{"class":15801},[28,21806,3217],{"class":15675},[28,21808,21644],{"class":15671},[28,21810,11404],{"class":15801},[28,21812,21790],{"class":15841},[28,21814,21816,21818,21820,21822,21824,21826,21828,21830,21832],{"class":15660,"line":21815},481,[28,21817,21771],{"class":15671},[28,21819,21798],{"class":15841},[28,21821,11385],{"class":15801},[28,21823,21779],{"class":15675},[28,21825,848],{"class":15801},[28,21827,3217],{"class":15675},[28,21829,21644],{"class":15671},[28,21831,11404],{"class":15801},[28,21833,21834],{"class":15841},")\"\n",[28,21836,21838],{"class":15660,"line":21837},482,[28,21839,18606],{"class":15675},[28,21841,21843,21845],{"class":15660,"line":21842},483,[28,21844,21619],{"class":15682},[28,21846,16077],{"class":15675},[28,21848,21850],{"class":15660,"line":21849},484,[28,21851,21852],{"class":15841},"        \"Detuned control axis n_det: \"\n",[28,21854,21856,21858,21860,21862,21865,21867,21869,21871,21873],{"class":15660,"line":21855},485,[28,21857,21771],{"class":15671},[28,21859,21774],{"class":15841},[28,21861,11385],{"class":15801},[28,21863,21864],{"class":15675},"detuned_axis[",[28,21866,652],{"class":15801},[28,21868,3217],{"class":15675},[28,21870,21644],{"class":15671},[28,21872,11404],{"class":15801},[28,21874,21790],{"class":15841},[28,21876,21878,21880,21882,21884,21886,21888,21890,21892,21894],{"class":15660,"line":21877},486,[28,21879,21771],{"class":15671},[28,21881,21798],{"class":15841},[28,21883,11385],{"class":15801},[28,21885,21864],{"class":15675},[28,21887,3200],{"class":15801},[28,21889,3217],{"class":15675},[28,21891,21644],{"class":15671},[28,21893,11404],{"class":15801},[28,21895,21790],{"class":15841},[28,21897,21899,21901,21903,21905,21907,21909,21911,21913,21915],{"class":15660,"line":21898},487,[28,21900,21771],{"class":15671},[28,21902,21798],{"class":15841},[28,21904,11385],{"class":15801},[28,21906,21864],{"class":15675},[28,21908,848],{"class":15801},[28,21910,3217],{"class":15675},[28,21912,21644],{"class":15671},[28,21914,11404],{"class":15801},[28,21916,21834],{"class":15841},[28,21918,21920],{"class":15660,"line":21919},488,[28,21921,18606],{"class":15675},[28,21923,21925,21927],{"class":15660,"line":21924},489,[28,21926,21619],{"class":15682},[28,21928,16077],{"class":15675},[28,21930,21932],{"class":15660,"line":21931},490,[28,21933,21934],{"class":15841},"        \"Resonant final excited-state probability: \"\n",[28,21936,21938,21940,21942,21944,21946,21949,21952,21954,21956,21958,21960,21962],{"class":15660,"line":21937},491,[28,21939,21771],{"class":15671},[28,21941,21798],{"class":15841},[28,21943,11385],{"class":15801},[28,21945,20608],{"class":15675},[28,21947,21948],{"class":15841},"'excited_probability'",[28,21950,21951],{"class":15675},"][",[28,21953,15689],{"class":15682},[28,21955,3200],{"class":15801},[28,21957,3217],{"class":15675},[28,21959,21644],{"class":15671},[28,21961,11404],{"class":15801},[28,21963,20490],{"class":15841},[28,21965,21967],{"class":15660,"line":21966},492,[28,21968,18606],{"class":15675},[28,21970,21972,21974],{"class":15660,"line":21971},493,[28,21973,21619],{"class":15682},[28,21975,16077],{"class":15675},[28,21977,21979],{"class":15660,"line":21978},494,[28,21980,21981],{"class":15841},"        \"Detuned final excited-state probability: \"\n",[28,21983,21985,21987,21989,21991,21993,21995,21997,21999,22001,22003,22005,22007],{"class":15660,"line":21984},495,[28,21986,21771],{"class":15671},[28,21988,21798],{"class":15841},[28,21990,11385],{"class":15801},[28,21992,20667],{"class":15675},[28,21994,21948],{"class":15841},[28,21996,21951],{"class":15675},[28,21998,15689],{"class":15682},[28,22000,3200],{"class":15801},[28,22002,3217],{"class":15675},[28,22004,21644],{"class":15671},[28,22006,11404],{"class":15801},[28,22008,20490],{"class":15841},[28,22010,22012],{"class":15660,"line":22011},496,[28,22013,18606],{"class":15675},[28,22015,22017,22019],{"class":15660,"line":22016},497,[28,22018,21619],{"class":15682},[28,22020,16077],{"class":15675},[28,22022,22024],{"class":15660,"line":22023},498,[28,22025,22026],{"class":15841},"        \"Detuned generalized Rabi frequency: \"\n",[28,22028,22030,22032,22034,22036,22038,22041,22043,22045,22047],{"class":15660,"line":22029},499,[28,22031,21771],{"class":15671},[28,22033,21798],{"class":15841},[28,22035,11385],{"class":15801},[28,22037,20667],{"class":15675},[28,22039,22040],{"class":15841},"'generalized_rabi_mhz'",[28,22042,3217],{"class":15675},[28,22044,21644],{"class":15671},[28,22046,11404],{"class":15801},[28,22048,22049],{"class":15841}," MHz\"\n",[28,22051,22053],{"class":15660,"line":22052},500,[28,22054,18606],{"class":15675},[17041,22056],{"alt":22057,"src":22058},"Output 2 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-02.webp",[15650,22060,22063],{"className":22061,"code":22062,"language":16058,"meta":15655},[16056],"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",[8103,22064,22062],{"__ignoreMap":15655},[15650,22066,22068],{"className":15652,"code":22067,"language":15654,"meta":15655,"style":15655},"# --- 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",[8103,22069,22070,22075,22087,22097,22107,22111,22116,22133,22150,22171,22175,22180,22185,22211,22240,22273,22293,22308,22312,22317,22340,22366,22394,22412,22426,22430,22435,22457,22461,22466,22505,22534,22538,22552,22557,22581,22601,22612,22632,22636,22640,22653,22683,22698,22711,22715,22720,22759,22775,22779,22787],{"__ignoreMap":15655},[28,22071,22072],{"class":15660,"line":15661},[28,22073,22074],{"class":15664},"# --- High-resolution 3D surface with colormap & legend  ---\n",[28,22076,22077,22079,22082,22084],{"class":15660,"line":15668},[28,22078,15714],{"class":15671},[28,22080,22081],{"class":15675}," mpl_toolkits.mplot3d ",[28,22083,15672],{"class":15671},[28,22085,22086],{"class":15675}," Axes3D\n",[28,22088,22089,22091,22093,22095],{"class":15660,"line":15679},[28,22090,15672],{"class":15671},[28,22092,15730],{"class":15675},[28,22094,15733],{"class":15671},[28,22096,15736],{"class":15675},[28,22098,22099,22101,22103,22105],{"class":15660,"line":15725},[28,22100,15672],{"class":15671},[28,22102,15744],{"class":15675},[28,22104,15733],{"class":15671},[28,22106,15749],{"class":15675},[28,22108,22109],{"class":15660,"line":15739},[28,22110,15769],{"emptyLinePlaceholder":15768},[28,22112,22113],{"class":15660,"line":15752},[28,22114,22115],{"class":15664},"# Controls for surface quality and appearance\n",[28,22117,22118,22121,22123,22125,22127,22130],{"class":15660,"line":15765},[28,22119,22120],{"class":15801},"UPSAMPLE_FACTOR",[28,22122,15883],{"class":15675},[28,22124,15955],{"class":15682},[28,22126,15805],{"class":15682},[28,22128,22129],{"class":15801}," 2",[28,22131,22132],{"class":15664},"           # 2=~4× more faces; try 3 for even smoother\n",[28,22134,22135,22138,22140,22142,22144,22147],{"class":15660,"line":15772},[28,22136,22137],{"class":15801},"SURFACE_CMAP",[28,22139,15883],{"class":15675},[28,22141,17582],{"class":15682},[28,22143,15805],{"class":15682},[28,22145,22146],{"class":15841}," 'inferno'",[28,22148,22149],{"class":15664},"      # e.g., 'viridis', 'plasma', 'magma', 'inferno'\n",[28,22151,22152,22155,22157,22160,22162,22164,22166,22168],{"class":15660,"line":15781},[28,22153,22154],{"class":15801},"Z_MIN",[28,22156,16397],{"class":15675},[28,22158,22159],{"class":15801},"Z_MAX",[28,22161,15805],{"class":15682},[28,22163,15977],{"class":15801},[28,22165,16397],{"class":15675},[28,22167,16580],{"class":15801},[28,22169,22170],{"class":15664},"            # clamp\u002Fnormalize expected Pe range\n",[28,22172,22173],{"class":15660,"line":15798},[28,22174,15769],{"emptyLinePlaceholder":15768},[28,22176,22177],{"class":15660,"line":15811},[28,22178,22179],{"class":15664},"# --- Bilinear upsampling (SciPy-free) ---\n",[28,22181,22182],{"class":15660,"line":15820},[28,22183,22184],{"class":15664},"# 1) Upsample frequency axis for each fixed τ\n",[28,22186,22187,22190,22192,22194,22196,22199,22202,22205,22208],{"class":15660,"line":15830},[28,22188,22189],{"class":15675},"freq_hi ",[28,22191,845],{"class":15682},[28,22193,16298],{"class":15675},[28,22195,16599],{"class":16073},[28,22197,22198],{"class":15675},"(freq_ghz.",[28,22200,22201],{"class":16073},"min",[28,22203,22204],{"class":15675},"(), freq_ghz.",[28,22206,22207],{"class":16073},"max",[28,22209,22210],{"class":15675},"(),\n",[28,22212,22213,22216,22218,22220,22223,22226,22228,22231,22234,22236,22238],{"class":15660,"line":15835},[28,22214,22215],{"class":15801},"                      UPSAMPLE_FACTOR",[28,22217,16210],{"class":15682},[28,22219,17308],{"class":15675},[28,22221,22222],{"class":15682},"len",[28,22224,22225],{"class":15675},"(freq_ghz) ",[28,22227,15689],{"class":15682},[28,22229,22230],{"class":15801}," 1",[28,22232,22233],{"class":15675},") ",[28,22235,2606],{"class":15682},[28,22237,22230],{"class":15801},[28,22239,16052],{"class":15675},[28,22241,22242,22245,22247,22249,22252,22255,22257,22260,22262,22265,22268,22271],{"class":15660,"line":15853},[28,22243,22244],{"class":15675},"Pe_tau_f ",[28,22246,845],{"class":15682},[28,22248,16298],{"class":15675},[28,22250,22251],{"class":16073},"empty",[28,22253,22254],{"class":15675},"((",[28,22256,22222],{"class":15682},[28,22258,22259],{"class":15675},"(dur_ns), ",[28,22261,22222],{"class":15682},[28,22263,22264],{"class":15675},"(freq_hi)), ",[28,22266,22267],{"class":16307},"dtype",[28,22269,22270],{"class":15682},"=float",[28,22272,16052],{"class":15675},[28,22274,22275,22278,22281,22283,22286,22288,22290],{"class":15660,"line":15867},[28,22276,22277],{"class":15671},"for",[28,22279,22280],{"class":15675}," i ",[28,22282,17720],{"class":15671},[28,22284,22285],{"class":15682}," range",[28,22287,65],{"class":15675},[28,22289,22222],{"class":15682},[28,22291,22292],{"class":15675},"(dur_ns)):\n",[28,22294,22295,22298,22300,22302,22305],{"class":15660,"line":15872},[28,22296,22297],{"class":15675},"    Pe_tau_f[i, :] ",[28,22299,845],{"class":15682},[28,22301,16298],{"class":15675},[28,22303,22304],{"class":16073},"interp",[28,22306,22307],{"class":15675},"(freq_hi, freq_ghz, Pe[i, :])\n",[28,22309,22310],{"class":15660,"line":15878},[28,22311,15769],{"emptyLinePlaceholder":15768},[28,22313,22314],{"class":15660,"line":15897},[28,22315,22316],{"class":15664},"# 2) Upsample τ axis for each fixed frequency\n",[28,22318,22319,22322,22324,22326,22328,22331,22333,22336,22338],{"class":15660,"line":15914},[28,22320,22321],{"class":15675},"dur_hi ",[28,22323,845],{"class":15682},[28,22325,16298],{"class":15675},[28,22327,16599],{"class":16073},[28,22329,22330],{"class":15675},"(dur_ns.",[28,22332,22201],{"class":16073},[28,22334,22335],{"class":15675},"(), dur_ns.",[28,22337,22207],{"class":16073},[28,22339,22210],{"class":15675},[28,22341,22342,22345,22347,22349,22351,22354,22356,22358,22360,22362,22364],{"class":15660,"line":15931},[28,22343,22344],{"class":15801},"                     UPSAMPLE_FACTOR",[28,22346,16210],{"class":15682},[28,22348,17308],{"class":15675},[28,22350,22222],{"class":15682},[28,22352,22353],{"class":15675},"(dur_ns) ",[28,22355,15689],{"class":15682},[28,22357,22230],{"class":15801},[28,22359,22233],{"class":15675},[28,22361,2606],{"class":15682},[28,22363,22230],{"class":15801},[28,22365,16052],{"class":15675},[28,22367,22368,22371,22373,22375,22377,22379,22381,22384,22386,22388,22390,22392],{"class":15660,"line":15948},[28,22369,22370],{"class":15675},"Pe_hi ",[28,22372,845],{"class":15682},[28,22374,16298],{"class":15675},[28,22376,22251],{"class":16073},[28,22378,22254],{"class":15675},[28,22380,22222],{"class":15682},[28,22382,22383],{"class":15675},"(dur_hi), ",[28,22385,22222],{"class":15682},[28,22387,22264],{"class":15675},[28,22389,22267],{"class":16307},[28,22391,22270],{"class":15682},[28,22393,16052],{"class":15675},[28,22395,22396,22398,22401,22403,22405,22407,22409],{"class":15660,"line":15966},[28,22397,22277],{"class":15671},[28,22399,22400],{"class":15675}," j ",[28,22402,17720],{"class":15671},[28,22404,22285],{"class":15682},[28,22406,65],{"class":15675},[28,22408,22222],{"class":15682},[28,22410,22411],{"class":15675},"(freq_hi)):\n",[28,22413,22414,22417,22419,22421,22423],{"class":15660,"line":15983},[28,22415,22416],{"class":15675},"    Pe_hi[:, j] ",[28,22418,845],{"class":15682},[28,22420,16298],{"class":15675},[28,22422,22304],{"class":16073},[28,22424,22425],{"class":15675},"(dur_hi, dur_ns, Pe_tau_f[:, j])\n",[28,22427,22428],{"class":15660,"line":16000},[28,22429,15769],{"emptyLinePlaceholder":15768},[28,22431,22432],{"class":15660,"line":16016},[28,22433,22434],{"class":15664},"# Mesh for plotting\n",[28,22436,22437,22440,22442,22444,22446,22449,22451,22453,22455],{"class":15660,"line":16036},[28,22438,22439],{"class":15675},"F_hi, T_hi ",[28,22441,845],{"class":15682},[28,22443,16298],{"class":15675},[28,22445,16301],{"class":16073},[28,22447,22448],{"class":15675},"(freq_hi, dur_hi, ",[28,22450,16308],{"class":16307},[28,22452,845],{"class":15682},[28,22454,16313],{"class":15841},[28,22456,16052],{"class":15675},[28,22458,22459],{"class":15660,"line":16041},[28,22460,15769],{"emptyLinePlaceholder":15768},[28,22462,22463],{"class":15660,"line":16371},[28,22464,22465],{"class":15664},"# --- Plot ---\n",[28,22467,22468,22471,22473,22475,22477,22479,22481,22483,22485,22487,22489,22492,22495,22498,22500,22503],{"class":15660,"line":16376},[28,22469,22470],{"class":15675},"fig ",[28,22472,845],{"class":15682},[28,22474,16879],{"class":15675},[28,22476,20380],{"class":16073},[28,22478,65],{"class":15675},[28,22480,16887],{"class":16307},[28,22482,845],{"class":15682},[28,22484,65],{"class":15675},[28,22486,15567],{"class":15801},[28,22488,16397],{"class":15675},[28,22490,22491],{"class":15801},"6",[28,22493,22494],{"class":15675},"), ",[28,22496,22497],{"class":16307},"dpi",[28,22499,845],{"class":15682},[28,22501,22502],{"class":15801},"200",[28,22504,16052],{"class":15675},[28,22506,22507,22510,22512,22514,22517,22519,22522,22524,22527,22529,22532],{"class":15660,"line":16410},[28,22508,22509],{"class":15675},"ax ",[28,22511,845],{"class":15682},[28,22513,16937],{"class":15675},[28,22515,22516],{"class":16073},"add_subplot",[28,22518,65],{"class":15675},[28,22520,22521],{"class":15801},"111",[28,22523,16397],{"class":15675},[28,22525,22526],{"class":16307},"projection",[28,22528,845],{"class":15682},[28,22530,22531],{"class":15841},"'3d'",[28,22533,16052],{"class":15675},[28,22535,22536],{"class":15660,"line":16431},[28,22537,15769],{"emptyLinePlaceholder":15768},[28,22539,22540,22543,22545,22547,22550],{"class":15660,"line":16452},[28,22541,22542],{"class":15675},"surf ",[28,22544,845],{"class":15682},[28,22546,16911],{"class":15675},[28,22548,22549],{"class":16073},"plot_surface",[28,22551,16077],{"class":15675},[28,22553,22554],{"class":15660,"line":16476},[28,22555,22556],{"class":15675},"    F_hi, T_hi, Pe_hi,\n",[28,22558,22559,22562,22564,22566,22568,22571,22573,22575,22578],{"class":15660,"line":16481},[28,22560,22561],{"class":16307},"    rstride",[28,22563,845],{"class":15682},[28,22565,3200],{"class":15801},[28,22567,16397],{"class":15675},[28,22569,22570],{"class":16307},"cstride",[28,22572,845],{"class":15682},[28,22574,3200],{"class":15801},[28,22576,22577],{"class":15675},",             ",[28,22579,22580],{"class":15664},"# use every row\u002Fcolumn (after upsampling)\n",[28,22582,22583,22586,22588,22590,22592,22595,22597,22599],{"class":15660,"line":16510},[28,22584,22585],{"class":16307},"    linewidth",[28,22587,845],{"class":15682},[28,22589,652],{"class":15801},[28,22591,16397],{"class":15675},[28,22593,22594],{"class":16307},"antialiased",[28,22596,845],{"class":15682},[28,22598,15597],{"class":15801},[28,22600,16105],{"class":15675},[28,22602,22603,22606,22608,22610],{"class":15660,"line":16542},[28,22604,22605],{"class":16307},"    cmap",[28,22607,845],{"class":15682},[28,22609,22137],{"class":15801},[28,22611,16105],{"class":15675},[28,22613,22614,22617,22619,22621,22623,22626,22628,22630],{"class":15660,"line":16557},[28,22615,22616],{"class":16307},"    vmin",[28,22618,845],{"class":15682},[28,22620,22154],{"class":15801},[28,22622,16397],{"class":15675},[28,22624,22625],{"class":16307},"vmax",[28,22627,845],{"class":15682},[28,22629,22159],{"class":15801},[28,22631,16105],{"class":15675},[28,22633,22634],{"class":15660,"line":16562},[28,22635,16052],{"class":15675},[28,22637,22638],{"class":15660,"line":17382},[28,22639,15769],{"emptyLinePlaceholder":15768},[28,22641,22642,22644,22646,22648,22651],{"class":15660,"line":17392},[28,22643,16970],{"class":15675},[28,22645,16973],{"class":16073},[28,22647,65],{"class":15675},[28,22649,22650],{"class":15841},"'f (GHz)'",[28,22652,16052],{"class":15675},[28,22654,22655,22657,22659,22661,22663,22666,22669,22672,22674,22676,22678,22681],{"class":15660,"line":17402},[28,22656,16970],{"class":15675},[28,22658,16987],{"class":16073},[28,22660,65],{"class":15675},[28,22662,16959],{"class":15671},[28,22664,22665],{"class":16962},"'$",[28,22667,22668],{"class":15682},"\\t",[28,22670,22671],{"class":16962},"au$ ",[28,22673,65],{"class":15801},[28,22675,7191],{"class":16962},[28,22677,78],{"class":15801},[28,22679,22680],{"class":16962},"'",[28,22682,16052],{"class":15675},[28,22684,22685,22687,22690,22692,22694,22696],{"class":15660,"line":17407},[28,22686,16970],{"class":15675},[28,22688,22689],{"class":16073},"set_zlabel",[28,22691,65],{"class":15675},[28,22693,16959],{"class":15671},[28,22695,16963],{"class":16962},[28,22697,16052],{"class":15675},[28,22699,22700,22702,22704,22706,22709],{"class":15660,"line":17418},[28,22701,16970],{"class":15675},[28,22703,17001],{"class":16073},[28,22705,65],{"class":15675},[28,22707,22708],{"class":15841},"'Rabi $P_e$ surface'",[28,22710,16052],{"class":15675},[28,22712,22713],{"class":15660,"line":17429},[28,22714,15769],{"emptyLinePlaceholder":15768},[28,22716,22717],{"class":15660,"line":17440},[28,22718,22719],{"class":15664},"# Colorbar (legend for the colormap)\n",[28,22721,22722,22724,22726,22728,22730,22733,22735,22737,22739,22742,22744,22747,22749,22752,22754,22757],{"class":15660,"line":17451},[28,22723,16932],{"class":15675},[28,22725,845],{"class":15682},[28,22727,16937],{"class":15675},[28,22729,16940],{"class":16073},[28,22731,22732],{"class":15675},"(surf, ",[28,22734,16946],{"class":16307},[28,22736,845],{"class":15682},[28,22738,16951],{"class":15675},[28,22740,22741],{"class":16307},"shrink",[28,22743,845],{"class":15682},[28,22745,22746],{"class":15801},"0.7",[28,22748,16397],{"class":15675},[28,22750,22751],{"class":16307},"pad",[28,22753,845],{"class":15682},[28,22755,22756],{"class":15801},"0.08",[28,22758,16052],{"class":15675},[28,22760,22761,22764,22767,22769,22771,22773],{"class":15660,"line":17462},[28,22762,22763],{"class":15675},"cb.",[28,22765,22766],{"class":16073},"set_label",[28,22768,65],{"class":15675},[28,22770,16959],{"class":15671},[28,22772,16963],{"class":16962},[28,22774,16052],{"class":15675},[28,22776,22777],{"class":15660,"line":17467},[28,22778,15769],{"emptyLinePlaceholder":15768},[28,22780,22781,22783,22785],{"class":15660,"line":17478},[28,22782,17023],{"class":15675},[28,22784,17026],{"class":16073},[28,22786,17029],{"class":15675},[28,22788,22789,22791,22793],{"class":15660,"line":17489},[28,22790,17034],{"class":15675},[28,22792,17037],{"class":16073},[28,22794,17029],{"class":15675},[17041,22796],{"alt":22797,"src":22798},"Output 3 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-03.webp",[15650,22800,22802],{"className":15652,"code":22801,"language":15654,"meta":15655,"style":15655},"# @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",[8103,22803,22804,22809,22814,22829,22845,22849,22859,22869,22873,22910,22915,22941,22964,23013,23027,23052,23056,23061,23082,23092,23096,23123,23142,23168,23182,23187,23233,23250,23258,23266,23270,23275,23296,23306,23310,23336,23353,23365,23379,23425,23441,23449],{"__ignoreMap":15655},[28,22805,22806],{"class":15660,"line":15661},[28,22807,22808],{"class":15664},"# @title Cross-sections\n",[28,22810,22811],{"class":15660,"line":15668},[28,22812,22813],{"class":15664},"# Slicing controls\n",[28,22815,22816,22818,22820,22822,22824,22826],{"class":15660,"line":15679},[28,22817,15346],{"class":15801},[28,22819,15883],{"class":15675},[28,22821,15886],{"class":15682},[28,22823,15805],{"class":15682},[28,22825,15891],{"class":15801},[28,22827,22828],{"class":15664},"  # drive frequency slice [GHz]\n",[28,22830,22831,22833,22835,22837,22839,22842],{"class":15660,"line":15725},[28,22832,15445],{"class":15801},[28,22834,15883],{"class":15675},[28,22836,15886],{"class":15682},[28,22838,15805],{"class":15682},[28,22840,22841],{"class":15801}," 100.0",[28,22843,22844],{"class":15664},"   # duration slice [ns]\n",[28,22846,22847],{"class":15660,"line":15739},[28,22848,15769],{"emptyLinePlaceholder":15768},[28,22850,22851,22853,22855,22857],{"class":15660,"line":15752},[28,22852,15672],{"class":15671},[28,22854,15730],{"class":15675},[28,22856,15733],{"class":15671},[28,22858,15736],{"class":15675},[28,22860,22861,22863,22865,22867],{"class":15660,"line":15765},[28,22862,15672],{"class":15671},[28,22864,15744],{"class":15675},[28,22866,15733],{"class":15671},[28,22868,15749],{"class":15675},[28,22870,22871],{"class":15660,"line":15772},[28,22872,15769],{"emptyLinePlaceholder":15768},[28,22874,22875,22877,22880,22882,22885,22888,22891,22893,22895,22897,22900,22902,22904,22906,22908],{"class":15660,"line":15781},[28,22876,16070],{"class":15671},[28,22878,22879],{"class":16073}," _nearest_index",[28,22881,65],{"class":15675},[28,22883,22884],{"class":16082},"vec",[28,22886,22887],{"class":15675},": np.ndarray, ",[28,22889,22890],{"class":16082},"value",[28,22892,15883],{"class":15675},[28,22894,15886],{"class":15682},[28,22896,16397],{"class":15675},[28,22898,22899],{"class":16082},"name",[28,22901,15883],{"class":15675},[28,22903,17582],{"class":15682},[28,22905,17585],{"class":15675},[28,22907,15955],{"class":15682},[28,22909,15778],{"class":15675},[28,22911,22912],{"class":15660,"line":15798},[28,22913,22914],{"class":15841},"    \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n",[28,22916,22917,22920,22922,22924,22927,22929,22932,22934,22936,22938],{"class":15660,"line":15811},[28,22918,22919],{"class":15675},"    vmin, vmax ",[28,22921,845],{"class":15682},[28,22923,18370],{"class":15682},[28,22925,22926],{"class":15675},"(vec.",[28,22928,22201],{"class":16073},[28,22930,22931],{"class":15675},"()), ",[28,22933,15886],{"class":15682},[28,22935,22926],{"class":15675},[28,22937,22207],{"class":16073},[28,22939,22940],{"class":15675},"())\n",[28,22942,22943,22945,22948,22951,22954,22957,22959,22961],{"class":15660,"line":15820},[28,22944,16484],{"class":15671},[28,22946,22947],{"class":15675}," value ",[28,22949,22950],{"class":15682},"\u003C",[28,22952,22953],{"class":15675}," vmin ",[28,22955,22956],{"class":15671},"or",[28,22958,22947],{"class":15675},[28,22960,16503],{"class":15682},[28,22962,22963],{"class":15675}," vmax:\n",[28,22965,22966,22969,22971,22973,22976,22978,22980,22982,22984,22986,22988,22990,22993,22995,22998,23000,23002,23004,23006,23008,23011],{"class":15660,"line":15830},[28,22967,22968],{"class":15682},"        print",[28,22970,65],{"class":15675},[28,22972,68],{"class":15671},[28,22974,22975],{"class":15841},"\"[Note] ",[28,22977,11385],{"class":15801},[28,22979,22899],{"class":15675},[28,22981,11404],{"class":15801},[28,22983,845],{"class":15841},[28,22985,11385],{"class":15801},[28,22987,22890],{"class":15675},[28,22989,11404],{"class":15801},[28,22991,22992],{"class":15841}," is outside sweep; clamping to [",[28,22994,11385],{"class":15801},[28,22996,22997],{"class":15675},"vmin",[28,22999,11404],{"class":15801},[28,23001,16397],{"class":15841},[28,23003,11385],{"class":15801},[28,23005,22625],{"class":15675},[28,23007,11404],{"class":15801},[28,23009,23010],{"class":15841},"].\"",[28,23012,16052],{"class":15675},[28,23014,23015,23018,23020,23022,23024],{"class":15660,"line":15835},[28,23016,23017],{"class":15675},"        value ",[28,23019,845],{"class":15682},[28,23021,16298],{"class":15675},[28,23023,16570],{"class":16073},[28,23025,23026],{"class":15675},"(value, vmin, vmax)\n",[28,23028,23029,23031,23034,23036,23039,23041,23044,23047,23049],{"class":15660,"line":15853},[28,23030,16565],{"class":15671},[28,23032,23033],{"class":15682}," int",[28,23035,18670],{"class":15675},[28,23037,23038],{"class":16073},"argmin",[28,23040,18670],{"class":15675},[28,23042,23043],{"class":16073},"abs",[28,23045,23046],{"class":15675},"(vec ",[28,23048,15689],{"class":15682},[28,23050,23051],{"class":15675}," value)))\n",[28,23053,23054],{"class":15660,"line":15867},[28,23055,15769],{"emptyLinePlaceholder":15768},[28,23057,23058],{"class":15660,"line":15872},[28,23059,23060],{"class":15664},"# --- Slice at f = F_CROSS_GHZ ---\n",[28,23062,23063,23066,23068,23070,23073,23075,23077,23080],{"class":15660,"line":15878},[28,23064,23065],{"class":15675},"j ",[28,23067,845],{"class":15682},[28,23069,22879],{"class":16073},[28,23071,23072],{"class":15675},"(freq_ghz, ",[28,23074,15346],{"class":15801},[28,23076,16397],{"class":15675},[28,23078,23079],{"class":15841},"\"F_CROSS_GHZ\"",[28,23081,16052],{"class":15675},[28,23083,23084,23087,23089],{"class":15660,"line":15897},[28,23085,23086],{"class":15675},"Pe_vs_tau ",[28,23088,845],{"class":15682},[28,23090,23091],{"class":15675}," Pe[:, j]\n",[28,23093,23094],{"class":15660,"line":15914},[28,23095,15769],{"emptyLinePlaceholder":15768},[28,23097,23098,23100,23102,23104,23106,23108,23110,23112,23114,23116,23118,23121],{"class":15660,"line":15931},[28,23099,16874],{"class":15675},[28,23101,845],{"class":15682},[28,23103,16879],{"class":15675},[28,23105,16882],{"class":16073},[28,23107,65],{"class":15675},[28,23109,16887],{"class":16307},[28,23111,845],{"class":15682},[28,23113,65],{"class":15675},[28,23115,22491],{"class":15801},[28,23117,16397],{"class":15675},[28,23119,23120],{"class":15801},"3.5",[28,23122,16539],{"class":15675},[28,23124,23125,23127,23129,23132,23135,23137,23140],{"class":15660,"line":15948},[28,23126,16970],{"class":15675},[28,23128,19308],{"class":16073},[28,23130,23131],{"class":15675},"(dur_ns, Pe_vs_tau, ",[28,23133,23134],{"class":16307},"color",[28,23136,845],{"class":15682},[28,23138,23139],{"class":15841},"'blue'",[28,23141,16052],{"class":15675},[28,23143,23144,23146,23148,23150,23152,23154,23156,23158,23160,23162,23164,23166],{"class":15660,"line":15966},[28,23145,16970],{"class":15675},[28,23147,16973],{"class":16073},[28,23149,65],{"class":15675},[28,23151,16959],{"class":15671},[28,23153,22665],{"class":16962},[28,23155,22668],{"class":15682},[28,23157,22671],{"class":16962},[28,23159,65],{"class":15801},[28,23161,7191],{"class":16962},[28,23163,78],{"class":15801},[28,23165,22680],{"class":16962},[28,23167,16052],{"class":15675},[28,23169,23170,23172,23174,23176,23178,23180],{"class":15660,"line":15983},[28,23171,16970],{"class":15675},[28,23173,16987],{"class":16073},[28,23175,65],{"class":15675},[28,23177,16959],{"class":15671},[28,23179,16963],{"class":16962},[28,23181,16052],{"class":15675},[28,23183,23184],{"class":15660,"line":16000},[28,23185,23186],{"class":15664},"# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\n",[28,23188,23189,23191,23193,23195,23197,23200,23202,23204,23207,23209,23212,23214,23217,23219,23221,23224,23226,23228,23231],{"class":15660,"line":16016},[28,23190,16970],{"class":15675},[28,23192,17001],{"class":16073},[28,23194,65],{"class":15675},[28,23196,16959],{"class":15671},[28,23198,23199],{"class":16962},"'$P_e",[28,23201,65],{"class":15801},[28,23203,22668],{"class":15682},[28,23205,23206],{"class":16962},"au",[28,23208,78],{"class":15801},[28,23210,23211],{"class":16962},"$ at f = '",[28,23213,16359],{"class":15682},[28,23215,23216],{"class":15671}," f",[28,23218,22680],{"class":15841},[28,23220,11385],{"class":15801},[28,23222,23223],{"class":15675},"freq_ghz[j]",[28,23225,21644],{"class":15671},[28,23227,11404],{"class":15801},[28,23229,23230],{"class":15841}," GHz'",[28,23232,16052],{"class":15675},[28,23234,23235,23237,23240,23242,23244,23246,23248],{"class":15660,"line":16036},[28,23236,16970],{"class":15675},[28,23238,23239],{"class":16073},"set_ylim",[28,23241,65],{"class":15675},[28,23243,15271],{"class":15801},[28,23245,16397],{"class":15675},[28,23247,16580],{"class":15801},[28,23249,16052],{"class":15675},[28,23251,23252,23254,23256],{"class":15660,"line":16041},[28,23253,17023],{"class":15675},[28,23255,17026],{"class":16073},[28,23257,17029],{"class":15675},[28,23259,23260,23262,23264],{"class":15660,"line":16371},[28,23261,17034],{"class":15675},[28,23263,17037],{"class":16073},[28,23265,17029],{"class":15675},[28,23267,23268],{"class":15660,"line":16376},[28,23269,15769],{"emptyLinePlaceholder":15768},[28,23271,23272],{"class":15660,"line":16410},[28,23273,23274],{"class":15664},"# --- Slice at τ = T_CROSS_NS ---\n",[28,23276,23277,23280,23282,23284,23287,23289,23291,23294],{"class":15660,"line":16431},[28,23278,23279],{"class":15675},"i ",[28,23281,845],{"class":15682},[28,23283,22879],{"class":16073},[28,23285,23286],{"class":15675},"(dur_ns, ",[28,23288,15445],{"class":15801},[28,23290,16397],{"class":15675},[28,23292,23293],{"class":15841},"\"T_CROSS_NS\"",[28,23295,16052],{"class":15675},[28,23297,23298,23301,23303],{"class":15660,"line":16452},[28,23299,23300],{"class":15675},"Pe_vs_f ",[28,23302,845],{"class":15682},[28,23304,23305],{"class":15675}," Pe[i, :]\n",[28,23307,23308],{"class":15660,"line":16476},[28,23309,15769],{"emptyLinePlaceholder":15768},[28,23311,23312,23314,23316,23318,23320,23322,23324,23326,23328,23330,23332,23334],{"class":15660,"line":16481},[28,23313,16874],{"class":15675},[28,23315,845],{"class":15682},[28,23317,16879],{"class":15675},[28,23319,16882],{"class":16073},[28,23321,65],{"class":15675},[28,23323,16887],{"class":16307},[28,23325,845],{"class":15682},[28,23327,65],{"class":15675},[28,23329,22491],{"class":15801},[28,23331,16397],{"class":15675},[28,23333,23120],{"class":15801},[28,23335,16539],{"class":15675},[28,23337,23338,23340,23342,23345,23347,23349,23351],{"class":15660,"line":16510},[28,23339,16970],{"class":15675},[28,23341,19308],{"class":16073},[28,23343,23344],{"class":15675},"(freq_ghz, Pe_vs_f, ",[28,23346,23134],{"class":16307},[28,23348,845],{"class":15682},[28,23350,23139],{"class":15841},[28,23352,16052],{"class":15675},[28,23354,23355,23357,23359,23361,23363],{"class":15660,"line":16542},[28,23356,16970],{"class":15675},[28,23358,16973],{"class":16073},[28,23360,65],{"class":15675},[28,23362,22650],{"class":15841},[28,23364,16052],{"class":15675},[28,23366,23367,23369,23371,23373,23375,23377],{"class":15660,"line":16557},[28,23368,16970],{"class":15675},[28,23370,16987],{"class":16073},[28,23372,65],{"class":15675},[28,23374,16959],{"class":15671},[28,23376,16963],{"class":16962},[28,23378,16052],{"class":15675},[28,23380,23381,23383,23385,23387,23389,23391,23393,23395,23397,23400,23402,23405,23407,23409,23411,23413,23416,23418,23420,23423],{"class":15660,"line":16562},[28,23382,16970],{"class":15675},[28,23384,17001],{"class":16073},[28,23386,65],{"class":15675},[28,23388,16959],{"class":15671},[28,23390,23199],{"class":16962},[28,23392,65],{"class":15801},[28,23394,68],{"class":16962},[28,23396,78],{"class":15801},[28,23398,23399],{"class":16962},"$ at $",[28,23401,22668],{"class":15682},[28,23403,23404],{"class":16962},"au$ = '",[28,23406,16359],{"class":15682},[28,23408,23216],{"class":15671},[28,23410,22680],{"class":15841},[28,23412,11385],{"class":15801},[28,23414,23415],{"class":15675},"dur_ns[i]",[28,23417,21719],{"class":15671},[28,23419,11404],{"class":15801},[28,23421,23422],{"class":15841}," ns'",[28,23424,16052],{"class":15675},[28,23426,23427,23429,23431,23433,23435,23437,23439],{"class":15660,"line":17382},[28,23428,16970],{"class":15675},[28,23430,23239],{"class":16073},[28,23432,65],{"class":15675},[28,23434,15271],{"class":15801},[28,23436,16397],{"class":15675},[28,23438,16580],{"class":15801},[28,23440,16052],{"class":15675},[28,23442,23443,23445,23447],{"class":15660,"line":17392},[28,23444,17023],{"class":15675},[28,23446,17026],{"class":16073},[28,23448,17029],{"class":15675},[28,23450,23451,23453,23455],{"class":15660,"line":17402},[28,23452,17034],{"class":15675},[28,23454,17037],{"class":16073},[28,23456,17029],{"class":15675},[17041,23458],{"alt":23459,"src":23460},"Output 4 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-04.webp",[17041,23462],{"alt":23463,"src":23464},"Output 5 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-05.webp",[15650,23466,23468],{"className":15652,"code":23467,"language":15654,"meta":15655,"style":15655},"# @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",[8103,23469,23470,23475,23485,23495,23499,23504,23520,23536,23552,23568,23584,23600,23604,23609,23635,23655,23668,23690,23694,23703,23710,23717,23732,23747,23762,23777,23792,23807,23812,23817,23821,23825,23830,23835,23840,23845,23850,23855,23860,23865,23869,23873,23878,23883,23887,23904,23921,23941,23945,23950,23983,23993,24036,24048,24052,24057,24067,24082,24089,24126,24140,24144,24149,24184,24210,24214,24219,24237,24255,24261,24274,24278,24283,24301,24341,24355,24361,24384,24388,24405,24436,24440,24445,24464,24478,24484,24497,24501,24508,24512,24517,24528,24533,24543,24553,24563,24573,24583,24593,24597],{"__ignoreMap":15655},[28,23471,23472],{"class":15660,"line":15661},[28,23473,23474],{"class":15664},"# @title FFT of Rabi oscillations along tau (controls + computation)\n",[28,23476,23477,23479,23481,23483],{"class":15660,"line":15668},[28,23478,15672],{"class":15671},[28,23480,15730],{"class":15675},[28,23482,15733],{"class":15671},[28,23484,15736],{"class":15675},[28,23486,23487,23489,23491,23493],{"class":15660,"line":15679},[28,23488,15672],{"class":15671},[28,23490,15744],{"class":15675},[28,23492,15733],{"class":15671},[28,23494,15749],{"class":15675},[28,23496,23497],{"class":15660,"line":15725},[28,23498,15769],{"emptyLinePlaceholder":15768},[28,23500,23501],{"class":15660,"line":15739},[28,23502,23503],{"class":15664},"# -------- Control knobs (FFT & display) --------\n",[28,23505,23506,23508,23510,23512,23514,23517],{"class":15660,"line":15752},[28,23507,15562],{"class":15801},[28,23509,15883],{"class":15675},[28,23511,15955],{"class":15682},[28,23513,15805],{"class":15682},[28,23515,23516],{"class":15801}," 8",[28,23518,23519],{"class":15664},"                 # zero-padding factor (2, 4, 8, ...)\n",[28,23521,23522,23524,23526,23528,23530,23533],{"class":15660,"line":15765},[28,23523,15577],{"class":15801},[28,23525,15883],{"class":15675},[28,23527,17582],{"class":15682},[28,23529,15805],{"class":15682},[28,23531,23532],{"class":15841}," \"hann\"",[28,23534,23535],{"class":15664},"             # \"hann\", \"boxcar\"\n",[28,23537,23538,23540,23542,23544,23546,23549],{"class":15660,"line":15772},[28,23539,15592],{"class":15801},[28,23541,15883],{"class":15675},[28,23543,17588],{"class":15682},[28,23545,15805],{"class":15682},[28,23547,23548],{"class":15801}," True",[28,23550,23551],{"class":15664},"        # subtract mean along τ before FFT\n",[28,23553,23554,23556,23558,23560,23562,23565],{"class":15660,"line":15781},[28,23555,15607],{"class":15801},[28,23557,15883],{"class":15675},[28,23559,17582],{"class":15682},[28,23561,15805],{"class":15682},[28,23563,23564],{"class":15841}," \"magnitude\"",[28,23566,23567],{"class":15664},"      # \"magnitude\" | \"power\" | \"db\"\n",[28,23569,23570,23572,23574,23576,23578,23581],{"class":15660,"line":15798},[28,23571,15622],{"class":15801},[28,23573,15883],{"class":15675},[28,23575,17582],{"class":15682},[28,23577,15805],{"class":15682},[28,23579,23580],{"class":15841}," \"global\"",[28,23582,23583],{"class":15664},"        # \"global\" | \"per_f\" (column-wise)\n",[28,23585,23586,23588,23590,23592,23594,23597],{"class":15660,"line":15811},[28,23587,15637],{"class":15801},[28,23589,15883],{"class":15675},[28,23591,17582],{"class":15682},[28,23593,15805],{"class":15682},[28,23595,23596],{"class":15841}," \"MHz\"",[28,23598,23599],{"class":15664},"          # \"MHz\" or \"GHz\" for Fourier axis\n",[28,23601,23602],{"class":15660,"line":15820},[28,23603,15769],{"emptyLinePlaceholder":15768},[28,23605,23606],{"class":15660,"line":15830},[28,23607,23608],{"class":15664},"# -------- Helpers --------\n",[28,23610,23611,23613,23616,23618,23620,23622,23624,23626,23629,23631,23633],{"class":15660,"line":15835},[28,23612,16070],{"class":15671},[28,23614,23615],{"class":16073}," _make_window",[28,23617,65],{"class":15675},[28,23619,7106],{"class":16082},[28,23621,15883],{"class":15675},[28,23623,15955],{"class":15682},[28,23625,16397],{"class":15675},[28,23627,23628],{"class":16082},"kind",[28,23630,15883],{"class":15675},[28,23632,17582],{"class":15682},[28,23634,16139],{"class":15675},[28,23636,23637,23639,23642,23645,23648,23651,23653],{"class":15660,"line":15853},[28,23638,16484],{"class":15671},[28,23640,23641],{"class":15675}," kind.",[28,23643,23644],{"class":16073},"lower",[28,23646,23647],{"class":15675},"() ",[28,23649,23650],{"class":15682},"==",[28,23652,23532],{"class":15841},[28,23654,15778],{"class":15675},[28,23656,23657,23660,23662,23665],{"class":15660,"line":15867},[28,23658,23659],{"class":15671},"        return",[28,23661,16298],{"class":15675},[28,23663,23664],{"class":16073},"hanning",[28,23666,23667],{"class":15675},"(n)\n",[28,23669,23670,23672,23674,23677,23680,23682,23684,23687],{"class":15660,"line":15872},[28,23671,16565],{"class":15671},[28,23673,16298],{"class":15675},[28,23675,23676],{"class":16073},"ones",[28,23678,23679],{"class":15675},"(n, ",[28,23681,22267],{"class":16307},[28,23683,22270],{"class":15682},[28,23685,23686],{"class":15675},")  ",[28,23688,23689],{"class":15664},"# boxcar\n",[28,23691,23692],{"class":15660,"line":15878},[28,23693,15769],{"emptyLinePlaceholder":15768},[28,23695,23696,23698,23701],{"class":15660,"line":15897},[28,23697,16070],{"class":15671},[28,23699,23700],{"class":16073}," rfft_spectrogram",[28,23702,16077],{"class":15675},[28,23704,23705,23708],{"class":15660,"line":15914},[28,23706,23707],{"class":16082},"    Pe",[28,23709,16086],{"class":15675},[28,23711,23712,23715],{"class":15660,"line":15931},[28,23713,23714],{"class":16082},"    dur_ns",[28,23716,16086],{"class":15675},[28,23718,23719,23722,23724,23726,23728,23730],{"class":15660,"line":15948},[28,23720,23721],{"class":16082},"    pad",[28,23723,15883],{"class":15675},[28,23725,15955],{"class":15682},[28,23727,15805],{"class":15682},[28,23729,23516],{"class":15801},[28,23731,16105],{"class":15675},[28,23733,23734,23737,23739,23741,23743,23745],{"class":15660,"line":15966},[28,23735,23736],{"class":16082},"    window",[28,23738,15883],{"class":15675},[28,23740,17582],{"class":15682},[28,23742,15805],{"class":15682},[28,23744,23532],{"class":15841},[28,23746,16105],{"class":15675},[28,23748,23749,23752,23754,23756,23758,23760],{"class":15660,"line":15983},[28,23750,23751],{"class":16082},"    detrend_mean",[28,23753,15883],{"class":15675},[28,23755,17588],{"class":15682},[28,23757,15805],{"class":15682},[28,23759,23548],{"class":15801},[28,23761,16105],{"class":15675},[28,23763,23764,23767,23769,23771,23773,23775],{"class":15660,"line":16000},[28,23765,23766],{"class":16082},"    amp_mode",[28,23768,15883],{"class":15675},[28,23770,17582],{"class":15682},[28,23772,15805],{"class":15682},[28,23774,23564],{"class":15841},[28,23776,16105],{"class":15675},[28,23778,23779,23782,23784,23786,23788,23790],{"class":15660,"line":16016},[28,23780,23781],{"class":16082},"    norm_mode",[28,23783,15883],{"class":15675},[28,23785,17582],{"class":15682},[28,23787,15805],{"class":15682},[28,23789,23580],{"class":15841},[28,23791,16105],{"class":15675},[28,23793,23794,23797,23799,23801,23803,23805],{"class":15660,"line":16036},[28,23795,23796],{"class":16082},"    freq_units",[28,23798,15883],{"class":15675},[28,23800,17582],{"class":15682},[28,23802,15805],{"class":15682},[28,23804,23596],{"class":15841},[28,23806,16105],{"class":15675},[28,23808,23809],{"class":15660,"line":16041},[28,23810,23811],{"class":15675},") -> tuple[np.ndarray, np.ndarray]:\n",[28,23813,23814],{"class":15660,"line":16371},[28,23815,23816],{"class":15841},"    \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n",[28,23818,23819],{"class":15660,"line":16376},[28,23820,15769],{"emptyLinePlaceholder":15768},[28,23822,23823],{"class":15660,"line":16410},[28,23824,16153],{"class":15841},[28,23826,23827],{"class":15660,"line":16431},[28,23828,23829],{"class":15841},"        Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n",[28,23831,23832],{"class":15660,"line":16452},[28,23833,23834],{"class":15841},"        dur_ns: τ grid [ns], length N_tau, evenly spaced.\n",[28,23836,23837],{"class":15660,"line":16476},[28,23838,23839],{"class":15841},"        pad: zero-padding factor (≥1).\n",[28,23841,23842],{"class":15660,"line":16481},[28,23843,23844],{"class":15841},"        window: \"hann\" or \"boxcar\".\n",[28,23846,23847],{"class":15660,"line":16510},[28,23848,23849],{"class":15841},"        detrend_mean: subtract column-wise mean before FFT.\n",[28,23851,23852],{"class":15660,"line":16542},[28,23853,23854],{"class":15841},"        amp_mode: \"magnitude\", \"power\", or \"db\".\n",[28,23856,23857],{"class":15660,"line":16557},[28,23858,23859],{"class":15841},"        norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n",[28,23861,23862],{"class":15660,"line":16562},[28,23863,23864],{"class":15841},"        freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n",[28,23866,23867],{"class":15660,"line":17382},[28,23868,15769],{"emptyLinePlaceholder":15768},[28,23870,23871],{"class":15660,"line":17392},[28,23872,16187],{"class":15841},[28,23874,23875],{"class":15660,"line":17402},[28,23876,23877],{"class":15841},"        f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n",[28,23879,23880],{"class":15660,"line":17407},[28,23881,23882],{"class":15841},"        S:     2D nonnegative spectrogram (N_w, N_f).\n",[28,23884,23885],{"class":15660,"line":17418},[28,23886,16197],{"class":15841},[28,23888,23889,23892,23894,23896,23898,23900,23902],{"class":15660,"line":17429},[28,23890,23891],{"class":15675},"    Pe ",[28,23893,845],{"class":15682},[28,23895,16298],{"class":15675},[28,23897,18673],{"class":16073},[28,23899,16573],{"class":15675},[28,23901,15886],{"class":15682},[28,23903,16052],{"class":15675},[28,23905,23906,23909,23911,23913,23915,23917,23919],{"class":15660,"line":17440},[28,23907,23908],{"class":15675},"    dur_ns ",[28,23910,845],{"class":15682},[28,23912,16298],{"class":15675},[28,23914,18673],{"class":16073},[28,23916,23286],{"class":15675},[28,23918,15886],{"class":15682},[28,23920,16052],{"class":15675},[28,23922,23923,23926,23929,23931,23933,23935,23938],{"class":15660,"line":17451},[28,23924,23925],{"class":15671},"    assert",[28,23927,23928],{"class":15675}," Pe.shape[",[28,23930,652],{"class":15801},[28,23932,15845],{"class":15675},[28,23934,23650],{"class":15682},[28,23936,23937],{"class":15675}," dur_ns.size, ",[28,23939,23940],{"class":15841},"\"Pe must be shaped (N_tau, N_f).\"\n",[28,23942,23943],{"class":15660,"line":17462},[28,23944,15769],{"emptyLinePlaceholder":15768},[28,23946,23947],{"class":15660,"line":17467},[28,23948,23949],{"class":15664},"    # Uniform sampling assumption (true for np.linspace)\n",[28,23951,23952,23955,23957,23959,23962,23964,23966,23968,23971,23973,23976,23978,23980],{"class":15660,"line":17478},[28,23953,23954],{"class":15675},"    dt ",[28,23956,845],{"class":15682},[28,23958,18370],{"class":15682},[28,23960,23961],{"class":15675},"(dur_ns[",[28,23963,3200],{"class":15801},[28,23965,15845],{"class":15675},[28,23967,15689],{"class":15682},[28,23969,23970],{"class":15675}," dur_ns[",[28,23972,652],{"class":15801},[28,23974,23975],{"class":15675},"]) ",[28,23977,16226],{"class":15682},[28,23979,16536],{"class":15801},[28,23981,23982],{"class":15664},"               # [s]\n",[28,23984,23985,23988,23990],{"class":15660,"line":17489},[28,23986,23987],{"class":15675},"    n_tau ",[28,23989,845],{"class":15682},[28,23991,23992],{"class":15675}," dur_ns.size\n",[28,23994,23995,23998,24000,24002,24004,24006,24009,24011,24013,24016,24018,24021,24024,24026,24029,24031,24033],{"class":15660,"line":17500},[28,23996,23997],{"class":15675},"    nfft ",[28,23999,845],{"class":15682},[28,24001,23033],{"class":15682},[28,24003,65],{"class":15675},[28,24005,848],{"class":15801},[28,24007,24008],{"class":15682}," **",[28,24010,23033],{"class":15682},[28,24012,18670],{"class":15675},[28,24014,24015],{"class":16073},"ceil",[28,24017,18670],{"class":15675},[28,24019,24020],{"class":16073},"log2",[28,24022,24023],{"class":15675},"(n_tau))) ",[28,24025,16226],{"class":15682},[28,24027,24028],{"class":15682}," max",[28,24030,65],{"class":15675},[28,24032,3200],{"class":15801},[28,24034,24035],{"class":15675},", pad))\n",[28,24037,24038,24041,24043,24045],{"class":15660,"line":17511},[28,24039,24040],{"class":15675},"    w ",[28,24042,845],{"class":15682},[28,24044,23615],{"class":16073},[28,24046,24047],{"class":15675},"(n_tau, window)\n",[28,24049,24050],{"class":15660,"line":17522},[28,24051,15769],{"emptyLinePlaceholder":15768},[28,24053,24054],{"class":15660,"line":17527},[28,24055,24056],{"class":15664},"    # Prepare output\n",[28,24058,24059,24062,24064],{"class":15660,"line":17538},[28,24060,24061],{"class":15675},"    S_list ",[28,24063,845],{"class":15682},[28,24065,24066],{"class":15675}," []\n",[28,24068,24069,24072,24074,24077,24080],{"class":15660,"line":17557},[28,24070,24071],{"class":15675},"    X ",[28,24073,845],{"class":15682},[28,24075,24076],{"class":15675}," Pe.",[28,24078,24079],{"class":16073},"copy",[28,24081,17029],{"class":15675},[28,24083,24084,24086],{"class":15660,"line":17562},[28,24085,16484],{"class":15671},[28,24087,24088],{"class":15675}," detrend_mean:\n",[28,24090,24091,24094,24096,24099,24101,24104,24107,24109,24111,24113,24115,24117,24120,24122,24124],{"class":15660,"line":17567},[28,24092,24093],{"class":15675},"        X ",[28,24095,845],{"class":15682},[28,24097,24098],{"class":15675}," X ",[28,24100,15689],{"class":15682},[28,24102,24103],{"class":15675}," X.",[28,24105,24106],{"class":16073},"mean",[28,24108,65],{"class":15675},[28,24110,19934],{"class":16307},[28,24112,845],{"class":15682},[28,24114,652],{"class":15801},[28,24116,16397],{"class":15675},[28,24118,24119],{"class":16307},"keepdims",[28,24121,845],{"class":15682},[28,24123,15597],{"class":15801},[28,24125,16052],{"class":15675},[28,24127,24128,24130,24133,24136,24138],{"class":15660,"line":17593},[28,24129,24071],{"class":15675},[28,24131,24132],{"class":15682},"*=",[28,24134,24135],{"class":15675}," w[:, ",[28,24137,15321],{"class":15801},[28,24139,16849],{"class":15675},[28,24141,24142],{"class":15660,"line":17599},[28,24143,15769],{"emptyLinePlaceholder":15768},[28,24145,24146],{"class":15660,"line":17619},[28,24147,24148],{"class":15664},"    # Real FFT along τ (axis=0)\n",[28,24150,24151,24154,24156,24159,24162,24165,24167,24169,24172,24174,24176,24178,24181],{"class":15660,"line":17624},[28,24152,24153],{"class":15675},"    Xf ",[28,24155,845],{"class":15682},[28,24157,24158],{"class":15675}," np.fft.",[28,24160,24161],{"class":16073},"rfft",[28,24163,24164],{"class":15675},"(X, ",[28,24166,7106],{"class":16307},[28,24168,845],{"class":15682},[28,24170,24171],{"class":15675},"nfft, ",[28,24173,19934],{"class":16307},[28,24175,845],{"class":15682},[28,24177,652],{"class":15801},[28,24179,24180],{"class":15675},")                     ",[28,24182,24183],{"class":15664},"# shape (N_w, N_f)\n",[28,24185,24186,24189,24191,24193,24196,24199,24202,24204,24207],{"class":15660,"line":17629},[28,24187,24188],{"class":15675},"    f_fft_hz ",[28,24190,845],{"class":15682},[28,24192,24158],{"class":15675},[28,24194,24195],{"class":16073},"rfftfreq",[28,24197,24198],{"class":15675},"(nfft, ",[28,24200,24201],{"class":16307},"d",[28,24203,845],{"class":15682},[28,24205,24206],{"class":15675},"dt)                  ",[28,24208,24209],{"class":15664},"# [Hz]\n",[28,24211,24212],{"class":15660,"line":17639},[28,24213,15769],{"emptyLinePlaceholder":15768},[28,24215,24216],{"class":15660,"line":17656},[28,24217,24218],{"class":15664},"    # Amplitude selection\n",[28,24220,24221,24223,24226,24228,24230,24232,24235],{"class":15660,"line":17665},[28,24222,16484],{"class":15671},[28,24224,24225],{"class":15675}," amp_mode.",[28,24227,23644],{"class":16073},[28,24229,23647],{"class":15675},[28,24231,23650],{"class":15682},[28,24233,24234],{"class":15841}," \"power\"",[28,24236,15778],{"class":15675},[28,24238,24239,24242,24244,24246,24248,24251,24253],{"class":15660,"line":17671},[28,24240,24241],{"class":15675},"        S ",[28,24243,845],{"class":15682},[28,24245,16298],{"class":15675},[28,24247,23043],{"class":16073},[28,24249,24250],{"class":15675},"(Xf) ",[28,24252,16354],{"class":15682},[28,24254,16428],{"class":15801},[28,24256,24257,24259],{"class":15660,"line":17683},[28,24258,19102],{"class":15671},[28,24260,15778],{"class":15675},[28,24262,24263,24265,24267,24269,24271],{"class":15660,"line":17689},[28,24264,24241],{"class":15675},[28,24266,845],{"class":15682},[28,24268,16298],{"class":15675},[28,24270,23043],{"class":16073},[28,24272,24273],{"class":15675},"(Xf)\n",[28,24275,24276],{"class":15660,"line":17694},[28,24277,15769],{"emptyLinePlaceholder":15768},[28,24279,24280],{"class":15660,"line":17705},[28,24281,24282],{"class":15664},"    # Normalization\n",[28,24284,24285,24287,24290,24292,24294,24296,24299],{"class":15660,"line":17711},[28,24286,16484],{"class":15671},[28,24288,24289],{"class":15675}," norm_mode.",[28,24291,23644],{"class":16073},[28,24293,23647],{"class":15675},[28,24295,23650],{"class":15682},[28,24297,24298],{"class":15841}," \"per_f\"",[28,24300,15778],{"class":15675},[28,24302,24303,24306,24308,24310,24313,24316,24318,24320,24322,24324,24326,24328,24330,24332,24334,24336,24339],{"class":15660,"line":17731},[28,24304,24305],{"class":15675},"        col_max ",[28,24307,845],{"class":15682},[28,24309,16298],{"class":15675},[28,24311,24312],{"class":16073},"maximum",[28,24314,24315],{"class":15675},"(S.",[28,24317,22207],{"class":16073},[28,24319,65],{"class":15675},[28,24321,19934],{"class":16307},[28,24323,845],{"class":15682},[28,24325,652],{"class":15801},[28,24327,16397],{"class":15675},[28,24329,24119],{"class":16307},[28,24331,845],{"class":15682},[28,24333,15597],{"class":15801},[28,24335,22494],{"class":15675},[28,24337,24338],{"class":15801},"1e-15",[28,24340,16052],{"class":15675},[28,24342,24343,24345,24347,24350,24352],{"class":15660,"line":17744},[28,24344,24241],{"class":15675},[28,24346,845],{"class":15682},[28,24348,24349],{"class":15675}," S ",[28,24351,1129],{"class":15682},[28,24353,24354],{"class":15675}," col_max\n",[28,24356,24357,24359],{"class":15660,"line":17750},[28,24358,19102],{"class":15671},[28,24360,15778],{"class":15675},[28,24362,24363,24365,24367,24369,24371,24373,24375,24377,24380,24382],{"class":15660,"line":17755},[28,24364,24241],{"class":15675},[28,24366,845],{"class":15682},[28,24368,24349],{"class":15675},[28,24370,1129],{"class":15682},[28,24372,24028],{"class":15682},[28,24374,24315],{"class":15675},[28,24376,22207],{"class":16073},[28,24378,24379],{"class":15675},"(), ",[28,24381,24338],{"class":15801},[28,24383,16052],{"class":15675},[28,24385,24386],{"class":15660,"line":17765},[28,24387,15769],{"emptyLinePlaceholder":15768},[28,24389,24390,24392,24394,24396,24398,24400,24403],{"class":15660,"line":17770},[28,24391,16484],{"class":15671},[28,24393,24225],{"class":15675},[28,24395,23644],{"class":16073},[28,24397,23647],{"class":15675},[28,24399,23650],{"class":15682},[28,24401,24402],{"class":15841}," \"db\"",[28,24404,15778],{"class":15675},[28,24406,24407,24409,24411,24413,24415,24417,24420,24422,24424,24427,24430,24433],{"class":15660,"line":17775},[28,24408,24241],{"class":15675},[28,24410,845],{"class":15682},[28,24412,15908],{"class":15801},[28,24414,16210],{"class":15682},[28,24416,16298],{"class":15675},[28,24418,24419],{"class":16073},"log10",[28,24421,18670],{"class":15675},[28,24423,24312],{"class":16073},[28,24425,24426],{"class":15675},"(S, ",[28,24428,24429],{"class":15801},"1e-12",[28,24431,24432],{"class":15675},"))  ",[28,24434,24435],{"class":15664},"# dB, clipped floor\n",[28,24437,24438],{"class":15660,"line":17783},[28,24439,15769],{"emptyLinePlaceholder":15768},[28,24441,24442],{"class":15660,"line":17794},[28,24443,24444],{"class":15664},"    # Units\n",[28,24446,24447,24449,24452,24455,24457,24459,24462],{"class":15660,"line":17825},[28,24448,16484],{"class":15671},[28,24450,24451],{"class":15675}," freq_units.",[28,24453,24454],{"class":16073},"upper",[28,24456,23647],{"class":15675},[28,24458,23650],{"class":15682},[28,24460,24461],{"class":15841}," \"GHz\"",[28,24463,15778],{"class":15675},[28,24465,24466,24469,24471,24474,24476],{"class":15660,"line":17837},[28,24467,24468],{"class":15675},"        f_fft ",[28,24470,845],{"class":15682},[28,24472,24473],{"class":15675}," f_fft_hz ",[28,24475,16226],{"class":15682},[28,24477,16258],{"class":15801},[28,24479,24480,24482],{"class":15660,"line":17843},[28,24481,19102],{"class":15671},[28,24483,15778],{"class":15675},[28,24485,24486,24488,24490,24492,24494],{"class":15660,"line":17852},[28,24487,24468],{"class":15675},[28,24489,845],{"class":15682},[28,24491,24473],{"class":15675},[28,24493,16226],{"class":15682},[28,24495,24496],{"class":15801}," 1e-6\n",[28,24498,24499],{"class":15660,"line":17858},[28,24500,15769],{"emptyLinePlaceholder":15768},[28,24502,24503,24505],{"class":15660,"line":17864},[28,24504,16565],{"class":15671},[28,24506,24507],{"class":15675}," f_fft, S\n",[28,24509,24510],{"class":15660,"line":17872},[28,24511,15769],{"emptyLinePlaceholder":15768},[28,24513,24514],{"class":15660,"line":17880},[28,24515,24516],{"class":15664},"# Compute spectrogram with current grid\n",[28,24518,24519,24522,24524,24526],{"class":15660,"line":17888},[28,24520,24521],{"class":15675},"f_fft_axis, S_fft ",[28,24523,845],{"class":15682},[28,24525,23700],{"class":16073},[28,24527,16077],{"class":15675},[28,24529,24530],{"class":15660,"line":17896},[28,24531,24532],{"class":15675},"    Pe, dur_ns,\n",[28,24534,24535,24537,24539,24541],{"class":15660,"line":17904},[28,24536,23721],{"class":16307},[28,24538,845],{"class":15682},[28,24540,15562],{"class":15801},[28,24542,16105],{"class":15675},[28,24544,24545,24547,24549,24551],{"class":15660,"line":17912},[28,24546,23736],{"class":16307},[28,24548,845],{"class":15682},[28,24550,15577],{"class":15801},[28,24552,16105],{"class":15675},[28,24554,24555,24557,24559,24561],{"class":15660,"line":17921},[28,24556,23751],{"class":16307},[28,24558,845],{"class":15682},[28,24560,15592],{"class":15801},[28,24562,16105],{"class":15675},[28,24564,24565,24567,24569,24571],{"class":15660,"line":17927},[28,24566,23766],{"class":16307},[28,24568,845],{"class":15682},[28,24570,15607],{"class":15801},[28,24572,16105],{"class":15675},[28,24574,24575,24577,24579,24581],{"class":15660,"line":17938},[28,24576,23781],{"class":16307},[28,24578,845],{"class":15682},[28,24580,15622],{"class":15801},[28,24582,16105],{"class":15675},[28,24584,24585,24587,24589,24591],{"class":15660,"line":17943},[28,24586,23796],{"class":16307},[28,24588,845],{"class":15682},[28,24590,15637],{"class":15801},[28,24592,16105],{"class":15675},[28,24594,24595],{"class":15660,"line":17952},[28,24596,16052],{"class":15675},[28,24598,24599,24601,24603,24606,24609,24612,24615,24617,24620,24623,24625,24627,24629,24631,24633],{"class":15660,"line":17961},[28,24600,16044],{"class":15682},[28,24602,65],{"class":15675},[28,24604,24605],{"class":15841},"\"FFT grid:\"",[28,24607,24608],{"class":15675},", S_fft.shape, ",[28,24610,24611],{"class":15841},"\"| Fourier freq range:\"",[28,24613,24614],{"class":15675},", f_fft_axis[",[28,24616,652],{"class":15801},[28,24618,24619],{"class":15675},"], ",[28,24621,24622],{"class":15841},"\"→\"",[28,24624,24614],{"class":15675},[28,24626,15689],{"class":15682},[28,24628,3200],{"class":15801},[28,24630,24619],{"class":15675},[28,24632,15637],{"class":15801},[28,24634,16052],{"class":15675},[15650,24636,24639],{"className":24637,"code":24638,"language":16058,"meta":15655},[16056],"FFT grid: (2049, 401) | Fourier freq range: 0.0 → 999.9999999999999 MHz\n",[8103,24640,24638],{"__ignoreMap":15655},[15650,24642,24644],{"className":15652,"code":24643,"language":15654,"meta":15655,"style":15655},"# @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",[8103,24645,24646,24651,24657,24670,24684,24712,24732,24764,24787,24806,24820,24826],{"__ignoreMap":15655},[28,24647,24648],{"class":15660,"line":15661},[28,24649,24650],{"class":15664},"# @title Theoretical Rabi ridge\n",[28,24652,24653,24655],{"class":15660,"line":15668},[28,24654,15672],{"class":15671},[28,24656,15709],{"class":15675},[28,24658,24659,24662,24664,24666,24668],{"class":15660,"line":15679},[28,24660,24661],{"class":15675},"two_pi ",[28,24663,845],{"class":15682},[28,24665,16207],{"class":15801},[28,24667,16210],{"class":15682},[28,24669,16213],{"class":15675},[28,24671,24672,24675,24677,24680,24682],{"class":15660,"line":15725},[28,24673,24674],{"class":15675},"f0 ",[28,24676,845],{"class":15682},[28,24678,24679],{"class":15801}," F0_GHZ",[28,24681,16210],{"class":15682},[28,24683,16229],{"class":15801},[28,24685,24686,24689,24691,24693,24695,24698,24700,24703,24706,24709],{"class":15660,"line":15739},[28,24687,24688],{"class":15675},"delta_hz ",[28,24690,845],{"class":15682},[28,24692,16268],{"class":15675},[28,24694,16226],{"class":15682},[28,24696,24697],{"class":15675}," (freq_ghz",[28,24699,16226],{"class":15682},[28,24701,24702],{"class":15801},"1e9",[28,24704,24705],{"class":15682}," -",[28,24707,24708],{"class":15675}," f0)   ",[28,24710,24711],{"class":15664},"# rad\u002Fs detuning\n",[28,24713,24714,24717,24719,24721,24723,24726,24728,24730],{"class":15660,"line":15752},[28,24715,24716],{"class":15675},"omega_onres ",[28,24718,845],{"class":15682},[28,24720,16268],{"class":15675},[28,24722,16226],{"class":15682},[28,24724,24725],{"class":15801}," OMEGA_ONRESONANCE_MHZ",[28,24727,16210],{"class":15682},[28,24729,16278],{"class":15801},[28,24731,16281],{"class":15664},[28,24733,24734,24737,24739,24741,24743,24746,24748,24750,24752,24755,24757,24759,24762],{"class":15660,"line":15765},[28,24735,24736],{"class":15675},"Omega_R ",[28,24738,845],{"class":15682},[28,24740,16298],{"class":15675},[28,24742,5750],{"class":16073},[28,24744,24745],{"class":15675},"(omega_onres",[28,24747,16354],{"class":15682},[28,24749,848],{"class":15801},[28,24751,16359],{"class":15682},[28,24753,24754],{"class":15675}," delta_hz",[28,24756,16354],{"class":15682},[28,24758,848],{"class":15801},[28,24760,24761],{"class":15675},")     ",[28,24763,16337],{"class":15664},[28,24765,24766,24769,24771,24773,24775,24777,24779,24781,24784],{"class":15660,"line":15772},[28,24767,24768],{"class":15675},"ridge_hz ",[28,24770,845],{"class":15682},[28,24772,16444],{"class":15675},[28,24774,1129],{"class":15682},[28,24776,17308],{"class":15675},[28,24778,18292],{"class":15801},[28,24780,16210],{"class":15682},[28,24782,24783],{"class":15675}," math.pi)                ",[28,24785,24786],{"class":15664},"# Hz\n",[28,24788,24789,24791,24794,24796,24798,24800,24802,24804],{"class":15660,"line":15781},[28,24790,20133],{"class":15671},[28,24792,24793],{"class":15801}," FREQ_UNITS",[28,24795,2106],{"class":15675},[28,24797,24454],{"class":16073},[28,24799,23647],{"class":15675},[28,24801,23650],{"class":15682},[28,24803,24461],{"class":15841},[28,24805,15778],{"class":15675},[28,24807,24808,24811,24813,24816,24818],{"class":15660,"line":15798},[28,24809,24810],{"class":15675},"    ridge_axis ",[28,24812,845],{"class":15682},[28,24814,24815],{"class":15675}," ridge_hz ",[28,24817,16226],{"class":15682},[28,24819,16258],{"class":15801},[28,24821,24822,24824],{"class":15660,"line":15811},[28,24823,20171],{"class":15671},[28,24825,15778],{"class":15675},[28,24827,24828,24830,24832,24834,24836,24839],{"class":15660,"line":15820},[28,24829,24810],{"class":15675},[28,24831,845],{"class":15682},[28,24833,24815],{"class":15675},[28,24835,16226],{"class":15682},[28,24837,24838],{"class":15801}," 1e-6",[28,24840,24841],{"class":15664},"  # MHz\n",[15650,24843,24845],{"className":15652,"code":24844,"language":15654,"meta":15655,"style":15655},"# @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",[8103,24846,24847,24852,24880,24913,24933,24977,24981,25027,25040,25061,25099,25117,25125],{"__ignoreMap":15655},[28,24848,24849],{"class":15660,"line":15661},[28,24850,24851],{"class":15664},"# @title 2D heatmap\n",[28,24853,24854,24856,24858,24860,24862,24864,24866,24868,24870,24873,24875,24878],{"class":15660,"line":15668},[28,24855,16874],{"class":15675},[28,24857,845],{"class":15682},[28,24859,16879],{"class":15675},[28,24861,16882],{"class":16073},[28,24863,65],{"class":15675},[28,24865,16887],{"class":16307},[28,24867,845],{"class":15682},[28,24869,65],{"class":15675},[28,24871,24872],{"class":15801},"7.5",[28,24874,16397],{"class":15675},[28,24876,24877],{"class":15801},"5.5",[28,24879,16539],{"class":15675},[28,24881,24882,24885,24887,24889,24891,24894,24896,24898,24901,24903,24906,24908,24911],{"class":15660,"line":15679},[28,24883,24884],{"class":15675},"im ",[28,24886,845],{"class":15682},[28,24888,16911],{"class":15675},[28,24890,16914],{"class":16073},[28,24892,24893],{"class":15675},"(freq_ghz, f_fft_axis, S_fft, ",[28,24895,16920],{"class":16307},[28,24897,845],{"class":15682},[28,24899,24900],{"class":15841},"\"auto\"",[28,24902,16397],{"class":15675},[28,24904,24905],{"class":16307},"cmap",[28,24907,845],{"class":15682},[28,24909,24910],{"class":15841},"\"inferno\"",[28,24912,16052],{"class":15675},[28,24914,24915,24917,24919,24921,24923,24926,24928,24930],{"class":15660,"line":15725},[28,24916,16932],{"class":15675},[28,24918,845],{"class":15682},[28,24920,16937],{"class":15675},[28,24922,16940],{"class":16073},[28,24924,24925],{"class":15675},"(im, ",[28,24927,16946],{"class":16307},[28,24929,845],{"class":15682},[28,24931,24932],{"class":15675},"ax)\n",[28,24934,24935,24937,24939,24941,24944,24946,24948,24951,24954,24957,24959,24961,24964,24966,24969,24972,24975],{"class":15660,"line":15739},[28,24936,22763],{"class":15675},[28,24938,22766],{"class":16073},[28,24940,65],{"class":15675},[28,24942,24943],{"class":15841},"\"FFT amplitude\"",[28,24945,16359],{"class":15682},[28,24947,17308],{"class":15675},[28,24949,24950],{"class":15841},"\" (dB)\"",[28,24952,24953],{"class":15671}," if",[28,24955,24956],{"class":15801}," AMP_MODE",[28,24958,2106],{"class":15675},[28,24960,23644],{"class":16073},[28,24962,24963],{"class":15675},"()",[28,24965,23650],{"class":15682},[28,24967,24968],{"class":15841},"\"db\"",[28,24970,24971],{"class":15671}," else",[28,24973,24974],{"class":15841}," \" (arb.)\"",[28,24976,16539],{"class":15675},[28,24978,24979],{"class":15660,"line":15752},[28,24980,15769],{"emptyLinePlaceholder":15768},[28,24982,24983,24985,24987,24990,24993,24995,24998,25000,25003,25005,25007,25009,25011,25013,25016,25019,25022,25025],{"class":15660,"line":15765},[28,24984,16970],{"class":15675},[28,24986,19308],{"class":16073},[28,24988,24989],{"class":15675},"(freq_ghz, ridge_axis, ",[28,24991,24992],{"class":15841},"\"--\"",[28,24994,16397],{"class":15675},[28,24996,24997],{"class":16307},"lw",[28,24999,845],{"class":15682},[28,25001,25002],{"class":15801},"1.5",[28,25004,16397],{"class":15675},[28,25006,16954],{"class":16307},[28,25008,845],{"class":15682},[28,25010,16959],{"class":15671},[28,25012,20444],{"class":16962},[28,25014,25015],{"class":15682},"\\O",[28,25017,25018],{"class":16962},"mega_R\u002F2",[28,25020,25021],{"class":15682},"\\p",[28,25023,25024],{"class":16962},"i$\"",[28,25026,16052],{"class":15675},[28,25028,25029,25031,25033,25035,25038],{"class":15660,"line":15772},[28,25030,16970],{"class":15675},[28,25032,16973],{"class":16073},[28,25034,65],{"class":15675},[28,25036,25037],{"class":15841},"\"Drive frequency f (GHz)\"",[28,25039,16052],{"class":15675},[28,25041,25042,25044,25046,25048,25050,25053,25056,25059],{"class":15660,"line":15781},[28,25043,16970],{"class":15675},[28,25045,16987],{"class":16073},[28,25047,65],{"class":15675},[28,25049,68],{"class":15671},[28,25051,25052],{"class":15841},"\"Fourier frequency (",[28,25054,25055],{"class":15801},"{FREQ_UNITS}",[28,25057,25058],{"class":15841},")\"",[28,25060,16052],{"class":15675},[28,25062,25063,25065,25067,25069,25072,25074,25076,25079,25082,25084,25087,25089,25092,25094,25097],{"class":15660,"line":15798},[28,25064,16970],{"class":15675},[28,25066,17001],{"class":16073},[28,25068,65],{"class":15675},[28,25070,25071],{"class":15841},"\"Fourier transform along pulse length: $|",[28,25073,17009],{"class":15682},[28,25075,11469],{"class":15841},[28,25077,25078],{"class":15801},"{F}",[28,25080,25081],{"class":15841},"_",[28,25083,17009],{"class":15682},[28,25085,25086],{"class":15841},"tau",[28,25088,17009],{"class":15682},[28,25090,25091],{"class":15841},"{P_e",[28,25093,17009],{"class":15682},[28,25095,25096],{"class":15841},"}|$\"",[28,25098,16052],{"class":15675},[28,25100,25101,25103,25105,25107,25110,25112,25115],{"class":15660,"line":15811},[28,25102,16970],{"class":15675},[28,25104,21483],{"class":16073},[28,25106,65],{"class":15675},[28,25108,25109],{"class":16307},"loc",[28,25111,845],{"class":15682},[28,25113,25114],{"class":15841},"\"upper right\"",[28,25116,16052],{"class":15675},[28,25118,25119,25121,25123],{"class":15660,"line":15820},[28,25120,17023],{"class":15675},[28,25122,17026],{"class":16073},[28,25124,17029],{"class":15675},[28,25126,25127,25129,25131],{"class":15660,"line":15830},[28,25128,17034],{"class":15675},[28,25130,17037],{"class":16073},[28,25132,17029],{"class":15675},[17041,25134],{"alt":25135,"src":25136},"Output 6 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-06.webp",[15650,25138,25140],{"className":15652,"code":25139,"language":15654,"meta":15655,"style":15655},"# --- 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",[8103,25141,25142,25147,25157,25167,25177,25181,25186,25203,25220,25237,25254,25258,25263,25280,25296,25300,25305,25318,25322,25327,25366,25387,25391,25411,25424,25438,25442,25447,25452,25488,25517,25521,25526,25560,25570,25580,25584,25588,25614,25618,25623,25657,25682,25694,25704,25722,25740,25750,25766,25770,25774,25787,25806,25842,25855,25894,25898,25903,25931,25935,25943],{"__ignoreMap":15655},[28,25143,25144],{"class":15660,"line":15661},[28,25145,25146],{"class":15664},"# --- Adaptive high-res 3D surface ---\n",[28,25148,25149,25151,25153,25155],{"class":15660,"line":15668},[28,25150,15714],{"class":15671},[28,25152,22081],{"class":15675},[28,25154,15672],{"class":15671},[28,25156,22086],{"class":15675},[28,25158,25159,25161,25163,25165],{"class":15660,"line":15679},[28,25160,15672],{"class":15671},[28,25162,15730],{"class":15675},[28,25164,15733],{"class":15671},[28,25166,15736],{"class":15675},[28,25168,25169,25171,25173,25175],{"class":15660,"line":15725},[28,25170,15672],{"class":15671},[28,25172,15744],{"class":15675},[28,25174,15733],{"class":15671},[28,25176,15749],{"class":15675},[28,25178,25179],{"class":15660,"line":15739},[28,25180,15769],{"emptyLinePlaceholder":15768},[28,25182,25183],{"class":15660,"line":15752},[28,25184,25185],{"class":15664},"# ------ Control knobs ------\n",[28,25187,25188,25191,25193,25195,25197,25200],{"class":15660,"line":15765},[28,25189,25190],{"class":15801},"UPSAMPLE_F_FREQ",[28,25192,15883],{"class":15675},[28,25194,15955],{"class":15682},[28,25196,15805],{"class":15682},[28,25198,25199],{"class":15801}," 3",[28,25201,25202],{"class":15664},"       # try 2–3; upsample the drive-frequency axis only\n",[28,25204,25205,25208,25210,25212,25214,25217],{"class":15660,"line":15772},[28,25206,25207],{"class":15801},"MAX_FACES",[28,25209,15883],{"class":15675},[28,25211,15955],{"class":15682},[28,25213,15805],{"class":15682},[28,25215,25216],{"class":15801}," 300_000",[28,25218,25219],{"class":15664},"       # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\n",[28,25221,25222,25225,25227,25229,25231,25234],{"class":15660,"line":15781},[28,25223,25224],{"class":15801},"MIN_ROWS",[28,25226,15883],{"class":15675},[28,25228,15955],{"class":15682},[28,25230,15805],{"class":15682},[28,25232,25233],{"class":15801}," 128",[28,25235,25236],{"class":15664},"            # keep at least this many Fourier-frequency rows\n",[28,25238,25239,25242,25244,25246,25248,25251],{"class":15660,"line":15798},[28,25240,25241],{"class":15801},"MAX_ROWS",[28,25243,15883],{"class":15675},[28,25245,15955],{"class":15682},[28,25247,15805],{"class":15682},[28,25249,25250],{"class":15801}," 1024",[28,25252,25253],{"class":15664},"           # hard cap for rows to keep plotting fast\n",[28,25255,25256],{"class":15660,"line":15811},[28,25257,15769],{"emptyLinePlaceholder":15768},[28,25259,25260],{"class":15660,"line":15820},[28,25261,25262],{"class":15664},"# Preserve global color limits so the legend stays consistent\n",[28,25264,25265,25268,25270,25272,25274,25277],{"class":15660,"line":15830},[28,25266,25267],{"class":15675},"vmin ",[28,25269,845],{"class":15682},[28,25271,18370],{"class":15682},[28,25273,18670],{"class":15675},[28,25275,25276],{"class":16073},"nanmin",[28,25278,25279],{"class":15675},"(S_fft))\n",[28,25281,25282,25285,25287,25289,25291,25294],{"class":15660,"line":15835},[28,25283,25284],{"class":15675},"vmax ",[28,25286,845],{"class":15682},[28,25288,18370],{"class":15682},[28,25290,18670],{"class":15675},[28,25292,25293],{"class":16073},"nanmax",[28,25295,25279],{"class":15675},[28,25297,25298],{"class":15660,"line":15853},[28,25299,15769],{"emptyLinePlaceholder":15768},[28,25301,25302],{"class":15660,"line":15867},[28,25303,25304],{"class":15664},"# Shapes\n",[28,25306,25307,25310,25312,25315],{"class":15660,"line":15872},[28,25308,25309],{"class":15675},"n_rows, n_cols ",[28,25311,845],{"class":15682},[28,25313,25314],{"class":15675}," S_fft.shape        ",[28,25316,25317],{"class":15664},"# rows=Fourier freq (ν), cols=drive freq (f)\n",[28,25319,25320],{"class":15660,"line":15878},[28,25321,15769],{"emptyLinePlaceholder":15768},[28,25323,25324],{"class":15660,"line":15897},[28,25325,25326],{"class":15664},"# 1) Upsample along the typically smaller axis (drive frequency) only\n",[28,25328,25329,25332,25334,25337,25339,25341,25343,25346,25348,25350,25352,25354,25356,25358,25361,25363],{"class":15660,"line":15914},[28,25330,25331],{"class":15675},"n_cols_up ",[28,25333,845],{"class":15682},[28,25335,25336],{"class":15682}," min",[28,25338,65],{"class":15675},[28,25340,25190],{"class":15801},[28,25342,16210],{"class":15682},[28,25344,25345],{"class":15675}," (n_cols ",[28,25347,15689],{"class":15682},[28,25349,22230],{"class":15801},[28,25351,22233],{"class":15675},[28,25353,2606],{"class":15682},[28,25355,22230],{"class":15801},[28,25357,16397],{"class":15675},[28,25359,25360],{"class":15801},"1200",[28,25362,23686],{"class":15675},[28,25364,25365],{"class":15664},"# sensible upper bound\n",[28,25367,25368,25370,25372,25374,25376,25378,25380,25382,25384],{"class":15660,"line":15931},[28,25369,22189],{"class":15675},[28,25371,845],{"class":15682},[28,25373,16298],{"class":15675},[28,25375,16599],{"class":16073},[28,25377,22198],{"class":15675},[28,25379,22201],{"class":16073},[28,25381,22204],{"class":15675},[28,25383,22207],{"class":16073},[28,25385,25386],{"class":15675},"(), n_cols_up)\n",[28,25388,25389],{"class":15660,"line":15948},[28,25390,15769],{"emptyLinePlaceholder":15768},[28,25392,25393,25396,25398,25400,25402,25405,25407,25409],{"class":15660,"line":15966},[28,25394,25395],{"class":15675},"S_up_f ",[28,25397,845],{"class":15682},[28,25399,16298],{"class":15675},[28,25401,22251],{"class":16073},[28,25403,25404],{"class":15675},"((n_rows, n_cols_up), ",[28,25406,22267],{"class":16307},[28,25408,22270],{"class":15682},[28,25410,16052],{"class":15675},[28,25412,25413,25415,25417,25419,25421],{"class":15660,"line":15983},[28,25414,22277],{"class":15671},[28,25416,22280],{"class":15675},[28,25418,17720],{"class":15671},[28,25420,22285],{"class":15682},[28,25422,25423],{"class":15675},"(n_rows):\n",[28,25425,25426,25429,25431,25433,25435],{"class":15660,"line":16000},[28,25427,25428],{"class":15675},"    S_up_f[i, :] ",[28,25430,845],{"class":15682},[28,25432,16298],{"class":15675},[28,25434,22304],{"class":16073},[28,25436,25437],{"class":15675},"(freq_hi, freq_ghz, S_fft[i, :])\n",[28,25439,25440],{"class":15660,"line":16016},[28,25441,15769],{"emptyLinePlaceholder":15768},[28,25443,25444],{"class":15660,"line":16036},[28,25445,25446],{"class":15664},"# 2) Determine how many Fourier rows we can afford for MAX_FACES\n",[28,25448,25449],{"class":15660,"line":16041},[28,25450,25451],{"class":15664},"# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\n",[28,25453,25454,25457,25459,25461,25463,25465,25467,25469,25472,25474,25476,25478,25480,25483,25485],{"class":15660,"line":16371},[28,25455,25456],{"class":15675},"n_rows_cap ",[28,25458,845],{"class":15682},[28,25460,23033],{"class":15682},[28,25462,65],{"class":15675},[28,25464,25207],{"class":15801},[28,25466,16528],{"class":15682},[28,25468,24028],{"class":15682},[28,25470,25471],{"class":15675},"(n_cols_up ",[28,25473,15689],{"class":15682},[28,25475,22230],{"class":15801},[28,25477,16397],{"class":15675},[28,25479,3200],{"class":15801},[28,25481,25482],{"class":15675},")) ",[28,25484,2606],{"class":15682},[28,25486,25487],{"class":15801}," 1\n",[28,25489,25490,25493,25495,25497,25499,25501,25504,25506,25508,25510,25512,25514],{"class":15660,"line":16376},[28,25491,25492],{"class":15675},"n_rows_plot ",[28,25494,845],{"class":15682},[28,25496,23033],{"class":15682},[28,25498,18670],{"class":15675},[28,25500,16570],{"class":16073},[28,25502,25503],{"class":15675},"(n_rows_cap, ",[28,25505,25224],{"class":15801},[28,25507,16397],{"class":15675},[28,25509,22201],{"class":15682},[28,25511,65],{"class":15675},[28,25513,25241],{"class":15801},[28,25515,25516],{"class":15675},", n_rows)))\n",[28,25518,25519],{"class":15660,"line":16410},[28,25520,15769],{"emptyLinePlaceholder":15768},[28,25522,25523],{"class":15660,"line":16431},[28,25524,25525],{"class":15664},"# Down-sample ν-axis evenly to n_rows_plot\n",[28,25527,25528,25531,25533,25535,25537,25539,25541,25544,25546,25548,25551,25554,25556,25558],{"class":15660,"line":16452},[28,25529,25530],{"class":15675},"row_idx ",[28,25532,845],{"class":15682},[28,25534,16298],{"class":15675},[28,25536,16599],{"class":16073},[28,25538,65],{"class":15675},[28,25540,652],{"class":15801},[28,25542,25543],{"class":15675},", n_rows ",[28,25545,15689],{"class":15682},[28,25547,22230],{"class":15801},[28,25549,25550],{"class":15675},", n_rows_plot).",[28,25552,25553],{"class":16073},"astype",[28,25555,65],{"class":15675},[28,25557,15955],{"class":15682},[28,25559,16052],{"class":15675},[28,25561,25562,25565,25567],{"class":15660,"line":16476},[28,25563,25564],{"class":15675},"f_fft_plot ",[28,25566,845],{"class":15682},[28,25568,25569],{"class":15675}," f_fft_axis[row_idx]\n",[28,25571,25572,25575,25577],{"class":15660,"line":16481},[28,25573,25574],{"class":15675},"S_final ",[28,25576,845],{"class":15682},[28,25578,25579],{"class":15675}," S_up_f[row_idx, :]\n",[28,25581,25582],{"class":15660,"line":16510},[28,25583,15769],{"emptyLinePlaceholder":15768},[28,25585,25586],{"class":15660,"line":16542},[28,25587,22434],{"class":15664},[28,25589,25590,25593,25596,25598,25600,25602,25605,25607,25609,25612],{"class":15660,"line":16557},[28,25591,25592],{"class":15675},"F2_hi, ",[28,25594,25595],{"class":15801},"FFOUR_hi",[28,25597,15805],{"class":15682},[28,25599,16298],{"class":15675},[28,25601,16301],{"class":16073},[28,25603,25604],{"class":15675},"(freq_hi, f_fft_plot, ",[28,25606,16308],{"class":16307},[28,25608,845],{"class":15682},[28,25610,25611],{"class":15841},"\"xy\"",[28,25613,16052],{"class":15675},[28,25615,25616],{"class":15660,"line":16562},[28,25617,15769],{"emptyLinePlaceholder":15768},[28,25619,25620],{"class":15660,"line":17382},[28,25621,25622],{"class":15664},"# 3) Plot (keeps 'inferno')\n",[28,25624,25625,25627,25629,25631,25633,25635,25637,25639,25641,25643,25645,25647,25649,25651,25653,25655],{"class":15660,"line":17392},[28,25626,22470],{"class":15675},[28,25628,845],{"class":15682},[28,25630,16879],{"class":15675},[28,25632,20380],{"class":16073},[28,25634,65],{"class":15675},[28,25636,16887],{"class":16307},[28,25638,845],{"class":15682},[28,25640,65],{"class":15675},[28,25642,15567],{"class":15801},[28,25644,16397],{"class":15675},[28,25646,22491],{"class":15801},[28,25648,22494],{"class":15675},[28,25650,22497],{"class":16307},[28,25652,845],{"class":15682},[28,25654,22502],{"class":15801},[28,25656,16052],{"class":15675},[28,25658,25659,25661,25663,25665,25667,25669,25671,25673,25675,25677,25680],{"class":15660,"line":17402},[28,25660,22509],{"class":15675},[28,25662,845],{"class":15682},[28,25664,16937],{"class":15675},[28,25666,22516],{"class":16073},[28,25668,65],{"class":15675},[28,25670,22521],{"class":15801},[28,25672,16397],{"class":15675},[28,25674,22526],{"class":16307},[28,25676,845],{"class":15682},[28,25678,25679],{"class":15841},"\"3d\"",[28,25681,16052],{"class":15675},[28,25683,25684,25686,25688,25690,25692],{"class":15660,"line":17407},[28,25685,22542],{"class":15675},[28,25687,845],{"class":15682},[28,25689,16911],{"class":15675},[28,25691,22549],{"class":16073},[28,25693,16077],{"class":15675},[28,25695,25696,25699,25701],{"class":15660,"line":17418},[28,25697,25698],{"class":15675},"    F2_hi, ",[28,25700,25595],{"class":15801},[28,25702,25703],{"class":15675},", S_final,\n",[28,25705,25706,25708,25710,25712,25714,25716,25718,25720],{"class":15660,"line":17429},[28,25707,22561],{"class":16307},[28,25709,845],{"class":15682},[28,25711,3200],{"class":15801},[28,25713,16397],{"class":15675},[28,25715,22570],{"class":16307},[28,25717,845],{"class":15682},[28,25719,3200],{"class":15801},[28,25721,16105],{"class":15675},[28,25723,25724,25726,25728,25730,25732,25734,25736,25738],{"class":15660,"line":17440},[28,25725,22585],{"class":16307},[28,25727,845],{"class":15682},[28,25729,652],{"class":15801},[28,25731,16397],{"class":15675},[28,25733,22594],{"class":16307},[28,25735,845],{"class":15682},[28,25737,15597],{"class":15801},[28,25739,16105],{"class":15675},[28,25741,25742,25744,25746,25748],{"class":15660,"line":17451},[28,25743,22605],{"class":16307},[28,25745,845],{"class":15682},[28,25747,24910],{"class":15841},[28,25749,16105],{"class":15675},[28,25751,25752,25754,25756,25759,25761,25763],{"class":15660,"line":17462},[28,25753,22616],{"class":16307},[28,25755,845],{"class":15682},[28,25757,25758],{"class":15675},"vmin, ",[28,25760,22625],{"class":16307},[28,25762,845],{"class":15682},[28,25764,25765],{"class":15675},"vmax\n",[28,25767,25768],{"class":15660,"line":17467},[28,25769,16052],{"class":15675},[28,25771,25772],{"class":15660,"line":17478},[28,25773,15769],{"emptyLinePlaceholder":15768},[28,25775,25776,25778,25780,25782,25785],{"class":15660,"line":17489},[28,25777,16970],{"class":15675},[28,25779,16973],{"class":16073},[28,25781,65],{"class":15675},[28,25783,25784],{"class":15841},"\"f (GHz)\"",[28,25786,16052],{"class":15675},[28,25788,25789,25791,25793,25795,25797,25800,25802,25804],{"class":15660,"line":17500},[28,25790,16970],{"class":15675},[28,25792,16987],{"class":16073},[28,25794,65],{"class":15675},[28,25796,68],{"class":15671},[28,25798,25799],{"class":15841},"\"Fourier freq (",[28,25801,25055],{"class":15801},[28,25803,25058],{"class":15841},[28,25805,16052],{"class":15675},[28,25807,25808,25810,25812,25814,25816,25818,25820,25822,25824,25826,25828,25830,25832,25834,25836,25838,25840],{"class":15660,"line":17511},[28,25809,16970],{"class":15675},[28,25811,22689],{"class":16073},[28,25813,65],{"class":15675},[28,25815,24943],{"class":15841},[28,25817,16359],{"class":15682},[28,25819,17308],{"class":15675},[28,25821,24950],{"class":15841},[28,25823,24953],{"class":15671},[28,25825,24956],{"class":15801},[28,25827,2106],{"class":15675},[28,25829,23644],{"class":16073},[28,25831,23647],{"class":15675},[28,25833,23650],{"class":15682},[28,25835,24402],{"class":15841},[28,25837,24971],{"class":15671},[28,25839,24974],{"class":15841},[28,25841,16539],{"class":15675},[28,25843,25844,25846,25848,25850,25853],{"class":15660,"line":17522},[28,25845,16970],{"class":15675},[28,25847,17001],{"class":16073},[28,25849,65],{"class":15675},[28,25851,25852],{"class":15841},"\"Rabi FFT surface (adaptive resolution)\"",[28,25854,16052],{"class":15675},[28,25856,25857,25859,25861,25863,25865,25867,25869,25871,25873,25875,25877,25879,25881,25883,25885,25887,25889,25892],{"class":15660,"line":17527},[28,25858,17023],{"class":15675},[28,25860,16940],{"class":16073},[28,25862,22732],{"class":15675},[28,25864,16946],{"class":16307},[28,25866,845],{"class":15682},[28,25868,16951],{"class":15675},[28,25870,22741],{"class":16307},[28,25872,845],{"class":15682},[28,25874,22746],{"class":15801},[28,25876,16397],{"class":15675},[28,25878,22751],{"class":16307},[28,25880,845],{"class":15682},[28,25882,22756],{"class":15801},[28,25884,16397],{"class":15675},[28,25886,16954],{"class":16307},[28,25888,845],{"class":15682},[28,25890,25891],{"class":15841},"\"Amplitude\"",[28,25893,16052],{"class":15675},[28,25895,25896],{"class":15660,"line":17538},[28,25897,15769],{"emptyLinePlaceholder":15768},[28,25899,25900],{"class":15660,"line":17557},[28,25901,25902],{"class":15664},"# A clearer default view\n",[28,25904,25905,25907,25910,25912,25915,25917,25919,25921,25924,25927,25929],{"class":15660,"line":17562},[28,25906,16970],{"class":15675},[28,25908,25909],{"class":16073},"view_init",[28,25911,65],{"class":15675},[28,25913,25914],{"class":16307},"elev",[28,25916,845],{"class":15682},[28,25918,7099],{"class":15801},[28,25920,16397],{"class":15675},[28,25922,25923],{"class":16307},"azim",[28,25925,25926],{"class":15682},"=-",[28,25928,17342],{"class":15801},[28,25930,16052],{"class":15675},[28,25932,25933],{"class":15660,"line":17567},[28,25934,15769],{"emptyLinePlaceholder":15768},[28,25936,25937,25939,25941],{"class":15660,"line":17593},[28,25938,17023],{"class":15675},[28,25940,17026],{"class":16073},[28,25942,17029],{"class":15675},[28,25944,25945,25947,25949],{"class":15660,"line":17599},[28,25946,17034],{"class":15675},[28,25948,17037],{"class":16073},[28,25950,17029],{"class":15675},[17041,25952],{"alt":25953,"src":25954},"Output 7 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-07.webp",[15650,25956,25958],{"className":15652,"code":25957,"language":15654,"meta":15655,"style":15655},"# @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",[8103,25959,25960,25965,25988,26020,26024,26029,26054,26082,26099,26144,26162,26198,26225,26233,26246,26250,26255,26281,26307,26324,26336,26372,26401],{"__ignoreMap":15655},[28,25961,25962],{"class":15660,"line":15661},[28,25963,25964],{"class":15664},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\n",[28,25966,25967,25970,25972,25974,25976,25978,25980,25982,25985],{"class":15660,"line":15668},[28,25968,25969],{"class":15801},"F_STAR_GHZ",[28,25971,15883],{"class":15675},[28,25973,15886],{"class":15682},[28,25975,15805],{"class":15682},[28,25977,18370],{"class":15682},[28,25979,65],{"class":15675},[28,25981,15084],{"class":15801},[28,25983,25984],{"class":15675},")   ",[28,25986,25987],{"class":15664},"# slice near resonance\n",[28,25989,25990,25993,25995,25997,26000,26002,26004,26006,26009,26011,26014,26017],{"class":15660,"line":15679},[28,25991,25992],{"class":15801},"NU_STAR",[28,25994,15805],{"class":15682},[28,25996,18370],{"class":15682},[28,25998,25999],{"class":15675},"(ridge_axis[np.",[28,26001,23038],{"class":16073},[28,26003,18670],{"class":15675},[28,26005,23043],{"class":16073},[28,26007,26008],{"class":15675},"(freq_ghz ",[28,26010,15689],{"class":15682},[28,26012,26013],{"class":15801}," F_STAR_GHZ",[28,26015,26016],{"class":15675},"))])  ",[28,26018,26019],{"class":15664},"# near Ω_R\u002F2π\n",[28,26021,26022],{"class":15660,"line":15725},[28,26023,15769],{"emptyLinePlaceholder":15768},[28,26025,26026],{"class":15660,"line":15739},[28,26027,26028],{"class":15664},"# Slice at f = F_STAR_GHZ\n",[28,26030,26031,26033,26035,26037,26039,26041,26043,26045,26047,26049,26051],{"class":15660,"line":15752},[28,26032,23065],{"class":15675},[28,26034,845],{"class":15682},[28,26036,23033],{"class":15682},[28,26038,18670],{"class":15675},[28,26040,23038],{"class":16073},[28,26042,18670],{"class":15675},[28,26044,23043],{"class":16073},[28,26046,26008],{"class":15675},[28,26048,15689],{"class":15682},[28,26050,26013],{"class":15801},[28,26052,26053],{"class":15675},")))\n",[28,26055,26056,26058,26060,26062,26064,26066,26068,26070,26072,26075,26077,26080],{"class":15660,"line":15765},[28,26057,16874],{"class":15675},[28,26059,845],{"class":15682},[28,26061,16879],{"class":15675},[28,26063,16882],{"class":16073},[28,26065,65],{"class":15675},[28,26067,16887],{"class":16307},[28,26069,845],{"class":15682},[28,26071,65],{"class":15675},[28,26073,26074],{"class":15801},"6.0",[28,26076,16397],{"class":15675},[28,26078,26079],{"class":15801},"3.8",[28,26081,16539],{"class":15675},[28,26083,26084,26086,26088,26091,26093,26095,26097],{"class":15660,"line":15772},[28,26085,16970],{"class":15675},[28,26087,19308],{"class":16073},[28,26089,26090],{"class":15675},"(f_fft_axis, S_fft[:, j], ",[28,26092,23134],{"class":16307},[28,26094,845],{"class":15682},[28,26096,23139],{"class":15841},[28,26098,16052],{"class":15675},[28,26100,26101,26103,26106,26109,26112,26114,26116,26118,26120,26122,26124,26126,26128,26130,26132,26134,26136,26138,26140,26142],{"class":15660,"line":15781},[28,26102,16970],{"class":15675},[28,26104,26105],{"class":16073},"axvline",[28,26107,26108],{"class":15675},"(ridge_axis[j], ",[28,26110,26111],{"class":16307},"ls",[28,26113,845],{"class":15682},[28,26115,24992],{"class":15841},[28,26117,16397],{"class":15675},[28,26119,24997],{"class":16307},[28,26121,845],{"class":15682},[28,26123,16580],{"class":15801},[28,26125,16397],{"class":15675},[28,26127,16954],{"class":16307},[28,26129,845],{"class":15682},[28,26131,16959],{"class":15671},[28,26133,20444],{"class":16962},[28,26135,25015],{"class":15682},[28,26137,25018],{"class":16962},[28,26139,25021],{"class":15682},[28,26141,25024],{"class":16962},[28,26143,16052],{"class":15675},[28,26145,26146,26148,26150,26152,26154,26156,26158,26160],{"class":15660,"line":15798},[28,26147,16970],{"class":15675},[28,26149,16973],{"class":16073},[28,26151,65],{"class":15675},[28,26153,68],{"class":15671},[28,26155,25052],{"class":15841},[28,26157,25055],{"class":15801},[28,26159,25058],{"class":15841},[28,26161,16052],{"class":15675},[28,26163,26164,26166,26168,26170,26172,26174,26176,26178,26180,26182,26184,26186,26188,26190,26192,26194,26196],{"class":15660,"line":15811},[28,26165,16970],{"class":15675},[28,26167,16987],{"class":16073},[28,26169,65],{"class":15675},[28,26171,24943],{"class":15841},[28,26173,16359],{"class":15682},[28,26175,17308],{"class":15675},[28,26177,24950],{"class":15841},[28,26179,24953],{"class":15671},[28,26181,24956],{"class":15801},[28,26183,2106],{"class":15675},[28,26185,23644],{"class":16073},[28,26187,24963],{"class":15675},[28,26189,23650],{"class":15682},[28,26191,24968],{"class":15841},[28,26193,24971],{"class":15671},[28,26195,24974],{"class":15841},[28,26197,16539],{"class":15675},[28,26199,26200,26202,26204,26206,26209,26212,26214,26216,26218,26220,26223],{"class":15660,"line":15820},[28,26201,16970],{"class":15675},[28,26203,17001],{"class":16073},[28,26205,65],{"class":15675},[28,26207,26208],{"class":15671},"fr",[28,26210,26211],{"class":15841},"\"Slice at $f=",[28,26213,11385],{"class":15801},[28,26215,23223],{"class":15675},[28,26217,21644],{"class":15671},[28,26219,11404],{"class":15801},[28,26221,26222],{"class":15841},"$ GHz\"",[28,26224,16052],{"class":15675},[28,26226,26227,26229,26231],{"class":15660,"line":15830},[28,26228,16970],{"class":15675},[28,26230,21483],{"class":16073},[28,26232,17029],{"class":15675},[28,26234,26235,26237,26239,26242,26244],{"class":15660,"line":15835},[28,26236,17023],{"class":15675},[28,26238,17026],{"class":16073},[28,26240,26241],{"class":15675},"(); plt.",[28,26243,17037],{"class":16073},[28,26245,17029],{"class":15675},[28,26247,26248],{"class":15660,"line":15853},[28,26249,15769],{"emptyLinePlaceholder":15768},[28,26251,26252],{"class":15660,"line":15867},[28,26253,26254],{"class":15664},"# Slice at ν = NU_STAR\n",[28,26256,26257,26259,26261,26263,26265,26267,26269,26271,26274,26276,26279],{"class":15660,"line":15872},[28,26258,23279],{"class":15675},[28,26260,845],{"class":15682},[28,26262,23033],{"class":15682},[28,26264,18670],{"class":15675},[28,26266,23038],{"class":16073},[28,26268,18670],{"class":15675},[28,26270,23043],{"class":16073},[28,26272,26273],{"class":15675},"(f_fft_axis ",[28,26275,15689],{"class":15682},[28,26277,26278],{"class":15801}," NU_STAR",[28,26280,26053],{"class":15675},[28,26282,26283,26285,26287,26289,26291,26293,26295,26297,26299,26301,26303,26305],{"class":15660,"line":15878},[28,26284,16874],{"class":15675},[28,26286,845],{"class":15682},[28,26288,16879],{"class":15675},[28,26290,16882],{"class":16073},[28,26292,65],{"class":15675},[28,26294,16887],{"class":16307},[28,26296,845],{"class":15682},[28,26298,65],{"class":15675},[28,26300,26074],{"class":15801},[28,26302,16397],{"class":15675},[28,26304,26079],{"class":15801},[28,26306,16539],{"class":15675},[28,26308,26309,26311,26313,26316,26318,26320,26322],{"class":15660,"line":15897},[28,26310,16970],{"class":15675},[28,26312,19308],{"class":16073},[28,26314,26315],{"class":15675},"(freq_ghz, S_fft[i, :], ",[28,26317,23134],{"class":16307},[28,26319,845],{"class":15682},[28,26321,23139],{"class":15841},[28,26323,16052],{"class":15675},[28,26325,26326,26328,26330,26332,26334],{"class":15660,"line":15914},[28,26327,16970],{"class":15675},[28,26329,16973],{"class":16073},[28,26331,65],{"class":15675},[28,26333,25037],{"class":15841},[28,26335,16052],{"class":15675},[28,26337,26338,26340,26342,26344,26346,26348,26350,26352,26354,26356,26358,26360,26362,26364,26366,26368,26370],{"class":15660,"line":15931},[28,26339,16970],{"class":15675},[28,26341,16987],{"class":16073},[28,26343,65],{"class":15675},[28,26345,24943],{"class":15841},[28,26347,16359],{"class":15682},[28,26349,17308],{"class":15675},[28,26351,24950],{"class":15841},[28,26353,24953],{"class":15671},[28,26355,24956],{"class":15801},[28,26357,2106],{"class":15675},[28,26359,23644],{"class":16073},[28,26361,24963],{"class":15675},[28,26363,23650],{"class":15682},[28,26365,24968],{"class":15841},[28,26367,24971],{"class":15671},[28,26369,24974],{"class":15841},[28,26371,16539],{"class":15675},[28,26373,26374,26376,26378,26380,26382,26385,26387,26390,26392,26394,26397,26399],{"class":15660,"line":15948},[28,26375,16970],{"class":15675},[28,26377,17001],{"class":16073},[28,26379,65],{"class":15675},[28,26381,26208],{"class":15671},[28,26383,26384],{"class":15841},"\"Slice at Fourier freq ≈ ",[28,26386,11385],{"class":15801},[28,26388,26389],{"class":15675},"f_fft_axis[i]",[28,26391,21719],{"class":15671},[28,26393,11404],{"class":15801},[28,26395,26396],{"class":15801}," {FREQ_UNITS}",[28,26398,21798],{"class":15841},[28,26400,16052],{"class":15675},[28,26402,26403,26405,26407,26409,26411],{"class":15660,"line":15966},[28,26404,17023],{"class":15675},[28,26406,17026],{"class":16073},[28,26408,26241],{"class":15675},[28,26410,17037],{"class":16073},[28,26412,17029],{"class":15675},[17041,26414],{"alt":26415,"src":26416},"Output 8 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-08.webp",[17041,26418],{"alt":26419,"src":26420},"Output 9 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-09.webp",[251,26422,26424],{"id":26423},"run-it-yourself","Run it yourself",[12,26426,26427,26428,26430,26431,2106],{},"Get ",[19,26429,22],{"href":21}," or ",[19,26432,26434],{"href":26433},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FRabi_Oscillation_Visualization_for_Excited_State_Probability.ipynb","open it in Colab",[26436,26437,26438],"style",{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: 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.sVyAn{--shiki-default:#E06C75}",{"title":15655,"searchDepth":15668,"depth":15668,"links":26440},[26441,26442,26443,26444,26445,26446,26447,26448,26449,26450,26451,26452,26453,26454,26455],{"id":253,"depth":15668,"text":254},{"id":1769,"depth":15668,"text":1770},{"id":2568,"depth":15668,"text":2569},{"id":3945,"depth":15668,"text":3946},{"id":5089,"depth":15668,"text":5090},{"id":7588,"depth":15668,"text":7589},{"id":8109,"depth":15668,"text":8110},{"id":9208,"depth":15668,"text":9209},{"id":10833,"depth":15668,"text":10834},{"id":11206,"depth":15668,"text":11207},{"id":12203,"depth":15668,"text":12204},{"id":13381,"depth":15668,"text":13382},{"id":14561,"depth":15668,"text":14562},{"id":15056,"depth":15668,"text":15057},{"id":26423,"depth":15668,"text":26424},[26457,26458,26459,26460],"Qollab","Blog","Expert notes","Rabi oscillations",[26462],{"username":7,"name":26463,"role":26464,"bio":26465,"links":26466},"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.",[26467,26470],{"label":26468,"href":26469},"GitHub ↗","https:\u002F\u002Fgithub.com\u002FOJB-Quantum",{"label":26471,"href":21},"Original notebook ↗",{"username":7,"name":26463,"role":26473},"Quantum Hardware Engineer, IBM",null,"A primer on the quantum dynamics of a driven two-level system, visualizing excited-state probability under the rotating-wave approximation (RWA).",false,"md","Deep dive · Hardware","article",{},"\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Frabi-oscillation-visualization","2026-07-15","6 min read",[],{"title":5,"description":26475},"blog\u002Fexpert-notes\u002Frabi-oscillation-visualization",[26489,26490],"hardware","visualization","cLyZBkTx0Kr_9zY5k_7c4mZTRRqgWxllLNGfFbxk58w",[26493,26981,67231,91367],{"id":26494,"title":26495,"authors":26496,"body":26497,"breadcrumb":26940,"builders":26942,"byline":26947,"challenge":26474,"courseAuthor":26474,"courseLead":26474,"dek":26948,"description":26949,"draft":26476,"extension":26477,"eyebrow":26950,"finish":26474,"fork":26474,"hero":26474,"heroAlt":26474,"heroCta":26474,"heroImage":26474,"kind":26479,"lessonCount":26474,"meta":26951,"navigation":15768,"newsItems":26474,"next":26474,"ogImage":26952,"order":26474,"outcomes":26474,"path":26953,"publishDate":26483,"readingTime":26484,"related":26954,"relatedProjects":26955,"seo":26975,"stem":26977,"tags":26978,"track":26474,"trackName":26474,"__hash__":26980},"blog\u002Fblog\u002Fexpert-notes\u002F2-qubit-state-visualization.md","2-qubit states, visualized 8 ways",[7],{"type":9,"value":26498,"toc":26928},[26499,26502,26514,26518,26521,26797,26800,26804,26807,26813,26816,26822,26826,26829,26835,26839,26842,26848,26852,26855,26861,26865,26868,26872,26875,26881,26885,26888,26894,26898,26901,26907,26913,26915,26925],[12,26500,26501],{},"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.",[12,26503,26504,26505,26509,26510,2106],{},"Everything below comes from Onri's runnable notebook, shared openly on ",[19,26506,26508],{"href":26507},"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 ",[19,26511,26513],{"href":26512},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002F2_Qubit_State_Visualization_in_Python.ipynb","Google Colab",[251,26515,26517],{"id":26516},"one-circuit-two-states","One circuit, two states",[12,26519,26520],{},"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:",[26522,26523,26526],"code-block",{"name":26524,"tag":26525},"2_qubit_state_visualization.ipynb","Python",[15650,26527,26529],{"className":15652,"code":26528,"language":15654,"meta":15655,"style":15655},"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",[8103,26530,26531,26541,26553,26565,26569,26574,26590,26608,26629,26653,26668,26672,26685,26697,26701,26706,26719,26730,26742,26755,26762,26773,26785],{"__ignoreMap":15655},[28,26532,26533,26535,26537,26539],{"class":15660,"line":15661},[28,26534,15672],{"class":15671},[28,26536,15730],{"class":15675},[28,26538,15733],{"class":15671},[28,26540,15736],{"class":15675},[28,26542,26543,26545,26548,26550],{"class":15660,"line":15668},[28,26544,15714],{"class":15671},[28,26546,26547],{"class":15675}," qiskit ",[28,26549,15672],{"class":15671},[28,26551,26552],{"class":15675}," QuantumCircuit\n",[28,26554,26555,26557,26560,26562],{"class":15660,"line":15679},[28,26556,15714],{"class":15671},[28,26558,26559],{"class":15675}," qiskit.quantum_info ",[28,26561,15672],{"class":15671},[28,26563,26564],{"class":15675}," Statevector, DensityMatrix\n",[28,26566,26567],{"class":15660,"line":15725},[28,26568,15769],{"emptyLinePlaceholder":15768},[28,26570,26571],{"class":15660,"line":15739},[28,26572,26573],{"class":15664},"# Create a circuit with 2 qubits\n",[28,26575,26576,26579,26581,26584,26586,26588],{"class":15660,"line":15752},[28,26577,26578],{"class":15675},"qc ",[28,26580,845],{"class":15682},[28,26582,26583],{"class":16073}," QuantumCircuit",[28,26585,65],{"class":15675},[28,26587,848],{"class":15801},[28,26589,16052],{"class":15675},[28,26591,26592,26595,26598,26600,26602,26605],{"class":15660,"line":15765},[28,26593,26594],{"class":15675},"qc.",[28,26596,26597],{"class":16073},"h",[28,26599,65],{"class":15675},[28,26601,652],{"class":15801},[28,26603,26604],{"class":15675},")             ",[28,26606,26607],{"class":15664},"# Hadamard on q0\n",[28,26609,26610,26612,26615,26617,26619,26621,26623,26626],{"class":15660,"line":15772},[28,26611,26594],{"class":15675},[28,26613,26614],{"class":16073},"cx",[28,26616,65],{"class":15675},[28,26618,652],{"class":15801},[28,26620,16397],{"class":15675},[28,26622,3200],{"class":15801},[28,26624,26625],{"class":15675},")         ",[28,26627,26628],{"class":15664},"# CNOT with q0 as control, q1 as target\n",[28,26630,26631,26633,26636,26639,26641,26644,26646,26648,26650],{"class":15660,"line":15781},[28,26632,26594],{"class":15675},[28,26634,26635],{"class":16073},"ry",[28,26637,26638],{"class":15675},"(np.pi ",[28,26640,1129],{"class":15682},[28,26642,26643],{"class":15801}," 4",[28,26645,16397],{"class":15675},[28,26647,3200],{"class":15801},[28,26649,22233],{"class":15675},[28,26651,26652],{"class":15664},"# Y-rotation of pi\u002F4 on q1\n",[28,26654,26655,26657,26659,26661,26663,26665],{"class":15660,"line":15798},[28,26656,26594],{"class":15675},[28,26658,2266],{"class":16073},[28,26660,65],{"class":15675},[28,26662,3200],{"class":15801},[28,26664,26604],{"class":15675},[28,26666,26667],{"class":15664},"# Phase flip (Z gate) on q1\n",[28,26669,26670],{"class":15660,"line":15811},[28,26671,15769],{"emptyLinePlaceholder":15768},[28,26673,26674,26677,26679,26682],{"class":15660,"line":15820},[28,26675,26676],{"class":15675},"final_statevector ",[28,26678,845],{"class":15682},[28,26680,26681],{"class":16073}," Statevector",[28,26683,26684],{"class":15675},"(qc)\n",[28,26686,26687,26690,26692,26695],{"class":15660,"line":15830},[28,26688,26689],{"class":15675},"final_density_matrix ",[28,26691,845],{"class":15682},[28,26693,26694],{"class":16073}," DensityMatrix",[28,26696,26684],{"class":15675},[28,26698,26699],{"class":15660,"line":15835},[28,26700,15769],{"emptyLinePlaceholder":15768},[28,26702,26703],{"class":15660,"line":15853},[28,26704,26705],{"class":15664},"# Generate all 8 plots for the final state\n",[28,26707,26708,26711,26714,26717],{"class":15660,"line":15867},[28,26709,26710],{"class":16073},"plot_bloch_multivector_viz",[28,26712,26713],{"class":15675},"(final_density_matrix, ",[28,26715,26716],{"class":15841},"\"Final Entangled State\"",[28,26718,16052],{"class":15675},[28,26720,26721,26724,26726,26728],{"class":15660,"line":15872},[28,26722,26723],{"class":16073},"plot_q_sphere_viz",[28,26725,26713],{"class":15675},[28,26727,26716],{"class":15841},[28,26729,16052],{"class":15675},[28,26731,26732,26735,26737,26740],{"class":15660,"line":15878},[28,26733,26734],{"class":16073},"plot_steering_ellipsoid",[28,26736,26713],{"class":15675},[28,26738,26739],{"class":15841},"\"Steering Ellipsoid for Final State\"",[28,26741,16052],{"class":15675},[28,26743,26744,26747,26750,26753],{"class":15660,"line":15897},[28,26745,26746],{"class":16073},"visualize_schmidt_decomposition",[28,26748,26749],{"class":15675},"(final_statevector, ",[28,26751,26752],{"class":15841},"\"Schmidt Decomposition for Final State\"",[28,26754,16052],{"class":15675},[28,26756,26757,26760],{"class":15660,"line":15914},[28,26758,26759],{"class":16073},"explain_toroidal_geometry",[28,26761,17029],{"class":15675},[28,26763,26764,26767,26769,26771],{"class":15660,"line":15931},[28,26765,26766],{"class":16073},"plot_correlation_invariants",[28,26768,26713],{"class":15675},[28,26770,26716],{"class":15841},[28,26772,16052],{"class":15675},[28,26774,26775,26778,26780,26783],{"class":15660,"line":15948},[28,26776,26777],{"class":16073},"plot_wigner_function",[28,26779,26713],{"class":15675},[28,26781,26782],{"class":15841},"\"Wigner Functions for Final State\"",[28,26784,16052],{"class":15675},[28,26786,26787,26790,26792,26795],{"class":15660,"line":15966},[28,26788,26789],{"class":16073},"plot_density_matrix",[28,26791,26713],{"class":15675},[28,26793,26794],{"class":15841},"\"Density Matrix for Final State\"",[28,26796,16052],{"class":15675},[12,26798,26799],{},"Every method below shows the entangled state; where the contrast with |00⟩ is the whole point, both are shown.",[251,26801,26803],{"id":26802},"_1-two-bloch-spheres-one-giveaway","1. Two Bloch spheres, one giveaway",[12,26805,26806],{},"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.",[17041,26808],{"alt":26809,"caption":26810,"no":26811,"src":26812},"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",[12,26814,26815],{},"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.",[17041,26817],{"alt":26818,"caption":26819,"no":26820,"src":26821},"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",[251,26823,26825],{"id":26824},"_2-the-q-sphere","2. The Q-sphere",[12,26827,26828],{},"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.",[17041,26830],{"alt":26831,"caption":26832,"no":26833,"src":26834},"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",[251,26836,26838],{"id":26837},"_3-the-steering-ellipsoid","3. The steering ellipsoid",[12,26840,26841],{},"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.",[17041,26843],{"alt":26844,"caption":26845,"no":26846,"src":26847},"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",[251,26849,26851],{"id":26850},"_4-schmidt-decomposition","4. Schmidt decomposition",[12,26853,26854],{},"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.",[17041,26856],{"alt":26857,"caption":26858,"no":26859,"src":26860},"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",[251,26862,26864],{"id":26863},"_5-toroidal-geometry","5. Toroidal geometry",[12,26866,26867],{},"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.",[251,26869,26871],{"id":26870},"_6-correlation-invariants","6. Correlation invariants",[12,26873,26874],{},"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.",[17041,26876],{"alt":26877,"caption":26878,"no":26879,"src":26880},"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",[251,26882,26884],{"id":26883},"_7-the-wigner-function","7. The Wigner function",[12,26886,26887],{},"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.",[17041,26889],{"alt":26890,"caption":26891,"no":26892,"src":26893},"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",[251,26895,26897],{"id":26896},"_8-the-density-matrix","8. The density matrix",[12,26899,26900],{},"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.",[17041,26902],{"alt":26903,"caption":26904,"no":26905,"src":26906},"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",[17041,26908],{"alt":26909,"caption":26910,"no":26911,"src":26912},"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",[251,26914,26424],{"id":26423},[12,26916,26917,26918,26430,26921,26924],{},"Every figure above comes out of one script you can run top to bottom: ",[19,26919,26920],{"href":26507},"open the notebook on GitHub",[19,26922,26923],{"href":26512},"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.",[26436,26926,26927],{},"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":15655,"searchDepth":15668,"depth":15668,"links":26929},[26930,26931,26932,26933,26934,26935,26936,26937,26938,26939],{"id":26516,"depth":15668,"text":26517},{"id":26802,"depth":15668,"text":26803},{"id":26824,"depth":15668,"text":26825},{"id":26837,"depth":15668,"text":26838},{"id":26850,"depth":15668,"text":26851},{"id":26863,"depth":15668,"text":26864},{"id":26870,"depth":15668,"text":26871},{"id":26883,"depth":15668,"text":26884},{"id":26896,"depth":15668,"text":26897},{"id":26423,"depth":15668,"text":26424},[26457,26458,26459,26941],"2-qubit states",[26943],{"username":7,"name":26463,"role":26464,"bio":26465,"links":26944},[26945,26946],{"label":26468,"href":26469},{"label":26471,"href":26507},{"username":7,"name":26463,"role":26473},"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",[],[26956,26963,26970],{"username":26957,"project":26958,"title":26959,"category":26960,"thumb":26961,"to":26962},"q-inho","qave","QAVE","Visualization","\u002F_content\u002Fimages\u002Fqave\u002Fthumbnail.webp","\u002Fexplore\u002Fqave",{"username":26964,"project":26965,"title":26966,"category":26967,"thumb":26968,"to":26969},"AmberPincar","quantum-garden","Quantum Garden","Art","\u002F_content\u002Fimages\u002Fquantum-garden\u002Fthumbnail.webp","\u002Fexplore\u002Fquantum-garden",{"username":26971,"project":26972,"title":26973,"category":26967,"thumb":26974},"xinyi","quantum-butterfly-field","Quantum Butterfly Field","\u002F_content\u002Fimages\u002Fquantum-butterfly-field\u002Fthumbnail.webp",{"description":26976,"title":26495},"One entangled state, eight ways to see it. Onri Jay Benally's runnable notebook renders a 2-qubit state as Bloch spheres, Q-spheres, steering ellipsoids, and more.","blog\u002Fexpert-notes\u002F2-qubit-state-visualization",[26490,26979,15654],"quantum","hbpIuxS-SqH0GmOc1n1ds2B-L8OsTlp8DNHQ4vd3Ylw",{"id":26982,"title":26983,"authors":26984,"body":26986,"breadcrumb":67207,"builders":67209,"byline":67217,"challenge":26474,"courseAuthor":26474,"courseLead":26474,"dek":26474,"description":67219,"draft":26476,"extension":26477,"eyebrow":67220,"finish":26474,"fork":26474,"hero":26474,"heroAlt":26474,"heroCta":26474,"heroImage":26474,"kind":26479,"lessonCount":26474,"meta":67221,"navigation":15768,"newsItems":26474,"next":26474,"ogImage":26474,"order":26474,"outcomes":26474,"path":67222,"publishDate":26483,"readingTime":67223,"related":67224,"relatedProjects":26474,"seo":67225,"stem":67226,"tags":67227,"track":26474,"trackName":26474,"__hash__":67230},"blog\u002Fblog\u002Fexpert-notes\u002Fkrylov-quantum-diagonalization.md","Krylov Quantum Diagonalization of Lattice Hamiltonians",[26985],"mirko-amico",{"type":9,"value":26987,"toc":67178},[26988,26997,27002,27017,27021,27024,27052,27056,27059,27077,27081,28322,28326,28331,28547,28842,29494,29811,29814,29818,30117,30373,30644,30847,31621,31857,31860,32082,32404,32411,33353,33358,33841,34211,34217,34221,34318,34651,34657,34660,34708,34712,34794,34859,34863,34867,34975,35099,35105,35109,35115,36356,37183,37731,37734,38720,39111,39358,39763,39769,39772,39775,39842,39848,39852,39856,39859,39865,40507,43273,43481,44698,45932,45977,45981,46523,46954,46957,46961,47051,47054,47179,47182,47240,47465,47468,47523,47529,47532,47536,47539,47643,47647,47650,48036,48039,48249,48255,48259,48725,48729,48784,49114,49120,49212,49581,49585,49610,49614,49654,49743,49774,50100,50113,50176,50515,50527,50591,50763,50845,50938,50944,50947,50964,50970,51213,51217,51220,51594,51719,52315,52347,52812,52894,52900,52965,53680,53783,54046,54299,54630,54813,55151,55891,56205,57820,57852,59348,59825,60519,61302,61634,61961,62023,62032,62036,62234,62588,62590,63116,63119,63417,63473,63735,64228,64487,65429,66433,66873,67128,67132,67141,67146,67151,67156,67160,67163,67169,67171,67175],[12,26989,26990],{},[15,26991,17,26992,26996],{},[19,26993,26995],{"href":26994},"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.",[12,26998,26999],{},[15,27000,27001],{},"Usage estimate: 20 minutes on a Heron r2 (NOTE: This is an estimate only. Your runtime might vary.)",[15650,27003,27005],{"className":15652,"code":27004,"language":15654,"meta":15655,"style":15655},"# This cell is hidden from users – it disables some lint rules\n# ruff: noqa: E402 E722 F601\n",[8103,27006,27007,27012],{"__ignoreMap":15655},[28,27008,27009],{"class":15660,"line":15661},[28,27010,27011],{"class":15664},"# This cell is hidden from users – it disables some lint rules\n",[28,27013,27014],{"class":15660,"line":15668},[28,27015,27016],{"class":15664},"# ruff: noqa: E402 E722 F601\n",[251,27018,27020],{"id":27019},"background","Background",[12,27022,27023],{},"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.",[12,27025,27026,27027,27031,27032,27036,27037,27041,27042,27046,27047,27051],{},"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, ",[19,27028,27030],{"href":27029},"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, ",[19,27033,27035],{"href":27034},"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 ",[19,27038,27040],{"href":27039},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2312.00178","review paper","), the Krylov quantum diagonalization (KQD) algorithm. This algorithm performs well at large scale ",[19,27043,27045],{"href":27044},"#references","[1]"," on existing quantum hardware, shares similar ",[19,27048,27050],{"href":27049},"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.",[251,27053,27055],{"id":27054},"requirements","Requirements",[12,27057,27058],{},"Before starting this tutorial, be sure you have the following installed:",[27060,27061,27062,27070],"ul",{},[27063,27064,27065,27066,27069],"li",{},"Qiskit SDK v2.0 or later, with ",[19,27067,26490],{"href":27068},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fvisualization"," support",[27063,27071,27072,27073,27076],{},"Qiskit Runtime v0.22 or later ( ",[8103,27074,27075],{},"pip install qiskit-ibm-runtime"," )",[251,27078,27080],{"id":27079},"setup","Setup",[15650,27082,27084],{"className":15652,"code":27083,"language":15654,"meta":15655,"style":15655},"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",[8103,27085,27086,27096,27108,27119,27130,27142,27149,27161,27168,27172,27187,27191,27202,27214,27225,27237,27249,27261,27273,27277,27281,27292,27297,27307,27311,27315,27319,27328,27335,27342,27353,27368,27384,27399,27403,27408,27412,27416,27421,27426,27431,27436,27441,27446,27451,27456,27461,27466,27470,27474,27479,27483,27487,27491,27507,27517,27530,27558,27562,27588,27610,27623,27630,27655,27662,27680,27686,27692,27711,27717,27732,27736,27740,27759,27763,27768,27772,27781,27800,27839,27843,27852,27866,27898,27902,27912,27916,27928,27932,27941,27966,27984,27988,28010,28014,28023,28064,28072,28076,28086,28100,28113,28123,28127,28151,28166,28184,28220,28242,28247,28254,28262,28266,28281,28285,28303,28315],{"__ignoreMap":15655},[28,27087,27088,27090,27092,27094],{"class":15660,"line":15661},[28,27089,15672],{"class":15671},[28,27091,15730],{"class":15675},[28,27093,15733],{"class":15671},[28,27095,15736],{"class":15675},[28,27097,27098,27100,27103,27105],{"class":15660,"line":15668},[28,27099,15672],{"class":15671},[28,27101,27102],{"class":15675}," scipy ",[28,27104,15733],{"class":15671},[28,27106,27107],{"class":15675}," sp\n",[28,27109,27110,27112,27115,27117],{"class":15660,"line":15679},[28,27111,15672],{"class":15671},[28,27113,27114],{"class":15675}," matplotlib.pylab ",[28,27116,15733],{"class":15671},[28,27118,15749],{"class":15675},[28,27120,27121,27123,27125,27127],{"class":15660,"line":15725},[28,27122,15714],{"class":15671},[28,27124,15717],{"class":15675},[28,27126,15672],{"class":15671},[28,27128,27129],{"class":15675}," Union, List\n",[28,27131,27132,27134,27137,27139],{"class":15660,"line":15739},[28,27133,15672],{"class":15671},[28,27135,27136],{"class":15675}," itertools ",[28,27138,15733],{"class":15671},[28,27140,27141],{"class":15675}," it\n",[28,27143,27144,27146],{"class":15660,"line":15752},[28,27145,15672],{"class":15671},[28,27147,27148],{"class":15675}," copy\n",[28,27150,27151,27153,27156,27158],{"class":15660,"line":15765},[28,27152,15714],{"class":15671},[28,27154,27155],{"class":15675}," sympy ",[28,27157,15672],{"class":15671},[28,27159,27160],{"class":15675}," Matrix\n",[28,27162,27163,27165],{"class":15660,"line":15772},[28,27164,15672],{"class":15671},[28,27166,27167],{"class":15675}," warnings\n",[28,27169,27170],{"class":15660,"line":15781},[28,27171,15769],{"emptyLinePlaceholder":15768},[28,27173,27174,27177,27180,27182,27185],{"class":15660,"line":15798},[28,27175,27176],{"class":15675},"warnings.",[28,27178,27179],{"class":16073},"filterwarnings",[28,27181,65],{"class":15675},[28,27183,27184],{"class":15841},"\"ignore\"",[28,27186,16052],{"class":15675},[28,27188,27189],{"class":15660,"line":15811},[28,27190,15769],{"emptyLinePlaceholder":15768},[28,27192,27193,27195,27197,27199],{"class":15660,"line":15820},[28,27194,15714],{"class":15671},[28,27196,26559],{"class":15675},[28,27198,15672],{"class":15671},[28,27200,27201],{"class":15675}," SparsePauliOp, Pauli, StabilizerState\n",[28,27203,27204,27206,27209,27211],{"class":15660,"line":15830},[28,27205,15714],{"class":15671},[28,27207,27208],{"class":15675}," qiskit.circuit ",[28,27210,15672],{"class":15671},[28,27212,27213],{"class":15675}," Parameter, IfElseOp\n",[28,27215,27216,27218,27220,27222],{"class":15660,"line":15835},[28,27217,15714],{"class":15671},[28,27219,26547],{"class":15675},[28,27221,15672],{"class":15671},[28,27223,27224],{"class":15675}," QuantumCircuit, QuantumRegister\n",[28,27226,27227,27229,27232,27234],{"class":15660,"line":15853},[28,27228,15714],{"class":15671},[28,27230,27231],{"class":15675}," qiskit.circuit.library ",[28,27233,15672],{"class":15671},[28,27235,27236],{"class":15675}," PauliEvolutionGate\n",[28,27238,27239,27241,27244,27246],{"class":15660,"line":15867},[28,27240,15714],{"class":15671},[28,27242,27243],{"class":15675}," qiskit.synthesis ",[28,27245,15672],{"class":15671},[28,27247,27248],{"class":15675}," LieTrotter\n",[28,27250,27251,27253,27256,27258],{"class":15660,"line":15872},[28,27252,15714],{"class":15671},[28,27254,27255],{"class":15675}," qiskit.transpiler ",[28,27257,15672],{"class":15671},[28,27259,27260],{"class":15675}," Target, CouplingMap\n",[28,27262,27263,27265,27268,27270],{"class":15660,"line":15878},[28,27264,15714],{"class":15671},[28,27266,27267],{"class":15675}," qiskit.transpiler.preset_passmanagers ",[28,27269,15672],{"class":15671},[28,27271,27272],{"class":15675}," generate_preset_pass_manager\n",[28,27274,27275],{"class":15660,"line":15897},[28,27276,15769],{"emptyLinePlaceholder":15768},[28,27278,27279],{"class":15660,"line":15914},[28,27280,15769],{"emptyLinePlaceholder":15768},[28,27282,27283,27285,27288,27290],{"class":15660,"line":15931},[28,27284,15714],{"class":15671},[28,27286,27287],{"class":15675}," qiskit_ibm_runtime ",[28,27289,15672],{"class":15671},[28,27291,18567],{"class":15675},[28,27293,27294],{"class":15660,"line":15948},[28,27295,27296],{"class":15675},"    QiskitRuntimeService,\n",[28,27298,27299,27302,27304],{"class":15660,"line":15966},[28,27300,27301],{"class":15675},"    EstimatorV2 ",[28,27303,15733],{"class":15671},[28,27305,27306],{"class":15675}," Estimator,\n",[28,27308,27309],{"class":15660,"line":15983},[28,27310,16052],{"class":15675},[28,27312,27313],{"class":15660,"line":16000},[28,27314,15769],{"emptyLinePlaceholder":15768},[28,27316,27317],{"class":15660,"line":16016},[28,27318,15769],{"emptyLinePlaceholder":15768},[28,27320,27321,27323,27326],{"class":15660,"line":16036},[28,27322,16070],{"class":15671},[28,27324,27325],{"class":16073}," solve_regularized_gen_eig",[28,27327,16077],{"class":15675},[28,27329,27330,27333],{"class":15660,"line":16041},[28,27331,27332],{"class":16082},"    h",[28,27334,16086],{"class":15675},[28,27336,27337,27340],{"class":15660,"line":16371},[28,27338,27339],{"class":16082},"    s",[28,27341,16086],{"class":15675},[28,27343,27344,27347,27349,27351],{"class":15660,"line":16376},[28,27345,27346],{"class":16082},"    threshold",[28,27348,15883],{"class":15675},[28,27350,15886],{"class":15682},[28,27352,16105],{"class":15675},[28,27354,27355,27358,27360,27362,27364,27366],{"class":15660,"line":16410},[28,27356,27357],{"class":16082},"    k",[28,27359,15883],{"class":15675},[28,27361,15955],{"class":15682},[28,27363,15805],{"class":15682},[28,27365,22230],{"class":15801},[28,27367,16105],{"class":15675},[28,27369,27370,27373,27375,27377,27379,27382],{"class":15660,"line":16431},[28,27371,27372],{"class":16082},"    return_dimn",[28,27374,15883],{"class":15675},[28,27376,17588],{"class":15682},[28,27378,15805],{"class":15682},[28,27380,27381],{"class":15801}," False",[28,27383,16105],{"class":15675},[28,27385,27386,27389,27391,27394,27396],{"class":15660,"line":16452},[28,27387,27388],{"class":15675},") -> Union[",[28,27390,15886],{"class":15682},[28,27392,27393],{"class":15675},", List[",[28,27395,15886],{"class":15682},[28,27397,27398],{"class":15675},"]]:\n",[28,27400,27401],{"class":15660,"line":16476},[28,27402,16197],{"class":15841},[28,27404,27405],{"class":15660,"line":16481},[28,27406,27407],{"class":15841},"    Method for solving the generalized eigenvalue problem with regularization\n",[28,27409,27410],{"class":15660,"line":16510},[28,27411,15769],{"emptyLinePlaceholder":15768},[28,27413,27414],{"class":15660,"line":16542},[28,27415,16153],{"class":15841},[28,27417,27418],{"class":15660,"line":16557},[28,27419,27420],{"class":15841},"        h (numpy.ndarray):\n",[28,27422,27423],{"class":15660,"line":16562},[28,27424,27425],{"class":15841},"            The effective representation of the matrix in the Krylov subspace\n",[28,27427,27428],{"class":15660,"line":17382},[28,27429,27430],{"class":15841},"        s (numpy.ndarray):\n",[28,27432,27433],{"class":15660,"line":17392},[28,27434,27435],{"class":15841},"            The matrix of overlaps between vectors of the Krylov subspace\n",[28,27437,27438],{"class":15660,"line":17402},[28,27439,27440],{"class":15841},"        threshold (float):\n",[28,27442,27443],{"class":15660,"line":17407},[28,27444,27445],{"class":15841},"            Cut-off value for the eigenvalue of s\n",[28,27447,27448],{"class":15660,"line":17418},[28,27449,27450],{"class":15841},"        k (int):\n",[28,27452,27453],{"class":15660,"line":17429},[28,27454,27455],{"class":15841},"            Number of eigenvalues to return\n",[28,27457,27458],{"class":15660,"line":17440},[28,27459,27460],{"class":15841},"        return_dimn (bool):\n",[28,27462,27463],{"class":15660,"line":17451},[28,27464,27465],{"class":15841},"            Whether to return the size of the regularized subspace\n",[28,27467,27468],{"class":15660,"line":17462},[28,27469,15769],{"emptyLinePlaceholder":15768},[28,27471,27472],{"class":15660,"line":17467},[28,27473,16187],{"class":15841},[28,27475,27476],{"class":15660,"line":17478},[28,27477,27478],{"class":15841},"        lowest k-eigenvalue(s) that are the solution of the regularized generalized eigenvalue problem\n",[28,27480,27481],{"class":15660,"line":17489},[28,27482,15769],{"emptyLinePlaceholder":15768},[28,27484,27485],{"class":15660,"line":17500},[28,27486,15769],{"emptyLinePlaceholder":15768},[28,27488,27489],{"class":15660,"line":17511},[28,27490,16197],{"class":15841},[28,27492,27493,27496,27498,27501,27504],{"class":15660,"line":17522},[28,27494,27495],{"class":15675},"    s_vals, s_vecs ",[28,27497,845],{"class":15682},[28,27499,27500],{"class":15675}," sp.linalg.",[28,27502,27503],{"class":16073},"eigh",[28,27505,27506],{"class":15675},"(s)\n",[28,27508,27509,27512,27514],{"class":15660,"line":17527},[28,27510,27511],{"class":15675},"    s_vecs ",[28,27513,845],{"class":15682},[28,27515,27516],{"class":15675}," s_vecs.T\n",[28,27518,27519,27522,27524,27526,27528],{"class":15660,"line":17538},[28,27520,27521],{"class":15675},"    good_vecs ",[28,27523,845],{"class":15682},[28,27525,16298],{"class":15675},[28,27527,18851],{"class":16073},[28,27529,16077],{"class":15675},[28,27531,27532,27535,27537,27540,27542,27545,27548,27550,27553,27555],{"class":15660,"line":17557},[28,27533,27534],{"class":15675},"        [vec ",[28,27536,22277],{"class":15671},[28,27538,27539],{"class":15675}," val, vec ",[28,27541,17720],{"class":15671},[28,27543,27544],{"class":15682}," zip",[28,27546,27547],{"class":15675},"(s_vals, s_vecs) ",[28,27549,20133],{"class":15671},[28,27551,27552],{"class":15675}," val ",[28,27554,16503],{"class":15682},[28,27556,27557],{"class":15675}," threshold]\n",[28,27559,27560],{"class":15660,"line":17562},[28,27561,18606],{"class":15675},[28,27563,27564,27567,27569,27572,27575,27577,27580,27583,27585],{"class":15660,"line":17567},[28,27565,27566],{"class":15675},"    h_reg ",[28,27568,845],{"class":15682},[28,27570,27571],{"class":15675}," good_vecs.",[28,27573,27574],{"class":16073},"conj",[28,27576,23647],{"class":15675},[28,27578,27579],{"class":15682},"@",[28,27581,27582],{"class":15675}," h ",[28,27584,27579],{"class":15682},[28,27586,27587],{"class":15675}," good_vecs.T\n",[28,27589,27590,27593,27595,27597,27599,27601,27603,27606,27608],{"class":15660,"line":17593},[28,27591,27592],{"class":15675},"    s_reg ",[28,27594,845],{"class":15682},[28,27596,27571],{"class":15675},[28,27598,27574],{"class":16073},[28,27600,23647],{"class":15675},[28,27602,27579],{"class":15682},[28,27604,27605],{"class":15675}," s ",[28,27607,27579],{"class":15682},[28,27609,27587],{"class":15675},[28,27611,27612,27614,27617,27619,27621],{"class":15660,"line":17599},[28,27613,16484],{"class":15671},[28,27615,27616],{"class":15675}," k ",[28,27618,23650],{"class":15682},[28,27620,22230],{"class":15801},[28,27622,15778],{"class":15675},[28,27624,27625,27627],{"class":15660,"line":17619},[28,27626,17734],{"class":15671},[28,27628,27629],{"class":15675}," return_dimn:\n",[28,27631,27632,27635,27637,27639,27642,27644,27646,27648,27650,27652],{"class":15660,"line":17624},[28,27633,27634],{"class":15671},"            return",[28,27636,27500],{"class":15675},[28,27638,27503],{"class":16073},[28,27640,27641],{"class":15675},"(h_reg, s_reg)[",[28,27643,652],{"class":15801},[28,27645,21951],{"class":15675},[28,27647,652],{"class":15801},[28,27649,24619],{"class":15675},[28,27651,22222],{"class":15682},[28,27653,27654],{"class":15675},"(good_vecs)\n",[28,27656,27657,27660],{"class":15660,"line":17629},[28,27658,27659],{"class":15671},"        else",[28,27661,15778],{"class":15675},[28,27663,27664,27666,27668,27670,27672,27674,27676,27678],{"class":15660,"line":17639},[28,27665,27634],{"class":15671},[28,27667,27500],{"class":15675},[28,27669,27503],{"class":16073},[28,27671,27641],{"class":15675},[28,27673,652],{"class":15801},[28,27675,21951],{"class":15675},[28,27677,652],{"class":15801},[28,27679,16849],{"class":15675},[28,27681,27682,27684],{"class":15660,"line":17656},[28,27683,19102],{"class":15671},[28,27685,15778],{"class":15675},[28,27687,27688,27690],{"class":15660,"line":17665},[28,27689,17734],{"class":15671},[28,27691,27629],{"class":15675},[28,27693,27694,27696,27698,27700,27702,27704,27707,27709],{"class":15660,"line":17671},[28,27695,27634],{"class":15671},[28,27697,27500],{"class":15675},[28,27699,27503],{"class":16073},[28,27701,27641],{"class":15675},[28,27703,652],{"class":15801},[28,27705,27706],{"class":15675},"][:k], ",[28,27708,22222],{"class":15682},[28,27710,27654],{"class":15675},[28,27712,27713,27715],{"class":15660,"line":17683},[28,27714,27659],{"class":15671},[28,27716,15778],{"class":15675},[28,27718,27719,27721,27723,27725,27727,27729],{"class":15660,"line":17689},[28,27720,27634],{"class":15671},[28,27722,27500],{"class":15675},[28,27724,27503],{"class":16073},[28,27726,27641],{"class":15675},[28,27728,652],{"class":15801},[28,27730,27731],{"class":15675},"][:k]\n",[28,27733,27734],{"class":15660,"line":17694},[28,27735,15769],{"emptyLinePlaceholder":15768},[28,27737,27738],{"class":15660,"line":17705},[28,27739,15769],{"emptyLinePlaceholder":15768},[28,27741,27742,27744,27747,27749,27752,27754,27757],{"class":15660,"line":17711},[28,27743,16070],{"class":15671},[28,27745,27746],{"class":16073}," single_particle_gs",[28,27748,65],{"class":15675},[28,27750,27751],{"class":16082},"H_op",[28,27753,16397],{"class":15675},[28,27755,27756],{"class":16082},"n_qubits",[28,27758,16407],{"class":15675},[28,27760,27761],{"class":15660,"line":17731},[28,27762,16197],{"class":15841},[28,27764,27765],{"class":15660,"line":17744},[28,27766,27767],{"class":15841},"    Find the ground state of the single particle(excitation) sector\n",[28,27769,27770],{"class":15660,"line":17750},[28,27771,16197],{"class":15841},[28,27773,27774,27777,27779],{"class":15660,"line":17755},[28,27775,27776],{"class":15675},"    H_x ",[28,27778,845],{"class":15682},[28,27780,24066],{"class":15675},[28,27782,27783,27786,27789,27791,27794,27797],{"class":15660,"line":17765},[28,27784,27785],{"class":15671},"    for",[28,27787,27788],{"class":15675}," p, coeff ",[28,27790,17720],{"class":15671},[28,27792,27793],{"class":15675}," H_op.",[28,27795,27796],{"class":16073},"to_list",[28,27798,27799],{"class":15675},"():\n",[28,27801,27802,27805,27808,27810,27813,27816,27818,27821,27823,27826,27828,27831,27834,27836],{"class":15660,"line":17770},[28,27803,27804],{"class":15675},"        H_x.",[28,27806,27807],{"class":16073},"append",[28,27809,65],{"class":15675},[28,27811,27812],{"class":15682},"set",[28,27814,27815],{"class":15675},"([i ",[28,27817,22277],{"class":15671},[28,27819,27820],{"class":15675}," i, v ",[28,27822,17720],{"class":15671},[28,27824,27825],{"class":15682}," enumerate",[28,27827,65],{"class":15675},[28,27829,27830],{"class":16073},"Pauli",[28,27832,27833],{"class":15675},"(p).x) ",[28,27835,20133],{"class":15671},[28,27837,27838],{"class":15675}," v]))\n",[28,27840,27841],{"class":15660,"line":17775},[28,27842,15769],{"emptyLinePlaceholder":15768},[28,27844,27845,27848,27850],{"class":15660,"line":17783},[28,27846,27847],{"class":15675},"    H_z ",[28,27849,845],{"class":15682},[28,27851,24066],{"class":15675},[28,27853,27854,27856,27858,27860,27862,27864],{"class":15660,"line":17794},[28,27855,27785],{"class":15671},[28,27857,27788],{"class":15675},[28,27859,17720],{"class":15671},[28,27861,27793],{"class":15675},[28,27863,27796],{"class":16073},[28,27865,27799],{"class":15675},[28,27867,27868,27871,27873,27875,27877,27879,27881,27883,27885,27887,27889,27891,27894,27896],{"class":15660,"line":17825},[28,27869,27870],{"class":15675},"        H_z.",[28,27872,27807],{"class":16073},[28,27874,65],{"class":15675},[28,27876,27812],{"class":15682},[28,27878,27815],{"class":15675},[28,27880,22277],{"class":15671},[28,27882,27820],{"class":15675},[28,27884,17720],{"class":15671},[28,27886,27825],{"class":15682},[28,27888,65],{"class":15675},[28,27890,27830],{"class":16073},[28,27892,27893],{"class":15675},"(p).z) ",[28,27895,20133],{"class":15671},[28,27897,27838],{"class":15675},[28,27899,27900],{"class":15660,"line":17837},[28,27901,15769],{"emptyLinePlaceholder":15768},[28,27903,27904,27907,27909],{"class":15660,"line":17843},[28,27905,27906],{"class":15675},"    H_c ",[28,27908,845],{"class":15682},[28,27910,27911],{"class":15675}," H_op.coeffs\n",[28,27913,27914],{"class":15660,"line":17852},[28,27915,15769],{"emptyLinePlaceholder":15768},[28,27917,27918,27920,27922,27925],{"class":15660,"line":17858},[28,27919,21619],{"class":15682},[28,27921,65],{"class":15675},[28,27923,27924],{"class":15841},"\"n_sys_qubits\"",[28,27926,27927],{"class":15675},", n_qubits)\n",[28,27929,27930],{"class":15660,"line":17864},[28,27931,15769],{"emptyLinePlaceholder":15768},[28,27933,27934,27937,27939],{"class":15660,"line":17872},[28,27935,27936],{"class":15675},"    n_exc ",[28,27938,845],{"class":15682},[28,27940,25487],{"class":15801},[28,27942,27943,27946,27948,27950,27953,27956,27959,27961,27963],{"class":15660,"line":17880},[28,27944,27945],{"class":15675},"    sub_dimn ",[28,27947,845],{"class":15682},[28,27949,23033],{"class":15682},[28,27951,27952],{"class":15675},"(sp.special.",[28,27954,27955],{"class":16073},"comb",[28,27957,27958],{"class":15675},"(n_qubits ",[28,27960,2606],{"class":15682},[28,27962,22230],{"class":15801},[28,27964,27965],{"class":15675},", n_exc))\n",[28,27967,27968,27970,27972,27975,27978,27981],{"class":15660,"line":17888},[28,27969,21619],{"class":15682},[28,27971,65],{"class":15675},[28,27973,27974],{"class":15841},"\"n_exc\"",[28,27976,27977],{"class":15675},", n_exc, ",[28,27979,27980],{"class":15841},"\", subspace dimension\"",[28,27982,27983],{"class":15675},", sub_dimn)\n",[28,27985,27986],{"class":15660,"line":17896},[28,27987,15769],{"emptyLinePlaceholder":15768},[28,27989,27990,27993,27995,27997,28000,28003,28005,28008],{"class":15660,"line":17904},[28,27991,27992],{"class":15675},"    few_particle_H ",[28,27994,845],{"class":15682},[28,27996,16298],{"class":15675},[28,27998,27999],{"class":16073},"zeros",[28,28001,28002],{"class":15675},"((sub_dimn, sub_dimn), ",[28,28004,22267],{"class":16307},[28,28006,28007],{"class":15682},"=complex",[28,28009,16052],{"class":15675},[28,28011,28012],{"class":15660,"line":17912},[28,28013,15769],{"emptyLinePlaceholder":15768},[28,28015,28016,28019,28021],{"class":15660,"line":17921},[28,28017,28018],{"class":15675},"    sparse_vecs ",[28,28020,845],{"class":15682},[28,28022,17702],{"class":15675},[28,28024,28025,28028,28031,28033,28036,28038,28041,28044,28046,28049,28051,28053,28055,28057,28059,28061],{"class":15660,"line":17927},[28,28026,28027],{"class":15682},"        set",[28,28029,28030],{"class":15675},"(vec) ",[28,28032,22277],{"class":15671},[28,28034,28035],{"class":15675}," vec ",[28,28037,17720],{"class":15671},[28,28039,28040],{"class":15675}," it.",[28,28042,28043],{"class":16073},"combinations",[28,28045,65],{"class":15675},[28,28047,28048],{"class":15682},"range",[28,28050,27958],{"class":15675},[28,28052,2606],{"class":15682},[28,28054,22230],{"class":15801},[28,28056,22494],{"class":15675},[28,28058,16959],{"class":16307},[28,28060,845],{"class":15682},[28,28062,28063],{"class":15675},"n_exc)\n",[28,28065,28066,28069],{"class":15660,"line":17938},[28,28067,28068],{"class":15675},"    ]  ",[28,28070,28071],{"class":15664},"# list all of the possible sets of n_exc indices of 1s in n_exc-particle states\n",[28,28073,28074],{"class":15660,"line":17943},[28,28075,15769],{"emptyLinePlaceholder":15768},[28,28077,28078,28081,28083],{"class":15660,"line":17952},[28,28079,28080],{"class":15675},"    m ",[28,28082,845],{"class":15682},[28,28084,28085],{"class":15801}," 0\n",[28,28087,28088,28090,28093,28095,28097],{"class":15660,"line":17961},[28,28089,27785],{"class":15671},[28,28091,28092],{"class":15675}," i, i_set ",[28,28094,17720],{"class":15671},[28,28096,27825],{"class":15682},[28,28098,28099],{"class":15675},"(sparse_vecs):\n",[28,28101,28102,28104,28107,28109,28111],{"class":15660,"line":17966},[28,28103,17714],{"class":15671},[28,28105,28106],{"class":15675}," j, j_set ",[28,28108,17720],{"class":15671},[28,28110,27825],{"class":15682},[28,28112,28099],{"class":15675},[28,28114,28115,28118,28121],{"class":15660,"line":17971},[28,28116,28117],{"class":15675},"            m ",[28,28119,28120],{"class":15682},"+=",[28,28122,25487],{"class":15801},[28,28124,28125],{"class":15660,"line":17978},[28,28126,15769],{"emptyLinePlaceholder":15768},[28,28128,28129,28132,28135,28138,28141,28144,28147,28149],{"class":15660,"line":17985},[28,28130,28131],{"class":15671},"            if",[28,28133,28134],{"class":15682}," len",[28,28136,28137],{"class":15675},"(i_set.",[28,28139,28140],{"class":16073},"symmetric_difference",[28,28142,28143],{"class":15675},"(j_set)) ",[28,28145,28146],{"class":15682},"\u003C=",[28,28148,22129],{"class":15801},[28,28150,15778],{"class":15675},[28,28152,28153,28156,28159,28161,28163],{"class":15660,"line":17992},[28,28154,28155],{"class":15671},"                for",[28,28157,28158],{"class":15675}," p_x, p_z, coeff ",[28,28160,17720],{"class":15671},[28,28162,27544],{"class":15682},[28,28164,28165],{"class":15675},"(H_x, H_z, H_c):\n",[28,28167,28168,28171,28174,28176,28179,28181],{"class":15660,"line":17999},[28,28169,28170],{"class":15671},"                    if",[28,28172,28173],{"class":15675}," i_set.",[28,28175,28140],{"class":16073},[28,28177,28178],{"class":15675},"(j_set) ",[28,28180,23650],{"class":15682},[28,28182,28183],{"class":15675}," p_x:\n",[28,28185,28186,28189,28191,28194,28196,28198,28201,28203,28205,28207,28210,28213,28216,28218],{"class":15660,"line":18006},[28,28187,28188],{"class":15675},"                        sgn ",[28,28190,845],{"class":15682},[28,28192,28193],{"class":15675}," ((",[28,28195,15689],{"class":15682},[28,28197,3200],{"class":15801},[28,28199,28200],{"class":15671},"j",[28,28202,22233],{"class":15675},[28,28204,16354],{"class":15682},[28,28206,28134],{"class":15682},[28,28208,28209],{"class":15675},"(p_x.",[28,28211,28212],{"class":16073},"intersection",[28,28214,28215],{"class":15675},"(p_z))) ",[28,28217,16226],{"class":15682},[28,28219,18567],{"class":15675},[28,28221,28222,28225,28227,28229,28231,28233,28235,28237,28239],{"class":15660,"line":18011},[28,28223,28224],{"class":15675},"                            (",[28,28226,15689],{"class":15682},[28,28228,3200],{"class":15801},[28,28230,22233],{"class":15675},[28,28232,16354],{"class":15682},[28,28234,28134],{"class":15682},[28,28236,28137],{"class":15675},[28,28238,28212],{"class":16073},[28,28240,28241],{"class":15675},"(p_z))\n",[28,28243,28244],{"class":15660,"line":18022},[28,28245,28246],{"class":15675},"                        )\n",[28,28248,28249,28252],{"class":15660,"line":18027},[28,28250,28251],{"class":15671},"                    else",[28,28253,15778],{"class":15675},[28,28255,28256,28258,28260],{"class":15660,"line":18032},[28,28257,28188],{"class":15675},[28,28259,845],{"class":15682},[28,28261,28085],{"class":15801},[28,28263,28264],{"class":15660,"line":18037},[28,28265,15769],{"emptyLinePlaceholder":15768},[28,28267,28268,28271,28273,28276,28278],{"class":15660,"line":18049},[28,28269,28270],{"class":15675},"                    few_particle_H[i, j] ",[28,28272,28120],{"class":15682},[28,28274,28275],{"class":15675}," sgn ",[28,28277,16226],{"class":15682},[28,28279,28280],{"class":15675}," coeff\n",[28,28282,28283],{"class":15660,"line":18054},[28,28284,15769],{"emptyLinePlaceholder":15768},[28,28286,28287,28290,28292,28294,28297,28300],{"class":15660,"line":18065},[28,28288,28289],{"class":15675},"    gs_en ",[28,28291,845],{"class":15682},[28,28293,25336],{"class":15682},[28,28295,28296],{"class":15675},"(np.linalg.",[28,28298,28299],{"class":16073},"eigvalsh",[28,28301,28302],{"class":15675},"(few_particle_H))\n",[28,28304,28305,28307,28309,28312],{"class":15660,"line":18076},[28,28306,21619],{"class":15682},[28,28308,65],{"class":15675},[28,28310,28311],{"class":15841},"\"single particle ground state energy: \"",[28,28313,28314],{"class":15675},", gs_en)\n",[28,28316,28317,28319],{"class":15660,"line":18089},[28,28318,16565],{"class":15671},[28,28320,28321],{"class":15675}," gs_en\n",[251,28323,28325],{"id":28324},"step-1-map-classical-inputs-to-a-quantum-problem","Step 1: Map classical inputs to a quantum problem",[28327,28328,28330],"h3",{"id":28329},"the-krylov-space","The Krylov space",[12,28332,28333,28334,28396,28397,28425,28426,28454,28455,28506,28507,2106],{},"The Krylov space ",[28,28335,28337,28356],{"className":28336},[35],[28,28338,28340],{"className":28339},[39],[41,28341,28342],{"xmlns":43},[46,28343,28344,28353],{},[49,28345,28346],{},[2776,28347,28348,28351],{},[55,28349,28350],{"mathvariant":11377},"K",[55,28352,16959],{},[82,28354,28355],{"encoding":84},"\\mathcal{K}^r",[28,28357,28359],{"className":28358,"ariaHidden":71},[89],[28,28360,28362,28365],{"className":28361},[93],[28,28363],{"className":28364,"style":1102},[97],[28,28366,28368,28372],{"className":28367},[102],[28,28369,28350],{"className":28370,"style":28371},[102,11469],"margin-right:0.0144em;",[28,28373,28375],{"className":28374},[111],[28,28376,28378],{"className":28377},[115],[28,28379,28381],{"className":28380},[120],[28,28382,28385],{"className":28383,"style":28384},[124],"height:0.6644em;",[28,28386,28387,28390],{"style":1726},[28,28388],{"className":28389,"style":133},[132],[28,28391,28393],{"className":28392},[137,138,139,140],[28,28394,16959],{"className":28395,"style":9403},[102,106,140]," of order ",[28,28398,28400,28413],{"className":28399},[35],[28,28401,28403],{"className":28402},[39],[41,28404,28405],{"xmlns":43},[46,28406,28407,28411],{},[49,28408,28409],{},[55,28410,16959],{},[82,28412,16959],{"encoding":84},[28,28414,28416],{"className":28415,"ariaHidden":71},[89],[28,28417,28419,28422],{"className":28418},[93],[28,28420],{"className":28421,"style":245},[97],[28,28423,16959],{"className":28424,"style":9403},[102,106]," is the space spanned by vectors obtained by multiplying higher powers of a matrix ",[28,28427,28429,28442],{"className":28428},[35],[28,28430,28432],{"className":28431},[39],[41,28433,28434],{"xmlns":43},[46,28435,28436,28440],{},[49,28437,28438],{},[55,28439,4455],{},[82,28441,4455],{"encoding":84},[28,28443,28445],{"className":28444,"ariaHidden":71},[89],[28,28446,28448,28451],{"className":28447},[93],[28,28449],{"className":28450,"style":1102},[97],[28,28452,4455],{"className":28453},[102,106],", up to ",[28,28456,28458,28476],{"className":28457},[35],[28,28459,28461],{"className":28460},[39],[41,28462,28463],{"xmlns":43},[46,28464,28465,28473],{},[49,28466,28467,28469,28471],{},[55,28468,16959],{},[62,28470,1213],{},[650,28472,3200],{},[82,28474,28475],{"encoding":84},"r-1",[28,28477,28479,28497],{"className":28478,"ariaHidden":71},[89],[28,28480,28482,28485,28488,28491,28494],{"className":28481},[93],[28,28483],{"className":28484,"style":1250},[97],[28,28486,16959],{"className":28487,"style":9403},[102,106],[28,28489],{"className":28490,"style":1257},[173],[28,28492,1213],{"className":28493},[1261],[28,28495],{"className":28496,"style":1257},[173],[28,28498,28500,28503],{"className":28499},[93],[28,28501],{"className":28502,"style":2914},[97],[28,28504,3200],{"className":28505},[102],", with a reference vector ",[28,28508,28510,28529],{"className":28509},[35],[28,28511,28513],{"className":28512},[39],[41,28514,28515],{"xmlns":43},[46,28516,28517,28526],{},[49,28518,28519,28521,28524],{},[55,28520,296],{"mathvariant":1088},[55,28522,28523],{},"v",[62,28525,302],{"stretchy":64},[82,28527,28528],{"encoding":84},"\\vert v \\rangle",[28,28530,28532],{"className":28531,"ariaHidden":71},[89],[28,28533,28535,28538,28541,28544],{"className":28534},[93],[28,28536],{"className":28537,"style":98},[97],[28,28539,296],{"className":28540},[102],[28,28542,28523],{"className":28543,"style":321},[102,106],[28,28545,302],{"className":28546},[182],[28,28548,28550],{"className":28549},[31],[28,28551,28553,28637],{"className":28552},[35],[28,28554,28556],{"className":28555},[39],[41,28557,28558],{"xmlns":43,"display":44},[46,28559,28560,28634],{},[49,28561,28562,28568,28570],{},[2776,28563,28564,28566],{},[55,28565,28350],{"mathvariant":11377},[55,28567,16959],{},[62,28569,845],{},[49,28571,28572,28574,28576,28578,28580,28582,28584,28586,28588,28590,28592,28598,28600,28602,28604,28606,28608,28610,28612,28614,28626,28628,28630,28632],{},[62,28573,11385],{"fence":71},[55,28575,296],{"mathvariant":1088},[55,28577,28523],{},[62,28579,302],{"stretchy":64},[62,28581,72],{"separator":71},[55,28583,4455],{},[55,28585,296],{"mathvariant":1088},[55,28587,28523],{},[62,28589,302],{"stretchy":64},[62,28591,72],{"separator":71},[2776,28593,28594,28596],{},[55,28595,4455],{},[650,28597,848],{},[55,28599,296],{"mathvariant":1088},[55,28601,28523],{},[62,28603,302],{"stretchy":64},[62,28605,72],{"separator":71},[55,28607,2106],{"mathvariant":1088},[55,28609,2106],{"mathvariant":1088},[55,28611,2106],{"mathvariant":1088},[62,28613,72],{"separator":71},[2776,28615,28616,28618],{},[55,28617,4455],{},[49,28619,28620,28622,28624],{},[55,28621,16959],{},[62,28623,1213],{},[650,28625,3200],{},[55,28627,296],{"mathvariant":1088},[55,28629,28523],{},[62,28631,302],{"stretchy":64},[62,28633,11404],{"fence":71},[82,28635,28636],{"encoding":84},"\\mathcal{K}^r = \\left\\{ \\vert v \\rangle, A \\vert v \\rangle, A^2 \\vert v \\rangle, ..., A^{r-1} \\vert v \\rangle \\right\\}",[28,28638,28640,28685],{"className":28639,"ariaHidden":71},[89],[28,28641,28643,28647,28676,28679,28682],{"className":28642},[93],[28,28644],{"className":28645,"style":28646},[97],"height:0.7144em;",[28,28648,28650,28653],{"className":28649},[102],[28,28651,28350],{"className":28652,"style":28371},[102,11469],[28,28654,28656],{"className":28655},[111],[28,28657,28659],{"className":28658},[115],[28,28660,28662],{"className":28661},[120],[28,28663,28665],{"className":28664,"style":28646},[124],[28,28666,28667,28670],{"style":2839},[28,28668],{"className":28669,"style":133},[132],[28,28671,28673],{"className":28672},[137,138,139,140],[28,28674,16959],{"className":28675,"style":9403},[102,106,140],[28,28677],{"className":28678,"style":913},[173],[28,28680,845],{"className":28681},[917],[28,28683],{"className":28684,"style":913},[173],[28,28686,28688,28692],{"className":28687},[93],[28,28689],{"className":28690,"style":28691},[97],"height:1.2141em;vertical-align:-0.35em;",[28,28693,28695,28702,28705,28708,28711,28714,28717,28720,28723,28726,28729,28732,28735,28764,28767,28770,28773,28776,28779,28783,28786,28789,28827,28830,28833,28836],{"className":28694},[3398],[28,28696,28698],{"className":28697,"style":3403},[161,3402],[28,28699,11385],{"className":28700},[3407,28701],"size1",[28,28703,296],{"className":28704},[102],[28,28706,28523],{"className":28707,"style":321},[102,106],[28,28709,302],{"className":28710},[182],[28,28712,72],{"className":28713},[169],[28,28715],{"className":28716,"style":174},[173],[28,28718,4455],{"className":28719},[102,106],[28,28721,296],{"className":28722},[102],[28,28724,28523],{"className":28725,"style":321},[102,106],[28,28727,302],{"className":28728},[182],[28,28730,72],{"className":28731},[169],[28,28733],{"className":28734,"style":174},[173],[28,28736,28738,28741],{"className":28737},[102],[28,28739,4455],{"className":28740},[102,106],[28,28742,28744],{"className":28743},[111],[28,28745,28747],{"className":28746},[115],[28,28748,28750],{"className":28749},[120],[28,28751,28753],{"className":28752,"style":2836},[124],[28,28754,28755,28758],{"style":2839},[28,28756],{"className":28757,"style":133},[132],[28,28759,28761],{"className":28760},[137,138,139,140],[28,28762,848],{"className":28763},[102,140],[28,28765,296],{"className":28766},[102],[28,28768,28523],{"className":28769,"style":321},[102,106],[28,28771,302],{"className":28772},[182],[28,28774,72],{"className":28775},[169],[28,28777],{"className":28778,"style":174},[173],[28,28780,28782],{"className":28781},[102],"...",[28,28784,72],{"className":28785},[169],[28,28787],{"className":28788,"style":174},[173],[28,28790,28792,28795],{"className":28791},[102],[28,28793,4455],{"className":28794},[102,106],[28,28796,28798],{"className":28797},[111],[28,28799,28801],{"className":28800},[115],[28,28802,28804],{"className":28803},[120],[28,28805,28807],{"className":28806,"style":2836},[124],[28,28808,28809,28812],{"style":2839},[28,28810],{"className":28811,"style":133},[132],[28,28813,28815],{"className":28814},[137,138,139,140],[28,28816,28818,28821,28824],{"className":28817},[102,140],[28,28819,16959],{"className":28820,"style":9403},[102,106,140],[28,28822,1213],{"className":28823},[1261,140],[28,28825,3200],{"className":28826},[102,140],[28,28828,296],{"className":28829},[102],[28,28831,28523],{"className":28832,"style":321},[102,106],[28,28834,302],{"className":28835},[182],[28,28837,28839],{"className":28838,"style":3403},[182,3402],[28,28840,11404],{"className":28841},[3407,28701],[12,28843,28844,28845,28873,28874,28902,28903,28974,28975,29003,29004,29108,29109,29179,29180,29208,29209,29310,29311,29314,29315,29343,29344,2106],{},"If the matrix ",[28,28846,28848,28861],{"className":28847},[35],[28,28849,28851],{"className":28850},[39],[41,28852,28853],{"xmlns":43},[46,28854,28855,28859],{},[49,28856,28857],{},[55,28858,4455],{},[82,28860,4455],{"encoding":84},[28,28862,28864],{"className":28863,"ariaHidden":71},[89],[28,28865,28867,28870],{"className":28866},[93],[28,28868],{"className":28869,"style":1102},[97],[28,28871,4455],{"className":28872},[102,106]," is the Hamiltonian ",[28,28875,28877,28890],{"className":28876},[35],[28,28878,28880],{"className":28879},[39],[41,28881,28882],{"xmlns":43},[46,28883,28884,28888],{},[49,28885,28886],{},[55,28887,2263],{},[82,28889,2263],{"encoding":84},[28,28891,28893],{"className":28892,"ariaHidden":71},[89],[28,28894,28896,28899],{"className":28895},[93],[28,28897],{"className":28898,"style":1102},[97],[28,28900,2263],{"className":28901,"style":4040},[102,106],", we'll refer to the corresponding space as the power Krylov space ",[28,28904,28906,28924],{"className":28905},[35],[28,28907,28909],{"className":28908},[39],[41,28910,28911],{"xmlns":43},[46,28912,28913,28921],{},[49,28914,28915],{},[52,28916,28917,28919],{},[55,28918,28350],{"mathvariant":11377},[55,28920,57],{},[82,28922,28923],{"encoding":84},"\\mathcal{K}_P",[28,28925,28927],{"className":28926,"ariaHidden":71},[89],[28,28928,28930,28933],{"className":28929},[93],[28,28931],{"className":28932,"style":585},[97],[28,28934,28936,28939],{"className":28935},[102],[28,28937,28350],{"className":28938,"style":28371},[102,11469],[28,28940,28942],{"className":28941},[111],[28,28943,28945,28966],{"className":28944},[115,116],[28,28946,28948,28963],{"className":28947},[120],[28,28949,28951],{"className":28950,"style":1364},[124],[28,28952,28954,28957],{"style":28953},"top:-2.55em;margin-left:-0.0144em;margin-right:0.05em;",[28,28955],{"className":28956,"style":133},[132],[28,28958,28960],{"className":28959},[137,138,139,140],[28,28961,57],{"className":28962,"style":107},[102,106,140],[28,28964,148],{"className":28965},[147],[28,28967,28969],{"className":28968},[120],[28,28970,28972],{"className":28971,"style":155},[124],[28,28973],{},". In the case where ",[28,28976,28978,28991],{"className":28977},[35],[28,28979,28981],{"className":28980},[39],[41,28982,28983],{"xmlns":43},[46,28984,28985,28989],{},[49,28986,28987],{},[55,28988,4455],{},[82,28990,4455],{"encoding":84},[28,28992,28994],{"className":28993,"ariaHidden":71},[89],[28,28995,28997,29000],{"className":28996},[93],[28,28998],{"className":28999,"style":1102},[97],[28,29001,4455],{"className":29002},[102,106]," is the time-evolution operator generated by the Hamiltonian ",[28,29005,29007,29038],{"className":29006},[35],[28,29008,29010],{"className":29009},[39],[41,29011,29012],{"xmlns":43},[46,29013,29014,29035],{},[49,29015,29016,29019,29021],{},[55,29017,29018],{},"U",[62,29020,845],{},[2776,29022,29023,29025],{},[55,29024,60],{},[49,29026,29027,29029,29031,29033],{},[62,29028,1213],{},[55,29030,3256],{},[55,29032,2263],{},[55,29034,3971],{},[82,29036,29037],{"encoding":84},"U=e^{-iHt}",[28,29039,29041,29060],{"className":29040,"ariaHidden":71},[89],[28,29042,29044,29047,29051,29054,29057],{"className":29043},[93],[28,29045],{"className":29046,"style":1102},[97],[28,29048,29018],{"className":29049,"style":29050},[102,106],"margin-right:0.109em;",[28,29052],{"className":29053,"style":913},[173],[28,29055,845],{"className":29056},[917],[28,29058],{"className":29059,"style":913},[173],[28,29061,29063,29067],{"className":29062},[93],[28,29064],{"className":29065,"style":29066},[97],"height:0.8413em;",[28,29068,29070,29073],{"className":29069},[102],[28,29071,60],{"className":29072},[102,106],[28,29074,29076],{"className":29075},[111],[28,29077,29079],{"className":29078},[115],[28,29080,29082],{"className":29081},[120],[28,29083,29085],{"className":29084,"style":29066},[124],[28,29086,29087,29090],{"style":1726},[28,29088],{"className":29089,"style":133},[132],[28,29091,29093],{"className":29092},[137,138,139,140],[28,29094,29096,29099,29102,29105],{"className":29095},[102,140],[28,29097,1213],{"className":29098},[102,140],[28,29100,3256],{"className":29101},[102,106,140],[28,29103,2263],{"className":29104,"style":4040},[102,106,140],[28,29106,3971],{"className":29107},[102,106,140],", we'll refer to the space as the unitary Krylov space ",[28,29110,29112,29130],{"className":29111},[35],[28,29113,29115],{"className":29114},[39],[41,29116,29117],{"xmlns":43},[46,29118,29119,29127],{},[49,29120,29121],{},[52,29122,29123,29125],{},[55,29124,28350],{"mathvariant":11377},[55,29126,29018],{},[82,29128,29129],{"encoding":84},"\\mathcal{K}_U",[28,29131,29133],{"className":29132,"ariaHidden":71},[89],[28,29134,29136,29139],{"className":29135},[93],[28,29137],{"className":29138,"style":585},[97],[28,29140,29142,29145],{"className":29141},[102],[28,29143,28350],{"className":29144,"style":28371},[102,11469],[28,29146,29148],{"className":29147},[111],[28,29149,29151,29171],{"className":29150},[115,116],[28,29152,29154,29168],{"className":29153},[120],[28,29155,29157],{"className":29156,"style":1364},[124],[28,29158,29159,29162],{"style":28953},[28,29160],{"className":29161,"style":133},[132],[28,29163,29165],{"className":29164},[137,138,139,140],[28,29166,29018],{"className":29167,"style":29050},[102,106,140],[28,29169,148],{"className":29170},[147],[28,29172,29174],{"className":29173},[120],[28,29175,29177],{"className":29176,"style":155},[124],[28,29178],{},". The power Krylov subspace that we use classically cannot be generated directly on a quantum computer as ",[28,29181,29183,29196],{"className":29182},[35],[28,29184,29186],{"className":29185},[39],[41,29187,29188],{"xmlns":43},[46,29189,29190,29194],{},[49,29191,29192],{},[55,29193,2263],{},[82,29195,2263],{"encoding":84},[28,29197,29199],{"className":29198,"ariaHidden":71},[89],[28,29200,29202,29205],{"className":29201},[93],[28,29203],{"className":29204,"style":1102},[97],[28,29206,2263],{"className":29207,"style":4040},[102,106]," is not a unitary operator. Instead, we can use the time-evolution operator ",[28,29210,29212,29242],{"className":29211},[35],[28,29213,29215],{"className":29214},[39],[41,29216,29217],{"xmlns":43},[46,29218,29219,29239],{},[49,29220,29221,29223,29225],{},[55,29222,29018],{},[62,29224,845],{},[2776,29226,29227,29229],{},[55,29228,60],{},[49,29230,29231,29233,29235,29237],{},[62,29232,1213],{},[55,29234,3256],{},[55,29236,2263],{},[55,29238,3971],{},[82,29240,29241],{"encoding":84},"U = e^{-iHt}",[28,29243,29245,29263],{"className":29244,"ariaHidden":71},[89],[28,29246,29248,29251,29254,29257,29260],{"className":29247},[93],[28,29249],{"className":29250,"style":1102},[97],[28,29252,29018],{"className":29253,"style":29050},[102,106],[28,29255],{"className":29256,"style":913},[173],[28,29258,845],{"className":29259},[917],[28,29261],{"className":29262,"style":913},[173],[28,29264,29266,29269],{"className":29265},[93],[28,29267],{"className":29268,"style":29066},[97],[28,29270,29272,29275],{"className":29271},[102],[28,29273,60],{"className":29274},[102,106],[28,29276,29278],{"className":29277},[111],[28,29279,29281],{"className":29280},[115],[28,29282,29284],{"className":29283},[120],[28,29285,29287],{"className":29286,"style":29066},[124],[28,29288,29289,29292],{"style":1726},[28,29290],{"className":29291,"style":133},[132],[28,29293,29295],{"className":29294},[137,138,139,140],[28,29296,29298,29301,29304,29307],{"className":29297},[102,140],[28,29299,1213],{"className":29300},[102,140],[28,29302,3256],{"className":29303},[102,106,140],[28,29305,2263],{"className":29306,"style":4040},[102,106,140],[28,29308,3971],{"className":29309},[102,106,140]," which can be shown to give similar ",[19,29312,29313],{"href":27049},"convergence guarantees"," as the power method. Powers of ",[28,29316,29318,29331],{"className":29317},[35],[28,29319,29321],{"className":29320},[39],[41,29322,29323],{"xmlns":43},[46,29324,29325,29329],{},[49,29326,29327],{},[55,29328,29018],{},[82,29330,29018],{"encoding":84},[28,29332,29334],{"className":29333,"ariaHidden":71},[89],[28,29335,29337,29340],{"className":29336},[93],[28,29338],{"className":29339,"style":1102},[97],[28,29341,29018],{"className":29342,"style":29050},[102,106]," then become different time steps ",[28,29345,29347,29388],{"className":29346},[35],[28,29348,29350],{"className":29349},[39],[41,29351,29352],{"xmlns":43},[46,29353,29354,29385],{},[49,29355,29356,29363,29365],{},[2776,29357,29358,29360],{},[55,29359,29018],{},[55,29361,29362],{},"k",[62,29364,845],{},[2776,29366,29367,29369],{},[55,29368,60],{},[49,29370,29371,29373,29375,29377,29379,29381,29383],{},[62,29372,1213],{},[55,29374,3256],{},[55,29376,2263],{},[62,29378,65],{"stretchy":64},[55,29380,29362],{},[55,29382,3971],{},[62,29384,78],{"stretchy":64},[82,29386,29387],{"encoding":84},"U^k = e^{-iH(kt)}",[28,29389,29391,29437],{"className":29390,"ariaHidden":71},[89],[28,29392,29394,29398,29428,29431,29434],{"className":29393},[93],[28,29395],{"className":29396,"style":29397},[97],"height:0.8491em;",[28,29399,29401,29404],{"className":29400},[102],[28,29402,29018],{"className":29403,"style":29050},[102,106],[28,29405,29407],{"className":29406},[111],[28,29408,29410],{"className":29409},[115],[28,29411,29413],{"className":29412},[120],[28,29414,29416],{"className":29415,"style":29397},[124],[28,29417,29418,29421],{"style":1726},[28,29419],{"className":29420,"style":133},[132],[28,29422,29424],{"className":29423},[137,138,139,140],[28,29425,29362],{"className":29426,"style":29427},[102,106,140],"margin-right:0.0315em;",[28,29429],{"className":29430,"style":913},[173],[28,29432,845],{"className":29433},[917],[28,29435],{"className":29436,"style":913},[173],[28,29438,29440,29444],{"className":29439},[93],[28,29441],{"className":29442,"style":29443},[97],"height:0.888em;",[28,29445,29447,29450],{"className":29446},[102],[28,29448,60],{"className":29449},[102,106],[28,29451,29453],{"className":29452},[111],[28,29454,29456],{"className":29455},[115],[28,29457,29459],{"className":29458},[120],[28,29460,29462],{"className":29461,"style":29443},[124],[28,29463,29464,29467],{"style":1726},[28,29465],{"className":29466,"style":133},[132],[28,29468,29470],{"className":29469},[137,138,139,140],[28,29471,29473,29476,29479,29482,29485,29488,29491],{"className":29472},[102,140],[28,29474,1213],{"className":29475},[102,140],[28,29477,3256],{"className":29478},[102,106,140],[28,29480,2263],{"className":29481,"style":4040},[102,106,140],[28,29483,65],{"className":29484},[161,140],[28,29486,29362],{"className":29487,"style":29427},[102,106,140],[28,29489,3971],{"className":29490},[102,106,140],[28,29492,78],{"className":29493},[182,140],[28,29495,29497],{"className":29496},[31],[28,29498,29500,29586],{"className":29499},[35],[28,29501,29503],{"className":29502},[39],[41,29504,29505],{"xmlns":43,"display":44},[46,29506,29507,29583],{},[49,29508,29509,29517,29519],{},[1670,29510,29511,29513,29515],{},[55,29512,28350],{"mathvariant":11377},[55,29514,29018],{},[55,29516,16959],{},[62,29518,845],{},[49,29520,29521,29523,29525,29527,29529,29531,29533,29535,29537,29539,29541,29547,29549,29551,29553,29555,29557,29559,29561,29563,29575,29577,29579,29581],{},[62,29522,11385],{"fence":71},[55,29524,296],{"mathvariant":1088},[55,29526,394],{},[62,29528,302],{"stretchy":64},[62,29530,72],{"separator":71},[55,29532,29018],{},[55,29534,296],{"mathvariant":1088},[55,29536,394],{},[62,29538,302],{"stretchy":64},[62,29540,72],{"separator":71},[2776,29542,29543,29545],{},[55,29544,29018],{},[650,29546,848],{},[55,29548,296],{"mathvariant":1088},[55,29550,394],{},[62,29552,302],{"stretchy":64},[62,29554,72],{"separator":71},[55,29556,2106],{"mathvariant":1088},[55,29558,2106],{"mathvariant":1088},[55,29560,2106],{"mathvariant":1088},[62,29562,72],{"separator":71},[2776,29564,29565,29567],{},[55,29566,29018],{},[49,29568,29569,29571,29573],{},[55,29570,16959],{},[62,29572,1213],{},[650,29574,3200],{},[55,29576,296],{"mathvariant":1088},[55,29578,394],{},[62,29580,302],{"stretchy":64},[62,29582,11404],{"fence":71},[82,29584,29585],{"encoding":84},"\\mathcal{K}_U^r = \\left\\{ \\vert \\psi \\rangle, U \\vert \\psi \\rangle, U^2 \\vert \\psi \\rangle, ..., U^{r-1} \\vert \\psi \\rangle \\right\\}",[28,29587,29589,29657],{"className":29588,"ariaHidden":71},[89],[28,29590,29592,29596,29648,29651,29654],{"className":29591},[93],[28,29593],{"className":29594,"style":29595},[97],"height:0.9614em;vertical-align:-0.247em;",[28,29597,29599,29602],{"className":29598},[102],[28,29600,28350],{"className":29601,"style":28371},[102,11469],[28,29603,29605],{"className":29604},[111],[28,29606,29608,29640],{"className":29607},[115,116],[28,29609,29611,29637],{"className":29610},[120],[28,29612,29614,29626],{"className":29613,"style":28646},[124],[28,29615,29617,29620],{"style":29616},"top:-2.453em;margin-left:-0.0144em;margin-right:0.05em;",[28,29618],{"className":29619,"style":133},[132],[28,29621,29623],{"className":29622},[137,138,139,140],[28,29624,29018],{"className":29625,"style":29050},[102,106,140],[28,29627,29628,29631],{"style":2839},[28,29629],{"className":29630,"style":133},[132],[28,29632,29634],{"className":29633},[137,138,139,140],[28,29635,16959],{"className":29636,"style":9403},[102,106,140],[28,29638,148],{"className":29639},[147],[28,29641,29643],{"className":29642},[120],[28,29644,29646],{"className":29645,"style":8066},[124],[28,29647],{},[28,29649],{"className":29650,"style":913},[173],[28,29652,845],{"className":29653},[917],[28,29655],{"className":29656,"style":913},[173],[28,29658,29660,29663],{"className":29659},[93],[28,29661],{"className":29662,"style":28691},[97],[28,29664,29666,29672,29675,29678,29681,29684,29687,29690,29693,29696,29699,29702,29705,29734,29737,29740,29743,29746,29749,29752,29755,29758,29796,29799,29802,29805],{"className":29665},[3398],[28,29667,29669],{"className":29668,"style":3403},[161,3402],[28,29670,11385],{"className":29671},[3407,28701],[28,29673,296],{"className":29674},[102],[28,29676,394],{"className":29677,"style":321},[102,106],[28,29679,302],{"className":29680},[182],[28,29682,72],{"className":29683},[169],[28,29685],{"className":29686,"style":174},[173],[28,29688,29018],{"className":29689,"style":29050},[102,106],[28,29691,296],{"className":29692},[102],[28,29694,394],{"className":29695,"style":321},[102,106],[28,29697,302],{"className":29698},[182],[28,29700,72],{"className":29701},[169],[28,29703],{"className":29704,"style":174},[173],[28,29706,29708,29711],{"className":29707},[102],[28,29709,29018],{"className":29710,"style":29050},[102,106],[28,29712,29714],{"className":29713},[111],[28,29715,29717],{"className":29716},[115],[28,29718,29720],{"className":29719},[120],[28,29721,29723],{"className":29722,"style":2836},[124],[28,29724,29725,29728],{"style":2839},[28,29726],{"className":29727,"style":133},[132],[28,29729,29731],{"className":29730},[137,138,139,140],[28,29732,848],{"className":29733},[102,140],[28,29735,296],{"className":29736},[102],[28,29738,394],{"className":29739,"style":321},[102,106],[28,29741,302],{"className":29742},[182],[28,29744,72],{"className":29745},[169],[28,29747],{"className":29748,"style":174},[173],[28,29750,28782],{"className":29751},[102],[28,29753,72],{"className":29754},[169],[28,29756],{"className":29757,"style":174},[173],[28,29759,29761,29764],{"className":29760},[102],[28,29762,29018],{"className":29763,"style":29050},[102,106],[28,29765,29767],{"className":29766},[111],[28,29768,29770],{"className":29769},[115],[28,29771,29773],{"className":29772},[120],[28,29774,29776],{"className":29775,"style":2836},[124],[28,29777,29778,29781],{"style":2839},[28,29779],{"className":29780,"style":133},[132],[28,29782,29784],{"className":29783},[137,138,139,140],[28,29785,29787,29790,29793],{"className":29786},[102,140],[28,29788,16959],{"className":29789,"style":9403},[102,106,140],[28,29791,1213],{"className":29792},[1261,140],[28,29794,3200],{"className":29795},[102,140],[28,29797,296],{"className":29798},[102],[28,29800,394],{"className":29801,"style":321},[102,106],[28,29803,302],{"className":29804},[182],[28,29806,29808],{"className":29807,"style":3403},[182,3402],[28,29809,11404],{"className":29810},[3407,28701],[12,29812,29813],{},"See the Appendix for a detailed derivation of how the unitary Krylov space allows to represents low-energy eigenstates accurately.",[28327,29815,29817],{"id":29816},"krylov-quantum-diagonalization-algorithm","Krylov quantum diagonalization algorithm",[12,29819,29820,29821,29849,29850,29919,29920,29989,29990,30055,30056,30116],{},"Given an Hamiltonian ",[28,29822,29824,29837],{"className":29823},[35],[28,29825,29827],{"className":29826},[39],[41,29828,29829],{"xmlns":43},[46,29830,29831,29835],{},[49,29832,29833],{},[55,29834,2263],{},[82,29836,2263],{"encoding":84},[28,29838,29840],{"className":29839,"ariaHidden":71},[89],[28,29841,29843,29846],{"className":29842},[93],[28,29844],{"className":29845,"style":1102},[97],[28,29847,2263],{"className":29848,"style":4040},[102,106]," that we wish to diagonalize, first we consider the corresponding unitary Krylov space ",[28,29851,29853,29870],{"className":29852},[35],[28,29854,29856],{"className":29855},[39],[41,29857,29858],{"xmlns":43},[46,29859,29860,29868],{},[49,29861,29862],{},[52,29863,29864,29866],{},[55,29865,28350],{"mathvariant":11377},[55,29867,29018],{},[82,29869,29129],{"encoding":84},[28,29871,29873],{"className":29872,"ariaHidden":71},[89],[28,29874,29876,29879],{"className":29875},[93],[28,29877],{"className":29878,"style":585},[97],[28,29880,29882,29885],{"className":29881},[102],[28,29883,28350],{"className":29884,"style":28371},[102,11469],[28,29886,29888],{"className":29887},[111],[28,29889,29891,29911],{"className":29890},[115,116],[28,29892,29894,29908],{"className":29893},[120],[28,29895,29897],{"className":29896,"style":1364},[124],[28,29898,29899,29902],{"style":28953},[28,29900],{"className":29901,"style":133},[132],[28,29903,29905],{"className":29904},[137,138,139,140],[28,29906,29018],{"className":29907,"style":29050},[102,106,140],[28,29909,148],{"className":29910},[147],[28,29912,29914],{"className":29913},[120],[28,29915,29917],{"className":29916,"style":155},[124],[28,29918],{},". The goal is to find a compact representation of the Hamiltonian in ",[28,29921,29923,29940],{"className":29922},[35],[28,29924,29926],{"className":29925},[39],[41,29927,29928],{"xmlns":43},[46,29929,29930,29938],{},[49,29931,29932],{},[52,29933,29934,29936],{},[55,29935,28350],{"mathvariant":11377},[55,29937,29018],{},[82,29939,29129],{"encoding":84},[28,29941,29943],{"className":29942,"ariaHidden":71},[89],[28,29944,29946,29949],{"className":29945},[93],[28,29947],{"className":29948,"style":585},[97],[28,29950,29952,29955],{"className":29951},[102],[28,29953,28350],{"className":29954,"style":28371},[102,11469],[28,29956,29958],{"className":29957},[111],[28,29959,29961,29981],{"className":29960},[115,116],[28,29962,29964,29978],{"className":29963},[120],[28,29965,29967],{"className":29966,"style":1364},[124],[28,29968,29969,29972],{"style":28953},[28,29970],{"className":29971,"style":133},[132],[28,29973,29975],{"className":29974},[137,138,139,140],[28,29976,29018],{"className":29977,"style":29050},[102,106,140],[28,29979,148],{"className":29980},[147],[28,29982,29984],{"className":29983},[120],[28,29985,29987],{"className":29986,"style":155},[124],[28,29988],{},", which we'll refer to as ",[28,29991,29993,30012],{"className":29992},[35],[28,29994,29996],{"className":29995},[39],[41,29997,29998],{"xmlns":43},[46,29999,30000,30009],{},[49,30001,30002],{},[10155,30003,30004,30006],{"accent":71},[55,30005,2263],{},[62,30007,30008],{},"~",[82,30010,30011],{"encoding":84},"\\tilde{H}",[28,30013,30015],{"className":30014,"ariaHidden":71},[89],[28,30016,30018,30022],{"className":30017},[93],[28,30019],{"className":30020,"style":30021},[97],"height:0.9202em;",[28,30023,30025],{"className":30024},[102,10255],[28,30026,30028],{"className":30027},[115],[28,30029,30031],{"className":30030},[120],[28,30032,30034,30042],{"className":30033,"style":30021},[124],[28,30035,30036,30039],{"style":10025},[28,30037],{"className":30038,"style":3434},[132],[28,30040,2263],{"className":30041,"style":4040},[102,106],[28,30043,30045,30048],{"style":30044},"top:-3.6023em;",[28,30046],{"className":30047,"style":3434},[132],[28,30049,30052],{"className":30050,"style":30051},[10282],"left:-0.1944em;",[28,30053,30008],{"className":30054},[102],". The matrix elements of ",[28,30057,30059,30076],{"className":30058},[35],[28,30060,30062],{"className":30061},[39],[41,30063,30064],{"xmlns":43},[46,30065,30066,30074],{},[49,30067,30068],{},[10155,30069,30070,30072],{"accent":71},[55,30071,2263],{},[62,30073,30008],{},[82,30075,30011],{"encoding":84},[28,30077,30079],{"className":30078,"ariaHidden":71},[89],[28,30080,30082,30085],{"className":30081},[93],[28,30083],{"className":30084,"style":30021},[97],[28,30086,30088],{"className":30087},[102,10255],[28,30089,30091],{"className":30090},[115],[28,30092,30094],{"className":30093},[120],[28,30095,30097,30105],{"className":30096,"style":30021},[124],[28,30098,30099,30102],{"style":10025},[28,30100],{"className":30101,"style":3434},[132],[28,30103,2263],{"className":30104,"style":4040},[102,106],[28,30106,30107,30110],{"style":30044},[28,30108],{"className":30109,"style":3434},[132],[28,30111,30113],{"className":30112,"style":30051},[10282],[28,30114,30008],{"className":30115},[102],", the projection of the Hamiltonian in the Krylov space, can be calculated by calculating the following expectation values",[28,30118,30120],{"className":30119},[31],[28,30121,30123,30176],{"className":30122},[35],[28,30124,30126],{"className":30125},[39],[41,30127,30128],{"xmlns":43,"display":44},[46,30129,30130,30173],{},[49,30131,30132,30147,30149,30151,30157,30159,30161,30163,30169,30171],{},[52,30133,30134,30140],{},[10155,30135,30136,30138],{"accent":71},[55,30137,2263],{},[62,30139,30008],{},[49,30141,30142,30145],{},[55,30143,30144],{},"m",[55,30146,7106],{},[62,30148,845],{},[62,30150,2948],{"stretchy":64},[52,30152,30153,30155],{},[55,30154,394],{},[55,30156,30144],{},[55,30158,296],{"mathvariant":1088},[55,30160,2263],{},[55,30162,296],{"mathvariant":1088},[52,30164,30165,30167],{},[55,30166,394],{},[55,30168,7106],{},[62,30170,302],{"stretchy":64},[62,30172,845],{},[82,30174,30175],{"encoding":84},"\\tilde{H}_{mn} = \\langle \\psi_m \\vert H \\vert \\psi_n \\rangle =",[28,30177,30179,30266],{"className":30178,"ariaHidden":71},[89],[28,30180,30182,30186,30257,30260,30263],{"className":30181},[93],[28,30183],{"className":30184,"style":30185},[97],"height:1.0702em;vertical-align:-0.15em;",[28,30187,30189,30220],{"className":30188},[102],[28,30190,30192],{"className":30191},[102,10255],[28,30193,30195],{"className":30194},[115],[28,30196,30198],{"className":30197},[120],[28,30199,30201,30209],{"className":30200,"style":30021},[124],[28,30202,30203,30206],{"style":10025},[28,30204],{"className":30205,"style":3434},[132],[28,30207,2263],{"className":30208,"style":4040},[102,106],[28,30210,30211,30214],{"style":30044},[28,30212],{"className":30213,"style":3434},[132],[28,30215,30217],{"className":30216,"style":30051},[10282],[28,30218,30008],{"className":30219},[102],[28,30221,30223],{"className":30222},[111],[28,30224,30226,30249],{"className":30225},[115,116],[28,30227,30229,30246],{"className":30228},[120],[28,30230,30232],{"className":30231,"style":125},[124],[28,30233,30234,30237],{"style":4522},[28,30235],{"className":30236,"style":133},[132],[28,30238,30240],{"className":30239},[137,138,139,140],[28,30241,30243],{"className":30242},[102,140],[28,30244,650],{"className":30245},[102,106,140],[28,30247,148],{"className":30248},[147],[28,30250,30252],{"className":30251},[120],[28,30253,30255],{"className":30254,"style":155},[124],[28,30256],{},[28,30258],{"className":30259,"style":913},[173],[28,30261,845],{"className":30262},[917],[28,30264],{"className":30265,"style":913},[173],[28,30267,30269,30272,30275,30315,30318,30321,30324,30364,30367,30370],{"className":30268},[93],[28,30270],{"className":30271,"style":98},[97],[28,30273,2948],{"className":30274},[161],[28,30276,30278,30281],{"className":30277},[102],[28,30279,394],{"className":30280,"style":321},[102,106],[28,30282,30284],{"className":30283},[111],[28,30285,30287,30307],{"className":30286},[115,116],[28,30288,30290,30304],{"className":30289},[120],[28,30291,30293],{"className":30292,"style":125},[124],[28,30294,30295,30298],{"style":801},[28,30296],{"className":30297,"style":133},[132],[28,30299,30301],{"className":30300},[137,138,139,140],[28,30302,30144],{"className":30303},[102,106,140],[28,30305,148],{"className":30306},[147],[28,30308,30310],{"className":30309},[120],[28,30311,30313],{"className":30312,"style":155},[124],[28,30314],{},[28,30316,296],{"className":30317},[102],[28,30319,2263],{"className":30320,"style":4040},[102,106],[28,30322,296],{"className":30323},[102],[28,30325,30327,30330],{"className":30326},[102],[28,30328,394],{"className":30329,"style":321},[102,106],[28,30331,30333],{"className":30332},[111],[28,30334,30336,30356],{"className":30335},[115,116],[28,30337,30339,30353],{"className":30338},[120],[28,30340,30342],{"className":30341,"style":125},[124],[28,30343,30344,30347],{"style":801},[28,30345],{"className":30346,"style":133},[132],[28,30348,30350],{"className":30349},[137,138,139,140],[28,30351,7106],{"className":30352},[102,106,140],[28,30354,148],{"className":30355},[147],[28,30357,30359],{"className":30358},[120],[28,30360,30362],{"className":30361,"style":155},[124],[28,30363],{},[28,30365,302],{"className":30366},[182],[28,30368],{"className":30369,"style":913},[173],[28,30371,845],{"className":30372},[917],[28,30374,30376],{"className":30375},[31],[28,30377,30379,30441],{"className":30378},[35],[28,30380,30382],{"className":30381},[39],[41,30383,30384],{"xmlns":43,"display":44},[46,30385,30386,30438],{},[49,30387,30388,30390,30392,30394,30396,30412,30414,30432,30434,30436],{},[62,30389,845],{},[62,30391,2948],{"stretchy":64},[55,30393,394],{},[55,30395,296],{"mathvariant":1088},[2776,30397,30398,30400],{},[55,30399,60],{},[49,30401,30402,30404,30406],{},[55,30403,3256],{},[55,30405,2263],{},[52,30407,30408,30410],{},[55,30409,3971],{},[55,30411,30144],{},[55,30413,2263],{},[2776,30415,30416,30418],{},[55,30417,60],{},[49,30419,30420,30422,30424,30426],{},[62,30421,1213],{},[55,30423,3256],{},[55,30425,2263],{},[52,30427,30428,30430],{},[55,30429,3971],{},[55,30431,7106],{},[55,30433,296],{"mathvariant":1088},[55,30435,394],{},[62,30437,302],{"stretchy":64},[82,30439,30440],{"encoding":84},"= \\langle \\psi \\vert  e^{i H t_m}   H e^{-i H t_n} \\vert \\psi \\rangle",[28,30442,30444,30457],{"className":30443,"ariaHidden":71},[89],[28,30445,30447,30451,30454],{"className":30446},[93],[28,30448],{"className":30449,"style":30450},[97],"height:0.3669em;",[28,30452,845],{"className":30453},[917],[28,30455],{"className":30456,"style":913},[173],[28,30458,30460,30464,30467,30470,30473,30554,30557,30635,30638,30641],{"className":30459},[93],[28,30461],{"className":30462,"style":30463},[97],"height:1.1413em;vertical-align:-0.25em;",[28,30465,2948],{"className":30466},[161],[28,30468,394],{"className":30469,"style":321},[102,106],[28,30471,296],{"className":30472},[102],[28,30474,30476,30479],{"className":30475},[102],[28,30477,60],{"className":30478},[102,106],[28,30480,30482],{"className":30481},[111],[28,30483,30485],{"className":30484},[115],[28,30486,30488],{"className":30487},[120],[28,30489,30492],{"className":30490,"style":30491},[124],"height:0.8913em;",[28,30493,30494,30497],{"style":2839},[28,30495],{"className":30496,"style":133},[132],[28,30498,30500],{"className":30499},[137,138,139,140],[28,30501,30503,30506,30509],{"className":30502},[102,140],[28,30504,3256],{"className":30505},[102,106,140],[28,30507,2263],{"className":30508,"style":4040},[102,106,140],[28,30510,30512,30515],{"className":30511},[102,140],[28,30513,3971],{"className":30514},[102,106,140],[28,30516,30518],{"className":30517},[111],[28,30519,30521,30545],{"className":30520},[115,116],[28,30522,30524,30542],{"className":30523},[120],[28,30525,30528],{"className":30526,"style":30527},[124],"height:0.1645em;",[28,30529,30531,30535],{"style":30530},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[28,30532],{"className":30533,"style":30534},[132],"height:2.5em;",[28,30536,30539],{"className":30537},[137,30538,28701,140],"reset-size3",[28,30540,30144],{"className":30541},[102,106,140],[28,30543,148],{"className":30544},[147],[28,30546,30548],{"className":30547},[120],[28,30549,30552],{"className":30550,"style":30551},[124],"height:0.143em;",[28,30553],{},[28,30555,2263],{"className":30556,"style":4040},[102,106],[28,30558,30560,30563],{"className":30559},[102],[28,30561,60],{"className":30562},[102,106],[28,30564,30566],{"className":30565},[111],[28,30567,30569],{"className":30568},[115],[28,30570,30572],{"className":30571},[120],[28,30573,30575],{"className":30574,"style":30491},[124],[28,30576,30577,30580],{"style":2839},[28,30578],{"className":30579,"style":133},[132],[28,30581,30583],{"className":30582},[137,138,139,140],[28,30584,30586,30589,30592,30595],{"className":30585},[102,140],[28,30587,1213],{"className":30588},[102,140],[28,30590,3256],{"className":30591},[102,106,140],[28,30593,2263],{"className":30594,"style":4040},[102,106,140],[28,30596,30598,30601],{"className":30597},[102,140],[28,30599,3971],{"className":30600},[102,106,140],[28,30602,30604],{"className":30603},[111],[28,30605,30607,30627],{"className":30606},[115,116],[28,30608,30610,30624],{"className":30609},[120],[28,30611,30613],{"className":30612,"style":30527},[124],[28,30614,30615,30618],{"style":30530},[28,30616],{"className":30617,"style":30534},[132],[28,30619,30621],{"className":30620},[137,30538,28701,140],[28,30622,7106],{"className":30623},[102,106,140],[28,30625,148],{"className":30626},[147],[28,30628,30630],{"className":30629},[120],[28,30631,30633],{"className":30632,"style":30551},[124],[28,30634],{},[28,30636,296],{"className":30637},[102],[28,30639,394],{"className":30640,"style":321},[102,106],[28,30642,302],{"className":30643},[182],[28,30645,30647],{"className":30646},[31],[28,30648,30650,30712],{"className":30649},[35],[28,30651,30653],{"className":30652},[39],[41,30654,30655],{"xmlns":43,"display":44},[46,30656,30657,30709],{},[49,30658,30659,30661,30663,30665,30667,30683,30685,30703,30705,30707],{},[62,30660,845],{},[62,30662,2948],{"stretchy":64},[55,30664,394],{},[55,30666,296],{"mathvariant":1088},[2776,30668,30669,30671],{},[55,30670,60],{},[49,30672,30673,30675,30677,30679,30681],{},[55,30674,3256],{},[55,30676,2263],{},[55,30678,30144],{},[55,30680,24201],{},[55,30682,3971],{},[55,30684,2263],{},[2776,30686,30687,30689],{},[55,30688,60],{},[49,30690,30691,30693,30695,30697,30699,30701],{},[62,30692,1213],{},[55,30694,3256],{},[55,30696,2263],{},[55,30698,7106],{},[55,30700,24201],{},[55,30702,3971],{},[55,30704,296],{"mathvariant":1088},[55,30706,394],{},[62,30708,302],{"stretchy":64},[82,30710,30711],{"encoding":84},"= \\langle \\psi \\vert  e^{i H m dt}   H e^{-i H n dt} \\vert \\psi \\rangle",[28,30713,30715,30727],{"className":30714,"ariaHidden":71},[89],[28,30716,30718,30721,30724],{"className":30717},[93],[28,30719],{"className":30720,"style":30450},[97],[28,30722,845],{"className":30723},[917],[28,30725],{"className":30726,"style":913},[173],[28,30728,30730,30734,30737,30740,30743,30788,30791,30838,30841,30844],{"className":30729},[93],[28,30731],{"className":30732,"style":30733},[97],"height:1.1491em;vertical-align:-0.25em;",[28,30735,2948],{"className":30736},[161],[28,30738,394],{"className":30739,"style":321},[102,106],[28,30741,296],{"className":30742},[102],[28,30744,30746,30749],{"className":30745},[102],[28,30747,60],{"className":30748},[102,106],[28,30750,30752],{"className":30751},[111],[28,30753,30755],{"className":30754},[115],[28,30756,30758],{"className":30757},[120],[28,30759,30762],{"className":30760,"style":30761},[124],"height:0.8991em;",[28,30763,30764,30767],{"style":2839},[28,30765],{"className":30766,"style":133},[132],[28,30768,30770],{"className":30769},[137,138,139,140],[28,30771,30773,30776,30779,30782,30785],{"className":30772},[102,140],[28,30774,3256],{"className":30775},[102,106,140],[28,30777,2263],{"className":30778,"style":4040},[102,106,140],[28,30780,30144],{"className":30781},[102,106,140],[28,30783,24201],{"className":30784},[102,106,140],[28,30786,3971],{"className":30787},[102,106,140],[28,30789,2263],{"className":30790,"style":4040},[102,106],[28,30792,30794,30797],{"className":30793},[102],[28,30795,60],{"className":30796},[102,106],[28,30798,30800],{"className":30799},[111],[28,30801,30803],{"className":30802},[115],[28,30804,30806],{"className":30805},[120],[28,30807,30809],{"className":30808,"style":30761},[124],[28,30810,30811,30814],{"style":2839},[28,30812],{"className":30813,"style":133},[132],[28,30815,30817],{"className":30816},[137,138,139,140],[28,30818,30820,30823,30826,30829,30832,30835],{"className":30819},[102,140],[28,30821,1213],{"className":30822},[102,140],[28,30824,3256],{"className":30825},[102,106,140],[28,30827,2263],{"className":30828,"style":4040},[102,106,140],[28,30830,7106],{"className":30831},[102,106,140],[28,30833,24201],{"className":30834},[102,106,140],[28,30836,3971],{"className":30837},[102,106,140],[28,30839,296],{"className":30840},[102],[28,30842,394],{"className":30843,"style":321},[102,106],[28,30845,302],{"className":30846},[182],[12,30848,30849,30850,31059,31060,31163,31164,31198,31199,31268,31269,31297,31298,31350,31351,31379,31380,31436,31437,31497,31498,31558,31559],{},"Where ",[28,30851,30853,30901],{"className":30852},[35],[28,30854,30856],{"className":30855},[39],[41,30857,30858],{"xmlns":43},[46,30859,30860,30898],{},[49,30861,30862,30864,30870,30872,30874,30892,30894,30896],{},[55,30863,296],{"mathvariant":1088},[52,30865,30866,30868],{},[55,30867,394],{},[55,30869,7106],{},[62,30871,302],{"stretchy":64},[62,30873,845],{},[2776,30875,30876,30878],{},[55,30877,60],{},[49,30879,30880,30882,30884,30886],{},[62,30881,1213],{},[55,30883,3256],{},[55,30885,2263],{},[52,30887,30888,30890],{},[55,30889,3971],{},[55,30891,7106],{},[55,30893,296],{"mathvariant":1088},[55,30895,394],{},[62,30897,302],{"stretchy":64},[82,30899,30900],{"encoding":84},"\\vert \\psi_n \\rangle = e^{-i H t_n} \\vert \\psi \\rangle",[28,30902,30904,30965],{"className":30903,"ariaHidden":71},[89],[28,30905,30907,30910,30913,30953,30956,30959,30962],{"className":30906},[93],[28,30908],{"className":30909,"style":98},[97],[28,30911,296],{"className":30912},[102],[28,30914,30916,30919],{"className":30915},[102],[28,30917,394],{"className":30918,"style":321},[102,106],[28,30920,30922],{"className":30921},[111],[28,30923,30925,30945],{"className":30924},[115,116],[28,30926,30928,30942],{"className":30927},[120],[28,30929,30931],{"className":30930,"style":125},[124],[28,30932,30933,30936],{"style":801},[28,30934],{"className":30935,"style":133},[132],[28,30937,30939],{"className":30938},[137,138,139,140],[28,30940,7106],{"className":30941},[102,106,140],[28,30943,148],{"className":30944},[147],[28,30946,30948],{"className":30947},[120],[28,30949,30951],{"className":30950,"style":155},[124],[28,30952],{},[28,30954,302],{"className":30955},[182],[28,30957],{"className":30958,"style":913},[173],[28,30960,845],{"className":30961},[917],[28,30963],{"className":30964,"style":913},[173],[28,30966,30968,30972,31050,31053,31056],{"className":30967},[93],[28,30969],{"className":30970,"style":30971},[97],"height:1.0913em;vertical-align:-0.25em;",[28,30973,30975,30978],{"className":30974},[102],[28,30976,60],{"className":30977},[102,106],[28,30979,30981],{"className":30980},[111],[28,30982,30984],{"className":30983},[115],[28,30985,30987],{"className":30986},[120],[28,30988,30990],{"className":30989,"style":29066},[124],[28,30991,30992,30995],{"style":1726},[28,30993],{"className":30994,"style":133},[132],[28,30996,30998],{"className":30997},[137,138,139,140],[28,30999,31001,31004,31007,31010],{"className":31000},[102,140],[28,31002,1213],{"className":31003},[102,140],[28,31005,3256],{"className":31006},[102,106,140],[28,31008,2263],{"className":31009,"style":4040},[102,106,140],[28,31011,31013,31016],{"className":31012},[102,140],[28,31014,3971],{"className":31015},[102,106,140],[28,31017,31019],{"className":31018},[111],[28,31020,31022,31042],{"className":31021},[115,116],[28,31023,31025,31039],{"className":31024},[120],[28,31026,31028],{"className":31027,"style":30527},[124],[28,31029,31030,31033],{"style":30530},[28,31031],{"className":31032,"style":30534},[132],[28,31034,31036],{"className":31035},[137,30538,28701,140],[28,31037,7106],{"className":31038},[102,106,140],[28,31040,148],{"className":31041},[147],[28,31043,31045],{"className":31044},[120],[28,31046,31048],{"className":31047,"style":30551},[124],[28,31049],{},[28,31051,296],{"className":31052},[102],[28,31054,394],{"className":31055,"style":321},[102,106],[28,31057,302],{"className":31058},[182]," are the vectors of the unitary Krylov space and ",[28,31061,31063,31089],{"className":31062},[35],[28,31064,31066],{"className":31065},[39],[41,31067,31068],{"xmlns":43},[46,31069,31070,31086],{},[49,31071,31072,31078,31080,31082,31084],{},[52,31073,31074,31076],{},[55,31075,3971],{},[55,31077,7106],{},[62,31079,845],{},[55,31081,7106],{},[55,31083,24201],{},[55,31085,3971],{},[82,31087,31088],{"encoding":84},"t_n = n dt",[28,31090,31092,31148],{"className":31091,"ariaHidden":71},[89],[28,31093,31095,31099,31139,31142,31145],{"className":31094},[93],[28,31096],{"className":31097,"style":31098},[97],"height:0.7651em;vertical-align:-0.15em;",[28,31100,31102,31105],{"className":31101},[102],[28,31103,3971],{"className":31104},[102,106],[28,31106,31108],{"className":31107},[111],[28,31109,31111,31131],{"className":31110},[115,116],[28,31112,31114,31128],{"className":31113},[120],[28,31115,31117],{"className":31116,"style":125},[124],[28,31118,31119,31122],{"style":1367},[28,31120],{"className":31121,"style":133},[132],[28,31123,31125],{"className":31124},[137,138,139,140],[28,31126,7106],{"className":31127},[102,106,140],[28,31129,148],{"className":31130},[147],[28,31132,31134],{"className":31133},[120],[28,31135,31137],{"className":31136,"style":155},[124],[28,31138],{},[28,31140],{"className":31141,"style":913},[173],[28,31143,845],{"className":31144},[917],[28,31146],{"className":31147,"style":913},[173],[28,31149,31151,31154,31157,31160],{"className":31150},[93],[28,31152],{"className":31153,"style":9597},[97],[28,31155,7106],{"className":31156},[102,106],[28,31158,24201],{"className":31159},[102,106],[28,31161,3971],{"className":31162},[102,106]," are the multiples of the time step ",[28,31165,31167,31183],{"className":31166},[35],[28,31168,31170],{"className":31169},[39],[41,31171,31172],{"xmlns":43},[46,31173,31174,31180],{},[49,31175,31176,31178],{},[55,31177,24201],{},[55,31179,3971],{},[82,31181,31182],{"encoding":84},"dt",[28,31184,31186],{"className":31185,"ariaHidden":71},[89],[28,31187,31189,31192,31195],{"className":31188},[93],[28,31190],{"className":31191,"style":9597},[97],[28,31193,24201],{"className":31194},[102,106],[28,31196,3971],{"className":31197},[102,106]," 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 ",[28,31200,31202,31219],{"className":31201},[35],[28,31203,31205],{"className":31204},[39],[41,31206,31207],{"xmlns":43},[46,31208,31209,31217],{},[49,31210,31211],{},[52,31212,31213,31215],{},[55,31214,28350],{"mathvariant":11377},[55,31216,29018],{},[82,31218,29129],{"encoding":84},[28,31220,31222],{"className":31221,"ariaHidden":71},[89],[28,31223,31225,31228],{"className":31224},[93],[28,31226],{"className":31227,"style":585},[97],[28,31229,31231,31234],{"className":31230},[102],[28,31232,28350],{"className":31233,"style":28371},[102,11469],[28,31235,31237],{"className":31236},[111],[28,31238,31240,31260],{"className":31239},[115,116],[28,31241,31243,31257],{"className":31242},[120],[28,31244,31246],{"className":31245,"style":1364},[124],[28,31247,31248,31251],{"style":28953},[28,31249],{"className":31250,"style":133},[132],[28,31252,31254],{"className":31253},[137,138,139,140],[28,31255,29018],{"className":31256,"style":29050},[102,106,140],[28,31258,148],{"className":31259},[147],[28,31261,31263],{"className":31262},[120],[28,31264,31266],{"className":31265,"style":155},[124],[28,31267],{}," has dimension ",[28,31270,31272,31285],{"className":31271},[35],[28,31273,31275],{"className":31274},[39],[41,31276,31277],{"xmlns":43},[46,31278,31279,31283],{},[49,31280,31281],{},[55,31282,16959],{},[82,31284,16959],{"encoding":84},[28,31286,31288],{"className":31287,"ariaHidden":71},[89],[28,31289,31291,31294],{"className":31290},[93],[28,31292],{"className":31293,"style":245},[97],[28,31295,16959],{"className":31296,"style":9403},[102,106],", the Hamiltonian projected into the subspace will have dimensions ",[28,31299,31301,31320],{"className":31300},[35],[28,31302,31304],{"className":31303},[39],[41,31305,31306],{"xmlns":43},[46,31307,31308,31317],{},[49,31309,31310,31312,31315],{},[55,31311,16959],{},[62,31313,31314],{},"×",[55,31316,16959],{},[82,31318,31319],{"encoding":84},"r \\times r",[28,31321,31323,31341],{"className":31322,"ariaHidden":71},[89],[28,31324,31326,31329,31332,31335,31338],{"className":31325},[93],[28,31327],{"className":31328,"style":1250},[97],[28,31330,16959],{"className":31331,"style":9403},[102,106],[28,31333],{"className":31334,"style":1257},[173],[28,31336,31314],{"className":31337},[1261],[28,31339],{"className":31340,"style":1257},[173],[28,31342,31344,31347],{"className":31343},[93],[28,31345],{"className":31346,"style":245},[97],[28,31348,16959],{"className":31349,"style":9403},[102,106],". With ",[28,31352,31354,31367],{"className":31353},[35],[28,31355,31357],{"className":31356},[39],[41,31358,31359],{"xmlns":43},[46,31360,31361,31365],{},[49,31362,31363],{},[55,31364,16959],{},[82,31366,16959],{"encoding":84},[28,31368,31370],{"className":31369,"ariaHidden":71},[89],[28,31371,31373,31376],{"className":31372},[93],[28,31374],{"className":31375,"style":245},[97],[28,31377,16959],{"className":31378,"style":9403},[102,106]," small enough (generally ",[28,31381,31383,31404],{"className":31382},[35],[28,31384,31386],{"className":31385},[39],[41,31387,31388],{"xmlns":43},[46,31389,31390,31401],{},[49,31391,31392,31394,31396,31398],{},[55,31393,16959],{},[62,31395,22950],{},[62,31397,22950],{},[650,31399,31400],{},"100",[82,31402,31403],{"encoding":84},"r\u003C\u003C100",[28,31405,31407,31427],{"className":31406,"ariaHidden":71},[89],[28,31408,31410,31414,31417,31420,31424],{"className":31409},[93],[28,31411],{"className":31412,"style":31413},[97],"height:0.5782em;vertical-align:-0.0391em;",[28,31415,16959],{"className":31416,"style":9403},[102,106],[28,31418],{"className":31419,"style":913},[173],[28,31421,31423],{"className":31422},[917],"\u003C\u003C",[28,31425],{"className":31426,"style":913},[173],[28,31428,31430,31433],{"className":31429},[93],[28,31431],{"className":31432,"style":2914},[97],[28,31434,31400],{"className":31435},[102]," is sufficient to obtain convergence of estimates of eigenenergies) we can then easily diagonalize the projected Hamiltonian ",[28,31438,31440,31457],{"className":31439},[35],[28,31441,31443],{"className":31442},[39],[41,31444,31445],{"xmlns":43},[46,31446,31447,31455],{},[49,31448,31449],{},[10155,31450,31451,31453],{"accent":71},[55,31452,2263],{},[62,31454,30008],{},[82,31456,30011],{"encoding":84},[28,31458,31460],{"className":31459,"ariaHidden":71},[89],[28,31461,31463,31466],{"className":31462},[93],[28,31464],{"className":31465,"style":30021},[97],[28,31467,31469],{"className":31468},[102,10255],[28,31470,31472],{"className":31471},[115],[28,31473,31475],{"className":31474},[120],[28,31476,31478,31486],{"className":31477,"style":30021},[124],[28,31479,31480,31483],{"style":10025},[28,31481],{"className":31482,"style":3434},[132],[28,31484,2263],{"className":31485,"style":4040},[102,106],[28,31487,31488,31491],{"style":30044},[28,31489],{"className":31490,"style":3434},[132],[28,31492,31494],{"className":31493,"style":30051},[10282],[28,31495,30008],{"className":31496},[102],". However, we cannot directly diagonalize ",[28,31499,31501,31518],{"className":31500},[35],[28,31502,31504],{"className":31503},[39],[41,31505,31506],{"xmlns":43},[46,31507,31508,31516],{},[49,31509,31510],{},[10155,31511,31512,31514],{"accent":71},[55,31513,2263],{},[62,31515,30008],{},[82,31517,30011],{"encoding":84},[28,31519,31521],{"className":31520,"ariaHidden":71},[89],[28,31522,31524,31527],{"className":31523},[93],[28,31525],{"className":31526,"style":30021},[97],[28,31528,31530],{"className":31529},[102,10255],[28,31531,31533],{"className":31532},[115],[28,31534,31536],{"className":31535},[120],[28,31537,31539,31547],{"className":31538,"style":30021},[124],[28,31540,31541,31544],{"style":10025},[28,31542],{"className":31543,"style":3434},[132],[28,31545,2263],{"className":31546,"style":4040},[102,106],[28,31548,31549,31552],{"style":30044},[28,31550],{"className":31551,"style":3434},[132],[28,31553,31555],{"className":31554,"style":30051},[10282],[28,31556,30008],{"className":31557},[102]," because of the non-orthogonality of the Krylov space vectors. We'll have to measure their overlaps and construct a matrix ",[28,31560,31562,31580],{"className":31561},[35],[28,31563,31565],{"className":31564},[39],[41,31566,31567],{"xmlns":43},[46,31568,31569,31577],{},[49,31570,31571],{},[10155,31572,31573,31575],{"accent":71},[55,31574,11356],{},[62,31576,30008],{},[82,31578,31579],{"encoding":84},"\\tilde{S}",[28,31581,31583],{"className":31582,"ariaHidden":71},[89],[28,31584,31586,31589],{"className":31585},[93],[28,31587],{"className":31588,"style":30021},[97],[28,31590,31592],{"className":31591},[102,10255],[28,31593,31595],{"className":31594},[115],[28,31596,31598],{"className":31597},[120],[28,31599,31601,31609],{"className":31600,"style":30021},[124],[28,31602,31603,31606],{"style":10025},[28,31604],{"className":31605,"style":3434},[132],[28,31607,11356],{"className":31608,"style":1913},[102,106],[28,31610,31611,31614],{"style":30044},[28,31612],{"className":31613,"style":3434},[132],[28,31615,31618],{"className":31616,"style":31617},[10282],"left:-0.1667em;",[28,31619,30008],{"className":31620},[102],[28,31622,31624],{"className":31623},[31],[28,31625,31627,31673],{"className":31626},[35],[28,31628,31630],{"className":31629},[39],[41,31631,31632],{"xmlns":43,"display":44},[46,31633,31634,31670],{},[49,31635,31636,31650,31652,31654,31660,31662,31668],{},[52,31637,31638,31644],{},[10155,31639,31640,31642],{"accent":71},[55,31641,11356],{},[62,31643,30008],{},[49,31645,31646,31648],{},[55,31647,30144],{},[55,31649,7106],{},[62,31651,845],{},[62,31653,2948],{"stretchy":64},[52,31655,31656,31658],{},[55,31657,394],{},[55,31659,30144],{},[55,31661,296],{"mathvariant":1088},[52,31663,31664,31666],{},[55,31665,394],{},[55,31667,7106],{},[62,31669,302],{"stretchy":64},[82,31671,31672],{"encoding":84},"\\tilde{S}_{mn} = \\langle \\psi_m \\vert \\psi_n \\rangle",[28,31674,31676,31762],{"className":31675,"ariaHidden":71},[89],[28,31677,31679,31682,31753,31756,31759],{"className":31678},[93],[28,31680],{"className":31681,"style":30185},[97],[28,31683,31685,31716],{"className":31684},[102],[28,31686,31688],{"className":31687},[102,10255],[28,31689,31691],{"className":31690},[115],[28,31692,31694],{"className":31693},[120],[28,31695,31697,31705],{"className":31696,"style":30021},[124],[28,31698,31699,31702],{"style":10025},[28,31700],{"className":31701,"style":3434},[132],[28,31703,11356],{"className":31704,"style":1913},[102,106],[28,31706,31707,31710],{"style":30044},[28,31708],{"className":31709,"style":3434},[132],[28,31711,31713],{"className":31712,"style":31617},[10282],[28,31714,30008],{"className":31715},[102],[28,31717,31719],{"className":31718},[111],[28,31720,31722,31745],{"className":31721},[115,116],[28,31723,31725,31742],{"className":31724},[120],[28,31726,31728],{"className":31727,"style":125},[124],[28,31729,31730,31733],{"style":1928},[28,31731],{"className":31732,"style":133},[132],[28,31734,31736],{"className":31735},[137,138,139,140],[28,31737,31739],{"className":31738},[102,140],[28,31740,650],{"className":31741},[102,106,140],[28,31743,148],{"className":31744},[147],[28,31746,31748],{"className":31747},[120],[28,31749,31751],{"className":31750,"style":155},[124],[28,31752],{},[28,31754],{"className":31755,"style":913},[173],[28,31757,845],{"className":31758},[917],[28,31760],{"className":31761,"style":913},[173],[28,31763,31765,31768,31771,31811,31814,31854],{"className":31764},[93],[28,31766],{"className":31767,"style":98},[97],[28,31769,2948],{"className":31770},[161],[28,31772,31774,31777],{"className":31773},[102],[28,31775,394],{"className":31776,"style":321},[102,106],[28,31778,31780],{"className":31779},[111],[28,31781,31783,31803],{"className":31782},[115,116],[28,31784,31786,31800],{"className":31785},[120],[28,31787,31789],{"className":31788,"style":125},[124],[28,31790,31791,31794],{"style":801},[28,31792],{"className":31793,"style":133},[132],[28,31795,31797],{"className":31796},[137,138,139,140],[28,31798,30144],{"className":31799},[102,106,140],[28,31801,148],{"className":31802},[147],[28,31804,31806],{"className":31805},[120],[28,31807,31809],{"className":31808,"style":155},[124],[28,31810],{},[28,31812,296],{"className":31813},[102],[28,31815,31817,31820],{"className":31816},[102],[28,31818,394],{"className":31819,"style":321},[102,106],[28,31821,31823],{"className":31822},[111],[28,31824,31826,31846],{"className":31825},[115,116],[28,31827,31829,31843],{"className":31828},[120],[28,31830,31832],{"className":31831,"style":125},[124],[28,31833,31834,31837],{"style":801},[28,31835],{"className":31836,"style":133},[132],[28,31838,31840],{"className":31839},[137,138,139,140],[28,31841,7106],{"className":31842},[102,106,140],[28,31844,148],{"className":31845},[147],[28,31847,31849],{"className":31848},[120],[28,31850,31852],{"className":31851,"style":155},[124],[28,31853],{},[28,31855,302],{"className":31856},[182],[12,31858,31859],{},"This allows us to solve the eigenvalue problem in a non-orthogonal space (also called generalized eigenvalue problem)",[28,31861,31863],{"className":31862},[31],[28,31864,31866,31913],{"className":31865},[35],[28,31867,31869],{"className":31868},[39],[41,31870,31871],{"xmlns":43,"display":44},[46,31872,31873,31910],{},[49,31874,31875,31881,31883,31890,31892,31894,31896,31902,31904],{},[10155,31876,31877,31879],{"accent":71},[55,31878,2263],{},[62,31880,30008],{},[2253,31882,2255],{},[10155,31884,31885,31888],{"accent":71},[55,31886,31887],{},"c",[62,31889,10205],{},[62,31891,845],{},[55,31893,1871],{},[2253,31895,2255],{},[10155,31897,31898,31900],{"accent":71},[55,31899,11356],{},[62,31901,30008],{},[2253,31903,2255],{},[10155,31905,31906,31908],{"accent":71},[55,31907,31887],{},[62,31909,10205],{},[82,31911,31912],{"encoding":84},"\\tilde{H} \\ \\vec{c} = E \\ \\tilde{S} \\ \\vec{c}",[28,31914,31916,32001],{"className":31915,"ariaHidden":71},[89],[28,31917,31919,31922,31953,31956,31992,31995,31998],{"className":31918},[93],[28,31920],{"className":31921,"style":30021},[97],[28,31923,31925],{"className":31924},[102,10255],[28,31926,31928],{"className":31927},[115],[28,31929,31931],{"className":31930},[120],[28,31932,31934,31942],{"className":31933,"style":30021},[124],[28,31935,31936,31939],{"style":10025},[28,31937],{"className":31938,"style":3434},[132],[28,31940,2263],{"className":31941,"style":4040},[102,106],[28,31943,31944,31947],{"style":30044},[28,31945],{"className":31946,"style":3434},[132],[28,31948,31950],{"className":31949,"style":30051},[10282],[28,31951,30008],{"className":31952},[102],[28,31954,2255],{"className":31955},[173],[28,31957,31959],{"className":31958},[102,10255],[28,31960,31962],{"className":31961},[115],[28,31963,31965],{"className":31964},[120],[28,31966,31968,31976],{"className":31967,"style":10457},[124],[28,31969,31970,31973],{"style":10025},[28,31971],{"className":31972,"style":3434},[132],[28,31974,31887],{"className":31975},[102,106],[28,31977,31978,31981],{"style":10025},[28,31979],{"className":31980,"style":3434},[132],[28,31982,31985],{"className":31983,"style":31984},[10282],"left:-0.1799em;",[28,31986,31988],{"className":31987,"style":10479},[10478],[5855,31989,31990],{"xmlns":5857,"width":10482,"height":10483,"style":10484,"viewBox":10485,"preserveAspectRatio":10486},[5863,31991],{"d":10489},[28,31993],{"className":31994,"style":913},[173],[28,31996,845],{"className":31997},[917],[28,31999],{"className":32000,"style":913},[173],[28,32002,32004,32007,32010,32013,32044,32047],{"className":32003},[93],[28,32005],{"className":32006,"style":30021},[97],[28,32008,1871],{"className":32009,"style":1913},[102,106],[28,32011,2255],{"className":32012},[173],[28,32014,32016],{"className":32015},[102,10255],[28,32017,32019],{"className":32018},[115],[28,32020,32022],{"className":32021},[120],[28,32023,32025,32033],{"className":32024,"style":30021},[124],[28,32026,32027,32030],{"style":10025},[28,32028],{"className":32029,"style":3434},[132],[28,32031,11356],{"className":32032,"style":1913},[102,106],[28,32034,32035,32038],{"style":30044},[28,32036],{"className":32037,"style":3434},[132],[28,32039,32041],{"className":32040,"style":31617},[10282],[28,32042,30008],{"className":32043},[102],[28,32045,2255],{"className":32046},[173],[28,32048,32050],{"className":32049},[102,10255],[28,32051,32053],{"className":32052},[115],[28,32054,32056],{"className":32055},[120],[28,32057,32059,32067],{"className":32058,"style":10457},[124],[28,32060,32061,32064],{"style":10025},[28,32062],{"className":32063,"style":3434},[132],[28,32065,31887],{"className":32066},[102,106],[28,32068,32069,32072],{"style":10025},[28,32070],{"className":32071,"style":3434},[132],[28,32073,32075],{"className":32074,"style":31984},[10282],[28,32076,32078],{"className":32077,"style":10479},[10478],[5855,32079,32080],{"xmlns":5857,"width":10482,"height":10483,"style":10484,"viewBox":10485,"preserveAspectRatio":10486},[5863,32081],{"d":10489},[12,32083,32084,32085,32113,32114,32174,32175,32203,32204,32269,32270,32334,32335,2106],{},"One can then obtain estimates of the eigenvalues and eigenstates of ",[28,32086,32088,32101],{"className":32087},[35],[28,32089,32091],{"className":32090},[39],[41,32092,32093],{"xmlns":43},[46,32094,32095,32099],{},[49,32096,32097],{},[55,32098,2263],{},[82,32100,2263],{"encoding":84},[28,32102,32104],{"className":32103,"ariaHidden":71},[89],[28,32105,32107,32110],{"className":32106},[93],[28,32108],{"className":32109,"style":1102},[97],[28,32111,2263],{"className":32112,"style":4040},[102,106]," by looking at the ones of ",[28,32115,32117,32134],{"className":32116},[35],[28,32118,32120],{"className":32119},[39],[41,32121,32122],{"xmlns":43},[46,32123,32124,32132],{},[49,32125,32126],{},[10155,32127,32128,32130],{"accent":71},[55,32129,2263],{},[62,32131,30008],{},[82,32133,30011],{"encoding":84},[28,32135,32137],{"className":32136,"ariaHidden":71},[89],[28,32138,32140,32143],{"className":32139},[93],[28,32141],{"className":32142,"style":30021},[97],[28,32144,32146],{"className":32145},[102,10255],[28,32147,32149],{"className":32148},[115],[28,32150,32152],{"className":32151},[120],[28,32153,32155,32163],{"className":32154,"style":30021},[124],[28,32156,32157,32160],{"style":10025},[28,32158],{"className":32159,"style":3434},[132],[28,32161,2263],{"className":32162,"style":4040},[102,106],[28,32164,32165,32168],{"style":30044},[28,32166],{"className":32167,"style":3434},[132],[28,32169,32171],{"className":32170,"style":30051},[10282],[28,32172,30008],{"className":32173},[102],". For example, the estimate of the ground state energy is obtained by taking the smallest eigenvalue ",[28,32176,32178,32191],{"className":32177},[35],[28,32179,32181],{"className":32180},[39],[41,32182,32183],{"xmlns":43},[46,32184,32185,32189],{},[49,32186,32187],{},[55,32188,31887],{},[82,32190,31887],{"encoding":84},[28,32192,32194],{"className":32193,"ariaHidden":71},[89],[28,32195,32197,32200],{"className":32196},[93],[28,32198],{"className":32199,"style":245},[97],[28,32201,31887],{"className":32202},[102,106]," and the ground state from the corresponding eigenvector ",[28,32205,32207,32225],{"className":32206},[35],[28,32208,32210],{"className":32209},[39],[41,32211,32212],{"xmlns":43},[46,32213,32214,32222],{},[49,32215,32216],{},[10155,32217,32218,32220],{"accent":71},[55,32219,31887],{},[62,32221,10205],{},[82,32223,32224],{"encoding":84},"\\vec{c}",[28,32226,32228],{"className":32227,"ariaHidden":71},[89],[28,32229,32231,32234],{"className":32230},[93],[28,32232],{"className":32233,"style":10457},[97],[28,32235,32237],{"className":32236},[102,10255],[28,32238,32240],{"className":32239},[115],[28,32241,32243],{"className":32242},[120],[28,32244,32246,32254],{"className":32245,"style":10457},[124],[28,32247,32248,32251],{"style":10025},[28,32249],{"className":32250,"style":3434},[132],[28,32252,31887],{"className":32253},[102,106],[28,32255,32256,32259],{"style":10025},[28,32257],{"className":32258,"style":3434},[132],[28,32260,32262],{"className":32261,"style":31984},[10282],[28,32263,32265],{"className":32264,"style":10479},[10478],[5855,32266,32267],{"xmlns":5857,"width":10482,"height":10483,"style":10484,"viewBox":10485,"preserveAspectRatio":10486},[5863,32268],{"d":10489},". The coefficients in ",[28,32271,32273,32290],{"className":32272},[35],[28,32274,32276],{"className":32275},[39],[41,32277,32278],{"xmlns":43},[46,32279,32280,32288],{},[49,32281,32282],{},[10155,32283,32284,32286],{"accent":71},[55,32285,31887],{},[62,32287,10205],{},[82,32289,32224],{"encoding":84},[28,32291,32293],{"className":32292,"ariaHidden":71},[89],[28,32294,32296,32299],{"className":32295},[93],[28,32297],{"className":32298,"style":10457},[97],[28,32300,32302],{"className":32301},[102,10255],[28,32303,32305],{"className":32304},[115],[28,32306,32308],{"className":32307},[120],[28,32309,32311,32319],{"className":32310,"style":10457},[124],[28,32312,32313,32316],{"style":10025},[28,32314],{"className":32315,"style":3434},[132],[28,32317,31887],{"className":32318},[102,106],[28,32320,32321,32324],{"style":10025},[28,32322],{"className":32323,"style":3434},[132],[28,32325,32327],{"className":32326,"style":31984},[10282],[28,32328,32330],{"className":32329,"style":10479},[10478],[5855,32331,32332],{"xmlns":5857,"width":10482,"height":10483,"style":10484,"viewBox":10485,"preserveAspectRatio":10486},[5863,32333],{"d":10489}," determines the contribution of the different vectors that span ",[28,32336,32338,32355],{"className":32337},[35],[28,32339,32341],{"className":32340},[39],[41,32342,32343],{"xmlns":43},[46,32344,32345,32353],{},[49,32346,32347],{},[52,32348,32349,32351],{},[55,32350,28350],{"mathvariant":11377},[55,32352,29018],{},[82,32354,29129],{"encoding":84},[28,32356,32358],{"className":32357,"ariaHidden":71},[89],[28,32359,32361,32364],{"className":32360},[93],[28,32362],{"className":32363,"style":585},[97],[28,32365,32367,32370],{"className":32366},[102],[28,32368,28350],{"className":32369,"style":28371},[102,11469],[28,32371,32373],{"className":32372},[111],[28,32374,32376,32396],{"className":32375},[115,116],[28,32377,32379,32393],{"className":32378},[120],[28,32380,32382],{"className":32381,"style":1364},[124],[28,32383,32384,32387],{"style":28953},[28,32385],{"className":32386,"style":133},[132],[28,32388,32390],{"className":32389},[137,138,139,140],[28,32391,29018],{"className":32392,"style":29050},[102,106,140],[28,32394,148],{"className":32395},[147],[28,32397,32399],{"className":32398},[120],[28,32400,32402],{"className":32401,"style":155},[124],[28,32403],{},[12,32405,32406],{},[32407,32408],"img",{"alt":32409,"src":32410},"fig1.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-06.avif",[12,32412,32413,32414,32534,32535,16397,32615,32696,32697,16397,32782,32866,32867,32927,32928,33010,33011,33090,33091,33173,33174,33202,33203,33352],{},"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 ",[28,32415,32417,32445],{"className":32416},[35],[28,32418,32420],{"className":32419},[39],[41,32421,32422],{"xmlns":43},[46,32423,32424,32442],{},[49,32425,32426],{},[52,32427,32428,32434],{},[10155,32429,32430,32432],{"accent":71},[55,32431,2263],{},[62,32433,30008],{},[49,32435,32436,32438,32440],{},[55,32437,3256],{},[62,32439,72],{"separator":71},[55,32441,28200],{},[82,32443,32444],{"encoding":84},"\\tilde{H}_{i,j}",[28,32446,32448],{"className":32447,"ariaHidden":71},[89],[28,32449,32451,32455],{"className":32450},[93],[28,32452],{"className":32453,"style":32454},[97],"height:1.2063em;vertical-align:-0.2861em;",[28,32456,32458,32489],{"className":32457},[102],[28,32459,32461],{"className":32460},[102,10255],[28,32462,32464],{"className":32463},[115],[28,32465,32467],{"className":32466},[120],[28,32468,32470,32478],{"className":32469,"style":30021},[124],[28,32471,32472,32475],{"style":10025},[28,32473],{"className":32474,"style":3434},[132],[28,32476,2263],{"className":32477,"style":4040},[102,106],[28,32479,32480,32483],{"style":30044},[28,32481],{"className":32482,"style":3434},[132],[28,32484,32486],{"className":32485,"style":30051},[10282],[28,32487,30008],{"className":32488},[102],[28,32490,32492],{"className":32491},[111],[28,32493,32495,32526],{"className":32494},[115,116],[28,32496,32498,32523],{"className":32497},[120],[28,32499,32502],{"className":32500,"style":32501},[124],"height:0.3117em;",[28,32503,32504,32507],{"style":4522},[28,32505],{"className":32506,"style":133},[132],[28,32508,32510],{"className":32509},[137,138,139,140],[28,32511,32513,32516,32519],{"className":32512},[102,140],[28,32514,3256],{"className":32515},[102,106,140],[28,32517,72],{"className":32518},[169,140],[28,32520,28200],{"className":32521,"style":32522},[102,106,140],"margin-right:0.0572em;",[28,32524,148],{"className":32525},[147],[28,32527,32529],{"className":32528},[120],[28,32530,32532],{"className":32531,"style":2003},[124],[28,32533],{},", a Hadamard test between the state ",[28,32536,32538,32560],{"className":32537},[35],[28,32539,32541],{"className":32540},[39],[41,32542,32543],{"xmlns":43},[46,32544,32545,32557],{},[49,32546,32547,32549,32555],{},[55,32548,296],{"mathvariant":1088},[52,32550,32551,32553],{},[55,32552,394],{},[55,32554,3256],{},[62,32556,302],{"stretchy":64},[82,32558,32559],{"encoding":84},"\\vert \\psi_i \\rangle",[28,32561,32563],{"className":32562,"ariaHidden":71},[89],[28,32564,32566,32569,32572,32612],{"className":32565},[93],[28,32567],{"className":32568,"style":98},[97],[28,32570,296],{"className":32571},[102],[28,32573,32575,32578],{"className":32574},[102],[28,32576,394],{"className":32577,"style":321},[102,106],[28,32579,32581],{"className":32580},[111],[28,32582,32584,32604],{"className":32583},[115,116],[28,32585,32587,32601],{"className":32586},[120],[28,32588,32590],{"className":32589,"style":32501},[124],[28,32591,32592,32595],{"style":801},[28,32593],{"className":32594,"style":133},[132],[28,32596,32598],{"className":32597},[137,138,139,140],[28,32599,3256],{"className":32600},[102,106,140],[28,32602,148],{"className":32603},[147],[28,32605,32607],{"className":32606},[120],[28,32608,32610],{"className":32609,"style":155},[124],[28,32611],{},[28,32613,302],{"className":32614},[182],[28,32616,32618,32640],{"className":32617},[35],[28,32619,32621],{"className":32620},[39],[41,32622,32623],{"xmlns":43},[46,32624,32625,32637],{},[49,32626,32627,32629,32635],{},[55,32628,296],{"mathvariant":1088},[52,32630,32631,32633],{},[55,32632,394],{},[55,32634,28200],{},[62,32636,302],{"stretchy":64},[82,32638,32639],{"encoding":84},"\\vert \\psi_j \\rangle",[28,32641,32643],{"className":32642,"ariaHidden":71},[89],[28,32644,32646,32650,32653,32693],{"className":32645},[93],[28,32647],{"className":32648,"style":32649},[97],"height:1.0361em;vertical-align:-0.2861em;",[28,32651,296],{"className":32652},[102],[28,32654,32656,32659],{"className":32655},[102],[28,32657,394],{"className":32658,"style":321},[102,106],[28,32660,32662],{"className":32661},[111],[28,32663,32665,32685],{"className":32664},[115,116],[28,32666,32668,32682],{"className":32667},[120],[28,32669,32671],{"className":32670,"style":32501},[124],[28,32672,32673,32676],{"style":801},[28,32674],{"className":32675,"style":133},[132],[28,32677,32679],{"className":32678},[137,138,139,140],[28,32680,28200],{"className":32681,"style":32522},[102,106,140],[28,32683,148],{"className":32684},[147],[28,32686,32688],{"className":32687},[120],[28,32689,32691],{"className":32690,"style":2003},[124],[28,32692],{},[28,32694,302],{"className":32695},[182]," is carried out. This is highlighted in the figure by the color scheme for the matrix elements and the corresponding ",[28,32698,32700,32724],{"className":32699},[35],[28,32701,32703],{"className":32702},[39],[41,32704,32705],{"xmlns":43},[46,32706,32707,32721],{},[49,32708,32709,32712,32715],{},[2253,32710,32711],{},"Prep",[2253,32713,32714],{},"  ",[52,32716,32717,32719],{},[55,32718,394],{},[55,32720,3256],{},[82,32722,32723],{"encoding":84},"\\text{Prep} \\; \\psi_i",[28,32725,32727],{"className":32726,"ariaHidden":71},[89],[28,32728,32730,32733,32739,32742],{"className":32729},[93],[28,32731],{"className":32732,"style":214},[97],[28,32734,32736],{"className":32735},[102,16058],[28,32737,32711],{"className":32738},[102],[28,32740],{"className":32741,"style":913},[173],[28,32743,32745,32748],{"className":32744},[102],[28,32746,394],{"className":32747,"style":321},[102,106],[28,32749,32751],{"className":32750},[111],[28,32752,32754,32774],{"className":32753},[115,116],[28,32755,32757,32771],{"className":32756},[120],[28,32758,32760],{"className":32759,"style":32501},[124],[28,32761,32762,32765],{"style":801},[28,32763],{"className":32764,"style":133},[132],[28,32766,32768],{"className":32767},[137,138,139,140],[28,32769,3256],{"className":32770},[102,106,140],[28,32772,148],{"className":32773},[147],[28,32775,32777],{"className":32776},[120],[28,32778,32780],{"className":32779,"style":155},[124],[28,32781],{},[28,32783,32785,32807],{"className":32784},[35],[28,32786,32788],{"className":32787},[39],[41,32789,32790],{"xmlns":43},[46,32791,32792,32804],{},[49,32793,32794,32796,32798],{},[2253,32795,32711],{},[2253,32797,32714],{},[52,32799,32800,32802],{},[55,32801,394],{},[55,32803,28200],{},[82,32805,32806],{"encoding":84},"\\text{Prep} \\; \\psi_j",[28,32808,32810],{"className":32809,"ariaHidden":71},[89],[28,32811,32813,32817,32823,32826],{"className":32812},[93],[28,32814],{"className":32815,"style":32816},[97],"height:0.9805em;vertical-align:-0.2861em;",[28,32818,32820],{"className":32819},[102,16058],[28,32821,32711],{"className":32822},[102],[28,32824],{"className":32825,"style":913},[173],[28,32827,32829,32832],{"className":32828},[102],[28,32830,394],{"className":32831,"style":321},[102,106],[28,32833,32835],{"className":32834},[111],[28,32836,32838,32858],{"className":32837},[115,116],[28,32839,32841,32855],{"className":32840},[120],[28,32842,32844],{"className":32843,"style":32501},[124],[28,32845,32846,32849],{"style":801},[28,32847],{"className":32848,"style":133},[132],[28,32850,32852],{"className":32851},[137,138,139,140],[28,32853,28200],{"className":32854,"style":32522},[102,106,140],[28,32856,148],{"className":32857},[147],[28,32859,32861],{"className":32860},[120],[28,32862,32864],{"className":32863,"style":2003},[124],[28,32865],{}," 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 ",[28,32868,32870,32887],{"className":32869},[35],[28,32871,32873],{"className":32872},[39],[41,32874,32875],{"xmlns":43},[46,32876,32877,32885],{},[49,32878,32879],{},[10155,32880,32881,32883],{"accent":71},[55,32882,2263],{},[62,32884,30008],{},[82,32886,30011],{"encoding":84},[28,32888,32890],{"className":32889,"ariaHidden":71},[89],[28,32891,32893,32896],{"className":32892},[93],[28,32894],{"className":32895,"style":30021},[97],[28,32897,32899],{"className":32898},[102,10255],[28,32900,32902],{"className":32901},[115],[28,32903,32905],{"className":32904},[120],[28,32906,32908,32916],{"className":32907,"style":30021},[124],[28,32909,32910,32913],{"style":10025},[28,32911],{"className":32912,"style":3434},[132],[28,32914,2263],{"className":32915,"style":4040},[102,106],[28,32917,32918,32921],{"style":30044},[28,32919],{"className":32920,"style":3434},[132],[28,32922,32924],{"className":32923,"style":30051},[10282],[28,32925,30008],{"className":32926},[102],". 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 ",[28,32929,32931,32952],{"className":32930},[35],[28,32932,32934],{"className":32933},[39],[41,32935,32936],{"xmlns":43},[46,32937,32938,32950],{},[49,32939,32940,32942,32944],{},[2253,32941,32711],{},[2253,32943,32714],{},[52,32945,32946,32948],{},[55,32947,394],{},[55,32949,3256],{},[82,32951,32723],{"encoding":84},[28,32953,32955],{"className":32954,"ariaHidden":71},[89],[28,32956,32958,32961,32967,32970],{"className":32957},[93],[28,32959],{"className":32960,"style":214},[97],[28,32962,32964],{"className":32963},[102,16058],[28,32965,32711],{"className":32966},[102],[28,32968],{"className":32969,"style":913},[173],[28,32971,32973,32976],{"className":32972},[102],[28,32974,394],{"className":32975,"style":321},[102,106],[28,32977,32979],{"className":32978},[111],[28,32980,32982,33002],{"className":32981},[115,116],[28,32983,32985,32999],{"className":32984},[120],[28,32986,32988],{"className":32987,"style":32501},[124],[28,32989,32990,32993],{"style":801},[28,32991],{"className":32992,"style":133},[132],[28,32994,32996],{"className":32995},[137,138,139,140],[28,32997,3256],{"className":32998},[102,106,140],[28,33000,148],{"className":33001},[147],[28,33003,33005],{"className":33004},[120],[28,33006,33008],{"className":33007,"style":155},[124],[28,33009],{}," operation prepares the system qubit in the state ",[28,33012,33014,33035],{"className":33013},[35],[28,33015,33017],{"className":33016},[39],[41,33018,33019],{"xmlns":43},[46,33020,33021,33033],{},[49,33022,33023,33025,33031],{},[55,33024,296],{"mathvariant":1088},[52,33026,33027,33029],{},[55,33028,394],{},[55,33030,3256],{},[62,33032,302],{"stretchy":64},[82,33034,32559],{"encoding":84},[28,33036,33038],{"className":33037,"ariaHidden":71},[89],[28,33039,33041,33044,33047,33087],{"className":33040},[93],[28,33042],{"className":33043,"style":98},[97],[28,33045,296],{"className":33046},[102],[28,33048,33050,33053],{"className":33049},[102],[28,33051,394],{"className":33052,"style":321},[102,106],[28,33054,33056],{"className":33055},[111],[28,33057,33059,33079],{"className":33058},[115,116],[28,33060,33062,33076],{"className":33061},[120],[28,33063,33065],{"className":33064,"style":32501},[124],[28,33066,33067,33070],{"style":801},[28,33068],{"className":33069,"style":133},[132],[28,33071,33073],{"className":33072},[137,138,139,140],[28,33074,3256],{"className":33075},[102,106,140],[28,33077,148],{"className":33078},[147],[28,33080,33082],{"className":33081},[120],[28,33083,33085],{"className":33084,"style":155},[124],[28,33086],{},[28,33088,302],{"className":33089},[182]," controlled by the state of the ancilla qubit (similarly for ",[28,33092,33094,33115],{"className":33093},[35],[28,33095,33097],{"className":33096},[39],[41,33098,33099],{"xmlns":43},[46,33100,33101,33113],{},[49,33102,33103,33105,33107],{},[2253,33104,32711],{},[2253,33106,32714],{},[52,33108,33109,33111],{},[55,33110,394],{},[55,33112,28200],{},[82,33114,32806],{"encoding":84},[28,33116,33118],{"className":33117,"ariaHidden":71},[89],[28,33119,33121,33124,33130,33133],{"className":33120},[93],[28,33122],{"className":33123,"style":32816},[97],[28,33125,33127],{"className":33126},[102,16058],[28,33128,32711],{"className":33129},[102],[28,33131],{"className":33132,"style":913},[173],[28,33134,33136,33139],{"className":33135},[102],[28,33137,394],{"className":33138,"style":321},[102,106],[28,33140,33142],{"className":33141},[111],[28,33143,33145,33165],{"className":33144},[115,116],[28,33146,33148,33162],{"className":33147},[120],[28,33149,33151],{"className":33150,"style":32501},[124],[28,33152,33153,33156],{"style":801},[28,33154],{"className":33155,"style":133},[132],[28,33157,33159],{"className":33158},[137,138,139,140],[28,33160,28200],{"className":33161,"style":32522},[102,106,140],[28,33163,148],{"className":33164},[147],[28,33166,33168],{"className":33167},[120],[28,33169,33171],{"className":33170,"style":2003},[124],[28,33172],{},") and the operation ",[28,33175,33177,33190],{"className":33176},[35],[28,33178,33180],{"className":33179},[39],[41,33181,33182],{"xmlns":43},[46,33183,33184,33188],{},[49,33185,33186],{},[55,33187,57],{},[82,33189,57],{"encoding":84},[28,33191,33193],{"className":33192,"ariaHidden":71},[89],[28,33194,33196,33199],{"className":33195},[93],[28,33197],{"className":33198,"style":1102},[97],[28,33200,57],{"className":33201,"style":107},[102,106]," represents Pauli decomposition of the system Hamiltonian ",[28,33204,33206,33235],{"className":33205},[35],[28,33207,33209],{"className":33208},[39],[41,33210,33211],{"xmlns":43},[46,33212,33213,33232],{},[49,33214,33215,33217,33219,33226],{},[55,33216,2263],{},[62,33218,845],{},[52,33220,33221,33224],{},[62,33222,33223],{},"∑",[55,33225,3256],{},[52,33227,33228,33230],{},[55,33229,57],{},[55,33231,3256],{},[82,33233,33234],{"encoding":84},"H = \\sum_i P_i",[28,33236,33238,33256],{"className":33237,"ariaHidden":71},[89],[28,33239,33241,33244,33247,33250,33253],{"className":33240},[93],[28,33242],{"className":33243,"style":1102},[97],[28,33245,2263],{"className":33246,"style":4040},[102,106],[28,33248],{"className":33249,"style":913},[173],[28,33251,845],{"className":33252},[917],[28,33254],{"className":33255,"style":913},[173],[28,33257,33259,33263,33309,33312],{"className":33258},[93],[28,33260],{"className":33261,"style":33262},[97],"height:1.0497em;vertical-align:-0.2997em;",[28,33264,33266,33272],{"className":33265},[4241],[28,33267,33223],{"className":33268,"style":33271},[4241,33269,33270],"op-symbol","small-op","position:relative;top:0em;",[28,33273,33275],{"className":33274},[111],[28,33276,33278,33300],{"className":33277},[115,116],[28,33279,33281,33297],{"className":33280},[120],[28,33282,33285],{"className":33283,"style":33284},[124],"height:0.162em;",[28,33286,33288,33291],{"style":33287},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[28,33289],{"className":33290,"style":133},[132],[28,33292,33294],{"className":33293},[137,138,139,140],[28,33295,3256],{"className":33296},[102,106,140],[28,33298,148],{"className":33299},[147],[28,33301,33303],{"className":33302},[120],[28,33304,33307],{"className":33305,"style":33306},[124],"height:0.2997em;",[28,33308],{},[28,33310],{"className":33311,"style":174},[173],[28,33313,33315,33318],{"className":33314},[102],[28,33316,57],{"className":33317,"style":107},[102,106],[28,33319,33321],{"className":33320},[111],[28,33322,33324,33344],{"className":33323},[115,116],[28,33325,33327,33341],{"className":33326},[120],[28,33328,33330],{"className":33329,"style":32501},[124],[28,33331,33332,33335],{"style":128},[28,33333],{"className":33334,"style":133},[132],[28,33336,33338],{"className":33337},[137,138,139,140],[28,33339,3256],{"className":33340},[102,106,140],[28,33342,148],{"className":33343},[147],[28,33345,33347],{"className":33346},[120],[28,33348,33350],{"className":33349,"style":155},[124],[28,33351],{},". A more detailed derivation of the operations calculated by the Hadamard test is given below.",[33354,33355,33357],"h4",{"id":33356},"define-hamiltonian","Define Hamiltonian",[12,33359,33360,33361,33390,33391],{},"Let's consider the Heisenberg Hamiltonian for ",[28,33362,33364,33378],{"className":33363},[35],[28,33365,33367],{"className":33366},[39],[41,33368,33369],{"xmlns":43},[46,33370,33371,33376],{},[49,33372,33373],{},[55,33374,33375],{},"N",[82,33377,33375],{"encoding":84},[28,33379,33381],{"className":33380,"ariaHidden":71},[89],[28,33382,33384,33387],{"className":33383},[93],[28,33385],{"className":33386,"style":1102},[97],[28,33388,33375],{"className":33389,"style":29050},[102,106]," qubits on a linear chain: ",[28,33392,33394,33469],{"className":33393},[35],[28,33395,33397],{"className":33396},[39],[41,33398,33399],{"xmlns":43},[46,33400,33401,33466],{},[49,33402,33403,33405,33407,33421,33427,33433,33435,33442,33448,33450,33453,33460],{},[55,33404,2263],{},[62,33406,845],{},[1670,33408,33409,33411,33419],{},[62,33410,33223],{},[49,33412,33413,33415,33417],{},[55,33414,3256],{},[62,33416,72],{"separator":71},[55,33418,28200],{},[55,33420,33375],{},[52,33422,33423,33425],{},[55,33424,9841],{},[55,33426,3256],{},[52,33428,33429,33431],{},[55,33430,9841],{},[55,33432,28200],{},[62,33434,2606],{},[52,33436,33437,33440],{},[55,33438,33439],{},"Y",[55,33441,3256],{},[52,33443,33444,33446],{},[55,33445,33439],{},[55,33447,28200],{},[62,33449,1213],{},[55,33451,33452],{},"J",[52,33454,33455,33458],{},[55,33456,33457],{},"Z",[55,33459,3256],{},[52,33461,33462,33464],{},[55,33463,33457],{},[55,33465,28200],{},[82,33467,33468],{"encoding":84},"H= \\sum_{i,j}^N X_i X_j + Y_i Y_j - J Z_i Z_j",[28,33470,33472,33490,33653,33749],{"className":33471,"ariaHidden":71},[89],[28,33473,33475,33478,33481,33484,33487],{"className":33474},[93],[28,33476],{"className":33477,"style":1102},[97],[28,33479,2263],{"className":33480,"style":4040},[102,106],[28,33482],{"className":33483,"style":913},[173],[28,33485,845],{"className":33486},[917],[28,33488],{"className":33489,"style":913},[173],[28,33491,33493,33497,33560,33563,33604,33644,33647,33650],{"className":33492},[93],[28,33494],{"className":33495,"style":33496},[97],"height:1.417em;vertical-align:-0.4358em;",[28,33498,33500,33503],{"className":33499},[4241],[28,33501,33223],{"className":33502,"style":33271},[4241,33269,33270],[28,33504,33506],{"className":33505},[111],[28,33507,33509,33551],{"className":33508},[115,116],[28,33510,33512,33548],{"className":33511},[120],[28,33513,33516,33536],{"className":33514,"style":33515},[124],"height:0.9812em;",[28,33517,33518,33521],{"style":33287},[28,33519],{"className":33520,"style":133},[132],[28,33522,33524],{"className":33523},[137,138,139,140],[28,33525,33527,33530,33533],{"className":33526},[102,140],[28,33528,3256],{"className":33529},[102,106,140],[28,33531,72],{"className":33532},[169,140],[28,33534,28200],{"className":33535,"style":32522},[102,106,140],[28,33537,33539,33542],{"style":33538},"top:-3.2029em;margin-right:0.05em;",[28,33540],{"className":33541,"style":133},[132],[28,33543,33545],{"className":33544},[137,138,139,140],[28,33546,33375],{"className":33547,"style":29050},[102,106,140],[28,33549,148],{"className":33550},[147],[28,33552,33554],{"className":33553},[120],[28,33555,33558],{"className":33556,"style":33557},[124],"height:0.4358em;",[28,33559],{},[28,33561],{"className":33562,"style":174},[173],[28,33564,33566,33569],{"className":33565},[102],[28,33567,9841],{"className":33568,"style":9856},[102,106],[28,33570,33572],{"className":33571},[111],[28,33573,33575,33596],{"className":33574},[115,116],[28,33576,33578,33593],{"className":33577},[120],[28,33579,33581],{"className":33580,"style":32501},[124],[28,33582,33584,33587],{"style":33583},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[28,33585],{"className":33586,"style":133},[132],[28,33588,33590],{"className":33589},[137,138,139,140],[28,33591,3256],{"className":33592},[102,106,140],[28,33594,148],{"className":33595},[147],[28,33597,33599],{"className":33598},[120],[28,33600,33602],{"className":33601,"style":155},[124],[28,33603],{},[28,33605,33607,33610],{"className":33606},[102],[28,33608,9841],{"className":33609,"style":9856},[102,106],[28,33611,33613],{"className":33612},[111],[28,33614,33616,33636],{"className":33615},[115,116],[28,33617,33619,33633],{"className":33618},[120],[28,33620,33622],{"className":33621,"style":32501},[124],[28,33623,33624,33627],{"style":33583},[28,33625],{"className":33626,"style":133},[132],[28,33628,33630],{"className":33629},[137,138,139,140],[28,33631,28200],{"className":33632,"style":32522},[102,106,140],[28,33634,148],{"className":33635},[147],[28,33637,33639],{"className":33638},[120],[28,33640,33642],{"className":33641,"style":2003},[124],[28,33643],{},[28,33645],{"className":33646,"style":1257},[173],[28,33648,2606],{"className":33649},[1261],[28,33651],{"className":33652,"style":1257},[173],[28,33654,33656,33659,33700,33740,33743,33746],{"className":33655},[93],[28,33657],{"className":33658,"style":1964},[97],[28,33660,33662,33665],{"className":33661},[102],[28,33663,33439],{"className":33664,"style":1257},[102,106],[28,33666,33668],{"className":33667},[111],[28,33669,33671,33692],{"className":33670},[115,116],[28,33672,33674,33689],{"className":33673},[120],[28,33675,33677],{"className":33676,"style":32501},[124],[28,33678,33680,33683],{"style":33679},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[28,33681],{"className":33682,"style":133},[132],[28,33684,33686],{"className":33685},[137,138,139,140],[28,33687,3256],{"className":33688},[102,106,140],[28,33690,148],{"className":33691},[147],[28,33693,33695],{"className":33694},[120],[28,33696,33698],{"className":33697,"style":155},[124],[28,33699],{},[28,33701,33703,33706],{"className":33702},[102],[28,33704,33439],{"className":33705,"style":1257},[102,106],[28,33707,33709],{"className":33708},[111],[28,33710,33712,33732],{"className":33711},[115,116],[28,33713,33715,33729],{"className":33714},[120],[28,33716,33718],{"className":33717,"style":32501},[124],[28,33719,33720,33723],{"style":33679},[28,33721],{"className":33722,"style":133},[132],[28,33724,33726],{"className":33725},[137,138,139,140],[28,33727,28200],{"className":33728,"style":32522},[102,106,140],[28,33730,148],{"className":33731},[147],[28,33733,33735],{"className":33734},[120],[28,33736,33738],{"className":33737,"style":2003},[124],[28,33739],{},[28,33741],{"className":33742,"style":1257},[173],[28,33744,1213],{"className":33745},[1261],[28,33747],{"className":33748,"style":1257},[173],[28,33750,33752,33755,33759,33801],{"className":33751},[93],[28,33753],{"className":33754,"style":1964},[97],[28,33756,33452],{"className":33757,"style":33758},[102,106],"margin-right:0.0962em;",[28,33760,33762,33766],{"className":33761},[102],[28,33763,33457],{"className":33764,"style":33765},[102,106],"margin-right:0.0715em;",[28,33767,33769],{"className":33768},[111],[28,33770,33772,33793],{"className":33771},[115,116],[28,33773,33775,33790],{"className":33774},[120],[28,33776,33778],{"className":33777,"style":32501},[124],[28,33779,33781,33784],{"style":33780},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[28,33782],{"className":33783,"style":133},[132],[28,33785,33787],{"className":33786},[137,138,139,140],[28,33788,3256],{"className":33789},[102,106,140],[28,33791,148],{"className":33792},[147],[28,33794,33796],{"className":33795},[120],[28,33797,33799],{"className":33798,"style":155},[124],[28,33800],{},[28,33802,33804,33807],{"className":33803},[102],[28,33805,33457],{"className":33806,"style":33765},[102,106],[28,33808,33810],{"className":33809},[111],[28,33811,33813,33833],{"className":33812},[115,116],[28,33814,33816,33830],{"className":33815},[120],[28,33817,33819],{"className":33818,"style":32501},[124],[28,33820,33821,33824],{"style":33780},[28,33822],{"className":33823,"style":133},[132],[28,33825,33827],{"className":33826},[137,138,139,140],[28,33828,28200],{"className":33829,"style":32522},[102,106,140],[28,33831,148],{"className":33832},[147],[28,33834,33836],{"className":33835},[120],[28,33837,33839],{"className":33838,"style":2003},[124],[28,33840],{},[15650,33842,33844],{"className":15652,"code":33843,"language":15654,"meta":15655,"style":15655},"# 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",[8103,33845,33846,33851,33861,33873,33877,33882,33928,33946,33956,33971,33989,34008,34031,34049,34075,34105,34134,34180,34184,34189,34205],{"__ignoreMap":15655},[28,33847,33848],{"class":15660,"line":15661},[28,33849,33850],{"class":15664},"# Define problem Hamiltonian.\n",[28,33852,33853,33856,33858],{"class":15660,"line":15668},[28,33854,33855],{"class":15675},"n_qubits ",[28,33857,845],{"class":15682},[28,33859,33860],{"class":15801}," 30\n",[28,33862,33863,33866,33868,33870],{"class":15660,"line":15679},[28,33864,33865],{"class":15675},"J ",[28,33867,845],{"class":15682},[28,33869,22230],{"class":15801},[28,33871,33872],{"class":15664},"  # coupling strength for ZZ interaction\n",[28,33874,33875],{"class":15660,"line":15725},[28,33876,15769],{"emptyLinePlaceholder":15768},[28,33878,33879],{"class":15660,"line":15739},[28,33880,33881],{"class":15664},"# Define the Hamiltonian:\n",[28,33883,33884,33887,33889,33892,33895,33897,33899,33902,33904,33907,33909,33911,33913,33916,33918,33921,33923,33925],{"class":15660,"line":15752},[28,33885,33886],{"class":15675},"H_int ",[28,33888,845],{"class":15682},[28,33890,33891],{"class":15675}," [[",[28,33893,33894],{"class":15841},"\"I\"",[28,33896,15845],{"class":15675},[28,33898,16226],{"class":15682},[28,33900,33901],{"class":15675}," n_qubits ",[28,33903,22277],{"class":15671},[28,33905,33906],{"class":15675}," _ ",[28,33908,17720],{"class":15671},[28,33910,22285],{"class":15682},[28,33912,65],{"class":15675},[28,33914,33915],{"class":15801},"3",[28,33917,16210],{"class":15682},[28,33919,33920],{"class":15675}," (n_qubits ",[28,33922,15689],{"class":15682},[28,33924,22230],{"class":15801},[28,33926,33927],{"class":15675},"))]\n",[28,33929,33930,33932,33934,33936,33938,33940,33942,33944],{"class":15660,"line":15765},[28,33931,22277],{"class":15671},[28,33933,22280],{"class":15675},[28,33935,17720],{"class":15671},[28,33937,22285],{"class":15682},[28,33939,27958],{"class":15675},[28,33941,15689],{"class":15682},[28,33943,22230],{"class":15801},[28,33945,16407],{"class":15675},[28,33947,33948,33951,33953],{"class":15660,"line":15772},[28,33949,33950],{"class":15675},"    H_int[i][i] ",[28,33952,845],{"class":15682},[28,33954,33955],{"class":15841}," \"Z\"\n",[28,33957,33958,33961,33963,33965,33967,33969],{"class":15660,"line":15781},[28,33959,33960],{"class":15675},"    H_int[i][i ",[28,33962,2606],{"class":15682},[28,33964,22230],{"class":15801},[28,33966,15845],{"class":15675},[28,33968,845],{"class":15682},[28,33970,33955],{"class":15841},[28,33972,33973,33975,33977,33979,33981,33983,33985,33987],{"class":15660,"line":15798},[28,33974,22277],{"class":15671},[28,33976,22280],{"class":15675},[28,33978,17720],{"class":15671},[28,33980,22285],{"class":15682},[28,33982,27958],{"class":15675},[28,33984,15689],{"class":15682},[28,33986,22230],{"class":15801},[28,33988,16407],{"class":15675},[28,33990,33991,33994,33996,33998,34000,34003,34005],{"class":15660,"line":15811},[28,33992,33993],{"class":15675},"    H_int[n_qubits ",[28,33995,15689],{"class":15682},[28,33997,22230],{"class":15801},[28,33999,16359],{"class":15682},[28,34001,34002],{"class":15675}," i][i] ",[28,34004,845],{"class":15682},[28,34006,34007],{"class":15841}," \"X\"\n",[28,34009,34010,34012,34014,34016,34018,34021,34023,34025,34027,34029],{"class":15660,"line":15820},[28,34011,33993],{"class":15675},[28,34013,15689],{"class":15682},[28,34015,22230],{"class":15801},[28,34017,16359],{"class":15682},[28,34019,34020],{"class":15675}," i][i ",[28,34022,2606],{"class":15682},[28,34024,22230],{"class":15801},[28,34026,15845],{"class":15675},[28,34028,845],{"class":15682},[28,34030,34007],{"class":15841},[28,34032,34033,34035,34037,34039,34041,34043,34045,34047],{"class":15660,"line":15830},[28,34034,22277],{"class":15671},[28,34036,22280],{"class":15675},[28,34038,17720],{"class":15671},[28,34040,22285],{"class":15682},[28,34042,27958],{"class":15675},[28,34044,15689],{"class":15682},[28,34046,22230],{"class":15801},[28,34048,16407],{"class":15675},[28,34050,34051,34054,34056,34058,34060,34062,34064,34066,34068,34070,34072],{"class":15660,"line":15835},[28,34052,34053],{"class":15675},"    H_int[",[28,34055,848],{"class":15801},[28,34057,16210],{"class":15682},[28,34059,33920],{"class":15675},[28,34061,15689],{"class":15682},[28,34063,22230],{"class":15801},[28,34065,22233],{"class":15675},[28,34067,2606],{"class":15682},[28,34069,34002],{"class":15675},[28,34071,845],{"class":15682},[28,34073,34074],{"class":15841}," \"Y\"\n",[28,34076,34077,34079,34081,34083,34085,34087,34089,34091,34093,34095,34097,34099,34101,34103],{"class":15660,"line":15853},[28,34078,34053],{"class":15675},[28,34080,848],{"class":15801},[28,34082,16210],{"class":15682},[28,34084,33920],{"class":15675},[28,34086,15689],{"class":15682},[28,34088,22230],{"class":15801},[28,34090,22233],{"class":15675},[28,34092,2606],{"class":15682},[28,34094,34020],{"class":15675},[28,34096,2606],{"class":15682},[28,34098,22230],{"class":15801},[28,34100,15845],{"class":15675},[28,34102,845],{"class":15682},[28,34104,34074],{"class":15841},[28,34106,34107,34109,34111,34113,34116,34118,34121,34124,34126,34129,34131],{"class":15660,"line":15867},[28,34108,33886],{"class":15675},[28,34110,845],{"class":15682},[28,34112,17337],{"class":15675},[28,34114,34115],{"class":15841},"\"\"",[28,34117,2106],{"class":15675},[28,34119,34120],{"class":16073},"join",[28,34122,34123],{"class":15675},"(term) ",[28,34125,22277],{"class":15671},[28,34127,34128],{"class":15675}," term ",[28,34130,17720],{"class":15671},[28,34132,34133],{"class":15675}," H_int]\n",[28,34135,34136,34139,34141,34144,34146,34149,34152,34154,34157,34159,34161,34163,34165,34168,34170,34172,34174,34176,34178],{"class":15660,"line":15872},[28,34137,34138],{"class":15675},"H_tot ",[28,34140,845],{"class":15682},[28,34142,34143],{"class":15675}," [(term, J) ",[28,34145,20133],{"class":15671},[28,34147,34148],{"class":15675}," term.",[28,34150,34151],{"class":16073},"count",[28,34153,65],{"class":15675},[28,34155,34156],{"class":15841},"\"Z\"",[28,34158,22233],{"class":15675},[28,34160,23650],{"class":15682},[28,34162,22129],{"class":15801},[28,34164,24971],{"class":15671},[28,34166,34167],{"class":15675}," (term, ",[28,34169,3200],{"class":15801},[28,34171,22233],{"class":15675},[28,34173,22277],{"class":15671},[28,34175,34128],{"class":15675},[28,34177,17720],{"class":15671},[28,34179,34133],{"class":15675},[28,34181,34182],{"class":15660,"line":15878},[28,34183,15769],{"emptyLinePlaceholder":15768},[28,34185,34186],{"class":15660,"line":15897},[28,34187,34188],{"class":15664},"# Get operator\n",[28,34190,34191,34194,34196,34199,34202],{"class":15660,"line":15914},[28,34192,34193],{"class":15675},"H_op ",[28,34195,845],{"class":15682},[28,34197,34198],{"class":15675}," SparsePauliOp.",[28,34200,34201],{"class":16073},"from_list",[28,34203,34204],{"class":15675},"(H_tot)\n",[28,34206,34207,34209],{"class":15660,"line":15931},[28,34208,16044],{"class":15682},[28,34210,34204],{"class":15675},[15650,34212,34215],{"className":34213,"code":34214,"language":16058,"meta":15655},[16056],"[('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",[8103,34216,34214],{"__ignoreMap":15655},[33354,34218,34220],{"id":34219},"set-parameters-for-the-algorithm","Set parameters for the algorithm",[12,34222,34223,34224,34226,34227,34230,34231,34283,34284,34317],{},"We heuristically choose a value for the time-step ",[8103,34225,31182],{}," (based on upper bounds on the Hamiltonian norm). Ref ",[19,34228,34229],{"href":27044},"[2]","  showed that a sufficiently small timestep is ",[28,34232,34234,34260],{"className":34233},[35],[28,34235,34237],{"className":34236},[39],[41,34238,34239],{"xmlns":43},[46,34240,34241,34257],{},[49,34242,34243,34245,34247,34249,34251,34253,34255],{},[55,34244,851],{},[55,34246,1129],{"mathvariant":1088},[55,34248,296],{"mathvariant":1088},[55,34250,296],{"mathvariant":1088},[55,34252,2263],{},[55,34254,296],{"mathvariant":1088},[55,34256,296],{"mathvariant":1088},[82,34258,34259],{"encoding":84},"\\pi\u002F\\vert \\vert H \\vert \\vert",[28,34261,34263],{"className":34262,"ariaHidden":71},[89],[28,34264,34266,34269,34272,34276,34279],{"className":34265},[93],[28,34267],{"className":34268,"style":98},[97],[28,34270,851],{"className":34271,"style":321},[102,106],[28,34273,34275],{"className":34274},[102],"\u002F∣∣",[28,34277,2263],{"className":34278,"style":4040},[102,106],[28,34280,34282],{"className":34281},[102],"∣∣",", 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 ",[28,34285,34287,34302],{"className":34286},[35],[28,34288,34290],{"className":34289},[39],[41,34291,34292],{"xmlns":43},[46,34293,34294,34300],{},[49,34295,34296,34298],{},[55,34297,24201],{},[55,34299,3971],{},[82,34301,31182],{"encoding":84},[28,34303,34305],{"className":34304,"ariaHidden":71},[89],[28,34306,34308,34311,34314],{"className":34307},[93],[28,34309],{"className":34310,"style":9597},[97],[28,34312,24201],{"className":34313},[102,106],[28,34315,3971],{"className":34316},[102,106]," to be too small leads to worse conditioning of the Krylov subspace, since the Krylov basis vectors differ less from timestep to timestep.",[15650,34319,34321],{"className":15652,"code":34320,"language":15654,"meta":15655,"style":15655},"# 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",[8103,34322,34323,34328,34342,34355,34374,34388,34401,34413,34422,34463,34468,34477,34515,34535,34542,34550,34563,34575,34594,34608,34612,34617,34646],{"__ignoreMap":15655},[28,34324,34325],{"class":15660,"line":15661},[28,34326,34327],{"class":15664},"# Get Hamiltonian restricted to single-particle states\n",[28,34329,34330,34333,34335,34337,34339],{"class":15660,"line":15668},[28,34331,34332],{"class":15675},"single_particle_H ",[28,34334,845],{"class":15682},[28,34336,16298],{"class":15675},[28,34338,27999],{"class":16073},[28,34340,34341],{"class":15675},"((n_qubits, n_qubits))\n",[28,34343,34344,34346,34348,34350,34352],{"class":15660,"line":15679},[28,34345,22277],{"class":15671},[28,34347,22280],{"class":15675},[28,34349,17720],{"class":15671},[28,34351,22285],{"class":15682},[28,34353,34354],{"class":15675},"(n_qubits):\n",[28,34356,34357,34359,34361,34363,34365,34368,34370,34372],{"class":15660,"line":15725},[28,34358,27785],{"class":15671},[28,34360,22400],{"class":15675},[28,34362,17720],{"class":15671},[28,34364,22285],{"class":15682},[28,34366,34367],{"class":15675},"(i ",[28,34369,2606],{"class":15682},[28,34371,22230],{"class":15801},[28,34373,16407],{"class":15675},[28,34375,34376,34378,34380,34382,34384,34386],{"class":15660,"line":15739},[28,34377,17714],{"class":15671},[28,34379,27788],{"class":15675},[28,34381,17720],{"class":15671},[28,34383,27793],{"class":15675},[28,34385,27796],{"class":16073},[28,34387,27799],{"class":15675},[28,34389,34390,34393,34395,34398],{"class":15660,"line":15752},[28,34391,34392],{"class":15675},"            p_x ",[28,34394,845],{"class":15682},[28,34396,34397],{"class":16073}," Pauli",[28,34399,34400],{"class":15675},"(p).x\n",[28,34402,34403,34406,34408,34410],{"class":15660,"line":15765},[28,34404,34405],{"class":15675},"            p_z ",[28,34407,845],{"class":15682},[28,34409,34397],{"class":16073},[28,34411,34412],{"class":15675},"(p).z\n",[28,34414,34415,34417,34420],{"class":15660,"line":15772},[28,34416,28131],{"class":15671},[28,34418,34419],{"class":15682}," all",[28,34421,16077],{"class":15675},[28,34423,34424,34427,34429,34432,34434,34437,34439,34442,34444,34447,34449,34451,34454,34456,34458,34460],{"class":15660,"line":15781},[28,34425,34426],{"class":15675},"                p_x[k] ",[28,34428,23650],{"class":15682},[28,34430,34431],{"class":15675}," ((i ",[28,34433,23650],{"class":15682},[28,34435,34436],{"class":15675}," k) ",[28,34438,2606],{"class":15682},[28,34440,34441],{"class":15675}," (j ",[28,34443,23650],{"class":15682},[28,34445,34446],{"class":15675}," k)) ",[28,34448,15683],{"class":15682},[28,34450,22129],{"class":15801},[28,34452,34453],{"class":15671}," for",[28,34455,27616],{"class":15675},[28,34457,17720],{"class":15671},[28,34459,22285],{"class":15682},[28,34461,34462],{"class":15675},"(n_qubits)\n",[28,34464,34465],{"class":15660,"line":15798},[28,34466,34467],{"class":15675},"            ):\n",[28,34469,34470,34473,34475],{"class":15660,"line":15811},[28,34471,34472],{"class":15675},"                sgn ",[28,34474,845],{"class":15682},[28,34476,18567],{"class":15675},[28,34478,34479,34482,34484,34486,34488,34490,34492,34495,34498,34501,34504,34506,34508,34510,34512],{"class":15660,"line":15820},[28,34480,34481],{"class":15675},"                    (",[28,34483,15689],{"class":15682},[28,34485,3200],{"class":15801},[28,34487,28200],{"class":15671},[28,34489,22233],{"class":15675},[28,34491,16354],{"class":15682},[28,34493,34494],{"class":15682}," sum",[28,34496,34497],{"class":15675},"(p_z[k] ",[28,34499,34500],{"class":15671},"and",[28,34502,34503],{"class":15675}," p_x[k] ",[28,34505,22277],{"class":15671},[28,34507,27616],{"class":15675},[28,34509,17720],{"class":15671},[28,34511,22285],{"class":15682},[28,34513,34514],{"class":15675},"(n_qubits))\n",[28,34516,34517,34520,34522,34524,34526,34528,34530,34532],{"class":15660,"line":15830},[28,34518,34519],{"class":15675},"                ) ",[28,34521,16226],{"class":15682},[28,34523,28193],{"class":15675},[28,34525,15689],{"class":15682},[28,34527,3200],{"class":15801},[28,34529,22233],{"class":15675},[28,34531,16354],{"class":15682},[28,34533,34534],{"class":15675}," p_z[i])\n",[28,34536,34537,34540],{"class":15660,"line":15835},[28,34538,34539],{"class":15671},"            else",[28,34541,15778],{"class":15675},[28,34543,34544,34546,34548],{"class":15660,"line":15853},[28,34545,34472],{"class":15675},[28,34547,845],{"class":15682},[28,34549,28085],{"class":15801},[28,34551,34552,34555,34557,34559,34561],{"class":15660,"line":15867},[28,34553,34554],{"class":15675},"            single_particle_H[i, j] ",[28,34556,28120],{"class":15682},[28,34558,28275],{"class":15675},[28,34560,16226],{"class":15682},[28,34562,28280],{"class":15675},[28,34564,34565,34567,34569,34571,34573],{"class":15660,"line":15872},[28,34566,22277],{"class":15671},[28,34568,22280],{"class":15675},[28,34570,17720],{"class":15671},[28,34572,22285],{"class":15682},[28,34574,34354],{"class":15675},[28,34576,34577,34579,34581,34583,34585,34587,34589,34591],{"class":15660,"line":15878},[28,34578,27785],{"class":15671},[28,34580,22400],{"class":15675},[28,34582,17720],{"class":15671},[28,34584,22285],{"class":15682},[28,34586,34367],{"class":15675},[28,34588,2606],{"class":15682},[28,34590,22230],{"class":15801},[28,34592,34593],{"class":15675},", n_qubits):\n",[28,34595,34596,34599,34601,34603,34605],{"class":15660,"line":15897},[28,34597,34598],{"class":15675},"        single_particle_H[i, j] ",[28,34600,845],{"class":15682},[28,34602,16298],{"class":15675},[28,34604,27574],{"class":16073},[28,34606,34607],{"class":15675},"(single_particle_H[j, i])\n",[28,34609,34610],{"class":15660,"line":15914},[28,34611,15769],{"emptyLinePlaceholder":15768},[28,34613,34614],{"class":15660,"line":15931},[28,34615,34616],{"class":15664},"# Set dt according to spectral norm\n",[28,34618,34619,34622,34624,34626,34628,34631,34634,34637,34640,34642,34644],{"class":15660,"line":15948},[28,34620,34621],{"class":15675},"dt ",[28,34623,845],{"class":15682},[28,34625,18394],{"class":15675},[28,34627,1129],{"class":15682},[28,34629,34630],{"class":15675}," np.linalg.",[28,34632,34633],{"class":16073},"norm",[28,34635,34636],{"class":15675},"(single_particle_H, ",[28,34638,34639],{"class":16307},"ord",[28,34641,845],{"class":15682},[28,34643,848],{"class":15801},[28,34645,16052],{"class":15675},[28,34647,34648],{"class":15660,"line":15966},[28,34649,34650],{"class":15675},"dt\n",[15650,34652,34655],{"className":34653,"code":34654,"language":16058,"meta":15655},[16056],"np.float64(0.10833078115826875)\n",[8103,34656,34654],{"__ignoreMap":15655},[12,34658,34659],{},"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.",[15650,34661,34663],{"className":15652,"code":34662,"language":15654,"meta":15655,"style":15655},"# 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",[8103,34664,34665,34670,34683,34693],{"__ignoreMap":15655},[28,34666,34667],{"class":15660,"line":15661},[28,34668,34669],{"class":15664},"# Set parameters for quantum Krylov algorithm\n",[28,34671,34672,34675,34677,34680],{"class":15660,"line":15668},[28,34673,34674],{"class":15675},"krylov_dim ",[28,34676,845],{"class":15682},[28,34678,34679],{"class":15801}," 5",[28,34681,34682],{"class":15664},"  # size of Krylov subspace\n",[28,34684,34685,34688,34690],{"class":15660,"line":15679},[28,34686,34687],{"class":15675},"num_trotter_steps ",[28,34689,845],{"class":15682},[28,34691,34692],{"class":15801}," 6\n",[28,34694,34695,34698,34700,34703,34705],{"class":15660,"line":15725},[28,34696,34697],{"class":15675},"dt_circ ",[28,34699,845],{"class":15682},[28,34701,34702],{"class":15675}," dt ",[28,34704,1129],{"class":15682},[28,34706,34707],{"class":15675}," num_trotter_steps\n",[33354,34709,34711],{"id":34710},"state-preparation","State preparation",[12,34713,34714,34715,34754,34755,34793],{},"Pick a reference state ",[28,34716,34718,34736],{"className":34717},[35],[28,34719,34721],{"className":34720},[39],[41,34722,34723],{"xmlns":43},[46,34724,34725,34733],{},[49,34726,34727,34729,34731],{},[55,34728,296],{"mathvariant":1088},[55,34730,394],{},[62,34732,302],{"stretchy":64},[82,34734,34735],{"encoding":84},"\\vert \\psi \\rangle",[28,34737,34739],{"className":34738,"ariaHidden":71},[89],[28,34740,34742,34745,34748,34751],{"className":34741},[93],[28,34743],{"className":34744,"style":98},[97],[28,34746,296],{"className":34747},[102],[28,34749,394],{"className":34750,"style":321},[102,106],[28,34752,302],{"className":34753},[182]," that has some overlap with the ground state. For this Hamiltonian, We use the a state with an excitation in the middle qubit ",[28,34756,34758,34777],{"className":34757},[35],[28,34759,34761],{"className":34760},[39],[41,34762,34763],{"xmlns":43},[46,34764,34765,34774],{},[49,34766,34767,34769,34772],{},[55,34768,296],{"mathvariant":1088},[650,34770,34771],{},"00..010...00",[62,34773,302],{"stretchy":64},[82,34775,34776],{"encoding":84},"\\vert 00..010...00 \\rangle",[28,34778,34780],{"className":34779,"ariaHidden":71},[89],[28,34781,34783,34786,34790],{"className":34782},[93],[28,34784],{"className":34785,"style":98},[97],[28,34787,34789],{"className":34788},[102],"∣00..010...00",[28,34791,302],{"className":34792},[182]," as our reference state.",[15650,34795,34797],{"className":15652,"code":34796,"language":15654,"meta":15655,"style":15655},"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",[8103,34798,34799,34810,34835],{"__ignoreMap":15655},[28,34800,34801,34804,34806,34808],{"class":15660,"line":15661},[28,34802,34803],{"class":15675},"qc_state_prep ",[28,34805,845],{"class":15682},[28,34807,26583],{"class":16073},[28,34809,34462],{"class":15675},[28,34811,34812,34815,34817,34819,34821,34823,34825,34827,34829,34831,34833],{"class":15660,"line":15668},[28,34813,34814],{"class":15675},"qc_state_prep.",[28,34816,3169],{"class":16073},[28,34818,65],{"class":15675},[28,34820,15955],{"class":15682},[28,34822,27958],{"class":15675},[28,34824,1129],{"class":15682},[28,34826,22129],{"class":15801},[28,34828,22233],{"class":15675},[28,34830,2606],{"class":15682},[28,34832,22230],{"class":15801},[28,34834,16052],{"class":15675},[28,34836,34837,34839,34842,34844,34847,34849,34852,34854,34857],{"class":15660,"line":15679},[28,34838,34814],{"class":15675},[28,34840,34841],{"class":16073},"draw",[28,34843,65],{"class":15675},[28,34845,34846],{"class":15841},"\"mpl\"",[28,34848,16397],{"class":15675},[28,34850,34851],{"class":16307},"scale",[28,34853,845],{"class":15682},[28,34855,34856],{"class":15801},"0.5",[28,34858,16052],{"class":15675},[17041,34860],{"alt":34861,"src":34862},"Output of the previous code cell","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-01.avif",[33354,34864,34866],{"id":34865},"time-evolution","Time evolution",[12,34868,34869,34870,34970,34971,2106],{},"We can realize the time-evolution operator generated by a given Hamiltonian: ",[28,34871,34873,34902],{"className":34872},[35],[28,34874,34876],{"className":34875},[39],[41,34877,34878],{"xmlns":43},[46,34879,34880,34900],{},[49,34881,34882,34884,34886],{},[55,34883,29018],{},[62,34885,845],{},[2776,34887,34888,34890],{},[55,34889,60],{},[49,34891,34892,34894,34896,34898],{},[62,34893,1213],{},[55,34895,3256],{},[55,34897,2263],{},[55,34899,3971],{},[82,34901,29037],{"encoding":84},[28,34903,34905,34923],{"className":34904,"ariaHidden":71},[89],[28,34906,34908,34911,34914,34917,34920],{"className":34907},[93],[28,34909],{"className":34910,"style":1102},[97],[28,34912,29018],{"className":34913,"style":29050},[102,106],[28,34915],{"className":34916,"style":913},[173],[28,34918,845],{"className":34919},[917],[28,34921],{"className":34922,"style":913},[173],[28,34924,34926,34929],{"className":34925},[93],[28,34927],{"className":34928,"style":29066},[97],[28,34930,34932,34935],{"className":34931},[102],[28,34933,60],{"className":34934},[102,106],[28,34936,34938],{"className":34937},[111],[28,34939,34941],{"className":34940},[115],[28,34942,34944],{"className":34943},[120],[28,34945,34947],{"className":34946,"style":29066},[124],[28,34948,34949,34952],{"style":1726},[28,34950],{"className":34951,"style":133},[132],[28,34953,34955],{"className":34954},[137,138,139,140],[28,34956,34958,34961,34964,34967],{"className":34957},[102,140],[28,34959,1213],{"className":34960},[102,140],[28,34962,3256],{"className":34963},[102,106,140],[28,34965,2263],{"className":34966,"style":4040},[102,106,140],[28,34968,3971],{"className":34969},[102,106,140]," via the ",[19,34972,34974],{"href":34973},"https:\u002F\u002Fquantum.cloud.ibm.com\u002Fdocs\u002Fapi\u002Fqiskit\u002Fqiskit.synthesis.LieTrotter","Lie-Trotter approximation",[15650,34976,34978],{"className":15652,"code":34977,"language":15654,"meta":15655,"style":15655},"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",[8103,34979,34980,34997,35001,35006,35018,35049,35053,35057,35069,35081],{"__ignoreMap":15655},[28,34981,34982,34985,34987,34990,34992,34995],{"class":15660,"line":15661},[28,34983,34984],{"class":15675},"t ",[28,34986,845],{"class":15682},[28,34988,34989],{"class":16073}," Parameter",[28,34991,65],{"class":15675},[28,34993,34994],{"class":15841},"\"t\"",[28,34996,16052],{"class":15675},[28,34998,34999],{"class":15660,"line":15668},[28,35000,15769],{"emptyLinePlaceholder":15768},[28,35002,35003],{"class":15660,"line":15679},[28,35004,35005],{"class":15664},"## Create the time-evo op circuit\n",[28,35007,35008,35011,35013,35016],{"class":15660,"line":15725},[28,35009,35010],{"class":15675},"evol_gate ",[28,35012,845],{"class":15682},[28,35014,35015],{"class":16073}," PauliEvolutionGate",[28,35017,16077],{"class":15675},[28,35019,35020,35023,35026,35028,35031,35034,35036,35039,35041,35044,35046],{"class":15660,"line":15739},[28,35021,35022],{"class":15675},"    H_op, ",[28,35024,35025],{"class":16307},"time",[28,35027,845],{"class":15682},[28,35029,35030],{"class":15675},"t, ",[28,35032,35033],{"class":16307},"synthesis",[28,35035,845],{"class":15682},[28,35037,35038],{"class":16073},"LieTrotter",[28,35040,65],{"class":15675},[28,35042,35043],{"class":16307},"reps",[28,35045,845],{"class":15682},[28,35047,35048],{"class":15675},"num_trotter_steps)\n",[28,35050,35051],{"class":15660,"line":15752},[28,35052,16052],{"class":15675},[28,35054,35055],{"class":15660,"line":15765},[28,35056,15769],{"emptyLinePlaceholder":15768},[28,35058,35059,35062,35064,35067],{"class":15660,"line":15772},[28,35060,35061],{"class":15675},"qr ",[28,35063,845],{"class":15682},[28,35065,35066],{"class":16073}," QuantumRegister",[28,35068,34462],{"class":15675},[28,35070,35071,35074,35076,35078],{"class":15660,"line":15781},[28,35072,35073],{"class":15675},"qc_evol ",[28,35075,845],{"class":15682},[28,35077,26583],{"class":16073},[28,35079,35080],{"class":15675},"(qr)\n",[28,35082,35083,35086,35088,35091,35094,35096],{"class":15660,"line":15798},[28,35084,35085],{"class":15675},"qc_evol.",[28,35087,27807],{"class":16073},[28,35089,35090],{"class":15675},"(evol_gate, ",[28,35092,35093],{"class":16307},"qargs",[28,35095,845],{"class":15682},[28,35097,35098],{"class":15675},"qr)\n",[15650,35100,35103],{"className":35101,"code":35102,"language":16058,"meta":15655},[16056],"\u003Cqiskit.circuit.instructionset.InstructionSet at 0x11eef9be0>\n",[8103,35104,35102],{"__ignoreMap":15655},[33354,35106,35108],{"id":35107},"hadamard-test","Hadamard test",[12,35110,35111],{},[32407,35112],{"alt":35113,"src":35114},"fig2.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-07.avif",[28,35116,35118],{"className":35117},[31],[28,35119,35121,35361],{"className":35120},[35],[28,35122,35124],{"className":35123},[39],[41,35125,35126],{"xmlns":43,"display":44},[46,35127,35128,35358],{},[3178,35129,35130],{"rowspacing":3180,"columnspacing":3182},[3184,35131,35132],{},[3187,35133,35134],{},[3190,35135,35136],{"scriptlevel":652,"displaystyle":71},[49,35137,35138,35140,35142,35144,35146,35148,35154,35156,35159,35161,35169,35172,35174,35176,35178,35180,35182,35184,35186,35188,35190,35192,35198,35200,35202,35204,35212,35214,35216,35218,35220,35222,35224,35230,35232,35234,35236,35238,35240,35246,35248,35250,35252,35254,35256,35264,35266,35268,35270,35272,35274,35276,35282,35284,35286,35288,35290,35292,35294,35300,35302,35304,35306,35308,35310,35318,35320,35322,35324,35326,35328,35334,35336,35338,35340,35342,35344,35346,35348,35354,35356],{},[55,35139,296],{"mathvariant":1088},[650,35141,652],{},[62,35143,302],{"stretchy":64},[55,35145,296],{"mathvariant":1088},[650,35147,652],{},[2776,35149,35150,35152],{},[62,35151,302],{"stretchy":64},[55,35153,33375],{},[173,35155],{"width":3182},[62,35157,35158],{},"⟶",[173,35160],{"width":3182},[3977,35162,35163,35165],{},[650,35164,3200],{},[5661,35166,35167],{},[650,35168,848],{},[62,35170,65],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},"1.8em",[55,35173,296],{"mathvariant":1088},[650,35175,652],{},[62,35177,302],{"stretchy":64},[62,35179,2606],{},[55,35181,296],{"mathvariant":1088},[650,35183,3200],{},[62,35185,302],{"stretchy":64},[62,35187,78],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[55,35189,296],{"mathvariant":1088},[650,35191,652],{},[2776,35193,35194,35196],{},[62,35195,302],{"stretchy":64},[55,35197,33375],{},[173,35199],{"width":3182},[62,35201,35158],{},[173,35203],{"width":3182},[3977,35205,35206,35208],{},[650,35207,3200],{},[5661,35209,35210],{},[650,35211,848],{},[62,35213,65],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[55,35215,296],{"mathvariant":1088},[650,35217,652],{},[62,35219,302],{"stretchy":64},[55,35221,296],{"mathvariant":1088},[650,35223,652],{},[2776,35225,35226,35228],{},[62,35227,302],{"stretchy":64},[55,35229,33375],{},[62,35231,2606],{},[55,35233,296],{"mathvariant":1088},[650,35235,3200],{},[62,35237,302],{"stretchy":64},[55,35239,296],{"mathvariant":1088},[52,35241,35242,35244],{},[55,35243,394],{},[55,35245,3256],{},[62,35247,302],{"stretchy":64},[62,35249,78],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[173,35251],{"width":3182},[62,35253,35158],{},[173,35255],{"width":3182},[3977,35257,35258,35260],{},[650,35259,3200],{},[5661,35261,35262],{},[650,35263,848],{},[62,35265,65],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[55,35267,296],{"mathvariant":1088},[650,35269,652],{},[62,35271,302],{"stretchy":64},[55,35273,296],{"mathvariant":1088},[650,35275,652],{},[2776,35277,35278,35280],{},[62,35279,302],{"stretchy":64},[55,35281,33375],{},[62,35283,2606],{},[55,35285,296],{"mathvariant":1088},[650,35287,3200],{},[62,35289,302],{"stretchy":64},[55,35291,57],{},[55,35293,296],{"mathvariant":1088},[52,35295,35296,35298],{},[55,35297,394],{},[55,35299,3256],{},[62,35301,302],{"stretchy":64},[62,35303,78],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[173,35305],{"width":3182},[62,35307,35158],{},[173,35309],{"width":3182},[3977,35311,35312,35314],{},[650,35313,3200],{},[5661,35315,35316],{},[650,35317,848],{},[62,35319,65],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[55,35321,296],{"mathvariant":1088},[650,35323,652],{},[62,35325,302],{"stretchy":64},[55,35327,296],{"mathvariant":1088},[52,35329,35330,35332],{},[55,35331,394],{},[55,35333,28200],{},[62,35335,302],{"stretchy":64},[62,35337,2606],{},[55,35339,296],{"mathvariant":1088},[650,35341,3200],{},[62,35343,302],{"stretchy":64},[55,35345,57],{},[55,35347,296],{"mathvariant":1088},[52,35349,35350,35352],{},[55,35351,394],{},[55,35353,3256],{},[62,35355,302],{"stretchy":64},[62,35357,78],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[82,35359,35360],{"encoding":84},"\\begin{equation*}\n    |0\\rangle|0\\rangle^N \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle + |1\\rangle \\Big)|0\\rangle^N \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle|0\\rangle^N+|1\\rangle |\\psi_i\\rangle\\Big) \\quad\\longrightarrow\\quad \\frac{1}{\\sqrt{2}}\\Big(|0\\rangle |0\\rangle^N+|1\\rangle P |\\psi_i\\rangle\\Big) \\quad\\longrightarrow\\quad\\frac{1}{\\sqrt{2}}\\Big(|0\\rangle |\\psi_j\\rangle+|1\\rangle 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is one of the terms in the decomposition of the Hamiltonian ",[28,36388,36390,36410],{"className":36389},[35],[28,36391,36393],{"className":36392},[39],[41,36394,36395],{"xmlns":43},[46,36396,36397,36407],{},[49,36398,36399,36401,36403,36405],{},[55,36400,2263],{},[62,36402,845],{},[62,36404,33223],{},[55,36406,57],{},[82,36408,36409],{"encoding":84},"H=\\sum P",[28,36411,36413,36431],{"className":36412,"ariaHidden":71},[89],[28,36414,36416,36419,36422,36425,36428],{"className":36415},[93],[28,36417],{"className":36418,"style":1102},[97],[28,36420,2263],{"className":36421,"style":4040},[102,106],[28,36423],{"className":36424,"style":913},[173],[28,36426,845],{"className":36427},[917],[28,36429],{"className":36430,"style":913},[173],[28,36432,36434,36437,36440,36443],{"className":36433},[93],[28,36435],{"className":36436,"style":98},[97],[28,36438,33223],{"className":36439,"style":33271},[4241,33269,33270],[28,36441],{"className":36442,"style":174},[173],[28,36444,57],{"className":36445,"style":107},[102,106],[28,36447,36449,36470],{"className":36448},[35],[28,36450,36452],{"className":36451},[39],[41,36453,36454],{"xmlns":43},[46,36455,36456,36468],{},[49,36457,36458,36460,36462],{},[2253,36459,32711],{},[2253,36461,32714],{},[52,36463,36464,36466],{},[55,36465,394],{},[55,36467,3256],{},[82,36469,32723],{"encoding":84},[28,36471,36473],{"className":36472,"ariaHidden":71},[89],[28,36474,36476,36479,36485,36488],{"className":36475},[93],[28,36477],{"className":36478,"style":214},[97],[28,36480,36482],{"className":36481},[102,16058],[28,36483,32711],{"className":36484},[102],[28,36486],{"className":36487,"style":913},[173],[28,36489,36491,36494],{"className":36490},[102],[28,36492,394],{"className":36493,"style":321},[102,106],[28,36495,36497],{"className":36496},[111],[28,36498,36500,36520],{"className":36499},[115,116],[28,36501,36503,36517],{"className":36502},[120],[28,36504,36506],{"className":36505,"style":32501},[124],[28,36507,36508,36511],{"style":801},[28,36509],{"className":36510,"style":133},[132],[28,36512,36514],{"className":36513},[137,138,139,140],[28,36515,3256],{"className":36516},[102,106,140],[28,36518,148],{"className":36519},[147],[28,36521,36523],{"className":36522},[120],[28,36524,36526],{"className":36525,"style":155},[124],[28,36527],{},[28,36529,36531,36552],{"className":36530},[35],[28,36532,36534],{"className":36533},[39],[41,36535,36536],{"xmlns":43},[46,36537,36538,36550],{},[49,36539,36540,36542,36544],{},[2253,36541,32711],{},[2253,36543,32714],{},[52,36545,36546,36548],{},[55,36547,394],{},[55,36549,28200],{},[82,36551,32806],{"encoding":84},[28,36553,36555],{"className":36554,"ariaHidden":71},[89],[28,36556,36558,36561,36567,36570],{"className":36557},[93],[28,36559],{"className":36560,"style":32816},[97],[28,36562,36564],{"className":36563},[102,16058],[28,36565,32711],{"className":36566},[102],[28,36568],{"className":36569,"style":913},[173],[28,36571,36573,36576],{"className":36572},[102],[28,36574,394],{"className":36575,"style":321},[102,106],[28,36577,36579],{"className":36578},[111],[28,36580,36582,36602],{"className":36581},[115,116],[28,36583,36585,36599],{"className":36584},[120],[28,36586,36588],{"className":36587,"style":32501},[124],[28,36589,36590,36593],{"style":801},[28,36591],{"className":36592,"style":133},[132],[28,36594,36596],{"className":36595},[137,138,139,140],[28,36597,28200],{"className":36598,"style":32522},[102,106,140],[28,36600,148],{"className":36601},[147],[28,36603,36605],{"className":36604},[120],[28,36606,36608],{"className":36607,"style":2003},[124],[28,36609],{}," 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",[28,36860,36862,36923,36995],{"className":36861,"ariaHidden":71},[89],[28,36863,36865,36868,36871,36911,36914,36917,36920],{"className":36864},[93],[28,36866],{"className":36867,"style":98},[97],[28,36869,296],{"className":36870},[102],[28,36872,36874,36877],{"className":36873},[102],[28,36875,394],{"className":36876,"style":321},[102,106],[28,36878,36880],{"className":36879},[111],[28,36881,36883,36903],{"className":36882},[115,116],[28,36884,36886,36900],{"className":36885},[120],[28,36887,36889],{"className":36888,"style":10240},[124],[28,36890,36891,36894],{"style":801},[28,36892],{"className":36893,"style":133},[132],[28,36895,36897],{"className":36896},[137,138,139,140],[28,36898,29362],{"className":36899,"style":29427},[102,106,140],[28,36901,148],{"className":36902},[147],[28,36904,36906],{"className":36905},[120],[28,36907,36909],{"className":36908,"style":155},[124],[28,36910],{},[28,36912,302],{"className":36913},[182],[28,36915],{"className":36916,"style":913},[173],[28,36918,845],{"className":36919},[917],[28,36921],{"className":36922,"style":913},[173],[28,36924,36926,36930,36977,36980,36983,36986,36989,36992],{"className":36925},[93],[28,36927],{"className":36928,"style":36929},[97],"height:1.0991em;vertical-align:-0.25em;",[28,36931,36933,36936],{"className":36932},[102],[28,36934,60],{"className":36935},[102,106],[28,36937,36939],{"className":36938},[111],[28,36940,36942],{"className":36941},[115],[28,36943,36945],{"className":36944},[120],[28,36946,36948],{"className":36947,"style":29397},[124],[28,36949,36950,36953],{"style":1726},[28,36951],{"className":36952,"style":133},[132],[28,36954,36956],{"className":36955},[137,138,139,140],[28,36957,36959,36962,36965,36968,36971,36974],{"className":36958},[102,140],[28,36960,1213],{"className":36961},[102,140],[28,36963,3256],{"className":36964},[102,106,140],[28,36966,2263],{"className":36967,"style":4040},[102,106,140],[28,36969,29362],{"className":36970,"style":29427},[102,106,140],[28,36972,24201],{"className":36973},[102,106,140],[28,36975,3971],{"className":36976},[102,106,140],[28,36978,296],{"className":36979},[102],[28,36981,394],{"className":36982,"style":321},[102,106],[28,36984,302],{"className":36985},[182],[28,36987],{"className":36988,"style":913},[173],[28,36990,845],{"className":36991},[917],[28,36993],{"className":36994,"style":913},[173],[28,36996,36998,37002,37049,37093,37096],{"className":36997},[93],[28,36999],{"className":37000,"style":37001},[97],"height:1.1352em;vertical-align:-0.2861em;",[28,37003,37005,37008],{"className":37004},[102],[28,37006,60],{"className":37007},[102,106],[28,37009,37011],{"className":37010},[111],[28,37012,37014],{"className":37013},[115],[28,37015,37017],{"className":37016},[120],[28,37018,37020],{"className":37019,"style":29397},[124],[28,37021,37022,37025],{"style":1726},[28,37023],{"className":37024,"style":133},[132],[28,37026,37028],{"className":37027},[137,138,139,140],[28,37029,37031,37034,37037,37040,37043,37046],{"className":37030},[102,140],[28,37032,1213],{"className":37033},[102,140],[28,37035,3256],{"className":37036},[102,106,140],[28,37038,2263],{"className":37039,"style":4040},[102,106,140],[28,37041,29362],{"className":37042,"style":29427},[102,106,140],[28,37044,24201],{"className":37045},[102,106,140],[28,37047,3971],{"className":37048},[102,106,140],[28,37050,37052,37055],{"className":37051},[102],[28,37053,29018],{"className":37054,"style":29050},[102,106],[28,37056,37058],{"className":37057},[111],[28,37059,37061,37085],{"className":37060},[115,116],[28,37062,37064,37082],{"className":37063},[120],[28,37065,37067],{"className":37066,"style":10240},[124],[28,37068,37070,37073],{"style":37069},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[28,37071],{"className":37072,"style":133},[132],[28,37074,37076],{"className":37075},[137,138,139,140],[28,37077,37079],{"className":37078},[102,140],[28,37080,394],{"className":37081,"style":321},[102,106,140],[28,37083,148],{"className":37084},[147],[28,37086,37088],{"className":37087},[120],[28,37089,37091],{"className":37090,"style":2003},[124],[28,37092],{},[28,37094,35403],{"className":37095},[102],[28,37097,37099,37102],{"className":37098},[182],[28,37100,302],{"className":37101},[182],[28,37103,37105],{"className":37104},[111],[28,37106,37108],{"className":37107},[115],[28,37109,37111],{"className":37110},[120],[28,37112,37114],{"className":37113,"style":29066},[124],[28,37115,37116,37119],{"style":1726},[28,37117],{"className":37118,"style":133},[132],[28,37120,37122],{"className":37121},[137,138,139,140],[28,37123,33375],{"className":37124,"style":29050},[102,106,140],". To measure ",[28,37127,37129,37142],{"className":37128},[35],[28,37130,37132],{"className":37131},[39],[41,37133,37134],{"xmlns":43},[46,37135,37136,37140],{},[49,37137,37138],{},[55,37139,9841],{},[82,37141,9841],{"encoding":84},[28,37143,37145],{"className":37144,"ariaHidden":71},[89],[28,37146,37148,37151],{"className":37147},[93],[28,37149],{"className":37150,"style":1102},[97],[28,37152,9841],{"className":37153,"style":9856},[102,106],", first apply ",[28,37156,37158,37171],{"className":37157},[35],[28,37159,37161],{"className":37160},[39],[41,37162,37163],{"xmlns":43},[46,37164,37165,37169],{},[49,37166,37167],{},[55,37168,2263],{},[82,37170,2263],{"encoding":84},[28,37172,37174],{"className":37173,"ariaHidden":71},[89],[28,37175,37177,37180],{"className":37176},[93],[28,37178],{"className":37179,"style":1102},[97],[28,37181,2263],{"className":37182,"style":4040},[102,106],[28,37184,37186],{"className":37185},[31],[28,37187,37189,37295],{"className":37188},[35],[28,37190,37192],{"className":37191},[39],[41,37193,37194],{"xmlns":43,"display":44},[46,37195,37196,37292],{},[3178,37197,37198],{"rowspacing":3180,"columnspacing":3182},[3184,37199,37200],{},[3187,37201,37202],{},[3190,37203,37204],{"scriptlevel":652,"displaystyle":71},[49,37205,37206,37208,37210,37216,37218,37220,37222,37224,37226,37232,37234,37236,37238,37240,37246,37248,37250,37252,37258,37260,37262,37264,37266,37268,37274,37276,37278,37280,37282,37288,37290],{},[62,37207,35158],{},[173,37209],{"width":3182},[3977,37211,37212,37214],{},[650,37213,3200],{},[650,37215,848],{},[55,37217,296],{"mathvariant":1088},[650,37219,652],{},[62,37221,302],{"stretchy":64},[62,37223,65],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[55,37225,296],{"mathvariant":1088},[52,37227,37228,37230],{},[55,37229,394],{},[55,37231,28200],{},[62,37233,302],{"stretchy":64},[62,37235,2606],{},[55,37237,57],{},[55,37239,296],{"mathvariant":1088},[52,37241,37242,37244],{},[55,37243,394],{},[55,37245,3256],{},[62,37247,302],{"stretchy":64},[62,37249,78],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[62,37251,2606],{},[3977,37253,37254,37256],{},[650,37255,3200],{},[650,37257,848],{},[55,37259,296],{"mathvariant":1088},[650,37261,3200],{},[62,37263,302],{"stretchy":64},[62,37265,65],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[55,37267,296],{"mathvariant":1088},[52,37269,37270,37272],{},[55,37271,394],{},[55,37273,28200],{},[62,37275,302],{"stretchy":64},[62,37277,1213],{},[55,37279,57],{},[55,37281,296],{"mathvariant":1088},[52,37283,37284,37286],{},[55,37285,394],{},[55,37287,3256],{},[62,37289,302],{"stretchy":64},[62,37291,78],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[82,37293,37294],{"encoding":84},"\\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*}",[28,37296,37298],{"className":37297,"ariaHidden":71},[89],[28,37299,37301,37305],{"className":37300},[93],[28,37302],{"className":37303,"style":37304},[97],"height:2.0074em;vertical-align:-0.7537em;",[28,37306,37308],{"className":37307},[102],[28,37309,37311],{"className":37310},[3178],[28,37312,37314],{"className":37313},[3417],[28,37315,37317,37722],{"className":37316},[115,116],[28,37318,37320,37719],{"className":37319},[120],[28,37321,37324],{"className":37322,"style":37323},[124],"height:1.2537em;",[28,37325,37327,37330],{"style":37326},"top:-3.2537em;",[28,37328],{"className":37329,"style":35396},[132],[28,37331,37333,37336,37339,37342,37404,37407,37410,37416,37419,37459,37462,37465,37468,37471,37474,37477,37517,37520,37526,37529,37532,37535,37597,37600,37603,37609,37612,37652,37655,37658,37661,37664,37667,37670,37710,37713],{"className":37332},[102],[28,37334,35158],{"className":37335},[917],[28,37337],{"className":37338,"style":35442},[173],[28,37340],{"className":37341,"style":913},[173],[28,37343,37345,37348,37401],{"className":37344},[102],[28,37346],{"className":37347},[161,4072],[28,37349,37351],{"className":37350},[3977],[28,37352,37354,37393],{"className":37353},[115,116],[28,37355,37357,37390],{"className":37356},[120],[28,37358,37360,37371,37379],{"className":37359,"style":6825},[124],[28,37361,37362,37365],{"style":4088},[28,37363],{"className":37364,"style":3434},[132],[28,37366,37368],{"className":37367},[102],[28,37369,848],{"className":37370},[102],[28,37372,37373,37376],{"style":4100},[28,37374],{"className":37375,"style":3434},[132],[28,37377],{"className":37378,"style":4108},[4107],[28,37380,37381,37384],{"style":4111},[28,37382],{"className":37383,"style":3434},[132],[28,37385,37387],{"className":37386},[102],[28,37388,3200],{"className":37389},[102],[28,37391,148],{"className":37392},[147],[28,37394,37396],{"className":37395},[120],[28,37397,37399],{"className":37398,"style":4170},[124],[28,37400],{},[28,37402],{"className":37403},[182,4072],[28,37405,35403],{"className":37406},[102],[28,37408,302],{"className":37409},[182],[28,37411,37413],{"className":37412},[102],[28,37414,65],{"className":37415},[3407,35567],[28,37417,296],{"className":37418},[102],[28,37420,37422,37425],{"className":37421},[102],[28,37423,394],{"className":37424,"style":321},[102,106],[28,37426,37428],{"className":37427},[111],[28,37429,37431,37451],{"className":37430},[115,116],[28,37432,37434,37448],{"className":37433},[120],[28,37435,37437],{"className":37436,"style":32501},[124],[28,37438,37439,37442],{"style":801},[28,37440],{"className":37441,"style":133},[132],[28,37443,37445],{"className":37444},[137,138,139,140],[28,37446,28200],{"className":37447,"style":32522},[102,106,140],[28,37449,148],{"className":37450},[147],[28,37452,37454],{"className":37453},[120],[28,37455,37457],{"className":37456,"style":2003},[124],[28,37458],{},[28,37460,302],{"className":37461},[182],[28,37463],{"className":37464,"style":1257},[173],[28,37466,2606],{"className":37467},[1261],[28,37469],{"className":37470,"style":1257},[173],[28,37472,57],{"className":37473,"style":107},[102,106],[28,37475,296],{"className":37476},[102],[28,37478,37480,37483],{"className":37479},[102],[28,37481,394],{"className":37482,"style":321},[102,106],[28,37484,37486],{"className":37485},[111],[28,37487,37489,37509],{"className":37488},[115,116],[28,37490,37492,37506],{"className":37491},[120],[28,37493,37495],{"className":37494,"style":32501},[124],[28,37496,37497,37500],{"style":801},[28,37498],{"className":37499,"style":133},[132],[28,37501,37503],{"className":37502},[137,138,139,140],[28,37504,3256],{"className":37505},[102,106,140],[28,37507,148],{"className":37508},[147],[28,37510,37512],{"className":37511},[120],[28,37513,37515],{"className":37514,"style":155},[124],[28,37516],{},[28,37518,302],{"className":37519},[182],[28,37521,37523],{"className":37522},[102],[28,37524,78],{"className":37525},[3407,35567],[28,37527],{"className":37528,"style":1257},[173],[28,37530,2606],{"className":37531},[1261],[28,37533],{"className":37534,"style":1257},[173],[28,37536,37538,37541,37594],{"className":37537},[102],[28,37539],{"className":37540},[161,4072],[28,37542,37544],{"className":37543},[3977],[28,37545,37547,37586],{"className":37546},[115,116],[28,37548,37550,37583],{"className":37549},[120],[28,37551,37553,37564,37572],{"className":37552,"style":6825},[124],[28,37554,37555,37558],{"style":4088},[28,37556],{"className":37557,"style":3434},[132],[28,37559,37561],{"className":37560},[102],[28,37562,848],{"className":37563},[102],[28,37565,37566,37569],{"style":4100},[28,37567],{"className":37568,"style":3434},[132],[28,37570],{"className":37571,"style":4108},[4107],[28,37573,37574,37577],{"style":4111},[28,37575],{"className":37576,"style":3434},[132],[28,37578,37580],{"className":37579},[102],[28,37581,3200],{"className":37582},[102],[28,37584,148],{"className":37585},[147],[28,37587,37589],{"className":37588},[120],[28,37590,37592],{"className":37591,"style":4170},[124],[28,37593],{},[28,37595],{"className":37596},[182,4072],[28,37598,35586],{"className":37599},[102],[28,37601,302],{"className":37602},[182],[28,37604,37606],{"className":37605},[102],[28,37607,65],{"className":37608},[3407,35567],[28,37610,296],{"className":37611},[102],[28,37613,37615,37618],{"className":37614},[102],[28,37616,394],{"className":37617,"style":321},[102,106],[28,37619,37621],{"className":37620},[111],[28,37622,37624,37644],{"className":37623},[115,116],[28,37625,37627,37641],{"className":37626},[120],[28,37628,37630],{"className":37629,"style":32501},[124],[28,37631,37632,37635],{"style":801},[28,37633],{"className":37634,"style":133},[132],[28,37636,37638],{"className":37637},[137,138,139,140],[28,37639,28200],{"className":37640,"style":32522},[102,106,140],[28,37642,148],{"className":37643},[147],[28,37645,37647],{"className":37646},[120],[28,37648,37650],{"className":37649,"style":2003},[124],[28,37651],{},[28,37653,302],{"className":37654},[182],[28,37656],{"className":37657,"style":1257},[173],[28,37659,1213],{"className":37660},[1261],[28,37662],{"className":37663,"style":1257},[173],[28,37665,57],{"className":37666,"style":107},[102,106],[28,37668,296],{"className":37669},[102],[28,37671,37673,37676],{"className":37672},[102],[28,37674,394],{"className":37675,"style":321},[102,106],[28,37677,37679],{"className":37678},[111],[28,37680,37682,37702],{"className":37681},[115,116],[28,37683,37685,37699],{"className":37684},[120],[28,37686,37688],{"className":37687,"style":32501},[124],[28,37689,37690,37693],{"style":801},[28,37691],{"className":37692,"style":133},[132],[28,37694,37696],{"className":37695},[137,138,139,140],[28,37697,3256],{"className":37698},[102,106,140],[28,37700,148],{"className":37701},[147],[28,37703,37705],{"className":37704},[120],[28,37706,37708],{"className":37707,"style":155},[124],[28,37709],{},[28,37711,302],{"className":37712},[182],[28,37714,37716],{"className":37715},[102],[28,37717,78],{"className":37718},[3407,35567],[28,37720,148],{"className":37721},[147],[28,37723,37725],{"className":37724},[120],[28,37726,37729],{"className":37727,"style":37728},[124],"height:0.7537em;",[28,37730],{},[12,37732,37733],{},"... then measure:",[28,37735,37737],{"className":37736},[31],[28,37738,37740,37920],{"className":37739},[35],[28,37741,37743],{"className":37742},[39],[41,37744,37745],{"xmlns":43,"display":44},[46,37746,37747,37917],{},[3178,37748,37749],{"rowspacing":3180,"columnspacing":3182},[3184,37750,37751],{},[3187,37752,37753],{},[3190,37754,37755],{"scriptlevel":652,"displaystyle":71},[3178,37756,37760,37868],{"rowspacing":37757,"columnalign":37758,"columnspacing":37759},"0.25em","right left","0em",[3184,37761,37762,37779],{},[3187,37763,37764],{},[3190,37765,37766],{"scriptlevel":652,"displaystyle":71},[49,37767,37768,37771,37773,37775,37777],{},[62,37769,37770],{},"⇒",[173,37772],{"width":3182},[62,37774,2948],{"stretchy":64},[55,37776,9841],{},[62,37778,302],{"stretchy":64},[3187,37780,37781],{},[3190,37782,37783],{"scriptlevel":652,"displaystyle":71},[49,37784,37785,37787,37789,37796,37799,37802,37804,37810,37812,37814,37816,37818,37824,37826,37832,37834,37836,37838,37844,37846,37848,37850,37852,37858,37860,37866],{},[49,37786],{},[62,37788,845],{},[3977,37790,37791,37793],{},[650,37792,3200],{},[650,37794,37795],{},"4",[62,37797,65],{"fence":64,"stretchy":71,"minsize":37798,"maxsize":37798},"3em",[62,37800,37801],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},"∥",[55,37803,296],{"mathvariant":1088},[52,37805,37806,37808],{},[55,37807,394],{},[55,37809,28200],{},[62,37811,302],{"stretchy":64},[62,37813,2606],{},[55,37815,57],{},[55,37817,296],{"mathvariant":1088},[52,37819,37820,37822],{},[55,37821,394],{},[55,37823,3256],{},[62,37825,302],{"stretchy":64},[2776,37827,37828,37830],{},[62,37829,37801],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[650,37831,848],{},[62,37833,1213],{},[62,37835,37801],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[55,37837,296],{"mathvariant":1088},[52,37839,37840,37842],{},[55,37841,394],{},[55,37843,28200],{},[62,37845,302],{"stretchy":64},[62,37847,1213],{},[55,37849,57],{},[55,37851,296],{"mathvariant":1088},[52,37853,37854,37856],{},[55,37855,394],{},[55,37857,3256],{},[62,37859,302],{"stretchy":64},[2776,37861,37862,37864],{},[62,37863,37801],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[650,37865,848],{},[62,37867,78],{"fence":64,"stretchy":71,"minsize":37798,"maxsize":37798},[3184,37869,37870,37876],{},[3187,37871,37872],{},[3190,37873,37874],{"scriptlevel":652,"displaystyle":71},[49,37875],{},[3187,37877,37878],{},[3190,37879,37880],{"scriptlevel":652,"displaystyle":71},[49,37881,37882,37884,37886,37889,37891,37893,37899,37901,37903,37905,37911,37913,37915],{},[49,37883],{},[62,37885,845],{},[2253,37887,37888],{},"Re",[62,37890,3176],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[62,37892,2948],{"stretchy":64},[52,37894,37895,37897],{},[55,37896,394],{},[55,37898,28200],{},[55,37900,296],{"mathvariant":1088},[55,37902,57],{},[55,37904,296],{"mathvariant":1088},[52,37906,37907,37909],{},[55,37908,394],{},[55,37910,3256],{},[62,37912,302],{"stretchy":64},[62,37914,3217],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[55,37916,2106],{"mathvariant":1088},[82,37918,37919],{"encoding":84},"\\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*}",[28,37921,37923],{"className":37922,"ariaHidden":71},[89],[28,37924,37926,37930],{"className":37925},[93],[28,37927],{"className":37928,"style":37929},[97],"height:5.1em;vertical-align:-2.3em;",[28,37931,37933],{"className":37932},[102],[28,37934,37936],{"className":37935},[3178],[28,37937,37939],{"className":37938},[3417],[28,37940,37942,38712],{"className":37941},[115,116],[28,37943,37945,38709],{"className":37944},[120],[28,37946,37949],{"className":37947,"style":37948},[124],"height:2.8em;",[28,37950,37952,37956],{"style":37951},"top:-4.8em;",[28,37953],{"className":37954,"style":37955},[132],"height:4.8em;",[28,37957,37959],{"className":37958},[102],[28,37960,37962],{"className":37961},[102],[28,37963,37965,38026],{"className":37964},[3178],[28,37966,37969],{"className":37967},[37968],"col-align-r",[28,37970,37972,38017],{"className":37971},[115,116],[28,37973,37975,38014],{"className":37974},[120],[28,37976,37978,38005],{"className":37977,"style":37948},[124],[28,37979,37980,37984],{"style":37951},[28,37981],{"className":37982,"style":37983},[132],"height:3.75em;",[28,37985,37987,37990,37993,37996,37999,38002],{"className":37986},[102],[28,37988,37770],{"className":37989},[917],[28,37991],{"className":37992,"style":35442},[173],[28,37994],{"className":37995,"style":913},[173],[28,37997,2948],{"className":37998},[161],[28,38000,9841],{"className":38001,"style":9856},[102,106],[28,38003,302],{"className":38004},[182],[28,38006,38008,38011],{"style":38007},"top:-2.1em;",[28,38009],{"className":38010,"style":37983},[132],[28,38012],{"className":38013},[102],[28,38015,148],{"className":38016},[147],[28,38018,38020],{"className":38019},[120],[28,38021,38024],{"className":38022,"style":38023},[124],"height:2.3em;",[28,38025],{},[28,38027,38030],{"className":38028},[38029],"col-align-l",[28,38031,38033,38701],{"className":38032},[115,116],[28,38034,38036,38698],{"className":38035},[120],[28,38037,38039,38562],{"className":38038,"style":37948},[124],[28,38040,38041,38044],{"style":37951},[28,38042],{"className":38043,"style":37983},[132],[28,38045,38047,38050,38053,38056,38059,38121,38128,38174,38177,38217,38220,38223,38226,38229,38232,38235,38275,38278,38343,38346,38349,38352,38389,38392,38432,38435,38438,38441,38444,38447,38450,38490,38493,38556],{"className":38046},[102],[28,38048],{"className":38049},[102],[28,38051],{"className":38052,"style":913},[173],[28,38054,845],{"className":38055},[917],[28,38057],{"className":38058,"style":913},[173],[28,38060,38062,38065,38118],{"className":38061},[102],[28,38063],{"className":38064},[161,4072],[28,38066,38068],{"className":38067},[3977],[28,38069,38071,38110],{"className":38070},[115,116],[28,38072,38074,38107],{"className":38073},[120],[28,38075,38077,38088,38096],{"className":38076,"style":6825},[124],[28,38078,38079,38082],{"style":4088},[28,38080],{"className":38081,"style":3434},[132],[28,38083,38085],{"className":38084},[102],[28,38086,37795],{"className":38087},[102],[28,38089,38090,38093],{"style":4100},[28,38091],{"className":38092,"style":3434},[132],[28,38094],{"className":38095,"style":4108},[4107],[28,38097,38098,38101],{"style":4111},[28,38099],{"className":38100,"style":3434},[132],[28,38102,38104],{"className":38103},[102],[28,38105,3200],{"className":38106},[102],[28,38108,148],{"className":38109},[147],[28,38111,38113],{"className":38112},[120],[28,38114,38116],{"className":38115,"style":4170},[124],[28,38117],{},[28,38119],{"className":38120},[182,4072],[28,38122,38124],{"className":38123},[102],[28,38125,65],{"className":38126},[3407,38127],"size4",[28,38129,38131],{"className":38130},[102],[28,38132,38135],{"className":38133},[3407,38134],"mult",[28,38136,38138,38165],{"className":38137},[115,116],[28,38139,38141,38162],{"className":38140},[120],[28,38142,38145],{"className":38143,"style":38144},[124],"height:1.15em;",[28,38146,38148,38152],{"style":38147},"top:-3.15em;",[28,38149],{"className":38150,"style":38151},[132],"height:3.8em;",[28,38153,38155],{"style":38154},"width:0.556em;height:1.8em;",[5855,38156,38159],{"xmlns":5857,"width":38157,"height":35171,"viewBox":38158},"0.556em","0 0 556 1800",[5863,38160],{"d":38161},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z\nM367 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M410 15 H367 v585 v600 v585 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",[28,38724,38726,38813],{"className":38725},[35],[28,38727,38729],{"className":38728},[39],[41,38730,38731],{"xmlns":43},[46,38732,38733,38810],{},[49,38734,38735,38737,38739,38741,38744,38750,38752,38754,38756,38758,38760,38762,38764,38766,38768,38770,38772,38774,38776,38782,38784,38786,38788,38794,38796,38798,38800,38802,38804,38806,38808],{},[55,38736,296],{"mathvariant":1088},[55,38738,19],{},[62,38740,2606],{},[55,38742,38743],{},"b",[2776,38745,38746,38748],{},[55,38747,37801],{"mathvariant":1088},[650,38749,848],{},[62,38751,845],{},[62,38753,2948],{"stretchy":64},[55,38755,19],{},[62,38757,2606],{},[55,38759,38743],{},[55,38761,296],{"mathvariant":1088},[55,38763,19],{},[62,38765,2606],{},[55,38767,38743],{},[62,38769,302],{"stretchy":64},[62,38771,845],{},[55,38773,37801],{"mathvariant":1088},[55,38775,19],{},[2776,38777,38778,38780],{},[55,38779,37801],{"mathvariant":1088},[650,38781,848],{},[62,38783,2606],{},[55,38785,37801],{"mathvariant":1088},[55,38787,38743],{},[2776,38789,38790,38792],{},[55,38791,37801],{"mathvariant":1088},[650,38793,848],{},[62,38795,2606],{},[650,38797,848],{},[2253,38799,37888],{},[62,38801,2948],{"stretchy":64},[55,38803,19],{},[55,38805,296],{"mathvariant":1088},[55,38807,38743],{},[62,38809,302],{"stretchy":64},[82,38811,38812],{"encoding":84},"|a + b\\|^2 = \\langle a + b | a + b \\rangle = \\|a\\|^2 + \\|b\\|^2 + 2\\text{Re}\\langle a | b \\rangle",[28,38814,38816,38837,38885,38906,38930,38951,39001,39051],{"className":38815,"ariaHidden":71},[89],[28,38817,38819,38822,38825,38828,38831,38834],{"className":38818},[93],[28,38820],{"className":38821,"style":98},[97],[28,38823,296],{"className":38824},[102],[28,38826,19],{"className":38827},[102,106],[28,38829],{"className":38830,"style":1257},[173],[28,38832,2606],{"className":38833},[1261],[28,38835],{"className":38836,"style":1257},[173],[28,38838,38840,38844,38847,38876,38879,38882],{"className":38839},[93],[28,38841],{"className":38842,"style":38843},[97],"height:1.0641em;vertical-align:-0.25em;",[28,38845,38743],{"className":38846},[102,106],[28,38848,38850,38853],{"className":38849},[102],[28,38851,37801],{"className":38852},[102],[28,38854,38856],{"className":38855},[111],[28,38857,38859],{"className":38858},[115],[28,38860,38862],{"className":38861},[120],[28,38863,38865],{"className":38864,"style":5446},[124],[28,38866,38867,38870],{"style":1726},[28,38868],{"className":38869,"style":133},[132],[28,38871,38873],{"className":38872},[137,138,139,140],[28,38874,848],{"className":38875},[102,140],[28,38877],{"className":38878,"style":913},[173],[28,38880,845],{"className":38881},[917],[28,38883],{"className":38884,"style":913},[173],[28,38886,38888,38891,38894,38897,38900,38903],{"className":38887},[93],[28,38889],{"className":38890,"style":98},[97],[28,38892,2948],{"className":38893},[161],[28,38895,19],{"className":38896},[102,106],[28,38898],{"className":38899,"style":1257},[173],[28,38901,2606],{"className":38902},[1261],[28,38904],{"className":38905,"style":1257},[173],[28,38907,38909,38912,38915,38918,38921,38924,38927],{"className":38908},[93],[28,38910],{"className":38911,"style":98},[97],[28,38913,38743],{"className":38914},[102,106],[28,38916,296],{"className":38917},[102],[28,38919,19],{"className":38920},[102,106],[28,38922],{"className":38923,"style":1257},[173],[28,38925,2606],{"className":38926},[1261],[28,38928],{"className":38929,"style":1257},[173],[28,38931,38933,38936,38939,38942,38945,38948],{"className":38932},[93],[28,38934],{"className":38935,"style":98},[97],[28,38937,38743],{"className":38938},[102,106],[28,38940,302],{"className":38941},[182],[28,38943],{"className":38944,"style":913},[173],[28,38946,845],{"className":38947},[917],[28,38949],{"className":38950,"style":913},[173],[28,38952,38954,38957,38960,38963,38992,38995,38998],{"className":38953},[93],[28,38955],{"className":38956,"style":38843},[97],[28,38958,37801],{"className":38959},[102],[28,38961,19],{"className":38962},[102,106],[28,38964,38966,38969],{"className":38965},[102],[28,38967,37801],{"className":38968},[102],[28,38970,38972],{"className":38971},[111],[28,38973,38975],{"className":38974},[115],[28,38976,38978],{"className":38977},[120],[28,38979,38981],{"className":38980,"style":5446},[124],[28,38982,38983,38986],{"style":1726},[28,38984],{"className":38985,"style":133},[132],[28,38987,38989],{"className":38988},[137,138,139,140],[28,38990,848],{"className":38991},[102,140],[28,38993],{"className":38994,"style":1257},[173],[28,38996,2606],{"className":38997},[1261],[28,38999],{"className":39000,"style":1257},[173],[28,39002,39004,39007,39010,39013,39042,39045,39048],{"className":39003},[93],[28,39005],{"className":39006,"style":38843},[97],[28,39008,37801],{"className":39009},[102],[28,39011,38743],{"className":39012},[102,106],[28,39014,39016,39019],{"className":39015},[102],[28,39017,37801],{"className":39018},[102],[28,39020,39022],{"className":39021},[111],[28,39023,39025],{"className":39024},[115],[28,39026,39028],{"className":39027},[120],[28,39029,39031],{"className":39030,"style":5446},[124],[28,39032,39033,39036],{"style":1726},[28,39034],{"className":39035,"style":133},[132],[28,39037,39039],{"className":39038},[137,138,139,140],[28,39040,848],{"className":39041},[102,140],[28,39043],{"className":39044,"style":1257},[173],[28,39046,2606],{"className":39047},[1261],[28,39049],{"className":39050,"style":1257},[173],[28,39052,39054,39057,39060,39066,39069,39072,39075,39078],{"className":39053},[93],[28,39055],{"className":39056,"style":98},[97],[28,39058,848],{"className":39059},[102],[28,39061,39063],{"className":39062},[102,16058],[28,39064,37888],{"className":39065},[102],[28,39067,2948],{"className":39068},[161],[28,39070,19],{"className":39071},[102,106],[28,39073,296],{"className":39074},[102],[28,39076,38743],{"className":39077},[102,106],[28,39079,302],{"className":39080},[182],". Similarly, measuring ",[28,39083,39085,39098],{"className":39084},[35],[28,39086,39088],{"className":39087},[39],[41,39089,39090],{"xmlns":43},[46,39091,39092,39096],{},[49,39093,39094],{},[55,39095,33439],{},[82,39097,33439],{"encoding":84},[28,39099,39101],{"className":39100,"ariaHidden":71},[89],[28,39102,39104,39107],{"className":39103},[93],[28,39105],{"className":39106,"style":1102},[97],[28,39108,33439],{"className":39109,"style":1257},[102,106]," yields",[28,39112,39114],{"className":39113},[31],[28,39115,39117,39176],{"className":39116},[35],[28,39118,39120],{"className":39119},[39],[41,39121,39122],{"xmlns":43,"display":44},[46,39123,39124,39173],{},[3178,39125,39126],{"rowspacing":3180,"columnspacing":3182},[3184,39127,39128],{},[3187,39129,39130],{},[3190,39131,39132],{"scriptlevel":652,"displaystyle":71},[49,39133,39134,39136,39138,39140,39142,39145,39147,39149,39155,39157,39159,39161,39167,39169,39171],{},[62,39135,2948],{"stretchy":64},[55,39137,33439],{},[62,39139,302],{"stretchy":64},[62,39141,845],{},[2253,39143,39144],{},"Im",[62,39146,3176],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[62,39148,2948],{"stretchy":64},[52,39150,39151,39153],{},[55,39152,394],{},[55,39154,28200],{},[55,39156,296],{"mathvariant":1088},[55,39158,57],{},[55,39160,296],{"mathvariant":1088},[52,39162,39163,39165],{},[55,39164,394],{},[55,39166,3256],{},[62,39168,302],{"stretchy":64},[62,39170,3217],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[55,39172,2106],{"mathvariant":1088},[82,39174,39175],{"encoding":84},"\\begin{equation*}\n    \\langle Y\\rangle = \\text{Im}\\Big[\\langle\\psi_j| P|\\psi_i\\rangle\\Big].\n\\end{equation*}",[28,39177,39179],{"className":39178,"ariaHidden":71},[89],[28,39180,39182,39186],{"className":39181},[93],[28,39183],{"className":39184,"style":39185},[97],"height:1.8em;vertical-align:-0.65em;",[28,39187,39189],{"className":39188},[102],[28,39190,39192],{"className":39191},[3178],[28,39193,39195],{"className":39194},[3417],[28,39196,39198,39350],{"className":39197},[115,116],[28,39199,39201,39347],{"className":39200},[120],[28,39202,39204],{"className":39203,"style":38144},[124],[28,39205,39206,39210],{"style":38147},[28,39207],{"className":39208,"style":39209},[132],"height:3.15em;",[28,39211,39213,39216,39219,39222,39225,39228,39231,39237,39243,39246,39286,39289,39292,39295,39335,39338,39344],{"className":39212},[102],[28,39214,2948],{"className":39215},[161],[28,39217,33439],{"className":39218,"style":1257},[102,106],[28,39220,302],{"className":39221},[182],[28,39223],{"className":39224,"style":913},[173],[28,39226,845],{"className":39227},[917],[28,39229],{"className":39230,"style":913},[173],[28,39232,39234],{"className":39233},[102,16058],[28,39235,39144],{"className":39236},[102],[28,39238,39240],{"className":39239},[102],[28,39241,3176],{"className":39242},[3407,35567],[28,39244,2948],{"className":39245},[161],[28,39247,39249,39252],{"className":39248},[102],[28,39250,394],{"className":39251,"style":321},[102,106],[28,39253,39255],{"className":39254},[111],[28,39256,39258,39278],{"className":39257},[115,116],[28,39259,39261,39275],{"className":39260},[120],[28,39262,39264],{"className":39263,"style":32501},[124],[28,39265,39266,39269],{"style":801},[28,39267],{"className":39268,"style":133},[132],[28,39270,39272],{"className":39271},[137,138,139,140],[28,39273,28200],{"className":39274,"style":32522},[102,106,140],[28,39276,148],{"className":39277},[147],[28,39279,39281],{"className":39280},[120],[28,39282,39284],{"className":39283,"style":2003},[124],[28,39285],{},[28,39287,296],{"className":39288},[102],[28,39290,57],{"className":39291,"style":107},[102,106],[28,39293,296],{"className":39294},[102],[28,39296,39298,39301],{"className":39297},[102],[28,39299,394],{"className":39300,"style":321},[102,106],[28,39302,39304],{"className":39303},[111],[28,39305,39307,39327],{"className":39306},[115,116],[28,39308,39310,39324],{"className":39309},[120],[28,39311,39313],{"className":39312,"style":32501},[124],[28,39314,39315,39318],{"style":801},[28,39316],{"className":39317,"style":133},[132],[28,39319,39321],{"className":39320},[137,138,139,140],[28,39322,3256],{"className":39323},[102,106,140],[28,39325,148],{"className":39326},[147],[28,39328,39330],{"className":39329},[120],[28,39331,39333],{"className":39332,"style":155},[124],[28,39334],{},[28,39336,302],{"className":39337},[182],[28,39339,39341],{"className":39340},[102],[28,39342,3217],{"className":39343},[3407,35567],[28,39345,2106],{"className":39346},[102],[28,39348,148],{"className":39349},[147],[28,39351,39353],{"className":39352},[120],[28,39354,39356],{"className":39355,"style":38171},[124],[28,39357],{},[15650,39359,39361],{"className":15652,"code":39360,"language":15654,"meta":15655,"style":15655},"## 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",[8103,39362,39363,39367,39377,39402,39406,39410,39415,39426,39450,39454,39468,39472,39477,39488,39500,39520,39533,39549,39561,39576,39597,39601,39606,39630,39634,39639,39655,39666,39679,39703,39715,39719,39725,39730,39734],{"__ignoreMap":15655},[28,39364,39365],{"class":15660,"line":15661},[28,39366,35005],{"class":15664},[28,39368,39369,39371,39373,39375],{"class":15660,"line":15668},[28,39370,35010],{"class":15675},[28,39372,845],{"class":15682},[28,39374,35015],{"class":16073},[28,39376,16077],{"class":15675},[28,39378,39379,39381,39383,39385,39388,39390,39392,39394,39396,39398,39400],{"class":15660,"line":15679},[28,39380,35022],{"class":15675},[28,39382,35025],{"class":16307},[28,39384,845],{"class":15682},[28,39386,39387],{"class":15675},"dt, ",[28,39389,35033],{"class":16307},[28,39391,845],{"class":15682},[28,39393,35038],{"class":16073},[28,39395,65],{"class":15675},[28,39397,35043],{"class":16307},[28,39399,845],{"class":15682},[28,39401,35048],{"class":15675},[28,39403,39404],{"class":15660,"line":15725},[28,39405,16052],{"class":15675},[28,39407,39408],{"class":15660,"line":15739},[28,39409,15769],{"emptyLinePlaceholder":15768},[28,39411,39412],{"class":15660,"line":15752},[28,39413,39414],{"class":15664},"## Create the time-evo op dagger circuit\n",[28,39416,39417,39420,39422,39424],{"class":15660,"line":15765},[28,39418,39419],{"class":15675},"evol_gate_d ",[28,39421,845],{"class":15682},[28,39423,35015],{"class":16073},[28,39425,16077],{"class":15675},[28,39427,39428,39430,39432,39434,39436,39438,39440,39442,39444,39446,39448],{"class":15660,"line":15772},[28,39429,35022],{"class":15675},[28,39431,35025],{"class":16307},[28,39433,845],{"class":15682},[28,39435,39387],{"class":15675},[28,39437,35033],{"class":16307},[28,39439,845],{"class":15682},[28,39441,35038],{"class":16073},[28,39443,65],{"class":15675},[28,39445,35043],{"class":16307},[28,39447,845],{"class":15682},[28,39449,35048],{"class":15675},[28,39451,39452],{"class":15660,"line":15781},[28,39453,16052],{"class":15675},[28,39455,39456,39458,39460,39463,39466],{"class":15660,"line":15798},[28,39457,39419],{"class":15675},[28,39459,845],{"class":15682},[28,39461,39462],{"class":15675}," evol_gate_d.",[28,39464,39465],{"class":16073},"inverse",[28,39467,17029],{"class":15675},[28,39469,39470],{"class":15660,"line":15811},[28,39471,15769],{"emptyLinePlaceholder":15768},[28,39473,39474],{"class":15660,"line":15820},[28,39475,39476],{"class":15664},"# Put pieces together\n",[28,39478,39479,39482,39484,39486],{"class":15660,"line":15830},[28,39480,39481],{"class":15675},"qc_reg ",[28,39483,845],{"class":15682},[28,39485,35066],{"class":16073},[28,39487,34462],{"class":15675},[28,39489,39490,39493,39495,39497],{"class":15660,"line":15835},[28,39491,39492],{"class":15675},"qc_temp ",[28,39494,845],{"class":15682},[28,39496,26583],{"class":16073},[28,39498,39499],{"class":15675},"(qc_reg)\n",[28,39501,39502,39505,39508,39511,39514,39516,39518],{"class":15660,"line":15853},[28,39503,39504],{"class":15675},"qc_temp.",[28,39506,39507],{"class":16073},"compose",[28,39509,39510],{"class":15675},"(qc_state_prep, ",[28,39512,39513],{"class":16307},"inplace",[28,39515,845],{"class":15682},[28,39517,15597],{"class":15801},[28,39519,16052],{"class":15675},[28,39521,39522,39524,39526,39528,39530],{"class":15660,"line":15867},[28,39523,22277],{"class":15671},[28,39525,33906],{"class":15675},[28,39527,17720],{"class":15671},[28,39529,22285],{"class":15682},[28,39531,39532],{"class":15675},"(num_trotter_steps):\n",[28,39534,39535,39538,39540,39542,39544,39546],{"class":15660,"line":15872},[28,39536,39537],{"class":15675},"    qc_temp.",[28,39539,27807],{"class":16073},[28,39541,35090],{"class":15675},[28,39543,35093],{"class":16307},[28,39545,845],{"class":15682},[28,39547,39548],{"class":15675},"qc_reg)\n",[28,39550,39551,39553,39555,39557,39559],{"class":15660,"line":15878},[28,39552,22277],{"class":15671},[28,39554,33906],{"class":15675},[28,39556,17720],{"class":15671},[28,39558,22285],{"class":15682},[28,39560,39532],{"class":15675},[28,39562,39563,39565,39567,39570,39572,39574],{"class":15660,"line":15897},[28,39564,39537],{"class":15675},[28,39566,27807],{"class":16073},[28,39568,39569],{"class":15675},"(evol_gate_d, ",[28,39571,35093],{"class":16307},[28,39573,845],{"class":15682},[28,39575,39548],{"class":15675},[28,39577,39578,39580,39582,39585,39587,39589,39591,39593,39595],{"class":15660,"line":15914},[28,39579,39504],{"class":15675},[28,39581,39507],{"class":16073},[28,39583,39584],{"class":15675},"(qc_state_prep.",[28,39586,39465],{"class":16073},[28,39588,24379],{"class":15675},[28,39590,39513],{"class":16307},[28,39592,845],{"class":15682},[28,39594,15597],{"class":15801},[28,39596,16052],{"class":15675},[28,39598,39599],{"class":15660,"line":15931},[28,39600,15769],{"emptyLinePlaceholder":15768},[28,39602,39603],{"class":15660,"line":15948},[28,39604,39605],{"class":15664},"# Create controlled version of the circuit\n",[28,39607,39608,39611,39613,39616,39619,39621,39624,39626,39628],{"class":15660,"line":15966},[28,39609,39610],{"class":15675},"controlled_U ",[28,39612,845],{"class":15682},[28,39614,39615],{"class":15675}," qc_temp.",[28,39617,39618],{"class":16073},"to_gate",[28,39620,17194],{"class":15675},[28,39622,39623],{"class":16073},"control",[28,39625,65],{"class":15675},[28,39627,3200],{"class":15801},[28,39629,16052],{"class":15675},[28,39631,39632],{"class":15660,"line":15983},[28,39633,15769],{"emptyLinePlaceholder":15768},[28,39635,39636],{"class":15660,"line":16000},[28,39637,39638],{"class":15664},"# Create hadamard test circuit for real part\n",[28,39640,39641,39643,39645,39647,39649,39651,39653],{"class":15660,"line":16016},[28,39642,35061],{"class":15675},[28,39644,845],{"class":15682},[28,39646,35066],{"class":16073},[28,39648,27958],{"class":15675},[28,39650,2606],{"class":15682},[28,39652,22230],{"class":15801},[28,39654,16052],{"class":15675},[28,39656,39657,39660,39662,39664],{"class":15660,"line":16036},[28,39658,39659],{"class":15675},"qc_real ",[28,39661,845],{"class":15682},[28,39663,26583],{"class":16073},[28,39665,35080],{"class":15675},[28,39667,39668,39671,39673,39675,39677],{"class":15660,"line":16041},[28,39669,39670],{"class":15675},"qc_real.",[28,39672,26597],{"class":16073},[28,39674,65],{"class":15675},[28,39676,652],{"class":15801},[28,39678,16052],{"class":15675},[28,39680,39681,39683,39685,39688,39691,39693,39695,39697,39699,39701],{"class":15660,"line":16371},[28,39682,39670],{"class":15675},[28,39684,27807],{"class":16073},[28,39686,39687],{"class":15675},"(controlled_U, ",[28,39689,39690],{"class":15682},"list",[28,39692,65],{"class":15675},[28,39694,28048],{"class":15682},[28,39696,27958],{"class":15675},[28,39698,2606],{"class":15682},[28,39700,22230],{"class":15801},[28,39702,26053],{"class":15675},[28,39704,39705,39707,39709,39711,39713],{"class":15660,"line":16376},[28,39706,39670],{"class":15675},[28,39708,26597],{"class":16073},[28,39710,65],{"class":15675},[28,39712,652],{"class":15801},[28,39714,16052],{"class":15675},[28,39716,39717],{"class":15660,"line":16410},[28,39718,15769],{"emptyLinePlaceholder":15768},[28,39720,39721,39723],{"class":15660,"line":16431},[28,39722,16044],{"class":15682},[28,39724,16077],{"class":15675},[28,39726,39727],{"class":15660,"line":16452},[28,39728,39729],{"class":15841},"    \"Circuit for calculating the real part of the overlap in S via Hadamard test\"\n",[28,39731,39732],{"class":15660,"line":16476},[28,39733,16052],{"class":15675},[28,39735,39736,39738,39740,39742,39744,39746,39749,39751,39753,39755,39757,39759,39761],{"class":15660,"line":16481},[28,39737,39670],{"class":15675},[28,39739,34841],{"class":16073},[28,39741,65],{"class":15675},[28,39743,34846],{"class":15841},[28,39745,16397],{"class":15675},[28,39747,39748],{"class":16307},"fold",[28,39750,25926],{"class":15682},[28,39752,3200],{"class":15801},[28,39754,16397],{"class":15675},[28,39756,34851],{"class":16307},[28,39758,845],{"class":15682},[28,39760,34856],{"class":15801},[28,39762,16052],{"class":15675},[15650,39764,39767],{"className":39765,"code":39766,"language":16058,"meta":15655},[16056],"Circuit for calculating the real part of the overlap in S via Hadamard test\n",[8103,39768,39766],{"__ignoreMap":15655},[17041,39770],{"alt":34861,"src":39771},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-02.avif",[12,39773,39774],{},"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)",[15650,39776,39778],{"className":15652,"code":39777,"language":15654,"meta":15655,"style":15655},"print(\n    \"Number of layers of 2Q operations\",\n    qc_real.decompose(reps=2).depth(lambda x: x[0].num_qubits == 2),\n)\n",[8103,39779,39780,39786,39793,39838],{"__ignoreMap":15655},[28,39781,39782,39784],{"class":15660,"line":15661},[28,39783,16044],{"class":15682},[28,39785,16077],{"class":15675},[28,39787,39788,39791],{"class":15660,"line":15668},[28,39789,39790],{"class":15841},"    \"Number of layers of 2Q operations\"",[28,39792,16105],{"class":15675},[28,39794,39795,39798,39801,39803,39805,39807,39809,39812,39815,39817,39820,39823,39826,39828,39831,39833,39835],{"class":15660,"line":15679},[28,39796,39797],{"class":15675},"    qc_real.",[28,39799,39800],{"class":16073},"decompose",[28,39802,65],{"class":15675},[28,39804,35043],{"class":16307},[28,39806,845],{"class":15682},[28,39808,848],{"class":15801},[28,39810,39811],{"class":15675},").",[28,39813,39814],{"class":16073},"depth",[28,39816,65],{"class":15675},[28,39818,39819],{"class":15671},"lambda",[28,39821,39822],{"class":16082}," x",[28,39824,39825],{"class":15675},": x[",[28,39827,652],{"class":15801},[28,39829,39830],{"class":15675},"].num_qubits ",[28,39832,23650],{"class":15682},[28,39834,22129],{"class":15801},[28,39836,39837],{"class":15675},"),\n",[28,39839,39840],{"class":15660,"line":15725},[28,39841,16052],{"class":15675},[15650,39843,39846],{"className":39844,"code":39845,"language":16058,"meta":15655},[16056],"Number of layers of 2Q operations 112753\n",[8103,39847,39845],{"__ignoreMap":15655},[251,39849,39851],{"id":39850},"step-2-optimize-problem-for-quantum-hardware-execution","Step 2: Optimize problem for quantum hardware execution",[28327,39853,39855],{"id":39854},"efficient-hadamard-test","Efficient Hadamard test",[12,39857,39858],{},"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:",[12,39860,39861],{},[32407,39862],{"alt":39863,"src":39864},"fig3.png","\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Fdocs-fig-08.avif",[12,39866,39867,39868,39938,39939,40005,40006,40034,40035,40100,40101,40143,40144,40422,40423,40464,40465,40506],{},"Assume we can classically calculate ",[28,39869,39871,39889],{"className":39870},[35],[28,39872,39874],{"className":39873},[39],[41,39875,39876],{"xmlns":43},[46,39877,39878,39886],{},[49,39879,39880],{},[52,39881,39882,39884],{},[55,39883,1871],{},[650,39885,652],{},[82,39887,39888],{"encoding":84},"E_0",[28,39890,39892],{"className":39891,"ariaHidden":71},[89],[28,39893,39895,39898],{"className":39894},[93],[28,39896],{"className":39897,"style":585},[97],[28,39899,39901,39904],{"className":39900},[102],[28,39902,1871],{"className":39903,"style":1913},[102,106],[28,39905,39907],{"className":39906},[111],[28,39908,39910,39930],{"className":39909},[115,116],[28,39911,39913,39927],{"className":39912},[120],[28,39914,39916],{"className":39915,"style":683},[124],[28,39917,39918,39921],{"style":1928},[28,39919],{"className":39920,"style":133},[132],[28,39922,39924],{"className":39923},[137,138,139,140],[28,39925,652],{"className":39926},[102,140],[28,39928,148],{"className":39929},[147],[28,39931,39933],{"className":39932},[120],[28,39934,39936],{"className":39935,"style":155},[124],[28,39937],{},", the eigenvalue of ",[28,39940,39942,39964],{"className":39941},[35],[28,39943,39945],{"className":39944},[39],[41,39946,39947],{"xmlns":43},[46,39948,39949,39961],{},[49,39950,39951,39953,39955],{},[55,39952,296],{"mathvariant":1088},[650,39954,652],{},[2776,39956,39957,39959],{},[62,39958,302],{"stretchy":64},[55,39960,33375],{},[82,39962,39963],{"encoding":84},"|0\\rangle^N",[28,39965,39967],{"className":39966,"ariaHidden":71},[89],[28,39968,39970,39973,39976],{"className":39969},[93],[28,39971],{"className":39972,"style":30971},[97],[28,39974,35403],{"className":39975},[102],[28,39977,39979,39982],{"className":39978},[182],[28,39980,302],{"className":39981},[182],[28,39983,39985],{"className":39984},[111],[28,39986,39988],{"className":39987},[115],[28,39989,39991],{"className":39990},[120],[28,39992,39994],{"className":39993,"style":29066},[124],[28,39995,39996,39999],{"style":1726},[28,39997],{"className":39998,"style":133},[132],[28,40000,40002],{"className":40001},[137,138,139,140],[28,40003,33375],{"className":40004,"style":29050},[102,106,140]," under the Hamiltonian ",[28,40007,40009,40022],{"className":40008},[35],[28,40010,40012],{"className":40011},[39],[41,40013,40014],{"xmlns":43},[46,40015,40016,40020],{},[49,40017,40018],{},[55,40019,2263],{},[82,40021,2263],{"encoding":84},[28,40023,40025],{"className":40024,"ariaHidden":71},[89],[28,40026,40028,40031],{"className":40027},[93],[28,40029],{"className":40030,"style":1102},[97],[28,40032,2263],{"className":40033,"style":4040},[102,106],".\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 ",[28,40036,40038,40059],{"className":40037},[35],[28,40039,40041],{"className":40040},[39],[41,40042,40043],{"xmlns":43},[46,40044,40045,40057],{},[49,40046,40047,40049,40051],{},[55,40048,296],{"mathvariant":1088},[650,40050,652],{},[2776,40052,40053,40055],{},[62,40054,302],{"stretchy":64},[55,40056,33375],{},[82,40058,39963],{"encoding":84},[28,40060,40062],{"className":40061,"ariaHidden":71},[89],[28,40063,40065,40068,40071],{"className":40064},[93],[28,40066],{"className":40067,"style":30971},[97],[28,40069,35403],{"className":40070},[102],[28,40072,40074,40077],{"className":40073},[182],[28,40075,302],{"className":40076},[182],[28,40078,40080],{"className":40079},[111],[28,40081,40083],{"className":40082},[115],[28,40084,40086],{"className":40085},[120],[28,40087,40089],{"className":40088,"style":29066},[124],[28,40090,40091,40094],{"style":1726},[28,40092],{"className":40093,"style":133},[132],[28,40095,40097],{"className":40096},[137,138,139,140],[28,40098,33375],{"className":40099,"style":29050},[102,106,140]," 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 ",[28,40102,40104,40122],{"className":40103},[35],[28,40105,40107],{"className":40106},[39],[41,40108,40109],{"xmlns":43},[46,40110,40111,40119],{},[49,40112,40113,40115,40117],{},[2253,40114,32711],{},[2253,40116,32714],{},[55,40118,394],{},[82,40120,40121],{"encoding":84},"\\text{Prep} \\; \\psi",[28,40123,40125],{"className":40124,"ariaHidden":71},[89],[28,40126,40128,40131,40137,40140],{"className":40127},[93],[28,40129],{"className":40130,"style":214},[97],[28,40132,40134],{"className":40133},[102,16058],[28,40135,32711],{"className":40136},[102],[28,40138],{"className":40139,"style":913},[173],[28,40141,394],{"className":40142,"style":321},[102,106],", prepares the desired reference state ",[28,40145,40147,40224],{"className":40146},[35],[28,40148,40150],{"className":40149},[39],[41,40151,40152],{"xmlns":43},[46,40153,40154,40221],{},[49,40155,40156,40171,40173,40175,40177,40179,40187,40189,40207,40213],{},[40157,40158,40159,40161,40169],"mpadded",{},[55,40160,296],{"mathvariant":1088},[49,40162,40163,40165,40167],{},[55,40164,12],{},[55,40166,7109],{},[55,40168,3256],{},[62,40170,302],{"stretchy":64},[62,40172,845],{},[2253,40174,32711],{},[2253,40176,32714],{},[55,40178,394],{},[40157,40180,40181,40183,40185],{},[55,40182,296],{"mathvariant":1088},[650,40184,652],{},[62,40186,302],{"stretchy":64},[62,40188,845],{},[2776,40190,40191,40193],{},[55,40192,60],{},[49,40194,40195,40197,40199,40201,40203,40205],{},[62,40196,1213],{},[55,40198,3256],{},[55,40200,2263],{},[650,40202,652],{},[55,40204,24201],{},[55,40206,3971],{},[52,40208,40209,40211],{},[55,40210,29018],{},[55,40212,394],{},[40157,40214,40215,40217,40219],{},[55,40216,296],{"mathvariant":1088},[650,40218,652],{},[62,40220,302],{"stretchy":64},[82,40222,40223],{"encoding":84},"\\ket{psi} = \\text{Prep} \\; \\psi \\ket{0} = e^{-i H 0 dt} U_{\\psi} \\ket{0}",[28,40225,40227,40263,40308],{"className":40226,"ariaHidden":71},[89],[28,40228,40230,40233,40254,40257,40260],{"className":40229},[93],[28,40231],{"className":40232,"style":98},[97],[28,40234,40236,40239,40251],{"className":40235},[3398],[28,40237,296],{"className":40238},[102],[28,40240,40242,40245,40248],{"className":40241},[102],[28,40243,12],{"className":40244},[102,106],[28,40246,7109],{"className":40247},[102,106],[28,40249,3256],{"className":40250},[102,106],[28,40252,302],{"className":40253},[182],[28,40255],{"className":40256,"style":913},[173],[28,40258,845],{"className":40259},[917],[28,40261],{"className":40262,"style":913},[173],[28,40264,40266,40269,40275,40278,40281,40284,40299,40302,40305],{"className":40265},[93],[28,40267],{"className":40268,"style":98},[97],[28,40270,40272],{"className":40271},[102,16058],[28,40273,32711],{"className":40274},[102],[28,40276],{"className":40277,"style":913},[173],[28,40279,394],{"className":40280,"style":321},[102,106],[28,40282],{"className":40283,"style":174},[173],[28,40285,40287,40290,40296],{"className":40286},[3398],[28,40288,296],{"className":40289},[102],[28,40291,40293],{"className":40292},[102],[28,40294,652],{"className":40295},[102],[28,40297,302],{"className":40298},[182],[28,40300],{"className":40301,"style":913},[173],[28,40303,845],{"className":40304},[917],[28,40306],{"className":40307,"style":913},[173],[28,40309,40311,40314,40361,40404,40407],{"className":40310},[93],[28,40312],{"className":40313,"style":37001},[97],[28,40315,40317,40320],{"className":40316},[102],[28,40318,60],{"className":40319},[102,106],[28,40321,40323],{"className":40322},[111],[28,40324,40326],{"className":40325},[115],[28,40327,40329],{"className":40328},[120],[28,40330,40332],{"className":40331,"style":29397},[124],[28,40333,40334,40337],{"style":1726},[28,40335],{"className":40336,"style":133},[132],[28,40338,40340],{"className":40339},[137,138,139,140],[28,40341,40343,40346,40349,40352,40355,40358],{"className":40342},[102,140],[28,40344,1213],{"className":40345},[102,140],[28,40347,3256],{"className":40348},[102,106,140],[28,40350,2263],{"className":40351,"style":4040},[102,106,140],[28,40353,652],{"className":40354},[102,140],[28,40356,24201],{"className":40357},[102,106,140],[28,40359,3971],{"className":40360},[102,106,140],[28,40362,40364,40367],{"className":40363},[102],[28,40365,29018],{"className":40366,"style":29050},[102,106],[28,40368,40370],{"className":40369},[111],[28,40371,40373,40396],{"className":40372},[115,116],[28,40374,40376,40393],{"className":40375},[120],[28,40377,40379],{"className":40378,"style":10240},[124],[28,40380,40381,40384],{"style":37069},[28,40382],{"className":40383,"style":133},[132],[28,40385,40387],{"className":40386},[137,138,139,140],[28,40388,40390],{"className":40389},[102,140],[28,40391,394],{"className":40392,"style":321},[102,106,140],[28,40394,148],{"className":40395},[147],[28,40397,40399],{"className":40398},[120],[28,40400,40402],{"className":40401,"style":2003},[124],[28,40403],{},[28,40405],{"className":40406,"style":174},[173],[28,40408,40410,40413,40419],{"className":40409},[3398],[28,40411,296],{"className":40412},[102],[28,40414,40416],{"className":40415},[102],[28,40417,652],{"className":40418},[102],[28,40420,302],{"className":40421},[182],", for example, to prepare the HF state for chemistry ",[28,40424,40426,40443],{"className":40425},[35],[28,40427,40429],{"className":40428},[39],[41,40430,40431],{"xmlns":43},[46,40432,40433,40441],{},[49,40434,40435,40437,40439],{},[2253,40436,32711],{},[2253,40438,32714],{},[55,40440,394],{},[82,40442,40121],{"encoding":84},[28,40444,40446],{"className":40445,"ariaHidden":71},[89],[28,40447,40449,40452,40458,40461],{"className":40448},[93],[28,40450],{"className":40451,"style":214},[97],[28,40453,40455],{"className":40454},[102,16058],[28,40456,32711],{"className":40457},[102],[28,40459],{"className":40460,"style":913},[173],[28,40462,394],{"className":40463,"style":321},[102,106]," would be a product of single-qubit NOTs, so controlled-",[28,40466,40468,40485],{"className":40467},[35],[28,40469,40471],{"className":40470},[39],[41,40472,40473],{"xmlns":43},[46,40474,40475,40483],{},[49,40476,40477,40479,40481],{},[2253,40478,32711],{},[2253,40480,32714],{},[55,40482,394],{},[82,40484,40121],{"encoding":84},[28,40486,40488],{"className":40487,"ariaHidden":71},[89],[28,40489,40491,40494,40500,40503],{"className":40490},[93],[28,40492],{"className":40493,"style":214},[97],[28,40495,40497],{"className":40496},[102,16058],[28,40498,32711],{"className":40499},[102],[28,40501],{"className":40502,"style":913},[173],[28,40504,394],{"className":40505,"style":321},[102,106]," is just a product of CNOTs.\nThen the circuit above implements the following state prior to measurement:",[28,40508,40510],{"className":40509},[31],[28,40511,40513,41132],{"className":40512},[35],[28,40514,40516],{"className":40515},[39],[41,40517,40518],{"xmlns":43,"display":44},[46,40519,40520,41129],{},[3178,40521,40522],{"rowspacing":3180,"columnspacing":3182},[3184,40523,40524,40528,41122,41125],{},[3187,40525],{"className":40526},[40527],"mtr-glue",[3187,40529,40530],{},[3190,40531,40532],{"scriptlevel":652,"displaystyle":71},[3178,40533,40534,40637,40704,40791,40874,40992],{"rowspacing":37757,"columnalign":37758,"columnspacing":37759},[3184,40535,40536,40573],{},[3187,40537,40538],{},[3190,40539,40540],{"scriptlevel":652,"displaystyle":71},[49,40541,40542,40550,40562],{},[40157,40543,40544,40546,40548],{},[55,40545,296],{"mathvariant":1088},[650,40547,652],{},[62,40549,302],{"stretchy":64},[2776,40551,40552,40560],{},[40157,40553,40554,40556,40558],{},[55,40555,296],{"mathvariant":1088},[650,40557,652],{},[62,40559,302],{"stretchy":64},[55,40561,33375],{},[10155,40563,40564,40567],{},[62,40565,7626],{"stretchy":71,"minsize":40566},"3.0em",[40157,40568,40571],{"width":40569,"lspace":40570},"+0.6em","0.3em",[55,40572,2263],{},[3187,40574,40575],{},[3190,40576,40577],{"scriptlevel":652,"displaystyle":71},[49,40578,40579,40581,40589],{},[49,40580],{},[3977,40582,40583,40585],{},[650,40584,3200],{},[5661,40586,40587],{},[650,40588,848],{},[49,40590,40591,40593,40601,40613,40615,40623,40635],{},[62,40592,65],{"fence":71},[40157,40594,40595,40597,40599],{},[55,40596,296],{"mathvariant":1088},[650,40598,652],{},[62,40600,302],{"stretchy":64},[2776,40602,40603,40611],{},[40157,40604,40605,40607,40609],{},[55,40606,296],{"mathvariant":1088},[650,40608,652],{},[62,40610,302],{"stretchy":64},[55,40612,33375],{},[62,40614,2606],{},[40157,40616,40617,40619,40621],{},[55,40618,296],{"mathvariant":1088},[650,40620,3200],{},[62,40622,302],{"stretchy":64},[2776,40624,40625,40633],{},[40157,40626,40627,40629,40631],{},[55,40628,296],{"mathvariant":1088},[650,40630,652],{},[62,40632,302],{"stretchy":64},[55,40634,33375],{},[62,40636,78],{"fence":71},[3184,40638,40639,40652],{},[3187,40640,40641],{},[3190,40642,40643],{"scriptlevel":652,"displaystyle":71},[10155,40644,40645,40647],{},[62,40646,7626],{"stretchy":71,"minsize":40566},[40157,40648,40649],{"width":40569,"lspace":40570},[2253,40650,40651],{},"1-ctrl-init",[3187,40653,40654],{},[3190,40655,40656],{"scriptlevel":652,"displaystyle":71},[49,40657,40658,40660,40668],{},[49,40659],{},[3977,40661,40662,40664],{},[650,40663,3200],{},[5661,40665,40666],{},[650,40667,848],{},[49,40669,40670,40672,40674,40676,40678,40680,40682,40688,40690,40692,40694,40696,40698,40700,40702],{},[62,40671,65],{"fence":71},[55,40673,296],{"mathvariant":1088},[650,40675,652],{},[62,40677,302],{"stretchy":64},[55,40679,296],{"mathvariant":1088},[650,40681,652],{},[2776,40683,40684,40686],{},[62,40685,302],{"stretchy":64},[55,40687,33375],{},[62,40689,2606],{},[55,40691,296],{"mathvariant":1088},[650,40693,3200],{},[62,40695,302],{"stretchy":64},[55,40697,296],{"mathvariant":1088},[55,40699,394],{},[62,40701,302],{"stretchy":64},[62,40703,78],{"fence":71},[3184,40705,40706,40718],{},[3187,40707,40708],{},[3190,40709,40710],{"scriptlevel":652,"displaystyle":71},[10155,40711,40712,40714],{},[62,40713,7626],{"stretchy":71,"minsize":40566},[40157,40715,40716],{"width":40569,"lspace":40570},[55,40717,29018],{},[3187,40719,40720],{},[3190,40721,40722],{"scriptlevel":652,"displaystyle":71},[49,40723,40724,40726,40734],{},[49,40725],{},[3977,40727,40728,40730],{},[650,40729,3200],{},[5661,40731,40732],{},[650,40733,848],{},[49,40735,40736,40738,40749,40757,40769,40771,40779,40781,40789],{},[62,40737,65],{"fence":71},[2776,40739,40740,40742],{},[55,40741,60],{},[49,40743,40744,40746],{},[55,40745,3256],{},[55,40747,40748],{},"ϕ",[40157,40750,40751,40753,40755],{},[55,40752,296],{"mathvariant":1088},[650,40754,652],{},[62,40756,302],{"stretchy":64},[2776,40758,40759,40767],{},[40157,40760,40761,40763,40765],{},[55,40762,296],{"mathvariant":1088},[650,40764,652],{},[62,40766,302],{"stretchy":64},[55,40768,33375],{},[62,40770,2606],{},[40157,40772,40773,40775,40777],{},[55,40774,296],{"mathvariant":1088},[650,40776,3200],{},[62,40778,302],{"stretchy":64},[55,40780,29018],{},[40157,40782,40783,40785,40787],{},[55,40784,296],{"mathvariant":1088},[55,40786,394],{},[62,40788,302],{"stretchy":64},[62,40790,78],{"fence":71},[3184,40792,40793,40806],{},[3187,40794,40795],{},[3190,40796,40797],{"scriptlevel":652,"displaystyle":71},[10155,40798,40799,40801],{},[62,40800,7626],{"stretchy":71,"minsize":40566},[40157,40802,40803],{"width":40569,"lspace":40570},[2253,40804,40805],{},"0-ctrl-init",[3187,40807,40808],{},[3190,40809,40810],{"scriptlevel":652,"displaystyle":71},[49,40811,40812,40814,40822],{},[49,40813],{},[3977,40815,40816,40818],{},[650,40817,3200],{},[5661,40819,40820],{},[650,40821,848],{},[49,40823,40824,40826,40836,40844,40852,40854,40862,40864,40872],{},[62,40825,65],{"fence":71},[2776,40827,40828,40830],{},[55,40829,60],{},[49,40831,40832,40834],{},[55,40833,3256],{},[55,40835,40748],{},[40157,40837,40838,40840,40842],{},[55,40839,296],{"mathvariant":1088},[650,40841,652],{},[62,40843,302],{"stretchy":64},[40157,40845,40846,40848,40850],{},[55,40847,296],{"mathvariant":1088},[55,40849,394],{},[62,40851,302],{"stretchy":64},[62,40853,2606],{},[40157,40855,40856,40858,40860],{},[55,40857,296],{"mathvariant":1088},[650,40859,3200],{},[62,40861,302],{"stretchy":64},[55,40863,29018],{},[40157,40865,40866,40868,40870],{},[55,40867,296],{"mathvariant":1088},[55,40869,394],{},[62,40871,302],{"stretchy":64},[62,40873,78],{"fence":71},[3184,40875,40876,40882],{},[3187,40877,40878],{},[3190,40879,40880],{"scriptlevel":652,"displaystyle":71},[62,40881,845],{"lspace":37759,"rspace":37759},[3187,40883,40884],{},[3190,40885,40886],{"scriptlevel":652,"displaystyle":71},[49,40887,40888,40890,40896],{},[49,40889],{},[3977,40891,40892,40894],{},[650,40893,3200],{},[650,40895,848],{},[49,40897,40898,40900,40908,40944,40946,40954,40990],{},[62,40899,65],{"fence":71},[40157,40901,40902,40904,40906],{},[55,40903,296],{"mathvariant":1088},[62,40905,2606],{"lspace":37759,"rspace":37759},[62,40907,302],{"stretchy":64},[49,40909,40910,40912,40922,40930,40932,40934,40942],{},[62,40911,65],{"fence":71},[2776,40913,40914,40916],{},[55,40915,60],{},[49,40917,40918,40920],{},[55,40919,3256],{},[55,40921,40748],{},[40157,40923,40924,40926,40928],{},[55,40925,296],{"mathvariant":1088},[55,40927,394],{},[62,40929,302],{"stretchy":64},[62,40931,2606],{},[55,40933,29018],{},[40157,40935,40936,40938,40940],{},[55,40937,296],{"mathvariant":1088},[55,40939,394],{},[62,40941,302],{"stretchy":64},[62,40943,78],{"fence":71},[62,40945,2606],{},[40157,40947,40948,40950,40952],{},[55,40949,296],{"mathvariant":1088},[62,40951,1213],{"lspace":37759,"rspace":37759},[62,40953,302],{"stretchy":64},[49,40955,40956,40958,40968,40976,40978,40980,40988],{},[62,40957,65],{"fence":71},[2776,40959,40960,40962],{},[55,40961,60],{},[49,40963,40964,40966],{},[55,40965,3256],{},[55,40967,40748],{},[40157,40969,40970,40972,40974],{},[55,40971,296],{"mathvariant":1088},[55,40973,394],{},[62,40975,302],{"stretchy":64},[62,40977,1213],{},[55,40979,29018],{},[40157,40981,40982,40984,40986],{},[55,40983,296],{"mathvariant":1088},[55,40985,394],{},[62,40987,302],{"stretchy":64},[62,40989,78],{"fence":71},[62,40991,78],{"fence":71},[3184,40993,40994,41000],{},[3187,40995,40996],{},[3190,40997,40998],{"scriptlevel":652,"displaystyle":71},[62,40999,845],{"lspace":37759,"rspace":37759},[3187,41001,41002],{},[3190,41003,41004],{"scriptlevel":652,"displaystyle":71},[49,41005,41006,41008,41014],{},[49,41007],{},[3977,41009,41010,41012],{},[650,41011,3200],{},[650,41013,848],{},[49,41015,41016,41018,41030,41068,41070,41082,41120],{},[62,41017,65],{"fence":71},[40157,41019,41020,41022,41028],{},[55,41021,296],{"mathvariant":1088},[49,41023,41024,41026],{},[62,41025,2606],{},[55,41027,3256],{},[62,41029,302],{"stretchy":64},[49,41031,41032,41034,41044,41052,41054,41056,41058,41066],{},[62,41033,65],{"fence":71},[2776,41035,41036,41038],{},[55,41037,60],{},[49,41039,41040,41042],{},[55,41041,3256],{},[55,41043,40748],{},[40157,41045,41046,41048,41050],{},[55,41047,296],{"mathvariant":1088},[55,41049,394],{},[62,41051,302],{"stretchy":64},[62,41053,1213],{},[55,41055,3256],{},[55,41057,29018],{},[40157,41059,41060,41062,41064],{},[55,41061,296],{"mathvariant":1088},[55,41063,394],{},[62,41065,302],{"stretchy":64},[62,41067,78],{"fence":71},[62,41069,2606],{},[40157,41071,41072,41074,41080],{},[55,41073,296],{"mathvariant":1088},[49,41075,41076,41078],{},[62,41077,1213],{},[55,41079,3256],{},[62,41081,302],{"stretchy":64},[49,41083,41084,41086,41096,41104,41106,41108,41110,41118],{},[62,41085,65],{"fence":71},[2776,41087,41088,41090],{},[55,41089,60],{},[49,41091,41092,41094],{},[55,41093,3256],{},[55,41095,40748],{},[40157,41097,41098,41100,41102],{},[55,41099,296],{"mathvariant":1088},[55,41101,394],{},[62,41103,302],{"stretchy":64},[62,41105,2606],{},[55,41107,3256],{},[55,41109,29018],{},[40157,41111,41112,41114,41116],{},[55,41113,296],{"mathvariant":1088},[55,41115,394],{},[62,41117,302],{"stretchy":64},[62,41119,78],{"fence":71},[62,41121,78],{"fence":71},[3187,41123],{"className":41124},[40527],[3187,41126],{"className":41127},[41128],"mml-eqn-num",[82,41130,41131],{"encoding":84},"\\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    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we have used the classical simulable phase shift ",[28,43277,43279,43329],{"className":43278},[35],[28,43280,43282],{"className":43281},[39],[41,43283,43284],{"xmlns":43},[46,43285,43286,43326],{},[49,43287,43288,43290,43302,43304,43314],{},[55,43289,29018],{},[2776,43291,43292,43300],{},[40157,43293,43294,43296,43298],{},[55,43295,296],{"mathvariant":1088},[650,43297,652],{},[62,43299,302],{"stretchy":64},[55,43301,33375],{},[62,43303,845],{},[2776,43305,43306,43308],{},[55,43307,60],{},[49,43309,43310,43312],{},[55,43311,3256],{},[55,43313,40748],{},[2776,43315,43316,43324],{},[40157,43317,43318,43320,43322],{},[55,43319,296],{"mathvariant":1088},[650,43321,652],{},[62,43323,302],{"stretchy":64},[55,43325,33375],{},[82,43327,43328],{"encoding":84}," U\\ket{0}^N = e^{i\\phi}\\ket{0}^N",[28,43330,43332,43395],{"className":43331,"ariaHidden":71},[89],[28,43333,43335,43339,43342,43345,43386,43389,43392],{"className":43334},[93],[28,43336],{"className":43337,"style":43338},[97],"height:1.2312em;vertical-align:-0.25em;",[28,43340,29018],{"className":43341,"style":29050},[102,106],[28,43343],{"className":43344,"style":174},[173],[28,43346,43348,43363],{"className":43347},[3398],[28,43349,43351,43354,43360],{"className":43350},[3398],[28,43352,296],{"className":43353},[102],[28,43355,43357],{"className":43356},[102],[28,43358,652],{"className":43359},[102],[28,43361,302],{"className":43362},[182],[28,43364,43366],{"className":43365},[111],[28,43367,43369],{"className":43368},[115],[28,43370,43372],{"className":43371},[120],[28,43373,43375],{"className":43374,"style":33515},[124],[28,43376,43377,43380],{"style":33538},[28,43378],{"className":43379,"style":133},[132],[28,43381,43383],{"className":43382},[137,138,139,140],[28,43384,33375],{"className":43385,"style":29050},[102,106,140],[28,43387],{"className":43388,"style":913},[173],[28,43390,845],{"className":43391},[917],[28,43393],{"className":43394,"style":913},[173],[28,43396,43398,43401,43436,43439],{"className":43397},[93],[28,43399],{"className":43400,"style":43338},[97],[28,43402,43404,43407],{"className":43403},[102],[28,43405,60],{"className":43406},[102,106],[28,43408,43410],{"className":43409},[111],[28,43411,43413],{"className":43412},[115],[28,43414,43416],{"className":43415},[120],[28,43417,43419],{"className":43418,"style":29397},[124],[28,43420,43421,43424],{"style":1726},[28,43422],{"className":43423,"style":133},[132],[28,43425,43427],{"className":43426},[137,138,139,140],[28,43428,43430,43433],{"className":43429},[102,140],[28,43431,3256],{"className":43432},[102,106,140],[28,43434,40748],{"className":43435},[102,106,140],[28,43437],{"className":43438,"style":174},[173],[28,43440,43442,43457],{"className":43441},[3398],[28,43443,43445,43448,43454],{"className":43444},[3398],[28,43446,296],{"className":43447},[102],[28,43449,43451],{"className":43450},[102],[28,43452,652],{"className":43453},[102],[28,43455,302],{"className":43456},[182],[28,43458,43460],{"className":43459},[111],[28,43461,43463],{"className":43462},[115],[28,43464,43466],{"className":43465},[120],[28,43467,43469],{"className":43468,"style":33515},[124],[28,43470,43471,43474],{"style":33538},[28,43472],{"className":43473,"style":133},[132],[28,43475,43477],{"className":43476},[137,138,139,140],[28,43478,33375],{"className":43479,"style":29050},[102,106,140]," in the third line. Therefore the expectation values are obtained as",[28,43482,43484],{"className":43483},[31],[28,43485,43487,43796],{"className":43486},[35],[28,43488,43490],{"className":43489},[39],[41,43491,43492],{"xmlns":43,"display":44},[46,43493,43494,43793],{},[3178,43495,43496],{"rowspacing":3180,"columnspacing":3182},[3184,43497,43498,43501,43787,43790],{},[3187,43499],{"className":43500},[40527],[3187,43502,43503],{},[3190,43504,43505],{"scriptlevel":652,"displaystyle":71},[3178,43506,43507,43625,43727],{"rowspacing":37757,"columnalign":37758,"columnspacing":37759},[3184,43508,43509,43526],{},[3187,43510,43511],{},[3190,43512,43513],{"scriptlevel":652,"displaystyle":71},[49,43514,43515,43517,43519,43522,43524],{},[62,43516,2948],{"stretchy":64},[55,43518,9841],{},[62,43520,43521],{},"⊗",[55,43523,57],{},[62,43525,302],{"stretchy":64},[3187,43527,43528],{},[3190,43529,43530],{"scriptlevel":652,"displaystyle":71},[49,43531,43532,43534,43536,43542,43544,43587,43589],{},[49,43533],{},[62,43535,845],{},[3977,43537,43538,43540],{},[650,43539,3200],{},[650,43541,37795],{},[62,43543,65],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[49,43545,43546,43548,43560,43568,43570,43578,43585],{},[62,43547,65],{"fence":71},[2776,43549,43550,43552],{},[55,43551,60],{},[49,43553,43554,43556,43558],{},[62,43555,1213],{},[55,43557,3256],{},[55,43559,40748],{},[40157,43561,43562,43564,43566],{},[62,43563,2948],{"stretchy":64},[55,43565,394],{},[55,43567,296],{"mathvariant":1088},[62,43569,2606],{},[40157,43571,43572,43574,43576],{},[62,43573,2948],{"stretchy":64},[55,43575,394],{},[55,43577,296],{"mathvariant":1088},[2776,43579,43580,43582],{},[55,43581,29018],{},[62,43583,43584],{},"†",[62,43586,78],{"fence":71},[55,43588,57],{},[49,43590,43591,43593,43603,43611,43613,43615,43623],{},[62,43592,65],{"fence":71},[2776,43594,43595,43597],{},[55,43596,60],{},[49,43598,43599,43601],{},[55,43600,3256],{},[55,43602,40748],{},[40157,43604,43605,43607,43609],{},[55,43606,296],{"mathvariant":1088},[55,43608,394],{},[62,43610,302],{"stretchy":64},[62,43612,2606],{},[55,43614,29018],{},[40157,43616,43617,43619,43621],{},[55,43618,296],{"mathvariant":1088},[55,43620,394],{},[62,43622,302],{"stretchy":64},[62,43624,78],{"fence":71},[3184,43626,43627,43633],{},[3187,43628,43629],{},[3190,43630,43631],{"scriptlevel":652,"displaystyle":71},[49,43632],{},[3187,43634,43635],{},[3190,43636,43637],{"scriptlevel":652,"displaystyle":71},[49,43638,43639,43641,43643,43645,43687,43689,43725],{},[49,43640],{},[173,43642],{"width":3222},[62,43644,1213],{},[49,43646,43647,43649,43661,43669,43671,43679,43685],{},[62,43648,65],{"fence":71},[2776,43650,43651,43653],{},[55,43652,60],{},[49,43654,43655,43657,43659],{},[62,43656,1213],{},[55,43658,3256],{},[55,43660,40748],{},[40157,43662,43663,43665,43667],{},[62,43664,2948],{"stretchy":64},[55,43666,394],{},[55,43668,296],{"mathvariant":1088},[62,43670,1213],{},[40157,43672,43673,43675,43677],{},[62,43674,2948],{"stretchy":64},[55,43676,394],{},[55,43678,296],{"mathvariant":1088},[2776,43680,43681,43683],{},[55,43682,29018],{},[62,43684,43584],{},[62,43686,78],{"fence":71},[55,43688,57],{},[49,43690,43691,43693,43703,43711,43713,43715,43723],{},[62,43692,65],{"fence":71},[2776,43694,43695,43697],{},[55,43696,60],{},[49,43698,43699,43701],{},[55,43700,3256],{},[55,43702,40748],{},[40157,43704,43705,43707,43709],{},[55,43706,296],{"mathvariant":1088},[55,43708,394],{},[62,43710,302],{"stretchy":64},[62,43712,1213],{},[55,43714,29018],{},[40157,43716,43717,43719,43721],{},[55,43718,296],{"mathvariant":1088},[55,43720,394],{},[62,43722,302],{"stretchy":64},[62,43724,78],{"fence":71},[62,43726,78],{"fence":64,"stretchy":71,"minsize":35171,"maxsize":35171},[3184,43728,43729,43735],{},[3187,43730,43731],{},[3190,43732,43733],{"scriptlevel":652,"displaystyle":71},[49,43734],{},[3187,43736,43737],{},[3190,43738,43739],{"scriptlevel":652,"displaystyle":71},[49,43740,43741,43743,43745,43747,43785],{},[49,43742],{},[62,43744,845],{},[2253,43746,37888],{},[49,43748,43749,43751,43763,43771,43773,43775,43783],{},[62,43750,3176],{"fence":71},[2776,43752,43753,43755],{},[55,43754,60],{},[49,43756,43757,43759,43761],{},[62,43758,1213],{},[55,43760,3256],{},[55,43762,40748],{},[40157,43764,43765,43767,43769],{},[62,43766,2948],{"stretchy":64},[55,43768,394],{},[55,43770,296],{"mathvariant":1088},[55,43772,57],{},[55,43774,29018],{},[40157,43776,43777,43779,43781],{},[55,43778,296],{"mathvariant":1088},[55,43780,394],{},[62,43782,302],{"stretchy":64},[62,43784,3217],{"fence":71},[62,43786,72],{"separator":71},[3187,43788],{"className":43789},[40527],[3187,43791],{"className":43792},[41128],[82,43794,43795],{"encoding":84},"\\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}",[28,43797,43799,44667],{"className":43798,"ariaHidden":71},[89],[28,43800,43802,43806],{"className":43801},[93],[28,43803],{"className":43804,"style":43805},[97],"height:5.6666em;vertical-align:-2.5833em;",[28,43807,43809],{"className":43808},[3178],[28,43810,43812],{"className":43811},[3417],[28,43813,43815,44659],{"className":43814},[115,116],[28,43816,43818,44656],{"className":43817},[120],[28,43819,43822],{"className":43820,"style":43821},[124],"height:3.0833em;",[28,43823,43825,43829],{"style":43824},"top:-5.0833em;",[28,43826],{"className":43827,"style":43828},[132],"height:5.0833em;",[28,43830,43832],{"className":43831},[102],[28,43833,43835],{"className":43834},[102],[28,43836,43838,43909],{"className":43837},[3178],[28,43839,43841],{"className":43840},[37968],[28,43842,43844,43900],{"className":43843},[115,116],[28,43845,43847,43897],{"className":43846},[120],[28,43848,43850,43879,43888],{"className":43849,"style":43821},[124],[28,43851,43852,43855],{"style":43824},[28,43853],{"className":43854,"style":35396},[132],[28,43856,43858,43861,43864,43867,43870,43873,43876],{"className":43857},[102],[28,43859,2948],{"className":43860},[161],[28,43862,9841],{"className":43863,"style":9856},[102,106],[28,43865],{"className":43866,"style":1257},[173],[28,43868,43521],{"className":43869},[1261],[28,43871],{"className":43872,"style":1257},[173],[28,43874,57],{"className":43875,"style":107},[102,106],[28,43877,302],{"className":43878},[182],[28,43880,43882,43885],{"style":43881},"top:-2.9473em;",[28,43883],{"className":43884,"style":35396},[132],[28,43886],{"className":43887},[102],[28,43889,43891,43894],{"style":43890},"top:-1.0982em;",[28,43892],{"className":43893,"style":35396},[132],[28,43895],{"className":43896},[102],[28,43898,148],{"className":43899},[147],[28,43901,43903],{"className":43902},[120],[28,43904,43907],{"className":43905,"style":43906},[124],"height:2.5833em;",[28,43908],{},[28,43910,43912],{"className":43911},[38029],[28,43913,43915,44648],{"className":43914},[115,116],[28,43916,43918,44645],{"className":43917},[120],[28,43919,43921,44246,44512],{"className":43920,"style":43821},[124],[28,43922,43923,43926],{"style":43824},[28,43924],{"className":43925,"style":35396},[132],[28,43927,43929,43932,43935,43938,43941,44003,44009,44012,44139,44142,44145,44148],{"className":43928},[102],[28,43930],{"className":43931},[102],[28,43933],{"className":43934,"style":913},[173],[28,43936,845],{"className":43937},[917],[28,43939],{"className":43940,"style":913},[173],[28,43942,43944,43947,44000],{"className":43943},[102],[28,43945],{"className":43946},[161,4072],[28,43948,43950],{"className":43949},[3977],[28,43951,43953,43992],{"className":43952},[115,116],[28,43954,43956,43989],{"className":43955},[120],[28,43957,43959,43970,43978],{"className":43958,"style":6825},[124],[28,43960,43961,43964],{"style":4088},[28,43962],{"className":43963,"style":3434},[132],[28,43965,43967],{"className":43966},[102],[28,43968,37795],{"className":43969},[102],[28,43971,43972,43975],{"style":4100},[28,43973],{"className":43974,"style":3434},[132],[28,43976],{"className":43977,"style":4108},[4107],[28,43979,43980,43983],{"style":4111},[28,43981],{"className":43982,"style":3434},[132],[28,43984,43986],{"className":43985},[102],[28,43987,3200],{"className":43988},[102],[28,43990,148],{"className":43991},[147],[28,43993,43995],{"className":43994},[120],[28,43996,43998],{"className":43997,"style":4170},[124],[28,43999],{},[28,44001],{"className":44002},[182,4072],[28,44004,44006],{"className":44005},[102],[28,44007,65],{"className":44008},[3407,35567],[28,44010],{"className":44011,"style":174},[173],[28,44013,44015,44021,44059,44062,44077,44080,44083,44086,44101,44104,44133],{"className":44014},[3398],[28,44016,44018],{"className":44017,"style":3403},[161,3402],[28,44019,65],{"className":44020},[3407,28701],[28,44022,44024,44027],{"className":44023},[102],[28,44025,60],{"className":44026},[102,106],[28,44028,44030],{"className":44029},[111],[28,44031,44033],{"className":44032},[115],[28,44034,44036],{"className":44035},[120],[28,44037,44039],{"className":44038,"style":30761},[124],[28,44040,44041,44044],{"style":2839},[28,44042],{"className":44043,"style":133},[132],[28,44045,44047],{"className":44046},[137,138,139,140],[28,44048,44050,44053,44056],{"className":44049},[102,140],[28,44051,1213],{"className":44052},[102,140],[28,44054,3256],{"className":44055},[102,106,140],[28,44057,40748],{"className":44058},[102,106,140],[28,44060],{"className":44061,"style":174},[173],[28,44063,44065,44068,44074],{"className":44064},[3398],[28,44066,2948],{"className":44067},[161],[28,44069,44071],{"className":44070},[102],[28,44072,394],{"className":44073,"style":321},[102,106],[28,44075,296],{"className":44076},[102],[28,44078],{"className":44079,"style":1257},[173],[28,44081,2606],{"className":44082},[1261],[28,44084],{"className":44085,"style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[40527],[3187,45014],{"className":45015},[41128],[82,45017,45018],{"encoding":84},"\\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}",[28,45020,45022,45901],{"className":45021,"ariaHidden":71},[89],[28,45023,45025,45028],{"className":45024},[93],[28,45026],{"className":45027,"style":43805},[97],[28,45029,45031],{"className":45030},[3178],[28,45032,45034],{"className":45033},[3417],[28,45035,45037,45893],{"className":45036},[115,116],[28,45038,45040,45890],{"className":45039},[120],[28,45041,45043],{"className":45042,"style":43821},[124],[28,45044,45045,45048],{"style":43824},[28,45046],{"className":45047,"style":43828},[132],[28,45049,45051],{"className":45050},[102],[28,45052,45054],{"className":45053},[102],[28,45055,45057,45125],{"className":45056},[3178],[28,45058,45060],{"className":45059},[37968],[28,45061,45063,45117],{"className":45062},[115,116],[28,45064,45066,45114],{"className":45065},[120],[28,45067,45069,45098,45106],{"className":45068,"style":43821},[124],[28,45070,45071,45074],{"style":43824},[28,45072],{"className":45073,"style":35396},[132],[28,45075,45077,45080,45083,45086,45089,45092,45095],{"className":45076},[102],[28,45078,2948],{"className":45079},[161],[28,45081,33439],{"className":45082,"style":1257},[102,106],[28,45084],{"className":45085,"style":1257},[173],[28,45087,43521],{"className":45088},[1261],[28,45090],{"className":45091,"style":1257},[173],[28,45093,57],{"className":45094,"style":107},[102,106],[28,45096,302],{"className":45097},[182],[28,45099,45100,45103],{"style":43881},[28,45101],{"className":45102,"style":35396},[132],[28,45104],{"className":45105},[102],[28,45107,45108,45111],{"style":43890},[28,45109],{"className":45110,"style":35396},[132],[28,45112],{"className":45113},[102],[28,45115,148],{"className":45116},[147],[28,45118,45120],{"className":45119},[120],[28,45121,45123],{"className":45122,"style":43906},[124],[28,45124],{},[28,45126,45128],{"className":45127},[38029],[28,45129,45131,45882],{"className":45130},[115,116],[28,45132,45134,45879],{"className":45133},[120],[28,45135,45137,45471,45746],{"className":45136,"style":43821},[124],[28,45138,45139,45142],{"style":43824},[28,45140],{"className":45141,"style":35396},[132],[28,45143,45145,45148,45151,45154,45157,45219,45225,45228,45361,45364,45367,45370],{"className":45144},[102],[28,45146],{"className":45147},[102],[28,45149],{"className":45150,"style":913},[173],[28,45152,845],{"className":45153},[917],[28,45155],{"className":45156,"style":913},[173],[28,45158,45160,45163,45216],{"className":45159},[102],[28,45161],{"className":45162},[161,4072],[28,45164,45166],{"className":45165},[3977],[28,45167,45169,45208],{"className":45168},[115,116],[28,45170,45172,45205],{"className":45171},[120],[28,45173,45175,45186,45194],{"className":45174,"style":6825},[124],[28,45176,45177,45180],{"style":4088},[28,45178],{"className":45179,"style":3434},[132],[28,45181,45183],{"className":45182},[102],[28,45184,37795],{"className":45185},[102],[28,45187,45188,45191],{"style":4100},[28,45189],{"className":45190,"style":3434},[132],[28,45192],{"className":45193,"style":4108},[4107],[28,45195,45196,45199],{"style":4111},[28,45197],{"className":45198,"style":3434},[132],[28,45200,45202],{"className":45201},[102],[28,45203,3200],{"className":45204},[102],[28,45206,148],{"className":45207},[147],[28,45209,45211],{"className":45210},[120],[28,45212,45214],{"className":45213,"style":4170},[124],[28,45215],{},[28,45217],{"className":45218},[182,4072],[28,45220,45222],{"className":45221},[102],[28,45223,65],{"className":45224},[3407,35567],[28,45226],{"className":45227,"style":174},[173],[28,45229,45231,45237,45275,45278,45293,45296,45299,45302,45305,45308,45323,45326,45355],{"className":45230},[3398],[28,45232,45234],{"className":45233,"style":3403},[161,3402],[28,45235,65],{"className":45236},[3407,28701],[28,45238,45240,45243],{"className":45239},[102],[28,45241,60],{"className":45242},[102,106],[28,45244,45246],{"className":45245},[111],[28,45247,45249],{"className":45248},[115],[28,45250,45252],{"className":45251},[120],[28,45253,45255],{"className":45254,"style":30761},[124],[28,45256,45257,45260],{"style":2839},[28,45258],{"className":45259,"style":133},[132],[28,45261,45263],{"className":45262},[137,138,139,140],[28,45264,45266,45269,45272],{"className":45265},[102,140],[28,45267,1213],{"className":45268},[102,140],[28,45270,3256],{"className":45271},[102,106,140],[28,45273,40748],{"className":45274},[102,106,140],[28,45276],{"className":45277,"style":174},[173],[28,45279,45281,45284,45290],{"className":45280},[3398],[28,45282,2948],{"className":45283},[161],[28,45285,45287],{"className":45286},[102],[28,45288,394],{"className":45289,"style":321},[102,106],[28,45291,296],{"className":45292},[102],[28,45294],{"className":45295,"style":1257},[173],[28,45297,2606],{"className":45298},[1261],[28,45300],{"className":45301,"style":1257},[173],[28,45303,3256],{"className":45304},[102,106],[28,45306],{"className":45307,"style":174},[173],[28,45309,45311,45314,45320],{"className":45310},[3398],[28,45312,2948],{"className":45313},[161],[28,45315,45317],{"className":45316},[102],[28,45318,394],{"className":45319,"style":321},[102,106],[28,45321,296],{"className":45322},[102],[28,45324],{"className":45325,"style":174},[173],[28,45327,45329,45332],{"className":45328},[102],[28,45330,29018],{"className":45331,"style":29050},[102,106],[28,45333,45335],{"className":45334},[111],[28,45336,45338],{"className":45337},[115],[28,45339,45341],{"className":45340},[120],[28,45342,45344],{"className":45343,"style":30761},[124],[28,45345,45346,45349],{"style":2839},[28,45347],{"className":45348,"style":133},[132],[28,45350,45352],{"className":45351},[137,138,139,140],[28,45353,43584],{"className":45354},[1261,140],[28,45356,45358],{"className":45357,"style":3403},[182,3402],[28,45359,78],{"className":45360},[3407,28701],[28,45362],{"className":45363,"style":174},[173],[28,45365,57],{"className":45366,"style":107},[102,106],[28,45368],{"className":45369,"style":174},[173],[28,45371,45373,45379,45414,45417,45432,45435,45438,45441,45444,45447,45450,45465],{"className":45372},[3398],[28,45374,45376],{"className":45375,"style":3403},[161,3402],[28,45377,65],{"className":45378},[3407,28701],[28,45380,45382,45385],{"className":45381},[102],[28,45383,60],{"className":45384},[102,106],[28,45386,45388],{"className":45387},[111],[28,45389,45391],{"className":45390},[115],[28,45392,45394],{"className":45393},[120],[28,45395,45397],{"className":45396,"style":30761},[124],[28,45398,45399,45402],{"style":2839},[28,45400],{"className":45401,"style":133},[132],[28,45403,45405],{"className":45404},[137,138,139,140],[28,45406,45408,45411],{"className":45407},[102,140],[28,45409,3256],{"className":45410},[102,106,140],[28,45412,40748],{"className":45413},[102,106,140],[28,45415],{"className":45416,"style":174},[173],[28,45418,45420,45423,45429],{"className":45419},[3398],[28,45421,296],{"className":45422},[102],[28,45424,45426],{"className":45425},[102],[28,45427,394],{"className":45428,"style":321},[102,106],[28,45430,302],{"className":45431},[182],[28,45433],{"className":45434,"style":1257},[173],[28,45436,1213],{"className":45437},[1261],[28,45439],{"className":45440,"style":1257},[173],[28,45442,3256],{"className":45443},[102,106],[28,45445,29018],{"className":45446,"style":29050},[102,106],[28,45448],{"className":45449,"style":174},[173],[28,45451,45453,45456,45462],{"className":45452},[3398],[28,45454,296],{"className":45455},[102],[28,45457,45459],{"className":45458},[102],[28,45460,394],{"className":45461,"style":321},[102,106],[28,45463,302],{"className":45464},[182],[28,45466,45468],{"className":45467,"style":3403},[182,3402],[28,45469,78],{"className":45470},[3407,28701],[28,45472,45473,45476],{"style":43881},[28,45474],{"className":45475,"style":35396},[132],[28,45477,45479,45482,45485,45488,45491,45494,45627,45630,45633,45636,45737,45740],{"className":45478},[102],[28,45480],{"className":45481},[102],[28,45483],{"className":45484,"style":3533},[173],[28,45486],{"className":45487,"style":1257},[173],[28,45489,1213],{"className":45490},[1261],[28,45492],{"className":45493,"style":1257},[173],[28,45495,45497,45503,45541,45544,45559,45562,45565,45568,45571,45574,45589,45592,45621],{"className":45496},[3398],[28,45498,45500],{"className":45499,"style":3403},[161,3402],[28,45501,65],{"className":45502},[3407,28701],[28,45504,45506,45509],{"className":45505},[102],[28,45507,60],{"className":45508},[102,106],[28,45510,45512],{"className":45511},[111],[28,45513,45515],{"className":45514},[115],[28,45516,45518],{"className":45517},[120],[28,45519,45521],{"className":45520,"style":30761},[124],[28,45522,45523,45526],{"style":2839},[28,45524],{"className":45525,"style":133},[132],[28,45527,45529],{"className":45528},[137,138,139,140],[28,45530,45532,45535,45538],{"className":45531},[102,140],[28,45533,1213],{"className":45534},[102,140],[28,45536,3256],{"className":45537},[102,106,140],[28,45539,40748],{"className":45540},[102,106,140],[28,45542],{"className":45543,"style":174},[173],[28,45545,45547,45550,45556],{"className":45546},[3398],[28,45548,2948],{"className":45549},[161],[28,45551,45553],{"className":45552},[102],[28,45554,394],{"className":45555,"style":321},[102,106],[28,45557,296],{"className":45558},[102],[28,45560],{"className":45561,"style":1257},[173],[28,45563,1213],{"className":45564},[1261],[28,45566],{"className":45567,"style":1257},[173],[28,45569,3256],{"className":45570},[102,106],[28,45572],{"className":45573,"style":174},[173],[28,45575,45577,45580,45586],{"className":45576},[3398],[28,45578,2948],{"className":45579},[161],[28,45581,45583],{"className":45582},[102],[28,45584,394],{"className":45585,"style":321},[102,106],[28,45587,296],{"className":45588},[102],[28,45590],{"className":45591,"style":174},[173],[28,45593,45595,45598],{"className":45594},[102],[28,45596,29018],{"className":45597,"style":29050},[102,106],[28,45599,45601],{"className":45600},[111],[28,45602,45604],{"cla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},[124],[28,45900],{},[28,45902,45904],{"className":45903},[43243],[28,45905,45907,45924],{"className":45906},[115,116],[28,45908,45910,45921],{"className":45909},[120],[28,45911,45913],{"className":45912,"style":43821},[124],[28,45914,45915,45918],{"style":43824},[28,45916],{"className":45917,"style":43828},[132],[28,45919],{"className":45920},[43261],[28,45922,148],{"className":45923},[147],[28,45925,45927],{"className":45926},[120],[28,45928,45930],{"className":45929,"style":43906},[124],[28,45931],{},[12,45933,45934,45935,45976],{},"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 ",[28,45936,45938,45955],{"className":45937},[35],[28,45939,45941],{"className":45940},[39],[41,45942,45943],{"xmlns":43},[46,45944,45945,45953],{},[49,45946,45947,45949,45951],{},[2253,45948,32711],{},[2253,45950,32714],{},[55,45952,394],{},[82,45954,40121],{"encoding":84},[28,45956,45958],{"className":45957,"ariaHidden":71},[89],[28,45959,45961,45964,45970,45973],{"className":45960},[93],[28,45962],{"className":45963,"style":214},[97],[28,45965,45967],{"className":45966},[102,16058],[28,45968,32711],{"className":45969},[102],[28,45971],{"className":45972,"style":913},[173],[28,45974,394],{"className":45975,"style":321},[102,106]," and not controlled time evolutions. Reframing our calculation as above will allow us to greatly reduce the depth of the resulting circuits.",[28327,45978,45980],{"id":45979},"decompose-time-evolution-operator-with-trotter-decomposition","Decompose time-evolution operator with Trotter decomposition",[12,45982,45983,45984,16397,46062,16397,46142,46222,46223,46252,46253,46377,46378,46434,46435,46468,46469,46518,46519,46522],{},"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 ",[28,45985,45987,46009],{"className":45986},[35],[28,45988,45990],{"className":45989},[39],[41,45991,45992],{"xmlns":43},[46,45993,45994,46006],{},[49,45995,45996],{},[52,45997,45998,46000],{},[55,45999,1333],{},[49,46001,46002,46004],{},[55,46003,3169],{},[55,46005,3169],{},[82,46007,46008],{"encoding":84},"R_{xx}",[28,46010,46012],{"className":46011,"ariaHidden":71},[89],[28,46013,46015,46018],{"className":46014},[93],[28,46016],{"className":46017,"style":585},[97],[28,46019,46021,46024],{"className":46020},[102],[28,46022,1333],{"className":46023,"style":1377},[102,106],[28,46025,46027],{"className":46026},[111],[28,46028,46030,46054],{"className":46029},[115,116],[28,46031,46033,46051],{"className":46032},[120],[28,46034,46036],{"className":46035,"style":125},[124],[28,46037,46038,46041],{"style":9376},[28,46039],{"className":46040,"style":133},[132],[28,46042,46044],{"className":46043},[137,138,139,140],[28,46045,46047],{"className":46046},[102,140],[28,46048,46050],{"className":46049},[102,106,140],"xx",[28,46052,148],{"className":46053},[147],[28,46055,46057],{"className":46056},[120],[28,46058,46060],{"className":46059,"style":155},[124],[28,46061],{},[28,46063,46065,46087],{"className":46064},[35],[28,46066,46068],{"className":46067},[39],[41,46069,46070],{"xmlns":43},[46,46071,46072,46084],{},[49,46073,46074],{},[52,46075,46076,46078],{},[55,46077,1333],{},[49,46079,46080,46082],{},[55,46081,3229],{},[55,46083,3229],{},[82,46085,46086],{"encoding":84},"R_{yy}",[28,46088,46090],{"className":46089,"ariaHidden":71},[89],[28,46091,46093,46096],{"className":46092},[93],[28,46094],{"className":46095,"style":1964},[97],[28,46097,46099,46102],{"className":46098},[102],[28,46100,1333],{"className":46101,"style":1377},[102,106],[28,46103,46105],{"className":46104},[111],[28,46106,46108,46134],{"className":46107},[115,116],[28,46109,46111,46131],{"className":46110},[120],[28,46112,46114],{"className":46113,"style":125},[124],[28,46115,46116,46119],{"style":9376},[28,46117],{"className":46118,"style":133},[132],[28,46120,46122],{"className":46121},[137,138,139,140],[28,46123,46125,46128],{"className":46124},[102,140],[28,46126,3229],{"className":46127,"style":321},[102,106,140],[28,46129,3229],{"className":46130,"style":321},[102,106,140],[28,46132,148],{"className":46133},[147],[28,46135,46137],{"className":46136},[120],[28,46138,46140],{"className":46139,"style":2003},[124],[28,46141],{},[28,46143,46145,46167],{"className":46144},[35],[28,46146,46148],{"className":46147},[39],[41,46149,46150],{"xmlns":43},[46,46151,46152,46164],{},[49,46153,46154],{},[52,46155,46156,46158],{},[55,46157,1333],{},[49,46159,46160,46162],{},[55,46161,2266],{},[55,46163,2266],{},[82,46165,46166],{"encoding":84},"R_{zz}",[28,46168,46170],{"className":46169,"ariaHidden":71},[89],[28,46171,46173,46176],{"className":46172},[93],[28,46174],{"className":46175,"style":585},[97],[28,46177,46179,46182],{"className":46178},[102],[28,46180,1333],{"className":46181,"style":1377},[102,106],[28,46183,46185],{"className":46184},[111],[28,46186,46188,46214],{"className":46187},[115,116],[28,46189,46191,46211],{"className":46190},[120],[28,46192,46194],{"className":46193,"style":125},[124],[28,46195,46196,46199],{"style":9376},[28,46197],{"className":46198,"style":133},[132],[28,46200,46202],{"className":46201},[137,138,139,140],[28,46203,46205,46208],{"className":46204},[102,140],[28,46206,2266],{"className":46207,"style":3753},[102,106,140],[28,46209,2266],{"className":46210,"style":3753},[102,106,140],[28,46212,148],{"className":46213},[147],[28,46215,46217],{"className":46216},[120],[28,46218,46220],{"className":46219,"style":155},[124],[28,46221],{}," with a parametrized angle ",[28,46224,46226,46239],{"className":46225},[35],[28,46227,46229],{"className":46228},[39],[41,46230,46231],{"xmlns":43},[46,46232,46233,46237],{},[49,46234,46235],{},[55,46236,3971],{},[82,46238,3971],{"encoding":84},[28,46240,46242],{"className":46241,"ariaHidden":71},[89],[28,46243,46245,46249],{"className":46244},[93],[28,46246],{"className":46247,"style":46248},[97],"height:0.6151em;",[28,46250,3971],{"className":46251},[102,106]," which correspond to the approximate implementation of ",[28,46254,46256,46300],{"className":46255},[35],[28,46257,46259],{"className":46258},[39],[41,46260,46261],{"xmlns":43},[46,46262,46263,46297],{},[49,46264,46265],{},[2776,46266,46267,46269],{},[55,46268,60],{},[49,46270,46271,46273,46275,46277,46279,46281,46283,46285,46287,46289,46291,46293,46295],{},[62,46272,1213],{},[55,46274,3256],{},[62,46276,65],{"stretchy":64},[55,46278,9841],{},[55,46280,9841],{},[62,46282,2606],{},[55,46284,33439],{},[55,46286,33439],{},[62,46288,2606],{},[55,46290,33457],{},[55,46292,33457],{},[62,46294,78],{"stretchy":64},[55,46296,3971],{},[82,46298,46299],{"encoding":84},"e^{-i (XX + YY + ZZ) t}",[28,46301,46303],{"className":46302,"ariaHidden":71},[89],[28,46304,46306,46309],{"className":46305},[93],[28,46307],{"className":46308,"style":29443},[97],[28,46310,46312,46315],{"className":46311},[102],[28,46313,60],{"className":46314},[102,106],[28,46316,46318],{"className":46317},[111],[28,46319,46321],{"className":46320},[115],[28,46322,46324],{"className":46323},[120],[28,46325,46327],{"className":46326,"style":29443},[124],[28,46328,46329,46332],{"style":1726},[28,46330],{"className":46331,"style":133},[132],[28,46333,46335],{"className":46334},[137,138,139,140],[28,46336,46338,46341,46344,46347,46350,46353,46356,46359,46362,46365,46368,46371,46374],{"className":46337},[102,140],[28,46339,1213],{"className":46340},[102,140],[28,46342,3256],{"className":46343},[102,106,140],[28,46345,65],{"className":46346},[161,140],[28,46348,9841],{"className":46349,"style":9856},[102,106,140],[28,46351,9841],{"className":46352,"style":9856},[102,106,140],[28,46354,2606],{"className":46355},[1261,140],[28,46357,33439],{"className":46358,"style":1257},[102,106,140],[28,46360,33439],{"className":46361,"style":1257},[102,106,140],[28,46363,2606],{"className":46364},[1261,140],[28,46366,33457],{"className":46367,"style":33765},[102,106,140],[28,46369,33457],{"className":46370,"style":33765},[102,106,140],[28,46372,78],{"className":46373},[182,140],[28,46375,3971],{"className":46376},[102,106,140],". 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 ",[28,46379,46381,46401],{"className":46380},[35],[28,46382,46384],{"className":46383},[39],[41,46385,46386],{"xmlns":43},[46,46387,46388,46398],{},[49,46389,46390,46392,46394,46396],{},[650,46391,848],{},[62,46393,1679],{},[55,46395,24201],{},[55,46397,3971],{},[82,46399,46400],{"encoding":84},"2*dt",[28,46402,46404,46422],{"className":46403,"ariaHidden":71},[89],[28,46405,46407,46410,46413,46416,46419],{"className":46406},[93],[28,46408],{"className":46409,"style":2914},[97],[28,46411,848],{"className":46412},[102],[28,46414],{"className":46415,"style":1257},[173],[28,46417,1679],{"className":46418},[1261],[28,46420],{"className":46421,"style":1257},[173],[28,46423,46425,46428,46431],{"className":46424},[93],[28,46426],{"className":46427,"style":9597},[97],[28,46429,24201],{"className":46430},[102,106],[28,46432,3971],{"className":46433},[102,106]," to achieve a time-evolution of ",[28,46436,46438,46453],{"className":46437},[35],[28,46439,46441],{"className":46440},[39],[41,46442,46443],{"xmlns":43},[46,46444,46445,46451],{},[49,46446,46447,46449],{},[55,46448,24201],{},[55,46450,3971],{},[82,46452,31182],{"encoding":84},[28,46454,46456],{"className":46455,"ariaHidden":71},[89],[28,46457,46459,46462,46465],{"className":46458},[93],[28,46460],{"className":46461,"style":9597},[97],[28,46463,24201],{"className":46464},[102,106],[28,46466,3971],{"className":46467},[102,106],". 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 ",[28,46470,46472,46494],{"className":46471},[35],[28,46473,46475],{"className":46474},[39],[41,46476,46477],{"xmlns":43},[46,46478,46479,46491],{},[49,46480,46481,46483,46485,46487,46489],{},[55,46482,11356],{},[55,46484,29018],{},[62,46486,65],{"stretchy":64},[650,46488,848],{},[62,46490,78],{"stretchy":64},[82,46492,46493],{"encoding":84},"SU(2)",[28,46495,46497],{"className":46496,"ariaHidden":71},[89],[28,46498,46500,46503,46506,46509,46512,46515],{"className":46499},[93],[28,46501],{"className":46502,"style":98},[97],[28,46504,11356],{"className":46505,"style":1913},[102,106],[28,46507,29018],{"className":46508,"style":29050},[102,106],[28,46510,65],{"className":46511},[161],[28,46513,848],{"className":46514},[102],[28,46516,78],{"className":46517},[182]," unitary. This gives a much shallower circuit than what is obtained using the generic ",[8103,46520,46521],{},"PauliEvolutionGate()"," functionality.",[15650,46524,46526],{"className":15652,"code":46525,"language":15654,"meta":15655,"style":15655},"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",[8103,46527,46528,46542,46546,46551,46566,46585,46602,46619,46643,46647,46656,46693,46727,46735,46739,46749,46760,46774,46786,46796,46815,46824,46839,46843,46853,46866,46884,46899,46905,46918,46922],{"__ignoreMap":15655},[28,46529,46530,46532,46534,46536,46538,46540],{"class":15660,"line":15661},[28,46531,34984],{"class":15675},[28,46533,845],{"class":15682},[28,46535,34989],{"class":16073},[28,46537,65],{"class":15675},[28,46539,34994],{"class":15841},[28,46541,16052],{"class":15675},[28,46543,46544],{"class":15660,"line":15668},[28,46545,15769],{"emptyLinePlaceholder":15768},[28,46547,46548],{"class":15660,"line":15679},[28,46549,46550],{"class":15664},"# Create instruction for rotation about XX+YY-ZZ:\n",[28,46552,46553,46556,46558,46560,46562,46564],{"class":15660,"line":15725},[28,46554,46555],{"class":15675},"Rxyz_circ ",[28,46557,845],{"class":15682},[28,46559,26583],{"class":16073},[28,46561,65],{"class":15675},[28,46563,848],{"class":15801},[28,46565,16052],{"class":15675},[28,46567,46568,46571,46574,46577,46579,46581,46583],{"class":15660,"line":15739},[28,46569,46570],{"class":15675},"Rxyz_circ.",[28,46572,46573],{"class":16073},"rxx",[28,46575,46576],{"class":15675},"(t, ",[28,46578,652],{"class":15801},[28,46580,16397],{"class":15675},[28,46582,3200],{"class":15801},[28,46584,16052],{"class":15675},[28,46586,46587,46589,46592,46594,46596,46598,46600],{"class":15660,"line":15752},[28,46588,46570],{"class":15675},[28,46590,46591],{"class":16073},"ryy",[28,46593,46576],{"class":15675},[28,46595,652],{"class":15801},[28,46597,16397],{"class":15675},[28,46599,3200],{"class":15801},[28,46601,16052],{"class":15675},[28,46603,46604,46606,46609,46611,46613,46615,46617],{"class":15660,"line":15765},[28,46605,46570],{"class":15675},[28,46607,46608],{"class":16073},"rzz",[28,46610,46576],{"class":15675},[28,46612,652],{"class":15801},[28,46614,16397],{"class":15675},[28,46616,3200],{"class":15801},[28,46618,16052],{"class":15675},[28,46620,46621,46624,46626,46629,46632,46634,46636,46638,46641],{"class":15660,"line":15772},[28,46622,46623],{"class":15675},"Rxyz_instr ",[28,46625,845],{"class":15682},[28,46627,46628],{"class":15675}," Rxyz_circ.",[28,46630,46631],{"class":16073},"to_instruction",[28,46633,65],{"class":15675},[28,46635,16954],{"class":16307},[28,46637,845],{"class":15682},[28,46639,46640],{"class":15841},"\"RXX+YY+ZZ\"",[28,46642,16052],{"class":15675},[28,46644,46645],{"class":15660,"line":15781},[28,46646,15769],{"emptyLinePlaceholder":15768},[28,46648,46649,46652,46654],{"class":15660,"line":15798},[28,46650,46651],{"class":15675},"interaction_list ",[28,46653,845],{"class":15682},[28,46655,17702],{"class":15675},[28,46657,46658,46661,46663,46665,46667,46669,46671,46673,46675,46677,46679,46682,46684,46686,46688,46690],{"class":15660,"line":15811},[28,46659,46660],{"class":15675},"    [[i, i ",[28,46662,2606],{"class":15682},[28,46664,22230],{"class":15801},[28,46666,15845],{"class":15675},[28,46668,22277],{"class":15671},[28,46670,22280],{"class":15675},[28,46672,17720],{"class":15671},[28,46674,22285],{"class":15682},[28,46676,65],{"class":15675},[28,46678,652],{"class":15801},[28,46680,46681],{"class":15675},", n_qubits ",[28,46683,15689],{"class":15682},[28,46685,22230],{"class":15801},[28,46687,16397],{"class":15675},[28,46689,848],{"class":15801},[28,46691,46692],{"class":15675},")],\n",[28,46694,46695,46697,46699,46701,46703,46705,46707,46709,46711,46713,46715,46717,46719,46721,46723,46725],{"class":15660,"line":15820},[28,46696,46660],{"class":15675},[28,46698,2606],{"class":15682},[28,46700,22230],{"class":15801},[28,46702,15845],{"class":15675},[28,46704,22277],{"class":15671},[28,46706,22280],{"class":15675},[28,46708,17720],{"class":15671},[28,46710,22285],{"class":15682},[28,46712,65],{"class":15675},[28,46714,3200],{"class":15801},[28,46716,46681],{"class":15675},[28,46718,15689],{"class":15682},[28,46720,22230],{"class":15801},[28,46722,16397],{"class":15675},[28,46724,848],{"class":15801},[28,46726,46692],{"class":15675},[28,46728,46729,46732],{"class":15660,"line":15830},[28,46730,46731],{"class":15675},"]  ",[28,46733,46734],{"class":15664},"# linear chain\n",[28,46736,46737],{"class":15660,"line":15835},[28,46738,15769],{"emptyLinePlaceholder":15768},[28,46740,46741,46743,46745,46747],{"class":15660,"line":15853},[28,46742,35061],{"class":15675},[28,46744,845],{"class":15682},[28,46746,35066],{"class":16073},[28,46748,34462],{"class":15675},[28,46750,46751,46754,46756,46758],{"class":15660,"line":15867},[28,46752,46753],{"class":15675},"trotter_step_circ ",[28,46755,845],{"class":15682},[28,46757,26583],{"class":16073},[28,46759,35080],{"class":15675},[28,46761,46762,46764,46767,46769,46771],{"class":15660,"line":15872},[28,46763,22277],{"class":15671},[28,46765,46766],{"class":15675}," i, color ",[28,46768,17720],{"class":15671},[28,46770,27825],{"class":15682},[28,46772,46773],{"class":15675},"(interaction_list):\n",[28,46775,46776,46778,46781,46783],{"class":15660,"line":15878},[28,46777,27785],{"class":15671},[28,46779,46780],{"class":15675}," interaction ",[28,46782,17720],{"class":15671},[28,46784,46785],{"class":15675}," color:\n",[28,46787,46788,46791,46793],{"class":15660,"line":15897},[28,46789,46790],{"class":15675},"        trotter_step_circ.",[28,46792,27807],{"class":16073},[28,46794,46795],{"class":15675},"(Rxyz_instr, interaction)\n",[28,46797,46798,46800,46802,46804,46806,46809,46811,46813],{"class":15660,"line":15914},[28,46799,16484],{"class":15671},[28,46801,22280],{"class":15675},[28,46803,22950],{"class":15682},[28,46805,28134],{"class":15682},[28,46807,46808],{"class":15675},"(interaction_list) ",[28,46810,15689],{"class":15682},[28,46812,22230],{"class":15801},[28,46814,15778],{"class":15675},[28,46816,46817,46819,46822],{"class":15660,"line":15931},[28,46818,46790],{"class":15675},[28,46820,46821],{"class":16073},"barrier",[28,46823,17029],{"class":15675},[28,46825,46826,46829,46831,46834,46837],{"class":15660,"line":15948},[28,46827,46828],{"class":15675},"reverse_trotter_step_circ ",[28,46830,845],{"class":15682},[28,46832,46833],{"class":15675}," trotter_step_circ.",[28,46835,46836],{"class":16073},"reverse_ops",[28,46838,17029],{"class":15675},[28,46840,46841],{"class":15660,"line":15966},[28,46842,15769],{"emptyLinePlaceholder":15768},[28,46844,46845,46847,46849,46851],{"class":15660,"line":15983},[28,46846,35073],{"class":15675},[28,46848,845],{"class":15682},[28,46850,26583],{"class":16073},[28,46852,35080],{"class":15675},[28,46854,46855,46857,46860,46862,46864],{"class":15660,"line":16000},[28,46856,22277],{"class":15671},[28,46858,46859],{"class":15675}," step ",[28,46861,17720],{"class":15671},[28,46863,22285],{"class":15682},[28,46865,39532],{"class":15675},[28,46867,46868,46870,46872,46874,46876,46879,46882],{"class":15660,"line":16016},[28,46869,16484],{"class":15671},[28,46871,46859],{"class":15675},[28,46873,15683],{"class":15682},[28,46875,22129],{"class":15801},[28,46877,46878],{"class":15682}," ==",[28,46880,46881],{"class":15801}," 0",[28,46883,15778],{"class":15675},[28,46885,46886,46889,46891,46894,46896],{"class":15660,"line":16036},[28,46887,46888],{"class":15675},"        qc_evol ",[28,46890,845],{"class":15682},[28,46892,46893],{"class":15675}," qc_evol.",[28,46895,39507],{"class":16073},[28,46897,46898],{"class":15675},"(trotter_step_circ)\n",[28,46900,46901,46903],{"class":15660,"line":16041},[28,46902,19102],{"class":15671},[28,46904,15778],{"class":15675},[28,46906,46907,46909,46911,46913,46915],{"class":15660,"line":16371},[28,46908,46888],{"class":15675},[28,46910,845],{"class":15682},[28,46912,46893],{"class":15675},[28,46914,39507],{"class":16073},[28,46916,46917],{"class":15675},"(reverse_trotter_step_circ)\n",[28,46919,46920],{"class":15660,"line":16376},[28,46921,15769],{"emptyLinePlaceholder":15768},[28,46923,46924,46926,46928,46930,46932,46934,46936,46938,46940,46942,46944,46946,46948,46950,46952],{"class":15660,"line":16410},[28,46925,35085],{"class":15675},[28,46927,39800],{"class":16073},[28,46929,17194],{"class":15675},[28,46931,34841],{"class":16073},[28,46933,65],{"class":15675},[28,46935,34846],{"class":15841},[28,46937,16397],{"class":15675},[28,46939,39748],{"class":16307},[28,46941,25926],{"class":15682},[28,46943,3200],{"class":15801},[28,46945,16397],{"class":15675},[28,46947,34851],{"class":16307},[28,46949,845],{"class":15682},[28,46951,34856],{"class":15801},[28,46953,16052],{"class":15675},[17041,46955],{"alt":34861,"src":46956},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-03.avif",[28327,46958,46960],{"id":46959},"use-an-optimized-circuit-for-state-preparation","Use an optimized circuit for state preparation",[15650,46962,46964],{"className":15652,"code":46963,"language":15654,"meta":15655,"style":15655},"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",[8103,46965,46966,46975,46996,47013,47023],{"__ignoreMap":15655},[28,46967,46968,46971,46973],{"class":15660,"line":15661},[28,46969,46970],{"class":15675},"control ",[28,46972,845],{"class":15682},[28,46974,28085],{"class":15801},[28,46976,46977,46980,46982,46984,46986,46988,46990,46992,46994],{"class":15660,"line":15668},[28,46978,46979],{"class":15675},"excitation ",[28,46981,845],{"class":15682},[28,46983,23033],{"class":15682},[28,46985,27958],{"class":15675},[28,46987,1129],{"class":15682},[28,46989,22129],{"class":15801},[28,46991,22233],{"class":15675},[28,46993,2606],{"class":15682},[28,46995,25487],{"class":15801},[28,46997,46998,47001,47003,47005,47007,47009,47011],{"class":15660,"line":15679},[28,46999,47000],{"class":15675},"controlled_state_prep ",[28,47002,845],{"class":15682},[28,47004,26583],{"class":16073},[28,47006,27958],{"class":15675},[28,47008,2606],{"class":15682},[28,47010,22230],{"class":15801},[28,47012,16052],{"class":15675},[28,47014,47015,47018,47020],{"class":15660,"line":15725},[28,47016,47017],{"class":15675},"controlled_state_prep.",[28,47019,26614],{"class":16073},[28,47021,47022],{"class":15675},"(control, excitation)\n",[28,47024,47025,47027,47029,47031,47033,47035,47037,47039,47041,47043,47045,47047,47049],{"class":15660,"line":15739},[28,47026,47017],{"class":15675},[28,47028,34841],{"class":16073},[28,47030,65],{"class":15675},[28,47032,34846],{"class":15841},[28,47034,16397],{"class":15675},[28,47036,39748],{"class":16307},[28,47038,25926],{"class":15682},[28,47040,3200],{"class":15801},[28,47042,16397],{"class":15675},[28,47044,34851],{"class":16307},[28,47046,845],{"class":15682},[28,47048,34856],{"class":15801},[28,47050,16052],{"class":15675},[17041,47052],{"alt":34861,"src":47053},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-04.avif",[28327,47055,47057,47058,512,47118,47178],{"id":47056},"template-circuits-for-calculating-matrix-elements-of-stildess-and-htildehh-via-hadamard-test","Template circuits for calculating matrix elements of ",[28,47059,47061,47078],{"className":47060},[35],[28,47062,47064],{"className":47063},[39],[41,47065,47066],{"xmlns":43},[46,47067,47068,47076],{},[49,47069,47070],{},[10155,47071,47072,47074],{"accent":71},[55,47073,11356],{},[62,47075,30008],{},[82,47077,31579],{"encoding":84},[28,47079,47081],{"className":47080,"ariaHidden":71},[89],[28,47082,47084,47087],{"className":47083},[93],[28,47085],{"className":47086,"style":30021},[97],[28,47088,47090],{"className":47089},[102,10255],[28,47091,47093],{"className":47092},[115],[28,47094,47096],{"className":47095},[120],[28,47097,47099,47107],{"className":47098,"style":30021},[124],[28,47100,47101,47104],{"style":10025},[28,47102],{"className":47103,"style":3434},[132],[28,47105,11356],{"className":47106,"style":1913},[102,106],[28,47108,47109,47112],{"style":30044},[28,47110],{"className":47111,"style":3434},[132],[28,47113,47115],{"className":47114,"style":31617},[10282],[28,47116,30008],{"className":47117},[102],[28,47119,47121,47138],{"className":47120},[35],[28,47122,47124],{"className":47123},[39],[41,47125,47126],{"xmlns":43},[46,47127,47128,47136],{},[49,47129,47130],{},[10155,47131,47132,47134],{"accent":71},[55,47133,2263],{},[62,47135,30008],{},[82,47137,30011],{"encoding":84},[28,47139,47141],{"className":47140,"ariaHidden":71},[89],[28,47142,47144,47147],{"className":47143},[93],[28,47145],{"className":47146,"style":30021},[97],[28,47148,47150],{"className":47149},[102,10255],[28,47151,47153],{"className":47152},[115],[28,47154,47156],{"className":47155},[120],[28,47157,47159,47167],{"className":47158,"style":30021},[124],[28,47160,47161,47164],{"style":10025},[28,47162],{"className":47163,"style":3434},[132],[28,47165,2263],{"className":47166,"style":4040},[102,106],[28,47168,47169,47172],{"style":30044},[28,47170],{"className":47171,"style":3434},[132],[28,47173,47175],{"className":47174,"style":30051},[10282],[28,47176,30008],{"className":47177},[102]," via Hadamard test",[12,47180,47181],{},"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.",[15650,47183,47185],{"className":15652,"code":47184,"language":15654,"meta":15655,"style":15655},"# Parameters for the template circuits\nparameters = []\nfor idx in range(1, krylov_dim):\n    parameters.append(2 * dt_circ * (idx))\n",[8103,47186,47187,47192,47201,47219],{"__ignoreMap":15655},[28,47188,47189],{"class":15660,"line":15661},[28,47190,47191],{"class":15664},"# Parameters for the template circuits\n",[28,47193,47194,47197,47199],{"class":15660,"line":15668},[28,47195,47196],{"class":15675},"parameters ",[28,47198,845],{"class":15682},[28,47200,24066],{"class":15675},[28,47202,47203,47205,47208,47210,47212,47214,47216],{"class":15660,"line":15679},[28,47204,22277],{"class":15671},[28,47206,47207],{"class":15675}," idx ",[28,47209,17720],{"class":15671},[28,47211,22285],{"class":15682},[28,47213,65],{"class":15675},[28,47215,3200],{"class":15801},[28,47217,47218],{"class":15675},", krylov_dim):\n",[28,47220,47221,47224,47226,47228,47230,47232,47235,47237],{"class":15660,"line":15725},[28,47222,47223],{"class":15675},"    parameters.",[28,47225,27807],{"class":16073},[28,47227,65],{"class":15675},[28,47229,848],{"class":15801},[28,47231,16210],{"class":15682},[28,47233,47234],{"class":15675}," dt_circ ",[28,47236,16226],{"class":15682},[28,47238,47239],{"class":15675}," (idx))\n",[15650,47241,47243],{"className":15652,"code":47242,"language":15654,"meta":15655,"style":15655},"# 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",[8103,47244,47245,47250,47266,47276,47288,47320,47328,47363,47371,47383,47391,47421,47425,47437,47441],{"__ignoreMap":15655},[28,47246,47247],{"class":15660,"line":15661},[28,47248,47249],{"class":15664},"# Create modified hadamard test circuit\n",[28,47251,47252,47254,47256,47258,47260,47262,47264],{"class":15660,"line":15668},[28,47253,35061],{"class":15675},[28,47255,845],{"class":15682},[28,47257,35066],{"class":16073},[28,47259,27958],{"class":15675},[28,47261,2606],{"class":15682},[28,47263,22230],{"class":15801},[28,47265,16052],{"class":15675},[28,47267,47268,47270,47272,47274],{"class":15660,"line":15679},[28,47269,26578],{"class":15675},[28,47271,845],{"class":15682},[28,47273,26583],{"class":16073},[28,47275,35080],{"class":15675},[28,47277,47278,47280,47282,47284,47286],{"class":15660,"line":15725},[28,47279,26594],{"class":15675},[28,47281,26597],{"class":16073},[28,47283,65],{"class":15675},[28,47285,652],{"class":15801},[28,47287,16052],{"class":15675},[28,47289,47290,47292,47294,47297,47299,47301,47303,47305,47307,47309,47312,47314,47316,47318],{"class":15660,"line":15739},[28,47291,26594],{"class":15675},[28,47293,39507],{"class":16073},[28,47295,47296],{"class":15675},"(controlled_state_prep, ",[28,47298,39690],{"class":15682},[28,47300,65],{"class":15675},[28,47302,28048],{"class":15682},[28,47304,27958],{"class":15675},[28,47306,2606],{"class":15682},[28,47308,22230],{"class":15801},[28,47310,47311],{"class":15675},")), ",[28,47313,39513],{"class":16307},[28,47315,845],{"class":15682},[28,47317,15597],{"class":15801},[28,47319,16052],{"class":15675},[28,47321,47322,47324,47326],{"class":15660,"line":15752},[28,47323,26594],{"class":15675},[28,47325,46821],{"class":16073},[28,47327,17029],{"class":15675},[28,47329,47330,47332,47334,47337,47339,47341,47343,47345,47347,47349,47351,47353,47355,47357,47359,47361],{"class":15660,"line":15765},[28,47331,26594],{"class":15675},[28,47333,39507],{"class":16073},[28,47335,47336],{"class":15675},"(qc_evol, ",[28,47338,39690],{"class":15682},[28,47340,65],{"class":15675},[28,47342,28048],{"class":15682},[28,47344,65],{"class":15675},[28,47346,3200],{"class":15801},[28,47348,46681],{"class":15675},[28,47350,2606],{"class":15682},[28,47352,22230],{"class":15801},[28,47354,47311],{"class":15675},[28,47356,39513],{"class":16307},[28,47358,845],{"class":15682},[28,47360,15597],{"class":15801},[28,47362,16052],{"class":15675},[28,47364,47365,47367,47369],{"class":15660,"line":15772},[28,47366,26594],{"class":15675},[28,47368,46821],{"class":16073},[28,47370,17029],{"class":15675},[28,47372,47373,47375,47377,47379,47381],{"class":15660,"line":15781},[28,47374,26594],{"class":15675},[28,47376,3169],{"class":16073},[28,47378,65],{"class":15675},[28,47380,652],{"class":15801},[28,47382,16052],{"class":15675},[28,47384,47385,47387,47389],{"class":15660,"line":15798},[28,47386,26594],{"class":15675},[28,47388,39507],{"class":16073},[28,47390,16077],{"class":15675},[28,47392,47393,47396,47398,47400,47402,47404,47406,47408,47410,47412,47414,47416,47418],{"class":15660,"line":15811},[28,47394,47395],{"class":15675},"    controlled_state_prep.",[28,47397,39465],{"class":16073},[28,47399,24379],{"class":15675},[28,47401,39690],{"class":15682},[28,47403,65],{"class":15675},[28,47405,28048],{"class":15682},[28,47407,27958],{"class":15675},[28,47409,2606],{"class":15682},[28,47411,22230],{"class":15801},[28,47413,47311],{"class":15675},[28,47415,39513],{"class":16307},[28,47417,845],{"class":15682},[28,47419,47420],{"class":15801},"True\n",[28,47422,47423],{"class":15660,"line":15820},[28,47424,16052],{"class":15675},[28,47426,47427,47429,47431,47433,47435],{"class":15660,"line":15830},[28,47428,26594],{"class":15675},[28,47430,3169],{"class":16073},[28,47432,65],{"class":15675},[28,47434,652],{"class":15801},[28,47436,16052],{"class":15675},[28,47438,47439],{"class":15660,"line":15835},[28,47440,15769],{"emptyLinePlaceholder":15768},[28,47442,47443,47445,47447,47449,47451,47453,47455,47457,47459,47461,47463],{"class":15660,"line":15853},[28,47444,26594],{"class":15675},[28,47446,39800],{"class":16073},[28,47448,17194],{"class":15675},[28,47450,34841],{"class":16073},[28,47452,65],{"class":15675},[28,47454,34846],{"class":15841},[28,47456,16397],{"class":15675},[28,47458,39748],{"class":16307},[28,47460,25926],{"class":15682},[28,47462,3200],{"class":15801},[28,47464,16052],{"class":15675},[17041,47466],{"alt":34861,"src":47467},"\u002F_content\u002Fimages\u002Fkrylov-quantum-diagonalization\u002Foutput-05.avif",[15650,47469,47471],{"className":15652,"code":47470,"language":15654,"meta":15655,"style":15655},"print(\n    \"The optimized circuit has 2Q gates depth: \",\n    qc.decompose().decompose().depth(lambda x: x[0].num_qubits == 2),\n)\n",[8103,47472,47473,47479,47486,47519],{"__ignoreMap":15655},[28,47474,47475,47477],{"class":15660,"line":15661},[28,47476,16044],{"class":15682},[28,47478,16077],{"class":15675},[28,47480,47481,47484],{"class":15660,"line":15668},[28,47482,47483],{"class":15841},"    \"The optimized circuit has 2Q gates depth: \"",[28,47485,16105],{"class":15675},[28,47487,47488,47491,47493,47495,47497,47499,47501,47503,47505,47507,47509,47511,47513,47515,47517],{"class":15660,"line":15679},[28,47489,47490],{"class":15675},"    qc.",[28,47492,39800],{"class":16073},[28,47494,17194],{"class":15675},[28,47496,39800],{"class":16073},[28,47498,17194],{"class":15675},[28,47500,39814],{"class":16073},[28,47502,65],{"class":15675},[28,47504,39819],{"class":15671},[28,47506,39822],{"class":16082},[28,47508,39825],{"class":15675},[28,47510,652],{"class":15801},[28,47512,39830],{"class":15675},[28,47514,23650],{"class":15682},[28,47516,22129],{"class":15801},[28,47518,39837],{"class":15675},[28,47520,47521],{"class":15660,"line":15725},[28,47522,16052],{"class":15675},[15650,47524,47527],{"className":47525,"code":47526,"language":16058,"meta":15655},[16056],"The optimized circuit has 2Q gates depth:  74\n",[8103,47528,47526],{"__ignoreMap":15655},[12,47530,47531],{},"We have considerably reduced the depth of the Hadamard test with a combination of Trotter approximation and uncontrolled unitaries",[251,47533,47535],{"id":47534},"step-3-execute-using-qiskit-primitives","Step 3: Execute using Qiskit primitives",[12,47537,47538],{},"Instantiate the backend and set runtime parameters",[15650,47540,47542],{"className":15652,"code":47541,"language":15654,"meta":15655,"style":15655},"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",[8103,47543,47544,47556,47589,47595,47608,47616,47636],{"__ignoreMap":15655},[28,47545,47546,47549,47551,47554],{"class":15660,"line":15661},[28,47547,47548],{"class":15675},"service ",[28,47550,845],{"class":15682},[28,47552,47553],{"class":16073}," QiskitRuntimeService",[28,47555,17029],{"class":15675},[28,47557,47558,47561,47563,47566,47569,47571,47574,47576,47578,47580,47583,47585,47587],{"class":15660,"line":15668},[28,47559,47560],{"class":15675},"backend ",[28,47562,845],{"class":15682},[28,47564,47565],{"class":15675}," service.",[28,47567,47568],{"class":16073},"least_busy",[28,47570,65],{"class":15675},[28,47572,47573],{"class":16307},"operational",[28,47575,845],{"class":15682},[28,47577,15597],{"class":15801},[28,47579,16397],{"class":15675},[28,47581,47582],{"class":16307},"simulator",[28,47584,845],{"class":15682},[28,47586,19707],{"class":15801},[28,47588,16052],{"class":15675},[28,47590,47591,47593],{"class":15660,"line":15679},[28,47592,20133],{"class":15671},[28,47594,18567],{"class":15675},[28,47596,47597,47600,47602,47605],{"class":15660,"line":15725},[28,47598,47599],{"class":15841},"    \"if_else\"",[28,47601,16493],{"class":15671},[28,47603,47604],{"class":15671}," in",[28,47606,47607],{"class":15675}," backend.target.operation_names\n",[28,47609,47610,47613],{"class":15660,"line":15739},[28,47611,47612],{"class":15675},"):  ",[28,47614,47615],{"class":15664},"# Needed as \"op_name\" could be \"if_else\"\n",[28,47617,47618,47621,47624,47627,47629,47631,47634],{"class":15660,"line":15752},[28,47619,47620],{"class":15675},"    backend.target.",[28,47622,47623],{"class":16073},"add_instruction",[28,47625,47626],{"class":15675},"(IfElseOp, ",[28,47628,22899],{"class":16307},[28,47630,845],{"class":15682},[28,47632,47633],{"class":15841},"\"if_else\"",[28,47635,16052],{"class":15675},[28,47637,47638,47640],{"class":15660,"line":15765},[28,47639,16044],{"class":15682},[28,47641,47642],{"class":15675},"(backend.name)\n",[28327,47644,47646],{"id":47645},"transpiling-to-a-qpu","Transpiling to a QPU",[12,47648,47649],{},"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",[15650,47651,47653],{"className":15652,"code":47652,"language":15654,"meta":15655,"style":15655},"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",[8103,47654,47655,47665,47689,47707,47711,47727,47741,47757,47772,47797,47809,47820,47827,47838,47845,47850,47854,47864,47886,47911,47925,47935,47945,47951,47960,47966,47970,47974,47978,47991,48006,48022,48032],{"__ignoreMap":15655},[28,47656,47657,47660,47662],{"class":15660,"line":15661},[28,47658,47659],{"class":15675},"target ",[28,47661,845],{"class":15682},[28,47663,47664],{"class":15675}," backend.target\n",[28,47666,47667,47670,47672,47675,47678,47680,47683,47685,47687],{"class":15660,"line":15668},[28,47668,47669],{"class":15675},"cmap ",[28,47671,845],{"class":15682},[28,47673,47674],{"class":15675}," target.",[28,47676,47677],{"class":16073},"build_coupling_map",[28,47679,65],{"class":15675},[28,47681,47682],{"class":16307},"filter_idle_qubits",[28,47684,845],{"class":15682},[28,47686,15597],{"class":15801},[28,47688,16052],{"class":15675},[28,47690,47691,47694,47696,47699,47702,47705],{"class":15660,"line":15679},[28,47692,47693],{"class":15675},"cmap_list ",[28,47695,845],{"class":15682},[28,47697,47698],{"class":15682}," list",[28,47700,47701],{"class":15675},"(cmap.",[28,47703,47704],{"class":16073},"get_edges",[28,47706,22940],{"class":15675},[28,47708,47709],{"class":15660,"line":15725},[28,47710,15769],{"emptyLinePlaceholder":15768},[28,47712,47713,47716,47718,47721,47724],{"class":15660,"line":15739},[28,47714,47715],{"class":15675},"cust_cmap_list ",[28,47717,845],{"class":15682},[28,47719,47720],{"class":15675}," copy.",[28,47722,47723],{"class":16073},"deepcopy",[28,47725,47726],{"class":15675},"(cmap_list)\n",[28,47728,47729,47731,47734,47736,47738],{"class":15660,"line":15752},[28,47730,22277],{"class":15671},[28,47732,47733],{"class":15675}," q ",[28,47735,17720],{"class":15671},[28,47737,22285],{"class":15682},[28,47739,47740],{"class":15675},"(target.num_qubits):\n",[28,47742,47743,47746,47748,47751,47754],{"class":15660,"line":15765},[28,47744,47745],{"class":15675},"    meas_err ",[28,47747,845],{"class":15682},[28,47749,47750],{"class":15675}," target[",[28,47752,47753],{"class":15841},"\"measure\"",[28,47755,47756],{"class":15675},"][(q,)].error\n",[28,47758,47759,47762,47764,47767,47769],{"class":15660,"line":15772},[28,47760,47761],{"class":15675},"    t2 ",[28,47763,845],{"class":15682},[28,47765,47766],{"class":15675}," target.qubit_properties[q].t2 ",[28,47768,16226],{"class":15682},[28,47770,47771],{"class":15801}," 1e6\n",[28,47773,47774,47776,47779,47781,47784,47787,47790,47792,47795],{"class":15660,"line":15781},[28,47775,16484],{"class":15671},[28,47777,47778],{"class":15675}," meas_err ",[28,47780,16503],{"class":15682},[28,47782,47783],{"class":15801}," 0.02",[28,47785,47786],{"class":15671}," or",[28,47788,47789],{"class":15675}," t2 ",[28,47791,22950],{"class":15682},[28,47793,47794],{"class":15801}," 100",[28,47796,15778],{"class":15675},[28,47798,47799,47801,47804,47806],{"class":15660,"line":15798},[28,47800,17714],{"class":15671},[28,47802,47803],{"class":15675}," q_pair ",[28,47805,17720],{"class":15671},[28,47807,47808],{"class":15675}," cmap_list:\n",[28,47810,47811,47813,47815,47817],{"class":15660,"line":15811},[28,47812,28131],{"class":15671},[28,47814,47733],{"class":15675},[28,47816,17720],{"class":15671},[28,47818,47819],{"class":15675}," q_pair:\n",[28,47821,47822,47825],{"class":15660,"line":15820},[28,47823,47824],{"class":15671},"                try",[28,47826,15778],{"class":15675},[28,47828,47829,47832,47835],{"class":15660,"line":15830},[28,47830,47831],{"class":15675},"                    cust_cmap_list.",[28,47833,47834],{"class":16073},"remove",[28,47836,47837],{"class":15675},"(q_pair)\n",[28,47839,47840,47843],{"class":15660,"line":15835},[28,47841,47842],{"class":15671},"                except",[28,47844,15778],{"class":15675},[28,47846,47847],{"class":15660,"line":15853},[28,47848,47849],{"class":15671},"                    continue\n",[28,47851,47852],{"class":15660,"line":15867},[28,47853,15769],{"emptyLinePlaceholder":15768},[28,47855,47856,47858,47860,47862],{"class":15660,"line":15872},[28,47857,22277],{"class":15671},[28,47859,47733],{"class":15675},[28,47861,17720],{"class":15671},[28,47863,47808],{"class":15675},[28,47865,47866,47869,47871,47873,47876,47879,47882,47884],{"class":15660,"line":15878},[28,47867,47868],{"class":15675},"    op_name ",[28,47870,845],{"class":15682},[28,47872,47698],{"class":15682},[28,47874,47875],{"class":15675},"(target.",[28,47877,47878],{"class":16073},"operation_names_for_qargs",[28,47880,47881],{"class":15675},"(q))[",[28,47883,652],{"class":15801},[28,47885,16849],{"class":15675},[28,47887,47888,47891,47893,47895,47897,47899,47901,47904,47906,47908],{"class":15660,"line":15897},[28,47889,47890],{"class":15675},"    twoq_gate_err ",[28,47892,845],{"class":15682},[28,47894,47750],{"class":15675},[28,47896,68],{"class":15671},[28,47898,21798],{"class":15841},[28,47900,11385],{"class":15801},[28,47902,47903],{"class":15675},"op_name",[28,47905,11404],{"class":15801},[28,47907,21798],{"class":15841},[28,47909,47910],{"class":15675},"][q].error\n",[28,47912,47913,47915,47918,47920,47923],{"class":15660,"line":15914},[28,47914,16484],{"class":15671},[28,47916,47917],{"class":15675}," twoq_gate_err ",[28,47919,16503],{"class":15682},[28,47921,47922],{"class":15801}," 0.005",[28,47924,15778],{"class":15675},[28,47926,47927,47929,47931,47933],{"class":15660,"line":15931},[28,47928,17714],{"class":15671},[28,47930,47803],{"class":15675},[28,47932,17720],{"class":15671},[28,47934,47808],{"class":15675},[28,47936,47937,47939,47941,47943],{"class":15660,"line":15948},[28,47938,28131],{"class":15671},[28,47940,47733],{"class":15675},[28,47942,23650],{"class":15682},[28,47944,47819],{"class":15675},[28,47946,47947,47949],{"class":15660,"line":15966},[28,47948,47824],{"class":15671},[28,47950,15778],{"class":15675},[28,47952,47953,47955,47957],{"class":15660,"line":15983},[28,47954,47831],{"class":15675},[28,47956,47834],{"class":16073},[28,47958,47959],{"class":15675},"(q)\n",[28,47961,47962,47964],{"class":15660,"line":16000},[28,47963,47842],{"class":15671},[28,47965,15778],{"class":15675},[28,47967,47968],{"class":15660,"line":16016},[28,47969,47849],{"class":15671},[28,47971,47972],{"class":15660,"line":16036},[28,47973,15769],{"emptyLinePlaceholder":15768},[28,47975,47976],{"class":15660,"line":16041},[28,47977,15769],{"emptyLinePlaceholder":15768},[28,47979,47980,47983,47985,47988],{"class":15660,"line":16371},[28,47981,47982],{"class":15675},"cust_cmap ",[28,47984,845],{"class":15682},[28,47986,47987],{"class":16073}," CouplingMap",[28,47989,47990],{"class":15675},"(cust_cmap_list)\n",[28,47992,47993,47996,47998,48001,48004],{"class":15660,"line":16376},[28,47994,47995],{"class":15675},"cust_target ",[28,47997,845],{"class":15682},[28,47999,48000],{"class":15675}," Target.",[28,48002,48003],{"class":16073},"from_configuration",[28,48005,16077],{"class":15675},[28,48007,48008,48011,48013,48016,48019],{"class":15660,"line":16410},[28,48009,48010],{"class":16307},"    basis_gates",[28,48012,845],{"class":15682},[28,48014,48015],{"class":15675},"backend.",[28,48017,48018],{"class":16073},"configuration",[28,48020,48021],{"class":15675},"().basis_gates,\n",[28,48023,48024,48027,48029],{"class":15660,"line":16431},[28,48025,48026],{"class":16307},"    coupling_map",[28,48028,845],{"class":15682},[28,48030,48031],{"class":15675},"cust_cmap,\n",[28,48033,48034],{"class":15660,"line":16452},[28,48035,16052],{"class":15675},[12,48037,48038],{},"Then transpile the virtual circuit to the best physical layout in this new target",[15650,48040,48042],{"className":15652,"code":48041,"language":15654,"meta":15655,"style":15655},"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",[8103,48043,48044,48056,48068,48079,48089,48098,48102,48106,48120,48124,48156,48172,48178,48185,48192,48196,48201,48223,48236,48240,48245],{"__ignoreMap":15655},[28,48045,48046,48049,48051,48053],{"class":15660,"line":15661},[28,48047,48048],{"class":15675},"basis_gates ",[28,48050,845],{"class":15682},[28,48052,47698],{"class":15682},[28,48054,48055],{"class":15675},"(target.operation_names)\n",[28,48057,48058,48061,48063,48066],{"class":15660,"line":15668},[28,48059,48060],{"class":15675},"pm ",[28,48062,845],{"class":15682},[28,48064,48065],{"class":16073}," generate_preset_pass_manager",[28,48067,16077],{"class":15675},[28,48069,48070,48073,48075,48077],{"class":15660,"line":15679},[28,48071,48072],{"class":16307},"    optimization_level",[28,48074,845],{"class":15682},[28,48076,33915],{"class":15801},[28,48078,16105],{"class":15675},[28,48080,48081,48084,48086],{"class":15660,"line":15725},[28,48082,48083],{"class":16307},"    target",[28,48085,845],{"class":15682},[28,48087,48088],{"class":15675},"cust_target,\n",[28,48090,48091,48093,48095],{"class":15660,"line":15739},[28,48092,48010],{"class":16307},[28,48094,845],{"class":15682},[28,48096,48097],{"class":15675},"basis_gates,\n",[28,48099,48100],{"class":15660,"line":15752},[28,48101,16052],{"class":15675},[28,48103,48104],{"class":15660,"line":15765},[28,48105,15769],{"emptyLinePlaceholder":15768},[28,48107,48108,48111,48113,48116,48118],{"class":15660,"line":15772},[28,48109,48110],{"class":15675},"qc_trans ",[28,48112,845],{"class":15682},[28,48114,48115],{"class":15675}," pm.",[28,48117,17789],{"class":16073},[28,48119,26684],{"class":15675},[28,48121,48122],{"class":15660,"line":15781},[28,48123,15769],{"emptyLinePlaceholder":15768},[28,48125,48126,48128,48130,48133,48136,48138,48140,48142,48144,48146,48148,48150,48152,48154],{"class":15660,"line":15798},[28,48127,16044],{"class":15682},[28,48129,65],{"class":15675},[28,48131,48132],{"class":15841},"\"depth\"",[28,48134,48135],{"class":15675},", qc_trans.",[28,48137,39814],{"class":16073},[28,48139,65],{"class":15675},[28,48141,39819],{"class":15671},[28,48143,39822],{"class":16082},[28,48145,39825],{"class":15675},[28,48147,652],{"class":15801},[28,48149,39830],{"class":15675},[28,48151,23650],{"class":15682},[28,48153,22129],{"class":15801},[28,48155,16539],{"class":15675},[28,48157,48158,48160,48162,48165,48167,48170],{"class":15660,"line":15811},[28,48159,16044],{"class":15682},[28,48161,65],{"class":15675},[28,48163,48164],{"class":15841},"\"num 2q ops\"",[28,48166,48135],{"class":15675},[28,48168,48169],{"class":16073},"count_ops",[28,48171,22940],{"class":15675},[28,48173,48174,48176],{"class":15660,"line":15820},[28,48175,16044],{"class":15682},[28,48177,16077],{"class":15675},[28,48179,48180,48183],{"class":15660,"line":15830},[28,48181,48182],{"class":15841},"    \"physical qubits\"",[28,48184,16105],{"class":15675},[28,48186,48187,48190],{"class":15660,"line":15835},[28,48188,48189],{"class":15682},"    sorted",[28,48191,16077],{"class":15675},[28,48193,48194],{"class":15660,"line":15853},[28,48195,18804],{"class":15675},[28,48197,48198],{"class":15660,"line":15867},[28,48199,48200],{"class":15675},"            idx\n",[28,48202,48203,48206,48209,48211,48214,48217,48219,48221],{"class":15660,"line":15872},[28,48204,48205],{"class":15671},"            for",[28,48207,48208],{"class":15675}," idx, qb ",[28,48210,17720],{"class":15671},[28,48212,48213],{"class":15675}," qc_trans.layout.initial_layout.",[28,48215,48216],{"class":16073},"get_physical_bits",[28,48218,17194],{"class":15675},[28,48220,17726],{"class":16073},[28,48222,17029],{"class":15675},[28,48224,48225,48227,48230,48233],{"class":15660,"line":15878},[28,48226,28131],{"class":15671},[28,48228,48229],{"class":15675}," qb._register.name ",[28,48231,48232],{"class":15682},"!=",[28,48234,48235],{"class":15841}," \"ancilla\"\n",[28,48237,48238],{"class":15660,"line":15897},[28,48239,18828],{"class":15675},[28,48241,48242],{"class":15660,"line":15914},[28,48243,48244],{"class":15675},"    ),\n",[28,48246,48247],{"class":15660,"line":15931},[28,48248,16052],{"class":15675},[15650,48250,48253],{"className":48251,"code":48252,"language":16058,"meta":15655},[16056],"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",[8103,48254,48252],{"__ignoreMap":15655},[28327,48256,48258],{"id":48257},"create-pubs-for-execution-with-estimator","Create PUBs for execution with Estimator",[15650,48260,48262],{"className":15652,"code":48261,"language":15654,"meta":15655,"style":15655},"# 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",[8103,48263,48264,48269,48288,48305,48309,48321,48326,48333,48345,48349,48362,48376,48381,48388,48402,48410,48420,48427,48440,48444,48454,48458,48462,48467,48472,48481,48495,48500,48526,48549,48559,48568,48572,48581,48585,48594,48606,48618,48631,48646,48650,48660,48664,48668,48673,48687,48691,48695,48700,48705,48710],{"__ignoreMap":15655},[28,48265,48266],{"class":15660,"line":15661},[28,48267,48268],{"class":15664},"# Define observables to measure for S\n",[28,48270,48271,48274,48276,48279,48281,48284,48286],{"class":15660,"line":15668},[28,48272,48273],{"class":15675},"observable_S_real ",[28,48275,845],{"class":15682},[28,48277,48278],{"class":15841}," \"I\"",[28,48280,16210],{"class":15682},[28,48282,48283],{"class":15675}," (n_qubits) ",[28,48285,2606],{"class":15682},[28,48287,34007],{"class":15841},[28,48289,48290,48293,48295,48297,48299,48301,48303],{"class":15660,"line":15679},[28,48291,48292],{"class":15675},"observable_S_imag ",[28,48294,845],{"class":15682},[28,48296,48278],{"class":15841},[28,48298,16210],{"class":15682},[28,48300,48283],{"class":15675},[28,48302,2606],{"class":15682},[28,48304,34074],{"class":15841},[28,48306,48307],{"class":15660,"line":15725},[28,48308,15769],{"emptyLinePlaceholder":15768},[28,48310,48311,48314,48316,48319],{"class":15660,"line":15739},[28,48312,48313],{"class":15675},"observable_op_real ",[28,48315,845],{"class":15682},[28,48317,48318],{"class":16073}," SparsePauliOp",[28,48320,16077],{"class":15675},[28,48322,48323],{"class":15660,"line":15752},[28,48324,48325],{"class":15675},"    observable_S_real\n",[28,48327,48328,48330],{"class":15660,"line":15765},[28,48329,23686],{"class":15675},[28,48331,48332],{"class":15664},"# define a sparse pauli operator for the observable\n",[28,48334,48335,48338,48340,48342],{"class":15660,"line":15772},[28,48336,48337],{"class":15675},"observable_op_imag ",[28,48339,845],{"class":15682},[28,48341,48318],{"class":16073},[28,48343,48344],{"class":15675},"(observable_S_imag)\n",[28,48346,48347],{"class":15660,"line":15781},[28,48348,15769],{"emptyLinePlaceholder":15768},[28,48350,48351,48354,48356,48359],{"class":15660,"line":15798},[28,48352,48353],{"class":15675},"layout ",[28,48355,845],{"class":15682},[28,48357,48358],{"class":15675}," qc_trans.layout  ",[28,48360,48361],{"class":15664},"# get layout of transpiled circuit\n",[28,48363,48364,48366,48368,48371,48374],{"class":15660,"line":15811},[28,48365,48313],{"class":15675},[28,48367,845],{"class":15682},[28,48369,48370],{"class":15675}," observable_op_real.",[28,48372,48373],{"class":16073},"apply_layout",[28,48375,16077],{"class":15675},[28,48377,48378],{"class":15660,"line":15820},[28,48379,48380],{"class":15675},"    layout\n",[28,48382,48383,48385],{"class":15660,"line":15830},[28,48384,23686],{"class":15675},[28,48386,48387],{"class":15664},"# apply physical layout to the observable\n",[28,48389,48390,48392,48394,48397,48399],{"class":15660,"line":15835},[28,48391,48337],{"class":15675},[28,48393,845],{"class":15682},[28,48395,48396],{"class":15675}," observable_op_imag.",[28,48398,48373],{"class":16073},[28,48400,48401],{"class":15675},"(layout)\n",[28,48403,48404,48406,48408],{"class":15660,"line":15853},[28,48405,48273],{"class":15675},[28,48407,845],{"class":15682},[28,48409,18567],{"class":15675},[28,48411,48412,48415,48418],{"class":15660,"line":15867},[28,48413,48414],{"class":15675},"    observable_op_real.paulis.",[28,48416,48417],{"class":16073},"to_labels",[28,48419,17029],{"class":15675},[28,48421,48422,48424],{"class":15660,"line":15872},[28,48423,23686],{"class":15675},[28,48425,48426],{"class":15664},"# get the label of the physical observable\n",[28,48428,48429,48431,48433,48436,48438],{"class":15660,"line":15878},[28,48430,48292],{"class":15675},[28,48432,845],{"class":15682},[28,48434,48435],{"class":15675}," observable_op_imag.paulis.",[28,48437,48417],{"class":16073},[28,48439,17029],{"class":15675},[28,48441,48442],{"class":15660,"line":15897},[28,48443,15769],{"emptyLinePlaceholder":15768},[28,48445,48446,48449,48451],{"class":15660,"line":15914},[28,48447,48448],{"class":15675},"observables_S ",[28,48450,845],{"class":15682},[28,48452,48453],{"class":15675}," [[observable_S_real], [observable_S_imag]]\n",[28,48455,48456],{"class":15660,"line":15931},[28,48457,15769],{"emptyLinePlaceholder":15768},[28,48459,48460],{"class":15660,"line":15948},[28,48461,15769],{"emptyLinePlaceholder":15768},[28,48463,48464],{"class":15660,"line":15966},[28,48465,48466],{"class":15664},"# Define observables to measure for H\n",[28,48468,48469],{"class":15660,"line":15983},[28,48470,48471],{"class":15664},"# Hamiltonian terms to measure\n",[28,48473,48474,48477,48479],{"class":15660,"line":16000},[28,48475,48476],{"class":15675},"observable_list ",[28,48478,845],{"class":15682},[28,48480,24066],{"class":15675},[28,48482,48483,48485,48488,48490,48492],{"class":15660,"line":16016},[28,48484,22277],{"class":15671},[28,48486,48487],{"class":15675}," pauli, coeff ",[28,48489,17720],{"class":15671},[28,48491,27544],{"class":15682},[28,48493,48494],{"class":15675},"(H_op.paulis, H_op.coeffs):\n",[28,48496,48497],{"class":15660,"line":16036},[28,48498,48499],{"class":15664},"    # print(pauli)\n",[28,48501,48502,48505,48507,48510,48512,48514,48517,48520,48522,48524],{"class":15660,"line":16041},[28,48503,48504],{"class":15675},"    observable_H_real ",[28,48506,845],{"class":15682},[28,48508,48509],{"class":15675}," pauli[::",[28,48511,15689],{"class":15682},[28,48513,3200],{"class":15801},[28,48515,48516],{"class":15675},"].",[28,48518,48519],{"class":16073},"to_label",[28,48521,23647],{"class":15675},[28,48523,2606],{"class":15682},[28,48525,34007],{"class":15841},[28,48527,48528,48531,48533,48535,48537,48539,48541,48543,48545,48547],{"class":15660,"line":16371},[28,48529,48530],{"class":15675},"    observable_H_imag ",[28,48532,845],{"class":15682},[28,48534,48509],{"class":15675},[28,48536,15689],{"class":15682},[28,48538,3200],{"class":15801},[28,48540,48516],{"class":15675},[28,48542,48519],{"class":16073},[28,48544,23647],{"class":15675},[28,48546,2606],{"class":15682},[28,48548,34074],{"class":15841},[28,48550,48551,48554,48556],{"class":15660,"line":16376},[28,48552,48553],{"class":15675},"    observable_list.",[28,48555,27807],{"class":16073},[28,48557,48558],{"class":15675},"([observable_H_real])\n",[28,48560,48561,48563,48565],{"class":15660,"line":16410},[28,48562,48553],{"class":15675},[28,48564,27807],{"class":16073},[28,48566,48567],{"class":15675},"([observable_H_imag])\n",[28,48569,48570],{"class":15660,"line":16431},[28,48571,15769],{"emptyLinePlaceholder":15768},[28,48573,48574,48576,48578],{"class":15660,"line":16452},[28,48575,48353],{"class":15675},[28,48577,845],{"class":15682},[28,48579,48580],{"class":15675}," qc_trans.layout\n",[28,48582,48583],{"class":15660,"line":16476},[28,48584,15769],{"emptyLinePlaceholder":15768},[28,48586,48587,48590,48592],{"class":15660,"line":16481},[28,48588,48589],{"class":15675},"observable_trans_list ",[28,48591,845],{"class":15682},[28,48593,24066],{"class":15675},[28,48595,48596,48598,48601,48603],{"class":15660,"line":16510},[28,48597,22277],{"class":15671},[28,48599,48600],{"class":15675}," observable ",[28,48602,17720],{"class":15671},[28,48604,48605],{"class":15675}," observable_list:\n",[28,48607,48608,48611,48613,48615],{"class":15660,"line":16542},[28,48609,48610],{"class":15675},"    observable_op ",[28,48612,845],{"class":15682},[28,48614,48318],{"class":16073},[28,48616,48617],{"class":15675},"(observable)\n",[28,48619,48620,48622,48624,48627,48629],{"class":15660,"line":16557},[28,48621,48610],{"class":15675},[28,48623,845],{"class":15682},[28,48625,48626],{"class":15675}," observable_op.",[28,48628,48373],{"class":16073},[28,48630,48401],{"class":15675},[28,48632,48633,48636,48638,48641,48643],{"class":15660,"line":16562},[28,48634,48635],{"class":15675},"    observable_trans_list.",[28,48637,27807],{"class":16073},[28,48639,48640],{"class":15675},"([observable_op.paulis.",[28,48642,48417],{"class":16073},[28,48644,48645],{"class":15675},"()])\n",[28,48647,48648],{"class":15660,"line":17382},[28,48649,15769],{"emptyLinePlaceholder":15768},[28,48651,48652,48655,48657],{"class":15660,"line":17392},[28,48653,48654],{"class":15675},"observables_H ",[28,48656,845],{"class":15682},[28,48658,48659],{"class":15675}," observable_trans_list\n",[28,48661,48662],{"class":15660,"line":17402},[28,48663,15769],{"emptyLinePlaceholder":15768},[28,48665,48666],{"class":15660,"line":17407},[28,48667,15769],{"emptyLinePlaceholder":15768},[28,48669,48670],{"class":15660,"line":17418},[28,48671,48672],{"class":15664},"# Define a sweep over parameter values\n",[28,48674,48675,48678,48680,48682,48684],{"class":15660,"line":17429},[28,48676,48677],{"class":15675},"params ",[28,48679,845],{"class":15682},[28,48681,16298],{"class":15675},[28,48683,18796],{"class":16073},[28,48685,48686],{"class":15675},"(parameters).T\n",[28,48688,48689],{"class":15660,"line":17440},[28,48690,15769],{"emptyLinePlaceholder":15768},[28,48692,48693],{"class":15660,"line":17451},[28,48694,15769],{"emptyLinePlaceholder":15768},[28,48696,48697],{"class":15660,"line":17462},[28,48698,48699],{"class":15664},"# Estimate the expectation value for all combinations of\n",[28,48701,48702],{"class":15660,"line":17467},[28,48703,48704],{"class":15664},"# observables and parameter values, where the pub result will have\n",[28,48706,48707],{"class":15660,"line":17478},[28,48708,48709],{"class":15664},"# shape (# observables, # parameter values).\n",[28,48711,48712,48715,48717,48720,48722],{"class":15660,"line":17489},[28,48713,48714],{"class":15675},"pub ",[28,48716,845],{"class":15682},[28,48718,48719],{"class":15675}," (qc_trans, observables_S ",[28,48721,2606],{"class":15682},[28,48723,48724],{"class":15675}," observables_H, params)\n",[28327,48726,48728],{"id":48727},"run-circuits","Run circuits",[12,48730,48731,48732,48783],{},"Circuits for ",[28,48733,48735,48753],{"className":48734},[35],[28,48736,48738],{"className":48737},[39],[41,48739,48740],{"xmlns":43},[46,48741,48742,48750],{},[49,48743,48744,48746,48748],{},[55,48745,3971],{},[62,48747,845],{},[650,48749,652],{},[82,48751,48752],{"encoding":84},"t=0",[28,48754,48756,48774],{"className":48755,"ariaHidden":71},[89],[28,48757,48759,48762,48765,48768,48771],{"className":48758},[93],[28,48760],{"className":48761,"style":46248},[97],[28,48763,3971],{"className":48764},[102,106],[28,48766],{"className":48767,"style":913},[173],[28,48769,845],{"className":48770},[917],[28,48772],{"className":48773,"style":913},[173],[28,48775,48777,48780],{"className":48776},[93],[28,48778],{"className":48779,"style":2914},[97],[28,48781,652],{"className":48782},[102]," are classically calculable",[15650,48785,48787],{"className":15652,"code":48786,"language":15654,"meta":15655,"style":15655},"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",[8103,48788,48789,48810,48814,48818,48823,48841,48861,48865,48880,48899,48903,48907,48912,48933,48937,48946,48965,48970,48985,49009,49013,49028,49050,49054,49075,49079,49084,49099,49103,49107],{"__ignoreMap":15655},[28,48790,48791,48794,48796,48799,48802,48805,48807],{"class":15660,"line":15661},[28,48792,48793],{"class":15675},"qc_cliff ",[28,48795,845],{"class":15682},[28,48797,48798],{"class":15675}," qc.",[28,48800,48801],{"class":16073},"assign_parameters",[28,48803,48804],{"class":15675},"({t: ",[28,48806,652],{"class":15801},[28,48808,48809],{"class":15675},"})\n",[28,48811,48812],{"class":15660,"line":15668},[28,48813,15769],{"emptyLinePlaceholder":15768},[28,48815,48816],{"class":15660,"line":15679},[28,48817,15769],{"emptyLinePlaceholder":15768},[28,48819,48820],{"class":15660,"line":15725},[28,48821,48822],{"class":15664},"# Get expectation values from experiment\n",[28,48824,48825,48828,48830,48833,48836,48839],{"class":15660,"line":15739},[28,48826,48827],{"class":15675},"S_expval_real ",[28,48829,845],{"class":15682},[28,48831,48832],{"class":16073}," StabilizerState",[28,48834,48835],{"class":15675},"(qc_cliff).",[28,48837,48838],{"class":16073},"expectation_value",[28,48840,16077],{"class":15675},[28,48842,48843,48846,48848,48850,48852,48854,48856,48859],{"class":15660,"line":15752},[28,48844,48845],{"class":16073},"    Pauli",[28,48847,65],{"class":15675},[28,48849,33894],{"class":15841},[28,48851,16210],{"class":15682},[28,48853,48283],{"class":15675},[28,48855,2606],{"class":15682},[28,48857,48858],{"class":15841}," \"X\"",[28,48860,16052],{"class":15675},[28,48862,48863],{"class":15660,"line":15765},[28,48864,16052],{"class":15675},[28,48866,48867,48870,48872,48874,48876,48878],{"class":15660,"line":15772},[28,48868,48869],{"class":15675},"S_expval_imag ",[28,48871,845],{"class":15682},[28,48873,48832],{"class":16073},[28,48875,48835],{"class":15675},[28,48877,48838],{"class":16073},[28,48879,16077],{"class":15675},[28,48881,48882,48884,48886,48888,48890,48892,48894,48897],{"class":15660,"line":15781},[28,48883,48845],{"class":16073},[28,48885,65],{"class":15675},[28,48887,33894],{"class":15841},[28,48889,16210],{"class":15682},[28,48891,48283],{"class":15675},[28,48893,2606],{"class":15682},[28,48895,48896],{"class":15841}," \"Y\"",[28,48898,16052],{"class":15675},[28,48900,48901],{"class":15660,"line":15798},[28,48902,16052],{"class":15675},[28,48904,48905],{"class":15660,"line":15811},[28,48906,15769],{"emptyLinePlaceholder":15768},[28,48908,48909],{"class":15660,"line":15820},[28,48910,48911],{"class":15664},"# Get expectation values\n",[28,48913,48914,48917,48919,48922,48924,48926,48928,48930],{"class":15660,"line":15830},[28,48915,48916],{"class":15675},"S_expval ",[28,48918,845],{"class":15682},[28,48920,48921],{"class":15675}," S_expval_real ",[28,48923,2606],{"class":15682},[28,48925,22230],{"class":15801},[28,48927,28200],{"class":15671},[28,48929,16210],{"class":15682},[28,48931,48932],{"class":15675}," S_expval_imag\n",[28,48934,48935],{"class":15660,"line":15835},[28,48936,15769],{"emptyLinePlaceholder":15768},[28,48938,48939,48942,48944],{"class":15660,"line":15853},[28,48940,48941],{"class":15675},"H_expval ",[28,48943,845],{"class":15682},[28,48945,28085],{"class":15801},[28,48947,48948,48950,48953,48955,48957,48959,48962],{"class":15660,"line":15867},[28,48949,22277],{"class":15671},[28,48951,48952],{"class":15675}," obs_idx, (pauli, coeff) ",[28,48954,17720],{"class":15671},[28,48956,27825],{"class":15682},[28,48958,65],{"class":15675},[28,48960,48961],{"class":15682},"zip",[28,48963,48964],{"class":15675},"(H_op.paulis, H_op.coeffs)):\n",[28,48966,48967],{"class":15660,"line":15872},[28,48968,48969],{"class":15664},"    # Get expectation values from experiment\n",[28,48971,48972,48975,48977,48979,48981,48983],{"class":15660,"line":15878},[28,48973,48974],{"class":15675},"    expval_real ",[28,48976,845],{"class":15682},[28,48978,48832],{"class":16073},[28,48980,48835],{"class":15675},[28,48982,48838],{"class":16073},[28,48984,16077],{"class":15675},[28,48986,48987,48990,48993,48995,48997,48999,49001,49003,49005,49007],{"class":15660,"line":15897},[28,48988,48989],{"class":16073},"        Pauli",[28,48991,48992],{"class":15675},"(pauli[::",[28,48994,15689],{"class":15682},[28,48996,3200],{"class":15801},[28,48998,48516],{"class":15675},[28,49000,48519],{"class":16073},[28,49002,23647],{"class":15675},[28,49004,2606],{"class":15682},[28,49006,48858],{"class":15841},[28,49008,16052],{"class":15675},[28,49010,49011],{"class":15660,"line":15914},[28,49012,18606],{"class":15675},[28,49014,49015,49018,49020,49022,49024,49026],{"class":15660,"line":15931},[28,49016,49017],{"class":15675},"    expval_imag ",[28,49019,845],{"class":15682},[28,49021,48832],{"class":16073},[28,49023,48835],{"class":15675},[28,49025,48838],{"class":16073},[28,49027,16077],{"class":15675},[28,49029,49030,49032,49034,49036,49038,49040,49042,49044,49046,49048],{"class":15660,"line":15948},[28,49031,48989],{"class":16073},[28,49033,48992],{"class":15675},[28,49035,15689],{"class":15682},[28,49037,3200],{"class":15801},[28,49039,48516],{"class":15675},[28,49041,48519],{"class":16073},[28,49043,23647],{"class":15675},[28,49045,2606],{"class":15682},[28,49047,48896],{"class":15841},[28,49049,16052],{"class":15675},[28,49051,49052],{"class":15660,"line":15966},[28,49053,18606],{"class":15675},[28,49055,49056,49059,49061,49064,49066,49068,49070,49072],{"class":15660,"line":15983},[28,49057,49058],{"class":15675},"    expval ",[28,49060,845],{"class":15682},[28,49062,49063],{"class":15675}," expval_real ",[28,49065,2606],{"class":15682},[28,49067,22230],{"class":15801},[28,49069,28200],{"class":15671},[28,49071,16210],{"class":15682},[28,49073,49074],{"class":15675}," expval_imag\n",[28,49076,49077],{"class":15660,"line":16000},[28,49078,15769],{"emptyLinePlaceholder":15768},[28,49080,49081],{"class":15660,"line":16016},[28,49082,49083],{"class":15664},"    # Fill-in matrix elements\n",[28,49085,49086,49089,49091,49094,49096],{"class":15660,"line":16036},[28,49087,49088],{"class":15675},"    H_expval ",[28,49090,28120],{"class":15682},[28,49092,49093],{"class":15675}," coeff ",[28,49095,16226],{"class":15682},[28,49097,49098],{"class":15675}," expval\n",[28,49100,49101],{"class":15660,"line":16041},[28,49102,15769],{"emptyLinePlaceholder":15768},[28,49104,49105],{"class":15660,"line":16371},[28,49106,15769],{"emptyLinePlaceholder":15768},[28,49108,49109,49111],{"class":15660,"line":16376},[28,49110,16044],{"class":15682},[28,49112,49113],{"class":15675},"(H_expval)\n",[15650,49115,49118],{"className":49116,"code":49117,"language":16058,"meta":15655},[16056],"(25+0j)\n",[8103,49119,49117],{"__ignoreMap":15655},[12,49121,49122,49123,512,49151,49211],{},"Execute circuits for ",[28,49124,49126,49139],{"className":49125},[35],[28,49127,49129],{"className":49128},[39],[41,49130,49131],{"xmlns":43},[46,49132,49133,49137],{},[49,49134,49135],{},[55,49136,11356],{},[82,49138,11356],{"encoding":84},[28,49140,49142],{"className":49141,"ariaHidden":71},[89],[28,49143,49145,49148],{"className":49144},[93],[28,49146],{"className":49147,"style":1102},[97],[28,49149,11356],{"className":49150,"style":1913},[102,106],[28,49152,49154,49171],{"className":49153},[35],[28,49155,49157],{"className":49156},[39],[41,49158,49159],{"xmlns":43},[46,49160,49161,49169],{},[49,49162,49163],{},[10155,49164,49165,49167],{"accent":71},[55,49166,2263],{},[62,49168,30008],{},[82,49170,30011],{"encoding":84},[28,49172,49174],{"className":49173,"ariaHidden":71},[89],[28,49175,49177,49180],{"className":49176},[93],[28,49178],{"className":49179,"style":30021},[97],[28,49181,49183],{"className":49182},[102,10255],[28,49184,49186],{"className":49185},[115],[28,49187,49189],{"className":49188},[120],[28,49190,49192,49200],{"className":49191,"style":30021},[124],[28,49193,49194,49197],{"style":10025},[28,49195],{"className":49196,"style":3434},[132],[28,49198,2263],{"className":49199,"style":4040},[102,106],[28,49201,49202,49205],{"style":30044},[28,49203],{"className":49204,"style":3434},[132],[28,49206,49208],{"className":49207,"style":30051},[10282],[28,49209,30008],{"className":49210},[102]," with Estimator",[15650,49213,49215],{"className":15652,"code":49214,"language":15654,"meta":15655,"style":15655},"# 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",[8103,49216,49217,49222,49231,49240,49250,49273,49300,49304,49308,49318,49332,49343,49351,49359,49367,49372,49383,49397,49404,49415,49423,49429,49435,49439,49445,49457,49474,49478,49482,49495,49503,49510,49522,49526,49530,49558,49562,49566],{"__ignoreMap":15655},[28,49218,49219],{"class":15660,"line":15661},[28,49220,49221],{"class":15664},"# Experiment options\n",[28,49223,49224,49227,49229],{"class":15660,"line":15668},[28,49225,49226],{"class":15675},"num_randomizations ",[28,49228,845],{"class":15682},[28,49230,17298],{"class":15801},[28,49232,49233,49236,49238],{"class":15660,"line":15679},[28,49234,49235],{"class":15675},"num_randomizations_learning ",[28,49237,845],{"class":15682},[28,49239,33860],{"class":15801},[28,49241,49242,49245,49247],{"class":15660,"line":15725},[28,49243,49244],{"class":15675},"shots_per_randomization ",[28,49246,845],{"class":15682},[28,49248,49249],{"class":15801}," 100\n",[28,49251,49252,49255,49257,49259,49261,49263,49266,49268,49271],{"class":15660,"line":15739},[28,49253,49254],{"class":15675},"noise_factors ",[28,49256,845],{"class":15682},[28,49258,17337],{"class":15675},[28,49260,3200],{"class":15801},[28,49262,16397],{"class":15675},[28,49264,49265],{"class":15801},"1.2",[28,49267,16397],{"class":15675},[28,49269,49270],{"class":15801},"1.4",[28,49272,16849],{"class":15675},[28,49274,49275,49278,49280,49282,49284,49286,49288,49290,49293,49295,49298],{"class":15660,"line":15752},[28,49276,49277],{"class":15675},"learning_pair_depths ",[28,49279,845],{"class":15682},[28,49281,17337],{"class":15675},[28,49283,652],{"class":15801},[28,49285,16397],{"class":15675},[28,49287,37795],{"class":15801},[28,49289,16397],{"class":15675},[28,49291,49292],{"class":15801},"24",[28,49294,16397],{"class":15675},[28,49296,49297],{"class":15801},"48",[28,49299,16849],{"class":15675},[28,49301,49302],{"class":15660,"line":15765},[28,49303,15769],{"emptyLinePlaceholder":15768},[28,49305,49306],{"class":15660,"line":15772},[28,49307,15769],{"emptyLinePlaceholder":15768},[28,49309,49310,49313,49315],{"class":15660,"line":15781},[28,49311,49312],{"class":15675},"experimental_opts ",[28,49314,845],{"class":15682},[28,49316,49317],{"class":15675}," {}\n",[28,49319,49320,49323,49326,49328,49330],{"class":15660,"line":15798},[28,49321,49322],{"class":15675},"experimental_opts[",[28,49324,49325],{"class":15841},"\"resilience\"",[28,49327,15845],{"class":15675},[28,49329,845],{"class":15682},[28,49331,17124],{"class":15675},[28,49333,49334,49337,49339,49341],{"class":15660,"line":15811},[28,49335,49336],{"class":15841},"    \"measure_mitigation\"",[28,49338,15883],{"class":15675},[28,49340,15597],{"class":15801},[28,49342,16105],{"class":15675},[28,49344,49345,49348],{"class":15660,"line":15820},[28,49346,49347],{"class":15841},"    \"measure_noise_learning\"",[28,49349,49350],{"class":15675},": {\n",[28,49352,49353,49356],{"class":15660,"line":15830},[28,49354,49355],{"class":15841},"        \"num_randomizations\"",[28,49357,49358],{"class":15675},": num_randomizations_learning,\n",[28,49360,49361,49364],{"class":15660,"line":15835},[28,49362,49363],{"class":15841},"        \"shots_per_randomization\"",[28,49365,49366],{"class":15675},": shots_per_randomization,\n",[28,49368,49369],{"class":15660,"line":15853},[28,49370,49371],{"class":15675},"    },\n",[28,49373,49374,49377,49379,49381],{"class":15660,"line":15867},[28,49375,49376],{"class":15841},"    \"zne_mitigation\"",[28,49378,15883],{"class":15675},[28,49380,15597],{"class":15801},[28,49382,16105],{"class":15675},[28,49384,49385,49388,49391,49394],{"class":15660,"line":15872},[28,49386,49387],{"class":15841},"    \"zne\"",[28,49389,49390],{"class":15675},": {",[28,49392,49393],{"class":15841},"\"noise_factors\"",[28,49395,49396],{"class":15675},": noise_factors},\n",[28,49398,49399,49402],{"class":15660,"line":15878},[28,49400,49401],{"class":15841},"    \"layer_noise_learning\"",[28,49403,49350],{"class":15675},[28,49405,49406,49409,49411,49413],{"class":15660,"line":15897},[28,49407,49408],{"class":15841},"        \"max_layers_to_learn\"",[28,49410,15883],{"class":15675},[28,49412,19879],{"class":15801},[28,49414,16105],{"class":15675},[28,49416,49417,49420],{"class":15660,"line":15914},[28,49418,49419],{"class":15841},"        \"layer_pair_depths\"",[28,49421,49422],{"class":15675},": learning_pair_depths,\n",[28,49424,49425,49427],{"class":15660,"line":15931},[28,49426,49363],{"class":15841},[28,49428,49366],{"class":15675},[28,49430,49431,49433],{"class":15660,"line":15948},[28,49432,49355],{"class":15841},[28,49434,49358],{"class":15675},[28,49436,49437],{"class":15660,"line":15966},[28,49438,49371],{"class":15675},[28,49440,49441,49443],{"class":15660,"line":15983},[28,49442,49387],{"class":15841},[28,49444,49350],{"class":15675},[28,49446,49447,49450,49452,49455],{"class":15660,"line":16000},[28,49448,49449],{"class":15841},"        \"amplifier\"",[28,49451,15883],{"class":15675},[28,49453,49454],{"class":15841},"\"pea\"",[28,49456,16105],{"class":15675},[28,49458,49459,49462,49465,49467,49469,49471],{"class":15660,"line":16016},[28,49460,49461],{"class":15841},"        \"extrapolated_noise_factors\"",[28,49463,49464],{"class":15675},": [",[28,49466,652],{"class":15801},[28,49468,15845],{"class":15675},[28,49470,2606],{"class":15682},[28,49472,49473],{"class":15675}," noise_factors,\n",[28,49475,49476],{"class":15660,"line":16036},[28,49477,49371],{"class":15675},[28,49479,49480],{"class":15660,"line":16041},[28,49481,17177],{"class":15675},[28,49483,49484,49486,49489,49491,49493],{"class":15660,"line":16371},[28,49485,49322],{"class":15675},[28,49487,49488],{"class":15841},"\"twirling\"",[28,49490,15845],{"class":15675},[28,49492,845],{"class":15682},[28,49494,17124],{"class":15675},[28,49496,49497,49500],{"class":15660,"line":16376},[28,49498,49499],{"class":15841},"    \"num_randomizations\"",[28,49501,49502],{"class":15675},": num_randomizations,\n",[28,49504,49505,49508],{"class":15660,"line":16410},[28,49506,49507],{"class":15841},"    \"shots_per_randomization\"",[28,49509,49366],{"class":15675},[28,49511,49512,49515,49517,49520],{"class":15660,"line":16431},[28,49513,49514],{"class":15841},"    \"strategy\"",[28,49516,15883],{"class":15675},[28,49518,49519],{"class":15841},"\"all\"",[28,49521,16105],{"class":15675},[28,49523,49524],{"class":15660,"line":16452},[28,49525,17177],{"class":15675},[28,49527,49528],{"class":15660,"line":16476},[28,49529,15769],{"emptyLinePlaceholder":15768},[28,49531,49532,49535,49537,49540,49542,49545,49547,49550,49553,49555],{"class":15660,"line":16481},[28,49533,49534],{"class":15675},"estimator ",[28,49536,845],{"class":15682},[28,49538,49539],{"class":16073}," Estimator",[28,49541,65],{"class":15675},[28,49543,49544],{"class":16307},"mode",[28,49546,845],{"class":15682},[28,49548,49549],{"class":15675},"backend, ",[28,49551,49552],{"class":16307},"options",[28,49554,845],{"class":15682},[28,49556,49557],{"class":15675},"experimental_opts)\n",[28,49559,49560],{"class":15660,"line":16510},[28,49561,15769],{"emptyLinePlaceholder":15768},[28,49563,49564],{"class":15660,"line":16542},[28,49565,15769],{"emptyLinePlaceholder":15768},[28,49567,49568,49571,49573,49576,49578],{"class":15660,"line":16557},[28,49569,49570],{"class":15675},"job ",[28,49572,845],{"class":15682},[28,49574,49575],{"class":15675}," estimator.",[28,49577,17789],{"class":16073},[28,49579,49580],{"class":15675},"([pub])\n",[251,49582,49584],{"id":49583},"step-4-post-process-and-return-result-in-desired-classical-format","Step 4: Post-process and return result in desired classical format",[15650,49586,49588],{"className":15652,"code":49587,"language":15654,"meta":15655,"style":15655},"results = job.result()[0]\n",[8103,49589,49590],{"__ignoreMap":15655},[28,49591,49592,49595,49597,49600,49603,49606,49608],{"class":15660,"line":15661},[28,49593,49594],{"class":15675},"results ",[28,49596,845],{"class":15682},[28,49598,49599],{"class":15675}," job.",[28,49601,49602],{"class":16073},"result",[28,49604,49605],{"class":15675},"()[",[28,49607,652],{"class":15801},[28,49609,16849],{"class":15675},[28327,49611,49613],{"id":49612},"calculate-effective-hamiltonian-and-overlap-matrices","Calculate Effective Hamiltonian and Overlap matrices",[12,49615,49616,49617,49653],{},"First calculate the phase accumulated by the ",[28,49618,49620,49638],{"className":49619},[35],[28,49621,49623],{"className":49622},[39],[41,49624,49625],{"xmlns":43},[46,49626,49627,49635],{},[49,49628,49629,49631,49633],{},[55,49630,296],{"mathvariant":1088},[650,49632,652],{},[62,49634,302],{"stretchy":64},[82,49636,49637],{"encoding":84},"\\vert 0 \\rangle",[28,49639,49641],{"className":49640,"ariaHidden":71},[89],[28,49642,49644,49647,49650],{"className":49643},[93],[28,49645],{"className":49646,"style":98},[97],[28,49648,35403],{"className":49649},[102],[28,49651,302],{"className":49652},[182]," state during the uncontrolled time evolution",[15650,49655,49657],{"className":15652,"code":49656,"language":15654,"meta":15655,"style":15655},"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",[8103,49658,49659,49668,49722,49739],{"__ignoreMap":15655},[28,49660,49661,49664,49666],{"class":15660,"line":15661},[28,49662,49663],{"class":15675},"prefactors ",[28,49665,845],{"class":15682},[28,49667,17702],{"class":15675},[28,49669,49670,49673,49675,49677,49679,49681,49683,49685,49687,49690,49692,49695,49697,49699,49701,49703,49705,49708,49710,49713,49715,49717,49719],{"class":15660,"line":15668},[28,49671,49672],{"class":15675},"    np.",[28,49674,7645],{"class":16073},[28,49676,65],{"class":15675},[28,49678,15689],{"class":15682},[28,49680,3200],{"class":15801},[28,49682,28200],{"class":15671},[28,49684,16210],{"class":15682},[28,49686,34494],{"class":15682},[28,49688,49689],{"class":15675},"([c ",[28,49691,22277],{"class":15671},[28,49693,49694],{"class":15675}," p, c ",[28,49696,17720],{"class":15671},[28,49698,27793],{"class":15675},[28,49700,27796],{"class":16073},[28,49702,23647],{"class":15675},[28,49704,20133],{"class":15671},[28,49706,49707],{"class":15841}," \"Z\"",[28,49709,47604],{"class":15671},[28,49711,49712],{"class":15675}," p]) ",[28,49714,16226],{"class":15682},[28,49716,22280],{"class":15675},[28,49718,16226],{"class":15682},[28,49720,49721],{"class":15675}," dt)\n",[28,49723,49724,49726,49728,49730,49732,49734,49736],{"class":15660,"line":15679},[28,49725,27785],{"class":15671},[28,49727,22280],{"class":15675},[28,49729,17720],{"class":15671},[28,49731,22285],{"class":15682},[28,49733,65],{"class":15675},[28,49735,3200],{"class":15801},[28,49737,49738],{"class":15675},", krylov_dim)\n",[28,49740,49741],{"class":15660,"line":15725},[28,49742,16849],{"class":15675},[12,49744,49745,49746],{},"Once we have the results of the circuit executions we can post-process the data to calculate the matrix elements of ",[28,49747,49749,49762],{"className":49748},[35],[28,49750,49752],{"className":49751},[39],[41,49753,49754],{"xmlns":43},[46,49755,49756,49760],{},[49,49757,49758],{},[55,49759,11356],{},[82,49761,11356],{"encoding":84},[28,49763,49765],{"className":49764,"ariaHidden":71},[89],[28,49766,49768,49771],{"className":49767},[93],[28,49769],{"className":49770,"style":1102},[97],[28,49772,11356],{"className":49773,"style":1913},[102,106],[15650,49775,49777],{"className":15652,"code":49776,"language":15654,"meta":15655,"style":15655},"# 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",[8103,49778,49779,49784,49804,49824,49828,49833,49852,49856,49874,49879,49886,49902,49906,49912,49916,49921,49939,49959,49963,49982,49986,49990,50010,50014,50019,50049,50061,50076,50082],{"__ignoreMap":15655},[28,49780,49781],{"class":15660,"line":15661},[28,49782,49783],{"class":15664},"# Assemble S, the overlap matrix of dimension D:\n",[28,49785,49786,49789,49791,49793,49795,49798,49800,49802],{"class":15660,"line":15668},[28,49787,49788],{"class":15675},"S_first_row ",[28,49790,845],{"class":15682},[28,49792,16298],{"class":15675},[28,49794,27999],{"class":16073},[28,49796,49797],{"class":15675},"(krylov_dim, ",[28,49799,22267],{"class":16307},[28,49801,28007],{"class":15682},[28,49803,16052],{"class":15675},[28,49805,49806,49809,49811,49813,49815,49817,49819,49821],{"class":15660,"line":15679},[28,49807,49808],{"class":15675},"S_first_row[",[28,49810,652],{"class":15801},[28,49812,15845],{"class":15675},[28,49814,845],{"class":15682},[28,49816,22230],{"class":15801},[28,49818,16359],{"class":15682},[28,49820,46881],{"class":15801},[28,49822,49823],{"class":15671},"j\n",[28,49825,49826],{"class":15660,"line":15725},[28,49827,15769],{"emptyLinePlaceholder":15768},[28,49829,49830],{"class":15660,"line":15739},[28,49831,49832],{"class":15664},"# Add in ancilla-only measurements:\n",[28,49834,49835,49837,49839,49841,49843,49846,49848,49850],{"class":15660,"line":15752},[28,49836,22277],{"class":15671},[28,49838,22280],{"class":15675},[28,49840,17720],{"class":15671},[28,49842,22285],{"class":15682},[28,49844,49845],{"class":15675},"(krylov_dim ",[28,49847,15689],{"class":15682},[28,49849,22230],{"class":15801},[28,49851,16407],{"class":15675},[28,49853,49854],{"class":15660,"line":15765},[28,49855,48969],{"class":15664},[28,49857,49858,49860,49862,49865,49867,49869,49871],{"class":15660,"line":15772},[28,49859,48974],{"class":15675},[28,49861,845],{"class":15682},[28,49863,49864],{"class":15675}," results.data.evs[",[28,49866,652],{"class":15801},[28,49868,21951],{"class":15675},[28,49870,652],{"class":15801},[28,49872,49873],{"class":15675},"][\n",[28,49875,49876],{"class":15660,"line":15781},[28,49877,49878],{"class":15675},"        i\n",[28,49880,49881,49883],{"class":15660,"line":15798},[28,49882,28068],{"class":15675},[28,49884,49885],{"class":15664},"# automatic extrapolated evs if ZNE is used\n",[28,49887,49888,49890,49892,49894,49896,49898,49900],{"class":15660,"line":15811},[28,49889,49017],{"class":15675},[28,49891,845],{"class":15682},[28,49893,49864],{"class":15675},[28,49895,3200],{"class":15801},[28,49897,21951],{"class":15675},[28,49899,652],{"class":15801},[28,49901,49873],{"class":15675},[28,49903,49904],{"class":15660,"line":15820},[28,49905,49878],{"class":15675},[28,49907,49908,49910],{"class":15660,"line":15830},[28,49909,28068],{"class":15675},[28,49911,49885],{"class":15664},[28,49913,49914],{"class":15660,"line":15835},[28,49915,15769],{"emptyLinePlaceholder":15768},[28,49917,49918],{"class":15660,"line":15853},[28,49919,49920],{"class":15664},"    # Get expectation values\n",[28,49922,49923,49925,49927,49929,49931,49933,49935,49937],{"class":15660,"line":15867},[28,49924,49058],{"class":15675},[28,49926,845],{"class":15682},[28,49928,49063],{"class":15675},[28,49930,2606],{"class":15682},[28,49932,22230],{"class":15801},[28,49934,28200],{"class":15671},[28,49936,16210],{"class":15682},[28,49938,49074],{"class":15675},[28,49940,49941,49944,49946,49948,49950,49952,49955,49957],{"class":15660,"line":15872},[28,49942,49943],{"class":15675},"    S_first_row[i ",[28,49945,2606],{"class":15682},[28,49947,22230],{"class":15801},[28,49949,15845],{"class":15675},[28,49951,28120],{"class":15682},[28,49953,49954],{"class":15675}," prefactors[i] ",[28,49956,16226],{"class":15682},[28,49958,49098],{"class":15675},[28,49960,49961],{"class":15660,"line":15878},[28,49962,15769],{"emptyLinePlaceholder":15768},[28,49964,49965,49968,49970,49973,49976,49979],{"class":15660,"line":15897},[28,49966,49967],{"class":15675},"S_first_row_list ",[28,49969,845],{"class":15682},[28,49971,49972],{"class":15675}," S_first_row.",[28,49974,49975],{"class":16073},"tolist",[28,49977,49978],{"class":15675},"()  ",[28,49980,49981],{"class":15664},"# for saving purposes\n",[28,49983,49984],{"class":15660,"line":15914},[28,49985,15769],{"emptyLinePlaceholder":15768},[28,49987,49988],{"class":15660,"line":15931},[28,49989,15769],{"emptyLinePlaceholder":15768},[28,49991,49992,49995,49997,49999,50001,50004,50006,50008],{"class":15660,"line":15948},[28,49993,49994],{"class":15675},"S_circ ",[28,49996,845],{"class":15682},[28,49998,16298],{"class":15675},[28,50000,27999],{"class":16073},[28,50002,50003],{"class":15675},"((krylov_dim, krylov_dim), ",[28,50005,22267],{"class":16307},[28,50007,28007],{"class":15682},[28,50009,16052],{"class":15675},[28,50011,50012],{"class":15660,"line":15966},[28,50013,15769],{"emptyLinePlaceholder":15768},[28,50015,50016],{"class":15660,"line":15983},[28,50017,50018],{"class":15664},"# Distribute entries from first row across matrix:\n",[28,50020,50021,50023,50026,50028,50030,50033,50035,50037,50040,50043,50045,50047],{"class":15660,"line":16000},[28,50022,22277],{"class":15671},[28,50024,50025],{"class":15675}," i, j ",[28,50027,17720],{"class":15671},[28,50029,28040],{"class":15675},[28,50031,50032],{"class":16073},"product",[28,50034,65],{"class":15675},[28,50036,28048],{"class":15682},[28,50038,50039],{"class":15675},"(krylov_dim), ",[28,50041,50042],{"class":16307},"repeat",[28,50044,845],{"class":15682},[28,50046,848],{"class":15801},[28,50048,16407],{"class":15675},[28,50050,50051,50053,50055,50058],{"class":15660,"line":16016},[28,50052,16484],{"class":15671},[28,50054,22280],{"class":15675},[28,50056,50057],{"class":15682},">=",[28,50059,50060],{"class":15675}," j:\n",[28,50062,50063,50066,50068,50071,50073],{"class":15660,"line":16036},[28,50064,50065],{"class":15675},"        S_circ[j, i] ",[28,50067,845],{"class":15682},[28,50069,50070],{"class":15675}," S_first_row[i ",[28,50072,15689],{"class":15682},[28,50074,50075],{"class":15675}," j]\n",[28,50077,50078,50080],{"class":15660,"line":16041},[28,50079,19102],{"class":15671},[28,50081,15778],{"class":15675},[28,50083,50084,50086,50088,50090,50092,50095,50097],{"class":15660,"line":16371},[28,50085,50065],{"class":15675},[28,50087,845],{"class":15682},[28,50089,16298],{"class":15675},[28,50091,27574],{"class":16073},[28,50093,50094],{"class":15675},"(S_first_row[j ",[28,50096,15689],{"class":15682},[28,50098,50099],{"class":15675}," i])\n",[15650,50101,50103],{"className":15652,"code":50102,"language":15654,"meta":15655,"style":15655},"Matrix(S_circ)\n",[8103,50104,50105],{"__ignoreMap":15655},[28,50106,50107,50110],{"class":15660,"line":15661},[28,50108,50109],{"class":16073},"Matrix",[28,50111,50112],{"class":15675},"(S_circ)\n",[12,50114,50115,50116],{},"And the matrix elements of ",[28,50117,50119,50136],{"className":50118},[35],[28,50120,50122],{"className":50121},[39],[41,50123,50124],{"xmlns":43},[46,50125,50126,50134],{},[49,50127,50128],{},[10155,50129,50130,50132],{"accent":71},[55,50131,2263],{},[62,50133,30008],{},[82,50135,30011],{"encoding":84},[28,50137,50139],{"className":50138,"ariaHidden":71},[89],[28,50140,50142,50145],{"className":50141},[93],[28,50143],{"className":50144,"style":30021},[97],[28,50146,50148],{"className":50147},[102,10255],[28,50149,50151],{"className":50150},[115],[28,50152,50154],{"className":50153},[120],[28,50155,50157,50165],{"className":50156,"style":30021},[124],[28,50158,50159,50162],{"style":10025},[28,50160],{"className":50161,"style":3434},[132],[28,50163,2263],{"className":50164,"style":4040},[102,106],[28,50166,50167,50170],{"style":30044},[28,50168],{"className":50169,"style":3434},[132],[28,50171,50173],{"className":50172,"style":30051},[10282],[28,50174,30008],{"className":50175},[102],[15650,50177,50179],{"className":15652,"code":50178,"language":15654,"meta":15655,"style":15655},"# 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",[8103,50180,50181,50185,50204,50218,50222,50238,50243,50261,50266,50290,50295,50302,50332,50336,50342,50346,50351,50370,50393,50397,50411,50415,50434,50438,50442,50468,50478,50492,50498],{"__ignoreMap":15655},[28,50182,50183],{"class":15660,"line":15661},[28,50184,49783],{"class":15664},[28,50186,50187,50190,50192,50194,50196,50198,50200,50202],{"class":15660,"line":15668},[28,50188,50189],{"class":15675},"H_first_row ",[28,50191,845],{"class":15682},[28,50193,16298],{"class":15675},[28,50195,27999],{"class":16073},[28,50197,49797],{"class":15675},[28,50199,22267],{"class":16307},[28,50201,28007],{"class":15682},[28,50203,16052],{"class":15675},[28,50205,50206,50209,50211,50213,50215],{"class":15660,"line":15679},[28,50207,50208],{"class":15675},"H_first_row[",[28,50210,652],{"class":15801},[28,50212,15845],{"class":15675},[28,50214,845],{"class":15682},[28,50216,50217],{"class":15675}," H_expval\n",[28,50219,50220],{"class":15660,"line":15725},[28,50221,15769],{"emptyLinePlaceholder":15768},[28,50223,50224,50226,50228,50230,50232,50234,50236],{"class":15660,"line":15739},[28,50225,22277],{"class":15671},[28,50227,48952],{"class":15675},[28,50229,17720],{"class":15671},[28,50231,27825],{"class":15682},[28,50233,65],{"class":15675},[28,50235,48961],{"class":15682},[28,50237,48964],{"class":15675},[28,50239,50240],{"class":15660,"line":15752},[28,50241,50242],{"class":15664},"    # Add in ancilla-only measurements:\n",[28,50244,50245,50247,50249,50251,50253,50255,50257,50259],{"class":15660,"line":15765},[28,50246,27785],{"class":15671},[28,50248,22280],{"class":15675},[28,50250,17720],{"class":15671},[28,50252,22285],{"class":15682},[28,50254,49845],{"class":15675},[28,50256,15689],{"class":15682},[28,50258,22230],{"class":15801},[28,50260,16407],{"class":15675},[28,50262,50263],{"class":15660,"line":15772},[28,50264,50265],{"class":15664},"        # Get expectation values from experiment\n",[28,50267,50268,50271,50273,50275,50277,50279,50281,50283,50286,50288],{"class":15660,"line":15781},[28,50269,50270],{"class":15675},"        expval_real ",[28,50272,845],{"class":15682},[28,50274,49864],{"class":15675},[28,50276,848],{"class":15801},[28,50278,16359],{"class":15682},[28,50280,22129],{"class":15801},[28,50282,16210],{"class":15682},[28,50284,50285],{"class":15675}," obs_idx][",[28,50287,652],{"class":15801},[28,50289,49873],{"class":15675},[28,50291,50292],{"class":15660,"line":15798},[28,50293,50294],{"class":15675},"            i\n",[28,50296,50297,50300],{"class":15660,"line":15811},[28,50298,50299],{"class":15675},"        ]  ",[28,50301,49885],{"class":15664},[28,50303,50304,50307,50309,50311,50313,50315,50317,50319,50322,50324,50326,50328,50330],{"class":15660,"line":15820},[28,50305,50306],{"class":15675},"        expval_imag ",[28,50308,845],{"class":15682},[28,50310,49864],{"class":15675},[28,50312,848],{"class":15801},[28,50314,16359],{"class":15682},[28,50316,22129],{"class":15801},[28,50318,16210],{"class":15682},[28,50320,50321],{"class":15675}," obs_idx ",[28,50323,2606],{"class":15682},[28,50325,22230],{"class":15801},[28,50327,21951],{"class":15675},[28,50329,652],{"class":15801},[28,50331,49873],{"class":15675},[28,50333,50334],{"class":15660,"line":15830},[28,50335,50294],{"class":15675},[28,50337,50338,50340],{"class":15660,"line":15835},[28,50339,50299],{"class":15675},[28,50341,49885],{"class":15664},[28,50343,50344],{"class":15660,"line":15853},[28,50345,15769],{"emptyLinePlaceholder":15768},[28,50347,50348],{"class":15660,"line":15867},[28,50349,50350],{"class":15664},"        # Get expectation values\n",[28,50352,50353,50356,50358,50360,50362,50364,50366,50368],{"class":15660,"line":15872},[28,50354,50355],{"class":15675},"        expval ",[28,50357,845],{"class":15682},[28,50359,49063],{"class":15675},[28,50361,2606],{"class":15682},[28,50363,22230],{"class":15801},[28,50365,28200],{"class":15671},[28,50367,16210],{"class":15682},[28,50369,49074],{"class":15675},[28,50371,50372,50375,50377,50379,50381,50383,50385,50387,50389,50391],{"class":15660,"line":15878},[28,50373,50374],{"class":15675},"        H_first_row[i ",[28,50376,2606],{"class":15682},[28,50378,22230],{"class":15801},[28,50380,15845],{"class":15675},[28,50382,28120],{"class":15682},[28,50384,49954],{"class":15675},[28,50386,16226],{"class":15682},[28,50388,49093],{"class":15675},[28,50390,16226],{"class":15682},[28,50392,49098],{"class":15675},[28,50394,50395],{"class":15660,"line":15897},[28,50396,15769],{"emptyLinePlaceholder":15768},[28,50398,50399,50402,50404,50407,50409],{"class":15660,"line":15914},[28,50400,50401],{"class":15675},"H_first_row_list ",[28,50403,845],{"class":15682},[28,50405,50406],{"class":15675}," H_first_row.",[28,50408,49975],{"class":16073},[28,50410,17029],{"class":15675},[28,50412,50413],{"class":15660,"line":15931},[28,50414,15769],{"emptyLinePlaceholder":15768},[28,50416,50417,50420,50422,50424,50426,50428,50430,50432],{"class":15660,"line":15948},[28,50418,50419],{"class":15675},"H_eff_circ ",[28,50421,845],{"class":15682},[28,50423,16298],{"class":15675},[28,50425,27999],{"class":16073},[28,50427,50003],{"class":15675},[28,50429,22267],{"class":16307},[28,50431,28007],{"class":15682},[28,50433,16052],{"class":15675},[28,50435,50436],{"class":15660,"line":15966},[28,50437,15769],{"emptyLinePlaceholder":15768},[28,50439,50440],{"class":15660,"line":15983},[28,50441,50018],{"class":15664},[28,50443,50444,50446,50448,50450,50452,50454,50456,50458,50460,50462,50464,50466],{"class":15660,"line":16000},[28,50445,22277],{"class":15671},[28,50447,50025],{"class":15675},[28,50449,17720],{"class":15671},[28,50451,28040],{"class":15675},[28,50453,50032],{"class":16073},[28,50455,65],{"class":15675},[28,50457,28048],{"class":15682},[28,50459,50039],{"class":15675},[28,50461,50042],{"class":16307},[28,50463,845],{"class":15682},[28,50465,848],{"class":15801},[28,50467,16407],{"class":15675},[28,50469,50470,50472,50474,50476],{"class":15660,"line":16016},[28,50471,16484],{"class":15671},[28,50473,22280],{"class":15675},[28,50475,50057],{"class":15682},[28,50477,50060],{"class":15675},[28,50479,50480,50483,50485,50488,50490],{"class":15660,"line":16036},[28,50481,50482],{"class":15675},"        H_eff_circ[j, i] ",[28,50484,845],{"class":15682},[28,50486,50487],{"class":15675}," H_first_row[i ",[28,50489,15689],{"class":15682},[28,50491,50075],{"class":15675},[28,50493,50494,50496],{"class":15660,"line":16041},[28,50495,19102],{"class":15671},[28,50497,15778],{"class":15675},[28,50499,50500,50502,50504,50506,50508,50511,50513],{"class":15660,"line":16371},[28,50501,50482],{"class":15675},[28,50503,845],{"class":15682},[28,50505,16298],{"class":15675},[28,50507,27574],{"class":16073},[28,50509,50510],{"class":15675},"(H_first_row[j ",[28,50512,15689],{"class":15682},[28,50514,50099],{"class":15675},[15650,50516,50518],{"className":15652,"code":50517,"language":15654,"meta":15655,"style":15655},"Matrix(H_eff_circ)\n",[8103,50519,50520],{"__ignoreMap":15655},[28,50521,50522,50524],{"class":15660,"line":15661},[28,50523,50109],{"class":16073},[28,50525,50526],{"class":15675},"(H_eff_circ)\n",[12,50528,50529,50530,50590],{},"Finally, we can solve the generalized eigenvalue problem for ",[28,50531,50533,50550],{"className":50532},[35],[28,50534,50536],{"className":50535},[39],[41,50537,50538],{"xmlns":43},[46,50539,50540,50548],{},[49,50541,50542],{},[10155,50543,50544,50546],{"accent":71},[55,50545,2263],{},[62,50547,30008],{},[82,50549,30011],{"encoding":84},[28,50551,50553],{"className":50552,"ariaHidden":71},[89],[28,50554,50556,50559],{"className":50555},[93],[28,50557],{"className":50558,"style":30021},[97],[28,50560,50562],{"className":50561},[102,10255],[28,50563,50565],{"className":50564},[115],[28,50566,50568],{"className":50567},[120],[28,50569,50571,50579],{"className":50570,"style":30021},[124],[28,50572,50573,50576],{"style":10025},[28,50574],{"className":50575,"style":3434},[132],[28,50577,2263],{"className":50578,"style":4040},[102,106],[28,50580,50581,50584],{"style":30044},[28,50582],{"className":50583,"style":3434},[132],[28,50585,50587],{"className":50586,"style":30051},[10282],[28,50588,30008],{"className":50589},[102],":",[12,50592,50593],{},[28,50594,50596,50632],{"className":50595},[35],[28,50597,50599],{"className":50598},[39],[41,50600,50601],{"xmlns":43},[46,50602,50603,50629],{},[49,50604,50605,50611,50617,50619,50621,50623],{},[10155,50606,50607,50609],{"accent":71},[55,50608,2263],{},[62,50610,30008],{},[10155,50612,50613,50615],{"accent":71},[55,50614,31887],{},[62,50616,10205],{},[62,50618,845],{},[55,50620,31887],{},[55,50622,11356],{},[10155,50624,50625,50627],{"accent":71},[55,50626,31887],{},[62,50628,10205],{},[82,50630,50631],{"encoding":84},"\\tilde{H} \\vec{c} = c S \\vec{c}",[28,50633,50635,50716],{"className":50634,"ariaHidden":71},[89],[28,50636,50638,50641,50672,50707,50710,50713],{"className":50637},[93],[28,50639],{"className":50640,"style":30021},[97],[28,50642,50644],{"className":50643},[102,10255],[28,50645,50647],{"className":50646},[115],[28,50648,50650],{"className":50649},[120],[28,50651,50653,50661],{"className":50652,"style":30021},[124],[28,50654,50655,50658],{"style":10025},[28,50656],{"className":50657,"style":3434},[132],[28,50659,2263],{"className":50660,"style":4040},[102,106],[28,50662,50663,50666],{"style":30044},[28,50664],{"className":50665,"style":3434},[132],[28,50667,50669],{"className":50668,"style":30051},[10282],[28,50670,30008],{"className":50671},[102],[28,50673,50675],{"className":50674},[102,10255],[28,50676,50678],{"className":50677},[115],[28,50679,50681],{"className":50680},[120],[28,50682,50684,50692],{"className":50683,"style":10457},[124],[28,50685,50686,50689],{"style":10025},[28,50687],{"className":50688,"style":3434},[132],[28,50690,31887],{"className":50691},[102,106],[28,50693,50694,50697],{"style":10025},[28,50695],{"className":50696,"style":3434},[132],[28,50698,50700],{"className":50699,"style":31984},[10282],[28,50701,50703],{"className":50702,"style":10479},[10478],[5855,50704,50705],{"xmlns":5857,"width":10482,"height":10483,"style":10484,"viewBox":10485,"preserveAspectRatio":10486},[5863,50706],{"d":10489},[28,50708],{"className":50709,"style":913},[173],[28,50711,845],{"className":50712},[917],[28,50714],{"className":50715,"style":913},[173],[28,50717,50719,50722,50725,50728],{"className":50718},[93],[28,50720],{"className":50721,"style":10457},[97],[28,50723,31887],{"className":50724},[102,106],[28,50726,11356],{"className":50727,"style":1913},[102,106],[28,50729,50731],{"className":50730},[102,10255],[28,50732,50734],{"className":50733},[115],[28,50735,50737],{"className":50736},[120],[28,50738,50740,50748],{"className":50739,"style":10457},[124],[28,50741,50742,50745],{"style":10025},[28,50743],{"className":50744,"style":3434},[132],[28,50746,31887],{"className":50747},[102,106],[28,50749,50750,50753],{"style":10025},[28,50751],{"className":50752,"style":3434},[132],[28,50754,50756],{"className":50755,"style":31984},[10282],[28,50757,50759],{"className":50758,"style":10479},[10478],[5855,50760,50761],{"xmlns":5857,"width":10482,"height":10483,"style":10484,"viewBox":10485,"preserveAspectRatio":10486},[5863,50762],{"d":10489},[12,50764,50765,50766],{},"and get an estimate of the ground state energy ",[28,50767,50769,50793],{"className":50768},[35],[28,50770,50772],{"className":50771},[39],[41,50773,50774],{"xmlns":43},[46,50775,50776,50790],{},[49,50777,50778],{},[52,50779,50780,50782],{},[55,50781,31887],{},[49,50783,50784,50786,50788],{},[55,50785,30144],{},[55,50787,3256],{},[55,50789,7106],{},[82,50791,50792],{"encoding":84},"c_{min}",[28,50794,50796],{"className":50795,"ariaHidden":71},[89],[28,50797,50799,50802],{"className":50798},[93],[28,50800],{"className":50801,"style":780},[97],[28,50803,50805,50808],{"className":50804},[102],[28,50806,31887],{"className":50807},[102,106],[28,50809,50811],{"className":50810},[111],[28,50812,50814,50837],{"className":50813},[115,116],[28,50815,50817,50834],{"className":50816},[120],[28,50818,50820],{"className":50819,"style":32501},[124],[28,50821,50822,50825],{"style":1367},[28,50823],{"className":50824,"style":133},[132],[28,50826,50828],{"className":50827},[137,138,139,140],[28,50829,50831],{"className":50830},[102,140],[28,50832,22201],{"className":50833},[102,106,140],[28,50835,148],{"className":50836},[147],[28,50838,50840],{"className":50839},[120],[28,50841,50843],{"className":50842,"style":155},[124],[28,50844],{},[15650,50846,50848],{"className":15652,"code":50847,"language":15654,"meta":15655,"style":15655},"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",[8103,50849,50850,50859,50883,50888,50899,50912,50916,50926],{"__ignoreMap":15655},[28,50851,50852,50855,50857],{"class":15660,"line":15661},[28,50853,50854],{"class":15675},"gnd_en_circ_est_list ",[28,50856,845],{"class":15682},[28,50858,24066],{"class":15675},[28,50860,50861,50863,50866,50868,50870,50872,50874,50877,50879,50881],{"class":15660,"line":15668},[28,50862,22277],{"class":15671},[28,50864,50865],{"class":15675}," d ",[28,50867,17720],{"class":15671},[28,50869,22285],{"class":15682},[28,50871,65],{"class":15675},[28,50873,3200],{"class":15801},[28,50875,50876],{"class":15675},", krylov_dim ",[28,50878,2606],{"class":15682},[28,50880,22230],{"class":15801},[28,50882,16407],{"class":15675},[28,50884,50885],{"class":15660,"line":15679},[28,50886,50887],{"class":15664},"    # Solve generalized eigenvalue problem for different size of the Krylov space\n",[28,50889,50890,50893,50895,50897],{"class":15660,"line":15725},[28,50891,50892],{"class":15675},"    gnd_en_circ_est ",[28,50894,845],{"class":15682},[28,50896,27325],{"class":16073},[28,50898,16077],{"class":15675},[28,50900,50901,50904,50907,50909],{"class":15660,"line":15739},[28,50902,50903],{"class":15675},"        H_eff_circ[:d, :d], S_circ[:d, :d], ",[28,50905,50906],{"class":16307},"threshold",[28,50908,845],{"class":15682},[28,50910,50911],{"class":15801},"9e-1\n",[28,50913,50914],{"class":15660,"line":15752},[28,50915,18606],{"class":15675},[28,50917,50918,50921,50923],{"class":15660,"line":15765},[28,50919,50920],{"class":15675},"    gnd_en_circ_est_list.",[28,50922,27807],{"class":16073},[28,50924,50925],{"class":15675},"(gnd_en_circ_est)\n",[28,50927,50928,50930,50932,50935],{"class":15660,"line":15772},[28,50929,21619],{"class":15682},[28,50931,65],{"class":15675},[28,50933,50934],{"class":15841},"\"The estimated ground state energy is: \"",[28,50936,50937],{"class":15675},", gnd_en_circ_est)\n",[15650,50939,50942],{"className":50940,"code":50941,"language":16058,"meta":15655},[16056],"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",[8103,50943,50941],{"__ignoreMap":15655},[12,50945,50946],{},"For a single-particle sector, we can efficiently calculate the ground state of this sector of the Hamiltonian classically",[15650,50948,50950],{"className":15652,"code":50949,"language":15654,"meta":15655,"style":15655},"gs_en = single_particle_gs(H_op, n_qubits)\n",[8103,50951,50952],{"__ignoreMap":15655},[28,50953,50954,50957,50959,50961],{"class":15660,"line":15661},[28,50955,50956],{"class":15675},"gs_en ",[28,50958,845],{"class":15682},[28,50960,27746],{"class":16073},[28,50962,50963],{"class":15675},"(H_op, n_qubits)\n",[15650,50965,50968],{"className":50966,"code":50967,"language":16058,"meta":15655},[16056],"n_sys_qubits 30\nn_exc 1 , subspace dimension 31\nsingle particle ground state energy:  21.021912418526906\n",[8103,50969,50967],{"__ignoreMap":15655},[15650,50971,50973],{"className":15652,"code":50972,"language":15654,"meta":15655,"style":15655},"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",[8103,50974,50975,50983,51000,51005,51016,51028,51039,51043,51051,51067,51077,51088,51099,51110,51114,51151,51159,51173,51187,51196,51201,51205],{"__ignoreMap":15655},[28,50976,50977,50979,50981],{"class":15660,"line":15661},[28,50978,17034],{"class":15675},[28,50980,19308],{"class":16073},[28,50982,16077],{"class":15675},[28,50984,50985,50988,50990,50992,50994,50996,50998],{"class":15660,"line":15668},[28,50986,50987],{"class":15682},"    range",[28,50989,65],{"class":15675},[28,50991,3200],{"class":15801},[28,50993,50876],{"class":15675},[28,50995,2606],{"class":15682},[28,50997,22230],{"class":15801},[28,50999,39837],{"class":15675},[28,51001,51002],{"class":15660,"line":15679},[28,51003,51004],{"class":15675},"    gnd_en_circ_est_list,\n",[28,51006,51007,51009,51011,51014],{"class":15660,"line":15725},[28,51008,19266],{"class":16307},[28,51010,845],{"class":15682},[28,51012,51013],{"class":15841},"\"blue\"",[28,51015,16105],{"class":15675},[28,51017,51018,51021,51023,51026],{"class":15660,"line":15739},[28,51019,51020],{"class":16307},"    linestyle",[28,51022,845],{"class":15682},[28,51024,51025],{"class":15841},"\"-.\"",[28,51027,16105],{"class":15675},[28,51029,51030,51032,51034,51037],{"class":15660,"line":15752},[28,51031,19278],{"class":16307},[28,51033,845],{"class":15682},[28,51035,51036],{"class":15841},"\"KQD estimate\"",[28,51038,16105],{"class":15675},[28,51040,51041],{"class":15660,"line":15765},[28,51042,16052],{"class":15675},[28,51044,51045,51047,51049],{"class":15660,"line":15772},[28,51046,17034],{"class":15675},[28,51048,19308],{"class":16073},[28,51050,16077],{"class":15675},[28,51052,51053,51055,51057,51059,51061,51063,51065],{"class":15660,"line":15781},[28,51054,50987],{"class":15682},[28,51056,65],{"class":15675},[28,51058,3200],{"class":15801},[28,51060,50876],{"class":15675},[28,51062,2606],{"class":15682},[28,51064,22230],{"class":15801},[28,51066,39837],{"class":15675},[28,51068,51069,51072,51074],{"class":15660,"line":15798},[28,51070,51071],{"class":15675},"    [gs_en] ",[28,51073,16226],{"class":15682},[28,51075,51076],{"class":15675}," krylov_dim,\n",[28,51078,51079,51081,51083,51086],{"class":15660,"line":15811},[28,51080,19266],{"class":16307},[28,51082,845],{"class":15682},[28,51084,51085],{"class":15841},"\"red\"",[28,51087,16105],{"class":15675},[28,51089,51090,51092,51094,51097],{"class":15660,"line":15820},[28,51091,51020],{"class":16307},[28,51093,845],{"class":15682},[28,51095,51096],{"class":15841},"\"-\"",[28,51098,16105],{"class":15675},[28,51100,51101,51103,51105,51108],{"class":15660,"line":15830},[28,51102,19278],{"class":16307},[28,51104,845],{"class":15682},[28,51106,51107],{"class":15841},"\"exact\"",[28,51109,16105],{"class":15675},[28,51111,51112],{"class":15660,"line":15835},[28,51113,16052],{"class":15675},[28,51115,51116,51118,51121,51123,51125,51127,51129,51131,51133,51135,51137,51139,51141,51143,51145,51147,51149],{"class":15660,"line":15853},[28,51117,17034],{"class":15675},[28,51119,51120],{"class":16073},"xticks",[28,51122,65],{"class":15675},[28,51124,28048],{"class":15682},[28,51126,65],{"class":15675},[28,51128,3200],{"class":15801},[28,51130,50876],{"class":15675},[28,51132,2606],{"class":15682},[28,51134,22230],{"class":15801},[28,51136,22494],{"class":15675},[28,51138,28048],{"class":15682},[28,51140,65],{"class":15675},[28,51142,3200],{"class":15801},[28,51144,50876],{"class":15675},[28,51146,2606],{"class":15682},[28,51148,22230],{"class":15801},[28,51150,16539],{"class":15675},[28,51152,51153,51155,51157],{"class":15660,"line":15867},[28,51154,17034],{"class":15675},[28,51156,21483],{"class":16073},[28,51158,17029],{"class":15675},[28,51160,51161,51163,51166,51168,51171],{"class":15660,"line":15872},[28,51162,17034],{"class":15675},[28,51164,51165],{"class":16073},"xlabel",[28,51167,65],{"class":15675},[28,51169,51170],{"class":15841},"\"Krylov space dimension\"",[28,51172,16052],{"class":15675},[28,51174,51175,51177,51180,51182,51185],{"class":15660,"line":15878},[28,51176,17034],{"class":15675},[28,51178,51179],{"class":16073},"ylabel",[28,51181,65],{"class":15675},[28,51183,51184],{"class":15841},"\"Energy\"",[28,51186,16052],{"class":15675},[28,51188,51189,51191,51194],{"class":15660,"line":15897},[28,51190,17034],{"class":15675},[28,51192,51193],{"class":16073},"title",[28,51195,16077],{"class":15675},[28,51197,51198],{"class":15660,"line":15914},[28,51199,51200],{"class":15841},"    \"Estimating Ground state energy with Krylov Quantum Diagonalization\"\n",[28,51202,51203],{"class":15660,"line":15931},[28,51204,16052],{"class":15675},[28,51206,51207,51209,51211],{"class":15660,"line":15948},[28,51208,17034],{"class":15675},[28,51210,17037],{"class":16073},[28,51212,17029],{"class":15675},[251,51214,51216],{"id":51215},"appendix-krylov-subspace-from-real-time-evolutions","Appendix: Krylov subspace from real time-evolutions",[12,51218,51219],{},"The unitary Krylov space is defined as",[28,51221,51223],{"className":51222},[31],[28,51224,51226,51334],{"className":51225},[35],[28,51227,51229],{"className":51228},[39],[41,51230,51231],{"xmlns":43,"display":44},[46,51232,51233,51331],{},[49,51234,51235,51241,51243,51245,51247,51249,51251,51253,51255,51257,51259],{},[52,51236,51237,51239],{},[55,51238,28350],{"mathvariant":11377},[55,51240,29018],{},[62,51242,65],{"stretchy":64},[55,51244,2263],{},[62,51246,72],{"separator":71},[55,51248,296],{"mathvariant":1088},[55,51250,394],{},[62,51252,302],{"stretchy":64},[62,51254,78],{"stretchy":64},[62,51256,845],{},[2253,51258,28],{},[49,51260,51261,51263,51265,51267,51269,51271,51290,51292,51294,51296,51298,51301,51303,51323,51325,51327,51329],{},[62,51262,11385],{"fence":71},[55,51264,296],{"mathvariant":1088},[55,51266,394],{},[62,51268,302],{"stretchy":64},[62,51270,72],{"separator":71},[2776,51272,51273,51275],{},[55,51274,60],{},[49,51276,51277,51279,51281,51283,51286,51288],{},[62,51278,1213],{},[55,51280,3256],{},[55,51282,2263],{},[2253,51284,51285],{}," ",[55,51287,24201],{},[55,51289,3971],{},[55,51291,296],{"mathvariant":1088},[55,51293,394],{},[62,51295,302],{"stretchy":64},[62,51297,72],{"separator":71},[62,51299,51300],{},"…",[62,51302,72],{"separator":71},[2776,51304,51305,51307],{},[55,51306,60],{},[49,51308,51309,51311,51313,51315,51317,51319,51321],{},[62,51310,1213],{},[55,51312,3256],{},[55,51314,16959],{},[55,51316,2263],{},[2253,51318,51285],{},[55,51320,24201],{},[55,51322,3971],{},[55,51324,296],{"mathvariant":1088},[55,51326,394],{},[62,51328,302],{"stretchy":64},[62,51330,11404],{"fence":71},[82,51332,51333],{"encoding":84},"\\mathcal{K}_U(H, |\\psi\\rangle) = \\text{span}\\left\\{ |\\psi\\rangle,  e^{-iH\\,dt} |\\psi\\rangle, \\dots, e^{-irH\\,dt} |\\psi\\rangle \\right\\}",[28,51335,51337,51414],{"className":51336,"ariaHidden":71},[89],[28,51338,51340,51343,51383,51386,51389,51392,51395,51398,51401,51405,51408,51411],{"className":51339},[93],[28,51341],{"className":51342,"style":98},[97],[28,51344,51346,51349],{"className":51345},[102],[28,51347,28350],{"className":51348,"style":28371},[102,11469],[28,51350,51352],{"className":51351},[111],[28,51353,51355,51375],{"className":51354},[115,116],[28,51356,51358,51372],{"className":51357},[120],[28,51359,51361],{"className":51360,"style":1364},[124],[28,51362,51363,51366],{"style":28953},[28,51364],{"className":51365,"style":133},[132],[28,51367,51369],{"className":51368},[137,138,139,140],[28,51370,29018],{"className":51371,"style":29050},[102,106,140],[28,51373,148],{"className":51374},[147],[28,51376,51378],{"className":51377},[120],[28,51379,51381],{"className":51380,"style":155},[124],[28,51382],{},[28,51384,65],{"className":51385},[161],[28,51387,2263],{"className":51388,"style":4040},[102,106],[28,51390,72],{"className":51391},[169],[28,51393],{"className":51394,"style":174},[173],[28,51396,296],{"className":51397},[102],[28,51399,394],{"className":51400,"style":321},[102,106],[28,51402,51404],{"className":51403},[182],"⟩)",[28,51406],{"className":51407,"style":913},[173],[28,51409,845],{"className":51410},[917],[28,51412],{"className":51413,"style":913},[173],[28,51415,51417,51421,51427,51430],{"className":51416},[93],[28,51418],{"className":51419,"style":51420},[97],"height:1.2491em;vertical-align:-0.35em;",[28,51422,51424],{"className":51423},[102,16058],[28,51425,28],{"className":51426},[102],[28,51428],{"className":51429,"style":174},[173],[28,51431,51433,51439,51442,51445,51448,51451,51454,51502,51505,51508,51511,51514,51517,51520,51523,51526,51529,51579,51582,51585,51588],{"className":51432},[3398],[28,51434,51436],{"className":51435,"style":3403},[161,3402],[28,51437,11385],{"className":51438},[3407,28701],[28,51440,296],{"className":51441},[102],[28,51443,394],{"className":51444,"style":321},[102,106],[28,51446,302],{"className":51447},[182],[28,51449,72],{"className":51450},[169],[28,51452],{"className":51453,"style":174},[173],[28,51455,51457,51460],{"className":51456},[102],[28,51458,60],{"className":51459},[102,106],[28,51461,51463],{"className":51462},[111],[28,51464,51466],{"className":51465},[115],[28,51467,51469],{"className":51468},[120],[28,51470,51472],{"className":51471,"style":30761},[124],[28,51473,51474,51477],{"style":2839},[28,51475],{"className":51476,"style":133},[132],[28,51478,51480],{"className":51479},[137,138,139,140],[28,51481,51483,51486,51489,51492,51496,51499],{"className":51482},[102,140],[28,51484,1213],{"className":51485},[102,140],[28,51487,3256],{"className":51488},[102,106,140],[28,51490,2263],{"className":51491,"style":4040},[102,106,140],[28,51493],{"className":51494,"style":51495},[173,140],"margin-right:0.1952em;",[28,51497,24201],{"className":51498},[102,106,140],[28,51500,3971],{"className":51501},[102,106,140],[28,51503,296],{"className":51504},[102],[28,51506,394],{"className":51507,"style":321},[102,106],[28,51509,302],{"className":51510},[182],[28,51512,72],{"className":51513},[169],[28,51515],{"className":51516,"style":174},[173],[28,51518,51300],{"className":51519},[3398],[28,51521],{"className":51522,"style":174},[173],[28,51524,72],{"className":51525},[169],[28,51527],{"className":51528,"style":174},[173],[28,51530,51532,51535],{"className":51531},[102],[28,51533,60],{"className":51534},[102,106],[28,51536,51538],{"className":51537},[111],[28,51539,51541],{"className":51540},[115],[28,51542,51544],{"className":51543},[120],[28,51545,51547],{"className":51546,"style":30761},[124],[28,51548,51549,51552],{"style":2839},[28,51550],{"className":51551,"style":133},[132],[28,51553,51555],{"className":51554},[137,138,139,140],[28,51556,51558,51561,51564,51567,51570,51573,51576],{"className":51557},[102,140],[28,51559,1213],{"className":51560},[102,140],[28,51562,3256],{"className":51563},[102,106,140],[28,51565,16959],{"className":51566,"style":9403},[102,106,140],[28,51568,2263],{"className":51569,"style":4040},[102,106,140],[28,51571],{"className":51572,"style":51495},[173,140],[28,51574,24201],{"className":51575},[102,106,140],[28,51577,3971],{"className":51578},[102,106,140],[28,51580,296],{"className":51581},[102],[28,51583,394],{"className":51584,"style":321},[102,106],[28,51586,302],{"className":51587},[182],[28,51589,51591],{"className":51590,"style":3403},[182,3402],[28,51592,11404],{"className":51593},[3407,28701],[12,51595,51596,51597,51630,51631,51659,51660,51718],{},"for some timestep ",[28,51598,51600,51615],{"className":51599},[35],[28,51601,51603],{"className":51602},[39],[41,51604,51605],{"xmlns":43},[46,51606,51607,51613],{},[49,51608,51609,51611],{},[55,51610,24201],{},[55,51612,3971],{},[82,51614,31182],{"encoding":84},[28,51616,51618],{"className":51617,"ariaHidden":71},[89],[28,51619,51621,51624,51627],{"className":51620},[93],[28,51622],{"className":51623,"style":9597},[97],[28,51625,24201],{"className":51626},[102,106],[28,51628,3971],{"className":51629},[102,106]," that we will determine later. Temporarily assume ",[28,51632,51634,51647],{"className":51633},[35],[28,51635,51637],{"className":51636},[39],[41,51638,51639],{"xmlns":43},[46,51640,51641,51645],{},[49,51642,51643],{},[55,51644,16959],{},[82,51646,16959],{"encoding":84},[28,51648,51650],{"className":51649,"ariaHidden":71},[89],[28,51651,51653,51656],{"className":51652},[93],[28,51654],{"className":51655,"style":245},[97],[28,51657,16959],{"className":51658,"style":9403},[102,106]," is even: then define ",[28,51661,51663,51685],{"className":51662},[35],[28,51664,51666],{"className":51665},[39],[41,51667,51668],{"xmlns":43},[46,51669,51670,51682],{},[49,51671,51672,51674,51676,51678,51680],{},[55,51673,24201],{},[62,51675,845],{},[55,51677,16959],{},[55,51679,1129],{"mathvariant":1088},[650,51681,848],{},[82,51683,51684],{"encoding":84},"d=r\u002F2",[28,51686,51688,51706],{"className":51687,"ariaHidden":71},[89],[28,51689,51691,51694,51697,51700,51703],{"className":51690},[93],[28,51692],{"className":51693,"style":9597},[97],[28,51695,24201],{"className":51696},[102,106],[28,51698],{"className":51699,"style":913},[173],[28,51701,845],{"className":51702},[917],[28,51704],{"className":51705,"style":913},[173],[28,51707,51709,51712,51715],{"className":51708},[93],[28,51710],{"className":51711,"style":98},[97],[28,51713,16959],{"className":51714,"style":9403},[102,106],[28,51716,1614],{"className":51717},[102],". Notice that when we project the Hamiltonian into the Krylov space above, it is indistinguishable from the Krylov space",[28,51720,51722],{"className":51721},[31],[28,51723,51725,51903],{"className":51724},[35],[28,51726,51728],{"className":51727},[39],[41,51729,51730],{"xmlns":43,"display":44},[46,51731,51732,51900],{},[49,51733,51734,51740,51742,51744,51746,51748,51750,51752,51754,51756,51758,51898],{},[52,51735,51736,51738],{},[55,51737,28350],{"mathvariant":11377},[55,51739,29018],{},[62,51741,65],{"stretchy":64},[55,51743,2263],{},[62,51745,72],{"separator":71},[55,51747,296],{"mathvariant":1088},[55,51749,394],{},[62,51751,302],{"stretchy":64},[62,51753,78],{"stretchy":64},[62,51755,845],{},[2253,51757,28],{},[49,51759,51760,51762,51784,51786,51788,51790,51792,51818,51820,51822,51824,51826,51828,51830,51858,51860,51862,51864,51866,51890,51892,51894,51896],{},[62,51761,11385],{"fence":71},[2776,51763,51764,51766],{},[55,51765,60],{},[49,51767,51768,51770,51772,51774,51776,51778,51780,51782],{},[55,51769,3256],{},[2253,51771,51285],{},[55,51773,24201],{},[2253,51775,51285],{},[55,51777,2263],{},[2253,51779,51285],{},[55,51781,24201],{},[55,51783,3971],{},[55,51785,296],{"mathvariant":1088},[55,51787,394],{},[62,51789,302],{"stretchy":64},[62,51791,72],{"separator":71},[2776,51793,51794,51796],{},[55,51795,60],{},[49,51797,51798,51800,51802,51804,51806,51808,51810,51812,51814,51816],{},[55,51799,3256],{},[62,51801,65],{"stretchy":64},[55,51803,24201],{},[62,51805,1213],{},[650,51807,3200],{},[62,51809,78],{"stretchy":64},[55,51811,2263],{},[2253,51813,51285],{},[55,51815,24201],{},[55,51817,3971],{},[55,51819,296],{"mathvariant":1088},[55,51821,394],{},[62,51823,302],{"stretchy":64},[62,51825,72],{"separator":71},[62,51827,51300],{},[62,51829,72],{"separator":71},[2776,51831,51832,51834],{},[55,51833,60],{},[49,51835,51836,51838,51840,51842,51844,51846,51848,51850,51852,51854,51856],{},[62,51837,1213],{},[55,51839,3256],{},[62,51841,65],{"stretchy":64},[55,51843,24201],{},[62,51845,1213],{},[650,51847,3200],{},[62,51849,78],{"stretchy":64},[55,51851,2263],{},[2253,51853,51285],{},[55,51855,24201],{},[55,51857,3971],{},[55,51859,296],{"mathvariant":1088},[55,51861,394],{},[62,51863,302],{"stretchy":64},[62,51865,72],{"separator":71},[2776,51867,51868,51870],{},[55,51869,60],{},[49,51871,51872,51874,51876,51878,51880,51882,51884,51886,51888],{},[62,51873,1213],{},[55,51875,3256],{},[2253,51877,51285],{},[55,51879,24201],{},[2253,51881,51285],{},[55,51883,2263],{},[2253,51885,51285],{},[55,51887,24201],{},[55,51889,3971],{},[55,51891,296],{"mathvariant":1088},[55,51893,394],{},[62,51895,302],{"stretchy":64},[62,51897,11404],{"fence":71},[62,51899,72],{"separator":71},[82,51901,51902],{"encoding":84},"\\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\\},",[28,51904,51906,51982],{"className":51905,"ariaHidden":71},[89],[28,51907,51909,51912,51952,51955,51958,51961,51964,51967,51970,51973,51976,51979],{"className":51908},[93],[28,51910],{"className":51911,"style":98},[97],[28,51913,51915,51918],{"className":51914},[102],[28,51916,28350],{"className":51917,"style":28371},[102,11469],[28,51919,51921],{"className":51920},[111],[28,51922,51924,51944],{"className":51923},[115,116],[28,51925,51927,51941],{"className":51926},[120],[28,51928,51930],{"className":51929,"style":1364},[124],[28,51931,51932,51935],{"style":28953},[28,51933],{"className":51934,"style":133},[132],[28,51936,51938],{"className":51937},[137,138,139,140],[28,51939,29018],{"className":51940,"style":29050},[102,106,140],[28,51942,148],{"className":51943},[147],[28,51945,51947],{"className":51946},[120],[28,51948,51950],{"className":51949,"style":155},[124],[28,51951],{},[28,51953,65],{"className":51954},[161],[28,51956,2263],{"className":51957,"style":4040},[102,106],[28,51959,72],{"className":51960},[169],[28,51962],{"className":51963,"style":174},[173],[28,51965,296],{"className":51966},[102],[28,51968,394],{"className":51969,"style":321},[102,106],[28,51971,51404],{"className":51972},[182],[28,51974],{"className":51975,"style":913},[173],[28,51977,845],{"className":51978},[917],[28,51980],{"className":51981,"style":913},[173],[28,51983,51985,51988,51994,51997,52309,52312],{"className":51984},[93],[28,51986],{"className":51987,"style":39185},[97],[28,51989,51991],{"className":51990},[102,16058],[28,51992,28],{"className":51993},[102],[28,51995],{"className":51996,"style":174},[173],[28,51998,52000,52006,52059,52062,52065,52068,52071,52074,52134,52137,52140,52143,52146,52149,52152,52155,52158,52161,52223,52226,52229,52232,52235,52238,52294,52297,52300,52303],{"className":51999},[3398],[28,52001,52003],{"className":52002,"style":3403},[161,3402],[28,52004,11385],{"className":52005},[3407,35567],[28,52007,52009,52012],{"className":52008},[102],[28,52010,60],{"className":52011},[102,106],[28,52013,52015],{"className":52014},[111],[28,52016,52018],{"className":52017},[115],[28,52019,52021],{"className":52020},[120],[28,52022,52024],{"className":52023,"style":30761},[124],[28,52025,52026,52029],{"style":2839},[28,52027],{"className":52028,"style":133},[132],[28,52030,52032],{"className":52031},[137,138,139,140],[28,52033,52035,52038,52041,52044,52047,52050,52053,52056],{"className":52034},[102,140],[28,52036,3256],{"className":52037},[102,106,140],[28,52039],{"className":52040,"style":51495},[173,140],[28,52042,24201],{"className":52043},[102,106,140],[28,52045],{"className":52046,"style":51495},[173,140],[28,52048,2263],{"className":52049,"style":4040},[102,106,140],[28,52051],{"className":52052,"style":51495},[173,140],[28,52054,24201],{"className":52055},[102,106,140],[28,52057,3971],{"className":52058},[102,106,140],[28,52060,296],{"className":52061},[102],[28,52063,394],{"className":52064,"style":321},[102,106],[28,52066,302],{"className":52067},[182],[28,52069,72],{"className":52070},[169],[28,52072],{"className":52073,"style":174},[173],[28,52075,52077,52080],{"className":52076},[102],[28,52078,60],{"className":52079},[102,106],[28,52081,52083],{"className":52082},[111],[28,52084,52086],{"className":52085},[115],[28,52087,52089],{"className":52088},[120],[28,52090,52093],{"className":52091,"style":52092},[124],"height:0.938em;",[28,52094,52095,52098],{"style":2839},[28,52096],{"className":52097,"style":133},[132],[28,52099,52101],{"className":52100},[137,138,139,140],[28,52102,52104,52107,52110,52113,52116,52119,52122,52125,52128,52131],{"className":52103},[102,140],[28,52105,3256],{"className":52106},[102,106,140],[28,52108,65],{"className":52109},[161,140],[28,52111,24201],{"className":52112},[102,106,140],[28,52114,1213],{"className":52115},[1261,140],[28,52117,3200],{"className":52118},[102,140],[28,52120,78],{"className":52121},[182,140],[28,52123,2263],{"className":52124,"style":4040},[102,106,140],[28,52126],{"className":52127,"style":51495},[173,140],[28,52129,24201],{"className":52130},[102,106,140],[28,52132,3971],{"className":52133},[102,106,140],[28,52135,296],{"className":52136},[102],[28,52138,394],{"className":52139,"style":321},[102,106],[28,52141,302],{"className":52142},[182],[28,52144,72],{"className":52145},[169],[28,52147],{"className":52148,"style":174},[173],[28,52150,51300],{"className":52151},[3398],[28,52153],{"className":52154,"style":174},[173],[28,52156,72],{"className":52157},[169],[28,52159],{"className":52160,"style":174},[173],[28,52162,52164,52167],{"className":52163},[102],[28,52165,60],{"className":52166},[102,106],[28,52168,52170],{"className":52169},[111],[28,52171,52173],{"className":52172},[115],[28,52174,52176],{"className":52175},[120],[28,52177,52179],{"className":52178,"style":52092},[124],[28,52180,52181,52184],{"style":2839},[28,52182],{"className":52183,"style":133},[132],[28,52185,52187],{"className":52186},[137,138,139,140],[28,52188,52190,52193,52196,52199,52202,52205,52208,52211,52214,52217,52220],{"className":52189},[102,140],[28,52191,1213],{"className":52192},[102,140],[28,52194,3256],{"className":52195},[102,106,140],[28,52197,65],{"className":52198},[161,140],[28,52200,24201],{"className":52201},[102,106,140],[28,52203,1213],{"className":52204},[1261,140],[28,52206,3200],{"className":52207},[102,140],[28,52209,78],{"className":52210},[182,140],[28,52212,2263],{"className":52213,"style":4040},[102,106,140],[28,52215],{"className":52216,"style":51495},[173,140],[28,52218,24201],{"className":52219},[102,106,140],[28,52221,3971],{"className":52222},[102,106,140],[28,52224,296],{"className":52225},[102],[28,52227,394],{"className":52228,"style":321},[102,106],[28,52230,302],{"className":52231},[182],[28,52233,72],{"className":52234},[169],[28,52236],{"className":52237,"style":174},[173],[28,52239,52241,52244],{"className":52240},[102],[28,52242,60],{"className":52243},[102,106],[28,52245,52247],{"className":52246},[111],[28,52248,52250],{"className":52249},[115],[28,52251,52253],{"className":52252},[120],[28,52254,52256],{"className":52255,"style":30761},[124],[28,52257,52258,52261],{"style":2839},[28,52259],{"className":52260,"style":133},[132],[28,52262,52264],{"className":52263},[137,138,139,140],[28,52265,52267,52270,52273,52276,52279,52282,52285,52288,52291],{"className":52266},[102,140],[28,52268,1213],{"className":52269},[102,140],[28,52271,3256],{"className":52272},[102,106,140],[28,52274],{"className":52275,"style":51495},[173,140],[28,52277,24201],{"className":52278},[102,106,140],[28,52280],{"className":52281,"style":51495},[173,140],[28,52283,2263],{"className":52284,"style":4040},[102,106,140],[28,52286],{"className":52287,"style":51495},[173,140],[28,52289,24201],{"className":52290},[102,106,140],[28,52292,3971],{"className":52293},[102,106,140],[28,52295,296],{"className":52296},[102],[28,52298,394],{"className":52299,"style":321},[102,106],[28,52301,302],{"className":52302},[182],[28,52304,52306],{"className":52305,"style":3403},[182,3402],[28,52307,11404],{"className":52308},[3407,35567],[28,52310],{"className":52311,"style":174},[173],[28,52313,72],{"className":52314},[169],[12,52316,52317,52318,52346],{},"that is, where all the time-evolutions are shifted backward by ",[28,52319,52321,52334],{"className":52320},[35],[28,52322,52324],{"className":52323},[39],[41,52325,52326],{"xmlns":43},[46,52327,52328,52332],{},[49,52329,52330],{},[55,52331,24201],{},[82,52333,24201],{"encoding":84},[28,52335,52337],{"className":52336,"ariaHidden":71},[89],[28,52338,52340,52343],{"className":52339},[93],[28,52341],{"className":52342,"style":9597},[97],[28,52344,24201],{"className":52345},[102,106]," timesteps.\nThe reason it is indistinguishable is because the matrix elements",[28,52348,52350],{"className":52349},[31],[28,52351,52353,52485],{"className":52352},[35],[28,52354,52356],{"className":52355},[39],[41,52357,52358],{"xmlns":43,"display":44},[46,52359,52360,52482],{},[49,52361,52362,52378,52380,52382,52384,52386,52408,52410,52434,52436,52438,52440,52442,52444,52446,52448,52450,52476,52478,52480],{},[52,52363,52364,52370],{},[10155,52365,52366,52368],{"accent":71},[55,52367,2263],{},[62,52369,30008],{},[49,52371,52372,52374,52376],{},[55,52373,28200],{},[62,52375,72],{"separator":71},[55,52377,29362],{},[62,52379,845],{},[62,52381,2948],{"stretchy":64},[55,52383,394],{},[55,52385,296],{"mathvariant":1088},[2776,52387,52388,52390],{},[55,52389,60],{},[49,52391,52392,52394,52396,52398,52400,52402,52404,52406],{},[55,52393,3256],{},[2253,52395,51285],{},[55,52397,28200],{},[2253,52399,51285],{},[55,52401,2263],{},[2253,52403,51285],{},[55,52405,24201],{},[55,52407,3971],{},[55,52409,2263],{},[2776,52411,52412,52414],{},[55,52413,60],{},[49,52415,52416,52418,52420,52422,52424,52426,52428,52430,52432],{},[62,52417,1213],{},[55,52419,3256],{},[2253,52421,51285],{},[55,52423,29362],{},[2253,52425,51285],{},[55,52427,2263],{},[2253,52429,51285],{},[55,52431,24201],{},[55,52433,3971],{},[55,52435,296],{"mathvariant":1088},[55,52437,394],{},[62,52439,302],{"stretchy":64},[62,52441,845],{},[62,52443,2948],{"stretchy":64},[55,52445,394],{},[55,52447,296],{"mathvariant":1088},[55,52449,2263],{},[2776,52451,52452,52454],{},[55,52453,60],{},[49,52455,52456,52458,52460,52462,52464,52466,52468,52470,52472,52474],{},[55,52457,3256],{},[62,52459,65],{"stretchy":64},[55,52461,28200],{},[62,52463,1213],{},[55,52465,29362],{},[62,52467,78],{"stretchy":64},[55,52469,2263],{},[2253,52471,51285],{},[55,52473,24201],{},[55,52475,3971],{},[55,52477,296],{"mathvariant":1088},[55,52479,394],{},[62,52481,302],{"stretchy":64},[82,52483,52484],{"encoding":84},"\\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",[28,52486,52488,52580,52725],{"className":52487,"ariaHidden":71},[89],[28,52489,52491,52494,52571,52574,52577],{"className":52490},[93],[28,52492],{"className":52493,"style":32454},[97],[28,52495,52497,52528],{"className":52496},[102],[28,52498,52500],{"className":52499},[102,10255],[28,52501,52503],{"className":52502},[115],[28,52504,52506],{"className":52505},[120],[28,52507,52509,52517],{"className":52508,"style":30021},[124],[28,52510,52511,52514],{"style":10025},[28,52512],{"className":52513,"style":3434},[132],[28,52515,2263],{"className":52516,"style":4040},[102,106],[28,52518,52519,52522],{"style":30044},[28,52520],{"className":52521,"style":3434},[132],[28,52523,52525],{"className":52524,"style":30051},[10282],[28,52526,30008],{"className":52527},[102],[28,52529,52531],{"className":52530},[111],[28,52532,52534,52563],{"className":52533},[115,116],[28,52535,52537,52560],{"className":52536},[120],[28,52538,52540],{"className":52539,"style":10240},[124],[28,52541,52542,52545],{"style":4522},[28,52543],{"className":52544,"style":133},[132],[28,52546,52548],{"className":52547},[137,138,139,140],[28,52549,52551,52554,52557],{"className":52550},[102,140],[28,52552,28200],{"className":52553,"style":32522},[102,106,140],[28,52555,72],{"className":52556},[169,140],[28,52558,29362],{"className":52559,"style":29427},[102,106,140],[28,52561,148],{"className":52562},[147],[28,52564,52566],{"className":52565},[120],[28,52567,52569],{"className":52568,"style":2003},[124],[28,52570],{},[28,52572],{"className":52573,"style":913},[173],[28,52575,845],{"className":52576},[917],[28,52578],{"className":52579,"style":913},[173],[28,52581,52583,52586,52589,52592,52595,52648,52651,52707,52710,52713,52716,52719,52722],{"className":52582},[93],[28,52584],{"className":52585,"style":30733},[97],[28,52587,2948],{"className":52588},[161],[28,52590,394],{"className":52591,"style":321},[102,106],[28,52593,296],{"className":52594},[102],[28,52596,52598,52601],{"className":52597},[102],[28,52599,60],{"className":52600},[102,106],[28,52602,52604],{"className":52603},[111],[28,52605,52607],{"className":52606},[115],[28,52608,52610],{"className":52609},[120],[28,52611,52613],{"className":52612,"style":30761},[124],[28,52614,52615,52618],{"style":2839},[28,52616],{"className":52617,"style":133},[132],[28,52619,52621],{"className":52620},[137,138,139,140],[28,52622,52624,52627,52630,52633,52636,52639,52642,52645],{"className":52623},[102,140],[28,52625,3256],{"className":52626},[102,106,140],[28,52628],{"className":52629,"style":51495},[173,140],[28,52631,28200],{"className":52632,"style":32522},[102,106,140],[28,52634],{"className":52635,"style":51495},[173,140],[28,52637,2263],{"className":52638,"style":4040},[102,106,140],[28,52640],{"className":52641,"style":51495},[173,140],[28,52643,24201],{"className":52644},[102,106,140],[28,52646,3971],{"className":52647},[102,106,140],[28,52649,2263],{"className":52650,"style":4040},[102,106],[28,52652,52654,52657],{"className":52653},[102],[28,52655,60],{"className":52656},[102,106],[28,52658,52660],{"className":52659},[111],[28,52661,52663],{"className":52662},[115],[28,52664,52666],{"className":52665},[120],[28,52667,52669],{"className":52668,"style":30761},[124],[28,52670,52671,52674],{"style":2839},[28,52672],{"className":52673,"style":133},[132],[28,52675,52677],{"className":52676},[137,138,139,140],[28,52678,52680,52683,52686,52689,52692,52695,52698,52701,52704],{"className":52679},[102,140],[28,52681,1213],{"className":52682},[102,140],[28,52684,3256],{"className":52685},[102,106,140],[28,52687],{"className":52688,"style":51495},[173,140],[28,52690,29362],{"className":52691,"style":29427},[102,106,140],[28,52693],{"className":52694,"style":51495},[173,140],[28,52696,2263],{"className":52697,"style":4040},[102,106,140],[28,52699],{"className":52700,"style":51495},[173,140],[28,52702,24201],{"className":52703},[102,106,140],[28,52705,3971],{"className":52706},[102,106,140],[28,52708,296],{"className":52709},[102],[28,52711,394],{"className":52712,"style":321},[102,106],[28,52714,302],{"className":52715},[182],[28,52717],{"className":52718,"style":913},[173],[28,52720,845],{"className":52721},[917],[28,52723],{"className":52724,"style":913},[173],[28,52726,52728,52732,52735,52738,52741,52744,52803,52806,52809],{"className":52727},[93],[28,52729],{"className":52730,"style":52731},[97],"height:1.188em;vertical-align:-0.25em;",[28,52733,2948],{"className":52734},[161],[28,52736,394],{"className":52737,"style":321},[102,106],[28,52739,296],{"className":52740},[102],[28,52742,2263],{"className":52743,"style":4040},[102,106],[28,52745,52747,52750],{"className":52746},[102],[28,52748,60],{"className":52749},[102,106],[28,52751,52753],{"className":52752},[111],[28,52754,52756],{"className":52755},[115],[28,52757,52759],{"className":52758},[120],[28,52760,52762],{"className":52761,"style":52092},[124],[28,52763,52764,52767],{"style":2839},[28,52765],{"className":52766,"style":133},[132],[28,52768,52770],{"className":52769},[137,138,139,140],[28,52771,52773,52776,52779,52782,52785,52788,52791,52794,52797,52800],{"className":52772},[102,140],[28,52774,3256],{"className":52775},[102,106,140],[28,52777,65],{"className":52778},[161,140],[28,52780,28200],{"className":52781,"style":32522},[102,106,140],[28,52783,1213],{"className":52784},[1261,140],[28,52786,29362],{"className":52787,"style":29427},[102,106,140],[28,52789,78],{"className":52790},[182,140],[28,52792,2263],{"className":52793,"style":4040},[102,106,140],[28,52795],{"className":52796,"style":51495},[173,140],[28,52798,24201],{"className":52799},[102,106,140],[28,52801,3971],{"className":52802},[102,106,140],[28,52804,296],{"className":52805},[102],[28,52807,394],{"className":52808,"style":321},[102,106],[28,52810,302],{"className":52811},[182],[12,52813,52814,52815,52843,52844,2106],{},"are invariant under overall shifts of the evolution time, since the time-evolutions commute with the Hamiltonian. For odd ",[28,52816,52818,52831],{"className":52817},[35],[28,52819,52821],{"className":52820},[39],[41,52822,52823],{"xmlns":43},[46,52824,52825,52829],{},[49,52826,52827],{},[55,52828,16959],{},[82,52830,16959],{"encoding":84},[28,52832,52834],{"className":52833,"ariaHidden":71},[89],[28,52835,52837,52840],{"className":52836},[93],[28,52838],{"className":52839,"style":245},[97],[28,52841,16959],{"className":52842,"style":9403},[102,106],", we can use the analysis for ",[28,52845,52847,52864],{"className":52846},[35],[28,52848,52850],{"className":52849},[39],[41,52851,52852],{"xmlns":43},[46,52853,52854,52862],{},[49,52855,52856,52858,52860],{},[55,52857,16959],{},[62,52859,1213],{},[650,52861,3200],{},[82,52863,28475],{"encoding":84},[28,52865,52867,52885],{"className":52866,"ariaHidden":71},[89],[28,52868,52870,52873,52876,52879,52882],{"className":52869},[93],[28,52871],{"className":52872,"style":1250},[97],[28,52874,16959],{"className":52875,"style":9403},[102,106],[28,52877],{"className":52878,"style":1257},[173],[28,52880,1213],{"className":52881},[1261],[28,52883],{"className":52884,"style":1257},[173],[28,52886,52888,52891],{"className":52887},[93],[28,52889],{"className":52890,"style":2914},[97],[28,52892,3200],{"className":52893},[102],[12,52895,52896,52897,50590],{},"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 ",[19,52898,52899],{"href":27044},"[3]",[12,52901,52902,52906,52907,52935,52936,52964],{},[52903,52904,52905],"strong",{},"Claim 1:"," there exists a function ",[28,52908,52910,52923],{"className":52909},[35],[28,52911,52913],{"className":52912},[39],[41,52914,52915],{"xmlns":43},[46,52916,52917,52921],{},[49,52918,52919],{},[55,52920,68],{},[82,52922,68],{"encoding":84},[28,52924,52926],{"className":52925,"ariaHidden":71},[89],[28,52927,52929,52932],{"className":52928},[93],[28,52930],{"className":52931,"style":214},[97],[28,52933,68],{"className":52934,"style":165},[102,106]," such that for energies ",[28,52937,52939,52952],{"className":52938},[35],[28,52940,52942],{"className":52941},[39],[41,52943,52944],{"xmlns":43},[46,52945,52946,52950],{},[49,52947,52948],{},[55,52949,1871],{},[82,52951,1871],{"encoding":84},[28,52953,52955],{"className":52954,"ariaHidden":71},[89],[28,52956,52958,52961],{"className":52957},[93],[28,52959],{"className":52960,"style":1102},[97],[28,52962,1871],{"className":52963,"style":1913},[102,106]," in the spectral range of the Hamiltonian (that is, between the ground state energy and the maximum energy)...",[52966,52967,52968,53077,53384],"ol",{},[27063,52969,52970],{},[28,52971,52973,53001],{"className":52972},[35],[28,52974,52976],{"className":52975},[39],[41,52977,52978],{"xmlns":43},[46,52979,52980,52998],{},[49,52981,52982,52984,52986,52992,52994,52996],{},[55,52983,68],{},[62,52985,65],{"stretchy":64},[52,52987,52988,52990],{},[55,52989,1871],{},[650,52991,652],{},[62,52993,78],{"stretchy":64},[62,52995,845],{},[650,52997,3200],{},[82,52999,53000],{"encoding":84},"f(E_0)=1",[28,53002,53004,53068],{"className":53003,"ariaHidden":71},[89],[28,53005,53007,53010,53013,53016,53056,53059,53062,53065],{"className":53006},[93],[28,53008],{"className":53009,"style":98},[97],[28,53011,68],{"className":53012,"style":165},[102,106],[28,53014,65],{"className":53015},[161],[28,53017,53019,53022],{"className":53018},[102],[28,53020,1871],{"className":53021,"style":1913},[102,106],[28,53023,53025],{"className":53024},[111],[28,53026,53028,53048],{"className":53027},[115,116],[28,53029,53031,53045],{"className":53030},[120],[28,53032,53034],{"className":53033,"style":683},[124],[28,53035,53036,53039],{"style":1928},[28,53037],{"className":53038,"style":133},[132],[28,53040,53042],{"className":53041},[137,138,139,140],[28,53043,652],{"className":53044},[102,140],[28,53046,148],{"className":53047},[147],[28,53049,53051],{"className":53050},[120],[28,53052,53054],{"className":53053,"style":155},[124],[28,53055],{},[28,53057,78],{"className":53058},[182],[28,53060],{"className":53061,"style":913},[173],[28,53063,845],{"className":53064},[917],[28,53066],{"className":53067,"style":913},[173],[28,53069,53071,53074],{"className":53070},[93],[28,53072],{"className":53073,"style":2914},[97],[28,53075,3200],{"className":53076},[102],[27063,53078,53079,53238,53239,53267,53268,53313,53314,53383],{},[28,53080,53082,53131],{"className":53081},[35],[28,53083,53085],{"className":53084},[39],[41,53086,53087],{"xmlns":43},[46,53088,53089,53128],{},[49,53090,53091,53093,53095,53097,53099,53101,53103,53105,53107],{},[55,53092,296],{"mathvariant":1088},[55,53094,68],{},[62,53096,65],{"stretchy":64},[55,53098,1871],{},[62,53100,78],{"stretchy":64},[55,53102,296],{"mathvariant":1088},[62,53104,13276],{},[650,53106,848],{},[2776,53108,53109,53122],{},[49,53110,53111,53113,53115,53117,53120],{},[62,53112,65],{"fence":71},[650,53114,3200],{},[62,53116,2606],{},[55,53118,53119],{},"δ",[62,53121,78],{"fence":71},[49,53123,53124,53126],{},[62,53125,1213],{},[55,53127,24201],{},[82,53129,53130],{"encoding":84},"|f(E)|\\le2\\left(1 + \\delta\\right)^{-d}",[28,53132,53134,53167],{"className":53133,"ariaHidden":71},[89],[28,53135,53137,53140,53143,53146,53149,53152,53155,53158,53161,53164],{"className":53136},[93],[28,53138],{"className":53139,"style":98},[97],[28,53141,296],{"className":53142},[102],[28,53144,68],{"className":53145,"style":165},[102,106],[28,53147,65],{"className":53148},[161],[28,53150,1871],{"className":53151,"style":1913},[102,106],[28,53153,78],{"className":53154},[182],[28,53156,296],{"className":53157},[102],[28,53159],{"className":53160,"style":913},[173],[28,53162,13276],{"className":53163},[917],[28,53165],{"className":53166,"style":913},[173],[28,53168,53170,53174,53177,53180],{"className":53169},[93],[28,53171],{"className":53172,"style":53173},[97],"height:1.239em;vertical-align:-0.25em;",[28,53175,848],{"className":53176},[102],[28,53178],{"className":53179,"style":174},[173],[28,53181,53183,53208],{"className":53182},[3398],[28,53184,53186,53189,53192,53195,53198,53201,53205],{"className":53185},[3398],[28,53187,65],{"className":53188,"style":3403},[161,3402],[28,53190,3200],{"className":53191},[102],[28,53193],{"className":53194,"style":1257},[173],[28,53196,2606],{"className":53197},[1261],[28,53199],{"className":53200,"style":1257},[173],[28,53202,53119],{"className":53203,"style":53204},[102,106],"margin-right:0.0379em;",[28,53206,78],{"className":53207,"style":3403},[182,3402],[28,53209,53211],{"className":53210},[111],[28,53212,53214],{"className":53213},[115],[28,53215,53217],{"className":53216},[120],[28,53218,53221],{"className":53219,"style":53220},[124],"height:0.989em;",[28,53222,53223,53226],{"style":33538},[28,53224],{"className":53225,"style":133},[132],[28,53227,53229],{"className":53228},[137,138,139,140],[28,53230,53232,53235],{"className":53231},[102,140],[28,53233,1213],{"className":53234},[102,140],[28,53236,24201],{"className":53237},[102,106,140]," for all values of ",[28,53240,53242,53255],{"className":53241},[35],[28,53243,53245],{"className":53244},[39],[41,53246,53247],{"xmlns":43},[46,53248,53249,53253],{},[49,53250,53251],{},[55,53252,1871],{},[82,53254,1871],{"encoding":84},[28,53256,53258],{"className":53257,"ariaHidden":71},[89],[28,53259,53261,53264],{"className":53260},[93],[28,53262],{"className":53263,"style":1102},[97],[28,53265,1871],{"className":53266,"style":1913},[102,106]," that lie ",[28,53269,53271,53288],{"className":53270},[35],[28,53272,53274],{"className":53273},[39],[41,53275,53276],{"xmlns":43},[46,53277,53278,53285],{},[49,53279,53280,53283],{},[62,53281,53282],{},"≥",[55,53284,53119],{},[82,53286,53287],{"encoding":84},"\\ge\\delta",[28,53289,53291,53304],{"className":53290,"ariaHidden":71},[89],[28,53292,53294,53298,53301],{"className":53293},[93],[28,53295],{"className":53296,"style":53297},[97],"height:0.7719em;vertical-align:-0.136em;",[28,53299,53282],{"className":53300},[917],[28,53302],{"className":53303,"style":913},[173],[28,53305,53307,53310],{"className":53306},[93],[28,53308],{"className":53309,"style":9597},[97],[28,53311,53119],{"className":53312,"style":53204},[102,106]," away from ",[28,53315,53317,53334],{"className":53316},[35],[28,53318,53320],{"className":53319},[39],[41,53321,53322],{"xmlns":43},[46,53323,53324,53332],{},[49,53325,53326],{},[52,53327,53328,53330],{},[55,53329,1871],{},[650,53331,652],{},[82,53333,39888],{"encoding":84},[28,53335,53337],{"className":53336,"ariaHidden":71},[89],[28,53338,53340,53343],{"className":53339},[93],[28,53341],{"className":53342,"style":585},[97],[28,53344,53346,53349],{"className":53345},[102],[28,53347,1871],{"className":53348,"style":1913},[102,106],[28,53350,53352],{"className":53351},[111],[28,53353,53355,53375],{"className":53354},[115,116],[28,53356,53358,53372],{"className":53357},[120],[28,53359,53361],{"className":53360,"style":683},[124],[28,53362,53363,53366],{"style":1928},[28,53364],{"className":53365,"style":133},[132],[28,53367,53369],{"className":53368},[137,138,139,140],[28,53370,652],{"className":53371},[102,140],[28,53373,148],{"className":53374},[147],[28,53376,53378],{"className":53377},[120],[28,53379,53381],{"className":53380,"style":155},[124],[28,53382],{},", that is, it is exponentially suppressed",[27063,53385,53386,53430,53431,53518,53519],{},[28,53387,53389,53409],{"className":53388},[35],[28,53390,53392],{"className":53391},[39],[41,53393,53394],{"xmlns":43},[46,53395,53396,53406],{},[49,53397,53398,53400,53402,53404],{},[55,53399,68],{},[62,53401,65],{"stretchy":64},[55,53403,1871],{},[62,53405,78],{"stretchy":64},[82,53407,53408],{"encoding":84},"f(E)",[28,53410,53412],{"className":53411,"ariaHidden":71},[89],[28,53413,53415,53418,53421,53424,53427],{"className":53414},[93],[28,53416],{"className":53417,"style":98},[97],[28,53419,68],{"className":53420,"style":165},[102,106],[28,53422,65],{"className":53423},[161],[28,53425,1871],{"className":53426,"style":1913},[102,106],[28,53428,78],{"className":53429},[182]," is a linear combination of ",[28,53432,53434,53464],{"className":53433},[35],[28,53435,53437],{"className":53436},[39],[41,53438,53439],{"xmlns":43},[46,53440,53441,53461],{},[49,53442,53443],{},[2776,53444,53445,53447],{},[55,53446,60],{},[49,53448,53449,53451,53453,53455,53457,53459],{},[55,53450,3256],{},[55,53452,28200],{},[55,53454,1871],{},[2253,53456,51285],{},[55,53458,24201],{},[55,53460,3971],{},[82,53462,53463],{"encoding":84},"e^{ijE\\,dt}",[28,53465,53467],{"className":53466,"ariaHidden":71},[89],[28,53468,53470,53473],{"className":53469},[93],[28,53471],{"className":53472,"style":29397},[97],[28,53474,53476,53479],{"className":53475},[102],[28,53477,60],{"className":53478},[102,106],[28,53480,53482],{"className":53481},[111],[28,53483,53485],{"className":53484},[115],[28,53486,53488],{"className":53487},[120],[28,53489,53491],{"className":53490,"style":29397},[124],[28,53492,53493,53496],{"style":1726},[28,53494],{"className":53495,"style":133},[132],[28,53497,53499],{"className":53498},[137,138,139,140],[28,53500,53502,53506,53509,53512,53515],{"className":53501},[102,140],[28,53503,53505],{"className":53504,"style":32522},[102,106,140],"ij",[28,53507,1871],{"className":53508,"style":1913},[102,106,140],[28,53510],{"className":53511,"style":51495},[173,140],[28,53513,24201],{"className":53514},[102,106,140],[28,53516,3971],{"className":53517},[102,106,140]," for ",[28,53520,53522,53572],{"className":53521},[35],[28,53523,53525],{"className":53524},[39],[41,53526,53527],{"xmlns":43},[46,53528,53529,53569],{},[49,53530,53531,53533,53535,53537,53539,53541,53543,53545,53547,53549,53551,53553,53555,53557,53559,53561,53563,53565,53567],{},[55,53532,28200],{},[62,53534,845],{},[62,53536,1213],{},[55,53538,24201],{},[62,53540,72],{"separator":71},[62,53542,1213],{},[55,53544,24201],{},[62,53546,2606],{},[650,53548,3200],{},[62,53550,72],{"separator":71},[55,53552,2106],{"mathvariant":1088},[55,53554,2106],{"mathvariant":1088},[55,53556,2106],{"mathvariant":1088},[62,53558,72],{"separator":71},[55,53560,24201],{},[62,53562,1213],{},[650,53564,3200],{},[62,53566,72],{"separator":71},[55,53568,24201],{},[82,53570,53571],{"encoding":84},"j=-d,-d+1,...,d-1,d",[28,53573,53575,53593,53626,53662],{"className":53574,"ariaHidden":71},[89],[28,53576,53578,53581,53584,53587,53590],{"className":53577},[93],[28,53579],{"className":53580,"style":9774},[97],[28,53582,28200],{"className":53583,"style":32522},[102,106],[28,53585],{"className":53586,"style":913},[173],[28,53588,845],{"className":53589},[917],[28,53591],{"className":53592,"style":913},[173],[28,53594,53596,53599,53602,53605,53608,53611,53614,53617,53620,53623],{"className":53595},[93],[28,53597],{"className":53598,"style":214},[97],[28,53600,1213],{"className":53601},[102],[28,53603,24201],{"className":53604},[102,106],[28,53606,72],{"className":53607},[169],[28,53609],{"className":53610,"style":174},[173],[28,53612,1213],{"className":53613},[102],[28,53615,24201],{"className":53616},[102,106],[28,53618],{"className":53619,"style":1257},[173],[28,53621,2606],{"className":53622},[1261],[28,53624],{"className":53625,"style":1257},[173],[28,53627,53629,53632,53635,53638,53641,53644,53647,53650,53653,53656,53659],{"className":53628},[93],[28,53630],{"className":53631,"style":214},[97],[28,53633,3200],{"className":53634},[102],[28,53636,72],{"className":53637},[169],[28,53639],{"className":53640,"style":174},[173],[28,53642,28782],{"className":53643},[102],[28,53645,72],{"className":53646},[169],[28,53648],{"className":53649,"style":174},[173],[28,53651,24201],{"className":53652},[102,106],[28,53654],{"className":53655,"style":1257},[173],[28,53657,1213],{"className":53658},[1261],[28,53660],{"className":53661,"style":1257},[173],[28,53663,53665,53668,53671,53674,53677],{"className":53664},[93],[28,53666],{"className":53667,"style":214},[97],[28,53669,3200],{"className":53670},[102],[28,53672,72],{"className":53673},[169],[28,53675],{"className":53676,"style":174},[173],[28,53678,24201],{"className":53679},[102,106],[12,53681,53682,53683,53742,53743,53782],{},"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 ",[28,53684,53686,53712],{"className":53685},[35],[28,53687,53689],{"className":53688},[39],[41,53690,53691],{"xmlns":43},[46,53692,53693,53709],{},[49,53694,53695,53697,53699,53701,53703,53705,53707],{},[55,53696,68],{},[62,53698,65],{"stretchy":64},[55,53700,2263],{},[62,53702,78],{"stretchy":64},[55,53704,296],{"mathvariant":1088},[55,53706,394],{},[62,53708,302],{"stretchy":64},[82,53710,53711],{"encoding":84},"f(H)|\\psi\\rangle",[28,53713,53715],{"className":53714,"ariaHidden":71},[89],[28,53716,53718,53721,53724,53727,53730,53733,53736,53739],{"className":53717},[93],[28,53719],{"className":53720,"style":98},[97],[28,53722,68],{"className":53723,"style":165},[102,106],[28,53725,65],{"className":53726},[161],[28,53728,2263],{"className":53729,"style":4040},[102,106],[28,53731,78],{"className":53732},[182],[28,53734,296],{"className":53735},[102],[28,53737,394],{"className":53738,"style":321},[102,106],[28,53740,302],{"className":53741},[182],". This is our low-energy state. To see why, write ",[28,53744,53746,53764],{"className":53745},[35],[28,53747,53749],{"className":53748},[39],[41,53750,53751],{"xmlns":43},[46,53752,53753,53761],{},[49,53754,53755,53757,53759],{},[55,53756,296],{"mathvariant":1088},[55,53758,394],{},[62,53760,302],{"stretchy":64},[82,53762,53763],{"encoding":84},"|\\psi\\rangle",[28,53765,53767],{"className":53766,"ariaHidden":71},[89],[28,53768,53770,53773,53776,53779],{"className":53769},[93],[28,53771],{"className":53772,"style":98},[97],[28,53774,296],{"className":53775},[102],[28,53777,394],{"className":53778,"style":321},[102,106],[28,53780,302],{"className":53781},[182]," in the energy eigenbasis:",[28,53784,53786],{"className":53785},[31],[28,53787,53789,53843],{"className":53788},[35],[28,53790,53792],{"className":53791},[39],[41,53793,53794],{"xmlns":43,"display":44},[46,53795,53796,53840],{},[49,53797,53798,53800,53802,53804,53806,53821,53828,53830,53836,53838],{},[55,53799,296],{"mathvariant":1088},[55,53801,394],{},[62,53803,302],{"stretchy":64},[62,53805,845],{},[53807,53808,53809,53811,53819],"munderover",{},[62,53810,33223],{},[49,53812,53813,53815,53817],{},[55,53814,29362],{},[62,53816,845],{},[650,53818,652],{},[55,53820,33375],{},[52,53822,53823,53826],{},[55,53824,53825],{},"γ",[55,53827,29362],{},[55,53829,296],{"mathvariant":1088},[52,53831,53832,53834],{},[55,53833,1871],{},[55,53835,29362],{},[62,53837,302],{"stretchy":64},[62,53839,72],{"separator":71},[82,53841,53842],{"encoding":84},"|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_k|E_k\\rangle,",[28,53844,53846,53870],{"className":53845,"ariaHidden":71},[89],[28,53847,53849,53852,53855,53858,53861,53864,53867],{"className":53848},[93],[28,53850],{"className":53851,"style":98},[97],[28,53853,296],{"className":53854},[102],[28,53856,394],{"className":53857,"style":321},[102,106],[28,53859,302],{"className":53860},[182],[28,53862],{"className":53863,"style":913},[173],[28,53865,845],{"className":53866},[917],[28,53868],{"className":53869,"style":913},[173],[28,53871,53873,53877,53952,53955,53997,54000,54040,54043],{"className":53872},[93],[28,53874],{"className":53875,"style":53876},[97],"height:3.1304em;vertical-align:-1.3021em;",[28,53878,53881],{"className":53879},[4241,53880],"op-limits",[28,53882,53884,53943],{"className":53883},[115,116],[28,53885,53887,53940],{"className":53886},[120],[28,53888,53891,53913,53925],{"className":53889,"style":53890},[124],"height:1.8283em;",[28,53892,53894,53898],{"style":53893},"top:-1.8479em;margin-left:0em;",[28,53895],{"className":53896,"style":53897},[132],"height:3.05em;",[28,53899,53901],{"className":53900},[137,138,139,140],[28,53902,53904,53907,53910],{"className":53903},[102,140],[28,53905,29362],{"className":53906,"style":29427},[102,106,140],[28,53908,845],{"className":53909},[917,140],[28,53911,652],{"className":53912},[102,140],[28,53914,53916,53919],{"style":53915},"top:-3.05em;",[28,53917],{"className":53918,"style":53897},[132],[28,53920,53921],{},[28,53922,33223],{"className":53923},[4241,33269,53924],"large-op",[28,53926,53928,53931],{"style":53927},"top:-4.3em;margin-left:0em;",[28,53929],{"className":53930,"style":53897},[132],[28,53932,53934],{"className":53933},[137,138,139,140],[28,53935,53937],{"className":53936},[102,140],[28,53938,33375],{"className":53939,"style":29050},[102,106,140],[28,53941,148],{"className":53942},[147],[28,53944,53946],{"className":53945},[120],[28,53947,53950],{"className":53948,"style":53949},[124],"height:1.3021em;",[28,53951],{},[28,53953],{"className":53954,"style":174},[173],[28,53956,53958,53962],{"className":53957},[102],[28,53959,53825],{"className":53960,"style":53961},[102,106],"margin-right:0.0556em;",[28,53963,53965],{"className":53964},[111],[28,53966,53968,53989],{"className":53967},[115,116],[28,53969,53971,53986],{"className":53970},[120],[28,53972,53974],{"className":53973,"style":10240},[124],[28,53975,53977,53980],{"style":53976},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[28,53978],{"className":53979,"style":133},[132],[28,53981,53983],{"className":53982},[137,138,139,140],[28,53984,29362],{"className":53985,"style":29427},[102,106,140],[28,53987,148],{"className":53988},[147],[28,53990,53992],{"className":53991},[120],[28,53993,53995],{"className":53994,"style":155},[124],[28,53996],{},[28,53998,296],{"className":53999},[102],[28,54001,54003,54006],{"className":54002},[102],[28,54004,1871],{"className":54005,"style":1913},[102,106],[28,54007,54009],{"className":54008},[111],[28,54010,54012,54032],{"className":54011},[115,116],[28,54013,54015,54029],{"className":54014},[120],[28,54016,54018],{"className":54017,"style":10240},[124],[28,54019,54020,54023],{"style":1928},[28,54021],{"className":54022,"style":133},[132],[28,54024,54026],{"className":54025},[137,138,139,140],[28,54027,29362],{"className":54028,"style":29427},[102,106,140],[28,54030,148],{"className":54031},[147],[28,54033,54035],{"className":54034},[120],[28,54036,54038],{"className":54037,"style":155},[124],[28,54039],{},[28,54041,302],{"className":54042},[182],[28,54044,72],{"className":54045},[169],[12,54047,188,54048,54128,54129,54200,54201,54239,54240,54298],{},[28,54049,54051,54073],{"className":54050},[35],[28,54052,54054],{"className":54053},[39],[41,54055,54056],{"xmlns":43},[46,54057,54058,54070],{},[49,54059,54060,54062,54068],{},[55,54061,296],{"mathvariant":1088},[52,54063,54064,54066],{},[55,54065,1871],{},[55,54067,29362],{},[62,54069,302],{"stretchy":64},[82,54071,54072],{"encoding":84},"|E_k\\rangle",[28,54074,54076],{"className":54075,"ariaHidden":71},[89],[28,54077,54079,54082,54085,54125],{"className":54078},[93],[28,54080],{"className":54081,"style":98},[97],[28,54083,296],{"className":54084},[102],[28,54086,54088,54091],{"className":54087},[102],[28,54089,1871],{"className":54090,"style":1913},[102,106],[28,54092,54094],{"className":54093},[111],[28,54095,54097,54117],{"className":54096},[115,116],[28,54098,54100,54114],{"className":54099},[120],[28,54101,54103],{"className":54102,"style":10240},[124],[28,54104,54105,54108],{"style":1928},[28,54106],{"className":54107,"style":133},[132],[28,54109,54111],{"className":54110},[137,138,139,140],[28,54112,29362],{"className":54113,"style":29427},[102,106,140],[28,54115,148],{"className":54116},[147],[28,54118,54120],{"className":54119},[120],[28,54121,54123],{"className":54122,"style":155},[124],[28,54124],{},[28,54126,302],{"className":54127},[182]," is the kth energy eigenstate and ",[28,54130,54132,54150],{"className":54131},[35],[28,54133,54135],{"className":54134},[39],[41,54136,54137],{"xmlns":43},[46,54138,54139,54147],{},[49,54140,54141],{},[52,54142,54143,54145],{},[55,54144,53825],{},[55,54146,29362],{},[82,54148,54149],{"encoding":84},"\\gamma_k",[28,54151,54153],{"className":54152,"ariaHidden":71},[89],[28,54154,54156,54160],{"className":54155},[93],[28,54157],{"className":54158,"style":54159},[97],"height:0.625em;vertical-align:-0.1944em;",[28,54161,54163,54166],{"className":54162},[102],[28,54164,53825],{"className":54165,"style":53961},[102,106],[28,54167,54169],{"className":54168},[111],[28,54170,54172,54192],{"className":54171},[115,116],[28,54173,54175,54189],{"className":54174},[120],[28,54176,54178],{"className":54177,"style":10240},[124],[28,54179,54180,54183],{"style":53976},[28,54181],{"className":54182,"style":133},[132],[28,54184,54186],{"className":54185},[137,138,139,140],[28,54187,29362],{"className":54188,"style":29427},[102,106,140],[28,54190,148],{"className":54191},[147],[28,54193,54195],{"className":54194},[120],[28,54196,54198],{"className":54197,"style":155},[124],[28,54199],{}," is its amplitude in the initial state ",[28,54202,54204,54221],{"className":54203},[35],[28,54205,54207],{"className":54206},[39],[41,54208,54209],{"xmlns":43},[46,54210,54211,54219],{},[49,54212,54213,54215,54217],{},[55,54214,296],{"mathvariant":1088},[55,54216,394],{},[62,54218,302],{"stretchy":64},[82,54220,53763],{"encoding":84},[28,54222,54224],{"className":54223,"ariaHidden":71},[89],[28,54225,54227,54230,54233,54236],{"className":54226},[93],[28,54228],{"className":54229,"style":98},[97],[28,54231,296],{"className":54232},[102],[28,54234,394],{"className":54235,"style":321},[102,106],[28,54237,302],{"className":54238},[182],". Expressed in terms of this, ",[28,54241,54243,54268],{"className":54242},[35],[28,54244,54246],{"className":54245},[39],[41,54247,54248],{"xmlns":43},[46,54249,54250,54266],{},[49,54251,54252,54254,54256,54258,54260,54262,54264],{},[55,54253,68],{},[62,54255,65],{"stretchy":64},[55,54257,2263],{},[62,54259,78],{"stretchy":64},[55,54261,296],{"mathvariant":1088},[55,54263,394],{},[62,54265,302],{"stretchy":64},[82,54267,53711],{"encoding":84},[28,54269,54271],{"className":54270,"ariaHidden":71},[89],[28,54272,54274,54277,54280,54283,54286,54289,54292,54295],{"className":54273},[93],[28,54275],{"className":54276,"style":98},[97],[28,54278,68],{"className":54279,"style":165},[102,106],[28,54281,65],{"className":54282},[161],[28,54284,2263],{"className":54285,"style":4040},[102,106],[28,54287,78],{"className":54288},[182],[28,54290,296],{"className":54291},[102],[28,54293,394],{"className":54294,"style":321},[102,106],[28,54296,302],{"className":54297},[182]," is given by",[28,54300,54302],{"className":54301},[31],[28,54303,54305,54377],{"className":54304},[35],[28,54306,54308],{"className":54307},[39],[41,54309,54310],{"xmlns":43,"display":44},[46,54311,54312,54374],{},[49,54313,54314,54316,54318,54320,54322,54324,54326,54328,54330,54344,54350,54352,54354,54360,54362,54364,54370,54372],{},[55,54315,68],{},[62,54317,65],{"stretchy":64},[55,54319,2263],{},[62,54321,78],{"stretchy":64},[55,54323,296],{"mathvariant":1088},[55,54325,394],{},[62,54327,302],{"stretchy":64},[62,54329,845],{},[53807,54331,54332,54334,54342],{},[62,54333,33223],{},[49,54335,54336,54338,54340],{},[55,54337,29362],{},[62,54339,845],{},[650,54341,652],{},[55,54343,33375],{},[52,54345,54346,54348],{},[55,54347,53825],{},[55,54349,29362],{},[55,54351,68],{},[62,54353,65],{"stretchy":64},[52,54355,54356,54358],{},[55,54357,1871],{},[55,54359,29362],{},[62,54361,78],{"stretchy":64},[55,54363,296],{"mathvariant":1088},[52,54365,54366,54368],{},[55,54367,1871],{},[55,54369,29362],{},[62,54371,302],{"stretchy":64},[62,54373,72],{"separator":71},[82,54375,54376],{"encoding":84},"f(H)|\\psi\\rangle = \\sum_{k=0}^{N}\\gamma_kf(E_k)|E_k\\rangle,",[28,54378,54380,54416],{"className":54379,"ariaHidden":71},[89],[28,54381,54383,54386,54389,54392,54395,54398,54401,54404,54407,54410,54413],{"className":54382},[93],[28,54384],{"className":54385,"style":98},[97],[28,54387,68],{"className":54388,"style":165},[102,106],[28,54390,65],{"className":54391},[161],[28,54393,2263],{"className":54394,"style":4040},[102,106],[28,54396,78],{"className":54397},[182],[28,54399,296],{"className":54400},[102],[28,54402,394],{"className":54403,"style":321},[102,106],[28,54405,302],{"className":54406},[182],[28,54408],{"className":54409,"style":913},[173],[28,54411,845],{"className":54412},[917],[28,54414],{"className":54415,"style":913},[173],[28,54417,54419,54422,54489,54492,54532,54535,54538,54578,54581,54584,54624,54627],{"className":54418},[93],[28,54420],{"className":54421,"style":53876},[97],[28,54423,54425],{"className":54424},[4241,53880],[28,54426,54428,54481],{"className":54427},[115,116],[28,54429,54431,54478],{"className":54430},[120],[28,54432,54434,54454,54464],{"className":54433,"style":53890},[124],[28,54435,54436,54439],{"style":53893},[28,54437],{"className":54438,"style":53897},[132],[28,54440,54442],{"className":54441},[137,138,139,140],[28,54443,54445,54448,54451],{"className":54444},[102,140],[28,54446,29362],{"className":54447,"style":29427},[102,106,140],[28,54449,845],{"className":54450},[917,140],[28,54452,652],{"className":54453},[102,140],[28,54455,54456,54459],{"style":53915},[28,54457],{"className":54458,"style":53897},[132],[28,54460,54461],{},[28,54462,33223],{"className":54463},[4241,33269,53924],[28,54465,54466,54469],{"style":53927},[28,54467],{"className":54468,"style":53897},[132],[28,54470,54472],{"className":54471},[137,138,139,140],[28,54473,54475],{"className":54474},[102,140],[28,54476,33375],{"className":54477,"style":29050},[102,106,140],[28,54479,148],{"className":54480},[147],[28,54482,54484],{"className":54483},[120],[28,54485,54487],{"className":54486,"style":53949},[124],[28,54488],{},[28,54490],{"className":54491,"style":174},[173],[28,54493,54495,54498],{"className":54494},[102],[28,54496,53825],{"className":54497,"style":53961},[102,106],[28,54499,54501],{"className":54500},[111],[28,54502,54504,54524],{"className":54503},[115,116],[28,54505,54507,54521],{"className":54506},[120],[28,54508,54510],{"className":54509,"style":10240},[124],[28,54511,54512,54515],{"style":53976},[28,54513],{"className":54514,"style":133},[132],[28,54516,54518],{"className":54517},[137,138,139,140],[28,54519,29362],{"className":54520,"style":29427},[102,106,140],[28,54522,148],{"className":54523},[147],[28,54525,54527],{"className":54526},[120],[28,54528,54530],{"className":54529,"style":155},[124],[28,54531],{},[28,54533,68],{"className":54534,"style":165},[102,106],[28,54536,65],{"className":54537},[161],[28,54539,54541,54544],{"className":54540},[102],[28,54542,1871],{"className":54543,"style":1913},[102,106],[28,54545,54547],{"className":54546},[111],[28,54548,54550,54570],{"className":54549},[115,116],[28,54551,54553,54567],{"className":54552},[120],[28,54554,54556],{"className":54555,"style":10240},[124],[28,54557,54558,54561],{"style":1928},[28,54559],{"className":54560,"style":133},[132],[28,54562,54564],{"className":54563},[137,138,139,140],[28,54565,29362],{"className":54566,"style":29427},[102,106,140],[28,54568,148],{"className":54569},[147],[28,54571,54573],{"className":54572},[120],[28,54574,54576],{"className":54575,"style":155},[124],[28,54577],{},[28,54579,78],{"className":54580},[182],[28,54582,296],{"className":54583},[102],[28,54585,54587,54590],{"className":54586},[102],[28,54588,1871],{"className":54589,"style":1913},[102,106],[28,54591,54593],{"className":54592},[111],[28,54594,54596,54616],{"className":54595},[115,116],[28,54597,54599,54613],{"className":54598},[120],[28,54600,54602],{"className":54601,"style":10240},[124],[28,54603,54604,54607],{"style":1928},[28,54605],{"className":54606,"style":133},[132],[28,54608,54610],{"className":54609},[137,138,139,140],[28,54611,29362],{"className":54612,"style":29427},[102,106,140],[28,54614,148],{"className":54615},[147],[28,54617,54619],{"className":54618},[120],[28,54620,54622],{"className":54621,"style":155},[124],[28,54623],{},[28,54625,302],{"className":54626},[182],[28,54628,72],{"className":54629},[169],[12,54631,54632,54633,54661,54662,54732,54733,54812],{},"using the fact that we can replace ",[28,54634,54636,54649],{"className":54635},[35],[28,54637,54639],{"className":54638},[39],[41,54640,54641],{"xmlns":43},[46,54642,54643,54647],{},[49,54644,54645],{},[55,54646,2263],{},[82,54648,2263],{"encoding":84},[28,54650,54652],{"className":54651,"ariaHidden":71},[89],[28,54653,54655,54658],{"className":54654},[93],[28,54656],{"className":54657,"style":1102},[97],[28,54659,2263],{"className":54660,"style":4040},[102,106]," by ",[28,54663,54665,54683],{"className":54664},[35],[28,54666,54668],{"className":54667},[39],[41,54669,54670],{"xmlns":43},[46,54671,54672,54680],{},[49,54673,54674],{},[52,54675,54676,54678],{},[55,54677,1871],{},[55,54679,29362],{},[82,54681,54682],{"encoding":84},"E_k",[28,54684,54686],{"className":54685,"ariaHidden":71},[89],[28,54687,54689,54692],{"className":54688},[93],[28,54690],{"className":54691,"style":585},[97],[28,54693,54695,54698],{"className":54694},[102],[28,54696,1871],{"className":54697,"style":1913},[102,106],[28,54699,54701],{"className":54700},[111],[28,54702,54704,54724],{"className":54703},[115,116],[28,54705,54707,54721],{"className":54706},[120],[28,54708,54710],{"className":54709,"style":10240},[124],[28,54711,54712,54715],{"style":1928},[28,54713],{"className":54714,"style":133},[132],[28,54716,54718],{"className":54717},[137,138,139,140],[28,54719,29362],{"className":54720,"style":29427},[102,106,140],[28,54722,148],{"className":54723},[147],[28,54725,54727],{"className":54726},[120],[28,54728,54730],{"className":54729,"style":155},[124],[28,54731],{}," when it acts on the eigenstate ",[28,54734,54736,54757],{"className":54735},[35],[28,54737,54739],{"className":54738},[39],[41,54740,54741],{"xmlns":43},[46,54742,54743,54755],{},[49,54744,54745,54747,54753],{},[55,54746,296],{"mathvariant":1088},[52,54748,54749,54751],{},[55,54750,1871],{},[55,54752,29362],{},[62,54754,302],{"stretchy":64},[82,54756,54072],{"encoding":84},[28,54758,54760],{"className":54759,"ariaHidden":71},[89],[28,54761,54763,54766,54769,54809],{"className":54762},[93],[28,54764],{"className":54765,"style":98},[97],[28,54767,296],{"className":54768},[102],[28,54770,54772,54775],{"className":54771},[102],[28,54773,1871],{"className":54774,"style":1913},[102,106],[28,54776,54778],{"className":54777},[111],[28,54779,54781,54801],{"className":54780},[115,116],[28,54782,54784,54798],{"className":54783},[120],[28,54785,54787],{"className":54786,"style":10240},[124],[28,54788,54789,54792],{"style":1928},[28,54790],{"className":54791,"style":133},[132],[28,54793,54795],{"className":54794},[137,138,139,140],[28,54796,29362],{"className":54797,"style":29427},[102,106,140],[28,54799,148],{"className":54800},[147],[28,54802,54804],{"className":54803},[120],[28,54805,54807],{"className":54806,"style":155},[124],[28,54808],{},[28,54810,302],{"className":54811},[182],". The energy error of this state is therefore",[28,54814,54816],{"className":54815},[31],[28,54817,54819,54908],{"className":54818},[35],[28,54820,54822],{"className":54821},[39],[41,54823,54824],{"xmlns":43,"display":44},[46,54825,54826,54905],{},[49,54827,54828,54831,54833],{},[2253,54829,54830],{},"energy error",[62,54832,845],{},[3977,54834,54835,54879],{},[49,54836,54837,54839,54841,54843,54845,54847,54849,54851,54853,54855,54857,54863,54865,54867,54869,54871,54873,54875,54877],{},[62,54838,2948],{"stretchy":64},[55,54840,394],{},[55,54842,296],{"mathvariant":1088},[55,54844,68],{},[62,54846,65],{"stretchy":64},[55,54848,2263],{},[62,54850,78],{"stretchy":64},[62,54852,65],{"stretchy":64},[55,54854,2263],{},[62,54856,1213],{},[52,54858,54859,54861],{},[55,54860,1871],{},[650,54862,652],{},[62,54864,78],{"stretchy":64},[55,54866,68],{},[62,54868,65],{"stretchy":64},[55,54870,2263],{},[62,54872,78],{"stretchy":64},[55,54874,296],{"mathvariant":1088},[55,54876,394],{},[62,54878,302],{"stretchy":64},[49,54880,54881,54883,54885,54887,54889,54891,54893,54899,54901,54903],{},[62,54882,2948],{"stretchy":64},[55,54884,394],{},[55,54886,296],{"mathvariant":1088},[55,54888,68],{},[62,54890,65],{"stretchy":64},[55,54892,2263],{},[2776,54894,54895,54897],{},[62,54896,78],{"stretchy":64},[650,54898,848],{},[55,54900,296],{"mathvariant":1088},[55,54902,394],{},[62,54904,302],{"stretchy":64},[82,54906,54907],{"encoding":84},"\\text{energy error} = 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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 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the first step follows because ",[28,59352,59354,59383],{"className":59353},[35],[28,59355,59357],{"className":59356},[39],[41,59358,59359],{"xmlns":43},[46,59360,59361,59381],{},[49,59362,59363,59369,59371,59377,59379],{},[52,59364,59365,59367],{},[55,59366,1871],{},[55,59368,29362],{},[62,59370,1213],{},[52,59372,59373,59375],{},[55,59374,1871],{},[650,59376,652],{},[62,59378,13276],{},[55,59380,53119],{},[82,59382,55926],{"encoding":84},[28,59384,59386,59441,59496],{"className":59385,"ariaHidden":71},[89],[28,59387,59389,59392,59432,59435,59438],{"className":59388},[93],[28,59390],{"className":59391,"style":585},[97],[28,59393,59395,59398],{"className":59394},[102],[28,59396,1871],{"className":59397,"style":1913},[102,106],[28,59399,59401],{"className":59400},[111],[28,59402,59404,59424],{"className":59403},[115,116],[28,59405,59407,59421],{"className":59406},[120],[28,59408,59410],{"className":59409,"style":10240},[124],[28,59411,59412,59415],{"style":1928},[28,59413],{"className":59414,"style":133},[132],[28,59416,59418],{"className":59417},[137,138,139,140],[28,59419,29362],{"className":59420,"style":29427},[102,106,140],[28,59422,148],{"className":59423},[147],[28,59425,59427],{"className":59426},[120],[28,59428,59430],{"className":59429,"style":155},[124],[28,59431],{},[28,59433],{"className":59434,"style":1257},[173],[28,59436,1213],{"className":59437},[1261],[28,59439],{"className":59440,"style":1257},[173],[28,59442,59444,59447,59487,59490,59493],{"className":59443},[93],[28,59445],{"className":59446,"style":585},[97],[28,59448,59450,59453],{"className":59449},[102],[28,59451,1871],{"className":59452,"style":1913},[102,106],[28,59454,59456],{"className":59455},[111],[28,59457,59459,59479],{"className":59458},[115,116],[28,59460,59462,59476],{"className":59461},[120],[28,59463,59465],{"className":59464,"style":683},[124],[28,59466,59467,59470],{"style":1928},[28,59468],{"className":59469,"style":133},[132],[28,59471,59473],{"className":59472},[137,138,139,140],[28,59474,652],{"className":59475},[102,140],[28,59477,148],{"className":59478},[147],[28,59480,59482],{"className":59481},[120],[28,59483,59485],{"className":59484,"style":155},[124],[28,59486],{},[28,59488],{"className":59489,"style":913},[173],[28,59491,13276],{"className":59492},[917],[28,59494],{"className":59495,"style":913},[173],[28,59497,59499,59502],{"className":59498},[93],[28,59500],{"className":59501,"style":9597},[97],[28,59503,53119],{"className":59504,"style":53204},[102,106]," for every ",[28,59507,59509,59526],{"className":59508},[35],[28,59510,59512],{"className":59511},[39],[41,59513,59514],{"xmlns":43},[46,59515,59516,59524],{},[49,59517,59518],{},[52,59519,59520,59522],{},[55,59521,1871],{},[55,59523,29362],{},[82,59525,54682],{"encoding":84},[28,59527,59529],{"className":59528,"ariaHidden":71},[89],[28,59530,59532,59535],{"className":59531},[93],[28,59533],{"className":59534,"style":585},[97],[28,59536,59538,59541],{"className":59537},[102],[28,59539,1871],{"className":59540,"style":1913},[102,106],[28,59542,59544],{"className":59543},[111],[28,59545,59547,59567],{"className":59546},[115,116],[28,59548,59550,59564],{"className":59549},[120],[28,59551,59553],{"className":59552,"style":10240},[124],[28,59554,59555,59558],{"style":1928},[28,59556],{"className":59557,"style":133},[132],[28,59559,59561],{"className":59560},[137,138,139,140],[28,59562,29362],{"className":59563,"style":29427},[102,106,140],[28,59565,148],{"className":59566},[147],[28,59568,59570],{"className":59569},[120],[28,59571,59573],{"className":59572,"style":155},[124],[28,59574],{}," 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 ",[28,59577,59579,59605],{"className":59578},[35],[28,59580,59582],{"className":59581},[39],[41,59583,59584],{"xmlns":43},[46,59585,59586,59602],{},[49,59587,59588,59590,59596],{},[55,59589,296],{"mathvariant":1088},[52,59591,59592,59594],{},[55,59593,53825],{},[650,59595,652],{},[2776,59597,59598,59600],{},[55,59599,296],{"mathvariant":1088},[650,59601,848],{},[82,59603,59604],{"encoding":84},"|\\gamma_0|^2",[28,59606,59608],{"className":59607,"ariaHidden":71},[89],[28,59609,59611,59614,59617,59657],{"className":59610},[93],[28,59612],{"className":59613,"style":38843},[97],[28,59615,296],{"className":59616},[102],[28,59618,59620,59623],{"className":59619},[102],[28,59621,53825],{"className":59622,"style":53961},[102,106],[28,59624,59626],{"className":59625},[111],[28,59627,59629,59649],{"className":59628},[115,116],[28,59630,59632,59646],{"className":59631},[120],[28,59633,59635],{"className":59634,"style":683},[124],[28,59636,59637,59640],{"style":53976},[28,59638],{"className":59639,"style":133},[132],[28,59641,59643],{"className":59642},[137,138,139,140],[28,59644,652],{"className":59645},[102,140],[28,59647,148],{"className":59648},[147],[28,59650,59652],{"className":59651},[120],[28,59653,59655],{"className":59654,"style":155},[124],[28,59656],{},[28,59658,59660,59663],{"className":59659},[102],[28,59661,296],{"className":59662},[102],[28,59664,59666],{"className":59665},[111],[28,59667,59669],{"className":59668},[115],[28,59670,59672],{"className":59671},[120],[28,59673,59675],{"className":59674,"style":5446},[124],[28,59676,59677,59680],{"style":1726},[28,59678],{"className":59679,"style":133},[132],[28,59681,59683],{"className":59682},[137,138,139,140],[28,59684,848],{"className":59685},[102,140],", since ",[28,59688,59690,59722],{"className":59689},[35],[28,59691,59693],{"className":59692},[39],[41,59694,59695],{"xmlns":43},[46,59696,59697,59719],{},[49,59698,59699,59701,59703,59709,59715,59717],{},[55,59700,68],{},[62,59702,65],{"stretchy":64},[52,59704,59705,59707],{},[55,59706,1871],{},[650,59708,652],{},[2776,59710,59711,59713],{},[62,59712,78],{"stretchy":64},[650,59714,848],{},[62,59716,845],{},[650,59718,3200],{},[82,59720,59721],{"encoding":84},"f(E_0)^2=1",[28,59723,59725,59815],{"className":59724,"ariaHidden":71},[89],[28,59726,59728,59731,59734,59737,59777,59806,59809,59812],{"className":59727},[93],[28,59729],{"className":59730,"style":38843},[97],[28,59732,68],{"className":59733,"style":165},[102,106],[28,59735,65],{"className":59736},[161],[28,59738,59740,59743],{"className":59739},[102],[28,59741,1871],{"className":59742,"style":1913},[102,106],[28,59744,59746],{"className":59745},[111],[28,59747,59749,59769],{"className":59748},[115,116],[28,59750,59752,59766],{"className":59751},[120],[28,59753,59755],{"className":59754,"style":683},[124],[28,59756,59757,59760],{"style":1928},[28,59758],{"className":59759,"style":133},[132],[28,59761,59763],{"className":59762},[137,138,139,140],[28,59764,652],{"className":59765},[102,140],[28,59767,148],{"className":59768},[147],[28,59770,59772],{"className":59771},[120],[28,59773,59775],{"className":59774,"style":155},[124],[28,59776],{},[28,59778,59780,59783],{"className":59779},[182],[28,59781,78],{"className":59782},[182],[28,59784,59786],{"className":59785},[111],[28,59787,59789],{"className":59788},[115],[28,59790,59792],{"className":59791},[120],[28,59793,59795],{"className":59794,"style":5446},[124],[28,59796,59797,59800],{"style":1726},[28,59798],{"className":59799,"style":133},[132],[28,59801,59803],{"className":59802},[137,138,139,140],[28,59804,848],{"className":59805},[102,140],[28,59807],{"className":59808,"style":913},[173],[28,59810,845],{"className":59811},[917],[28,59813],{"className":59814,"style":913},[173],[28,59816,59818,59821],{"className":59817},[93],[28,59819],{"className":59820,"style":2914},[97],[28,59822,3200],{"className":59823},[102],": adding everything back together, this gives",[28,59826,59828],{"className":59827},[31],[28,59829,59831,59945],{"className":59830},[35],[28,59832,59834],{"className":59833},[39],[41,59835,59836],{"xmlns":43,"display":44},[46,59837,59838,59942],{},[49,59839,59840,59842,59844,59846,59848,59868,59892,59894,59900,59906,59908,59910,59916,59922,59924,59930,59932,59938,59940],{},[2253,59841,54830],{},[62,59843,13276],{},[55,59845,53119],{},[62,59847,2606],{},[3977,59849,59850,59852],{},[650,59851,3200],{},[49,59853,59854,59856,59862],{},[55,59855,296],{"mathvariant":1088},[52,59857,59858,59860],{},[55,59859,53825],{},[650,59861,652],{},[2776,59863,59864,59866],{},[55,59865,296],{"mathvariant":1088},[650,59867,848],{},[56229,59869,59870,59872],{},[62,59871,33223],{},[49,59873,59874,59880,59882,59888,59890],{},[52,59875,59876,59878],{},[55,59877,1871],{},[55,59879,29362],{},[62,59881,16503],{},[52,59883,59884,59886],{},[55,59885,1871],{},[650,59887,652],{},[62,59889,2606],{},[55,59891,53119],{},[55,59893,296],{"mathvariant":1088},[52,59895,59896,59898],{},[55,59897,53825],{},[55,59899,29362],{},[2776,59901,59902,59904],{},[55,59903,296],{"mathvariant":1088},[650,59905,848],{},[55,59907,68],{},[62,59909,65],{"stretchy":64},[52,59911,59912,59914],{},[55,59913,1871],{},[55,59915,29362],{},[2776,59917,59918,59920],{},[62,59919,78],{"stretchy":64},[650,59921,848],{},[62,59923,65],{"stretchy":64},[52,59925,59926,59928],{},[55,59927,1871],{},[55,59929,29362],{},[62,59931,1213],{},[52,59933,59934,59936],{},[55,59935,1871],{},[650,59937,652],{},[62,59939,78],{"stretchy":64},[55,59941,2106],{"mathvariant":1088},[82,59943,59944],{"encoding":84},"\\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).",[28,59946,59948,59970,59989,60467],{"className":59947,"ariaHidden":71},[89],[28,59949,59951,59955,59961,59964,59967],{"className":59950},[93],[28,59952],{"className":59953,"style":59954},[97],"height:0.8304em;vertical-align:-0.1944em;",[28,59956,59958],{"className":59957},[102,16058],[28,59959,54830],{"className":59960},[102],[28,59962],{"className":59963,"style":913},[173],[28,59965,13276],{"className":59966},[917],[28,59968],{"className":59969,"style":913},[173],[28,59971,59973,59977,59980,59983,59986],{"className":59972},[93],[28,59974],{"className":59975,"style":59976},[97],"height:0.7778em;vertical-align:-0.0833em;",[28,59978,53119],{"className":59979,"style":53204},[102,106],[28,59981],{"className":59982,"style":1257},[173],[28,59984,2606],{"className":59985},[1261],[28,59987],{"className":59988,"style":1257},[173],[28,59990,59992,59996,60127,60130,60265,60268,60271,60311,60340,60343,60346,60386,60415,60418,60458,60461,60464],{"className":59991},[93],[28,59993],{"className":59994,"style":59995},[97],"height:2.7294em;vertical-align:-1.408em;",[28,59997,59999,60002,60124],{"className":59998},[102],[28,60000],{"className":60001},[161,4072],[28,60003,60005],{"className":60004},[3977],[28,60006,60008,60116],{"className":60007},[115,116],[28,60009,60011,60113],{"className":60010},[120],[28,60012,60014,60094,60102],{"className":60013,"style":6825},[124],[28,60015,60016,60019],{"style":4088},[28,60017],{"className":60018,"style":3434},[132],[28,60020,60022,60025,60065],{"className":60021},[102],[28,60023,296],{"className":60024},[102],[28,60026,60028,60031],{"className":60027},[102],[28,60029,53825],{"className":60030,"style":53961},[102,106],[28,60032,60034],{"className":60033},[111],[28,60035,60037,60057],{"className":60036},[115,116],[28,60038,60040,60054],{"className":60039},[120],[28,60041,60043],{"className":60042,"style":683},[124],[28,60044,60045,60048],{"style":53976},[28,60046],{"className":60047,"style":133},[132],[28,60049,60051],{"className":60050},[137,138,139,140],[28,60052,652],{"className":60053},[102,140],[28,60055,148],{"className":60056},[147],[28,60058,60060],{"className":60059},[120],[28,60061,60063],{"className":60062,"style":155},[124],[28,60064],{},[28,60066,60068,60071],{"className":60067},[102],[28,60069,296],{"className":60070},[102],[28,60072,60074],{"className":60073},[111],[28,60075,60077],{"className":60076},[115],[28,60078,60080],{"className":60079},[120],[28,60081,60083],{"className":60082,"style":5792},[124],[28,60084,60085,60088],{"style":5795},[28,60086],{"className":60087,"style":133},[132],[28,60089,60091],{"className":60090},[137,138,139,140],[28,60092,848],{"className":60093},[102,140],[28,60095,60096,60099],{"style":4100},[28,60097],{"className":60098,"style":3434},[132],[28,60100],{"className":60101,"style":4108},[4107],[28,60103,60104,60107],{"style":4111},[28,60105],{"className":60106,"style":3434},[132],[28,60108,60110],{"className":60109},[102],[28,60111,3200],{"className":60112},[102],[28,60114,148],{"className":60115},[147],[28,60117,60119],{"className":60118},[120],[28,60120,60122],{"className":60121,"style":6877},[124],[28,60123],{},[28,60125],{"className":60126},[182,4072],[28,60128],{"className":60129,"style":174},[173],[28,60131,60133],{"className":60132},[4241,53880],[28,60134,60136,60256],{"className":60135},[115,116],[28,60137,60139,60253],{"className":60138},[120],[28,60140,60143,60243],{"className":60141,"style":60142},[124],"height:1.05em;",[28,60144,60145,60148],{"style":53893},[28,60146],{"className":60147,"style":53897},[132],[28,60149,60151],{"className":60150},[137,138,139,140],[28,60152,60154,60194,60197,60237,60240],{"className":60153},[102,140],[28,60155,60157,60160],{"className":60156},[102,140],[28,60158,1871],{"className":60159,"style":1913},[102,106,140],[28,60161,60163],{"className":60162},[111],[28,60164,60166,60186],{"className":60165},[115,116],[28,60167,60169,60183],{"className":60168},[120],[28,60170,60172],{"className":60171,"style":56812},[124],[28,60173,60174,60177],{"style":56815},[28,60175],{"className":60176,"style":30534},[132],[28,60178,60180],{"className":60179},[137,30538,28701,140],[28,60181,29362],{"className":60182,"style":29427},[102,106,140],[28,60184,148],{"className":60185},[147],[28,60187,60189],{"className":60188},[120],[28,60190,60192],{"className":60191,"style":56834},[124],[28,60193],{},[28,60195,16503],{"className":60196},[917,140],[28,60198,60200,60203],{"className":60199},[102,140],[28,60201,1871],{"className":60202,"style":1913},[102,106,140],[28,60204,60206],{"className":60205},[111],[28,60207,60209,60229],{"className":60208},[115,116],[28,60210,60212,60226],{"className":60211},[120],[28,60213,60215],{"className":60214,"style":56858},[124],[28,60216,60217,60220],{"style":56861},[28,60218],{"className":60219,"style":30534},[132],[28,60221,60223],{"className":60222},[137,30538,28701,140],[28,60224,652],{"className":60225},[102,140],[28,60227,148],{"className":60228},[147],[28,60230,60232],{"className":60231},[120],[28,60233,60235],{"className":60234,"style":30551},[124],[28,60236],{},[28,60238,2606],{"className":60239},[1261,140],[28,60241,53119],{"className":60242,"style":53204},[102,106,140],[28,60244,60245,60248],{"style":53915},[28,60246],{"className":60247,"style":53897},[132],[28,60249,60250],{},[28,60251,33223],{"className":60252},[4241,33269,53924],[28,60254,148],{"className":60255},[147],[28,60257,60259],{"className":60258},[120],[28,60260,60263],{"className":60261,"style":60262},[124],"height:1.408em;",[28,60264],{},[28,60266],{"className":60267,"style":174},[173],[28,60269,296],{"className":60270},[102],[28,60272,60274,60277],{"className":60273},[102],[28,60275,53825],{"className":60276,"style":53961},[102,106],[28,60278,60280],{"className":60279},[111],[28,60281,60283,60303],{"className":60282},[115,116],[28,60284,60286,60300],{"className":60285},[120],[28,60287,60289],{"className":60288,"style":10240},[124],[28,60290,60291,60294],{"style":53976},[28,60292],{"className":60293,"style":133},[132],[28,60295,60297],{"className":60296},[137,138,139,140],[28,60298,29362],{"className":60299,"style":29427},[102,106,140],[28,60301,148],{"className":60302},[147],[28,60304,60306],{"className":60305},[120],[28,60307,60309],{"className":60308,"style":155},[124],[28,60310],{},[28,60312,60314,60317],{"className":60313},[102],[28,60315,296],{"className":60316},[102],[28,60318,60320],{"className":60319},[111],[28,60321,60323],{"className":60322},[115],[28,60324,60326],{"className":60325},[120],[28,60327,60329],{"className":60328,"style":2836},[124],[28,60330,60331,60334],{"style":2839},[28,60332],{"className":60333,"style":133},[132],[28,60335,60337],{"className":60336},[137,138,139,140],[28,60338,848],{"className":60339},[102,140],[28,60341,68],{"className":60342,"style":165},[102,106],[28,60344,65],{"className":60345},[161],[28,60347,60349,60352],{"className":60348},[102],[28,60350,1871],{"className":60351,"style":1913},[102,106],[28,60353,60355],{"className":60354},[111],[28,60356,60358,60378],{"className":60357},[115,116],[28,60359,60361,60375],{"className":60360},[120],[28,60362,60364],{"className":60363,"style":10240},[124],[28,60365,60366,60369],{"style":1928},[28,60367],{"className":60368,"style":133},[132],[28,60370,60372],{"className":60371},[137,138,139,140],[28,60373,29362],{"className":60374,"style":29427},[102,106,140],[28,60376,148],{"className":60377},[147],[28,60379,60381],{"className":60380},[120],[28,60382,60384],{"className":60383,"style":155},[124],[28,60385],{},[28,60387,60389,60392],{"className":60388},[182],[28,60390,78],{"className":60391},[182],[28,60393,60395],{"className":60394},[111],[28,60396,60398],{"className":60397},[115],[28,60399,60401],{"className":60400},[120],[28,60402,60404],{"className":60403,"style":2836},[124],[28,60405,60406,60409],{"style":2839},[28,60407],{"className":60408,"style":133},[132],[28,60410,60412],{"className":60411},[137,138,139,140],[28,60413,848],{"className":60414},[102,140],[28,60416,65],{"className":60417},[161],[28,60419,60421,60424],{"className":60420},[102],[28,60422,1871],{"className":60423,"style":1913},[102,106],[28,60425,60427],{"className":60426},[111],[28,60428,60430,60450],{"className":60429},[115,116],[28,60431,60433,60447],{"className":60432},[120],[28,60434,60436],{"className":60435,"style":10240},[124],[28,60437,60438,60441],{"style":1928},[28,60439],{"className":60440,"style":133},[132],[28,60442,60444],{"className":60443},[137,138,139,140],[28,60445,29362],{"className":60446,"style":29427},[102,106,140],[28,60448,148],{"className":60449},[147],[28,60451,60453],{"className":60452},[120],[28,60454,60456],{"className":60455,"style":155},[124],[28,60457],{},[28,60459],{"className":60460,"style":1257},[173],[28,60462,1213],{"className":60463},[1261],[28,60465],{"className":60466,"style":1257},[173],[28,60468,60470,60473,60513,60516],{"className":60469},[93],[28,60471],{"className":60472,"style":98},[97],[28,60474,60476,60479],{"className":60475},[102],[28,60477,1871],{"className":60478,"style":1913},[102,106],[28,60480,60482],{"className":60481},[111],[28,60483,60485,60505],{"className":60484},[115,116],[28,60486,60488,60502],{"className":60487},[120],[28,60489,60491],{"className":60490,"style":683},[124],[28,60492,60493,60496],{"style":1928},[28,60494],{"className":60495,"style":133},[132],[28,60497,60499],{"className":60498},[137,138,139,140],[28,60500,652],{"className":60501},[102,140],[28,60503,148],{"className":60504},[147],[28,60506,60508],{"className":60507},[120],[28,60509,60511],{"className":60510,"style":155},[124],[28,60512],{},[28,60514,78],{"className":60515},[182],[28,60517,2106],{"className":60518},[102],[12,60520,60521,60522,60591,60592,60620,60621,60842,60843,61011,61012,61301],{},"To simplify what is left, notice that for all these ",[28,60523,60525,60542],{"className":60524},[35],[28,60526,60528],{"className":60527},[39],[41,60529,60530],{"xmlns":43},[46,60531,60532,60540],{},[49,60533,60534],{},[52,60535,60536,60538],{},[55,60537,1871],{},[55,60539,29362],{},[82,60541,54682],{"encoding":84},[28,60543,60545],{"className":60544,"ariaHidden":71},[89],[28,60546,60548,60551],{"className":60547},[93],[28,60549],{"className":60550,"style":585},[97],[28,60552,60554,60557],{"className":60553},[102],[28,60555,1871],{"className":60556,"style":1913},[102,106],[28,60558,60560],{"className":60559},[111],[28,60561,60563,60583],{"className":60562},[115,116],[28,60564,60566,60580],{"className":60565},[120],[28,60567,60569],{"className":60568,"style":10240},[124],[28,60570,60571,60574],{"style":1928},[28,60572],{"className":60573,"style":133},[132],[28,60575,60577],{"className":60576},[137,138,139,140],[28,60578,29362],{"className":60579,"style":29427},[102,106,140],[28,60581,148],{"className":60582},[147],[28,60584,60586],{"className":60585},[120],[28,60587,60589],{"className":60588,"style":155},[124],[28,60590],{},", by the definition of ",[28,60593,60595,60608],{"className":60594},[35],[28,60596,60598],{"className":60597},[39],[41,60599,60600],{"xmlns":43},[46,60601,60602,60606],{},[49,60603,60604],{},[55,60605,68],{},[82,60607,68],{"encoding":84},[28,60609,60611],{"className":60610,"ariaHidden":71},[89],[28,60612,60614,60617],{"className":60613},[93],[28,60615],{"className":60616,"style":214},[97],[28,60618,68],{"className":60619,"style":165},[102,106]," we know that ",[28,60622,60624,60678],{"className":60623},[35],[28,60625,60627],{"className":60626},[39],[41,60628,60629],{"xmlns":43},[46,60630,60631,60675],{},[49,60632,60633,60635,60637,60643,60649,60651,60653],{},[55,60634,68],{},[62,60636,65],{"stretchy":64},[52,60638,60639,60641],{},[55,60640,1871],{},[55,60642,29362],{},[2776,60644,60645,60647],{},[62,60646,78],{"stretchy":64},[650,60648,848],{},[62,60650,13276],{},[650,60652,37795],{},[2776,60654,60655,60667],{},[49,60656,60657,60659,60661,60663,60665],{},[62,60658,65],{"fence":71},[650,60660,3200],{},[62,60662,2606],{},[55,60664,53119],{},[62,60666,78],{"fence":71},[49,60668,60669,60671,60673],{},[62,60670,1213],{},[650,60672,848],{},[55,60674,24201],{},[82,60676,60677],{"encoding":84},"f(E_k)^2 \\le 4\\left(1 + \\delta\\right)^{-2d}",[28,60679,60681,60771],{"className":60680,"ariaHidden":71},[89],[28,60682,60684,60687,60690,60693,60733,60762,60765,60768],{"className":60683},[93],[28,60685],{"className":60686,"style":38843},[97],[28,60688,68],{"className":60689,"style":165},[102,106],[28,60691,65],{"className":60692},[161],[28,60694,60696,60699],{"className":60695},[102],[28,60697,1871],{"className":60698,"style":1913},[102,106],[28,60700,60702],{"className":60701},[111],[28,60703,60705,60725],{"className":60704},[115,116],[28,60706,60708,60722],{"className":60707},[120],[28,60709,60711],{"className":60710,"style":10240},[124],[28,60712,60713,60716],{"style":1928},[28,60714],{"className":60715,"style":133},[132],[28,60717,60719],{"className":60718},[137,138,139,140],[28,60720,29362],{"className":60721,"style":29427},[102,106,140],[28,60723,148],{"className":60724},[147],[28,60726,60728],{"className":60727},[120],[28,60729,60731],{"className":60730,"style":155},[124],[28,60732],{},[28,60734,60736,60739],{"className":60735},[182],[28,60737,78],{"className":60738},[182],[28,60740,60742],{"className":60741},[111],[28,60743,60745],{"className":60744},[115],[28,60746,60748],{"className":60747},[120],[28,60749,60751],{"className":60750,"style":5446},[124],[28,60752,60753,60756],{"style":1726},[28,60754],{"className":60755,"style":133},[132],[28,60757,60759],{"className":60758},[137,138,139,140],[28,60760,848],{"className":60761},[102,140],[28,60763],{"className":60764,"style":913},[173],[28,60766,13276],{"className":60767},[917],[28,60769],{"className":60770,"style":913},[173],[28,60772,60774,60777,60780,60783],{"className":60773},[93],[28,60775],{"className":60776,"style":53173},[97],[28,60778,37795],{"className":60779},[102],[28,60781],{"className":60782,"style":174},[173],[28,60784,60786,60810],{"className":60785},[3398],[28,60787,60789,60792,60795,60798,60801,60804,60807],{"className":60788},[3398],[28,60790,65],{"className":60791,"style":3403},[161,3402],[28,60793,3200],{"className":60794},[102],[28,60796],{"className":60797,"style":1257},[173],[28,60799,2606],{"className":60800},[1261],[28,60802],{"className":60803,"style":1257},[173],[28,60805,53119],{"className":60806,"style":53204},[102,106],[28,60808,78],{"className":60809,"style":3403},[182,3402],[28,60811,60813],{"className":60812},[111],[28,60814,60816],{"className":60815},[115],[28,60817,60819],{"className":60818},[120],[28,60820,60822],{"className":60821,"style":53220},[124],[28,60823,60824,60827],{"style":33538},[28,60825],{"className":60826,"style":133},[132],[28,60828,60830],{"className":60829},[137,138,139,140],[28,60831,60833,60836,60839],{"className":60832},[102,140],[28,60834,1213],{"className":60835},[102,140],[28,60837,848],{"className":60838},[102,140],[28,60840,24201],{"className":60841},[102,106,140],". Additionally upper bounding ",[28,60844,60846,60882],{"className":60845},[35],[28,60847,60849],{"className":60848},[39],[41,60850,60851],{"xmlns":43},[46,60852,60853,60879],{},[49,60854,60855,60861,60863,60869,60871,60873,60875,60877],{},[52,60856,60857,60859],{},[55,60858,1871],{},[55,60860,29362],{},[62,60862,1213],{},[52,60864,60865,60867],{},[55,60866,1871],{},[650,60868,652],{},[62,60870,22950],{},[650,60872,848],{},[55,60874,37801],{"mathvariant":1088},[55,60876,2263],{},[55,60878,37801],{"mathvariant":1088},[82,60880,60881],{"encoding":84},"E_k-E_0\u003C2\\|H\\|",[28,60883,60885,60940,60995],{"className":60884,"ariaHidden":71},[89],[28,60886,60888,60891,60931,60934,60937],{"className":60887},[93],[28,60889],{"className":60890,"style":585},[97],[28,60892,60894,60897],{"className":60893},[102],[28,60895,1871],{"className":60896,"style":1913},[102,106],[28,60898,60900],{"className":60899},[111],[28,60901,60903,60923],{"className":60902},[115,116],[28,60904,60906,60920],{"className":60905},[120],[28,60907,60909],{"className":60908,"style":10240},[124],[28,60910,60911,60914],{"style":1928},[28,60912],{"className":60913,"style":133},[132],[28,60915,60917],{"className":60916},[137,138,139,140],[28,60918,29362],{"className":60919,"style":29427},[102,106,140],[28,60921,148],{"className":60922},[147],[28,60924,60926],{"className":60925},[120],[28,60927,60929],{"className":60928,"style":155},[124],[28,60930],{},[28,60932],{"className":60933,"style":1257},[173],[28,60935,1213],{"className":60936},[1261],[28,60938],{"className":60939,"style":1257},[173],[28,60941,60943,60946,60986,60989,60992],{"className":60942},[93],[28,60944],{"className":60945,"style":585},[97],[28,60947,60949,60952],{"className":60948},[102],[28,60950,1871],{"className":60951,"style":1913},[102,106],[28,60953,60955],{"className":60954},[111],[28,60956,60958,60978],{"className":60957},[115,116],[28,60959,60961,60975],{"className":60960},[120],[28,60962,60964],{"className":60963,"style":683},[124],[28,60965,60966,60969],{"style":1928},[28,60967],{"className":60968,"style":133},[132],[28,60970,60972],{"className":60971},[137,138,139,140],[28,60973,652],{"className":60974},[102,140],[28,60976,148],{"className":60977},[147],[28,60979,60981],{"className":60980},[120],[28,60982,60984],{"className":60983,"style":155},[124],[28,60985],{},[28,60987],{"className":60988,"style":913},[173],[28,60990,22950],{"className":60991},[917],[28,60993],{"className":60994,"style":913},[173],[28,60996,60998,61001,61005,61008],{"className":60997},[93],[28,60999],{"className":61000,"style":98},[97],[28,61002,61004],{"className":61003},[102],"2∥",[28,61006,2263],{"className":61007,"style":4040},[102,106],[28,61009,37801],{"className":61010},[102]," and upper bounding ",[28,61013,61015,61069],{"className":61014},[35],[28,61016,61018],{"className":61017},[39],[41,61019,61020],{"xmlns":43},[46,61021,61022,61066],{},[49,61023,61024,61048,61050,61056,61062,61064],{},[52,61025,61026,61028],{},[62,61027,33223],{},[49,61029,61030,61036,61038,61044,61046],{},[52,61031,61032,61034],{},[55,61033,1871],{},[55,61035,29362],{},[62,61037,16503],{},[52,61039,61040,61042],{},[55,61041,1871],{},[650,61043,652],{},[62,61045,2606],{},[55,61047,53119],{},[55,61049,296],{"mathvariant":1088},[52,61051,61052,61054],{},[55,61053,53825],{},[55,61055,29362],{},[2776,61057,61058,61060],{},[55,61059,296],{"mathvariant":1088},[650,61061,848],{},[62,61063,22950],{},[650,61065,3200],{},[82,61067,61068],{"encoding":84},"\\sum_{E_k>E_0+\\delta}|\\gamma_k|^2\u003C1",[28,61070,61072,61292],{"className":61071,"ariaHidden":71},[89],[28,61073,61075,61079,61208,61211,61214,61254,61283,61286,61289],{"className":61074},[93],[28,61076],{"className":61077,"style":61078},[97],"height:1.2197em;vertical-align:-0.4056em;",[28,61080,61082,61085],{"className":61081},[4241],[28,61083,33223],{"className":61084,"style":33271},[4241,33269,33270],[28,61086,61088],{"className":61087},[111],[28,61089,61091,61200],{"className":61090},[115,116],[28,61092,61094,61197],{"className":61093},[120],[28,61095,61097],{"className":61096,"style":56782},[124],[28,61098,61099,61102],{"style":33287},[28,61100],{"className":61101,"style":133},[132],[28,61103,61105],{"className":61104},[137,138,139,140],[28,61106,61108,61148,61151,61191,61194],{"className":61107},[102,140],[28,61109,61111,61114],{"className":61110},[102,140],[28,61112,1871],{"className":61113,"style":1913},[102,106,140],[28,61115,61117],{"className":61116},[111],[28,61118,61120,61140],{"className":61119},[115,116],[28,61121,61123,61137],{"className":61122},[120],[28,61124,61126],{"className":61125,"style":56812},[124],[28,61127,61128,61131],{"style":56815},[28,61129],{"className":61130,"style":30534},[132],[28,61132,61134],{"className":61133},[137,30538,28701,140],[28,61135,29362],{"className":61136,"style":29427},[102,106,140],[28,61138,148],{"className":61139},[147],[28,61141,61143],{"className":61142},[120],[28,61144,61146],{"className":61145,"style":56834},[124],[28,61147],{},[28,61149,16503],{"className":61150},[917,140],[28,61152,61154,61157],{"className":61153},[102,140],[28,61155,1871],{"className":61156,"style":1913},[102,106,140],[28,61158,61160],{"className":61159},[111],[28,61161,61163,61183],{"className":61162},[115,116],[28,61164,61166,61180],{"className":61165},[120],[28,61167,61169],{"className":61168,"style":56858},[124],[28,61170,61171,61174],{"style":56861},[28,61172],{"className":61173,"style":30534},[132],[28,61175,61177],{"className":61176},[137,30538,28701,140],[28,61178,652],{"className":61179},[102,140],[28,61181,148],{"className":61182},[147],[28,61184,61186],{"className":61185},[120],[28,61187,61189],{"className":61188,"style":30551},[124],[28,61190],{},[28,61192,2606],{"className":61193},[1261,140],[28,61195,53119],{"className":61196,"style":53204},[102,106,140],[28,61198,148],{"className":61199},[147],[28,61201,61203],{"className":61202},[120],[28,61204,61206],{"className":61205,"style":56897},[124],[28,61207],{},[28,61209],{"className":61210,"style":174},[173],[28,61212,296],{"className":61213},[102],[28,61215,61217,61220],{"className":61216},[102],[28,61218,53825],{"className":61219,"style":53961},[102,106],[28,61221,61223],{"className":61222},[111],[28,61224,61226,61246],{"className":61225},[115,116],[28,61227,61229,61243],{"className":61228},[120],[28,61230,61232],{"className":61231,"style":10240},[124],[28,61233,61234,61237],{"style":53976},[28,61235],{"className":61236,"style":133},[132],[28,61238,61240],{"className":61239},[137,138,139,140],[28,61241,29362],{"className":61242,"style":29427},[102,106,140],[28,61244,148],{"className":61245},[147],[28,61247,61249],{"className":61248},[120],[28,61250,61252],{"className":61251,"style":155},[124],[28,61253],{},[28,61255,61257,61260],{"className":61256},[102],[28,61258,296],{"className":61259},[102],[28,61261,61263],{"className":61262},[111],[28,61264,61266],{"className":61265},[115],[28,61267,61269],{"className":61268},[120],[28,61270,61272],{"className":61271,"style":5446},[124],[28,61273,61274,61277],{"style":1726},[28,61275],{"className":61276,"style":133},[132],[28,61278,61280],{"className":61279},[137,138,139,140],[28,61281,848],{"className":61282},[102,140],[28,61284],{"className":61285,"style":913},[173],[28,61287,22950],{"className":61288},[917],[28,61290],{"className":61291,"style":913},[173],[28,61293,61295,61298],{"className":61294},[93],[28,61296],{"className":61297,"style":2914},[97],[28,61299,3200],{"className":61300},[102]," gives",[28,61303,61305],{"className":61304},[31],[28,61306,61308,61378],{"className":61307},[35],[28,61309,61311],{"className":61310},[39],[41,61312,61313],{"xmlns":43,"display":44},[46,61314,61315,61375],{},[49,61316,61317,61319,61321,61323,61325,61345,61347,61349,61351,61373],{},[2253,61318,54830],{},[62,61320,13276],{},[55,61322,53119],{},[62,61324,2606],{},[3977,61326,61327,61329],{},[650,61328,15567],{},[49,61330,61331,61333,61339],{},[55,61332,296],{"mathvariant":1088},[52,61334,61335,61337],{},[55,61336,53825],{},[650,61338,652],{},[2776,61340,61341,61343],{},[55,61342,296],{"mathvariant":1088},[650,61344,848],{},[55,61346,37801],{"mathvariant":1088},[55,61348,2263],{},[55,61350,37801],{"mathvariant":1088},[2776,61352,61353,61365],{},[49,61354,61355,61357,61359,61361,61363],{},[62,61356,65],{"fence":71},[650,61358,3200],{},[62,61360,2606],{},[55,61362,53119],{},[62,61364,78],{"fence":71},[49,61366,61367,61369,61371],{},[62,61368,1213],{},[650,61370,848],{},[55,61372,24201],{},[55,61374,2106],{"mathvariant":1088},[82,61376,61377],{"encoding":84},"\\text{energy error} \\le \\delta + \\frac{8}{|\\gamma_0|^2}\\|H\\|\\left(1 + \\delta\\right)^{-2d}.",[28,61379,61381,61402,61420],{"className":61380,"ariaHidden":71},[89],[28,61382,61384,61387,61393,61396,61399],{"className":61383},[93],[28,61385],{"className":61386,"style":59954},[97],[28,61388,61390],{"className":61389},[102,16058],[28,61391,54830],{"className":61392},[102],[28,61394],{"className":61395,"style":913},[173],[28,61397,13276],{"className":61398},[917],[28,61400],{"className":61401,"style":913},[173],[28,61403,61405,61408,61411,61414,61417],{"className":61404},[93],[28,61406],{"className":61407,"style":59976},[97],[28,61409,53119],{"className":61410,"style":53204},[102,106],[28,61412],{"className":61413,"style":1257},[173],[28,61415,2606],{"className":61416},[1261],[28,61418],{"className":61419,"style":1257},[173],[28,61421,61423,61426,61557,61560,61563,61566,61569,61628,61631],{"className":61422},[93],[28,61424],{"className":61425,"style":6806},[97],[28,61427,61429,61432,61554],{"className":61428},[102],[28,61430],{"className":61431},[161,4072],[28,61433,61435],{"className":61434},[3977],[28,61436,61438,61546],{"className":61437},[115,116],[28,61439,61441,61543],{"className":61440},[120],[28,61442,61444,61524,61532],{"className":61443,"style":6825},[124],[28,61445,61446,61449],{"style":4088},[28,61447],{"className":61448,"style":3434},[132],[28,61450,61452,61455,61495],{"className":61451},[102],[28,61453,296],{"className":61454},[102],[28,61456,61458,61461],{"className":61457},[102],[28,61459,53825],{"className":61460,"style":53961},[102,106],[28,61462,61464],{"className":61463},[111],[28,61465,61467,61487],{"className":61466},[115,116],[28,61468,61470,61484],{"className":61469},[120],[28,61471,61473],{"className":61472,"style":683},[124],[28,61474,61475,61478],{"style":53976},[28,61476],{"className":61477,"style":133},[132],[28,61479,61481],{"className":61480},[137,138,139,140],[28,61482,652],{"className":61483},[102,140],[28,61485,148],{"className":61486},[147],[28,61488,61490],{"className":61489},[120],[28,61491,61493],{"className":61492,"style":155},[124],[28,61494],{},[28,61496,61498,61501],{"className":61497},[102],[28,61499,296],{"className":61500},[102],[28,61502,61504],{"className":61503},[111],[28,61505,61507],{"className":61506},[115],[28,61508,61510],{"className":61509},[120],[28,61511,61513],{"className":61512,"style":5792},[124],[28,61514,61515,61518],{"style":5795},[28,61516],{"className":61517,"style":133},[132],[28,61519,61521],{"className":61520},[137,138,139,140],[28,61522,848],{"className":61523},[102,140],[28,61525,61526,61529],{"style":4100},[28,61527],{"className":61528,"style":3434},[132],[28,61530],{"className":61531,"style":4108},[4107],[28,61533,61534,61537],{"style":4111},[28,61535],{"className":61536,"style":3434},[132],[28,61538,61540],{"className":61539},[102],[28,61541,15567],{"className":61542},[102],[28,61544,148],{"className":61545},[147],[28,61547,61549],{"className":61548},[120],[28,61550,61552],{"className":61551,"style":6877},[124],[28,61553],{},[28,61555],{"className":61556},[182,4072],[28,61558,37801],{"className":61559},[102],[28,61561,2263],{"className":61562,"style":4040},[102,106],[28,61564,37801],{"className":61565},[102],[28,61567],{"className":61568,"style":174},[173],[28,61570,61572,61596],{"className":61571},[3398],[28,61573,61575,61578,61581,61584,61587,61590,61593],{"className":61574},[3398],[28,61576,65],{"className":61577,"style":3403},[161,3402],[28,61579,3200],{"className":61580},[102],[28,61582],{"className":61583,"style":1257},[173],[28,61585,2606],{"className":61586},[1261],[28,61588],{"className":61589,"style":1257},[173],[28,61591,53119],{"className":61592,"style":53204},[102,106],[28,61594,78],{"className":61595,"style":3403},[182,3402],[28,61597,61599],{"className":61598},[111],[28,61600,61602],{"className":61601},[115],[28,61603,61605],{"className":61604},[120],[28,61606,61608],{"className":61607,"style":53220},[124],[28,61609,61610,61613],{"style":33538},[28,61611],{"className":61612,"style":133},[132],[28,61614,61616],{"className":61615},[137,138,139,140],[28,61617,61619,61622,61625],{"className":61618},[102,140],[28,61620,1213],{"className":61621},[102,140],[28,61623,848],{"className":61624},[102,140],[28,61626,24201],{"className":61627},[102,106,140],[28,61629],{"className":61630,"style":174},[173],[28,61632,2106],{"className":61633},[102],[12,61635,61636,61637,61689,61690,61718,61719,61775,61776,61931,61932,61960],{},"This holds for any ",[28,61638,61640,61658],{"className":61639},[35],[28,61641,61643],{"className":61642},[39],[41,61644,61645],{"xmlns":43},[46,61646,61647,61655],{},[49,61648,61649,61651,61653],{},[55,61650,53119],{},[62,61652,16503],{},[650,61654,652],{},[82,61656,61657],{"encoding":84},"\\delta>0",[28,61659,61661,61680],{"className":61660,"ariaHidden":71},[89],[28,61662,61664,61668,61671,61674,61677],{"className":61663},[93],[28,61665],{"className":61666,"style":61667},[97],"height:0.7335em;vertical-align:-0.0391em;",[28,61669,53119],{"className":61670,"style":53204},[102,106],[28,61672],{"className":61673,"style":913},[173],[28,61675,16503],{"className":61676},[917],[28,61678],{"className":61679,"style":913},[173],[28,61681,61683,61686],{"className":61682},[93],[28,61684],{"className":61685,"style":2914},[97],[28,61687,652],{"className":61688},[102],", so if we set ",[28,61691,61693,61706],{"className":61692},[35],[28,61694,61696],{"className":61695},[39],[41,61697,61698],{"xmlns":43},[46,61699,61700,61704],{},[49,61701,61702],{},[55,61703,53119],{},[82,61705,57839],{"encoding":84},[28,61707,61709],{"className":61708,"ariaHidden":71},[89],[28,61710,61712,61715],{"className":61711},[93],[28,61713],{"className":61714,"style":9597},[97],[28,61716,53119],{"className":61717,"style":53204},[102,106]," equal to our goal error, then the error bound above converges towards that exponentially with the Krylov dimension ",[28,61720,61722,61742],{"className":61721},[35],[28,61723,61725],{"className":61724},[39],[41,61726,61727],{"xmlns":43},[46,61728,61729,61739],{},[49,61730,61731,61733,61735,61737],{},[650,61732,848],{},[55,61734,24201],{},[62,61736,845],{},[55,61738,16959],{},[82,61740,61741],{"encoding":84},"2d=r",[28,61743,61745,61766],{"className":61744,"ariaHidden":71},[89],[28,61746,61748,61751,61754,61757,61760,61763],{"className":61747},[93],[28,61749],{"className":61750,"style":9597},[97],[28,61752,848],{"className":61753},[102],[28,61755,24201],{"className":61756},[102,106],[28,61758],{"className":61759,"style":913},[173],[28,61761,845],{"className":61762},[917],[28,61764],{"className":61765,"style":913},[173],[28,61767,61769,61772],{"className":61768},[93],[28,61770],{"className":61771,"style":245},[97],[28,61773,16959],{"className":61774,"style":9403},[102,106],". Also note that if ",[28,61777,61779,61809],{"className":61778},[35],[28,61780,61782],{"className":61781},[39],[41,61783,61784],{"xmlns":43},[46,61785,61786,61806],{},[49,61787,61788,61790,61792,61798,61800],{},[55,61789,53119],{},[62,61791,22950],{},[52,61793,61794,61796],{},[55,61795,1871],{},[650,61797,3200],{},[62,61799,1213],{},[52,61801,61802,61804],{},[55,61803,1871],{},[650,61805,652],{},[82,61807,61808],{"encoding":84},"\\delta\u003CE_1-E_0",[28,61810,61812,61830,61885],{"className":61811,"ariaHidden":71},[89],[28,61813,61815,61818,61821,61824,61827],{"className":61814},[93],[28,61816],{"className":61817,"style":61667},[97],[28,61819,53119],{"className":61820,"style":53204},[102,106],[28,61822],{"className":61823,"style":913},[173],[28,61825,22950],{"className":61826},[917],[28,61828],{"className":61829,"style":913},[173],[28,61831,61833,61836,61876,61879,61882],{"className":61832},[93],[28,61834],{"className":61835,"style":585},[97],[28,61837,61839,61842],{"className":61838},[102],[28,61840,1871],{"className":61841,"style":1913},[102,106],[28,61843,61845],{"className":61844},[111],[28,61846,61848,61868],{"className":61847},[115,116],[28,61849,61851,61865],{"className":61850},[120],[28,61852,61854],{"className":61853,"style":683},[124],[28,61855,61856,61859],{"style":1928},[28,61857],{"className":61858,"style":133},[132],[28,61860,61862],{"className":61861},[137,138,139,140],[28,61863,3200],{"className":61864},[102,140],[28,61866,148],{"className":61867},[147],[28,61869,61871],{"className":61870},[120],[28,61872,61874],{"className":61873,"style":155},[124],[28,61875],{},[28,61877],{"className":61878,"style":1257},[173],[28,61880,1213],{"className":61881},[1261],[28,61883],{"className":61884,"style":1257},[173],[28,61886,61888,61891],{"className":61887},[93],[28,61889],{"className":61890,"style":585},[97],[28,61892,61894,61897],{"className":61893},[102],[28,61895,1871],{"className":61896,"style":1913},[102,106],[28,61898,61900],{"className":61899},[111],[28,61901,61903,61923],{"className":61902},[115,116],[28,61904,61906,61920],{"className":61905},[120],[28,61907,61909],{"className":61908,"style":683},[124],[28,61910,61911,61914],{"style":1928},[28,61912],{"className":61913,"style":133},[132],[28,61915,61917],{"className":61916},[137,138,139,140],[28,61918,652],{"className":61919},[102,140],[28,61921,148],{"className":61922},[147],[28,61924,61926],{"className":61925},[120],[28,61927,61929],{"className":61928,"style":155},[124],[28,61930],{}," then the ",[28,61933,61935,61948],{"className":61934},[35],[28,61936,61938],{"className":61937},[39],[41,61939,61940],{"xmlns":43},[46,61941,61942,61946],{},[49,61943,61944],{},[55,61945,53119],{},[82,61947,57839],{"encoding":84},[28,61949,61951],{"className":61950,"ariaHidden":71},[89],[28,61952,61954,61957],{"className":61953},[93],[28,61955],{"className":61956,"style":9597},[97],[28,61958,53119],{"className":61959,"style":53204},[102,106]," term actually goes away entirely in the above bound.",[12,61962,61963,61964,62022],{},"To complete the argument, we first note that the above is just the energy error of the particular state ",[28,61965,61967,61992],{"className":61966},[35],[28,61968,61970],{"className":61969},[39],[41,61971,61972],{"xmlns":43},[46,61973,61974,61990],{},[49,61975,61976,61978,61980,61982,61984,61986,61988],{},[55,61977,68],{},[62,61979,65],{"stretchy":64},[55,61981,2263],{},[62,61983,78],{"stretchy":64},[55,61985,296],{"mathvariant":1088},[55,61987,394],{},[62,61989,302],{"stretchy":64},[82,61991,53711],{"encoding":84},[28,61993,61995],{"className":61994,"ariaHidden":71},[89],[28,61996,61998,62001,62004,62007,62010,62013,62016,62019],{"className":61997},[93],[28,61999],{"className":62000,"style":98},[97],[28,62002,68],{"className":62003,"style":165},[102,106],[28,62005,65],{"className":62006},[161],[28,62008,2263],{"className":62009,"style":4040},[102,106],[28,62011,78],{"className":62012},[182],[28,62014,296],{"className":62015},[102],[28,62017,394],{"className":62018,"style":321},[102,106],[28,62020,302],{"className":62021},[182],", 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.",[12,62024,62025,62026,512,62028,62031],{},"A similar analysis as the above can be carried out that additionally accounts for noise and the thresholding procedure discussed in the notebook. See ",[19,62027,34229],{"href":27044},[19,62029,62030],{"href":27044},"[4]"," for this analysis.",[251,62033,62035],{"id":62034},"appendix-proof-of-claim-1","Appendix: proof of Claim 1",[12,62037,62038,62039,62041,62042,62116,62117,62204,62205,62233],{},"The following is mostly derived from ",[19,62040,52899],{"href":27044},", Theorem 3.1: let ",[28,62043,62045,62067],{"className":62044},[35],[28,62046,62048],{"className":62047},[39],[41,62049,62050],{"xmlns":43},[46,62051,62052,62064],{},[49,62053,62054,62056,62058,62060,62062],{},[650,62055,652],{},[62,62057,22950],{},[55,62059,19],{},[62,62061,22950],{},[55,62063,38743],{},[82,62065,62066],{"encoding":84},"0 \u003C a \u003C b",[28,62068,62070,62089,62107],{"className":62069,"ariaHidden":71},[89],[28,62071,62073,62077,62080,62083,62086],{"className":62072},[93],[28,62074],{"className":62075,"style":62076},[97],"height:0.6835em;vertical-align:-0.0391em;",[28,62078,652],{"className":62079},[102],[28,62081],{"className":62082,"style":913},[173],[28,62084,22950],{"className":62085},[917],[28,62087],{"className":62088,"style":913},[173],[28,62090,62092,62095,62098,62101,62104],{"className":62091},[93],[28,62093],{"className":62094,"style":31413},[97],[28,62096,19],{"className":62097},[102,106],[28,62099],{"className":62100,"style":913},[173],[28,62102,22950],{"className":62103},[917],[28,62105],{"className":62106,"style":913},[173],[28,62108,62110,62113],{"className":62109},[93],[28,62111],{"className":62112,"style":9597},[97],[28,62114,38743],{"className":62115},[102,106]," and let ",[28,62118,62120,62141],{"className":62119},[35],[28,62121,62123],{"className":62122},[39],[41,62124,62125],{"xmlns":43},[46,62126,62127,62138],{},[49,62128,62129],{},[1670,62130,62131,62134,62136],{},[55,62132,62133],{"mathvariant":1088},"Π",[55,62135,24201],{},[62,62137,1679],{},[82,62139,62140],{"encoding":84},"\\Pi^*_d",[28,62142,62144],{"className":62143,"ariaHidden":71},[89],[28,62145,62147,62151],{"className":62146},[93],[28,62148],{"className":62149,"style":62150},[97],"height:0.9718em;vertical-align:-0.2831em;",[28,62152,62154,62157],{"className":62153},[102],[28,62155,62133],{"className":62156},[102],[28,62158,62160],{"className":62159},[111],[28,62161,62163,62195],{"className":62162},[115,116],[28,62164,62166,62192],{"className":62165},[120],[28,62167,62169,62181],{"className":62168,"style":1711},[124],[28,62170,62172,62175],{"style":62171},"top:-2.4169em;margin-left:0em;margin-right:0.05em;",[28,62173],{"className":62174,"style":133},[132],[28,62176,62178],{"className":62177},[137,138,139,140],[28,62179,24201],{"className":62180},[102,106,140],[28,62182,62183,62186],{"style":1726},[28,62184],{"className":62185,"style":133},[132],[28,62187,62189],{"className":62188},[137,138,139,140],[28,62190,1679],{"className":62191},[1261,140],[28,62193,148],{"className":62194},[147],[28,62196,62198],{"className":62197},[120],[28,62199,62202],{"className":62200,"style":62201},[124],"height:0.2831em;",[28,62203],{}," be the space of residual polynomials (polynomials whose value at 0 is 1) of degree at most ",[28,62206,62208,62221],{"className":62207},[35],[28,62209,62211],{"className":62210},[39],[41,62212,62213],{"xmlns":43},[46,62214,62215,62219],{},[49,62216,62217],{},[55,62218,24201],{},[82,62220,24201],{"encoding":84},[28,62222,62224],{"className":62223,"ariaHidden":71},[89],[28,62225,62227,62230],{"className":62226},[93],[28,62228],{"className":62229,"style":9597},[97],[28,62231,24201],{"className":62232},[102,106],". The solution to",[28,62235,62237],{"className":62236},[31],[28,62238,62240,62331],{"className":62239},[35],[28,62241,62243],{"className":62242},[39],[41,62244,62245],{"xmlns":43,"display":44},[46,62246,62247,62328],{},[49,62248,62249,62251,62253,62255,62257,62259,62261,62263,62265,62267,62290,62314,62316,62318,62320,62322,62324,62326],{},[55,62250,444],{},[62,62252,65],{"stretchy":64},[55,62254,19],{},[62,62256,72],{"separator":71},[55,62258,38743],{},[62,62260,72],{"separator":71},[55,62262,24201],{},[62,62264,78],{"stretchy":64},[62,62266,845],{},[56229,62268,62269,62275],{},[49,62270,62271,62273],{},[55,62272,22201],{},[62,62274,4008],{},[49,62276,62277,62279,62282],{},[55,62278,12],{},[62,62280,62281],{},"∈",[1670,62283,62284,62286,62288],{},[55,62285,62133],{"mathvariant":1088},[55,62287,24201],{},[62,62289,1679],{},[56229,62291,62292,62298],{},[49,62293,62294,62296],{},[55,62295,22207],{},[62,62297,4008],{},[49,62299,62300,62302,62304,62306,62308,62310,62312],{},[55,62301,3169],{},[62,62303,62281],{},[62,62305,3176],{"stretchy":64},[55,62307,19],{},[62,62309,72],{"separator":71},[55,62311,38743],{},[62,62313,3217],{"stretchy":64},[55,62315,296],{"mathvariant":1088},[55,62317,12],{},[62,62319,65],{"stretchy":64},[55,62321,3169],{},[62,62323,78],{"stretchy":64},[55,62325,296],{"mathvariant":1088},[173,62327],{"width":3182},[82,62329,62330],{"encoding":84},"\\beta(a, b, d) = \\min_{p \\in \\Pi^*_d} \\max_{x \\in [a, b]} |p(x)| \\quad",[28,62332,62334,62379],{"className":62333,"ariaHidden":71},[89],[28,62335,62337,62340,62343,62346,62349,62352,62355,62358,62361,62364,62367,62370,62373,62376],{"className":62336},[93],[28,62338],{"className":62339,"style":98},[97],[28,62341,444],{"className":62342,"style":470},[102,106],[28,62344,65],{"className":62345},[161],[28,62347,19],{"className":62348},[102,106],[28,62350,72],{"className":62351},[169],[28,62353],{"className":62354,"style":174},[173],[28,62356,38743],{"className":62357},[102,106],[28,62359,72],{"className":62360},[169],[28,62362],{"className":62363,"style":174},[173],[28,62365,24201],{"className":62366},[102,106],[28,62368,78],{"className":62369},[182],[28,62371],{"className":62372,"style":913},[173],[28,62374,845],{"className":62375},[917],[28,62377],{"className":62378,"style":913},[173],[28,62380,62382,62386,62494,62497,62564,62567,62570,62573,62576,62579,62582,62585],{"className":62381},[93],[28,62383],{"className":62384,"style":62385},[97],"height:1.7374em;vertical-align:-0.9874em;",[28,62387,62389],{"className":62388},[4241,53880],[28,62390,62392,62485],{"className":62391},[115,116],[28,62393,62395,62482],{"className":62394},[120],[28,62396,62399,62472],{"className":62397,"style":62398},[124],"height:0.6679em;",[28,62400,62402,62405],{"style":62401},"top:-2.3557em;margin-left:0em;",[28,62403],{"className":62404,"style":3434},[132],[28,62406,62408],{"className":62407},[137,138,139,140],[28,62409,62411,62414,62417],{"className":62410},[102,140],[28,62412,12],{"className":62413},[102,106,140],[28,62415,62281],{"className":62416},[917,140],[28,62418,62420,62423],{"className":62419},[102,140],[28,62421,62133],{"className":62422},[102,140],[28,62424,62426],{"className":62425},[111],[28,62427,62429,62463],{"className":62428},[115,116],[28,62430,62432,62460],{"className":62431},[120],[28,62433,62436,62448],{"className":62434,"style":62435},[124],"height:0.6771em;",[28,62437,62439,62442],{"style":62438},"top:-2.1528em;margin-left:0em;margin-right:0.0714em;",[28,62440],{"className":62441,"style":30534},[132],[28,62443,62445],{"className":62444},[137,30538,28701,140],[28,62446,24201],{"className":62447},[102,106,140],[28,62449,62451,62454],{"style":62450},"top:-2.8448em;margin-right:0.0714em;",[28,62452],{"className":62453,"style":30534},[132],[28,62455,62457],{"className":62456},[137,30538,28701,140],[28,62458,1679],{"className":62459},[1261,140],[28,62461,148],{"className":62462},[147],[28,62464,62466],{"className":62465},[120],[28,62467,62470],{"className":62468,"style":62469},[124],"height:0.3472em;",[28,62471],{},[28,62473,62474,62477],{"style":10025},[28,62475],{"className":62476,"style":3434},[132],[28,62478,62479],{},[28,62480,22201],{"className":62481},[4241],[28,62483,148],{"className":62484},[147],[28,62486,62488],{"className":62487},[120],[28,62489,62492],{"className":62490,"style":62491},[124],"height:0.9874em;",[28,62493],{},[28,62495],{"className":62496,"style":174},[173],[28,62498,62500],{"className":62499},[4241,53880],[28,62501,62503,62555],{"className":62502},[115,116],[28,62504,62506,62552],{"className":62505},[120],[28,62507,62509,62542],{"className":62508,"style":245},[124],[28,62510,62512,62515],{"style":62511},"top:-2.309em;margin-left:0em;",[28,62513],{"className":62514,"style":3434},[132],[28,62516,62518],{"className":62517},[137,138,139,140],[28,62519,62521,62524,62527,62530,62533,62536,62539],{"className":62520},[102,140],[28,62522,3169],{"className":62523},[102,106,140],[28,62525,62281],{"className":62526},[917,140],[28,62528,3176],{"className":62529},[161,140],[28,62531,19],{"className":62532},[102,106,140],[28,62534,72],{"className":62535},[169,140],[28,62537,38743],{"className":62538},[102,106,140],[28,62540,3217],{"className":62541},[182,140],[28,62543,62544,62547],{"style":10025},[28,62545],{"className":62546,"style":3434},[132],[28,62548,62549],{},[28,62550,22207],{"className":62551},[4241],[28,62553,148],{"className":62554},[147],[28,62556,62558],{"className":62557},[120],[28,62559,62562],{"className":62560,"style":62561},[124],"height:0.966em;",[28,62563],{},[28,62565],{"className":62566,"style":174},[173],[28,62568,296],{"className":62569},[102],[28,62571,12],{"className":62572},[102,106],[28,62574,65],{"className":62575},[161],[28,62577,3169],{"className":62578},[102,106],[28,62580,78],{"className":62581},[182],[28,62583,296],{"className":62584},[102],[28,62586],{"className":62587,"style":35442},[173],[12,62589,16490],{},[28,62591,62593],{"className":62592},[31],[28,62594,62596,62698],{"className":62595},[35],[28,62597,62599],{"className":62598},[39],[41,62600,62601],{"xmlns":43,"display":44},[46,62602,62603,62695],{},[49,62604,62605,62611,62613,62615,62617,62619,62691,62693],{},[2776,62606,62607,62609],{},[55,62608,12],{},[62,62610,1679],{},[62,62612,65],{"stretchy":64},[55,62614,3169],{},[62,62616,78],{"stretchy":64},[62,62618,845],{},[3977,62620,62621,62659],{},[49,62622,62623,62629],{},[52,62624,62625,62627],{},[55,62626,1674],{},[55,62628,24201],{},[49,62630,62631,62633,62657],{},[62,62632,65],{"fence":71},[3977,62634,62635,62649],{},[49,62636,62637,62639,62641,62643,62645,62647],{},[55,62638,38743],{},[62,62640,2606],{},[55,62642,19],{},[62,62644,1213],{},[650,62646,848],{},[55,62648,3169],{},[49,62650,62651,62653,62655],{},[55,62652,38743],{},[62,62654,1213],{},[55,62656,19],{},[62,62658,78],{"fence":71},[49,62660,62661,62667],{},[52,62662,62663,62665],{},[55,62664,1674],{},[55,62666,24201],{},[49,62668,62669,62671,62689],{},[62,62670,65],{"fence":71},[3977,62672,62673,62681],{},[49,62674,62675,62677,62679],{},[55,62676,38743],{},[62,62678,2606],{},[55,62680,19],{},[49,62682,62683,62685,62687],{},[55,62684,38743],{},[62,62686,1213],{},[55,62688,19],{},[62,62690,78],{"fence":71},[62,62692,72],{"separator":71},[173,62694],{"width":3182},[82,62696,62697],{"encoding":84},"p^*(x) = \\frac{T_d\\left(\\frac{b + a - 2x}{b - a}\\right)}{T_d\\left(\\frac{b + a}{b - a}\\right)}, \\quad",[28,62699,62701,62754],{"className":62700,"ariaHidden":71},[89],[28,62702,62704,62707,62736,62739,62742,62745,62748,62751],{"className":62703},[93],[28,62705],{"className":62706,"style":98},[97],[28,62708,62710,62713],{"className":62709},[102],[28,62711,12],{"className":62712},[102,106],[28,62714,62716],{"className":62715},[111],[28,62717,62719],{"className":62718},[115],[28,62720,62722],{"className":62721},[120],[28,62723,62725],{"className":62724,"style":8033},[124],[28,62726,62727,62730],{"style":2839},[28,62728],{"className":62729,"style":133},[132],[28,62731,62733],{"className":62732},[137,138,139,140],[28,62734,1679],{"className":62735},[1261,140],[28,62737,65],{"className":62738},[161],[28,62740,3169],{"className":62741},[102,106],[28,62743,78],{"className":62744},[182],[28,62746],{"className":62747,"style":913},[173],[28,62749,845],{"className":62750},[917],[28,62752],{"className":62753,"style":913},[173],[28,62755,62757,62761,63110,63113],{"className":62756},[93],[28,62758],{"className":62759,"style":62760},[97],"height:2.8469em;vertical-align:-1.1734em;",[28,62762,62764,62767,63107],{"className":62763},[102],[28,62765],{"className":62766},[161,4072],[28,62768,62770],{"className":62769},[3977],[28,62771,62773,63098],{"className":62772},[115,116],[28,62774,62776,63095],{"className":62775},[120],[28,62777,62780,62931,62939],{"className":62778,"style":62779},[124],"height:1.6734em;",[28,62781,62783,62786],{"style":62782},"top:-2.2299em;",[28,62784],{"className":62785,"style":3434},[132],[28,62787,62789,62829,62832],{"className":62788},[102],[28,62790,62792,62795],{"className":62791},[102],[28,62793,1674],{"className":62794,"style":107},[102,106],[28,62796,62798],{"className":62797},[111],[28,62799,62801,62821],{"className":62800},[115,116],[28,62802,62804,62818],{"className":62803},[120],[28,62805,62807],{"className":62806,"style":10240},[124],[28,62808,62809,62812],{"style":128},[28,62810],{"className":62811,"style":133},[132],[28,62813,62815],{"className":62814},[137,138,139,140],[28,62816,24201],{"className":62817},[102,106,140],[28,62819,148],{"className":62820},[147],[28,62822,62824],{"className":62823},[120],[28,62825,62827],{"className":62826,"style":155},[124],[28,62828],{},[28,62830],{"className":62831,"style":174},[173],[28,62833,62835,62841,62925],{"className":62834},[3398],[28,62836,62838],{"className":62837,"style":3403},[161,3402],[28,62839,65],{"className":62840},[3407,28701],[28,62842,62844,62847,62922],{"className":62843},[102],[28,62845],{"className":62846},[161,4072],[28,62848,62850],{"className":62849},[3977],[28,62851,62853,62913],{"className":62852},[115,116],[28,62854,62856,62910],{"className":62855},[120],[28,62857,62860,62881,62889],{"className":62858,"style":62859},[124],"height:0.8801em;",[28,62861,62863,62866],{"style":62862},"top:-2.655em;",[28,62864],{"className":62865,"style":3434},[132],[28,62867,62869],{"className":62868},[137,138,139,140],[28,62870,62872,62875,62878],{"className":62871},[102,140],[28,62873,38743],{"className":62874},[102,106,140],[28,62876,1213],{"className":62877},[1261,140],[28,62879,19],{"className":62880},[102,106,140],[28,62882,62883,62886],{"style":4100},[28,62884],{"className":62885,"style":3434},[132],[28,62887],{"className":62888,"style":4108},[4107],[28,62890,62892,62895],{"style":62891},"top:-3.394em;",[28,62893],{"className":62894,"style":3434},[132],[28,62896,62898],{"className":62897},[137,138,139,140],[28,62899,62901,62904,62907],{"className":62900},[102,140],[28,62902,38743],{"className":62903},[102,106,140],[28,62905,2606],{"className":62906},[1261,140],[28,62908,19],{"className":62909},[102,106,140],[28,62911,148],{"className":62912},[147],[28,62914,62916],{"className":62915},[120],[28,62917,62920],{"className":62918,"style":62919},[124],"height:0.4033em;",[28,62921],{},[28,62923],{"className":62924},[182,4072],[28,62926,62928],{"className":62927,"style":3403},[182,3402],[28,62929,78],{"className":62930},[3407,28701],[28,62932,62933,62936],{"style":4100},[28,62934],{"className":62935,"style":3434},[132],[28,62937],{"className":62938,"style":4108},[4107],[28,62940,62942,62945],{"style":62941},"top:-3.7933em;",[28,62943],{"className":62944,"style":3434},[132],[28,62946,62948,62988,62991],{"className":62947},[102],[28,62949,62951,62954],{"className":62950},[102],[28,62952,1674],{"className":62953,"style":107},[102,106],[28,62955,62957],{"className":62956},[111],[28,62958,62960,62980],{"className":62959},[115,116],[28,62961,62963,62977],{"className":62962},[120],[28,62964,62966],{"className":62965,"style":10240},[124],[28,62967,62968,62971],{"style":128},[28,62969],{"className":62970,"style":133},[132],[28,62972,62974],{"className":62973},[137,138,139,140],[28,62975,24201],{"className":62976},[102,106,140],[28,62978,148],{"className":62979},[147],[28,62981,62983],{"className":62982},[120],[28,62984,62986],{"className":62985,"style":155},[124],[28,62987],{},[28,62989],{"className":62990,"style":174},[173],[28,62992,62994,63000,630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the corresponding minimal value is",[28,63120,63122],{"className":63121},[31],[28,63123,63125,63193],{"className":63124},[35],[28,63126,63128],{"className":63127},[39],[41,63129,63130],{"xmlns":43,"display":44},[46,63131,63132,63190],{},[49,63133,63134,63136,63138,63140,63142,63144,63146,63148,63150,63152,63164,63188],{},[55,63135,444],{},[62,63137,65],{"stretchy":64},[55,63139,19],{},[62,63141,72],{"separator":71},[55,63143,38743],{},[62,63145,72],{"separator":71},[55,63147,24201],{},[62,63149,78],{"stretchy":64},[62,63151,845],{},[1670,63153,63154,63156,63158],{},[55,63155,1674],{},[55,63157,24201],{},[49,63159,63160,63162],{},[62,63161,1213],{},[650,63163,3200],{},[49,63165,63166,63168,63186],{},[62,63167,65],{"fence":71},[3977,63169,63170,63178],{},[49,63171,63172,63174,63176],{},[55,63173,38743],{},[62,63175,2606],{},[55,63177,19],{},[49,63179,63180,63182,63184],{},[55,63181,38743],{},[62,63183,1213],{},[55,63185,19],{},[62,63187,78],{"fence":71},[55,63189,2106],{"mathvariant":1088},[82,63191,63192],{"encoding":84},"\\beta(a, b, d) = T_d^{-1}\\left(\\frac{b + a}{b - a}\\right).",[28,63194,63196,63241],{"className":63195,"ariaHidden":71},[89],[28,63197,63199,63202,63205,63208,63211,63214,63217,63220,63223,63226,63229,63232,63235,63238],{"className":63198},[93],[28,63200],{"className":63201,"style":98},[97],[28,63203,444],{"className":63204,"style":470},[102,106],[28,63206,65],{"className":63207},[161],[28,63209,19],{"className":63210},[102,106],[28,63212,72],{"className":63213},[169],[28,63215],{"className":63216,"style":174},[173],[28,63218,38743],{"className":63219},[102,106],[28,63221,72],{"className":63222},[169],[28,63224],{"className":63225,"style":174},[173],[28,63227,24201],{"className":63228},[102,106],[28,63230,78],{"className":63231},[182],[28,63233],{"className":63234,"style":913},[173],[28,63236,845],{"className":63237},[917],[28,63239],{"className":63240,"style":913},[173],[28,63242,63244,63247,63306,63309,63411,63414],{"className":63243},[93],[28,63245],{"className":63246,"style":3394},[97],[28,63248,63250,63253],{"className":63249},[102],[28,63251,1674],{"className":63252,"style":107},[102,106],[28,63254,63256],{"className":63255},[111],[28,63257,63259,63297],{"className":63258},[115,116],[28,63260,63262,63294],{"className":63261},[120],[28,63263,63265,63277],{"className":63264,"style":2836},[124],[28,63266,63268,63271],{"style":63267},"top:-2.4086em;margin-left:-0.1389em;margin-right:0.05em;",[28,63269],{"className":63270,"style":133},[132],[28,63272,63274],{"className":63273},[137,138,139,140],[28,63275,24201],{"className":63276},[102,106,140],[28,63278,63279,63282],{"style":2839},[28,63280],{"className":63281,"style":133},[132],[28,63283,63285],{"className":63284},[137,138,139,140],[28,63286,63288,63291],{"className":63287},[102,140],[28,63289,1213],{"className":63290},[102,140],[28,63292,3200],{"className":63293},[102,140],[28,63295,148],{"className":63296},[147],[28,63298,63300],{"className":63299},[120],[28,63301,63304],{"className":63302,"style":63303},[124],"height:0.2914em;",[28,63305],{},[28,63307],{"className":63308,"style":174},[173],[28,63310,63312,63318,63405],{"className":63311},[3398],[28,63313,63315],{"className":63314,"style":3403},[161,3402],[28,63316,65],{"className":63317},[3407,139],[28,63319,63321,63324,63402],{"className":63320},[102],[28,63322],{"className":63323},[161,4072],[28,63325,63327],{"className":63326},[3977],[28,63328,63330,63393],{"className":63329},[115,116],[28,63331,63333,63390],{"className":63332},[120],[28,63334,63336,63359,63367],{"className":63335,"style":9458},[124],[28,63337,63338,63341],{"style":4088},[28,63339],{"className":63340,"style":3434},[132],[28,63342,63344,63347,63350,63353,63356],{"className":63343},[102],[28,63345,38743],{"className":63346},[102,106],[28,63348],{"className":63349,"style":1257},[173],[28,63351,1213],{"className":63352},[1261],[28,63354],{"className":63355,"style":1257},[173],[28,63357,19],{"className":63358},[102,106],[28,63360,63361,63364],{"style":4100},[28,63362],{"className":63363,"style":3434},[132],[28,63365],{"className":63366,"style":4108},[4107],[28,63368,63369,63372],{"style":4111},[28,63370],{"className":63371,"style":3434},[132],[28,63373,63375,63378,63381,63384,63387],{"className":63374},[102],[28,63376,38743],{"className":63377},[102,106],[28,63379],{"className":63380,"style":1257},[173],[28,63382,2606],{"className":63383},[1261],[28,63385],{"className":63386,"style":1257},[173],[28,63388,19],{"className":63389},[102,106],[28,63391,148],{"className":63392},[147],[28,63394,63396],{"className":63395},[120],[28,63397,63400],{"className":63398,"style":63399},[124],"height:0.7693em;",[28,63401],{},[28,63403],{"className":63404},[182,4072],[28,63406,63408],{"className":63407,"style":3403},[182,3402],[28,63409,78],{"className":63410},[3407,139],[28,63412],{"className":63413,"style":174},[173],[28,63415,2106],{"className":63416},[102],[12,63418,63419,63420,63472],{},"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 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= \\frac{1-\\cos\\big((E-E_0)dt\\big)}{2},",[28,63541,63543,63570],{"className":63542,"ariaHidden":71},[89],[28,63544,63546,63549,63552,63555,63558,63561,63564,63567],{"className":63545},[93],[28,63547],{"className":63548,"style":98},[97],[28,63550,299],{"className":63551,"style":321},[102,106],[28,63553,65],{"className":63554},[161],[28,63556,1871],{"className":63557,"style":1913},[102,106],[28,63559,78],{"className":63560},[182],[28,63562],{"className":63563,"style":913},[173],[28,63565,845],{"className":63566},[917],[28,63568],{"className":63569,"style":913},[173],[28,63571,63573,63577,63732],{"className":63572},[93],[28,63574],{"className":63575,"style":63576},[97],"height:2.276em;vertical-align:-0.686em;",[28,63578,63580,63583,63729],{"className":63579},[102],[28,63581],{"className":63582},[161,4072],[28,63584,63586],{"className":63585},[3977],[28,63587,63589,63721],{"className":63588},[115,116],[28,63590,63592,63718],{"className":63591},[120],[28,63593,63596,63607,63615],{"className":63594,"style":63595},[124],"height:1.59em;",[28,63597,63598,63601],{"style":4088},[28,63599],{"className":63600,"style":3434},[132],[28,63602,63604],{"className":63603},[102],[28,63605,848],{"className":63606},[102],[28,63608,63609,63612],{"style":4100},[28,63610],{"className":63611,"style":3434},[132],[28,63613],{"className":63614,"style":4108},[4107],[28,63616,63618,63621],{"style":63617},"top:-3.74em;",[28,63619],{"className":63620,"style":3434},[132],[28,63622,63624,63627,63630,63633,63636,63639,63642,63648,63651,63654,63657,63660,63663,63703,63706,63709,63712],{"className":63623},[102],[28,63625,3200],{"className":63626},[102],[28,63628],{"className":63629,"style":1257},[173],[28,63631,1213],{"className":63632},[1261],[28,63634],{"className":63635,"style":1257},[173],[28,63637,4005],{"className":63638},[4241],[28,63640],{"className":63641,"style":174},[173],[28,63643,63645],{"className":63644},[102],[28,63646,65],{"className":63647},[3407,28701],[28,63649,65],{"className":63650},[161],[28,63652,1871],{"className":63653,"style":1913},[102,106],[28,63655],{"className":63656,"style":1257},[173],[28,63658,1213],{"className":63659},[1261],[28,63661],{"className":63662,"style":1257},[173],[28,63664,63666,63669],{"className":63665},[102],[28,63667,1871],{"className":63668,"style":1913},[102,106],[28,63670,63672],{"className":63671},[111],[28,63673,63675,63695],{"className":63674},[115,116],[28,63676,63678,63692],{"className":63677},[120],[28,63679,63681],{"className":63680,"style":683},[124],[28,63682,63683,63686],{"style":1928},[28,63684],{"className":63685,"style":133},[132],[28,63687,63689],{"className":63688},[137,138,139,140],[28,63690,652],{"className":63691},[102,140],[28,63693,148],{"className":63694},[147],[28,63696,63698],{"className":63697},[120],[28,63699,63701],{"className":63700,"style":155},[124],[28,63702],{},[28,63704,78],{"className":63705},[182],[28,63707,24201],{"className":63708},[102,106],[28,63710,3971],{"className":63711},[102,106],[28,63713,63715],{"className":63714},[102],[28,63716,78],{"className":63717},[3407,28701],[28,63719,148],{"className":63720},[147],[28,63722,63724],{"className":63723},[120],[28,63725,63727],{"className":63726,"style":4170},[124],[28,63728],{},[28,63730],{"className":63731},[182,4072],[28,63733,72],{"className":63734},[169],[12,63736,188,63737,63770,63771,63977,63978,512,64085,64129,64130,64158,64159,2106],{},[28,63738,63740,63755],{"className":63739},[35],[28,63741,63743],{"className":63742},[39],[41,63744,63745],{"xmlns":43},[46,63746,63747,63753],{},[49,63748,63749,63751],{},[55,63750,24201],{},[55,63752,3971],{},[82,63754,31182],{"encoding":84},[28,63756,63758],{"className":63757,"ariaHidden":71},[89],[28,63759,63761,63764,63767],{"className":63760},[93],[28,63762],{"className":63763,"style":9597},[97],[28,63765,24201],{"className":63766},[102,106],[28,63768,3971],{"className":63769},[102,106]," is a timestep such that ",[28,63772,63774,63818],{"className":63773},[35],[28,63775,63777],{"className":63776},[39],[41,63778,63779],{"xmlns":43},[46,63780,63781,63815],{},[49,63782,63783,63785,63787,63789,63795,63797,63799,63801,63807,63809,63811,63813],{},[62,63784,1213],{},[55,63786,851],{},[62,63788,22950],{},[52,63790,63791,63793],{},[55,63792,1871],{},[650,63794,652],{},[55,63796,24201],{},[55,63798,3971],{},[62,63800,22950],{},[52,63802,63803,63805],{},[55,63804,1871],{},[2253,63806,22207],{},[55,63808,24201],{},[55,63810,3971],{},[62,63812,22950],{},[55,63814,851],{},[82,63816,63817],{"encoding":84},"-\\pi \u003C E_0dt \u003C E_\\text{max}dt \u003C \\pi",[28,63819,63821,63842,63904,63968],{"className":63820,"ariaHidden":71},[89],[28,63822,63824,63827,63830,63833,63836,63839],{"className":63823},[93],[28,63825],{"className":63826,"style":1250},[97],[28,63828,1213],{"className":63829},[102],[28,63831,851],{"className":63832,"style":321},[102,106],[28,63834],{"className":63835,"style":913},[173],[28,63837,22950],{"className":63838},[917],[28,63840],{"className":63841,"style":913},[173],[28,63843,63845,63849,63889,63892,63895,63898,63901],{"className":63844},[93],[28,63846],{"className":63847,"style":63848},[97],"height:0.8444em;vertical-align:-0.15em;",[28,63850,63852,63855],{"className":63851},[102],[28,63853,1871],{"className":63854,"style":1913},[102,106],[28,63856,63858],{"className":63857},[111],[28,63859,63861,63881],{"className":63860},[115,116],[28,63862,63864,63878],{"className":63863},[120],[28,63865,63867],{"className":63866,"style":683},[124],[28,63868,63869,63872],{"style":1928},[28,63870],{"className":63871,"style":133},[132],[28,63873,63875],{"className":63874},[137,138,139,140],[28,63876,652],{"className":63877},[102,140],[28,63879,148],{"className":63880},[147],[28,63882,63884],{"className":63883},[120],[28,63885,63887],{"className":63886,"style":155},[124],[28,63888],{},[28,63890,24201],{"className":63891},[102,106],[28,63893,3971],{"className":63894},[102,106],[28,63896],{"className":63897,"style":913},[173],[28,63899,22950],{"className":63900},[917],[28,63902],{"className":63903,"style":913},[173],[28,63905,63907,63910,63953,63956,63959,63962,63965],{"className":63906},[93],[28,63908],{"className":63909,"style":63848},[97],[28,63911,63913,63916],{"className":63912},[102],[28,63914,1871],{"className":63915,"style":1913},[102,106],[28,63917,63919],{"className":63918},[111],[28,63920,63922,63945],{"className":63921},[115,116],[28,63923,63925,63942],{"className":63924},[120],[28,63926,63928],{"className":63927,"style":125},[124],[28,63929,63930,63933],{"style":1928},[28,63931],{"className":63932,"style":133},[132],[28,63934,63936],{"className":63935},[137,138,139,140],[28,63937,63939],{"className":63938},[102,16058,140],[28,63940,22207],{"className":63941},[102,140],[28,63943,148],{"className":63944},[147],[28,63946,63948],{"className":63947},[120],[28,63949,63951],{"className":63950,"style":155},[124],[28,63952],{},[28,63954,24201],{"className":63955},[102,106],[28,63957,3971],{"className":63958},[102,106],[28,63960],{"className":63961,"style":913},[173],[28,63963,22950],{"className":63964},[917],[28,63966],{"className":63967,"style":913},[173],[28,63969,63971,63974],{"className":63970},[93],[28,63972],{"className":63973,"style":245},[97],[28,63975,851],{"className":63976,"style":321},[102,106],".\nNotice that ",[28,63979,63981,64009],{"className":63980},[35],[28,63982,63984],{"className":63983},[39],[41,63985,63986],{"xmlns":43},[46,63987,63988,64006],{},[49,63989,63990,63992,63994,64000,64002,64004],{},[55,63991,299],{},[62,63993,65],{"stretchy":64},[52,63995,63996,63998],{},[55,63997,1871],{},[650,63999,652],{},[62,64001,78],{"stretchy":64},[62,64003,845],{},[650,64005,652],{},[82,64007,64008],{"encoding":84},"g(E_0)=0",[28,64010,64012,64076],{"className":64011,"ariaHidden":71},[89],[28,64013,64015,64018,64021,64024,64064,64067,64070,64073],{"className":64014},[93],[28,64016],{"className":64017,"style":98},[97],[28,64019,299],{"className":64020,"style":321},[102,106],[28,64022,65],{"className":64023},[161],[28,64025,64027,64030],{"className":64026},[102],[28,64028,1871],{"className":64029,"style":1913},[102,106],[28,64031,64033],{"className":64032},[111],[28,64034,64036,64056],{"className":64035},[115,116],[28,64037,64039,64053],{"className":64038},[120],[28,64040,64042],{"className":64041,"style":683},[124],[28,64043,64044,64047],{"style":1928},[28,64045],{"className":64046,"style":133},[132],[28,64048,64050],{"className":64049},[137,138,139,140],[28,64051,652],{"className":64052},[102,140],[28,64054,148],{"className":64055},[147],[28,64057,64059],{"className":64058},[120],[28,64060,64062],{"className":64061,"style":155},[124],[28,64063],{},[28,64065,78],{"className":64066},[182],[28,64068],{"className":64069,"style":913},[173],[28,64071,845],{"className":64072},[917],[28,64074],{"className":64075,"style":913},[173],[28,64077,64079,64082],{"className":64078},[93],[28,64080],{"className":64081,"style":2914},[97],[28,64083,652],{"className":64084},[102],[28,64086,64088,64108],{"className":64087},[35],[28,64089,64091],{"className":64090},[39],[41,64092,64093],{"xmlns":43},[46,64094,64095,64105],{},[49,64096,64097,64099,64101,64103],{},[55,64098,299],{},[62,64100,65],{"stretchy":64},[55,64102,1871],{},[62,64104,78],{"stretchy":64},[82,64106,64107],{"encoding":84},"g(E)",[28,64109,64111],{"className":64110,"ariaHidden":71},[89],[28,64112,64114,64117,64120,64123,64126],{"className":64113},[93],[28,64115],{"className":64116,"style":98},[97],[28,64118,299],{"className":64119,"style":321},[102,106],[28,64121,65],{"className":64122},[161],[28,64124,1871],{"className":64125,"style":1913},[102,106],[28,64127,78],{"className":64128},[182]," grows as ",[28,64131,64133,64146],{"className":64132},[35],[28,64134,64136],{"className":64135},[39],[41,64137,64138],{"xmlns":43},[46,64139,64140,64144],{},[49,64141,64142],{},[55,64143,1871],{},[82,64145,1871],{"encoding":84},[28,64147,64149],{"className":64148,"ariaHidden":71},[89],[28,64150,64152,64155],{"className":64151},[93],[28,64153],{"className":64154,"style":1102},[97],[28,64156,1871],{"className":64157,"style":1913},[102,106]," moves away from ",[28,64160,64162,64179],{"className":64161},[35],[28,64163,64165],{"className":64164},[39],[41,64166,64167],{"xmlns":43},[46,64168,64169,64177],{},[49,64170,64171],{},[52,64172,64173,64175],{},[55,64174,1871],{},[650,64176,652],{},[82,64178,39888],{"encoding":84},[28,64180,64182],{"className":64181,"ariaHidden":71},[89],[28,64183,64185,64188],{"className":64184},[93],[28,64186],{"className":64187,"style":585},[97],[28,64189,64191,64194],{"className":64190},[102],[28,64192,1871],{"className":64193,"style":1913},[102,106],[28,64195,64197],{"className":64196},[111],[28,64198,64200,64220],{"className":64199},[115,116],[28,64201,64203,64217],{"className":64202},[120],[28,64204,64206],{"className":64205,"style":683},[124],[28,64207,64208,64211],{"style":1928},[28,64209],{"className":64210,"style":133},[132],[28,64212,64214],{"className":64213},[137,138,139,140],[28,64215,652],{"className":64216},[102,140],[28,64218,148],{"className":64219},[147],[28,64221,64223],{"className":64222},[120],[28,64224,64226],{"className":64225,"style":155},[124],[28,64227],{},[12,64229,64230,64231,64305,64306,16397,64435,64486],{},"Now using the polynomial ",[28,64232,64234,64258],{"className":64233},[35],[28,64235,64237],{"className":64236},[39],[41,64238,64239],{"xmlns":43},[46,64240,64241,64255],{},[49,64242,64243,64249,64251,64253],{},[2776,64244,64245,64247],{},[55,64246,12],{},[62,64248,1679],{},[62,64250,65],{"stretchy":64},[55,64252,3169],{},[62,64254,78],{"stretchy":64},[82,64256,64257],{"encoding":84},"p^*(x)",[28,64259,64261],{"className":64260,"ariaHidden":71},[89],[28,64262,64264,64267,64296,64299,64302],{"className":64263},[93],[28,64265],{"className":64266,"style":98},[97],[28,64268,64270,64273],{"className":64269},[102],[28,64271,12],{"className":64272},[102,106],[28,64274,64276],{"className":64275},[111],[28,64277,64279],{"className":64278},[115],[28,64280,64282],{"className":64281},[120],[28,64283,64285],{"className":64284,"style":1711},[124],[28,64286,64287,64290],{"style":1726},[28,64288],{"className":64289,"style":133},[132],[28,64291,64293],{"className":64292},[137,138,139,140],[28,64294,1679],{"className":64295},[1261,140],[28,64297,65],{"className":64298},[161],[28,64300,3169],{"className":64301},[102,106],[28,64303,78],{"className":64304},[182]," with the parameters a, b, d set to ",[28,64307,64309,64341],{"className":64308},[35],[28,64310,64312],{"className":64311},[39],[41,64313,64314],{"xmlns":43},[46,64315,64316,64338],{},[49,64317,64318,64320,64322,64324,64326,64332,64334,64336],{},[55,64319,19],{},[62,64321,845],{},[55,64323,299],{},[62,64325,65],{"stretchy":64},[52,64327,64328,64330],{},[55,64329,1871],{},[650,64331,652],{},[62,64333,2606],{},[55,64335,53119],{},[62,64337,78],{"stretchy":64},[82,64339,64340],{"encoding":84},"a = g(E_0 + \\delta)",[28,64342,64344,64362,64423],{"className":64343,"ariaHidden":71},[89],[28,64345,64347,64350,64353,64356,64359],{"className":64346},[93],[28,64348],{"className":64349,"style":245},[97],[28,64351,19],{"className":64352},[102,106],[28,64354],{"className":64355,"style":913},[173],[28,64357,845],{"className":64358},[917],[28,64360],{"className":64361,"style":913},[173],[28,64363,64365,64368,64371,64374,64414,64417,64420],{"className":64364},[93],[28,64366],{"className":64367,"style":98},[97],[28,64369,299],{"className":64370,"style":321},[102,106],[28,64372,65],{"className":64373},[161],[28,64375,64377,64380],{"className":64376},[102],[28,64378,1871],{"className":64379,"style":1913},[102,106],[28,64381,64383],{"className":64382},[111],[28,64384,64386,64406],{"className":64385},[115,116],[28,64387,64389,64403],{"className":64388},[120],[28,64390,64392],{"className":64391,"style":683},[124],[28,64393,64394,64397],{"style":1928},[28,64395],{"className":64396,"style":133},[132],[28,64398,64400],{"className":64399},[137,138,139,140],[28,64401,652],{"className":64402},[102,140],[28,64404,148],{"className":64405},[147],[28,64407,64409],{"className":64408},[120],[28,64410,64412],{"className":64411,"style":155},[124],[28,64413],{},[28,64415],{"className":64416,"style":1257},[173],[28,64418,2606],{"className":64419},[1261],[28,64421],{"className":64422,"style":1257},[173],[28,64424,64426,64429,64432],{"className":64425},[93],[28,64427],{"className":64428,"style":98},[97],[28,64430,53119],{"className":64431,"style":53204},[102,106],[28,64433,78],{"className":64434},[182],[28,64436,64438,64456],{"className":64437},[35],[28,64439,64441],{"className":64440},[39],[41,64442,64443],{"xmlns":43},[46,64444,64445,64453],{},[49,64446,64447,64449,64451],{},[55,64448,38743],{},[62,64450,845],{},[650,64452,3200],{},[82,64454,64455],{"encoding":84},"b = 1",[28,64457,64459,64477],{"className":64458,"ariaHidden":71},[89],[28,64460,64462,64465,64468,64471,64474],{"className":64461},[93],[28,64463],{"className":64464,"style":9597},[97],[28,64466,38743],{"className":64467},[102,106],[28,64469],{"className":64470,"style":913},[173],[28,64472,845],{"className":64473},[917],[28,64475],{"className":64476,"style":913},[173],[28,64478,64480,64483],{"className":64479},[93],[28,64481],{"className":64482,"style":2914},[97],[28,64484,3200],{"className":64485},[102],", and d = int(r\u002F2), we define the function:",[28,64488,64490],{"className":64489},[31],[28,64491,64493,64695],{"className":64492},[35],[28,64494,64496],{"className":64495},[39],[41,64497,64498],{"xmlns":43,"display":44},[46,64499,64500,64692],{},[49,64501,64502,64504,64506,64508,64510,64512,64518,64532,64534],{},[55,64503,68],{},[62,64505,65],{"stretchy":64},[55,64507,1871],{},[62,64509,78],{"stretchy":64},[62,64511,845],{},[2776,64513,64514,64516],{},[55,64515,12],{},[62,64517,1679],{},[49,64519,64520,64522,64524,64526,64528,64530],{},[62,64521,65],{"fence":71},[55,64523,299],{},[62,64525,65],{"stretchy":64},[55,64527,1871],{},[62,64529,78],{"stretchy":64},[62,64531,78],{"fence":71},[62,64533,845],{},[3977,64535,64536,64626],{},[49,64537,64538,64544],{},[52,64539,64540,64542],{},[55,64541,1674],{},[55,64543,24201],{},[49,64545,64546,64548,64550,64552,64554,64624],{},[62,64547,65],{"fence":71},[650,64549,3200],{},[62,64551,2606],{},[650,64553,848],{},[3977,64555,64556,64602],{},[49,64557,64558,64560,64562,64564,64566,64568,64570,64576,64578,64580,64582,64584,64586,64588,64590,64592,64594,64596,64598,64600],{},[55,64559,4005],{},[62,64561,4008],{},[62,64563,65],{"fence":64,"stretchy":71,"minsize":63512,"maxsize":63512},[62,64565,65],{"stretchy":64},[55,64567,1871],{},[62,64569,1213],{},[52,64571,64572,64574],{},[55,64573,1871],{},[650,64575,652],{},[62,64577,78],{"stretchy":64},[55,64579,24201],{},[55,64581,3971],{},[62,64583,78],{"fence":64,"stretchy":71,"minsize":63512,"maxsize":63512},[62,64585,1213],{},[55,64587,4005],{},[62,64589,4008],{},[62,64591,65],{"fence":64,"stretchy":71,"minsize":63512,"maxsize":63512},[55,64593,53119],{},[2253,64595,51285],{},[55,64597,24201],{},[55,64599,3971],{},[62,64601,78],{"fence":64,"stretchy":71,"minsize":63512,"maxsize":63512},[49,64603,64604,64606,64608,64610,64612,64614,64616,64618,64620,64622],{},[650,64605,3200],{},[62,64607,2606],{},[55,64609,4005],{},[62,64611,4008],{},[62,64613,65],{"fence":64,"stretchy":71,"minsize":63512,"maxsize":63512},[55,64615,53119],{},[2253,64617,51285],{},[55,64619,24201],{},[55,64621,3971],{},[62,64623,78],{"fence":64,"stretchy":71,"minsize":63512,"maxsize":63512},[62,64625,78],{"fence":71},[49,64627,64628,64634],{},[52,64629,64630,64632],{},[55,64631,1674],{},[55,64633,24201],{},[49,64635,64636,64638,64640,64642,64644,64690],{},[62,64637,65],{"fence":71},[650,64639,3200],{},[62,64641,2606],{},[650,64643,848],{},[3977,64645,64646,64668],{},[49,64647,64648,64650,64652,64654,64656,64658,64660,64662,64664,64666],{},[650,64649,3200],{},[62,64651,1213],{},[55,64653,4005],{},[62,64655,4008],{},[62,64657,65],{"fence":64,"stretchy":71,"minsize":63512,"maxsize":63512},[55,64659,53119],{},[2253,64661,51285],{},[55,64663,24201],{},[55,64665,3971],{},[62,64667,78],{"fence":64,"stretchy":71,"minsize":63512,"maxsize":63512},[49,64669,64670,64672,64674,64676,64678,64680,64682,64684,64686,64688],{},[650,64671,3200],{},[62,64673,2606],{},[55,64675,4005],{},[62,64677,4008],{},[62,64679,65],{"fence":64,"stretchy":71,"minsize":63512,"maxsize":63512},[55,64681,53119],{},[2253,64683,51285],{},[55,64685,24201],{},[55,64687,3971],{},[62,64689,78],{"fence":64,"stretchy":71,"minsize":63512,"maxsize":63512},[62,64691,78],{"fence":71},[82,64693,64694],{"encoding":84},"f(E) = p^* \\left( g(E) \\right) = \\frac{T_d\\left(1 + 2\\frac{\\cos\\big((E-E_0)dt\\big) - \\cos\\big(\\delta\\,dt\\big)}{1 +\\cos\\big(\\delta\\,dt\\big)}\\right)}{T_d\\left(1 + 2\\frac{1-\\cos\\big(\\delta\\,dt\\big)}{1 + \\cos\\big(\\delta\\,dt\\big)}\\right)}",[28,64696,64698,64725,64793],{"className":64697,"ariaHidden":71},[89],[28,64699,64701,64704,64707,64710,64713,64716,64719,64722],{"className":64700},[93],[28,64702],{"className":64703,"style":98},[97],[28,64705,68],{"className":64706,"style":165},[102,106],[28,64708,65],{"className":64709},[161],[28,64711,1871],{"className":64712,"style":1913},[102,106],[28,64714,78],{"className":64715},[182],[28,64717],{"className":64718,"style":913},[173],[28,64720,845],{"className":64721},[917],[28,64723],{"className":64724,"style":913},[173],[28,64726,64728,64731,64760,64763,64784,64787,64790],{"className":64727},[93],[28,64729],{"className":64730,"style":98},[97],[28,64732,64734,64737],{"className":64733},[102],[28,64735,12],{"className":64736},[102,106],[28,64738,64740],{"className":64739},[111],[28,64741,64743],{"className":64742},[115],[28,64744,64746],{"className":64745},[120],[28,64747,64749],{"className":64748,"style":8033},[124],[28,64750,64751,64754],{"style":2839},[28,64752],{"className":64753,"style":133},[132],[28,64755,64757],{"className":64756},[137,138,139,140],[28,64758,1679],{"className":64759},[1261,140],[28,64761],{"className":64762,"style":174},[173],[28,64764,64766,64769,64772,64775,64778,64781],{"className":64765},[3398],[28,64767,65],{"className":64768,"style":3403},[161,3402],[28,64770,299],{"className":64771,"style":321},[102,106],[28,64773,65],{"className":64774},[161],[28,64776,1871],{"className":64777,"style":1913},[102,106],[28,64779,78],{"className":64780},[182],[28,64782,78],{"className":64783,"style":3403},[182,3402],[28,64785],{"className":64786,"style":913},[173],[28,64788,845],{"className":64789},[917],[28,64791],{"className":64792,"style":913},[173],[28,64794,64796,64800],{"className":64795},[93],[28,64797],{"className":64798,"style":64799},[97],"height:5.32em;vertical-align:-2.41em;",[28,64801,64803,64806,65426],{"className":64802},[102],[28,64804],{"className":64805},[161,4072],[28,64807,64809],{"className":64808},[3977],[28,64810,64812,65417],{"className":64811},[115,116],[28,64813,64815,65414],{"className":64814},[120],[28,64816,64819,65072,65080],{"className":64817,"style":64818},[124],"height:2.91em;",[28,64820,64822,64826],{"style":64821},"top:-2.11em;",[28,64823],{"className":64824,"style":64825},[132],"height:3.51em;",[28,64827,64829,64869,64872],{"className":64828},[102],[28,64830,64832,64835],{"className":64831},[102],[28,64833,1674],{"className":64834,"style":107},[102,106],[28,64836,64838],{"className":64837},[111],[28,64839,64841,64861],{"className":64840},[115,116],[28,64842,64844,64858],{"className":64843},[120],[28,64845,64847],{"className":64846,"style":10240},[124],[28,64848,64849,64852],{"style":128},[28,64850],{"className":64851,"style":133},[132],[28,64853,64855],{"className":64854},[137,138,139,140],[28,64856,24201],{"className":64857},[102,106,140],[28,64859,148],{"className":64860},[147],[28,64862,64864],{"className":64863},[120],[28,64865,64867],{"className":64866,"style":155},[124],[28,64868],{},[28,64870],{"className":64871,"style":174},[173],[28,64873,64875,64881,64884,64887,64890,64893,64896,65066],{"className":64874},[3398],[28,64876,64878],{"className":64877,"style":3403},[161,3402],[28,64879,65],{"className":64880},[3407,139],[28,64882,3200],{"className":64883},[102],[28,64885],{"className":64886,"style":1257},[173],[28,64888,2606],{"className":64889},[1261],[28,64891],{"className":64892,"style":1257},[173],[28,64894,848],{"className":64895},[102],[28,64897,64899,64902,65063],{"className":64898},[102],[28,64900],{"className":64901},[161,4072],[28,64903,64905],{"className":64904},[3977],[28,64906,64908,65054],{"className":64907},[115,116],[28,64909,64911,65051],{"className":64910},[120],[28,64912,64915,64980,64988],{"className":64913,"style":64914},[124],"height:1.51em;",[28,64916,64918,64921],{"style":64917},"top:-2.34em;",[28,64919],{"className":64920,"style":3434},[132],[28,64922,64924],{"className":64923},[137,138,139,140],[28,64925,64927,64930,64933,64946,64949,64959,64962,64965,64968,64971],{"className":64926},[102,140],[28,64928,3200],{"className":64929},[102,140],[28,64931,2606],{"className":64932},[1261,140],[28,64934,64936,64939,64943],{"className":64935},[4241,140],[28,64937,31887],{"className":64938},[140],[28,64940,64942],{"className":64941},[140],"o",[28,64944,7109],{"className":64945},[140],[28,64947],{"className":64948,"style":51495},[173,140],[28,64950,64953],{"className":64951},[102,137,30538,64952,140],"size6",[28,64954,64956],{"className":64955},[3407,28701,140],[28,64957,65],{"className":64958},[140],[28,64960,53119],{"className":64961,"style":53204},[102,106,140],[28,64963],{"className":64964,"style":51495},[173,140],[28,64966,24201],{"className":64967},[102,106,140],[28,64969,3971],{"className":64970},[102,106,140],[28,64972,64974],{"className":64973},[102,137,30538,64952,140],[28,64975,64977],{"className":64976},[3407,28701,140],[28,64978,78],{"className":64979},[140],[28,64981,64982,64985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is the ground state energy. We can see by inserting ",[28,65502,65504,65558],{"className":65503},[35],[28,65505,65507],{"className":65506},[39],[41,65508,65509],{"xmlns":43},[46,65510,65511,65555],{},[49,65512,65513,65515,65517,65519,65521,65523,65525],{},[55,65514,4005],{},[62,65516,4008],{},[62,65518,65],{"stretchy":64},[55,65520,3169],{},[62,65522,78],{"stretchy":64},[62,65524,845],{},[3977,65526,65527,65553],{},[49,65528,65529,65539,65541],{},[2776,65530,65531,65533],{},[55,65532,60],{},[49,65534,65535,65537],{},[55,65536,3256],{},[55,65538,3169],{},[62,65540,2606],{},[2776,65542,65543,65545],{},[55,65544,60],{},[49,65546,65547,65549,65551],{},[62,65548,1213],{},[55,65550,3256],{},[55,65552,3169],{},[650,65554,848],{},[82,65556,65557],{"encoding":84},"\\cos(x)=\\frac{e^{ix}+e^{-ix}}{2}",[28,65559,65561,65588],{"className":65560,"ariaHidden":71},[89],[28,65562,65564,65567,65570,65573,65576,65579,65582,65585],{"className":65563},[93],[28,65565],{"className":65566,"style":98},[97],[28,65568,4005],{"className":65569},[4241],[28,65571,65],{"className":65572},[161],[28,65574,3169],{"className":65575},[102,106],[28,65577,78],{"className":65578},[182],[28,65580],{"className":65581,"style":913},[173],[28,65583,845],{"className":65584},[917],[28,65586],{"className":65587,"style":913},[173],[28,65589,65591,65595],{"className":65590},[93],[28,65592],{"className":65593,"style":65594},[97],"height:1.3705em;vertical-align:-0.345em;",[28,65596,65598,65601,65737],{"className":65597},[102],[28,65599],{"className":65600},[161,4072],[28,65602,65604],{"className":65603},[3977],[28,65605,65607,65728],{"className":65606},[115,116],[28,65608,65610,65725],{"className":65609},[120],[28,65611,65614,65628,65636],{"className":65612,"style":65613},[124],"height:1.0255em;",[28,65615,65616,65619],{"style":62862},[28,65617],{"className":65618,"style":3434},[132],[28,65620,65622],{"className":65621},[137,138,139,140],[28,65623,65625],{"className":65624},[102,140],[28,65626,848],{"className":65627},[102,140],[28,65629,65630,65633],{"style":4100},[28,65631],{"className":65632,"style":3434},[132],[28,65634],{"className":65635,"style":4108},[4107],[28,65637,65638,65641],{"style":62891},[28,65639],{"className":65640,"style":3434},[132],[28,65642,65644],{"className":65643},[137,138,139,140],[28,65645,65647,65684,65687],{"className":65646},[102,140],[28,65648,65650,65653],{"className":65649},[102,140],[28,65651,60],{"className":65652},[102,106,140],[28,65654,65656],{"className":65655},[111],[28,65657,65659],{"className":65658},[115],[28,65660,65662],{"className":65661},[120],[28,65663,65666],{"className":65664,"style":65665},[124],"height:0.9021em;",[28,65667,65669,65672],{"style":65668},"top:-2.931em;margin-right:0.0714em;",[28,65670],{"className":65671,"style":30534},[132],[28,65673,65675],{"className":65674},[137,30538,28701,140],[28,65676,65678,65681],{"className":65677},[102,140],[28,65679,3256],{"className":65680},[102,106,140],[28,65682,3169],{"className":65683},[102,106,140],[28,65685,2606],{"className":65686},[1261,140],[28,65688,65690,65693],{"className":65689},[102,140],[28,65691,60],{"className":65692},[102,106,140],[28,65694,65696],{"className":65695},[111],[28,65697,65699],{"className":65698},[115],[28,65700,65702],{"className":65701},[120],[28,65703,65705],{"className":65704,"style":65665},[124],[28,65706,65707,65710],{"style":65668},[28,65708],{"className":65709,"style":30534},[132],[28,65711,65713],{"className":65712},[137,30538,28701,140],[28,65714,65716,65719,65722],{"className":65715},[102,140],[28,65717,1213],{"className":65718},[102,140],[28,65720,3256],{"className":65721},[102,106,140],[28,65723,3169],{"className":65724},[102,106,140],[28,65726,148],{"className":65727},[147],[28,65729,65731],{"className":65730},[120],[28,65732,65735],{"className":65733,"style":65734},[124],"height:0.345em;",[28,65736],{},[28,65738],{"className":65739},[182,4072]," that ",[28,65742,65744,65763],{"className":65743},[35],[28,65745,65747],{"className":65746},[39],[41,65748,65749],{"xmlns":43},[46,65750,65751,65761],{},[49,65752,65753,65755,65757,65759],{},[55,65754,68],{},[62,65756,65],{"stretchy":64},[55,65758,1871],{},[62,65760,78],{"stretchy":64},[82,65762,53408],{"encoding":84},[28,65764,65766],{"className":65765,"ariaHidden":71},[89],[28,65767,65769,65772,65775,65778,65781],{"className":65768},[93],[28,65770],{"className":65771,"style":98},[97],[28,65773,68],{"className":65774,"style":165},[102,106],[28,65776,65],{"className":65777},[161],[28,65779,1871],{"className":65780,"style":1913},[102,106],[28,65782,78],{"className":65783},[182]," is a trigonometric polynomial of degree ",[28,65786,65788,65801],{"className":65787},[35],[28,65789,65791],{"className":65790},[39],[41,65792,65793],{"xmlns":43},[46,65794,65795,65799],{},[49,65796,65797],{},[55,65798,24201],{},[82,65800,24201],{"encoding":84},[28,65802,65804],{"className":65803,"ariaHidden":71},[89],[28,65805,65807,65810],{"className":65806},[93],[28,65808],{"className":65809,"style":9597},[97],[28,65811,24201],{"className":65812},[102,106],", that is, a linear combination of ",[28,65815,65817,65846],{"className":65816},[35],[28,65818,65820],{"className":65819},[39],[41,65821,65822],{"xmlns":43},[46,65823,65824,65844],{},[49,65825,65826],{},[2776,65827,65828,65830],{},[55,65829,60],{},[49,65831,65832,65834,65836,65838,65840,65842],{},[55,65833,3256],{},[55,65835,28200],{},[55,65837,1871],{},[2253,65839,51285],{},[55,65841,24201],{},[55,65843,3971],{},[82,65845,53463],{"encoding":84},[28,65847,65849],{"className":65848,"ariaHidden":71},[89],[28,65850,65852,65855],{"className":65851},[93],[28,65853],{"className":65854,"style":29397},[97],[28,65856,65858,65861],{"className":65857},[102],[28,65859,60],{"className":65860},[102,106],[28,65862,65864],{"className":65863},[111],[28,65865,65867],{"className":65866},[115],[28,65868,65870],{"className":65869},[120],[28,65871,65873],{"className":65872,"style":29397},[124],[28,65874,65875,65878],{"style":1726},[28,65876],{"className":65877,"style":133},[132],[28,65879,65881],{"className":65880},[137,138,139,140],[28,65882,65884,65887,65890,65893,65896],{"className":65883},[102,140],[28,65885,53505],{"className":65886,"style":32522},[102,106,140],[28,65888,1871],{"className":65889,"style":1913},[102,106,140],[28,65891],{"className":65892,"style":51495},[173,140],[28,65894,24201],{"className":65895},[102,106,140],[28,65897,3971],{"className":65898},[102,106,140],[28,65900,65902,65951],{"className":65901},[35],[28,65903,65905],{"className":65904},[39],[41,65906,65907],{"xmlns":43},[46,65908,65909,65949],{},[49,65910,65911,65913,65915,65917,65919,65921,65923,65925,65927,65929,65931,65933,65935,65937,65939,65941,65943,65945,65947],{},[55,65912,28200],{},[62,65914,845],{},[62,65916,1213],{},[55,65918,24201],{},[62,65920,72],{"separator":71},[62,65922,1213],{},[55,65924,24201],{},[62,65926,2606],{},[650,65928,3200],{},[62,65930,72],{"separator":71},[55,65932,2106],{"mathvariant":1088},[55,65934,2106],{"mathvariant":1088},[55,65936,2106],{"mathvariant":1088},[62,65938,72],{"separator":71},[55,65940,24201],{},[62,65942,1213],{},[650,65944,3200],{},[62,65946,72],{"separator":71},[55,65948,24201],{},[82,65950,53571],{"encoding":84},[28,65952,65954,65972,66005,66041],{"className":65953,"ariaHidden":71},[89],[28,65955,65957,65960,65963,65966,65969],{"className":65956},[93],[28,65958],{"className":65959,"style":9774},[97],[28,65961,28200],{"className":65962,"style":32522},[102,106],[28,65964],{"className":65965,"style":913},[173],[28,65967,845],{"className":65968},[917],[28,65970],{"className":65971,"style":913},[173],[28,65973,65975,65978,65981,65984,65987,65990,65993,65996,65999,66002],{"className":65974},[93],[28,65976],{"className":65977,"style":214},[97],[28,65979,1213],{"className":65980},[102],[28,65982,24201],{"className":65983},[102,106],[28,65985,72],{"className":65986},[169],[28,65988],{"className":65989,"style":174},[173],[28,65991,1213],{"className":65992},[102],[28,65994,24201],{"className":65995},[102,106],[28,65997],{"className":65998,"style":1257},[173],[28,66000,2606],{"className":66001},[1261],[28,66003],{"className":66004,"style":1257},[173],[28,66006,66008,66011,66014,66017,66020,66023,66026,66029,66032,66035,66038],{"className":66007},[93],[28,66009],{"className":66010,"style":214},[97],[28,66012,3200],{"className":66013},[102],[28,66015,72],{"className":66016},[169],[28,66018],{"className":66019,"style":174},[173],[28,66021,28782],{"className":66022},[102],[28,66024,72],{"className":66025},[169],[28,66027],{"className":66028,"style":174},[173],[28,66030,24201],{"className":66031},[102,106],[28,66033],{"className":66034,"style":1257},[173],[28,66036,1213],{"className":66037},[1261],[28,66039],{"className":66040,"style":1257},[173],[28,66042,66044,66047,66050,66053,66056],{"className":66043},[93],[28,66045],{"className":66046,"style":214},[97],[28,66048,3200],{"className":66049},[102],[28,66051,72],{"className":66052},[169],[28,66054],{"className":66055,"style":174},[173],[28,66057,24201],{"className":66058},[102,106],". Furthermore, from the definition of ",[28,66061,66063,66086],{"className":66062},[35],[28,66064,66066],{"className":66065},[39],[41,66067,66068],{"xmlns":43},[46,66069,66070,66084],{},[49,66071,66072,66078,66080,66082],{},[2776,66073,66074,66076],{},[55,66075,12],{},[62,66077,1679],{},[62,66079,65],{"stretchy":64},[55,66081,3169],{},[62,66083,78],{"stretchy":64},[82,66085,64257],{"encoding":84},[28,66087,66089],{"className":66088,"ariaHidden":71},[89],[28,66090,66092,66095,66124,66127,66130],{"className":66091},[93],[28,66093],{"className":66094,"style":98},[97],[28,66096,66098,66101],{"className":66097},[102],[28,66099,12],{"className":66100},[102,106],[28,66102,66104],{"className":66103},[111],[28,66105,66107],{"className":66106},[115],[28,66108,66110],{"className":66109},[120],[28,66111,66113],{"className":66112,"style":1711},[124],[28,66114,66115,66118],{"style":1726},[28,66116],{"className":66117,"style":133},[132],[28,66119,66121],{"className":66120},[137,138,139,140],[28,66122,1679],{"className":66123},[1261,140],[28,66125,65],{"className":66126},[161],[28,66128,3169],{"className":66129},[102,106],[28,66131,78],{"className":66132},[182]," above we have that ",[28,66135,66137,66175],{"className":66136},[35],[28,66138,66140],{"className":66139},[39],[41,66141,66142],{"xmlns":43},[46,66143,66144,66172],{},[49,66145,66146,66148,66150,66156,66158,66160,66162,66164,66166,66168,66170],{},[55,66147,68],{},[62,66149,65],{"stretchy":64},[52,66151,66152,66154],{},[55,66153,1871],{},[650,66155,652],{},[62,66157,78],{"stretchy":64},[62,66159,845],{},[55,66161,12],{},[62,66163,65],{"stretchy":64},[650,66165,652],{},[62,66167,78],{"stretchy":64},[62,66169,845],{},[650,66171,3200],{},[82,66173,66174],{"encoding":84},"f(E_0)=p(0)=1",[28,66176,66178,66242,66269],{"className":66177,"ariaHidden":71},[89],[28,66179,66181,66184,66187,66190,66230,66233,66236,66239],{"className":66180},[93],[28,66182],{"className":66183,"style":98},[97],[28,66185,68],{"className":66186,"style":165},[102,106],[28,66188,65],{"className":66189},[161],[28,66191,66193,66196],{"className":66192},[102],[28,66194,1871],{"className":66195,"style":1913},[102,106],[28,66197,66199],{"className":66198},[111],[28,66200,66202,66222],{"className":66201},[115,116],[28,66203,66205,66219],{"className":66204},[120],[28,66206,66208],{"className":66207,"style":683},[124],[28,66209,66210,66213],{"style":1928},[28,66211],{"className":66212,"style":133},[132],[28,66214,66216],{"className":66215},[137,138,139,140],[28,66217,652],{"className":66218},[102,140],[28,66220,148],{"className":66221},[147],[28,66223,66225],{"className":66224},[120],[28,66226,66228],{"className":66227,"style":155},[124],[28,66229],{},[28,66231,78],{"className":66232},[182],[28,66234],{"className":66235,"style":913},[173],[28,66237,845],{"className":66238},[917],[28,66240],{"className":66241,"style":913},[173],[28,66243,66245,66248,66251,66254,66257,66260,66263,66266],{"className":66244},[93],[28,66246],{"className":66247,"style":98},[97],[28,66249,12],{"className":66250},[102,106],[28,66252,65],{"className":66253},[161],[28,66255,652],{"className":66256},[102],[28,66258,78],{"className":66259},[182],[28,66261],{"className":66262,"style":913},[173],[28,66264,845],{"className":66265},[917],[28,66267],{"className":66268,"style":913},[173],[28,66270,66272,66275],{"className":66271},[93],[28,66273],{"className":66274,"style":2914},[97],[28,66276,3200],{"className":66277},[102]," and for any ",[28,66280,66282,66295],{"className":66281},[35],[28,66283,66285],{"className":66284},[39],[41,66286,66287],{"xmlns":43},[46,66288,66289,66293],{},[49,66290,66291],{},[55,66292,1871],{},[82,66294,1871],{"encoding":84},[28,66296,66298],{"className":66297,"ariaHidden":71},[89],[28,66299,66301,66304],{"className":66300},[93],[28,66302],{"className":66303,"style":1102},[97],[28,66305,1871],{"className":66306,"style":1913},[102,106]," in the spectral range such that ",[28,66309,66311,66341],{"className":66310},[35],[28,66312,66314],{"className":66313},[39],[41,66315,66316],{"xmlns":43},[46,66317,66318,66338],{},[49,66319,66320,66322,66324,66326,66332,66334,66336],{},[55,66321,296],{"mathvariant":1088},[55,66323,1871],{},[62,66325,1213],{},[52,66327,66328,66330],{},[55,66329,1871],{},[650,66331,652],{},[55,66333,296],{"mathvariant":1088},[62,66335,16503],{},[55,66337,53119],{},[82,66339,66340],{"encoding":84},"\\vert E-E_0 \\vert > \\delta",[28,66342,66344,66365,66423],{"className":66343,"ariaHidden":71},[89],[28,66345,66347,66350,66353,66356,66359,66362],{"className":66346},[93],[28,66348],{"className":66349,"style":98},[97],[28,66351,296],{"className":66352},[102],[28,66354,1871],{"className":66355,"style":1913},[102,106],[28,66357],{"className":66358,"style":1257},[173],[28,66360,1213],{"className":66361},[1261],[28,66363],{"className":66364,"style":1257},[173],[28,66366,66368,66371,66411,66414,66417,66420],{"className":66367},[93],[28,66369],{"className":66370,"style":98},[97],[28,66372,66374,66377],{"className":66373},[102],[28,66375,1871],{"className":66376,"style":1913},[102,106],[28,66378,66380],{"className":66379},[111],[28,66381,66383,66403],{"className":66382},[115,116],[28,66384,66386,66400],{"className":66385},[120],[28,66387,66389],{"className":66388,"style":683},[124],[28,66390,66391,66394],{"style":1928},[28,66392],{"className":66393,"style":133},[132],[28,66395,66397],{"className":66396},[137,138,139,140],[28,66398,652],{"className":66399},[102,140],[28,66401,148],{"className":66402},[147],[28,66404,66406],{"className":66405},[120],[28,66407,66409],{"className":66408,"style":155},[124],[28,66410],{},[28,66412,296],{"className":66413},[102],[28,66415],{"className":66416,"style":913},[173],[28,66418,16503],{"className":66419},[917],[28,66421],{"className":66422,"style":913},[173],[28,66424,66426,66429],{"className":66425},[93],[28,66427],{"className":66428,"style":9597},[97],[28,66430,53119],{"className":66431,"style":53204},[102,106]," we 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\\le \\beta(a, b, d) = T_d^{-1}\\left(1 + 2\\frac{1-\\cos\\big(\\delta\\,dt\\big)}{1 + 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N. Yoshioka, M. Amico, W. Kirby et al. \"Diagonalization of large many-body Hamiltonians on a quantum processor\". ",[19,67138,67140],{"href":67139},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2407.14431","arXiv:2407.14431",[12,67142,67143,67145],{},[28,67144,848],{}," 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).",[12,67147,67148,67150],{},[28,67149,33915],{}," Å. Björck. \"Numerical methods in matrix computations\". Texts in Applied Mathematics. Springer International Publishing. (2014).",[12,67152,67153,67155],{},[28,67154,37795],{}," William Kirby. \"Analysis of quantum Krylov algorithms with errors\". Quantum 8, 1457 (2024).",[251,67157,67159],{"id":67158},"tutorial-survey","Tutorial survey",[12,67161,67162],{},"Please take this short survey to provide feedback on this tutorial. Your insights will help us improve our content offerings and user experience.",[12,67164,67165],{},[19,67166,67168],{"href":67167},"https:\u002F\u002Fyour.feedback.ibm.com\u002Fjfe\u002Fform\u002FSV_82nennpKIjjD8rQ","Link to survey",[251,67170,26424],{"id":26423},[12,67172,26427,67173,2106],{},[19,67174,26995],{"href":26994},[26436,67176,67177],{},"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 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.s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}",{"title":15655,"searchDepth":15668,"depth":15668,"links":67179},[67180,67181,67182,67183,67187,67194,67199,67202,67203,67204,67205,67206],{"id":27019,"depth":15668,"text":27020},{"id":27054,"depth":15668,"text":27055},{"id":27079,"depth":15668,"text":27080},{"id":28324,"depth":15668,"text":28325,"children":67184},[67185,67186],{"id":28329,"depth":15679,"text":28330},{"id":29816,"depth":15679,"text":29817},{"id":39850,"depth":15668,"text":39851,"children":67188},[67189,67190,67191,67192],{"id":39854,"depth":15679,"text":39855},{"id":45979,"depth":15679,"text":45980},{"id":46959,"depth":15679,"text":46960},{"id":47056,"depth":15679,"text":67193},"Template circuits for calculating matrix elements of S~\\tilde{S}S~ and H~\\tilde{H}H~ via Hadamard 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Sampling-Budget & Bit-String Analogy for Qubit Classifications",[7],{"type":9,"value":91372,"toc":95954},[91373,91380,95795,95801,95804,95807,95813,95819,95822,95824,95942,95944,95951],[12,91374,91375],{},[15,91376,17,91377,23],{},[19,91378,22],{"href":91379},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[15650,91381,91383],{"className":15652,"code":91382,"language":15654,"meta":15655,"style":15655},"# 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",[8103,91384,91385,91390,91395,91400,91405,91410,91415,91420,91425,91430,91435,91439,91446,91452,91464,91475,91479,91489,91499,91509,91521,91533,91537,91550,91554,91559,91563,91610,91614,91619,91624,91628,91641,91655,91669,91694,91730,91796,91829,91833,91837,91884,91888,91893,91898,91902,91926,91947,91960,91966,91996,92018,92022,92026,92031,92035,92051,92062,92070,92080,92087,92091,92095,92107,92114,92121,92128,92133,92137,92143,92150,92157,92162,92166,92172,92179,92186,92191,92195,92201,92208,92215,92220,92224,92230,92237,92244,92249,92253,92257,92261,92266,92270,92282,92294,92306,92332,92339,92351,92361,92365,92372,92382,92386,92390,92394,92419,92423,92427,92432,92436,92468,92472,92477,92481,92514,92518,92522,92553,92557,92562,92566,92577,92594,92611,92644,92656,92660,92669,92677,92685,92693,92701,92709,92717,92725,92739,92752,92765,92769,92781,92818,92825,92829,92833,92872,92876,92881,92885,92902,92906,92910,92915,92919,92933,92942,92946,92957,92971,92993,93003,93028,93032,93046,93050,93055,93077,93097,93119,93133,93151,93168,93172,93176,93181,93185,93190,93206,93243,93274,93278,93288,93306,93310,93315,93340,93364,93368,93379,93383,93388,93392,93397,93412,93434,93453,93473,93477,93482,93487,93492,93519,93545,93569,93574,93578,93607,93661,93698,93711,93734,93742,93752,93780,93788,93792,93797,93819,93829,93849,93873,93877,93899,93924,93981,93985,94024,94037,94058,94072,94089,94097,94107,94132,94140,94144,94155,94159,94164,94168,94185,94198,94218,94238,94261,94283,94302,94319,94324,94328,94332,94337,94341,94354,94359,94363,94394,94398,94479,94483,94487,94498,94503,94508,94513,94518,94522,94530,94537,94548,94555,94562,94566,94577,94582,94587,94592,94597,94601,94615,94620,94625,94630,94635,94640,94644,94648,94660,94664,94668,94679,94684,94689,94694,94699,94703,94708,94713,94717,94737,94749,94759,94769,94789,94849,94880,94884,94899,94915,94966,94970,94985,94996,95007,95019,95024,95039,95054,95059,95063,95068,95086,95091,95095,95100,95108,95120,95125,95149,95153,95168,95195,95199,95204,95215,95242,95246,95251,95264,95272,95283,95291,95295,95313,95335,95347,95351,95356,95361,95377,95385,95399,95407,95433,95437,95454,95475,95486,95498,95509,95521,95525,95532,95539,95543,95547,95562,95576,95587,95597,95606,95610,95614,95624,95651,95662,95666,95671,95694,95705,95720,95742,95764],{"__ignoreMap":15655},[28,91386,91387],{"class":15660,"line":15661},[28,91388,91389],{"class":15664},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n",[28,91391,91392],{"class":15660,"line":15668},[28,91393,91394],{"class":15664},"# generates tables and figures, so you can iterate further.\n",[28,91396,91397],{"class":15660,"line":15679},[28,91398,91399],{"class":15664},"#\n",[28,91401,91402],{"class":15660,"line":15725},[28,91403,91404],{"class":15664},"# Contents created in the current directory:\n",[28,91406,91407],{"class":15660,"line":15739},[28,91408,91409],{"class":15664},"#  - Sampling_Budget_Qubit_Bitstrings.ipynb      (Colab notebook)\n",[28,91411,91412],{"class":15660,"line":15752},[28,91413,91414],{"class":15664},"#  - summary.csv                                 (summary table)\n",[28,91416,91417],{"class":15660,"line":15765},[28,91418,91419],{"class":15664},"#  - bitstrings_\u003CMODEL>.csv                      (raw bit-strings per model)\n",[28,91421,91422],{"class":15660,"line":15772},[28,91423,91424],{"class":15664},"#  - fig_ones_fraction.png                       (bar chart with Wilson CI)\n",[28,91426,91427],{"class":15660,"line":15781},[28,91428,91429],{"class":15664},"#  - fig_required_shots.png                      (required shots for ±ε)\n",[28,91431,91432],{"class":15660,"line":15798},[28,91433,91434],{"class":15664},"#  - simulation_config.json                      (run parameters)\n",[28,91436,91437],{"class":15660,"line":15811},[28,91438,15769],{"emptyLinePlaceholder":15768},[28,91440,91441,91443],{"class":15660,"line":15820},[28,91442,15672],{"class":15671},[28,91444,91445],{"class":15675}," json\n",[28,91447,91448,91450],{"class":15660,"line":15830},[28,91449,15672],{"class":15671},[28,91451,15709],{"class":15675},[28,91453,91454,91456,91459,91461],{"class":15660,"line":15835},[28,91455,15714],{"class":15671},[28,91457,91458],{"class":15675}," dataclasses ",[28,91460,15672],{"class":15671},[28,91462,91463],{"class":15675}," dataclass\n",[28,91465,91466,91468,91470,91472],{"class":15660,"line":15853},[28,91467,15714],{"class":15671},[28,91469,15717],{"class":15675},[28,91471,15672],{"class":15671},[28,91473,91474],{"class":15675}," List, Dict, Tuple\n",[28,91476,91477],{"class":15660,"line":15867},[28,91478,15769],{"emptyLinePlaceholder":15768},[28,91480,91481,91483,91485,91487],{"class":15660,"line":15872},[28,91482,15672],{"class":15671},[28,91484,15730],{"class":15675},[28,91486,15733],{"class":15671},[28,91488,15736],{"class":15675},[28,91490,91491,91493,91495,91497],{"class":15660,"line":15878},[28,91492,15672],{"class":15671},[28,91494,15757],{"class":15675},[28,91496,15733],{"class":15671},[28,91498,15762],{"class":15675},[28,91500,91501,91503,91505,91507],{"class":15660,"line":15897},[28,91502,15672],{"class":15671},[28,91504,15744],{"class":15675},[28,91506,15733],{"class":15671},[28,91508,15749],{"class":15675},[28,91510,91511,91513,91516,91518],{"class":15660,"line":15914},[28,91512,15714],{"class":15671},[28,91514,91515],{"class":15675}," IPython.display ",[28,91517,15672],{"class":15671},[28,91519,91520],{"class":15675}," display, Markdown\n",[28,91522,91523,91525,91528,91530],{"class":15660,"line":15931},[28,91524,15672],{"class":15671},[28,91526,91527],{"class":15675}," nbformat ",[28,91529,15733],{"class":15671},[28,91531,91532],{"class":15675}," nbf\n",[28,91534,91535],{"class":15660,"line":15948},[28,91536,15769],{"emptyLinePlaceholder":15768},[28,91538,91539,91541,91543,91545,91547],{"class":15660,"line":15966},[28,91540,15838],{"class":15675},[28,91542,15842],{"class":15841},[28,91544,15845],{"class":15675},[28,91546,845],{"class":15682},[28,91548,91549],{"class":15801}," 200\n",[28,91551,91552],{"class":15660,"line":15983},[28,91553,15769],{"emptyLinePlaceholder":15768},[28,91555,91556],{"class":15660,"line":16000},[28,91557,91558],{"class":15664},"# ---------- Helper math ----------\n",[28,91560,91561],{"class":15660,"line":16016},[28,91562,15769],{"emptyLinePlaceholder":15768},[28,91564,91565,91567,91570,91572,91574,91576,91578,91580,91582,91584,91586,91588,91590,91592,91594,91596,91599,91602,91604,91606,91608],{"class":15660,"line":16036},[28,91566,16070],{"class":15671},[28,91568,91569],{"class":16073}," wilson_ci",[28,91571,65],{"class":15675},[28,91573,29362],{"class":16082},[28,91575,15883],{"class":15675},[28,91577,15955],{"class":15682},[28,91579,16397],{"class":15675},[28,91581,7106],{"class":16082},[28,91583,15883],{"class":15675},[28,91585,15955],{"class":15682},[28,91587,16397],{"class":15675},[28,91589,2266],{"class":16082},[28,91591,15883],{"class":15675},[28,91593,15886],{"class":15682},[28,91595,15805],{"class":15682},[28,91597,91598],{"class":15801}," 1.96",[28,91600,91601],{"class":15675},") -> Tuple[",[28,91603,15886],{"class":15682},[28,91605,16397],{"class":15675},[28,91607,15886],{"class":15682},[28,91609,18373],{"class":15675},[28,91611,91612],{"class":15660,"line":16041},[28,91613,16197],{"class":15841},[28,91615,91616],{"class":15660,"line":16371},[28,91617,91618],{"class":15841},"    Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n",[28,91620,91621],{"class":15660,"line":16376},[28,91622,91623],{"class":15841},"    Returns (low, high). Works for edge cases k=0 or k=n.\n",[28,91625,91626],{"class":15660,"line":16410},[28,91627,16197],{"class":15841},[28,91629,91630,91632,91635,91637,91639],{"class":15660,"line":16431},[28,91631,16484],{"class":15671},[28,91633,91634],{"class":15675}," n ",[28,91636,23650],{"class":15682},[28,91638,46881],{"class":15801},[28,91640,15778],{"class":15675},[28,91642,91643,91645,91647,91649,91651,91653],{"class":15660,"line":16452},[28,91644,23659],{"class":15671},[28,91646,17308],{"class":15675},[28,91648,15271],{"class":15801},[28,91650,16397],{"class":15675},[28,91652,16580],{"class":15801},[28,91654,16052],{"class":15675},[28,91656,91657,91660,91662,91664,91666],{"class":15660,"line":16476},[28,91658,91659],{"class":15675},"    p_hat ",[28,91661,845],{"class":15682},[28,91663,27616],{"class":15675},[28,91665,1129],{"class":15682},[28,91667,91668],{"class":15675}," n\n",[28,91670,91671,91674,91676,91679,91681,91684,91686,91688,91690,91692],{"class":15660,"line":16481},[28,91672,91673],{"class":15675},"    denom ",[28,91675,845],{"class":15682},[28,91677,91678],{"class":15801}," 1.0",[28,91680,16359],{"class":15682},[28,91682,91683],{"class":15675}," (z ",[28,91685,16354],{"class":15682},[28,91687,22129],{"class":15801},[28,91689,22233],{"class":15675},[28,91691,1129],{"class":15682},[28,91693,91668],{"class":15675},[28,91695,91696,91699,91701,91704,91706,91708,91710,91712,91714,91716,91718,91720,91722,91725,91727],{"class":15660,"line":16510},[28,91697,91698],{"class":15675},"    center ",[28,91700,845],{"class":15682},[28,91702,91703],{"class":15675}," (p_hat ",[28,91705,2606],{"class":15682},[28,91707,91683],{"class":15675},[28,91709,16354],{"class":15682},[28,91711,22129],{"class":15801},[28,91713,22233],{"class":15675},[28,91715,1129],{"class":15682},[28,91717,17308],{"class":15675},[28,91719,848],{"class":15801},[28,91721,16210],{"class":15682},[28,91723,91724],{"class":15675}," n)) ",[28,91726,1129],{"class":15682},[28,91728,91729],{"class":15675}," denom\n",[28,91731,91732,91735,91737,91739,91741,91744,91746,91749,91751,91753,91755,91757,91760,91762,91765,91767,91769,91771,91773,91775,91777,91779,91781,91783,91785,91787,91789,91792,91794],{"class":15660,"line":16542},[28,91733,91734],{"class":15675},"    span ",[28,91736,845],{"class":15682},[28,91738,91683],{"class":15675},[28,91740,16226],{"class":15682},[28,91742,91743],{"class":15675}," math.",[28,91745,5750],{"class":16073},[28,91747,91748],{"class":15675},"((p_hat ",[28,91750,16226],{"class":15682},[28,91752,17308],{"class":15675},[28,91754,3200],{"class":15801},[28,91756,24705],{"class":15682},[28,91758,91759],{"class":15675}," p_hat) ",[28,91761,1129],{"class":15682},[28,91763,91764],{"class":15675}," n) ",[28,91766,2606],{"class":15682},[28,91768,91683],{"class":15675},[28,91770,16354],{"class":15682},[28,91772,22129],{"class":15801},[28,91774,22233],{"class":15675},[28,91776,1129],{"class":15682},[28,91778,17308],{"class":15675},[28,91780,37795],{"class":15801},[28,91782,16210],{"class":15682},[28,91784,91634],{"class":15675},[28,91786,16354],{"class":15682},[28,91788,22129],{"class":15801},[28,91790,91791],{"class":15675},"))) ",[28,91793,1129],{"class":15682},[28,91795,91729],{"class":15675},[28,91797,91798,91800,91802,91804,91806,91808,91811,91813,91816,91818,91820,91822,91824,91826],{"class":15660,"line":16557},[28,91799,16565],{"class":15671},[28,91801,17308],{"class":15675},[28,91803,22207],{"class":15682},[28,91805,65],{"class":15675},[28,91807,15271],{"class":15801},[28,91809,91810],{"class":15675},", center ",[28,91812,15689],{"class":15682},[28,91814,91815],{"class":15675}," span), ",[28,91817,22201],{"class":15682},[28,91819,65],{"class":15675},[28,91821,16580],{"class":15801},[28,91823,91810],{"class":15675},[28,91825,2606],{"class":15682},[28,91827,91828],{"class":15675}," span))\n",[28,91830,91831],{"class":15660,"line":16562},[28,91832,15769],{"emptyLinePlaceholder":15768},[28,91834,91835],{"class":15660,"line":17382},[28,91836,15769],{"emptyLinePlaceholder":15768},[28,91838,91839,91841,91844,91846,91848,91850,91852,91854,91857,91859,91861,91863,91866,91868,91870,91872,91874,91876,91878,91880,91882],{"class":15660,"line":17392},[28,91840,16070],{"class":15671},[28,91842,91843],{"class":16073}," shots_for_epsilon",[28,91845,65],{"class":15675},[28,91847,12],{"class":16082},[28,91849,15883],{"class":15675},[28,91851,15886],{"class":15682},[28,91853,16397],{"class":15675},[28,91855,91856],{"class":16082},"epsilon",[28,91858,15883],{"class":15675},[28,91860,15886],{"class":15682},[28,91862,15805],{"class":15682},[28,91864,91865],{"class":15801}," 0.01",[28,91867,16397],{"class":15675},[28,91869,2266],{"class":16082},[28,91871,15883],{"class":15675},[28,91873,15886],{"class":15682},[28,91875,15805],{"class":15682},[28,91877,91598],{"class":15801},[28,91879,17585],{"class":15675},[28,91881,15955],{"class":15682},[28,91883,15778],{"class":15675},[28,91885,91886],{"class":15660,"line":17402},[28,91887,16197],{"class":15841},[28,91889,91890],{"class":15660,"line":17407},[28,91891,91892],{"class":15841},"    Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n",[28,91894,91895],{"class":15660,"line":17418},[28,91896,91897],{"class":15841},"    at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n",[28,91899,91900],{"class":15660,"line":17429},[28,91901,16197],{"class":15841},[28,91903,91904,91907,91909,91911,91913,91915,91918,91920,91922,91924],{"class":15660,"line":17440},[28,91905,91906],{"class":15675},"    p ",[28,91908,845],{"class":15682},[28,91910,25336],{"class":15682},[28,91912,65],{"class":15675},[28,91914,22207],{"class":15682},[28,91916,91917],{"class":15675},"(p, ",[28,91919,15271],{"class":15801},[28,91921,22494],{"class":15675},[28,91923,16580],{"class":15801},[28,91925,16052],{"class":15675},[28,91927,91928,91931,91933,91936,91938,91940,91942,91944],{"class":15660,"line":17451},[28,91929,91930],{"class":15675},"    var ",[28,91932,845],{"class":15682},[28,91934,91935],{"class":15675}," p ",[28,91937,16226],{"class":15682},[28,91939,17308],{"class":15675},[28,91941,16580],{"class":15801},[28,91943,24705],{"class":15682},[28,91945,91946],{"class":15675}," p)\n",[28,91948,91949,91951,91954,91956,91958],{"class":15660,"line":17462},[28,91950,16484],{"class":15671},[28,91952,91953],{"class":15675}," var ",[28,91955,23650],{"class":15682},[28,91957,15977],{"class":15801},[28,91959,15778],{"class":15675},[28,91961,91962,91964],{"class":15660,"line":17467},[28,91963,23659],{"class":15671},[28,91965,25487],{"class":15801},[28,91967,91968,91971,91973,91975,91977,91979,91981,91983,91985,91987,91990,91992,91994],{"class":15660,"line":17478},[28,91969,91970],{"class":15675},"    n ",[28,91972,845],{"class":15682},[28,91974,91683],{"class":15675},[28,91976,16354],{"class":15682},[28,91978,22129],{"class":15801},[28,91980,22233],{"class":15675},[28,91982,16226],{"class":15682},[28,91984,91953],{"class":15675},[28,91986,1129],{"class":15682},[28,91988,91989],{"class":15675}," (epsilon ",[28,91991,16354],{"class":15682},[28,91993,22129],{"class":15801},[28,91995,16052],{"class":15675},[28,91997,91998,92000,92002,92004,92006,92008,92010,92013,92015],{"class":15660,"line":17489},[28,91999,16565],{"class":15671},[28,92001,24028],{"class":15682},[28,92003,65],{"class":15675},[28,92005,3200],{"class":15801},[28,92007,16397],{"class":15675},[28,92009,15955],{"class":15682},[28,92011,92012],{"class":15675},"(math.",[28,92014,24015],{"class":16073},[28,92016,92017],{"class":15675},"(n)))\n",[28,92019,92020],{"class":15660,"line":17500},[28,92021,15769],{"emptyLinePlaceholder":15768},[28,92023,92024],{"class":15660,"line":17511},[28,92025,15769],{"emptyLinePlaceholder":15768},[28,92027,92028],{"class":15660,"line":17522},[28,92029,92030],{"class":15664},"# ---------- Domain model ----------\n",[28,92032,92033],{"class":15660,"line":17527},[28,92034,15769],{"emptyLinePlaceholder":15768},[28,92036,92037,92040,92042,92045,92047,92049],{"class":15660,"line":17538},[28,92038,92039],{"class":16073},"@dataclass",[28,92041,65],{"class":15675},[28,92043,92044],{"class":16307},"frozen",[28,92046,845],{"class":15682},[28,92048,15597],{"class":15801},[28,92050,16052],{"class":15675},[28,92052,92053,92056,92060],{"class":15660,"line":17557},[28,92054,92055],{"class":15671},"class",[28,92057,92059],{"class":92058},"sU0A5"," QubitModel",[28,92061,15778],{"class":15675},[28,92063,92064,92067],{"class":15660,"line":17562},[28,92065,92066],{"class":15675},"    name: ",[28,92068,92069],{"class":15682},"str\n",[28,92071,92072,92075,92077],{"class":15660,"line":17567},[28,92073,92074],{"class":15675},"    satisfaction_prob: ",[28,92076,15886],{"class":15682},[28,92078,92079],{"class":15664},"  # Probability a single shot meets the analytical criterion (bit = 1)\n",[28,92081,92082,92085],{"class":15660,"line":17593},[28,92083,92084],{"class":15675},"    description: ",[28,92086,92069],{"class":15682},[28,92088,92089],{"class":15660,"line":17599},[28,92090,15769],{"emptyLinePlaceholder":15768},[28,92092,92093],{"class":15660,"line":17619},[28,92094,15769],{"emptyLinePlaceholder":15768},[28,92096,92097,92100,92103,92105],{"class":15660,"line":17624},[28,92098,92099],{"class":15801},"DEFAULT_MODELS",[28,92101,92102],{"class":15675},": List[QubitModel] ",[28,92104,845],{"class":15682},[28,92106,17702],{"class":15675},[28,92108,92109,92112],{"class":15660,"line":17629},[28,92110,92111],{"class":16073},"    QubitModel",[28,92113,16077],{"class":15675},[28,92115,92116,92119],{"class":15660,"line":17639},[28,92117,92118],{"class":15841},"        \"Toy working qubit\"",[28,92120,16105],{"class":15675},[28,92122,92123,92126],{"class":15660,"line":17656},[28,92124,92125],{"class":15801},"        0.50",[28,92127,16105],{"class":15675},[28,92129,92130],{"class":15660,"line":17665},[28,92131,92132],{"class":15841},"        \"Simple demonstrator; about half shots meet the criterion.\"\n",[28,92134,92135],{"class":15660,"line":17671},[28,92136,48244],{"class":15675},[28,92138,92139,92141],{"class":15660,"line":17683},[28,92140,92111],{"class":16073},[28,92142,16077],{"class":15675},[28,92144,92145,92148],{"class":15660,"line":17689},[28,92146,92147],{"class":15841},"        \"NISQ qubit\"",[28,92149,16105],{"class":15675},[28,92151,92152,92155],{"class":15660,"line":17694},[28,92153,92154],{"class":15801},"        0.60",[28,92156,16105],{"class":15675},[28,92158,92159],{"class":15660,"line":17705},[28,92160,92161],{"class":15841},"        \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n",[28,92163,92164],{"class":15660,"line":17711},[28,92165,48244],{"class":15675},[28,92167,92168,92170],{"class":15660,"line":17731},[28,92169,92111],{"class":16073},[28,92171,16077],{"class":15675},[28,92173,92174,92177],{"class":15660,"line":17744},[28,92175,92176],{"class":15841},"        \"Fault-tolerant (logical)\"",[28,92178,16105],{"class":15675},[28,92180,92181,92184],{"class":15660,"line":17750},[28,92182,92183],{"class":15801},"        0.90",[28,92185,16105],{"class":15675},[28,92187,92188],{"class":15660,"line":17755},[28,92189,92190],{"class":15841},"        \"Error-corrected logical qubit; high single-shot success probability.\"\n",[28,92192,92193],{"class":15660,"line":17765},[28,92194,48244],{"class":15675},[28,92196,92197,92199],{"class":15660,"line":17770},[28,92198,92111],{"class":16073},[28,92200,16077],{"class":15675},[28,92202,92203,92206],{"class":15660,"line":17775},[28,92204,92205],{"class":15841},"        \"Topologically protected\"",[28,92207,16105],{"class":15675},[28,92209,92210,92213],{"class":15660,"line":17783},[28,92211,92212],{"class":15801},"        0.98",[28,92214,16105],{"class":15675},[28,92216,92217],{"class":15660,"line":17794},[28,92218,92219],{"class":15841},"        \"Intrinsic protection; very high single-shot success probability.\"\n",[28,92221,92222],{"class":15660,"line":17825},[28,92223,48244],{"class":15675},[28,92225,92226,92228],{"class":15660,"line":17837},[28,92227,92111],{"class":16073},[28,92229,16077],{"class":15675},[28,92231,92232,92235],{"class":15660,"line":17843},[28,92233,92234],{"class":15841},"        \"Ideal qubit\"",[28,92236,16105],{"class":15675},[28,92238,92239,92242],{"class":15660,"line":17852},[28,92240,92241],{"class":15801},"        1.00",[28,92243,16105],{"class":15675},[28,92245,92246],{"class":15660,"line":17858},[28,92247,92248],{"class":15841},"        \"Theoretical perfect qubit; always successful in a single shot.\"\n",[28,92250,92251],{"class":15660,"line":17864},[28,92252,48244],{"class":15675},[28,92254,92255],{"class":15660,"line":17872},[28,92256,16849],{"class":15675},[28,92258,92259],{"class":15660,"line":17880},[28,92260,15769],{"emptyLinePlaceholder":15768},[28,92262,92263],{"class":15660,"line":17888},[28,92264,92265],{"class":15664},"# ---------- Simulation configuration ----------\n",[28,92267,92268],{"class":15660,"line":17896},[28,92269,15769],{"emptyLinePlaceholder":15768},[28,92271,92272,92275,92278,92280],{"class":15660,"line":17904},[28,92273,92274],{"class":15801},"CONFIG",[28,92276,92277],{"class":15675},": Dict ",[28,92279,845],{"class":15682},[28,92281,17124],{"class":15675},[28,92283,92284,92287,92289,92292],{"class":15660,"line":17912},[28,92285,92286],{"class":15841},"    \"random_seed\"",[28,92288,15883],{"class":15675},[28,92290,92291],{"class":15801},"123456",[28,92293,16105],{"class":15675},[28,92295,92296,92299,92301,92304],{"class":15660,"line":17921},[28,92297,92298],{"class":15841},"    \"shots_per_model\"",[28,92300,15883],{"class":15675},[28,92302,92303],{"class":15801},"10000",[28,92305,16105],{"class":15675},[28,92307,92308,92311,92313,92316,92318,92321,92323,92326,92329],{"class":15660,"line":17927},[28,92309,92310],{"class":15841},"    \"epsilons\"",[28,92312,49464],{"class":15675},[28,92314,92315],{"class":15801},"0.02",[28,92317,16397],{"class":15675},[28,92319,92320],{"class":15801},"0.01",[28,92322,16397],{"class":15675},[28,92324,92325],{"class":15801},"0.005",[28,92327,92328],{"class":15675},"],  ",[28,92330,92331],{"class":15664},"# ±2%, ±1%, ±0.5% targets\n",[28,92333,92334,92337],{"class":15660,"line":17938},[28,92335,92336],{"class":15841},"    \"cost_model\"",[28,92338,49350],{"class":15675},[28,92340,92341,92344,92346,92349],{"class":15660,"line":17943},[28,92342,92343],{"class":15841},"        \"default_cost_per_shot_usd\"",[28,92345,15883],{"class":15675},[28,92347,92348],{"class":15801},"0.00005",[28,92350,16105],{"class":15675},[28,92352,92353,92356,92358],{"class":15660,"line":17952},[28,92354,92355],{"class":15841},"        \"fixed_overhead_usd\"",[28,92357,15883],{"class":15675},[28,92359,92360],{"class":15801},"0.10\n",[28,92362,92363],{"class":15660,"line":17961},[28,92364,49371],{"class":15675},[28,92366,92367,92370],{"class":15660,"line":17966},[28,92368,92369],{"class":15841},"    \"timing\"",[28,92371,49350],{"class":15675},[28,92373,92374,92377,92379],{"class":15660,"line":17971},[28,92375,92376],{"class":15841},"        \"time_per_shot_seconds\"",[28,92378,15883],{"class":15675},[28,92380,92381],{"class":15801},"2e-6\n",[28,92383,92384],{"class":15660,"line":17978},[28,92385,18233],{"class":15675},[28,92387,92388],{"class":15660,"line":17985},[28,92389,17177],{"class":15675},[28,92391,92392],{"class":15660,"line":17992},[28,92393,15769],{"emptyLinePlaceholder":15768},[28,92395,92396,92399,92401,92404,92407,92409,92411,92413,92416],{"class":15660,"line":17999},[28,92397,92398],{"class":15675},"rng ",[28,92400,845],{"class":15682},[28,92402,92403],{"class":15675}," np.random.",[28,92405,92406],{"class":16073},"default_rng",[28,92408,65],{"class":15675},[28,92410,92274],{"class":15801},[28,92412,3176],{"class":15675},[28,92414,92415],{"class":15841},"\"random_seed\"",[28,92417,92418],{"class":15675},"])\n",[28,92420,92421],{"class":15660,"line":18006},[28,92422,15769],{"emptyLinePlaceholder":15768},[28,92424,92425],{"class":15660,"line":18011},[28,92426,15769],{"emptyLinePlaceholder":15768},[28,92428,92429],{"class":15660,"line":18022},[28,92430,92431],{"class":15664},"# ---------- Core simulation ----------\n",[28,92433,92434],{"class":15660,"line":18027},[28,92435,15769],{"emptyLinePlaceholder":15768},[28,92437,92438,92440,92443,92445,92447,92449,92451,92453,92456,92458,92460,92462,92465],{"class":15660,"line":18032},[28,92439,16070],{"class":15671},[28,92441,92442],{"class":16073}," simulate_bitstring",[28,92444,65],{"class":15675},[28,92446,12],{"class":16082},[28,92448,15883],{"class":15675},[28,92450,15886],{"class":15682},[28,92452,16397],{"class":15675},[28,92454,92455],{"class":16082},"shots",[28,92457,15883],{"class":15675},[28,92459,15955],{"class":15682},[28,92461,16397],{"class":15675},[28,92463,92464],{"class":16082},"rng_",[28,92466,92467],{"class":15675},": np.random.Generator) -> np.ndarray:\n",[28,92469,92470],{"class":15660,"line":18037},[28,92471,16197],{"class":15841},[28,92473,92474],{"class":15660,"line":18049},[28,92475,92476],{"class":15841},"    Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n",[28,92478,92479],{"class":15660,"line":18054},[28,92480,16197],{"class":15841},[28,92482,92483,92485,92488,92491,92493,92495,92498,92501,92503,92506,92508,92510,92512],{"class":15660,"line":18065},[28,92484,16565],{"class":15671},[28,92486,92487],{"class":15675}," rng_.",[28,92489,92490],{"class":16073},"binomial",[28,92492,65],{"class":15675},[28,92494,3200],{"class":15801},[28,92496,92497],{"class":15675},", p, ",[28,92499,92500],{"class":16307},"size",[28,92502,845],{"class":15682},[28,92504,92505],{"class":15675},"shots).",[28,92507,25553],{"class":16073},[28,92509,65],{"class":15675},[28,92511,15955],{"class":15682},[28,92513,16052],{"class":15675},[28,92515,92516],{"class":15660,"line":18076},[28,92517,15769],{"emptyLinePlaceholder":15768},[28,92519,92520],{"class":15660,"line":18089},[28,92521,15769],{"emptyLinePlaceholder":15768},[28,92523,92524,92526,92529,92531,92534,92537,92540,92542,92545,92548,92550],{"class":15660,"line":18102},[28,92525,16070],{"class":15671},[28,92527,92528],{"class":16073}," summarize_model",[28,92530,65],{"class":15675},[28,92532,92533],{"class":16082},"model",[28,92535,92536],{"class":15675},": QubitModel, ",[28,92538,92539],{"class":16082},"bits",[28,92541,22887],{"class":15675},[28,92543,92544],{"class":16082},"epsilons",[28,92546,92547],{"class":15675},": List[",[28,92549,15886],{"class":15682},[28,92551,92552],{"class":15675},"]) -> Dict:\n",[28,92554,92555],{"class":15660,"line":18107},[28,92556,16197],{"class":15841},[28,92558,92559],{"class":15660,"line":18114},[28,92560,92561],{"class":15841},"    Calculate summary statistics for a given model and its simulated bitstring.\n",[28,92563,92564],{"class":15660,"line":18128},[28,92565,16197],{"class":15841},[28,92567,92568,92570,92572,92574],{"class":15660,"line":18133},[28,92569,91970],{"class":15675},[28,92571,845],{"class":15682},[28,92573,23033],{"class":15682},[28,92575,92576],{"class":15675},"(bits.size)\n",[28,92578,92579,92582,92584,92586,92589,92592],{"class":15660,"line":18146},[28,92580,92581],{"class":15675},"    ones ",[28,92583,845],{"class":15682},[28,92585,23033],{"class":15682},[28,92587,92588],{"class":15675},"(bits.",[28,92590,92591],{"class":16073},"sum",[28,92593,22940],{"class":15675},[28,92595,92596,92599,92601,92603,92606,92608],{"class":15660,"line":18162},[28,92597,92598],{"class":15675},"    zeros ",[28,92600,845],{"class":15682},[28,92602,23033],{"class":15682},[28,92604,92605],{"class":15675},"(n ",[28,92607,15689],{"class":15682},[28,92609,92610],{"class":15675}," ones)\n",[28,92612,92613,92616,92618,92621,92623,92625,92627,92629,92631,92633,92635,92637,92639,92642],{"class":15660,"line":18182},[28,92614,92615],{"class":15675},"    frac ",[28,92617,845],{"class":15682},[28,92619,92620],{"class":15675}," ones ",[28,92622,1129],{"class":15682},[28,92624,91634],{"class":15675},[28,92626,20133],{"class":15671},[28,92628,91634],{"class":15675},[28,92630,16503],{"class":15682},[28,92632,46881],{"class":15801},[28,92634,24971],{"class":15671},[28,92636,18370],{"class":15682},[28,92638,65],{"class":15675},[28,92640,92641],{"class":15841},"\"nan\"",[28,92643,16052],{"class":15675},[28,92645,92646,92649,92651,92653],{"class":15660,"line":18190},[28,92647,92648],{"class":15675},"    lo, hi ",[28,92650,845],{"class":15682},[28,92652,91569],{"class":16073},[28,92654,92655],{"class":15675},"(ones, n)\n",[28,92657,92658],{"class":15660,"line":18196},[28,92659,15769],{"emptyLinePlaceholder":15768},[28,92661,92662,92665,92667],{"class":15660,"line":18201},[28,92663,92664],{"class":15675},"    row ",[28,92666,845],{"class":15682},[28,92668,17124],{"class":15675},[28,92670,92671,92674],{"class":15660,"line":18212},[28,92672,92673],{"class":15841},"        \"Model\"",[28,92675,92676],{"class":15675},": model.name,\n",[28,92678,92679,92682],{"class":15660,"line":18218},[28,92680,92681],{"class":15841},"        \"Description\"",[28,92683,92684],{"class":15675},": model.description,\n",[28,92686,92687,92690],{"class":15660,"line":18230},[28,92688,92689],{"class":15841},"        \"Configured p (target)\"",[28,92691,92692],{"class":15675},": model.satisfaction_prob,\n",[28,92694,92695,92698],{"class":15660,"line":18236},[28,92696,92697],{"class":15841},"        \"Shots\"",[28,92699,92700],{"class":15675},": n,\n",[28,92702,92703,92706],{"class":15660,"line":18241},[28,92704,92705],{"class":15841},"        \"Ones (successes)\"",[28,92707,92708],{"class":15675},": ones,\n",[28,92710,92711,92714],{"class":15660,"line":18246},[28,92712,92713],{"class":15841},"        \"Zeros (failures)\"",[28,92715,92716],{"class":15675},": zeros,\n",[28,92718,92719,92722],{"class":15660,"line":18251},[28,92720,92721],{"class":15841},"        \"Ones fraction (observed)\"",[28,92723,92724],{"class":15675},": frac,\n",[28,92726,92727,92730,92733,92736],{"class":15660,"line":18274},[28,92728,92729],{"class":15841},"        \"Wilson CI 95",[28,92731,92732],{"class":15801},"% lo",[28,92734,92735],{"class":15841},"w\"",[28,92737,92738],{"class":15675},": lo,\n",[28,92740,92741,92743,92746,92749],{"class":15660,"line":18280},[28,92742,92729],{"class":15841},[28,92744,92745],{"class":15801},"% hi",[28,92747,92748],{"class":15841},"gh\"",[28,92750,92751],{"class":15675},": hi,\n",[28,92753,92754,92757,92760,92762],{"class":15660,"line":18300},[28,92755,92756],{"class":15841},"        \"CI width (95%)\"",[28,92758,92759],{"class":15675},": hi ",[28,92761,15689],{"class":15682},[28,92763,92764],{"class":15675}," lo,\n",[28,92766,92767],{"class":15660,"line":18305},[28,92768,18233],{"class":15675},[28,92770,92771,92773,92776,92778],{"class":15660,"line":18310},[28,92772,27785],{"class":15671},[28,92774,92775],{"class":15675}," eps ",[28,92777,17720],{"class":15671},[28,92779,92780],{"class":15675}," epsilons:\n",[28,92782,92783,92786,92788,92791,92793,92795,92798,92800,92802,92804,92806,92809,92811,92813,92815],{"class":15660,"line":18320},[28,92784,92785],{"class":15675},"        row[",[28,92787,68],{"class":15671},[28,92789,92790],{"class":15841},"\"Required shots for ±",[28,92792,11385],{"class":15801},[28,92794,15955],{"class":15682},[28,92796,92797],{"class":15675},"(eps",[28,92799,16226],{"class":15682},[28,92801,31400],{"class":15801},[28,92803,78],{"class":15675},[28,92805,11404],{"class":15801},[28,92807,92808],{"class":15841},"%\"",[28,92810,15845],{"class":15675},[28,92812,845],{"class":15682},[28,92814,91843],{"class":16073},[28,92816,92817],{"class":15675},"(model.satisfaction_prob, eps)\n",[28,92819,92820,92822],{"class":15660,"line":18332},[28,92821,16565],{"class":15671},[28,92823,92824],{"class":15675}," row\n",[28,92826,92827],{"class":15660,"line":18344},[28,92828,15769],{"emptyLinePlaceholder":15768},[28,92830,92831],{"class":15660,"line":18356},[28,92832,15769],{"emptyLinePlaceholder":15768},[28,92834,92835,92837,92840,92842,92844,92846,92848,92850,92853,92855,92857,92859,92862,92864,92866,92868,92870],{"class":15660,"line":18376},[28,92836,16070],{"class":15671},[28,92838,92839],{"class":16073}," apply_costs",[28,92841,65],{"class":15675},[28,92843,92455],{"class":16082},[28,92845,15883],{"class":15675},[28,92847,15955],{"class":15682},[28,92849,16397],{"class":15675},[28,92851,92852],{"class":16082},"cost_per_shot",[28,92854,15883],{"class":15675},[28,92856,15886],{"class":15682},[28,92858,16397],{"class":15675},[28,92860,92861],{"class":16082},"fixed_overhead",[28,92863,15883],{"class":15675},[28,92865,15886],{"class":15682},[28,92867,17585],{"class":15675},[28,92869,15886],{"class":15682},[28,92871,15778],{"class":15675},[28,92873,92874],{"class":15660,"line":18382},[28,92875,16197],{"class":15841},[28,92877,92878],{"class":15660,"line":18407},[28,92879,92880],{"class":15841},"    Calculate illustrative cost based on shots and cost model.\n",[28,92882,92883],{"class":15660,"line":18430},[28,92884,16197],{"class":15841},[28,92886,92887,92889,92892,92894,92897,92899],{"class":15660,"line":18446},[28,92888,16565],{"class":15671},[28,92890,92891],{"class":15675}," fixed_overhead ",[28,92893,2606],{"class":15682},[28,92895,92896],{"class":15675}," shots ",[28,92898,16226],{"class":15682},[28,92900,92901],{"class":15675}," cost_per_shot\n",[28,92903,92904],{"class":15660,"line":18451},[28,92905,15769],{"emptyLinePlaceholder":15768},[28,92907,92908],{"class":15660,"line":18478},[28,92909,15769],{"emptyLinePlaceholder":15768},[28,92911,92912],{"class":15660,"line":18483},[28,92913,92914],{"class":15664},"# ---------- Run the simulation ----------\n",[28,92916,92917],{"class":15660,"line":18507},[28,92918,15769],{"emptyLinePlaceholder":15768},[28,92920,92921,92924,92926,92929,92931],{"class":15660,"line":18529},[28,92922,92923],{"class":15675},"all_bitstrings: Dict[",[28,92925,17582],{"class":15682},[28,92927,92928],{"class":15675},", np.ndarray] ",[28,92930,845],{"class":15682},[28,92932,49317],{"class":15675},[28,92934,92935,92938,92940],{"class":15660,"line":18550},[28,92936,92937],{"class":15675},"summary_rows: List[Dict] ",[28,92939,845],{"class":15682},[28,92941,24066],{"class":15675},[28,92943,92944],{"class":15660,"line":18555},[28,92945,15769],{"emptyLinePlaceholder":15768},[28,92947,92948,92950,92952,92955],{"class":15660,"line":18570},[28,92949,16044],{"class":15682},[28,92951,65],{"class":15675},[28,92953,92954],{"class":15841},"\"Running simulation...\"",[28,92956,16052],{"class":15675},[28,92958,92959,92961,92964,92966,92969],{"class":15660,"line":18585},[28,92960,22277],{"class":15671},[28,92962,92963],{"class":15675}," m ",[28,92965,17720],{"class":15671},[28,92967,92968],{"class":15801}," DEFAULT_MODELS",[28,92970,15778],{"class":15675},[28,92972,92973,92976,92978,92980,92983,92985,92987,92990],{"class":15660,"line":18603},[28,92974,92975],{"class":15675},"    bits ",[28,92977,845],{"class":15682},[28,92979,92442],{"class":16073},[28,92981,92982],{"class":15675},"(m.satisfaction_prob, ",[28,92984,92274],{"class":15801},[28,92986,3176],{"class":15675},[28,92988,92989],{"class":15841},"\"shots_per_model\"",[28,92991,92992],{"class":15675},"], rng)\n",[28,92994,92995,92998,93000],{"class":15660,"line":18609},[28,92996,92997],{"class":15675},"    all_bitstrings[m.name] ",[28,92999,845],{"class":15682},[28,93001,93002],{"class":15675}," bits\n",[28,93004,93005,93008,93010,93012,93015,93018,93020,93022,93025],{"class":15660,"line":18614},[28,93006,93007],{"class":15675},"    summary_rows.",[28,93009,27807],{"class":16073},[28,93011,65],{"class":15675},[28,93013,93014],{"class":16073},"summarize_model",[28,93016,93017],{"class":15675},"(m, bits, ",[28,93019,92274],{"class":15801},[28,93021,3176],{"class":15675},[28,93023,93024],{"class":15841},"\"epsilons\"",[28,93026,93027],{"class":15675},"]))\n",[28,93029,93030],{"class":15660,"line":18629},[28,93031,15769],{"emptyLinePlaceholder":15768},[28,93033,93034,93037,93039,93041,93043],{"class":15660,"line":18635},[28,93035,93036],{"class":15675},"summary_df ",[28,93038,845],{"class":15682},[28,93040,16695],{"class":15675},[28,93042,16698],{"class":16073},[28,93044,93045],{"class":15675},"(summary_rows)\n",[28,93047,93048],{"class":15660,"line":18641},[28,93049,15769],{"emptyLinePlaceholder":15768},[28,93051,93052],{"class":15660,"line":18647},[28,93053,93054],{"class":15664},"# Add cost estimates\n",[28,93056,93057,93060,93062,93065,93067,93070,93072,93075],{"class":15660,"line":18652},[28,93058,93059],{"class":15675},"cps ",[28,93061,845],{"class":15682},[28,93063,93064],{"class":15801}," CONFIG",[28,93066,3176],{"class":15675},[28,93068,93069],{"class":15841},"\"cost_model\"",[28,93071,21951],{"class":15675},[28,93073,93074],{"class":15841},"\"default_cost_per_shot_usd\"",[28,93076,16849],{"class":15675},[28,93078,93079,93082,93084,93086,93088,93090,93092,93095],{"class":15660,"line":18657},[28,93080,93081],{"class":15675},"over ",[28,93083,845],{"class":15682},[28,93085,93064],{"class":15801},[28,93087,3176],{"class":15675},[28,93089,93069],{"class":15841},[28,93091,21951],{"class":15675},[28,93093,93094],{"class":15841},"\"fixed_overhead_usd\"",[28,93096,16849],{"class":15675},[28,93098,93099,93101,93104,93106,93108,93110,93113,93115,93117],{"class":15660,"line":18689},[28,93100,20133],{"class":15671},[28,93102,93103],{"class":15675}," cps ",[28,93105,16503],{"class":15682},[28,93107,46881],{"class":15801},[28,93109,47786],{"class":15671},[28,93111,93112],{"class":15675}," over ",[28,93114,16503],{"class":15682},[28,93116,46881],{"class":15801},[28,93118,15778],{"class":15675},[28,93120,93121,93124,93127,93129,93131],{"class":15660,"line":18716},[28,93122,93123],{"class":15675},"    summary_df[",[28,93125,93126],{"class":15841},"\"Illustrative cost (USD)\"",[28,93128,15845],{"class":15675},[28,93130,845],{"class":15682},[28,93132,17702],{"class":15675},[28,93134,93135,93138,93140,93142,93145,93148],{"class":15660,"line":18742},[28,93136,93137],{"class":16073},"        apply_costs",[28,93139,65],{"class":15675},[28,93141,15955],{"class":15682},[28,93143,93144],{"class":15675},"(row[",[28,93146,93147],{"class":15841},"\"Shots\"",[28,93149,93150],{"class":15675},"]), cps, over)\n",[28,93152,93153,93155,93158,93160,93163,93166],{"class":15660,"line":18747},[28,93154,17714],{"class":15671},[28,93156,93157],{"class":15675}," _, row ",[28,93159,17720],{"class":15671},[28,93161,93162],{"class":15675}," summary_df.",[28,93164,93165],{"class":16073},"iterrows",[28,93167,17029],{"class":15675},[28,93169,93170],{"class":15660,"line":18761},[28,93171,17747],{"class":15675},[28,93173,93174],{"class":15660,"line":18776},[28,93175,15769],{"emptyLinePlaceholder":15768},[28,93177,93178],{"class":15660,"line":18781},[28,93179,93180],{"class":15664},"# ---------- Save artifacts ----------\n",[28,93182,93183],{"class":15660,"line":18786},[28,93184,15769],{"emptyLinePlaceholder":15768},[28,93186,93187],{"class":15660,"line":18801},[28,93188,93189],{"class":15664},"# Save raw bitstrings and summary CSV\n",[28,93191,93192,93194,93197,93199,93202,93204],{"class":15660,"line":18807},[28,93193,22277],{"class":15671},[28,93195,93196],{"class":15675}," name, arr ",[28,93198,17720],{"class":15671},[28,93200,93201],{"class":15675}," all_bitstrings.",[28,93203,17726],{"class":16073},[28,93205,27799],{"class":15675},[28,93207,93208,93211,93213,93215,93218,93220,93223,93226,93228,93231,93233,93236,93238,93240],{"class":15660,"line":18813},[28,93209,93210],{"class":15675},"    out_path ",[28,93212,845],{"class":15682},[28,93214,23216],{"class":15671},[28,93216,93217],{"class":15841},"\"bitstrings_",[28,93219,11385],{"class":15801},[28,93221,93222],{"class":15675},"name.",[28,93224,93225],{"class":16073},"replace",[28,93227,65],{"class":15675},[28,93229,93230],{"class":15841},"' '",[28,93232,16397],{"class":15675},[28,93234,93235],{"class":15841},"'_'",[28,93237,78],{"class":15675},[28,93239,11404],{"class":15801},[28,93241,93242],{"class":15841},".csv\"\n",[28,93244,93245,93248,93250,93253,93256,93259,93262,93265,93268,93270,93272],{"class":15660,"line":18819},[28,93246,93247],{"class":15675},"    pd.",[28,93249,16698],{"class":16073},[28,93251,93252],{"class":15675},"({",[28,93254,93255],{"class":15841},"\"bit\"",[28,93257,93258],{"class":15675},": arr}).",[28,93260,93261],{"class":16073},"to_csv",[28,93263,93264],{"class":15675},"(out_path, ",[28,93266,93267],{"class":16307},"index",[28,93269,845],{"class":15682},[28,93271,19707],{"class":15801},[28,93273,16052],{"class":15675},[28,93275,93276],{"class":15660,"line":18825},[28,93277,15769],{"emptyLinePlaceholder":15768},[28,93279,93280,93283,93285],{"class":15660,"line":18831},[28,93281,93282],{"class":15675},"summary_csv_path ",[28,93284,845],{"class":15682},[28,93286,93287],{"class":15841}," \"summary.csv\"\n",[28,93289,93290,93293,93295,93298,93300,93302,93304],{"class":15660,"line":18836},[28,93291,93292],{"class":15675},"summary_df.",[28,93294,93261],{"class":16073},[28,93296,93297],{"class":15675},"(summary_csv_path, ",[28,93299,93267],{"class":16307},[28,93301,845],{"class":15682},[28,93303,19707],{"class":15801},[28,93305,16052],{"class":15675},[28,93307,93308],{"class":15660,"line":18841},[28,93309,15769],{"emptyLinePlaceholder":15768},[28,93311,93312],{"class":15660,"line":18856},[28,93313,93314],{"class":15664},"# Save config JSON\n",[28,93316,93317,93320,93323,93325,93328,93330,93333,93335,93337],{"class":15660,"line":18861},[28,93318,93319],{"class":15671},"with",[28,93321,93322],{"class":15682}," open",[28,93324,65],{"class":15675},[28,93326,93327],{"class":15841},"\"simulation_config.json\"",[28,93329,16397],{"class":15675},[28,93331,93332],{"class":15841},"\"w\"",[28,93334,22233],{"class":15675},[28,93336,15733],{"class":15671},[28,93338,93339],{"class":15675}," f:\n",[28,93341,93342,93345,93348,93350,93352,93355,93358,93360,93362],{"class":15660,"line":18872},[28,93343,93344],{"class":15675},"    json.",[28,93346,93347],{"class":16073},"dump",[28,93349,65],{"class":15675},[28,93351,92274],{"class":15801},[28,93353,93354],{"class":15675},", f, ",[28,93356,93357],{"class":16307},"indent",[28,93359,845],{"class":15682},[28,93361,848],{"class":15801},[28,93363,16052],{"class":15675},[28,93365,93366],{"class":15660,"line":18880},[28,93367,15769],{"emptyLinePlaceholder":15768},[28,93369,93370,93372,93374,93377],{"class":15660,"line":18890},[28,93371,16044],{"class":15682},[28,93373,65],{"class":15675},[28,93375,93376],{"class":15841},"\"Saved data to CSV and JSON files.\"",[28,93378,16052],{"class":15675},[28,93380,93381],{"class":15660,"line":18895},[28,93382,15769],{"emptyLinePlaceholder":15768},[28,93384,93385],{"class":15660,"line":18900},[28,93386,93387],{"class":15664},"# ---------- Visualizations ----------\n",[28,93389,93390],{"class":15660,"line":18905},[28,93391,15769],{"emptyLinePlaceholder":15768},[28,93393,93394],{"class":15660,"line":18912},[28,93395,93396],{"class":15664},"# Figure 1: Observed ones fraction with Wilson CI\n",[28,93398,93399,93402,93405,93407,93410],{"class":15660,"line":18928},[28,93400,93401],{"class":15675},"plt.style.",[28,93403,93404],{"class":16073},"use",[28,93406,65],{"class":15675},[28,93408,93409],{"class":15841},"'seaborn-v0_8-whitegrid'",[28,93411,16052],{"class":15675},[28,93413,93414,93416,93418,93420,93422,93424,93426,93428,93430,93432],{"class":15660,"line":18937},[28,93415,17034],{"class":15675},[28,93417,20380],{"class":16073},[28,93419,65],{"class":15675},[28,93421,16887],{"class":16307},[28,93423,845],{"class":15682},[28,93425,65],{"class":15675},[28,93427,19879],{"class":15801},[28,93429,16397],{"class":15675},[28,93431,22491],{"class":15801},[28,93433,16539],{"class":15675},[28,93435,93436,93439,93441,93443,93446,93448,93450],{"class":15660,"line":18946},[28,93437,93438],{"class":15675},"x ",[28,93440,845],{"class":15682},[28,93442,16298],{"class":15675},[28,93444,93445],{"class":16073},"arange",[28,93447,65],{"class":15675},[28,93449,22222],{"class":15682},[28,93451,93452],{"class":15675},"(summary_df))\n",[28,93454,93455,93458,93460,93463,93466,93468,93471],{"class":15660,"line":18955},[28,93456,93457],{"class":15675},"y ",[28,93459,845],{"class":15682},[28,93461,93462],{"class":15675}," summary_df[",[28,93464,93465],{"class":15841},"\"Ones fraction (observed)\"",[28,93467,48516],{"class":15675},[28,93469,93470],{"class":16073},"to_numpy",[28,93472,17029],{"class":15675},[28,93474,93475],{"class":15660,"line":18964},[28,93476,15769],{"emptyLinePlaceholder":15768},[28,93478,93479],{"class":15660,"line":18987},[28,93480,93481],{"class":15664},"# --- FIX APPLIED HERE ---\n",[28,93483,93484],{"class":15660,"line":18993},[28,93485,93486],{"class":15664},"# Take the absolute value to prevent tiny negative numbers from floating-point\n",[28,93488,93489],{"class":15660,"line":18998},[28,93490,93491],{"class":15664},"# inaccuracies from causing a ValueError in the errorbar plot.\n",[28,93493,93494,93497,93499,93502,93504,93506,93509,93511,93513,93515,93517],{"class":15660,"line":19003},[28,93495,93496],{"class":15675},"lower_error ",[28,93498,845],{"class":15682},[28,93500,93501],{"class":15675}," y ",[28,93503,15689],{"class":15682},[28,93505,93462],{"class":15675},[28,93507,93508],{"class":15841},"\"Wilson CI 95",[28,93510,92732],{"class":15801},[28,93512,92735],{"class":15841},[28,93514,48516],{"class":15675},[28,93516,93470],{"class":16073},[28,93518,17029],{"class":15675},[28,93520,93521,93524,93526,93528,93530,93532,93534,93536,93538,93540,93542],{"class":15660,"line":19008},[28,93522,93523],{"class":15675},"upper_error ",[28,93525,845],{"class":15682},[28,93527,93462],{"class":15675},[28,93529,93508],{"class":15841},[28,93531,92745],{"class":15801},[28,93533,92748],{"class":15841},[28,93535,48516],{"class":15675},[28,93537,93470],{"class":16073},[28,93539,23647],{"class":15675},[28,93541,15689],{"class":15682},[28,93543,93544],{"class":15675}," y\n",[28,93546,93547,93550,93552,93554,93556,93559,93561,93564,93566],{"class":15660,"line":19018},[28,93548,93549],{"class":15675},"yerr ",[28,93551,845],{"class":15682},[28,93553,16298],{"class":15675},[28,93555,18796],{"class":16073},[28,93557,93558],{"class":15675},"([np.",[28,93560,23043],{"class":16073},[28,93562,93563],{"class":15675},"(lower_error), np.",[28,93565,23043],{"class":16073},[28,93567,93568],{"class":15675},"(upper_error)])\n",[28,93570,93571],{"class":15660,"line":19027},[28,93572,93573],{"class":15664},"# --- END OF FIX ---\n",[28,93575,93576],{"class":15660,"line":19036},[28,93577,15769],{"emptyLinePlaceholder":15768},[28,93579,93580,93582,93585,93588,93590,93592,93595,93597,93600,93602,93605],{"class":15660,"line":19041},[28,93581,17034],{"class":15675},[28,93583,93584],{"class":16073},"bar",[28,93586,93587],{"class":15675},"(x, y, ",[28,93589,23134],{"class":16307},[28,93591,845],{"class":15682},[28,93593,93594],{"class":15841},"'skyblue'",[28,93596,16397],{"class":15675},[28,93598,93599],{"class":16307},"edgecolor",[28,93601,845],{"class":15682},[28,93603,93604],{"class":15841},"'black'",[28,93606,16052],{"class":15675},[28,93608,93609,93611,93614,93616,93619,93621,93624,93627,93629,93632,93634,93637,93639,93641,93643,93645,93647,93650,93652,93655,93657,93659],{"class":15660,"line":19047},[28,93610,17034],{"class":15675},[28,93612,93613],{"class":16073},"errorbar",[28,93615,93587],{"class":15675},[28,93617,93618],{"class":16307},"yerr",[28,93620,845],{"class":15682},[28,93622,93623],{"class":15675},"yerr, ",[28,93625,93626],{"class":16307},"fmt",[28,93628,845],{"class":15682},[28,93630,93631],{"class":15841},"\"none\"",[28,93633,16397],{"class":15675},[28,93635,93636],{"class":16307},"capsize",[28,93638,845],{"class":15682},[28,93640,16899],{"class":15801},[28,93642,16397],{"class":15675},[28,93644,23134],{"class":16307},[28,93646,845],{"class":15682},[28,93648,93649],{"class":15841},"\"black\"",[28,93651,16397],{"class":15675},[28,93653,93654],{"class":16307},"elinewidth",[28,93656,845],{"class":15682},[28,93658,25002],{"class":15801},[28,93660,16052],{"class":15675},[28,93662,93663,93665,93667,93670,93673,93675,93677,93679,93682,93684,93686,93688,93691,93693,93696],{"class":15660,"line":19063},[28,93664,17034],{"class":15675},[28,93666,51120],{"class":16073},[28,93668,93669],{"class":15675},"(x, summary_df[",[28,93671,93672],{"class":15841},"\"Model\"",[28,93674,48516],{"class":15675},[28,93676,49975],{"class":16073},[28,93678,24379],{"class":15675},[28,93680,93681],{"class":16307},"rotation",[28,93683,845],{"class":15682},[28,93685,7099],{"class":15801},[28,93687,16397],{"class":15675},[28,93689,93690],{"class":16307},"ha",[28,93692,845],{"class":15682},[28,93694,93695],{"class":15841},"\"right\"",[28,93697,16052],{"class":15675},[28,93699,93700,93702,93704,93706,93709],{"class":15660,"line":19074},[28,93701,17034],{"class":15675},[28,93703,51179],{"class":16073},[28,93705,65],{"class":15675},[28,93707,93708],{"class":15841},"\"Observed Ones Fraction\"",[28,93710,16052],{"class":15675},[28,93712,93713,93715,93717,93719,93722,93724,93727,93729,93732],{"class":15660,"line":19089},[28,93714,17034],{"class":15675},[28,93716,51193],{"class":16073},[28,93718,65],{"class":15675},[28,93720,93721],{"class":15841},"\"Observed Single-Shot Success Rates with 95% Wilson CI\"",[28,93723,16397],{"class":15675},[28,93725,93726],{"class":16307},"fontsize",[28,93728,845],{"class":15682},[28,93730,93731],{"class":15801},"14",[28,93733,16052],{"class":15675},[28,93735,93736,93738,93740],{"class":15660,"line":19099},[28,93737,17034],{"class":15675},[28,93739,17026],{"class":16073},[28,93741,17029],{"class":15675},[28,93743,93744,93747,93749],{"class":15660,"line":19107},[28,93745,93746],{"class":15675},"fig1_path ",[28,93748,845],{"class":15682},[28,93750,93751],{"class":15841}," \"fig_ones_fraction.png\"\n",[28,93753,93754,93756,93759,93762,93764,93766,93768,93770,93773,93775,93778],{"class":15660,"line":19116},[28,93755,17034],{"class":15675},[28,93757,93758],{"class":16073},"savefig",[28,93760,93761],{"class":15675},"(fig1_path, ",[28,93763,22497],{"class":16307},[28,93765,845],{"class":15682},[28,93767,22502],{"class":15801},[28,93769,16397],{"class":15675},[28,93771,93772],{"class":16307},"bbox_inches",[28,93774,845],{"class":15682},[28,93776,93777],{"class":15841},"'tight'",[28,93779,16052],{"class":15675},[28,93781,93782,93784,93786],{"class":15660,"line":19121},[28,93783,17034],{"class":15675},[28,93785,17037],{"class":16073},[28,93787,17029],{"class":15675},[28,93789,93790],{"class":15660,"line":19143},[28,93791,15769],{"emptyLinePlaceholder":15768},[28,93793,93794],{"class":15660,"line":19148},[28,93795,93796],{"class":15664},"# Figure 2: Required shots at several ±epsilon targets\n",[28,93798,93799,93801,93803,93805,93807,93809,93811,93813,93815,93817],{"class":15660,"line":19165},[28,93800,17034],{"class":15675},[28,93802,20380],{"class":16073},[28,93804,65],{"class":15675},[28,93806,16887],{"class":16307},[28,93808,845],{"class":15682},[28,93810,65],{"class":15675},[28,93812,19879],{"class":15801},[28,93814,16397],{"class":15675},[28,93816,22491],{"class":15801},[28,93818,16539],{"class":15675},[28,93820,93821,93824,93826],{"class":15660,"line":19182},[28,93822,93823],{"class":15675},"width ",[28,93825,845],{"class":15682},[28,93827,93828],{"class":15801}," 0.2\n",[28,93830,93831,93833,93835,93837,93839,93841,93843,93845,93847],{"class":15660,"line":19187},[28,93832,93438],{"class":15675},[28,93834,845],{"class":15682},[28,93836,16298],{"class":15675},[28,93838,93445],{"class":16073},[28,93840,65],{"class":15675},[28,93842,22222],{"class":15682},[28,93844,65],{"class":15675},[28,93846,92099],{"class":15801},[28,93848,16539],{"class":15675},[28,93850,93851,93854,93856,93858,93861,93863,93866,93868,93871],{"class":15660,"line":19200},[28,93852,93853],{"class":15675},"colors ",[28,93855,845],{"class":15682},[28,93857,17337],{"class":15675},[28,93859,93860],{"class":15841},"'#ff9999'",[28,93862,72],{"class":15675},[28,93864,93865],{"class":15841},"'#66b3ff'",[28,93867,72],{"class":15675},[28,93869,93870],{"class":15841},"'#99ff99'",[28,93872,16849],{"class":15675},[28,93874,93875],{"class":15660,"line":19214},[28,93876,15769],{"emptyLinePlaceholder":15768},[28,93878,93879,93881,93884,93886,93888,93890,93892,93894,93896],{"class":15660,"line":19219},[28,93880,22277],{"class":15671},[28,93882,93883],{"class":15675}," i, eps ",[28,93885,17720],{"class":15671},[28,93887,27825],{"class":15682},[28,93889,65],{"class":15675},[28,93891,92274],{"class":15801},[28,93893,3176],{"class":15675},[28,93895,93024],{"class":15841},[28,93897,93898],{"class":15675},"]):\n",[28,93900,93901,93904,93906,93908,93911,93914,93916,93918,93920,93922],{"class":15660,"line":19227},[28,93902,93903],{"class":15675},"    req ",[28,93905,845],{"class":15682},[28,93907,17337],{"class":15675},[28,93909,93910],{"class":16073},"shots_for_epsilon",[28,93912,93913],{"class":15675},"(m.satisfaction_prob, eps) ",[28,93915,22277],{"class":15671},[28,93917,92963],{"class":15675},[28,93919,17720],{"class":15671},[28,93921,92968],{"class":15801},[28,93923,16849],{"class":15675},[28,93925,93926,93929,93931,93934,93936,93938,93940,93943,93945,93947,93949,93952,93954,93957,93960,93962,93964,93966,93968,93970,93973,93975,93977,93979],{"class":15660,"line":19232},[28,93927,93928],{"class":15675},"    plt.",[28,93930,93584],{"class":16073},[28,93932,93933],{"class":15675},"(x ",[28,93935,2606],{"class":15682},[28,93937,22280],{"class":15675},[28,93939,16226],{"class":15682},[28,93941,93942],{"class":15675}," width, req, width, ",[28,93944,16954],{"class":16307},[28,93946,845],{"class":15682},[28,93948,68],{"class":15671},[28,93950,93951],{"class":15841},"\"±",[28,93953,11385],{"class":15801},[28,93955,93956],{"class":15675},"eps",[28,93958,93959],{"class":15671},":.1%",[28,93961,11404],{"class":15801},[28,93963,21798],{"class":15841},[28,93965,16397],{"class":15675},[28,93967,23134],{"class":16307},[28,93969,845],{"class":15682},[28,93971,93972],{"class":15675},"colors[i], ",[28,93974,93599],{"class":16307},[28,93976,845],{"class":15682},[28,93978,93604],{"class":15841},[28,93980,16052],{"class":15675},[28,93982,93983],{"class":15660,"line":19237},[28,93984,15769],{"emptyLinePlaceholder":15768},[28,93986,93987,93989,93991,93993,93995,93998,94000,94002,94004,94006,94008,94010,94012,94014,94016,94018,94020,94022],{"class":15660,"line":19247},[28,93988,17034],{"class":15675},[28,93990,51120],{"class":16073},[28,93992,93933],{"class":15675},[28,93994,2606],{"class":15682},[28,93996,93997],{"class":15675}," width, [m.name ",[28,93999,22277],{"class":15671},[28,94001,92963],{"class":15675},[28,94003,17720],{"class":15671},[28,94005,92968],{"class":15801},[28,94007,24619],{"class":15675},[28,94009,93681],{"class":16307},[28,94011,845],{"class":15682},[28,94013,7099],{"class":15801},[28,94015,16397],{"class":15675},[28,94017,93690],{"class":16307},[28,94019,845],{"class":15682},[28,94021,93695],{"class":15841},[28,94023,16052],{"class":15675},[28,94025,94026,94028,94030,94032,94035],{"class":15660,"line":19255},[28,94027,17034],{"class":15675},[28,94029,51179],{"class":16073},[28,94031,65],{"class":15675},[28,94033,94034],{"class":15841},"\"Required Shots (Log Scale)\"",[28,94036,16052],{"class":15675},[28,94038,94039,94041,94043,94045,94048,94050,94052,94054,94056],{"class":15660,"line":19263},[28,94040,17034],{"class":15675},[28,94042,51193],{"class":16073},[28,94044,65],{"class":15675},[28,94046,94047],{"class":15841},"\"Approximate Shots to Estimate Success Rate within ±ε\"",[28,94049,16397],{"class":15675},[28,94051,93726],{"class":16307},[28,94053,845],{"class":15682},[28,94055,93731],{"class":15801},[28,94057,16052],{"class":15675},[28,94059,94060,94062,94065,94067,94070],{"class":15660,"line":19275},[28,94061,17034],{"class":15675},[28,94063,94064],{"class":16073},"yscale",[28,94066,65],{"class":15675},[28,94068,94069],{"class":15841},"'log'",[28,94071,16052],{"class":15675},[28,94073,94074,94076,94078,94080,94082,94084,94087],{"class":15660,"line":19287},[28,94075,17034],{"class":15675},[28,94077,21483],{"class":16073},[28,94079,65],{"class":15675},[28,94081,51193],{"class":16307},[28,94083,845],{"class":15682},[28,94085,94086],{"class":15841},"\"Error Tolerance\"",[28,94088,16052],{"class":15675},[28,94090,94091,94093,94095],{"class":15660,"line":19296},[28,94092,17034],{"class":15675},[28,94094,17026],{"class":16073},[28,94096,17029],{"class":15675},[28,94098,94099,94102,94104],{"class":15660,"line":19302},[28,94100,94101],{"class":15675},"fig2_path ",[28,94103,845],{"class":15682},[28,94105,94106],{"class":15841}," \"fig_required_shots.png\"\n",[28,94108,94109,94111,94113,94116,94118,94120,94122,94124,94126,94128,94130],{"class":15660,"line":19313},[28,94110,17034],{"class":15675},[28,94112,93758],{"class":16073},[28,94114,94115],{"class":15675},"(fig2_path, ",[28,94117,22497],{"class":16307},[28,94119,845],{"class":15682},[28,94121,22502],{"class":15801},[28,94123,16397],{"class":15675},[28,94125,93772],{"class":16307},[28,94127,845],{"class":15682},[28,94129,93777],{"class":15841},[28,94131,16052],{"class":15675},[28,94133,94134,94136,94138],{"class":15660,"line":19323},[28,94135,17034],{"class":15675},[28,94137,17037],{"class":16073},[28,94139,17029],{"class":15675},[28,94141,94142],{"class":15660,"line":19332},[28,94143,15769],{"emptyLinePlaceholder":15768},[28,94145,94146,94148,94150,94153],{"class":15660,"line":19341},[28,94147,16044],{"class":15682},[28,94149,65],{"class":15675},[28,94151,94152],{"class":15841},"\"Generated and saved plots.\"",[28,94154,16052],{"class":15675},[28,94156,94157],{"class":15660,"line":19352},[28,94158,15769],{"emptyLinePlaceholder":15768},[28,94160,94161],{"class":15660,"line":19364},[28,94162,94163],{"class":15664},"# ---------- Display interactive table ----------\n",[28,94165,94166],{"class":15660,"line":19375},[28,94167,15769],{"emptyLinePlaceholder":15768},[28,94169,94170,94173,94175,94178,94180,94183],{"class":15660,"line":19380},[28,94171,94172],{"class":16073},"display",[28,94174,65],{"class":15675},[28,94176,94177],{"class":16073},"Markdown",[28,94179,65],{"class":15675},[28,94181,94182],{"class":15841},"\"##  Qubit Sampling Budget - Summary\"",[28,94184,16539],{"class":15675},[28,94186,94187,94189,94192,94195],{"class":15660,"line":19385},[28,94188,94172],{"class":16073},[28,94190,94191],{"class":15675},"(summary_df.style.",[28,94193,94194],{"class":16073},"format",[28,94196,94197],{"class":15675},"({\n",[28,94199,94200,94203,94205,94207,94209,94212,94214,94216],{"class":15660,"line":19390},[28,94201,94202],{"class":15841},"    \"Configured p (target)\"",[28,94204,15883],{"class":15675},[28,94206,21798],{"class":15841},[28,94208,11385],{"class":15801},[28,94210,94211],{"class":15671},":.2f",[28,94213,11404],{"class":15801},[28,94215,21798],{"class":15841},[28,94217,16105],{"class":15675},[28,94219,94220,94223,94225,94227,94229,94232,94234,94236],{"class":15660,"line":19400},[28,94221,94222],{"class":15841},"    \"Ones fraction (observed)\"",[28,94224,15883],{"class":15675},[28,94226,21798],{"class":15841},[28,94228,11385],{"class":15801},[28,94230,94231],{"class":15671},":.4f",[28,94233,11404],{"class":15801},[28,94235,21798],{"class":15841},[28,94237,16105],{"class":15675},[28,94239,94240,94243,94245,94247,94249,94251,94253,94255,94257,94259],{"class":15660,"line":19407},[28,94241,94242],{"class":15841},"    \"Wilson CI 95",[28,94244,92732],{"class":15801},[28,94246,92735],{"class":15841},[28,94248,15883],{"class":15675},[28,94250,21798],{"class":15841},[28,94252,11385],{"class":15801},[28,94254,94231],{"class":15671},[28,94256,11404],{"class":15801},[28,94258,21798],{"class":15841},[28,94260,16105],{"class":15675},[28,94262,94263,94265,94267,94269,94271,94273,94275,94277,94279,94281],{"class":15660,"line":19415},[28,94264,94242],{"class":15841},[28,94266,92745],{"class":15801},[28,94268,92748],{"class":15841},[28,94270,15883],{"class":15675},[28,94272,21798],{"class":15841},[28,94274,11385],{"class":15801},[28,94276,94231],{"class":15671},[28,94278,11404],{"class":15801},[28,94280,21798],{"class":15841},[28,94282,16105],{"class":15675},[28,94284,94285,94288,94290,94292,94294,94296,94298,94300],{"class":15660,"line":19426},[28,94286,94287],{"class":15841},"    \"CI width (95%)\"",[28,94289,15883],{"class":15675},[28,94291,21798],{"class":15841},[28,94293,11385],{"class":15801},[28,94295,94231],{"class":15671},[28,94297,11404],{"class":15801},[28,94299,21798],{"class":15841},[28,94301,16105],{"class":15675},[28,94303,94304,94307,94309,94311,94313,94315,94317],{"class":15660,"line":19437},[28,94305,94306],{"class":15841},"    \"Illustrative cost (USD)\"",[28,94308,15883],{"class":15675},[28,94310,20444],{"class":15841},[28,94312,11385],{"class":15801},[28,94314,94211],{"class":15671},[28,94316,11404],{"class":15801},[28,94318,20490],{"class":15841},[28,94320,94321],{"class":15660,"line":19446},[28,94322,94323],{"class":15675},"}))\n",[28,94325,94326],{"class":15660,"line":19452},[28,94327,15769],{"emptyLinePlaceholder":15768},[28,94329,94330],{"class":15660,"line":19462},[28,94331,15769],{"emptyLinePlaceholder":15768},[28,94333,94334],{"class":15660,"line":19472},[28,94335,94336],{"class":15664},"# ---------- Build a ready-to-run Colab notebook ----------\n",[28,94338,94339],{"class":15660,"line":19481},[28,94340,15769],{"emptyLinePlaceholder":15768},[28,94342,94343,94346,94348,94351],{"class":15660,"line":19490},[28,94344,94345],{"class":15675},"md_intro ",[28,94347,845],{"class":15682},[28,94349,94350],{"class":15671}," r",[28,94352,94353],{"class":16962},"\"\"\"\n",[28,94355,94356],{"class":15660,"line":19499},[28,94357,94358],{"class":15664},"# Sampling Budget & Bit-String Analogy for Qubit Classes\n",[28,94360,94361],{"class":15660,"line":19511},[28,94362,15769],{"emptyLinePlaceholder":15768},[28,94364,94365,94367,94370,94372,94375,94377,94379,94381,94384,94386,94389,94391],{"class":15660,"line":19523},[28,94366,16354],{"class":15801},[28,94368,94369],{"class":16962},"High school level ",[28,94371,65],{"class":15801},[28,94373,94374],{"class":16962},"concise",[28,94376,78],{"class":15801},[28,94378,50590],{"class":16962},[28,94380,16354],{"class":15801},[28,94382,94383],{"class":16962}," We treat each measurement ",[28,94385,65],{"class":15801},[28,94387,94388],{"class":16962},"“single shot”",[28,94390,78],{"class":15801},[28,94392,94393],{"class":16962}," 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",[28,94395,94396],{"class":15660,"line":19532},[28,94397,15769],{"emptyLinePlaceholder":15768},[28,94399,94400,94402,94405,94407,94409,94411,94413,94415,94418,94420,94423,94425,94428,94430,94433,94435,94438,94440,94443,94445,94448,94451,94454,94457,94459,94461,94464,94466,94468,94471,94474,94476],{"class":15660,"line":19537},[28,94401,16354],{"class":15801},[28,94403,94404],{"class":16962},"Graduate level ",[28,94406,65],{"class":15801},[28,94408,94374],{"class":16962},[28,94410,78],{"class":15801},[28,94412,50590],{"class":16962},[28,94414,16354],{"class":15801},[28,94416,94417],{"class":16962}," We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ ",[28,94419,65],{"class":15801},[28,94421,94422],{"class":16962},"noisy intermediate-scale quantum",[28,94424,78],{"class":15801},[28,94426,94427],{"class":16962},", fault-tolerant ",[28,94429,65],{"class":15801},[28,94431,94432],{"class":16962},"logical",[28,94434,78],{"class":15801},[28,94436,94437],{"class":16962},", topologically protected, and ideal. We report Wilson score intervals ",[28,94439,65],{"class":15801},[28,94441,94442],{"class":16962},"95%",[28,94444,78],{"class":15801},[28,94446,94447],{"class":16962}," for observed proportions and estimate the required shot budget $n ",[28,94449,94450],{"class":15682},"\\a",[28,94452,94453],{"class":16962},"pprox z^2",[28,94455,94456],{"class":15682},"\\,",[28,94458,12],{"class":16962},[28,94460,65],{"class":15801},[28,94462,94463],{"class":16962},"1-p",[28,94465,78],{"class":15801},[28,94467,1129],{"class":16962},[28,94469,94470],{"class":15682},"\\e",[28,94472,94473],{"class":16962},"psilon^2$ to bound absolute error by ±$",[28,94475,94470],{"class":15682},[28,94477,94478],{"class":16962},"psilon$ at ~95% confidence.\n",[28,94480,94481],{"class":15660,"line":19542},[28,94482,94353],{"class":16962},[28,94484,94485],{"class":15660,"line":19547},[28,94486,15769],{"emptyLinePlaceholder":15768},[28,94488,94489,94492,94494,94496],{"class":15660,"line":19557},[28,94490,94491],{"class":15675},"code_config ",[28,94493,845],{"class":15682},[28,94495,94350],{"class":15671},[28,94497,94353],{"class":16962},[28,94499,94500],{"class":15660,"line":19564},[28,94501,94502],{"class":15664},"# @title Configuration\n",[28,94504,94505],{"class":15660,"line":19571},[28,94506,94507],{"class":16962},"from dataclasses import dataclass\n",[28,94509,94510],{"class":15660,"line":19582},[28,94511,94512],{"class":16962},"from typing import List, Dict\n",[28,94514,94515],{"class":15660,"line":19593},[28,94516,94517],{"class":16962},"import numpy as np\n",[28,94519,94520],{"class":15660,"line":19602},[28,94521,15769],{"emptyLinePlaceholder":15768},[28,94523,94524,94527],{"class":15660,"line":19608},[28,94525,94526],{"class":16962},"RANDOM_SEED = 579345  ",[28,94528,94529],{"class":15664},"# @param {type:\"number\"}\n",[28,94531,94532,94535],{"class":15660,"line":19618},[28,94533,94534],{"class":16962},"SHOTS_PER_MODEL = 10000  ",[28,94536,94529],{"class":15664},[28,94538,94539,94542,94545],{"class":15660,"line":19626},[28,94540,94541],{"class":16962},"EPSILONS = ",[28,94543,94544],{"class":15801},"[0.02, 0.01, 0.005]",[28,94546,94547],{"class":15664},"  # @param\n",[28,94549,94550,94553],{"class":15660,"line":19633},[28,94551,94552],{"class":16962},"COST_PER_SHOT_USD = 0.00005  ",[28,94554,94529],{"class":15664},[28,94556,94557,94560],{"class":15660,"line":19640},[28,94558,94559],{"class":16962},"FIXED_OVERHEAD_USD = 0.10    ",[28,94561,94529],{"class":15664},[28,94563,94564],{"class":15660,"line":19649},[28,94565,15769],{"emptyLinePlaceholder":15768},[28,94567,94568,94570,94572,94575],{"class":15660,"line":19658},[28,94569,92039],{"class":16962},[28,94571,65],{"class":15801},[28,94573,94574],{"class":16962},"frozen=True",[28,94576,16052],{"class":15801},[28,94578,94579],{"class":15660,"line":19667},[28,94580,94581],{"class":16962},"class QubitModel:\n",[28,94583,94584],{"class":15660,"line":19676},[28,94585,94586],{"class":16962},"    name: str\n",[28,94588,94589],{"class":15660,"line":19687},[28,94590,94591],{"class":16962},"    satisfaction_prob: float\n",[28,94593,94594],{"class":15660,"line":19699},[28,94595,94596],{"class":16962},"    description: str\n",[28,94598,94599],{"class":15660,"line":19712},[28,94600,15769],{"emptyLinePlaceholder":15768},[28,94602,94603,94606,94609,94612],{"class":15660,"line":19721},[28,94604,94605],{"class":16962},"MODELS: List",[28,94607,94608],{"class":15801},"[QubitModel]",[28,94610,94611],{"class":16962}," = ",[28,94613,94614],{"class":15801},"[\n",[28,94616,94617],{"class":15660,"line":19726},[28,94618,94619],{"class":15801},"    QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n",[28,94621,94622],{"class":15660,"line":19731},[28,94623,94624],{"class":15801},"    QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n",[28,94626,94627],{"class":15660,"line":19736},[28,94628,94629],{"class":15801},"    QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n",[28,94631,94632],{"class":15660,"line":19746},[28,94633,94634],{"class":15801},"    QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n",[28,94636,94637],{"class":15660,"line":19753},[28,94638,94639],{"class":15801},"    QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n",[28,94641,94642],{"class":15660,"line":19760},[28,94643,16849],{"class":15801},[28,94645,94646],{"class":15660,"line":19772},[28,94647,15769],{"emptyLinePlaceholder":15768},[28,94649,94650,94653,94655,94658],{"class":15660,"line":19783},[28,94651,94652],{"class":16962},"rng = np.random.default_rng",[28,94654,65],{"class":15801},[28,94656,94657],{"class":16962},"RANDOM_SEED",[28,94659,16052],{"class":15801},[28,94661,94662],{"class":15660,"line":19792},[28,94663,94353],{"class":16962},[28,94665,94666],{"class":15660,"line":19798},[28,94667,15769],{"emptyLinePlaceholder":15768},[28,94669,94670,94673,94675,94677],{"class":15660,"line":19814},[28,94671,94672],{"class":15675},"code_lib_and_run ",[28,94674,845],{"class":15682},[28,94676,94350],{"class":15671},[28,94678,94353],{"class":16962},[28,94680,94681],{"class":15660,"line":19819},[28,94682,94683],{"class":15664},"# @title Run Simulation & Display Results\n",[28,94685,94686],{"class":15660,"line":19828},[28,94687,94688],{"class":16962},"import math\n",[28,94690,94691],{"class":15660,"line":19838},[28,94692,94693],{"class":16962},"from typing import Tuple, Dict\n",[28,94695,94696],{"class":15660,"line":19847},[28,94697,94698],{"class":16962},"import pandas as pd\n",[28,94700,94701],{"class":15660,"line":19856},[28,94702,94517],{"class":16962},[28,94704,94705],{"class":15660,"line":19862},[28,94706,94707],{"class":16962},"import matplotlib.pyplot as plt\n",[28,94709,94710],{"class":15660,"line":19871},[28,94711,94712],{"class":16962},"from IPython.display import display, Markdown\n",[28,94714,94715],{"class":15660,"line":19884},[28,94716,15769],{"emptyLinePlaceholder":15768},[28,94718,94719,94722,94724,94727,94729,94732,94735],{"class":15660,"line":19897},[28,94720,94721],{"class":16962},"def wilson_ci",[28,94723,65],{"class":15801},[28,94725,94726],{"class":16962},"k: int, n: int, z: float = 1.96",[28,94728,78],{"class":15801},[28,94730,94731],{"class":16962}," -> Tuple",[28,94733,94734],{"class":15801},"[float, float]",[28,94736,15778],{"class":16962},[28,94738,94739,94742,94744,94747],{"class":15660,"line":19909},[28,94740,94741],{"class":16962},"    if n == 0: return ",[28,94743,65],{"class":15801},[28,94745,94746],{"class":16962},"0.0, 1.0",[28,94748,16052],{"class":15801},[28,94750,94751,94754,94756],{"class":15660,"line":19914},[28,94752,94753],{"class":16962},"    p_hat, z2_n = k \u002F n, z",[28,94755,16354],{"class":15801},[28,94757,94758],{"class":16962},"2 \u002F n\n",[28,94760,94761,94764,94766],{"class":15660,"line":19919},[28,94762,94763],{"class":16962},"    denom = 1.0 ",[28,94765,2606],{"class":15801},[28,94767,94768],{"class":16962}," z2_n\n",[28,94770,94771,94774,94776,94779,94781,94784,94786],{"class":15660,"line":19924},[28,94772,94773],{"class":16962},"    center = ",[28,94775,65],{"class":15801},[28,94777,94778],{"class":16962},"p_hat ",[28,94780,2606],{"class":15801},[28,94782,94783],{"class":16962}," z2_n \u002F 2",[28,94785,78],{"class":15801},[28,94787,94788],{"class":16962}," \u002F denom\n",[28,94790,94791,94794,94796,94799,94801,94804,94806,94808,94810,94812,94815,94817,94820,94822,94825,94827,94830,94832,94835,94837,94840,94842,94844,94847],{"class":15660,"line":19943},[28,94792,94793],{"class":16962},"    span = ",[28,94795,65],{"class":15801},[28,94797,94798],{"class":16962},"z ",[28,94800,16226],{"class":15801},[28,94802,94803],{"class":16962}," math.sqrt",[28,94805,65],{"class":15801},[28,94807,94778],{"class":16962},[28,94809,16226],{"class":15801},[28,94811,17308],{"class":15801},[28,94813,94814],{"class":16962},"1 - p_hat",[28,94816,78],{"class":15801},[28,94818,94819],{"class":16962}," \u002F n ",[28,94821,2606],{"class":15801},[28,94823,94824],{"class":16962}," z",[28,94826,16354],{"class":15801},[28,94828,94829],{"class":16962},"2 \u002F ",[28,94831,65],{"class":15801},[28,94833,94834],{"class":16962},"4 ",[28,94836,16226],{"class":15801},[28,94838,94839],{"class":16962}," n",[28,94841,16354],{"class":15801},[28,94843,848],{"class":16962},[28,94845,94846],{"class":15801},")))",[28,94848,94788],{"class":16962},[28,94850,94851,94854,94856,94858,94860,94863,94865,94868,94870,94873,94875,94878],{"class":15660,"line":19949},[28,94852,94853],{"class":16962},"    return ",[28,94855,65],{"class":15801},[28,94857,22207],{"class":16962},[28,94859,65],{"class":15801},[28,94861,94862],{"class":16962},"0.0, center - span",[28,94864,78],{"class":15801},[28,94866,94867],{"class":16962},", min",[28,94869,65],{"class":15801},[28,94871,94872],{"class":16962},"1.0, center ",[28,94874,2606],{"class":15801},[28,94876,94877],{"class":16962}," span",[28,94879,16539],{"class":15801},[28,94881,94882],{"class":15660,"line":19958},[28,94883,15769],{"emptyLinePlaceholder":15768},[28,94885,94886,94889,94891,94894,94896],{"class":15660,"line":19965},[28,94887,94888],{"class":16962},"def shots_for_epsilon",[28,94890,65],{"class":15801},[28,94892,94893],{"class":16962},"p: float, epsilon: float = 0.01, z: float = 1.96",[28,94895,78],{"class":15801},[28,94897,94898],{"class":16962}," -> int:\n",[28,94900,94901,94904,94906,94908,94910,94912],{"class":15660,"line":19972},[28,94902,94903],{"class":16962},"    if p ",[28,94905,16226],{"class":15801},[28,94907,17308],{"class":15801},[28,94909,94463],{"class":16962},[28,94911,78],{"class":15801},[28,94913,94914],{"class":16962}," == 0: return 1\n",[28,94916,94917,94920,94922,94925,94927,94930,94932,94934,94936,94939,94941,94943,94945,94947,94949,94952,94955,94957,94959,94961,94963],{"class":15660,"line":19980},[28,94918,94919],{"class":16962},"    return max",[28,94921,65],{"class":15801},[28,94923,94924],{"class":16962},"1, int",[28,94926,65],{"class":15801},[28,94928,94929],{"class":16962},"math.ceil",[28,94931,22254],{"class":15801},[28,94933,2266],{"class":16962},[28,94935,16354],{"class":15801},[28,94937,94938],{"class":16962},"2 ",[28,94940,16226],{"class":15801},[28,94942,91935],{"class":16962},[28,94944,16226],{"class":15801},[28,94946,17308],{"class":15801},[28,94948,94463],{"class":16962},[28,94950,94951],{"class":15801},"))",[28,94953,94954],{"class":16962}," \u002F ",[28,94956,65],{"class":15801},[28,94958,91856],{"class":16962},[28,94960,16354],{"class":15801},[28,94962,848],{"class":16962},[28,94964,94965],{"class":15801},"))))\n",[28,94967,94968],{"class":15660,"line":19991},[28,94969,15769],{"emptyLinePlaceholder":15768},[28,94971,94972,94975,94977,94980,94982],{"class":15660,"line":20002},[28,94973,94974],{"class":16962},"def summarize_model",[28,94976,65],{"class":15801},[28,94978,94979],{"class":16962},"model, bits: np.ndarray, epsilons",[28,94981,78],{"class":15801},[28,94983,94984],{"class":16962}," -> Dict:\n",[28,94986,94987,94990,94992,94994],{"class":15660,"line":20013},[28,94988,94989],{"class":16962},"    n = len",[28,94991,65],{"class":15801},[28,94993,92539],{"class":16962},[28,94995,16052],{"class":15801},[28,94997,94998,95001,95003,95005],{"class":15660,"line":20018},[28,94999,95000],{"class":16962},"    ones = sum",[28,95002,65],{"class":15801},[28,95004,92539],{"class":16962},[28,95006,16052],{"class":15801},[28,95008,95009,95012,95014,95017],{"class":15660,"line":20027},[28,95010,95011],{"class":16962},"    lo, hi = wilson_ci",[28,95013,65],{"class":15801},[28,95015,95016],{"class":16962},"ones, n",[28,95018,16052],{"class":15801},[28,95020,95021],{"class":15660,"line":20034},[28,95022,95023],{"class":16962},"    row = {\n",[28,95025,95026,95029,95031,95034,95036],{"class":15660,"line":20041},[28,95027,95028],{"class":16962},"        \"Model\": model.name, \"p ",[28,95030,65],{"class":15801},[28,95032,95033],{"class":16962},"target",[28,95035,78],{"class":15801},[28,95037,95038],{"class":16962},"\": model.satisfaction_prob, \"Shots\": n,\n",[28,95040,95041,95044,95046,95049,95051],{"class":15660,"line":20049},[28,95042,95043],{"class":16962},"        \"Ones\": ones, \"p ",[28,95045,65],{"class":15801},[28,95047,95048],{"class":16962},"observed",[28,95050,78],{"class":15801},[28,95052,95053],{"class":16962},"\": ones \u002F n if n > 0 else 0,\n",[28,95055,95056],{"class":15660,"line":20070},[28,95057,95058],{"class":16962},"        \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n",[28,95060,95061],{"class":15660,"line":20081},[28,95062,18233],{"class":16962},[28,95064,95065],{"class":15660,"line":20093},[28,95066,95067],{"class":16962},"    for eps in epsilons:\n",[28,95069,95070,95073,95076,95079,95081,95084],{"class":15660,"line":20104},[28,95071,95072],{"class":16962},"        row",[28,95074,95075],{"class":15801},"[f\"Shots for ±{eps:.1%}\"]",[28,95077,95078],{"class":16962}," = shots_for_epsilon",[28,95080,65],{"class":15801},[28,95082,95083],{"class":16962},"model.satisfaction_prob, eps",[28,95085,16052],{"class":15801},[28,95087,95088],{"class":15660,"line":20115},[28,95089,95090],{"class":16962},"    return row\n",[28,95092,95093],{"class":15660,"line":20120},[28,95094,15769],{"emptyLinePlaceholder":15768},[28,95096,95097],{"class":15660,"line":20125},[28,95098,95099],{"class":15664},"# --- Run Simulation ---\n",[28,95101,95102,95105],{"class":15660,"line":20130},[28,95103,95104],{"class":16962},"summary_rows = ",[28,95106,95107],{"class":15801},"[summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\n",[28,95109,95110,95113,95115,95118],{"class":15660,"line":20145},[28,95111,95112],{"class":16962},"summary_df = pd.DataFrame",[28,95114,65],{"class":15801},[28,95116,95117],{"class":16962},"summary_rows",[28,95119,16052],{"class":15801},[28,95121,95122],{"class":15660,"line":20157},[28,95123,95124],{"class":16962},"if COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n",[28,95126,95127,95130,95133,95136,95138,95141,95144,95146],{"class":15660,"line":20163},[28,95128,95129],{"class":16962},"    summary_df",[28,95131,95132],{"class":15801},"[\"Cost (USD)\"]",[28,95134,95135],{"class":16962}," = FIXED_OVERHEAD_USD ",[28,95137,2606],{"class":15801},[28,95139,95140],{"class":16962}," summary_df",[28,95142,95143],{"class":15801},"[\"Shots\"]",[28,95145,16210],{"class":15801},[28,95147,95148],{"class":16962}," COST_PER_SHOT_USD\n",[28,95150,95151],{"class":15660,"line":20168},[28,95152,15769],{"emptyLinePlaceholder":15768},[28,95154,95155,95157,95159,95161,95163,95165],{"class":15660,"line":20176},[28,95156,94172],{"class":16962},[28,95158,65],{"class":15801},[28,95160,94177],{"class":16962},[28,95162,65],{"class":15801},[28,95164,21798],{"class":16962},[28,95166,95167],{"class":15664},"### Simulation Summary\"))\n",[28,95169,95170,95172,95174,95177,95179,95182,95184,95187,95189,95192],{"class":15660,"line":20186},[28,95171,94172],{"class":16962},[28,95173,65],{"class":15801},[28,95175,95176],{"class":16962},"summary_df.style.format",[28,95178,65],{"class":15801},[28,95180,95181],{"class":16962},"precision=4",[28,95183,78],{"class":15801},[28,95185,95186],{"class":16962},".background_gradient",[28,95188,65],{"class":15801},[28,95190,95191],{"class":16962},"cmap='viridis', subset=",[28,95193,95194],{"class":15801},"['p (observed)', 'CI width']))\n",[28,95196,95197],{"class":15660,"line":20191},[28,95198,15769],{"emptyLinePlaceholder":15768},[28,95200,95201],{"class":15660,"line":20203},[28,95202,95203],{"class":15664},"# --- Plotting ---\n",[28,95205,95206,95209,95211,95213],{"class":15660,"line":20214},[28,95207,95208],{"class":16962},"plt.style.use",[28,95210,65],{"class":15801},[28,95212,93409],{"class":16962},[28,95214,16052],{"class":15801},[28,95216,95217,95220,95222,95225,95227,95230,95232,95235,95237,95240],{"class":15660,"line":20224},[28,95218,95219],{"class":16962},"fig, ",[28,95221,65],{"class":15801},[28,95223,95224],{"class":16962},"ax1, ax2",[28,95226,78],{"class":15801},[28,95228,95229],{"class":16962}," = plt.subplots",[28,95231,65],{"class":15801},[28,95233,95234],{"class":16962},"1, 2, figsize=",[28,95236,65],{"class":15801},[28,95238,95239],{"class":16962},"16, 6",[28,95241,16539],{"class":15801},[28,95243,95244],{"class":15660,"line":20235},[28,95245,15769],{"emptyLinePlaceholder":15768},[28,95247,95248],{"class":15660,"line":20240},[28,95249,95250],{"class":15664},"# Plot 1\n",[28,95252,95253,95256,95259,95262],{"class":15660,"line":20245},[28,95254,95255],{"class":16962},"y = summary_df",[28,95257,95258],{"class":15801},"[\"p (observed)\"]",[28,95260,95261],{"class":16962},".to_numpy",[28,95263,17029],{"class":15801},[28,95265,95266,95269],{"class":15660,"line":20257},[28,95267,95268],{"class":16962},"lower_error = y - summary_df",[28,95270,95271],{"class":15801},"[\"CI 95% low\"]\n",[28,95273,95274,95277,95280],{"class":15660,"line":20268},[28,95275,95276],{"class":16962},"upper_error = summary_df",[28,95278,95279],{"class":15801},"[\"CI 95% high\"]",[28,95281,95282],{"class":16962}," - y\n",[28,95284,95285,95288],{"class":15660,"line":20277},[28,95286,95287],{"class":16962},"yerr = np.vstack",[28,95289,95290],{"class":15801},"([np.abs(lower_error), np.abs(upper_error)])\n",[28,95292,95293],{"class":15660,"line":20288},[28,95294,15769],{"emptyLinePlaceholder":15768},[28,95296,95297,95300,95302,95305,95308,95311],{"class":15660,"line":20293},[28,95298,95299],{"class":16962},"ax1.bar",[28,95301,65],{"class":15801},[28,95303,95304],{"class":16962},"summary_df",[28,95306,95307],{"class":15801},"[\"Model\"]",[28,95309,95310],{"class":16962},", y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black'",[28,95312,16052],{"class":15801},[28,95314,95315,95318,95320,95323,95325,95328,95330,95333],{"class":15660,"line":20298},[28,95316,95317],{"class":16962},"ax1.set_title",[28,95319,65],{"class":15801},[28,95321,95322],{"class":16962},"\"Observed Success Rate ",[28,95324,65],{"class":15801},[28,95326,95327],{"class":16962},"95% Wilson CI",[28,95329,78],{"class":15801},[28,95331,95332],{"class":16962},"\", fontsize=14",[28,95334,16052],{"class":15801},[28,95336,95337,95340,95342,95345],{"class":15660,"line":20308},[28,95338,95339],{"class":16962},"ax1.tick_params",[28,95341,65],{"class":15801},[28,95343,95344],{"class":16962},"axis='x', rotation=30, labelsize=10",[28,95346,16052],{"class":15801},[28,95348,95349],{"class":15660,"line":20314},[28,95350,15769],{"emptyLinePlaceholder":15768},[28,95352,95353],{"class":15660,"line":20328},[28,95354,95355],{"class":15664},"# Plot 2\n",[28,95357,95358],{"class":15660,"line":20333},[28,95359,95360],{"class":16962},"width = 0.2\n",[28,95362,95363,95366,95368,95370,95372,95375],{"class":15660,"line":20343},[28,95364,95365],{"class":16962},"x = np.arange",[28,95367,65],{"class":15801},[28,95369,22222],{"class":16962},[28,95371,65],{"class":15801},[28,95373,95374],{"class":16962},"MODELS",[28,95376,16539],{"class":15801},[28,95378,95379,95382],{"class":15660,"line":20348},[28,95380,95381],{"class":16962},"colors = ",[28,95383,95384],{"class":15801},"['#ff9999','#66b3ff','#99ff99']\n",[28,95386,95387,95390,95392,95395,95397],{"class":15660,"line":20360},[28,95388,95389],{"class":16962},"for i, eps in enumerate",[28,95391,65],{"class":15801},[28,95393,95394],{"class":16962},"EPSILONS",[28,95396,78],{"class":15801},[28,95398,15778],{"class":16962},[28,95400,95401,95404],{"class":15660,"line":20365},[28,95402,95403],{"class":16962},"    req = summary_df",[28,95405,95406],{"class":15801},"[f\"Shots for ±{eps:.1%}\"]\n",[28,95408,95409,95412,95414,95416,95418,95420,95422,95425,95428,95431],{"class":15660,"line":20370},[28,95410,95411],{"class":16962},"    ax2.bar",[28,95413,65],{"class":15801},[28,95415,93438],{"class":16962},[28,95417,2606],{"class":15801},[28,95419,22280],{"class":16962},[28,95421,16226],{"class":15801},[28,95423,95424],{"class":16962}," width, req, width, label=f\"±{eps:.1%}\", color=colors",[28,95426,95427],{"class":15801},"[i]",[28,95429,95430],{"class":16962},", edgecolor='black'",[28,95432,16052],{"class":15801},[28,95434,95435],{"class":15660,"line":20393},[28,95436,15769],{"emptyLinePlaceholder":15768},[28,95438,95439,95442,95444,95446,95448,95451],{"class":15660,"line":20398},[28,95440,95441],{"class":16962},"ax2.set_xticks",[28,95443,65],{"class":15801},[28,95445,93438],{"class":16962},[28,95447,2606],{"class":15801},[28,95449,95450],{"class":16962}," width, summary_df",[28,95452,95453],{"class":15801},"[\"Model\"])\n",[28,95455,95456,95459,95461,95464,95466,95469,95471,95473],{"class":15660,"line":20421},[28,95457,95458],{"class":16962},"ax2.set_ylabel",[28,95460,65],{"class":15801},[28,95462,95463],{"class":16962},"\"Required Shots ",[28,95465,65],{"class":15801},[28,95467,95468],{"class":16962},"Log Scale",[28,95470,78],{"class":15801},[28,95472,21798],{"class":16962},[28,95474,16052],{"class":15801},[28,95476,95477,95480,95482,95484],{"class":15660,"line":20432},[28,95478,95479],{"class":16962},"ax2.set_yscale",[28,95481,65],{"class":15801},[28,95483,94069],{"class":16962},[28,95485,16052],{"class":15801},[28,95487,95488,95491,95493,95496],{"class":15660,"line":20471},[28,95489,95490],{"class":16962},"ax2.set_title",[28,95492,65],{"class":15801},[28,95494,95495],{"class":16962},"\"Shots Needed for ±ε Tolerance\", fontsize=14",[28,95497,16052],{"class":15801},[28,95499,95500,95503,95505,95507],{"class":15660,"line":20481},[28,95501,95502],{"class":16962},"ax2.tick_params",[28,95504,65],{"class":15801},[28,95506,95344],{"class":16962},[28,95508,16052],{"class":15801},[28,95510,95511,95514,95516,95519],{"class":15660,"line":20493},[28,95512,95513],{"class":16962},"ax2.legend",[28,95515,65],{"class":15801},[28,95517,95518],{"class":16962},"title=\"Tolerance\"",[28,95520,16052],{"class":15801},[28,95522,95523],{"class":15660,"line":20513},[28,95524,15769],{"emptyLinePlaceholder":15768},[28,95526,95527,95530],{"class":15660,"line":20531},[28,95528,95529],{"class":16962},"plt.tight_layout",[28,95531,17029],{"class":15801},[28,95533,95534,95537],{"class":15660,"line":20536},[28,95535,95536],{"class":16962},"plt.show",[28,95538,17029],{"class":15801},[28,95540,95541],{"class":15660,"line":20541},[28,95542,94353],{"class":16962},[28,95544,95545],{"class":15660,"line":20553},[28,95546,15769],{"emptyLinePlaceholder":15768},[28,95548,95549,95552,95554,95557,95560],{"class":15660,"line":20563},[28,95550,95551],{"class":15675},"nb ",[28,95553,845],{"class":15682},[28,95555,95556],{"class":15675}," nbf.v4.",[28,95558,95559],{"class":16073},"new_notebook",[28,95561,17029],{"class":15675},[28,95563,95564,95567,95570,95572,95574],{"class":15660,"line":20573},[28,95565,95566],{"class":15675},"nb[",[28,95568,95569],{"class":15841},"\"cells\"",[28,95571,15845],{"class":15675},[28,95573,845],{"class":15682},[28,95575,17702],{"class":15675},[28,95577,95578,95581,95584],{"class":15660,"line":20578},[28,95579,95580],{"class":15675},"    nbf.v4.",[28,95582,95583],{"class":16073},"new_markdown_cell",[28,95585,95586],{"class":15675},"(md_intro),\n",[28,95588,95589,95591,95594],{"class":15660,"line":20583},[28,95590,95580],{"class":15675},[28,95592,95593],{"class":16073},"new_code_cell",[28,95595,95596],{"class":15675},"(code_config),\n",[28,95598,95599,95601,95603],{"class":15660,"line":20591},[28,95600,95580],{"class":15675},[28,95602,95593],{"class":16073},[28,95604,95605],{"class":15675},"(code_lib_and_run),\n",[28,95607,95608],{"class":15660,"line":20601},[28,95609,16849],{"class":15675},[28,95611,95612],{"class":15660,"line":20616},[28,95613,15769],{"emptyLinePlaceholder":15768},[28,95615,95616,95619,95621],{"class":15660,"line":20627},[28,95617,95618],{"class":15675},"nb_path ",[28,95620,845],{"class":15682},[28,95622,95623],{"class":15841}," \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\n",[28,95625,95626,95628,95630,95633,95635,95637,95640,95642,95645,95647,95649],{"class":15660,"line":20639},[28,95627,93319],{"class":15671},[28,95629,93322],{"class":15682},[28,95631,95632],{"class":15675},"(nb_path, ",[28,95634,93332],{"class":15841},[28,95636,16397],{"class":15675},[28,95638,95639],{"class":16307},"encoding",[28,95641,845],{"class":15682},[28,95643,95644],{"class":15841},"\"utf-8\"",[28,95646,22233],{"class":15675},[28,95648,15733],{"class":15671},[28,95650,93339],{"class":15675},[28,95652,95653,95656,95659],{"class":15660,"line":20644},[28,95654,95655],{"class":15675},"    nbf.",[28,95657,95658],{"class":16073},"write",[28,95660,95661],{"class":15675},"(nb, f)\n",[28,95663,95664],{"class":15660,"line":20651},[28,95665,15769],{"emptyLinePlaceholder":15768},[28,95667,95668],{"class":15660,"line":20660},[28,95669,95670],{"class":15664},"# ---------- Final confirmation ----------\n",[28,95672,95673,95675,95677,95679,95681,95683,95685,95688,95690,95692],{"class":15660,"line":20674},[28,95674,16044],{"class":15682},[28,95676,65],{"class":15675},[28,95678,21798],{"class":15841},[28,95680,20487],{"class":15682},[28,95682,21798],{"class":15841},[28,95684,16359],{"class":15682},[28,95686,95687],{"class":15841}," \"=\"",[28,95689,16226],{"class":15682},[28,95691,7493],{"class":15801},[28,95693,16052],{"class":15675},[28,95695,95696,95698,95700,95703],{"class":15660,"line":20685},[28,95697,16044],{"class":15682},[28,95699,65],{"class":15675},[28,95701,95702],{"class":15841},"\"SCRIPT COMPLETE\"",[28,95704,16052],{"class":15675},[28,95706,95707,95709,95711,95714,95716,95718],{"class":15660,"line":20697},[28,95708,16044],{"class":15682},[28,95710,65],{"class":15675},[28,95712,95713],{"class":15841},"\"=\"",[28,95715,16226],{"class":15682},[28,95717,7493],{"class":15801},[28,95719,16052],{"class":15675},[28,95721,95722,95724,95726,95728,95731,95733,95736,95738,95740],{"class":15660,"line":20702},[28,95723,16044],{"class":15682},[28,95725,65],{"class":15675},[28,95727,68],{"class":15671},[28,95729,95730],{"class":15841},"\"Google Colab notebook saved to: ",[28,95732,11385],{"class":15801},[28,95734,95735],{"class":15675},"nb_path",[28,95737,11404],{"class":15801},[28,95739,21798],{"class":15841},[28,95741,16052],{"class":15675},[28,95743,95744,95746,95748,95750,95753,95755,95758,95760,95762],{"class":15660,"line":20707},[28,95745,16044],{"class":15682},[28,95747,65],{"class":15675},[28,95749,68],{"class":15671},[28,95751,95752],{"class":15841},"\"Summary data saved to: ",[28,95754,11385],{"class":15801},[28,95756,95757],{"class":15675},"summary_csv_path",[28,95759,11404],{"class":15801},[28,95761,21798],{"class":15841},[28,95763,16052],{"class":15675},[28,95765,95766,95768,95770,95772,95775,95777,95780,95782,95784,95786,95789,95791,95793],{"class":15660,"line":20717},[28,95767,16044],{"class":15682},[28,95769,65],{"class":15675},[28,95771,68],{"class":15671},[28,95773,95774],{"class":15841},"\"Plots saved to: ",[28,95776,11385],{"class":15801},[28,95778,95779],{"class":15675},"fig1_path",[28,95781,11404],{"class":15801},[28,95783,16397],{"class":15841},[28,95785,11385],{"class":15801},[28,95787,95788],{"class":15675},"fig2_path",[28,95790,11404],{"class":15801},[28,95792,21798],{"class":15841},[28,95794,16052],{"class":15675},[15650,95796,95799],{"className":95797,"code":95798,"language":16058,"meta":15655},[16056],"Running simulation...\nSaved data to CSV and JSON files.\n",[8103,95800,95798],{"__ignoreMap":15655},[17041,95802],{"alt":17043,"src":95803},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.webp",[17041,95805],{"alt":22057,"src":95806},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-02.webp",[15650,95808,95811],{"className":95809,"code":95810,"language":16058,"meta":15655},[16056],"Generated and saved plots.\n",[8103,95812,95810],{"__ignoreMap":15655},[15650,95814,95817],{"className":95815,"code":95816,"language":16058,"meta":15655},[16056],"==================================================\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",[8103,95818,95816],{"__ignoreMap":15655},[95820,95821],"hr",{},[251,95823,67131],{"id":67130},[52966,95825,95826,95833,95840,95847,95854,95861,95868,95875,95882,95889,95896,95903,95910,95921,95928,95935],{},[27063,95827,95828,95829,95832],{},"Agresti, A. and Coull, B.A. (1998) 'Approximate is better than “exact” for interval estimation of binomial proportions', ",[15,95830,95831],{},"The American Statistician",", 52(2), pp. 119-126.",[27063,95834,95835,95836,95839],{},"Brown, L.D., Cai, T.T. and DasGupta, A. (2001) 'Interval Estimation for a Binomial Proportion', ",[15,95837,95838],{},"Statistical Science",", 16(2), pp. 101-133.",[27063,95841,95842,95843,95846],{},"Cochran, W.G. (1977) ",[15,95844,95845],{},"Sampling Techniques",". 3rd edn. New York: John Wiley & Sons.",[27063,95848,95849,95850,95853],{},"Gottesman, D. (2009) 'An introduction to quantum error correction and fault-tolerant quantum computation', in ",[15,95851,95852],{},"Quantum Information Science and Its Contributions to Mathematics, Proceedings of Symposia in Applied Mathematics",", 72, pp. 13-58.",[27063,95855,95856,95857,95860],{},"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', ",[15,95858,95859],{},"Nature",", 585, pp. 357–362.",[27063,95862,95863,95864,95867],{},"Hunter, J.D. (2007) 'Matplotlib: A 2D Graphics Environment', ",[15,95865,95866],{},"Computing in Science & Engineering",", 9(3), pp. 90-95.",[27063,95869,95870,95871,95874],{},"Kitaev, A.Y. (2003) 'Fault-tolerant quantum computation by anyons', ",[15,95872,95873],{},"Annals of Physics",", 303(1), pp. 2-30.",[27063,95876,95877,95878,95881],{},"McKinney, W. (2010) 'Data Structures for Statistical Computing in Python', in ",[15,95879,95880],{},"Proceedings of the 9th Python in Science Conference",", pp. 56-61.",[27063,95883,95884,95885,95888],{},"Nayak, C., Simon, S.H., Stern, A., Freedman, M. and Das Sarma, S. (2008) 'Non-Abelian anyons and topological quantum computation', ",[15,95886,95887],{},"Reviews of Modern Physics",", 80(3), pp. 1083-1159.",[27063,95890,95891,95892,95895],{},"Newman, M. (2013) ",[15,95893,95894],{},"Computational Physics",". 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Available at: ",[19,95918,95920],{"href":95919},"https:\u002F\u002Fwww.google.com\u002Fsearch?q=https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134","https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134",[27063,95922,95923,95924,95927],{},"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', ",[15,95925,95926],{},"Nature Methods",", 17, pp. 261–272.",[27063,95929,95930,95931,95934],{},"Wilde, M.M. (2017) ",[15,95932,95933],{},"Quantum Information Theory",". 2nd edn. Cambridge: Cambridge University Press.",[27063,95936,95937,95938,95941],{},"Wilson, E.B. 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