[{"data":1,"prerenderedAt":1889},["ShallowReactive",2],{"blog-post-docs\u002Fcx-gate":3,"sibling-dives-docs\u002Fcx-gate":1723,"learn-track-docs\u002Fcx-gate":1724,"learn-docs-groups":1725},{"id":4,"title":5,"authors":6,"body":7,"breadcrumb":6,"builders":6,"byline":6,"category":1704,"categoryName":1705,"challenge":6,"courseAuthor":6,"courseLead":6,"dek":6,"description":1706,"draft":1707,"extension":1708,"eyebrow":6,"featured":1707,"finish":6,"fork":6,"group":1709,"groupName":1710,"hero":6,"heroAlt":6,"heroCta":6,"heroImage":6,"homepageFeatured":1707,"kind":1711,"lessonCount":6,"meta":1712,"navigation":1409,"newsItems":6,"next":6,"ogImage":6,"order":1713,"outcomes":6,"path":1714,"publishDate":1715,"readingTime":6,"related":1716,"relatedProjects":6,"seo":1717,"stem":1720,"tags":1721,"track":6,"trackName":6,"__hash__":1722},"blog\u002Fblog\u002Flearn\u002Fdocs\u002Fcx-gate.md","CX gate",null,{"type":8,"value":9,"toc":1695},"minimark",[10,120,165,170,220,224,398,984,991,995,1055,1363,1367,1370,1374,1377,1499,1618,1637,1641,1663,1667,1691],[11,12,13,14,75,76,114,115,119],"p",{},"The CX gate, the controlled-NOT or CNOT, flips the target qubit when the control qubit is ",[15,16,19,52],"span",{"className":17},[18],"katex",[15,20,23],{"className":21},[22],"katex-mathml",[24,25,27],"math",{"xmlns":26},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[28,29,30,47],"semantics",{},[31,32,33,38,42],"mrow",{},[34,35,37],"mi",{"mathvariant":36},"normal","∣",[39,40,41],"mn",{},"1",[43,44,46],"mo",{"stretchy":45},"false","⟩",[48,49,51],"annotation",{"encoding":50},"application\u002Fx-tex","\\vert 1 \\rangle",[15,53,57],{"className":54,"ariaHidden":56},[55],"katex-html","true",[15,58,61,66,71],{"className":59},[60],"base",[15,62],{"className":63,"style":65},[64],"strut","height:1em;vertical-align:-0.25em;",[15,67,70],{"className":68},[69],"mord","∣1",[15,72,46],{"className":73},[74],"mclose"," and leaves it unchanged when the control is ",[15,77,79,98],{"className":78},[18],[15,80,82],{"className":81},[22],[24,83,84],{"xmlns":26},[28,85,86,95],{},[31,87,88,90,93],{},[34,89,37],{"mathvariant":36},[39,91,92],{},"0",[43,94,46],{"stretchy":45},[48,96,97],{"encoding":50},"\\vert 0 \\rangle",[15,99,101],{"className":100,"ariaHidden":56},[55],[15,102,104,107,111],{"className":103},[60],[15,105],{"className":106,"style":65},[64],[15,108,110],{"className":109},[69],"∣0",[15,112,46],{"className":113},[74],". It acts on two qubits and takes no parameters. With the control in superposition, the two qubits come out entangled. The Circuit panel draws a dot on the control wire joined to an ",[116,117,118],"strong",{},"X"," box on the target wire.",[11,121,122,123,158,159,164],{},"Kets such as ",[15,124,126,143],{"className":125},[18],[15,127,129],{"className":128},[22],[24,130,131],{"xmlns":26},[28,132,133,141],{},[31,134,135,137,139],{},[34,136,37],{"mathvariant":36},[39,138,92],{},[43,140,46],{"stretchy":45},[48,142,97],{"encoding":50},[15,144,146],{"className":145,"ariaHidden":56},[55],[15,147,149,152,155],{"className":148},[60],[15,150],{"className":151,"style":65},[64],[15,153,110],{"className":154},[69],[15,156,46],{"className":157},[74]," name the basis states and rotations are described on the Bloch sphere; the ",[160,161,163],"a",{"href":162},"\u002Flearn\u002Fquantum-foundations\u002Fqubits","qubits lesson"," introduces both.",[166,167,169],"h2",{"id":168},"call","Call",[171,172,173,185],"table",{},[174,175,176],"thead",{},[177,178,179,183],"tr",{},[180,181,182],"th",{},"Where",[180,184,169],{},[186,187,188,200,210],"tbody",{},[177,189,190,194],{},[191,192,193],"td",{},"Python",[191,195,196],{},[197,198,199],"code",{},"qc.cx(control, target)",[177,201,202,205],{},[191,203,204],{},"JS",[191,206,207],{},[197,208,209],{},"circuit.cx(control, target)",[177,211,212,215],{},[191,213,214],{},"Qiskit class",[191,216,217],{},[197,218,219],{},"CXGate",[166,221,223],{"id":222},"matrix","Matrix",[11,225,226,227,289,290,397],{},"The matrix uses the basis order ",[15,228,230,256],{"className":229},[18],[15,231,233],{"className":232},[22],[24,234,235],{"xmlns":26},[28,236,237,253],{},[31,238,239,241,245,248,251],{},[34,240,37],{"mathvariant":36},[242,243,244],"mtext",{},"control",[242,246,247],{},"  ",[242,249,250],{},"target",[43,252,46],{"stretchy":45},[48,254,255],{"encoding":50},"\\vert \\text{control}\\;\\text{target} \\rangle",[15,257,259],{"className":258,"ariaHidden":56},[55],[15,260,262,265,268,275,280,286],{"className":261},[60],[15,263],{"className":264,"style":65},[64],[15,266,37],{"className":267},[69],[15,269,272],{"className":270},[69,271],"text",[15,273,244],{"className":274},[69],[15,276],{"className":277,"style":279},[278],"mspace","margin-right:0.2778em;",[15,281,283],{"className":282},[69,271],[15,284,250],{"className":285},[69],[15,287,46],{"className":288},[74],", that is ",[15,291,293,340],{"className":292},[18],[15,294,296],{"className":295},[22],[24,297,298],{"xmlns":26},[28,299,300,337],{},[31,301,302,304,307,309,312,314,317,319,321,323,326,328,330,332,335],{},[34,303,37],{"mathvariant":36},[39,305,306],{},"00",[43,308,46],{"stretchy":45},[43,310,311],{"separator":56},",",[34,313,37],{"mathvariant":36},[39,315,316],{},"01",[43,318,46],{"stretchy":45},[43,320,311],{"separator":56},[34,322,37],{"mathvariant":36},[39,324,325],{},"10",[43,327,46],{"stretchy":45},[43,329,311],{"separator":56},[34,331,37],{"mathvariant":36},[39,333,334],{},"11",[43,336,46],{"stretchy":45},[48,338,339],{"encoding":50},"\\vert 00 \\rangle, \\vert 01 \\rangle, \\vert 10 \\rangle, \\vert 11 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center center center","1em",[435,436,437,445,451,457],"mtr",{},[438,439,440],"mtd",{},[441,442,443],"mstyle",{"scriptlevel":92,"displaystyle":45},[39,444,41],{},[438,446,447],{},[441,448,449],{"scriptlevel":92,"displaystyle":45},[39,450,92],{},[438,452,453],{},[441,454,455],{"scriptlevel":92,"displaystyle":45},[39,456,92],{},[438,458,459],{},[441,460,461],{"scriptlevel":92,"displaystyle":45},[39,462,92],{},[435,464,465,471,477,483],{},[438,466,467],{},[441,468,469],{"scriptlevel":92,"displaystyle":45},[39,470,92],{},[438,472,473],{},[441,474,475],{"scriptlevel":92,"displaystyle":45},[39,476,41],{},[438,478,479],{},[441,480,481],{"scriptlevel":92,"displaystyle":45},[39,482,92],{},[438,484,485],{},[441,486,487],{"scriptlevel":92,"displaystyle":45},[39,488,92],{},[435,490,491,497,503,509],{},[438,492,493],{},[441,494,495],{"scriptlevel":92,"displaystyle":45},[39,496,92],{},[438,498,499],{},[441,500,501],{"scriptlevel":92,"displaystyle":45},[39,502,92],{},[438,504,505],{},[441,506,507],{"scriptlevel":92,"displaystyle":45},[39,508,92],{},[438,510,511],{},[441,512,513],{"scriptlevel":92,"displaystyle":45},[39,514,41],{},[435,516,517,523,529,535],{},[438,518,519],{},[441,520,521],{"scriptlevel":92,"displaystyle":45},[39,522,92],{},[438,524,525],{},[441,526,527],{"scriptlevel":92,"displaystyle":45},[39,528,92],{},[438,530,531],{},[441,532,533],{"scriptlevel":92,"displaystyle":45},[39,534,41],{},[438,536,537],{},[441,538,539],{"scriptlevel":92,"displaystyle":45},[39,540,92],{},[43,542,543],{"fence":56},")",[48,545,546],{"encoding":50},"CX 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numbers basis states with qubit 0 as the least significant bit, so ",[197,988,989],{},"Operator(CXGate())"," prints the same gate in a different row and column order.",[166,992,994],{"id":993},"effect-on-basis-states","Effect on basis states",[171,996,997,1013],{},[174,998,999],{},[177,1000,1001,1004,1007,1010],{},[180,1002,1003],{},"Control",[180,1005,1006],{},"Target",[180,1008,1009],{},"Control after",[180,1011,1012],{},"Target after",[186,1014,1015,1025,1035,1045],{},[177,1016,1017,1019,1021,1023],{},[191,1018,92],{},[191,1020,92],{},[191,1022,92],{},[191,1024,92],{},[177,1026,1027,1029,1031,1033],{},[191,1028,92],{},[191,1030,41],{},[191,1032,92],{},[191,1034,41],{},[177,1036,1037,1039,1041,1043],{},[191,1038,41],{},[191,1040,92],{},[191,1042,41],{},[191,1044,41],{},[177,1046,1047,1049,1051,1053],{},[191,1048,41],{},[191,1050,41],{},[191,1052,41],{},[191,1054,92],{},[11,1056,1057,1058,1098,1099,1134,1135,1362],{},"In the computational basis the control is never changed. With the control in ",[15,1059,1061,1080],{"className":1060},[18],[15,1062,1064],{"className":1063},[22],[24,1065,1066],{"xmlns":26},[28,1067,1068,1077],{},[31,1069,1070,1072,1075],{},[34,1071,37],{"mathvariant":36},[43,1073,1074],{},"+",[43,1076,46],{"stretchy":45},[48,1078,1079],{"encoding":50},"\\vert + \\rangle",[15,1081,1083],{"className":1082,"ariaHidden":56},[55],[15,1084,1086,1089,1092,1095],{"className":1085},[60],[15,1087],{"className":1088,"style":65},[64],[15,1090,37],{"className":1091},[69],[15,1093,1074],{"className":1094},[69],[15,1096,46],{"className":1097},[74]," and the target in 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\\rangle)",[15,1182,1184,1349],{"className":1183,"ariaHidden":56},[55],[15,1185,1187,1191,1329,1332,1335,1338,1342,1346],{"className":1186},[60],[15,1188],{"className":1189,"style":1190},[64],"height:1.3831em;vertical-align:-0.538em;",[15,1192,1194,1198,1326],{"className":1193},[69],[15,1195],{"className":1196},[590,1197],"nulldelimiter",[15,1199,1201],{"className":1200},[1150],[15,1202,1204,1317],{"className":1203},[599,600],[15,1205,1207,1314],{"className":1206},[604],[15,1208,1211,1288,1299],{"className":1209,"style":1210},[608],"height:0.8451em;",[15,1212,1214,1217],{"style":1213},"top:-2.551em;",[15,1215],{"className":1216,"style":671},[616],[15,1218,1224],{"className":1219},[1220,1221,1222,1223],"sizing","reset-size6","size3","mtight",[15,1225,1227],{"className":1226},[69,1223],[15,1228,1231],{"className":1229},[69,1230,1223],"sqrt",[15,1232,1234,1279],{"className":1233},[599,600],[15,1235,1237,1276],{"className":1236},[604],[15,1238,1241,1256],{"className":1239,"style":1240},[608],"height:0.9128em;",[15,1242,1246,1249],{"className":1243,"style":1245},[1244],"svg-align","top:-3em;",[15,1247],{"className":1248,"style":671},[616],[15,1250,1253],{"className":1251,"style":1252},[69,1223],"padding-left:0.833em;",[15,1254,1159],{"className":1255},[69,1223],[15,1257,1259,1262],{"style":1258},"top:-2.8728em;",[15,1260],{"className":1261,"style":671},[616],[15,1263,1267],{"className":1264,"style":1266},[1265,1223],"hide-tail","min-width:0.853em;height:1.08em;",[622,1268,1273],{"xmlns":624,"width":1269,"height":1270,"viewBox":1271,"preserveAspectRatio":1272},"400em","1.08em","0 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80h400000v40h-400000z",[15,1277,636],{"className":1278},[635],[15,1280,1282],{"className":1281},[604],[15,1283,1286],{"className":1284,"style":1285},[608],"height:0.1272em;",[15,1287],{},[15,1289,1291,1294],{"style":1290},"top:-3.23em;",[15,1292],{"className":1293,"style":671},[616],[15,1295],{"className":1296,"style":1298},[1297],"frac-line","border-bottom-width:0.04em;",[15,1300,1302,1305],{"style":1301},"top:-3.394em;",[15,1303],{"className":1304,"style":671},[616],[15,1306,1308],{"className":1307},[1220,1221,1222,1223],[15,1309,1311],{"className":1310},[69,1223],[15,1312,41],{"className":1313},[69,1223],[15,1315,636],{"className":1316},[635],[15,1318,1320],{"className":1319},[604],[15,1321,1324],{"className":1322,"style":1323},[608],"height:0.538em;",[15,1325],{},[15,1327],{"className":1328},[74,1197],[15,1330,427],{"className":1331},[590],[15,1333,352],{"className":1334},[69],[15,1336,46],{"className":1337},[74],[15,1339],{"className":1340,"style":1341},[278],"margin-right:0.2222em;",[15,1343,1074],{"className":1344},[1345],"mbin",[15,1347],{"className":1348,"style":1341},[278],[15,1350,1352,1355,1358],{"className":1351},[60],[15,1353],{"className":1354,"style":65},[64],[15,1356,393],{"className":1357},[69],[15,1359,1361],{"className":1360},[74],"⟩)",", a Bell state: measuring either qubit fixes the other.",[166,1364,1366],{"id":1365},"inverse","Inverse",[11,1368,1369],{},"CX is its own inverse. Two CX gates in a row on the same control and target cancel.",[166,1371,1373],{"id":1372},"usage","Usage",[11,1375,1376],{},"The Bell pair is an H on the control followed by a CX.",[1378,1379,1384],"pre",{"className":1380,"code":1381,"language":1382,"meta":1383,"style":1383},"language-python shiki shiki-themes one-dark-pro","from qiskit import QuantumCircuit\n\nqc = QuantumCircuit(2, 2)\nqc.h(0)\nqc.cx(0, 1)\nqc.measure([0, 1], [0, 1])\n","python","",[197,1385,1386,1404,1411,1437,1452,1470],{"__ignoreMap":1383},[15,1387,1390,1394,1398,1401],{"class":1388,"line":1389},"line",1,[15,1391,1393],{"class":1392},"seHd6","from",[15,1395,1397],{"class":1396},"sn6KH"," qiskit ",[15,1399,1400],{"class":1392},"import",[15,1402,1403],{"class":1396}," QuantumCircuit\n",[15,1405,1407],{"class":1388,"line":1406},2,[15,1408,1410],{"emptyLinePlaceholder":1409},true,"\n",[15,1412,1414,1417,1420,1424,1426,1429,1432,1434],{"class":1388,"line":1413},3,[15,1415,1416],{"class":1396},"qc ",[15,1418,422],{"class":1419},"sjrmR",[15,1421,1423],{"class":1422},"sVbv2"," QuantumCircuit",[15,1425,427],{"class":1396},[15,1427,1159],{"class":1428},"sVC51",[15,1430,1431],{"class":1396},", ",[15,1433,1159],{"class":1428},[15,1435,1436],{"class":1396},")\n",[15,1438,1440,1443,1446,1448,1450],{"class":1388,"line":1439},4,[15,1441,1442],{"class":1396},"qc.",[15,1444,1445],{"class":1422},"h",[15,1447,427],{"class":1396},[15,1449,92],{"class":1428},[15,1451,1436],{"class":1396},[15,1453,1455,1457,1460,1462,1464,1466,1468],{"class":1388,"line":1454},5,[15,1456,1442],{"class":1396},[15,1458,1459],{"class":1422},"cx",[15,1461,427],{"class":1396},[15,1463,92],{"class":1428},[15,1465,1431],{"class":1396},[15,1467,41],{"class":1428},[15,1469,1436],{"class":1396},[15,1471,1473,1475,1478,1481,1483,1485,1487,1490,1492,1494,1496],{"class":1388,"line":1472},6,[15,1474,1442],{"class":1396},[15,1476,1477],{"class":1422},"measure",[15,1479,1480],{"class":1396},"([",[15,1482,92],{"class":1428},[15,1484,1431],{"class":1396},[15,1486,41],{"class":1428},[15,1488,1489],{"class":1396},"], [",[15,1491,92],{"class":1428},[15,1493,1431],{"class":1396},[15,1495,41],{"class":1428},[15,1497,1498],{"class":1396},"])\n",[1378,1500,1504],{"className":1501,"code":1502,"language":1503,"meta":1383,"style":1383},"language-js shiki shiki-themes one-dark-pro","import { QuantumCircuit } from 'qiskit';\n\nconst circuit = QuantumCircuit(2, 2);\ncircuit.h(0);\ncircuit.cx(0, 1);\ncircuit.measure([0, 1], [0, 1]);\n","js",[197,1505,1506,1529,1533,1558,1573,1591],{"__ignoreMap":1383},[15,1507,1508,1510,1513,1517,1520,1522,1526],{"class":1388,"line":1389},[15,1509,1400],{"class":1392},[15,1511,1512],{"class":1396}," { ",[15,1514,1516],{"class":1515},"sVyAn","QuantumCircuit",[15,1518,1519],{"class":1396}," } ",[15,1521,1393],{"class":1392},[15,1523,1525],{"class":1524},"subq3"," 'qiskit'",[15,1527,1528],{"class":1396},";\n",[15,1530,1531],{"class":1388,"line":1406},[15,1532,1410],{"emptyLinePlaceholder":1409},[15,1534,1535,1538,1542,1545,1547,1549,1551,1553,1555],{"class":1388,"line":1413},[15,1536,1537],{"class":1392},"const",[15,1539,1541],{"class":1540},"sU0A5"," circuit",[15,1543,1544],{"class":1419}," =",[15,1546,1423],{"class":1422},[15,1548,427],{"class":1396},[15,1550,1159],{"class":1428},[15,1552,1431],{"class":1396},[15,1554,1159],{"class":1428},[15,1556,1557],{"class":1396},");\n",[15,1559,1560,1563,1565,1567,1569,1571],{"class":1388,"line":1439},[15,1561,1562],{"class":1540},"circuit",[15,1564,397],{"class":1396},[15,1566,1445],{"class":1422},[15,1568,427],{"class":1396},[15,1570,92],{"class":1428},[15,1572,1557],{"class":1396},[15,1574,1575,1577,1579,1581,1583,1585,1587,1589],{"class":1388,"line":1454},[15,1576,1562],{"class":1540},[15,1578,397],{"class":1396},[15,1580,1459],{"class":1422},[15,1582,427],{"class":1396},[15,1584,92],{"class":1428},[15,1586,1431],{"class":1396},[15,1588,41],{"class":1428},[15,1590,1557],{"class":1396},[15,1592,1593,1595,1597,1599,1601,1603,1605,1607,1609,1611,1613,1615],{"class":1388,"line":1472},[15,1594,1562],{"class":1540},[15,1596,397],{"class":1396},[15,1598,1477],{"class":1422},[15,1600,1480],{"class":1396},[15,1602,92],{"class":1428},[15,1604,1431],{"class":1396},[15,1606,41],{"class":1428},[15,1608,1489],{"class":1396},[15,1610,92],{"class":1428},[15,1612,1431],{"class":1396},[15,1614,41],{"class":1428},[15,1616,1617],{"class":1396},"]);\n",[11,1619,1620,1621,1623,1624,1626,1627,1629,1630,1632,1633,397],{},"On a noiseless simulator every shot reads ",[197,1622,306],{}," or ",[197,1625,334],{},", in close to equal numbers. On hardware, noise adds some ",[197,1628,316],{}," and ",[197,1631,325],{}," counts; see ",[160,1634,1636],{"href":1635},"\u002Flearn\u002Fdocs\u002Fwhy-your-results-look-wrong","Why your results look wrong",[166,1638,1640],{"id":1639},"on-ionq-hardware","On IonQ hardware",[11,1642,1643,1644,1647,1648,1651,1652,1431,1655,1658,1659,397],{},"CX is not a native gate. IonQ's reference decomposition, equal to CX up to a global phase, uses one entangling gate with single-qubit rotations around it: RY(π\u002F2) on the control, an XX(π\u002F4) entangler across both qubits, RX(−π\u002F2) on both, then RY(−π\u002F2) on the control. The entangler is ",[197,1645,1646],{},"MS"," on Aria and ",[197,1649,1650],{},"ZZ"," on Forte, the hardware behind Qollab's QPU backends; the rotations become ",[197,1653,1654],{},"GPi",[197,1656,1657],{},"GPi2"," and virtual Z. Each CX you write is one two-qubit native gate before IonQ's optimiser runs, and more gates means more opportunity for error. See ",[160,1660,1662],{"href":1661},"\u002Flearn\u002Fdocs\u002Fhow-your-circuit-is-compiled","How your circuit is compiled",[166,1664,1666],{"id":1665},"related","Related",[1668,1669,1670,1677,1683,1687],"ul",{},[1671,1672,1673],"li",{},[160,1674,1676],{"href":1675},"\u002Flearn\u002Fdocs\u002Fh-gate","H gate",[1671,1678,1679],{},[160,1680,1682],{"href":1681},"\u002Flearn\u002Fdocs\u002Fgates","Gate reference",[1671,1684,1685],{},[160,1686,1662],{"href":1661},[1671,1688,1689],{},[160,1690,1636],{"href":1635},[1692,1693,1694],"style",{},"html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .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 .default .shiki span {color: var(--shiki-default);background: 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