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is a ",[240,1068,1070],{"href":1069},"\u002Flearn\u002Fdocs\u002Fcz-gate","CZ gate",". CP is the controlled form of the ",[240,1073,1075],{"href":1074},"\u002Flearn\u002Fdocs\u002Fp-gate","P gate",[246,1077,1079],{"id":1078},"effect-on-basis-states","Effect on basis states",[251,1081,1082,1092],{},[254,1083,1084],{},[257,1085,1086,1089],{},[260,1087,1088],{},"Input",[260,1090,1091],{},"Output",[266,1093,1094,1171,1248,1325],{},[257,1095,1096,1134],{},[271,1097,1098],{},[15,1099,1101,1119],{"className":1100},[18],[15,1102,1104],{"className":1103},[22],[24,1105,1106],{"xmlns":26},[28,1107,1108,1116],{},[31,1109,1110,1112,1114],{},[34,1111,37],{"mathvariant":36},[39,1113,333],{},[43,1115,46],{"stretchy":45},[48,1117,1118],{"encoding":50},"\\vert 00 \\rangle",[15,1120,1122],{"className":1121,"ariaHidden":56},[55],[15,1123,1125,1128,1131],{"className":1124},[60],[15,1126],{"className":1127,"style":65},[64],[15,1129,378],{"className":1130},[69],[15,1132,46],{"className":1133},[74],[271,1135,1136],{},[15,1137,1139,1156],{"className":1138},[18],[15,1140,1142],{"className":1141},[22],[24,1143,1144],{"xmlns":26},[28,1145,1146,1154],{},[31,1147,1148,1150,1152],{},[34,1149,37],{"mathvariant":36},[39,1151,333],{},[43,1153,46],{"stretchy":45},[48,1155,1118],{"encoding":50},[15,1157,1159],{"className":1158,"ariaHidden":56},[55],[15,1160,1162,1165,1168],{"className":1161},[60],[15,1163],{"className":1164,"style":65},[64],[15,1166,378],{"className":1167},[69],[15,1169,46],{"className":1170},[74],[257,1172,1173,1211],{},[271,1174,1175],{},[15,1176,1178,1196],{"className":1177},[18],[15,1179,1181],{"className":1180},[22],[24,1182,1183],{"xmlns":26},[28,1184,1185,1193],{},[31,1186,1187,1189,1191],{},[34,1188,37],{"mathvariant":36},[39,1190,343],{},[43,1192,46],{"stretchy":45},[48,1194,1195],{"encoding":50},"\\vert 01 \\rangle",[15,1197,1199],{"className":1198,"ariaHidden":56},[55],[15,1200,1202,1205,1208],{"className":1201},[60],[15,1203],{"className":1204,"style":65},[64],[15,1206,394],{"className":1207},[69],[15,1209,46],{"className":1210},[74],[271,1212,1213],{},[15,1214,1216,1233],{"className":1215},[18],[15,1217,1219],{"className":1218},[22],[24,1220,1221],{"xmlns":26},[28,1222,1223,1231],{},[31,1224,1225,1227,1229],{},[34,1226,37],{"mathvariant":36},[39,1228,343],{},[43,1230,46],{"stretchy":45},[48,1232,1195],{"encoding":50},[15,1234,1236],{"className":1235,"ariaHidden":56},[55],[15,1237,1239,1242,1245],{"className":1238},[60],[15,1240],{"className":1241,"style":65},[64],[15,1243,394],{"className":1244},[69],[15,1246,46],{"className":1247},[74],[257,1249,1250,1288],{},[271,1251,1252],{},[15,1253,1255,1273],{"className":1254},[18],[15,1256,1258],{"className":1257},[22],[24,1259,1260],{"xmlns":26},[28,1261,1262,1270],{},[31,1263,1264,1266,1268],{},[34,1265,37],{"mathvariant":36},[39,1267,352],{},[43,1269,46],{"stretchy":45},[48,1271,1272],{"encoding":50},"\\vert 10 \\rangle",[15,1274,1276],{"className":1275,"ariaHidden":56},[55],[15,1277,1279,1282,1285],{"className":1278},[60],[15,1280],{"className":1281,"style":65},[64],[15,1283,407],{"className":1284},[69],[15,1286,46],{"className":1287},[74],[271,1289,1290],{},[15,1291,1293,1310],{"className":1292},[18],[15,1294,1296],{"className":1295},[22],[24,1297,1298],{"xmlns":26},[28,1299,1300,1308],{},[31,1301,1302,1304,1306],{},[34,1303,37],{"mathvariant":36},[39,1305,352],{},[43,1307,46],{"stretchy":45},[48,1309,1272],{"encoding":50},[15,1311,1313],{"className":1312,"ariaHidden":56},[55],[15,1314,1316,1319,1322],{"className":1315},[60],[15,1317],{"className":1318,"style":65},[64],[15,1320,407],{"className":1321},[69],[15,1323,46],{"className":1324},[74],[257,1326,1327,1364],{},[271,1328,1329],{},[15,1330,1332,1349],{"className":1331},[18],[15,1333,1335],{"className":1334},[22],[24,1336,1337],{"xmlns":26},[28,1338,1339,1347],{},[31,1340,1341,1343,1345],{},[34,1342,37],{"mathvariant":36},[39,1344,41],{},[43,1346,46],{"stretchy":45},[48,1348,51],{"encoding":50},[15,1350,1352],{"className":1351,"ariaHidden":56},[55],[15,1353,1355,1358,1361],{"className":1354},[60],[15,1356],{"className":1357,"style":65},[64],[15,1359,70],{"className":1360},[69],[15,1362,46],{"className":1363},[74],[271,1365,1366],{},[15,1367,1369,1397],{"className":1368},[18],[15,1370,1372],{"className":1371},[22],[24,1373,1374],{"xmlns":26},[28,1375,1376,1394],{},[31,1377,1378,1388,1390,1392],{},[89,1379,1380,1382],{},[34,1381,93],{},[31,1383,1384,1386],{},[34,1385,98],{},[34,1387,101],{},[34,1389,37],{"mathvariant":36},[39,1391,41],{},[43,1393,46],{"stretchy":45},[48,1395,1396],{"encoding":50},"e^{i\\lambda}\\vert 11 \\rangle",[15,1398,1400],{"className":1399,"ariaHidden":56},[55],[15,1401,1403,1407,1439,1442],{"className":1402},[60],[15,1404],{"className":1405,"style":1406},[64],"height:1.0991em;vertical-align:-0.25em;",[15,1408,1410,1413],{"className":1409},[69],[15,1411,93],{"className":1412},[69,121],[15,1414,1416],{"className":1415},[125],[15,1417,1419],{"className":1418},[129],[15,1420,1422],{"className":1421},[133],[15,1423,1425],{"className":1424,"style":114},[137],[15,1426,1427,1430],{"style":140},[15,1428],{"className":1429,"style":145},[144],[15,1431,1433],{"className":1432},[149,150,151,152],[15,1434,1436],{"className":1435},[69,152],[15,1437,159],{"className":1438},[69,121,152],[15,1440,70],{"className":1441},[69],[15,1443,46],{"className":1444},[74],[11,1446,1447],{},"A measurement straight after CP shows no change on any basis state, whatever the angle. The phase shows once the qubits are in superposition, which is what the quantum Fourier transform and phase-estimation circuits rely on.",[246,1449,1451],{"id":1450},"inverse","Inverse",[11,1453,1454,1455,1458,1459,423],{},"The inverse of ",[277,1456,1457],{},"cp(λ)"," is ",[277,1460,1461],{},"cp(-λ)",[246,1463,1465],{"id":1464},"usage","Usage",[1467,1468,1473],"pre",{"className":1469,"code":1470,"language":1471,"meta":1472,"style":1472},"language-python shiki shiki-themes one-dark-pro","from math import pi\nfrom qiskit import QuantumCircuit\n\nqc = QuantumCircuit(2, 2)\nqc.h(0)\nqc.h(1)\nqc.cp(pi, 0, 1)\nqc.h(1)\nqc.measure([0, 1], [0, 1])\n","python","",[277,1474,1475,1493,1506,1513,1540,1555,1568,1587,1600],{"__ignoreMap":1472},[15,1476,1479,1483,1487,1490],{"class":1477,"line":1478},"line",1,[15,1480,1482],{"class":1481},"seHd6","from",[15,1484,1486],{"class":1485},"sn6KH"," math ",[15,1488,1489],{"class":1481},"import",[15,1491,1492],{"class":1485}," pi\n",[15,1494,1496,1498,1501,1503],{"class":1477,"line":1495},2,[15,1497,1482],{"class":1481},[15,1499,1500],{"class":1485}," qiskit ",[15,1502,1489],{"class":1481},[15,1504,1505],{"class":1485}," QuantumCircuit\n",[15,1507,1509],{"class":1477,"line":1508},3,[15,1510,1512],{"emptyLinePlaceholder":1511},true,"\n",[15,1514,1516,1519,1522,1526,1528,1532,1535,1537],{"class":1477,"line":1515},4,[15,1517,1518],{"class":1485},"qc ",[15,1520,457],{"class":1521},"sjrmR",[15,1523,1525],{"class":1524},"sVbv2"," QuantumCircuit",[15,1527,449],{"class":1485},[15,1529,1531],{"class":1530},"sVC51","2",[15,1533,1534],{"class":1485},", ",[15,1536,1531],{"class":1530},[15,1538,1539],{"class":1485},")\n",[15,1541,1543,1546,1549,1551,1553],{"class":1477,"line":1542},5,[15,1544,1545],{"class":1485},"qc.",[15,1547,1548],{"class":1524},"h",[15,1550,449],{"class":1485},[15,1552,216],{"class":1530},[15,1554,1539],{"class":1485},[15,1556,1558,1560,1562,1564,1566],{"class":1477,"line":1557},6,[15,1559,1545],{"class":1485},[15,1561,1548],{"class":1524},[15,1563,449],{"class":1485},[15,1565,479],{"class":1530},[15,1567,1539],{"class":1485},[15,1569,1571,1573,1576,1579,1581,1583,1585],{"class":1477,"line":1570},7,[15,1572,1545],{"class":1485},[15,1574,1575],{"class":1524},"cp",[15,1577,1578],{"class":1485},"(pi, ",[15,1580,216],{"class":1530},[15,1582,1534],{"class":1485},[15,1584,479],{"class":1530},[15,1586,1539],{"class":1485},[15,1588,1590,1592,1594,1596,1598],{"class":1477,"line":1589},8,[15,1591,1545],{"class":1485},[15,1593,1548],{"class":1524},[15,1595,449],{"class":1485},[15,1597,479],{"class":1530},[15,1599,1539],{"class":1485},[15,1601,1603,1605,1608,1611,1613,1615,1617,1620,1622,1624,1626],{"class":1477,"line":1602},9,[15,1604,1545],{"class":1485},[15,1606,1607],{"class":1524},"measure",[15,1609,1610],{"class":1485},"([",[15,1612,216],{"class":1530},[15,1614,1534],{"class":1485},[15,1616,479],{"class":1530},[15,1618,1619],{"class":1485},"], [",[15,1621,216],{"class":1530},[15,1623,1534],{"class":1485},[15,1625,479],{"class":1530},[15,1627,1628],{"class":1485},"])\n",[1467,1630,1634],{"className":1631,"code":1632,"language":1633,"meta":1472,"style":1472},"language-js shiki shiki-themes one-dark-pro","import { QuantumCircuit } from 'qiskit';\n\nconst circuit = QuantumCircuit(2, 2);\ncircuit.h(0);\ncircuit.h(1);\ncircuit.cp(Math.PI, 0, 1);\ncircuit.h(1);\ncircuit.measure([0, 1], [0, 1]);\n","js",[277,1635,1636,1659,1663,1688,1703,1717,1745,1759],{"__ignoreMap":1472},[15,1637,1638,1640,1643,1647,1650,1652,1656],{"class":1477,"line":1478},[15,1639,1489],{"class":1481},[15,1641,1642],{"class":1485}," { ",[15,1644,1646],{"class":1645},"sVyAn","QuantumCircuit",[15,1648,1649],{"class":1485}," } ",[15,1651,1482],{"class":1481},[15,1653,1655],{"class":1654},"subq3"," 'qiskit'",[15,1657,1658],{"class":1485},";\n",[15,1660,1661],{"class":1477,"line":1495},[15,1662,1512],{"emptyLinePlaceholder":1511},[15,1664,1665,1668,1672,1675,1677,1679,1681,1683,1685],{"class":1477,"line":1508},[15,1666,1667],{"class":1481},"const",[15,1669,1671],{"class":1670},"sU0A5"," circuit",[15,1673,1674],{"class":1521}," =",[15,1676,1525],{"class":1524},[15,1678,449],{"class":1485},[15,1680,1531],{"class":1530},[15,1682,1534],{"class":1485},[15,1684,1531],{"class":1530},[15,1686,1687],{"class":1485},");\n",[15,1689,1690,1693,1695,1697,1699,1701],{"class":1477,"line":1515},[15,1691,1692],{"class":1670},"circuit",[15,1694,423],{"class":1485},[15,1696,1548],{"class":1524},[15,1698,449],{"class":1485},[15,1700,216],{"class":1530},[15,1702,1687],{"class":1485},[15,1704,1705,1707,1709,1711,1713,1715],{"class":1477,"line":1542},[15,1706,1692],{"class":1670},[15,1708,423],{"class":1485},[15,1710,1548],{"class":1524},[15,1712,449],{"class":1485},[15,1714,479],{"class":1530},[15,1716,1687],{"class":1485},[15,1718,1719,1721,1723,1725,1727,1730,1732,1735,1737,1739,1741,1743],{"class":1477,"line":1557},[15,1720,1692],{"class":1670},[15,1722,423],{"class":1485},[15,1724,1575],{"class":1524},[15,1726,449],{"class":1485},[15,1728,1729],{"class":1670},"Math",[15,1731,423],{"class":1485},[15,1733,1734],{"class":1645},"PI",[15,1736,1534],{"class":1485},[15,1738,216],{"class":1530},[15,1740,1534],{"class":1485},[15,1742,479],{"class":1530},[15,1744,1687],{"class":1485},[15,1746,1747,1749,1751,1753,1755,1757],{"class":1477,"line":1570},[15,1748,1692],{"class":1670},[15,1750,423],{"class":1485},[15,1752,1548],{"class":1524},[15,1754,449],{"class":1485},[15,1756,479],{"class":1530},[15,1758,1687],{"class":1485},[15,1760,1761,1763,1765,1767,1769,1771,1773,1775,1777,1779,1781,1783],{"class":1477,"line":1589},[15,1762,1692],{"class":1670},[15,1764,423],{"class":1485},[15,1766,1607],{"class":1524},[15,1768,1610],{"class":1485},[15,1770,216],{"class":1530},[15,1772,1534],{"class":1485},[15,1774,479],{"class":1530},[15,1776,1619],{"class":1485},[15,1778,216],{"class":1530},[15,1780,1534],{"class":1485},[15,1782,479],{"class":1530},[15,1784,1785],{"class":1485},"]);\n",[11,1787,1788,1789,1791,1792,1794,1795,1849],{},"On a noiseless simulator every shot reads ",[277,1790,333],{}," or ",[277,1793,41],{},", in close to equal numbers: at ",[15,1796,1798,1817],{"className":1797},[18],[15,1799,1801],{"className":1800},[22],[24,1802,1803],{"xmlns":26},[28,1804,1805,1814],{},[31,1806,1807,1809,1811],{},[34,1808,101],{},[43,1810,457],{},[34,1812,1813],{},"π",[48,1815,1816],{"encoding":50},"\\lambda = \\pi",[15,1818,1820,1838],{"className":1819,"ariaHidden":56},[55],[15,1821,1823,1826,1829,1832,1835],{"className":1822},[60],[15,1824],{"className":1825,"style":187},[64],[15,1827,101],{"className":1828},[69,121],[15,1830],{"className":1831,"style":619},[389],[15,1833,457],{"className":1834},[623],[15,1836],{"className":1837,"style":619},[389],[15,1839,1841,1845],{"className":1840},[60],[15,1842],{"className":1843,"style":1844},[64],"height:0.4306em;",[15,1846,1813],{"className":1847,"style":1848},[69,121],"margin-right:0.0359em;"," the CP is a CZ, and H, CZ, H on the target is a CX.",[246,1851,1853],{"id":1852},"on-ionq-hardware","On IonQ hardware",[11,1855,1856,1857,1859,1860,1863,1864,1867,1868,1871,1872,1875,1876,1879,1880,423],{},"CP is not a native gate, but it maps onto one entangling gate for any angle. Up to a global phase, ",[277,1858,1457],{}," is an ",[277,1861,1862],{},"rzz(-λ\u002F2)"," with an ",[277,1865,1866],{},"rz(λ\u002F2)"," on each qubit, and the ",[277,1869,1870],{},"rz"," rotations are virtual, so on Forte it is a single arbitrary-angle ",[277,1873,1874],{},"ZZ"," gate with no single-qubit pulses; on Aria it is one partially entangling ",[277,1877,1878],{},"MS"," gate with single-qubit pulses around it. See ",[240,1881,1883],{"href":1882},"\u002Flearn\u002Fdocs\u002Fhow-your-circuit-is-compiled","How your circuit is compiled",[246,1885,1887],{"id":1886},"related","Related",[1889,1890,1891,1896,1900,1906,1912],"ul",{},[1892,1893,1894],"li",{},[240,1895,1070],{"href":1069},[1892,1897,1898],{},[240,1899,1075],{"href":1074},[1892,1901,1902],{},[240,1903,1905],{"href":1904},"\u002Flearn\u002Fdocs\u002Frzz-gate","RZZ gate",[1892,1907,1908],{},[240,1909,1911],{"href":1910},"\u002Flearn\u002Fdocs\u002Fzz-gate","ZZ gate",[1892,1913,1914],{},[240,1915,1917],{"href":1916},"\u002Flearn\u002Fdocs\u002Fgates","Gate reference",[1919,1920,1921],"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: 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 .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sU0A5, html code.shiki .sU0A5{--shiki-default:#E5C07B}",{"title":1472,"searchDepth":1495,"depth":1495,"links":1923},[1924,1925,1926,1927,1928,1929,1930],{"id":248,"depth":1495,"text":249},{"id":312,"depth":1495,"text":313},{"id":1078,"depth":1495,"text":1079},{"id":1450,"depth":1495,"text":1451},{"id":1464,"depth":1495,"text":1465},{"id":1852,"depth":1495,"text":1853},{"id":1886,"depth":1495,"text":1887},"gates","Gates","The controlled-phase gate: matrix, effect on the basis states, the cp() call in Python and JS, and what it becomes on IonQ hardware.",false,"md","two-qubit","Two-qubit gates","doc",{},18,"\u002Fblog\u002Flearn\u002Fdocs\u002Fcp-gate","2026-09-16",[],{"title":1945,"description":1946},"CP gate (controlled phase) · Qollab docs","Reference for the controlled-phase gate CP(λ): matrix, action on the basis states, the Qiskit cp() call in Python and JS, and its IonQ native-gate 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