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\\rangle)",[15,1124,1126,1262],{"className":1125,"ariaHidden":56},[55],[15,1127,1129,1132,1244,1247,1250,1253,1256,1259],{"className":1128},[60],[15,1130],{"className":1131,"style":843},[64],[15,1133,1135,1138,1241],{"className":1134},[69],[15,1136],{"className":1137},[458,459],[15,1139,1141],{"className":1140},[355],[15,1142,1144,1233],{"className":1143},[466,467],[15,1145,1147,1230],{"className":1146},[471],[15,1148,1150,1208,1216],{"className":1149,"style":862},[475],[15,1151,1152,1155],{"style":865},[15,1153],{"className":1154,"style":484},[483],[15,1156,1158],{"className":1157},[872,873,609,874],[15,1159,1161],{"className":1160},[69,874],[15,1162,1164],{"className":1163},[69,491,874],[15,1165,1167,1200],{"className":1166},[466,467],[15,1168,1170,1197],{"className":1169},[471],[15,1171,1173,1185],{"className":1172,"style":890},[475],[15,1174,1176,1179],{"className":1175,"style":506},[505],[15,1177],{"className":1178,"style":484},[483],[15,1180,1182],{"className":1181,"style":513},[69,874],[15,1183,364],{"className":1184},[69,874],[15,1186,1187,1190],{"style":905},[15,1188],{"className":1189,"style":484},[483],[15,1191,1193],{"className":1192,"style":527},[526,874],[529,1194,1195],{"xmlns":531,"width":532,"height":533,"viewBox":534,"preserveAspectRatio":535},[537,1196],{"d":539},[15,1198,544],{"className":1199},[543],[15,1201,1203],{"className":1202},[471],[15,1204,1206],{"className":1205,"style":925},[475],[15,1207],{},[15,1209,1210,1213],{"style":556},[15,1211],{"className":1212,"style":484},[483],[15,1214],{"className":1215,"style":564},[563],[15,1217,1218,1221],{"style":938},[15,1219],{"className":1220,"style":484},[483],[15,1222,1224],{"className":1223},[872,873,609,874],[15,1225,1227],{"className":1226},[69,874],[15,1228,128],{"className":1229},[69,874],[15,1231,544],{"className":1232},[543],[15,1234,1236],{"className":1235},[471],[15,1237,1239],{"className":1238,"style":960},[475],[15,1240],{},[15,1242],{"className":1243},[74,459],[15,1245,369],{"className":1246},[458],[15,1248,70],{"className":1249},[69],[15,1251,46],{"className":1252},[74],[15,1254],{"className":1255,"style":978},[436],[15,1257,409],{"className":1258},[982],[15,1260],{"className":1261,"style":978},[436],[15,1263,1265,1268,1271],{"className":1264},[60],[15,1266],{"className":1267,"style":65},[64],[15,1269,146],{"className":1270},[69],[15,1272,998],{"className":1273},[74],[15,1275,1277,1295],{"className":1276},[18],[15,1278,1280],{"className":1279},[22],[24,1281,1282],{"xmlns":26},[28,1283,1284,1292],{},[31,1285,1286,1288,1290],{},[34,1287,37],{"mathvariant":36},[43,1289,409],{},[43,1291,46],{"stretchy":45},[48,1293,1294],{"encoding":50},"\\vert - \\rangle",[15,1296,1298],{"className":1297,"ariaHidden":56},[55],[15,1299,1301,1304,1307,1310],{"className":1300},[60],[15,1302],{"className":1303,"style":65},[64],[15,1305,37],{"className":1306},[69],[15,1308,409],{"className":1309},[69],[15,1311,46],{"className":1312},[74],[285,1314,1315,1355],{},[299,1316,1317],{},[15,1318,1320,1337],{"className":1319},[18],[15,1321,1323],{"className":1322},[22],[24,1324,1325],{"xmlns":26},[28,1326,1327,1335],{},[31,1328,1329,1331,1333],{},[34,1330,37],{"mathvariant":36},[43,1332,822],{},[43,1334,46],{"stretchy":45},[48,1336,1020],{"encoding":50},[15,1338,1340],{"className":1339,"ariaHidden":56},[55],[15,1341,1343,1346,1349,1352],{"className":1342},[60],[15,1344],{"className":1345,"style":65},[64],[15,1347,37],{"className":1348},[69],[15,1350,822],{"className":1351},[69],[15,1353,46],{"className":1354},[74],[299,1356,1357],{},[15,1358,1360,1377],{"className":1359},[18],[15,1361,1363],{"className":1362},[22],[24,1364,1365],{"xmlns":26},[28,1366,1367,1375],{},[31,1368,1369,1371,1373],{},[34,1370,37],{"mathvariant":36},[39,1372,41],{},[43,1374,46],{"stretchy":45},[48,1376,51],{"encoding":50},[15,1378,1380],{"className":1379,"ariaHidden":56},[55],[15,1381,1383,1386,1389],{"className":1382},[60],[15,1384],{"className":1385,"style":65},[64],[15,1387,70],{"className":1388},[69],[15,1390,46],{"className":1391},[74],[285,1393,1394,1434],{},[299,1395,1396],{},[15,1397,1399,1416],{"className":1398},[18],[15,1400,1402],{"className":1401},[22],[24,1403,1404],{"xmlns":26},[28,1405,1406,1414],{},[31,1407,1408,1410,1412],{},[34,1409,37],{"mathvariant":36},[43,1411,409],{},[43,1413,46],{"stretchy":45},[48,1415,1294],{"encoding":50},[15,1417,1419],{"className":1418,"ariaHidden":56},[55],[15,1420,1422,1425,1428,1431],{"className":1421},[60],[15,1423],{"className":1424,"style":65},[64],[15,1426,37],{"className":1427},[69],[15,1429,409],{"className":1430},[69],[15,1432,46],{"className":1433},[74],[299,1435,1436],{},[15,1437,1439,1456],{"className":1438},[18],[15,1440,1442],{"className":1441},[22],[24,1443,1444],{"xmlns":26},[28,1445,1446,1454],{},[31,1447,1448,1450,1452],{},[34,1449,37],{"mathvariant":36},[39,1451,128],{},[43,1453,46],{"stretchy":45},[48,1455,133],{"encoding":50},[15,1457,1459],{"className":1458,"ariaHidden":56},[55],[15,1460,1462,1465,1468],{"className":1461},[60],[15,1463],{"className":1464,"style":65},[64],[15,1466,146],{"className":1467},[69],[15,1469,46],{"className":1470},[74],[11,1472,1473,1474,1509,1510,1545,1546,111,1584,1622],{},"Measured straight after H, a qubit that started in ",[15,1475,1477,1494],{"className":1476},[18],[15,1478,1480],{"className":1479},[22],[24,1481,1482],{"xmlns":26},[28,1483,1484,1492],{},[31,1485,1486,1488,1490],{},[34,1487,37],{"mathvariant":36},[39,1489,41],{},[43,1491,46],{"stretchy":45},[48,1493,51],{"encoding":50},[15,1495,1497],{"className":1496,"ariaHidden":56},[55],[15,1498,1500,1503,1506],{"className":1499},[60],[15,1501],{"className":1502,"style":65},[64],[15,1504,70],{"className":1505},[69],[15,1507,46],{"className":1508},[74]," or ",[15,1511,1513,1530],{"className":1512},[18],[15,1514,1516],{"className":1515},[22],[24,1517,1518],{"xmlns":26},[28,1519,1520,1528],{},[31,1521,1522,1524,1526],{},[34,1523,37],{"mathvariant":36},[39,1525,128],{},[43,1527,46],{"stretchy":45},[48,1529,133],{"encoding":50},[15,1531,1533],{"className":1532,"ariaHidden":56},[55],[15,1534,1536,1539,1542],{"className":1535},[60],[15,1537],{"className":1538,"style":65},[64],[15,1540,146],{"className":1541},[69],[15,1543,46],{"className":1544},[74]," reads 0 or 1 with equal probability. The sign difference between ",[15,1547,1549,1566],{"className":1548},[18],[15,1550,1552],{"className":1551},[22],[24,1553,1554],{"xmlns":26},[28,1555,1556,1564],{},[31,1557,1558,1560,1562],{},[34,1559,37],{"mathvariant":36},[43,1561,822],{},[43,1563,46],{"stretchy":45},[48,1565,1020],{"encoding":50},[15,1567,1569],{"className":1568,"ariaHidden":56},[55],[15,1570,1572,1575,1578,1581],{"className":1571},[60],[15,1573],{"className":1574,"style":65},[64],[15,1576,37],{"className":1577},[69],[15,1579,822],{"className":1580},[69],[15,1582,46],{"className":1583},[74],[15,1585,1587,1604],{"className":1586},[18],[15,1588,1590],{"className":1589},[22],[24,1591,1592],{"xmlns":26},[28,1593,1594,1602],{},[31,1595,1596,1598,1600],{},[34,1597,37],{"mathvariant":36},[43,1599,409],{},[43,1601,46],{"stretchy":45},[48,1603,1294],{"encoding":50},[15,1605,1607],{"className":1606,"ariaHidden":56},[55],[15,1608,1610,1613,1616,1619],{"className":1609},[60],[15,1611],{"className":1612,"style":65},[64],[15,1614,37],{"className":1615},[69],[15,1617,409],{"className":1618},[69],[15,1620,46],{"className":1621},[74]," does not show in a measurement; it decides how later gates interfere.",[274,1624,1626],{"id":1625},"inverse","Inverse",[11,1628,1629,1630,1688],{},"H is its own inverse: ",[15,1631,1633,1654],{"className":1632},[18],[15,1634,1636],{"className":1635},[22],[24,1637,1638],{"xmlns":26},[28,1639,1640,1651],{},[31,1641,1642,1644,1646,1648],{},[34,1643,226],{},[34,1645,226],{},[43,1647,353],{},[34,1649,1650],{},"I",[48,1652,1653],{"encoding":50},"HH = I",[15,1655,1657,1678],{"className":1656,"ariaHidden":56},[55],[15,1658,1660,1663,1666,1669,1672,1675],{"className":1659},[60],[15,1661],{"className":1662,"style":427},[64],[15,1664,226],{"className":1665,"style":432},[69,431],[15,1667,226],{"className":1668,"style":432},[69,431],[15,1670],{"className":1671,"style":437},[436],[15,1673,353],{"className":1674},[441],[15,1676],{"className":1677,"style":437},[436],[15,1679,1681,1684],{"className":1680},[60],[15,1682],{"className":1683,"style":427},[64],[15,1685,1650],{"className":1686,"style":1687},[69,431],"margin-right:0.0785em;",". Two H gates in a row on the same qubit cancel.",[274,1690,1692],{"id":1691},"usage","Usage",[1694,1695,1700],"pre",{"className":1696,"code":1697,"language":1698,"meta":1699,"style":1699},"language-python shiki shiki-themes one-dark-pro","from qiskit import QuantumCircuit\n\nqc = QuantumCircuit(1, 1)\nqc.h(0)\nqc.measure(0, 0)\n","python","",[305,1701,1702,1720,1727,1753,1768],{"__ignoreMap":1699},[15,1703,1706,1710,1714,1717],{"class":1704,"line":1705},"line",1,[15,1707,1709],{"class":1708},"seHd6","from",[15,1711,1713],{"class":1712},"sn6KH"," qiskit ",[15,1715,1716],{"class":1708},"import",[15,1718,1719],{"class":1712}," QuantumCircuit\n",[15,1721,1723],{"class":1704,"line":1722},2,[15,1724,1726],{"emptyLinePlaceholder":1725},true,"\n",[15,1728,1730,1733,1736,1740,1742,1745,1748,1750],{"class":1704,"line":1729},3,[15,1731,1732],{"class":1712},"qc ",[15,1734,353],{"class":1735},"sjrmR",[15,1737,1739],{"class":1738},"sVbv2"," QuantumCircuit",[15,1741,369],{"class":1712},[15,1743,128],{"class":1744},"sVC51",[15,1746,1747],{"class":1712},", ",[15,1749,128],{"class":1744},[15,1751,1752],{"class":1712},")\n",[15,1754,1756,1759,1762,1764,1766],{"class":1704,"line":1755},4,[15,1757,1758],{"class":1712},"qc.",[15,1760,1761],{"class":1738},"h",[15,1763,369],{"class":1712},[15,1765,41],{"class":1744},[15,1767,1752],{"class":1712},[15,1769,1771,1773,1776,1778,1780,1782,1784],{"class":1704,"line":1770},5,[15,1772,1758],{"class":1712},[15,1774,1775],{"class":1738},"measure",[15,1777,369],{"class":1712},[15,1779,41],{"class":1744},[15,1781,1747],{"class":1712},[15,1783,41],{"class":1744},[15,1785,1752],{"class":1712},[1694,1787,1791],{"className":1788,"code":1789,"language":1790,"meta":1699,"style":1699},"language-js shiki shiki-themes one-dark-pro","import { QuantumCircuit } from 'qiskit';\n\nconst circuit = QuantumCircuit(1, 1);\ncircuit.h(0);\ncircuit.measure(0, 0);\n","js",[305,1792,1793,1816,1820,1845,1861],{"__ignoreMap":1699},[15,1794,1795,1797,1800,1804,1807,1809,1813],{"class":1704,"line":1705},[15,1796,1716],{"class":1708},[15,1798,1799],{"class":1712}," { ",[15,1801,1803],{"class":1802},"sVyAn","QuantumCircuit",[15,1805,1806],{"class":1712}," } ",[15,1808,1709],{"class":1708},[15,1810,1812],{"class":1811},"subq3"," 'qiskit'",[15,1814,1815],{"class":1712},";\n",[15,1817,1818],{"class":1704,"line":1722},[15,1819,1726],{"emptyLinePlaceholder":1725},[15,1821,1822,1825,1829,1832,1834,1836,1838,1840,1842],{"class":1704,"line":1729},[15,1823,1824],{"class":1708},"const",[15,1826,1828],{"class":1827},"sU0A5"," circuit",[15,1830,1831],{"class":1735}," =",[15,1833,1739],{"class":1738},[15,1835,369],{"class":1712},[15,1837,128],{"class":1744},[15,1839,1747],{"class":1712},[15,1841,128],{"class":1744},[15,1843,1844],{"class":1712},");\n",[15,1846,1847,1850,1853,1855,1857,1859],{"class":1704,"line":1755},[15,1848,1849],{"class":1827},"circuit",[15,1851,1852],{"class":1712},".",[15,1854,1761],{"class":1738},[15,1856,369],{"class":1712},[15,1858,41],{"class":1744},[15,1860,1844],{"class":1712},[15,1862,1863,1865,1867,1869,1871,1873,1875,1877],{"class":1704,"line":1770},[15,1864,1849],{"class":1827},[15,1866,1852],{"class":1712},[15,1868,1775],{"class":1738},[15,1870,369],{"class":1712},[15,1872,41],{"class":1744},[15,1874,1747],{"class":1712},[15,1876,41],{"class":1744},[15,1878,1844],{"class":1712},[11,1880,1881,1882,111,1884,1852],{},"Over many shots the counts split close to evenly between ",[305,1883,41],{},[305,1885,128],{},[274,1887,1889],{"id":1888},"on-ionq-hardware","On IonQ hardware",[11,1891,1892,1893,1896,1897,1852],{},"H is not a native gate. Up to a global phase it is a Y rotation by π\u002F2 applied after a Z rotation by π, so one ",[305,1894,1895],{},"GPi2"," pulse is enough, with the Z rotation carried as a virtual Z that costs no physical gate. What runs after IonQ's optimiser can still differ, because the compiler rewrites the whole circuit rather than one gate at a time. See ",[268,1898,1900],{"href":1899},"\u002Flearn\u002Fdocs\u002Fhow-your-circuit-is-compiled","How your circuit is compiled",[274,1902,1904],{"id":1903},"related","Related",[1906,1907,1908,1915,1921],"ul",{},[1909,1910,1911],"li",{},[268,1912,1914],{"href":1913},"\u002Flearn\u002Fdocs\u002Fcx-gate","CX gate",[1909,1916,1917],{},[268,1918,1920],{"href":1919},"\u002Flearn\u002Fdocs\u002Fgates","Gate reference",[1909,1922,1923],{},[268,1924,1900],{"href":1899},[1926,1927,1928],"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":1699,"searchDepth":1722,"depth":1722,"links":1930},[1931,1932,1933,1934,1935,1936,1937],{"id":276,"depth":1722,"text":277},{"id":330,"depth":1722,"text":331},{"id":733,"depth":1722,"text":734},{"id":1625,"depth":1722,"text":1626},{"id":1691,"depth":1722,"text":1692},{"id":1888,"depth":1722,"text":1889},{"id":1903,"depth":1722,"text":1904},"gates","Gates","The Hadamard gate: matrix, effect on |0⟩ and |1⟩, the h() call in Python and JS, and what it becomes on IonQ hardware.",false,"md","single-qubit","Single-qubit gates","doc",{},"\u002Fblog\u002Flearn\u002Fdocs\u002Fh-gate","2026-09-16",[],{"title":1951,"description":1952},"H gate (Hadamard) · Qollab docs","Reference for the Hadamard gate: matrix, action on the basis states, the Qiskit h() call in Python and JS, and its IonQ native-gate form.","blog\u002Flearn\u002Fdocs\u002Fh-gate",[],"lu8tr0xH4Db20G68__msKldvIsS0dKz638Xb1xBwUc0",[],[],[1959,1966,1982,2011,2101,2111],{"category":1960,"categoryName":1961,"articles":1962},"account","Account",[1963],{"title":1964,"href":1965},"Account and profile","\u002Flearn\u002Fdocs\u002Faccount-and-profile",{"category":1967,"categoryName":1968,"articles":1969},"projects","Projects",[1970,1973,1976,1979],{"title":1971,"href":1972},"The project page","\u002Flearn\u002Fdocs\u002Fproject-anatomy",{"title":1974,"href":1975},"Editing your 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