[{"data":1,"prerenderedAt":1580},["ShallowReactive",2],{"blog-post-docs\u002Fs-gate":3,"sibling-dives-docs\u002Fs-gate":1418,"learn-track-docs\u002Fs-gate":1419,"learn-docs-groups":1420},{"id":4,"title":5,"authors":6,"body":7,"breadcrumb":6,"builders":6,"byline":6,"category":1400,"categoryName":1401,"challenge":6,"courseAuthor":6,"courseLead":6,"dek":6,"description":1402,"draft":1403,"extension":1404,"eyebrow":6,"featured":1403,"finish":6,"fork":6,"group":1405,"groupName":1406,"hero":6,"heroAlt":6,"heroCta":6,"heroImage":6,"homepageFeatured":1403,"kind":1407,"lessonCount":6,"meta":1408,"navigation":1102,"newsItems":6,"next":6,"ogImage":6,"order":1146,"outcomes":6,"path":1409,"publishDate":1410,"readingTime":6,"related":1411,"relatedProjects":6,"seo":1412,"stem":1415,"tags":1416,"track":6,"trackName":6,"__hash__":1417},"blog\u002Fblog\u002Flearn\u002Fdocs\u002Fs-gate.md","S gate",null,{"type":8,"value":9,"toc":1391},"minimark",[10,152,197,202,252,256,497,519,523,975,1050,1054,1066,1070,1202,1337,1343,1347,1354,1358,1387],[11,12,13,14,75,76,107,108,146,147,151],"p",{},"The S gate multiplies the ",[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"," amplitude by ",[15,77,79,93],{"className":78},[18],[15,80,82],{"className":81},[22],[24,83,84],{"xmlns":26},[28,85,86,91],{},[31,87,88],{},[34,89,90],{},"i",[48,92,90],{"encoding":50},[15,94,96],{"className":95,"ariaHidden":56},[55],[15,97,99,103],{"className":98},[60],[15,100],{"className":101,"style":102},[64],"height:0.6595em;",[15,104,90],{"className":105},[69,106],"mathnormal",", a quarter turn of phase, and leaves ",[15,109,111,130],{"className":110},[18],[15,112,114],{"className":113},[22],[24,115,116],{"xmlns":26},[28,117,118,127],{},[31,119,120,122,125],{},[34,121,37],{"mathvariant":36},[39,123,124],{},"0",[43,126,46],{"stretchy":45},[48,128,129],{"encoding":50},"\\vert 0 \\rangle",[15,131,133],{"className":132,"ariaHidden":56},[55],[15,134,136,139,143],{"className":135},[60],[15,137],{"className":138,"style":65},[64],[15,140,142],{"className":141},[69],"∣0",[15,144,46],{"className":145},[74]," alone. 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S is the square root of ",[192,501,503],{"href":502},"\u002Flearn\u002Fdocs\u002Fz-gate","Z",": two S gates make one Z. 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measurement straight after S shows no change on ",[15,979,981,998],{"className":980},[18],[15,982,984],{"className":983},[22],[24,985,986],{"xmlns":26},[28,987,988,996],{},[31,989,990,992,994],{},[34,991,37],{"mathvariant":36},[39,993,124],{},[43,995,46],{"stretchy":45},[48,997,129],{"encoding":50},[15,999,1001],{"className":1000,"ariaHidden":56},[55],[15,1002,1004,1007,1010],{"className":1003},[60],[15,1005],{"className":1006,"style":65},[64],[15,1008,142],{"className":1009},[69],[15,1011,46],{"className":1012},[74]," or ",[15,1015,1017,1034],{"className":1016},[18],[15,1018,1020],{"className":1019},[22],[24,1021,1022],{"xmlns":26},[28,1023,1024,1032],{},[31,1025,1026,1028,1030],{},[34,1027,37],{"mathvariant":36},[39,1029,41],{},[43,1031,46],{"stretchy":45},[48,1033,51],{"encoding":50},[15,1035,1037],{"className":1036,"ariaHidden":56},[55],[15,1038,1040,1043,1046],{"className":1039},[60],[15,1041],{"className":1042,"style":65},[64],[15,1044,70],{"className":1045},[69],[15,1047,46],{"className":1048},[74],". The phase shows once the qubit is in superposition.",[198,1051,1053],{"id":1052},"inverse","Inverse",[11,1055,1056,1057,1061,1062,1065],{},"The inverse of S is ",[192,1058,1060],{"href":1059},"\u002Flearn\u002Fdocs\u002Fsdg-gate","Sdg",", ",[229,1063,1064],{},"sdg(qubit)",". S is not its own inverse: two S gates make a Z, not an identity.",[198,1067,1069],{"id":1068},"usage","Usage",[1071,1072,1077],"pre",{"className":1073,"code":1074,"language":1075,"meta":1076,"style":1076},"language-python shiki shiki-themes one-dark-pro","from qiskit import QuantumCircuit\n\nqc = QuantumCircuit(1, 1)\nqc.h(0)\nqc.s(0)\nqc.s(0)\nqc.h(0)\nqc.measure(0, 0)\n","python","",[229,1078,1079,1097,1104,1129,1144,1158,1171,1184],{"__ignoreMap":1076},[15,1080,1083,1087,1091,1094],{"class":1081,"line":1082},"line",1,[15,1084,1086],{"class":1085},"seHd6","from",[15,1088,1090],{"class":1089},"sn6KH"," qiskit ",[15,1092,1093],{"class":1085},"import",[15,1095,1096],{"class":1089}," QuantumCircuit\n",[15,1098,1100],{"class":1081,"line":1099},2,[15,1101,1103],{"emptyLinePlaceholder":1102},true,"\n",[15,1105,1107,1110,1113,1117,1119,1122,1124,1126],{"class":1081,"line":1106},3,[15,1108,1109],{"class":1089},"qc ",[15,1111,277],{"class":1112},"sjrmR",[15,1114,1116],{"class":1115},"sVbv2"," QuantumCircuit",[15,1118,282],{"class":1089},[15,1120,41],{"class":1121},"sVC51",[15,1123,1061],{"class":1089},[15,1125,41],{"class":1121},[15,1127,1128],{"class":1089},")\n",[15,1130,1132,1135,1138,1140,1142],{"class":1081,"line":1131},4,[15,1133,1134],{"class":1089},"qc.",[15,1136,1137],{"class":1115},"h",[15,1139,282],{"class":1089},[15,1141,124],{"class":1121},[15,1143,1128],{"class":1089},[15,1145,1147,1149,1152,1154,1156],{"class":1081,"line":1146},5,[15,1148,1134],{"class":1089},[15,1150,1151],{"class":1115},"s",[15,1153,282],{"class":1089},[15,1155,124],{"class":1121},[15,1157,1128],{"class":1089},[15,1159,1161,1163,1165,1167,1169],{"class":1081,"line":1160},6,[15,1162,1134],{"class":1089},[15,1164,1151],{"class":1115},[15,1166,282],{"class":1089},[15,1168,124],{"class":1121},[15,1170,1128],{"class":1089},[15,1172,1174,1176,1178,1180,1182],{"class":1081,"line":1173},7,[15,1175,1134],{"class":1089},[15,1177,1137],{"class":1115},[15,1179,282],{"class":1089},[15,1181,124],{"class":1121},[15,1183,1128],{"class":1089},[15,1185,1187,1189,1192,1194,1196,1198,1200],{"class":1081,"line":1186},8,[15,1188,1134],{"class":1089},[15,1190,1191],{"class":1115},"measure",[15,1193,282],{"class":1089},[15,1195,124],{"class":1121},[15,1197,1061],{"class":1089},[15,1199,124],{"class":1121},[15,1201,1128],{"class":1089},[1071,1203,1207],{"className":1204,"code":1205,"language":1206,"meta":1076,"style":1076},"language-js shiki shiki-themes one-dark-pro","import { QuantumCircuit } from 'qiskit';\n\nconst circuit = QuantumCircuit(1, 1);\ncircuit.h(0);\ncircuit.s(0);\ncircuit.s(0);\ncircuit.h(0);\ncircuit.measure(0, 0);\n","js",[229,1208,1209,1232,1236,1261,1277,1291,1305,1319],{"__ignoreMap":1076},[15,1210,1211,1213,1216,1220,1223,1225,1229],{"class":1081,"line":1082},[15,1212,1093],{"class":1085},[15,1214,1215],{"class":1089}," { ",[15,1217,1219],{"class":1218},"sVyAn","QuantumCircuit",[15,1221,1222],{"class":1089}," } ",[15,1224,1086],{"class":1085},[15,1226,1228],{"class":1227},"subq3"," 'qiskit'",[15,1230,1231],{"class":1089},";\n",[15,1233,1234],{"class":1081,"line":1099},[15,1235,1103],{"emptyLinePlaceholder":1102},[15,1237,1238,1241,1245,1248,1250,1252,1254,1256,1258],{"class":1081,"line":1106},[15,1239,1240],{"class":1085},"const",[15,1242,1244],{"class":1243},"sU0A5"," circuit",[15,1246,1247],{"class":1112}," =",[15,1249,1116],{"class":1115},[15,1251,282],{"class":1089},[15,1253,41],{"class":1121},[15,1255,1061],{"class":1089},[15,1257,41],{"class":1121},[15,1259,1260],{"class":1089},");\n",[15,1262,1263,1266,1269,1271,1273,1275],{"class":1081,"line":1131},[15,1264,1265],{"class":1243},"circuit",[15,1267,1268],{"class":1089},".",[15,1270,1137],{"class":1115},[15,1272,282],{"class":1089},[15,1274,124],{"class":1121},[15,1276,1260],{"class":1089},[15,1278,1279,1281,1283,1285,1287,1289],{"class":1081,"line":1146},[15,1280,1265],{"class":1243},[15,1282,1268],{"class":1089},[15,1284,1151],{"class":1115},[15,1286,282],{"class":1089},[15,1288,124],{"class":1121},[15,1290,1260],{"class":1089},[15,1292,1293,1295,1297,1299,1301,1303],{"class":1081,"line":1160},[15,1294,1265],{"class":1243},[15,1296,1268],{"class":1089},[15,1298,1151],{"class":1115},[15,1300,282],{"class":1089},[15,1302,124],{"class":1121},[15,1304,1260],{"class":1089},[15,1306,1307,1309,1311,1313,1315,1317],{"class":1081,"line":1173},[15,1308,1265],{"class":1243},[15,1310,1268],{"class":1089},[15,1312,1137],{"class":1115},[15,1314,282],{"class":1089},[15,1316,124],{"class":1121},[15,1318,1260],{"class":1089},[15,1320,1321,1323,1325,1327,1329,1331,1333,1335],{"class":1081,"line":1186},[15,1322,1265],{"class":1243},[15,1324,1268],{"class":1089},[15,1326,1191],{"class":1115},[15,1328,282],{"class":1089},[15,1330,124],{"class":1121},[15,1332,1061],{"class":1089},[15,1334,124],{"class":1121},[15,1336,1260],{"class":1089},[11,1338,1339,1340,1342],{},"Every shot reads ",[229,1341,41],{},": the two S gates make a Z, and H Z H is an X.",[198,1344,1346],{"id":1345},"on-ionq-hardware","On IonQ hardware",[11,1348,1349,1350,1268],{},"S is a virtual Z. The compiler folds the quarter-turn into the phase of the next pulse, so S costs no gate and no time on the machine. See ",[192,1351,1353],{"href":1352},"\u002Flearn\u002Fdocs\u002Fhow-your-circuit-is-compiled","How your circuit is compiled",[198,1355,1357],{"id":1356},"related","Related",[1359,1360,1361,1367,1372,1377,1381],"ul",{},[1362,1363,1364],"li",{},[192,1365,1366],{"href":1059},"Sdg gate",[1362,1368,1369],{},[192,1370,1371],{"href":516},"T gate",[1362,1373,1374],{},[192,1375,1376],{"href":502},"Z gate",[1362,1378,1379],{},[192,1380,512],{"href":511},[1362,1382,1383],{},[192,1384,1386],{"href":1385},"\u002Flearn\u002Fdocs\u002Fgates","Gate reference",[1388,1389,1390],"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: 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