[{"data":1,"prerenderedAt":1499},["ShallowReactive",2],{"blog-post-docs\u002Fz-gate":3,"sibling-dives-docs\u002Fz-gate":1339,"learn-track-docs\u002Fz-gate":1340,"learn-docs-groups":1341},{"id":4,"title":5,"authors":6,"body":7,"breadcrumb":6,"builders":6,"byline":6,"category":1321,"categoryName":1322,"challenge":6,"courseAuthor":6,"courseLead":6,"dek":6,"description":1323,"draft":1324,"extension":1325,"eyebrow":6,"featured":1324,"finish":6,"fork":6,"group":1326,"groupName":1327,"hero":6,"heroAlt":6,"heroCta":6,"heroImage":6,"homepageFeatured":1324,"kind":1328,"lessonCount":6,"meta":1329,"navigation":1048,"newsItems":6,"next":6,"ogImage":6,"order":1078,"outcomes":6,"path":1330,"publishDate":1331,"readingTime":6,"related":1332,"relatedProjects":6,"seo":1333,"stem":1336,"tags":1337,"track":6,"trackName":6,"__hash__":1338},"blog\u002Fblog\u002Flearn\u002Fdocs\u002Fz-gate.md","Z gate",null,{"type":8,"value":9,"toc":1312},"minimark",[10,120,165,170,233,237,487,504,508,847,1005,1009,1012,1016,1136,1257,1263,1267,1274,1278,1308],[11,12,13,14,75,76,114,115,119],"p",{},"The Z gate leaves ",[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",{},"0",[43,44,46],"mo",{"stretchy":45},"false","⟩",[48,49,51],"annotation",{"encoding":50},"application\u002Fx-tex","\\vert 0 \\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","∣0",[15,72,46],{"className":73},[74],"mclose"," alone and flips the sign of ",[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],{},"1",[43,94,46],{"stretchy":45},[48,96,97],{"encoding":50},"\\vert 1 \\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],"∣1",[15,112,46],{"className":113},[74],". It acts on one qubit and takes no parameters. The Circuit panel draws it as ",[116,117,118],"strong",{},"Z",". On its own it changes nothing a measurement can see; the sign shows once the qubit is in superposition.",[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},[39,138,41],{},[43,140,46],{"stretchy":45},[48,142,51],{"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],[15,153,70],{"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 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\\\\ 0 & -1 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the Bloch sphere this is a half-turn about the Z axis. Z is the special case ",[197,491,492],{},"p(π)"," of the ",[160,495,497],{"href":496},"\u002Flearn\u002Fdocs\u002Fp-gate","P gate",", and two ",[160,500,502],{"href":501},"\u002Flearn\u002Fdocs\u002Fs-gate","S gates"," make one Z.",[166,505,507],{"id":506},"effect-on-basis-states","Effect on basis states",[171,509,510,520],{},[174,511,512],{},[177,513,514,517],{},[180,515,516],{},"Input",[180,518,519],{},"Output",[186,521,522,598,680,765],{},[177,523,524,561],{},[191,525,526],{},[15,527,529,546],{"className":528},[18],[15,530,532],{"className":531},[22],[24,533,534],{"xmlns":26},[28,535,536,544],{},[31,537,538,540,542],{},[34,539,37],{"mathvariant":36},[39,541,41],{},[43,543,46],{"stretchy":45},[48,545,51],{"encoding":50},[15,547,549],{"className":548,"ariaHidden":56},[55],[15,550,552,555,558],{"className":551},[60],[15,553],{"className":554,"style":65},[64],[15,556,70],{"className":557},[69],[15,559,46],{"className":560},[74],[191,562,563],{},[15,564,566,583],{"className":565},[18],[15,567,569],{"className":568},[22],[24,570,571],{"xmlns":26},[28,572,573,581],{},[31,574,575,577,579],{},[34,576,37],{"mathvariant":36},[39,578,41],{},[43,580,46],{"stretchy":45},[48,582,51],{"encoding":50},[15,584,586],{"className":585,"ariaHidden":56},[55],[15,587,589,592,595],{"className":588},[60],[15,590],{"className":591,"style":65},[64],[15,593,70],{"className":594},[69],[15,596,46],{"className":597},[74],[177,599,600,637],{},[191,601,602],{},[15,603,605,622],{"className":604},[18],[15,606,608],{"className":607},[22],[24,609,610],{"xmlns":26},[28,611,612,620],{},[31,613,614,616,618],{},[34,615,37],{"mathvariant":36},[39,617,92],{},[43,619,46],{"stretchy":45},[48,621,97],{"encoding":50},[15,623,625],{"className":624,"ariaHidden":56},[55],[15,626,628,631,634],{"className":627},[60],[15,629],{"className":630,"style":65},[64],[15,632,110],{"className":633},[69],[15,635,46],{"className":636},[74],[191,638,639],{},[15,640,642,662],{"className":641},[18],[15,643,645],{"className":644},[22],[24,646,647],{"xmlns":26},[28,648,649,659],{},[31,650,651,653,655,657],{},[43,652,303],{},[34,654,37],{"mathvariant":36},[39,656,92],{},[43,658,46],{"stretchy":45},[48,660,661],{"encoding":50},"-\\vert 1 \\rangle",[15,663,665],{"className":664,"ariaHidden":56},[55],[15,666,668,671,674,677],{"className":667},[60],[15,669],{"className":670,"style":65},[64],[15,672,303],{"className":673},[69],[15,675,110],{"className":676},[69],[15,678,46],{"className":679},[74],[177,681,682,724],{},[191,683,684],{},[15,685,687,706],{"className":686},[18],[15,688,690],{"className":689},[22],[24,691,692],{"xmlns":26},[28,693,694,703],{},[31,695,696,698,701],{},[34,697,37],{"mathvariant":36},[43,699,700],{},"+",[43,702,46],{"stretchy":45},[48,704,705],{"encoding":50},"\\vert + \\rangle",[15,707,709],{"className":708,"ariaHidden":56},[55],[15,710,712,715,718,721],{"className":711},[60],[15,713],{"className":714,"style":65},[64],[15,716,37],{"className":717},[69],[15,719,700],{"className":720},[69],[15,722,46],{"className":723},[74],[191,725,726],{},[15,727,729,747],{"className":728},[18],[15,730,732],{"className":731},[22],[24,733,734],{"xmlns":26},[28,735,736,744],{},[31,737,738,740,742],{},[34,739,37],{"mathvariant":36},[43,741,303],{},[43,743,46],{"stretchy":45},[48,745,746],{"encoding":50},"\\vert - \\rangle",[15,748,750],{"className":749,"ariaHidden":56},[55],[15,751,753,756,759,762],{"className":752},[60],[15,754],{"className":755,"style":65},[64],[15,757,37],{"className":758},[69],[15,760,303],{"className":761},[69],[15,763,46],{"className":764},[74],[177,766,767,807],{},[191,768,769],{},[15,770,772,789],{"className":771},[18],[15,773,775],{"className":774},[22],[24,776,777],{"xmlns":26},[28,778,779,787],{},[31,780,781,783,785],{},[34,782,37],{"mathvariant":36},[43,784,303],{},[43,786,46],{"stretchy":45},[48,788,746],{"encoding":50},[15,790,792],{"className":791,"ariaHidden":56},[55],[15,793,795,798,801,804],{"className":794},[60],[15,796],{"className":797,"style":65},[64],[15,799,37],{"className":800},[69],[15,802,303],{"className":803},[69],[15,805,46],{"className":806},[74],[191,808,809],{},[15,810,812,829],{"className":811},[18],[15,813,815],{"className":814},[22],[24,816,817],{"xmlns":26},[28,818,819,827],{},[31,820,821,823,825],{},[34,822,37],{"mathvariant":36},[43,824,700],{},[43,826,46],{"stretchy":45},[48,828,705],{"encoding":50},[15,830,832],{"className":831,"ariaHidden":56},[55],[15,833,835,838,841,844],{"className":834},[60],[15,836],{"className":837,"style":65},[64],[15,839,37],{"className":840},[69],[15,842,700],{"className":843},[69],[15,845,46],{"className":846},[74],[11,848,849,850,885,886,921,922,960,961,999,1000,1004],{},"A measurement straight after Z shows no change on ",[15,851,853,870],{"className":852},[18],[15,854,856],{"className":855},[22],[24,857,858],{"xmlns":26},[28,859,860,868],{},[31,861,862,864,866],{},[34,863,37],{"mathvariant":36},[39,865,41],{},[43,867,46],{"stretchy":45},[48,869,51],{"encoding":50},[15,871,873],{"className":872,"ariaHidden":56},[55],[15,874,876,879,882],{"className":875},[60],[15,877],{"className":878,"style":65},[64],[15,880,70],{"className":881},[69],[15,883,46],{"className":884},[74]," or ",[15,887,889,906],{"className":888},[18],[15,890,892],{"className":891},[22],[24,893,894],{"xmlns":26},[28,895,896,904],{},[31,897,898,900,902],{},[34,899,37],{"mathvariant":36},[39,901,92],{},[43,903,46],{"stretchy":45},[48,905,97],{"encoding":50},[15,907,909],{"className":908,"ariaHidden":56},[55],[15,910,912,915,918],{"className":911},[60],[15,913],{"className":914,"style":65},[64],[15,916,110],{"className":917},[69],[15,919,46],{"className":920},[74],". The sign flip shows once the qubit is in superposition: Z turns ",[15,923,925,942],{"className":924},[18],[15,926,928],{"className":927},[22],[24,929,930],{"xmlns":26},[28,931,932,940],{},[31,933,934,936,938],{},[34,935,37],{"mathvariant":36},[43,937,700],{},[43,939,46],{"stretchy":45},[48,941,705],{"encoding":50},[15,943,945],{"className":944,"ariaHidden":56},[55],[15,946,948,951,954,957],{"className":947},[60],[15,949],{"className":950,"style":65},[64],[15,952,37],{"className":953},[69],[15,955,700],{"className":956},[69],[15,958,46],{"className":959},[74]," into ",[15,962,964,981],{"className":963},[18],[15,965,967],{"className":966},[22],[24,968,969],{"xmlns":26},[28,970,971,979],{},[31,972,973,975,977],{},[34,974,37],{"mathvariant":36},[43,976,303],{},[43,978,46],{"stretchy":45},[48,980,746],{"encoding":50},[15,982,984],{"className":983,"ariaHidden":56},[55],[15,985,987,990,993,996],{"className":986},[60],[15,988],{"className":989,"style":65},[64],[15,991,37],{"className":992},[69],[15,994,303],{"className":995},[69],[15,997,46],{"className":998},[74],", which an ",[160,1001,1003],{"href":1002},"\u002Flearn\u002Fdocs\u002Fh-gate","H gate"," then reads out as a 1.",[166,1006,1008],{"id":1007},"inverse","Inverse",[11,1010,1011],{},"Z is its own inverse. Two Z gates in a row on the same qubit cancel.",[166,1013,1015],{"id":1014},"usage","Usage",[1017,1018,1023],"pre",{"className":1019,"code":1020,"language":1021,"meta":1022,"style":1022},"language-python shiki shiki-themes one-dark-pro","from qiskit import QuantumCircuit\n\nqc = QuantumCircuit(1, 1)\nqc.h(0)\nqc.z(0)\nqc.h(0)\nqc.measure(0, 0)\n","python","",[197,1024,1025,1043,1050,1076,1091,1105,1118],{"__ignoreMap":1022},[15,1026,1029,1033,1037,1040],{"class":1027,"line":1028},"line",1,[15,1030,1032],{"class":1031},"seHd6","from",[15,1034,1036],{"class":1035},"sn6KH"," qiskit ",[15,1038,1039],{"class":1031},"import",[15,1041,1042],{"class":1035}," QuantumCircuit\n",[15,1044,1046],{"class":1027,"line":1045},2,[15,1047,1049],{"emptyLinePlaceholder":1048},true,"\n",[15,1051,1053,1056,1059,1063,1065,1068,1071,1073],{"class":1027,"line":1052},3,[15,1054,1055],{"class":1035},"qc ",[15,1057,258],{"class":1058},"sjrmR",[15,1060,1062],{"class":1061},"sVbv2"," QuantumCircuit",[15,1064,263],{"class":1035},[15,1066,92],{"class":1067},"sVC51",[15,1069,1070],{"class":1035},", ",[15,1072,92],{"class":1067},[15,1074,1075],{"class":1035},")\n",[15,1077,1079,1082,1085,1087,1089],{"class":1027,"line":1078},4,[15,1080,1081],{"class":1035},"qc.",[15,1083,1084],{"class":1061},"h",[15,1086,263],{"class":1035},[15,1088,41],{"class":1067},[15,1090,1075],{"class":1035},[15,1092,1094,1096,1099,1101,1103],{"class":1027,"line":1093},5,[15,1095,1081],{"class":1035},[15,1097,1098],{"class":1061},"z",[15,1100,263],{"class":1035},[15,1102,41],{"class":1067},[15,1104,1075],{"class":1035},[15,1106,1108,1110,1112,1114,1116],{"class":1027,"line":1107},6,[15,1109,1081],{"class":1035},[15,1111,1084],{"class":1061},[15,1113,263],{"class":1035},[15,1115,41],{"class":1067},[15,1117,1075],{"class":1035},[15,1119,1121,1123,1126,1128,1130,1132,1134],{"class":1027,"line":1120},7,[15,1122,1081],{"class":1035},[15,1124,1125],{"class":1061},"measure",[15,1127,263],{"class":1035},[15,1129,41],{"class":1067},[15,1131,1070],{"class":1035},[15,1133,41],{"class":1067},[15,1135,1075],{"class":1035},[1017,1137,1141],{"className":1138,"code":1139,"language":1140,"meta":1022,"style":1022},"language-js shiki shiki-themes one-dark-pro","import { QuantumCircuit } from 'qiskit';\n\nconst circuit = QuantumCircuit(1, 1);\ncircuit.h(0);\ncircuit.z(0);\ncircuit.h(0);\ncircuit.measure(0, 0);\n","js",[197,1142,1143,1166,1170,1195,1211,1225,1239],{"__ignoreMap":1022},[15,1144,1145,1147,1150,1154,1157,1159,1163],{"class":1027,"line":1028},[15,1146,1039],{"class":1031},[15,1148,1149],{"class":1035}," { ",[15,1151,1153],{"class":1152},"sVyAn","QuantumCircuit",[15,1155,1156],{"class":1035}," } ",[15,1158,1032],{"class":1031},[15,1160,1162],{"class":1161},"subq3"," 'qiskit'",[15,1164,1165],{"class":1035},";\n",[15,1167,1168],{"class":1027,"line":1045},[15,1169,1049],{"emptyLinePlaceholder":1048},[15,1171,1172,1175,1179,1182,1184,1186,1188,1190,1192],{"class":1027,"line":1052},[15,1173,1174],{"class":1031},"const",[15,1176,1178],{"class":1177},"sU0A5"," circuit",[15,1180,1181],{"class":1058}," =",[15,1183,1062],{"class":1061},[15,1185,263],{"class":1035},[15,1187,92],{"class":1067},[15,1189,1070],{"class":1035},[15,1191,92],{"class":1067},[15,1193,1194],{"class":1035},");\n",[15,1196,1197,1200,1203,1205,1207,1209],{"class":1027,"line":1078},[15,1198,1199],{"class":1177},"circuit",[15,1201,1202],{"class":1035},".",[15,1204,1084],{"class":1061},[15,1206,263],{"class":1035},[15,1208,41],{"class":1067},[15,1210,1194],{"class":1035},[15,1212,1213,1215,1217,1219,1221,1223],{"class":1027,"line":1093},[15,1214,1199],{"class":1177},[15,1216,1202],{"class":1035},[15,1218,1098],{"class":1061},[15,1220,263],{"class":1035},[15,1222,41],{"class":1067},[15,1224,1194],{"class":1035},[15,1226,1227,1229,1231,1233,1235,1237],{"class":1027,"line":1107},[15,1228,1199],{"class":1177},[15,1230,1202],{"class":1035},[15,1232,1084],{"class":1061},[15,1234,263],{"class":1035},[15,1236,41],{"class":1067},[15,1238,1194],{"class":1035},[15,1240,1241,1243,1245,1247,1249,1251,1253,1255],{"class":1027,"line":1120},[15,1242,1199],{"class":1177},[15,1244,1202],{"class":1035},[15,1246,1125],{"class":1061},[15,1248,263],{"class":1035},[15,1250,41],{"class":1067},[15,1252,1070],{"class":1035},[15,1254,41],{"class":1067},[15,1256,1194],{"class":1035},[11,1258,1259,1260,1262],{},"Every shot reads ",[197,1261,92],{},": the two H gates would cancel on their own, and the Z between them turns the pair into an X.",[166,1264,1266],{"id":1265},"on-ionq-hardware","On IonQ hardware",[11,1268,1269,1270,1202],{},"Z is a virtual Z. The hardware has no physical Z gate; the compiler folds the half-turn into the phase of whatever pulse comes next, so Z costs no gate and no time on the machine. See ",[160,1271,1273],{"href":1272},"\u002Flearn\u002Fdocs\u002Fhow-your-circuit-is-compiled","How your circuit is compiled",[166,1275,1277],{"id":1276},"related","Related",[1279,1280,1281,1287,1291,1297,1302],"ul",{},[1282,1283,1284],"li",{},[160,1285,1286],{"href":501},"S gate",[1282,1288,1289],{},[160,1290,497],{"href":496},[1282,1292,1293],{},[160,1294,1296],{"href":1295},"\u002Flearn\u002Fdocs\u002Frz-gate","RZ gate",[1282,1298,1299],{},[160,1300,1301],{"href":229},"CZ gate",[1282,1303,1304],{},[160,1305,1307],{"href":1306},"\u002Flearn\u002Fdocs\u002Fgates","Gate reference",[1309,1310,1311],"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 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