Single Qubit Gates - Qiskit

The Pauli Gates · Digression: The X, Y & Z-Bases · The Hadamard Gate · Digression: Measuring in Different Bases · The P-gate · The I, S and T-gates · The General U- ... Documentation Community Learn Overview LearnQuantumComputationusingQiskit LearnQuantumComputationusingQiskit WhatisQuantum? 0. Prerequisites 0.1 SettingUpYourEnvironment 0.2 PythonandJupyterNotebooks 1. QuantumStatesandQubits 1.1 Introduction 1.2 TheAtomsofComputation 1.3 RepresentingQubitStates 1.4 SingleQubitGates 1.5 TheCaseforQuantum 2. MultipleQubitsandEntanglement 2.1 Introduction 2.2 MultipleQubitsandEntangledStates 2.3 PhaseKickback 2.4 MoreCircuitIdentities 2.5 ProvingUniversality 2.6 ClassicalComputationonaQuantumComputer 3. QuantumProtocolsandQuantumAlgorithms 3.1 DefiningQuantumCircuits 3.2 Deutsch-JozsaAlgorithm 3.3 Bernstein-VaziraniAlgorithm 3.4 Simon'sAlgorithm 3.5 QuantumFourierTransform 3.6 QuantumPhaseEstimation 3.7 Shor'sAlgorithm 3.8 Grover'sAlgorithm 3.9 QuantumCounting 3.10 QuantumWalkSearchAlgorithm 3.11 QuantumTeleportation 3.12 SuperdenseCoding 3.13 QuantumKeyDistribution 4. QuantumAlgorithmsforApplications 4.1 AppliedQuantumAlgorithms 4.1.1 SolvingLinearSystemsofEquationsusingHHL 4.1.2 SimulatingMoleculesusingVQE 4.1.3 SolvingcombinatorialoptimizationproblemsusingQAOA 4.1.4 SolvingSatisfiabilityProblemsusingGrover'sAlgorithm 4.1.5 Hybridquantum-classicalNeuralNetworkswithPyTorchandQiskit 4.2 ImplementationsofRecentQuantumAlgorithms 4.2.1 VariationalQuantumLinearSolver 4.2.2 QuantumImageProcessing-FRQIandNEQRImageRepresentations 4.2.3 QuantumEdgeDetection-QHEDAlgorithmonSmallandLargeImages 4.2.4 SolvingtheTravellingSalesmanProblemusingPhaseEstimation 5. InvestigatingQuantumHardwareUsingQuantumCircuits 5.1 IntroductiontoQuantumErrorCorrectionusingRepetitionCodes 5.2 MeasurementErrorMitigation 5.3 RandomizedBenchmarking 5.4 MeasuringQuantumVolume 5.5 TheDensityMatrix&MixedStates 6. InvestigatingQuantumHardwareUsingMicrowavePulses 6.1 CalibratingQubitswithQiskitPulse 6.2 AccessingHigherEnergyStates 6.3 IntroductiontoTransmonPhysics 6.4 CircuitQuantumElectrodynamics 6.5 ExploringtheJaynes-CummingsHamiltonianwithQiskitPulse 6.6 MeasuringtheQubitac-StarkShift 6.7 HamiltonianTomography 7. QuantumComputingLabs Lab1.QuantumCircuits Lab2.QuantumMeasurement Lab3.AccuracyofQuantumPhaseEstimation Lab4.IterativeQuantumPhaseEstimation Lab5.ScalableShor’sAlgorithm Lab6.Grover'ssearchwithanunknownnumberofsolutions Lab7.QuantumSimulationasaSearchAlgorithm Lab8.QuantumErrorCorrection 8. Appendix 8.1 LinearAlgebra 8.2 Qiskit 9. Games&Demos HelloQiskitGame EstimatingPiUsingQuantumPhaseEstimationAlgorithm LocalRealityandtheCHSHInequality QuantumCoinGame VariationalQuantumRegression InteractivityIndex English Japanese OnThisPage OpeninIBMQuantumLab DownloadasJupyterNotebook ContributeonGithub Trythenewtextbookbeta ThenewQiskitTextbookbetaisnowavailable. Tryitoutnow SingleQubitGates Intheprevioussectionwelookedatallthepossiblestatesaqubitcouldbein.Wesawthatqubitscouldberepresentedby2Dvectors,andthattheirstatesarelimitedtotheform: $$|q\rangle=\cos{\tfrac{\theta}{2}}|0\rangle+e^{i\phi}\sin{\tfrac{\theta}{2}}|1\rangle$$Where$\theta$and$\phi$arerealnumbers.Inthissectionwewillcovergates,theoperationsthatchangeaqubitbetweenthesestates.Duetothenumberofgatesandthesimilaritiesbetweenthem,thischapterisatriskofbecomingalist.Tocounterthis,wehaveincludedafewdigressionstointroduceimportantideasatappropriateplacesthroughoutthechapter. Contents ThePauliGates 1.1TheX-Gate 1.2TheY&Z-Gates Digression:TheX,Y&Z-Bases TheHadamardGate Digression:MeasuringinDifferentBases TheP-gate TheI,SandT-gates 6.1TheI-Gate 6.2TheS-Gate 6.3TheT_Gate TheGeneralU-gate InTheAtomsofComputationwecameacrosssomegatesandusedthemtoperformaclassicalcomputation.Animportantfeatureofquantumcircuitsisthat,betweeninitialisingthequbitsandmeasuringthem,theoperations(gates)arealwaysreversible!Thesereversiblegatescanberepresentedasmatrices,andasrotationsaroundtheBlochsphere. fromqiskitimportQuantumCircuit,assemble,Aer frommathimportpi,sqrt fromqiskit.visualizationimportplot_bloch_multivector,plot_histogram sim=Aer.get_backend('aer_simulator') 1.ThePauliGatesYoushouldbefamiliarwiththePaulimatricesfromthelinearalgebrasection.Ifanyofthemathshereisnewtoyou,youshouldusethelinearalgebrasectiontobringyourselfuptospeed.WewillseeherethatthePaulimatricescanrepresentsomeverycommonlyusedquantumgates. 1.1TheX-GateTheX-gateisrepresentedbythePauli-Xmatrix: $$X=\begin{bmatrix}0&1\\1&0\end{bmatrix}=|0\rangle\langle1|+|1\rangle\langle0|$$Toseetheeffectagatehasonaqubit,wesimplymultiplythequbit’sstatevectorbythegate.WecanseethattheX-gateswitchestheamplitudesofthestates$|0\rangle$and$|1\rangle$: $$X|0\rangle=\begin{bmatrix}0&1\\1&0\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix}=\begin{bmatrix}0\\1\end{bmatrix}=|1\rangle$$ Reminder:MultiplyingVectorsbyMatrices(Clickheretoexpand) Matrixmultiplicationisageneralisationoftheinnerproductwesawinthelastchapter.Inthespecificcaseofmultiplyingavectorbyamatrix(asseenabove),wealwaysgetavectorback: $$M|v\rangle=\begin{bmatrix}a&b\\c&d\\\end{bmatrix}\begin{bmatrix}v_0\\v_1\\\end{bmatrix} =\begin{bmatrix}a\cdotv_0+b\cdotv_1\\c\cdotv_0+d\cdotv_1\end{bmatrix}$$ Inquantumcomputing,wecanwriteourmatricesintermsofbasisvectors: $$X=|0\rangle\langle1|+|1\rangle\langle0|$$ Thiscansometimesbeclearerthanusingaboxmatrixaswecanseewhatdifferentmultiplicationswillresultin: $$ \begin{aligned} X|1\rangle&=(|0\rangle\langle1|+|1\rangle\langle0|)|1\rangle\\ &=|0\rangle\langle1|1\rangle+|1\rangle\langle0|1\rangle\\ &=|0\rangle\times1+|1\rangle\times0\\ &=|0\rangle \end{aligned} $$ Infact,whenweseeaketandabramultipliedlikethis: $$|a\rangle\langleb|$$ thisiscalledtheouterproduct,whichfollowstherule: $$ |a\rangle\langleb|= \begin{bmatrix} a_0b_0^*&a_0b_1^*&\dots&a_0b_n^*\\ a_1b_0^*&\ddots&&\vdots\\ \vdots&&\ddots&\vdots\\ a_nb_0^*&\dots&\dots&a_nb_n^*\\ \end{bmatrix} $$ where$*$denotesthecomplexconjugation.WecanseethisdoesindeedresultintheX-matrixasseenabove: $$ |0\rangle\langle1|+|1\rangle\langle0|= \begin{bmatrix}0&1\\0&0\\\end{bmatrix}+ \begin{bmatrix}0&0\\1&0\\\end{bmatrix}= \begin{bmatrix}0&1\\1&0\\\end{bmatrix}=X $$ InQiskit,wecancreateashortcircuittoverifythis: #Let'sdoanX-gateona|0>qubit qc=QuantumCircuit(1) qc.x(0) qc.draw() Let'sseetheresultoftheabovecircuit.Note:Hereweuseplot_bloch_multivector()whichtakesaqubit'sstatevectorinsteadoftheBlochvector. #Let'sseetheresult qc.save_statevector() qobj=assemble(qc) state=sim.run(qobj).result().get_statevector() plot_bloch_multivector(state) Wecanindeedseethestateofthequbitis$|1\rangle$asexpected.Wecanthinkofthisasarotationby$\pi$radiansaroundthex-axisoftheBlochsphere.TheX-gateisalsooftencalledaNOT-gate,referringtoitsclassicalanalogue. 1.2TheY&Z-gatesSimilarlytotheX-gate,theY&ZPaulimatricesalsoactastheY&Z-gatesinourquantumcircuits: $$Y=\begin{bmatrix}0&-i\\i&0\end{bmatrix}\quad\quad\quad\quadZ=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$$$$Y=-i|0\rangle\langle1|+i|1\rangle\langle0|\quad\quadZ=|0\rangle\langle0|-|1\rangle\langle1|$$And,unsurprisingly,theyalsorespectivelyperformrotationsby$\pi$aroundtheyandz-axisoftheBlochsphere. Belowisawidgetthatdisplaysaqubit’sstateontheBlochsphere,pressingoneofthebuttonswillperformthegateonthequbit: #Runthecodeinthiscelltoseethewidget fromqiskit_textbook.widgetsimportgate_demo gate_demo(gates='pauli') InQiskit,wecanapplytheYandZ-gatestoourcircuitusing: qc.y(0)#DoY-gateonqubit0 qc.z(0)#DoZ-gateonqubit0 qc.draw() 2.Digression:TheX,Y&Z-Bases Reminder:EigenvectorsofMatrices(Clickheretoexpand) Wehaveseenthatmultiplyingavectorbyamatrixresultsinavector: $$ M|v\rangle=|v'\rangle\leftarrow\text{newvector} $$ Ifwechosetherightvectorsandmatrices,wecanfindacaseinwhichthismatrixmultiplicationisthesameasdoingamultiplicationbyascalar: $$ M|v\rangle=\lambda|v\rangle $$ (Above,$M$isamatrix,and$\lambda$isascalar).Foramatrix$M$,anyvectorthathasthispropertyiscalledaneigenvectorof$M$.Forexample,theeigenvectorsoftheZ-matrixarethestates$|0\rangle$and$|1\rangle$: $$ \begin{aligned} Z|0\rangle&=|0\rangle\\ Z|1\rangle&=-|1\rangle \end{aligned} $$ Sinceweusevectorstodescribethestateofourqubits,weoftencallthesevectorseigenstatesinthiscontext.Eigenvectorsareveryimportantinquantumcomputing,anditisimportantyouhaveasolidgraspofthem. YoumayalsonoticethattheZ-gateappearstohavenoeffectonourqubitwhenitisineitherofthesetwostates.Thisisbecausethestates$|0\rangle$and$|1\rangle$arethetwoeigenstatesoftheZ-gate.Infact,thecomputationalbasis(thebasisformedbythestates$|0\rangle$and$|1\rangle$)isoftencalledtheZ-basis.Thisisnottheonlybasiswecanuse,apopularbasisistheX-basis,formedbytheeigenstatesoftheX-gate.Wecallthesetwovectors$|+\rangle$and$|-\rangle$: $$|+\rangle=\tfrac{1}{\sqrt{2}}(|0\rangle+|1\rangle)=\tfrac{1}{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix}$$$$|-\rangle=\tfrac{1}{\sqrt{2}}(|0\rangle-|1\rangle)=\tfrac{1}{\sqrt{2}}\begin{bmatrix}1\\-1\end{bmatrix}$$AnotherlesscommonlyusedbasisisthatformedbytheeigenstatesoftheY-gate.Thesearecalled: $$|\circlearrowleft\rangle,\quad|\circlearrowright\rangle$$Weleaveitasanexercisetocalculatethese.Thereareinfactaninfinitenumberofbases;toformone,wesimplyneedtwoorthogonalvectors.TheeigenvectorsofbothHermitianandunitarymatricesformabasisforthevectorspace.Duetothisproperty,wecanbesurethattheeigenstatesoftheX-gateandtheY-gateindeedformabasisfor1-qubitstates(readmoreaboutthisinthelinearalgebrapageintheappendix) QuickExercises Verifythat$|+\rangle$and$|-\rangle$areinfacteigenstatesoftheX-gate. Whateigenvaluesdotheyhave? FindtheeigenstatesoftheY-gate,andtheirco-ordinatesontheBlochsphere. UsingonlythePauli-gatesitisimpossibletomoveourinitializedqubittoanystateotherthan$|0\rangle$or$|1\rangle$,i.e.wecannotachievesuperposition.Thismeanswecanseenobehaviourdifferenttothatofaclassicalbit.Tocreatemoreinterestingstateswewillneedmoregates! 3.TheHadamardGateTheHadamardgate(H-gate)isafundamentalquantumgate.ItallowsustomoveawayfromthepolesoftheBlochsphereandcreateasuperpositionof$|0\rangle$and$|1\rangle$.Ithasthematrix: $$H=\tfrac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1&-1\end{bmatrix}$$Wecanseethatthisperformsthetransformationsbelow: $$H|0\rangle=|+\rangle$$$$H|1\rangle=|-\rangle$$ThiscanbethoughtofasarotationaroundtheBlochvector[1,0,1](thelinebetweenthex&z-axis),orastransformingthestateofthequbitbetweentheXandZbases. Youcanplayaroundwiththesegatesusingthewidgetbelow: #Runthecodeinthiscelltoseethewidget fromqiskit_textbook.widgetsimportgate_demo gate_demo(gates='pauli+h') QuickExercise WritetheH-gateastheouterproductsofvectors$|0\rangle$,$|1\rangle$,$|+\rangle$and$|-\rangle$. Showthatapplyingthesequenceofgates:HZH,toanyqubitstateisequivalenttoapplyinganX-gate. FindacombinationofX,ZandH-gatesthatisequivalenttoaY-gate(ignoringglobalphase). 4.Digression:MeasuringinDifferentBasesWehaveseenthattheZ-axisisnotintrinsicallyspecial,andthatthereareinfinitelymanyotherbases.Similarlywithmeasurement,wedon’talwayshavetomeasureinthecomputationalbasis(theZ-basis),wecanmeasureourqubitsinanybasis. Asanexample,let’strymeasuringintheX-basis.Wecancalculatetheprobabilityofmeasuringeither$|+\rangle$or$|-\rangle$: $$p(|+\rangle)=|\langle+|q\rangle|^2,\quadp(|-\rangle)=|\langle-|q\rangle|^2$$Andaftermeasurement,thesuperpositionisdestroyed.SinceQiskitonlyallowsmeasuringintheZ-basis,wemustcreateourownusingHadamardgates: #CreatetheX-measurementfunction: defx_measurement(qc,qubit,cbit): """Measure'qubit'intheX-basis,andstoretheresultin'cbit'""" qc.h(qubit) qc.measure(qubit,cbit) returnqc initial_state=[1/sqrt(2),-1/sqrt(2)] #Initializeourqubitandmeasureit qc=QuantumCircuit(1,1) qc.initialize(initial_state,0) x_measurement(qc,0,0)#measurequbit0toclassicalbit0 qc.draw() Inthequickexercisesabove,wesawyoucouldcreateanX-gatebysandwichingourZ-gatebetweentwoH-gates: $$X=HZH$$StartingintheZ-basis,theH-gateswitchesourqubittotheX-basis,theZ-gateperformsaNOTintheX-basis,andthefinalH-gatereturnsourqubittotheZ-basis.WecanverifythisalwaysbehaveslikeanX-gatebymultiplyingthematrices: $$ HZH= \tfrac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1&-1\end{bmatrix} \begin{bmatrix}1&0\\0&-1\end{bmatrix} \tfrac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1&-1\end{bmatrix} = \begin{bmatrix}0&1\\1&0\end{bmatrix} =X $$Followingthesamelogic,wehavecreatedanX-measurementbytransformingfromtheX-basistotheZ-basisbeforeourmeasurement.Sincetheprocessofmeasuringcanhavedifferenteffectsdependingonthesystem(e.g.somesystemsalwaysreturnthequbitto$|0\rangle$aftermeasurement,whereasothersmayleaveitasthemeasuredstate),thestateofthequbitpost-measurementisundefinedandwemustresetitifwewanttouseitagain. ThereisanotherwaytoseewhytheHadamardgateindeedtakesusfromtheZ-basistotheX-basis.SupposethequbitwewanttomeasureintheX-basisisinthe(normalized)state$a|0\rangle+b|1\rangle$.TomeasureitinX-basis,wefirstexpressthestateasalinearcombinationof$|+\rangle$and$|-\rangle$.Usingtherelations$|0\rangle=\frac{|+\rangle+|-\rangle}{\sqrt{2}}$and$|1\rangle=\frac{|+\rangle-|-\rangle}{\sqrt{2}}$,thestatebecomes$\frac{a+b}{\sqrt{2}}|+\rangle+\frac{a-b}{\sqrt{2}}|-\rangle$.ObservethattheprobabilityamplitudesinX-basiscanbeobtainedbyapplyingaHadamardmatrixonthestatevectorexpressedinZ-basis. Let’snowseetheresults: qobj=assemble(qc)#AssemblecircuitintoaQobjthatcanberun counts=sim.run(qobj).result().get_counts()#Dothesimulation,returningthestatevector plot_histogram(counts)#Displaytheoutputonmeasurementofstatevector Weinitializedourqubitinthestate$|-\rangle$,butwecanseethat,afterthemeasurement,wehavecollapsedourqubittothestate$|1\rangle$.Ifyourunthecellagain,youwillseethesameresult,sincealongtheX-basis,thestate$|-\rangle$isabasisstateandmeasuringitalongXwillalwaysyieldthesameresult. QuickExercises Ifweinitializeourqubitinthestate$|+\rangle$,whatistheprobabilityofmeasuringitinstate$|-\rangle$? UseQiskittodisplaytheprobabilityofmeasuringa$|0\rangle$qubitinthestates$|+\rangle$and$|-\rangle$(Hint:youmightwanttouse.get_counts()andplot_histogram()). TrytocreateafunctionthatmeasuresintheY-basis. MeasuringindifferentbasesallowsustoseeHeisenberg’sfamousuncertaintyprincipleinaction.HavingcertaintyofmeasuringastateintheZ-basisremovesallcertaintyofmeasuringaspecificstateintheX-basis,andviceversa.Acommonmisconceptionisthattheuncertaintyisduetothelimitsinourequipment,butherewecanseetheuncertaintyisactuallypartofthenatureofthequbit. Forexample,ifweputourqubitinthestate$|0\rangle$,ourmeasurementintheZ-basisiscertaintobe$|0\rangle$,butourmeasurementintheX-basisiscompletelyrandom!Similarly,ifweputourqubitinthestate$|-\rangle$,ourmeasurementintheX-basisiscertaintobe$|-\rangle$,butnowanymeasurementintheZ-basiswillbecompletelyrandom. Moregenerally:Whateverstateourquantumsystemisin,thereisalwaysameasurementthathasadeterministicoutcome. TheintroductionoftheH-gatehasallowedustoexploresomeinterestingphenomena,butwearestillverylimitedinourquantumoperations.Letusnowintroduceanewtypeofgate: 5.TheP-gateTheP-gate(phasegate)isparametrised,thatis,itneedsanumber($\phi$)totellitexactlywhattodo.TheP-gateperformsarotationof$\phi$aroundtheZ-axisdirection.Ithasthematrixform: $$ P(\phi)=\begin{bmatrix}1&0\\0&e^{i\phi}\end{bmatrix} $$Where$\phi$isarealnumber. YoucanusethewidgetbelowtoplayaroundwiththeP-gate,specify$\phi$usingtheslider: #Runthecodeinthiscelltoseethewidget fromqiskit_textbook.widgetsimportgate_demo gate_demo(gates='pauli+h+p') InQiskit,wespecifyaP-gateusingp(phi,qubit): qc=QuantumCircuit(1) qc.p(pi/4,0) qc.draw() YoumaynoticethattheZ-gateisaspecialcaseoftheP-gate,with$\phi=\pi$.Infacttherearethreemorecommonlyreferencedgateswewillmentioninthischapter,allofwhicharespecialcasesoftheP-gate: 6.TheI,SandT-gates6.1TheI-gateFirstcomestheI-gate(aka‘Id-gate’or‘Identitygate’).Thisissimplyagatethatdoesnothing.Itsmatrixistheidentitymatrix: $$ I=\begin{bmatrix}1&0\\0&1\end{bmatrix} $$Applyingtheidentitygateanywhereinyourcircuitshouldhavenoeffectonthequbitstate,soit’sinterestingthisisevenconsideredagate.Therearetwomainreasonsbehindthis,oneisthatitisoftenusedincalculations,forexample:provingtheX-gateisitsowninverse: $$I=XX$$Thesecond,isthatitisoftenusefulwhenconsideringrealhardwaretospecifya‘do-nothing’or‘none’operation. QuickExercise WhataretheeigenstatesoftheI-gate? 6.2TheS-gatesThenextgatetomentionistheS-gate(sometimesknownasthe$\sqrt{Z}$-gate),thisisaP-gatewith$\phi=\pi/2$.Itdoesaquarter-turnaroundtheBlochsphere.Itisimportanttonotethatunlikeeverygateintroducedinthischaptersofar,theS-gateisnotitsowninverse!Asaresult,youwilloftenseetheS†-gate,(also“S-dagger”,“Sdg”or$\sqrt{Z}^\dagger$-gate).TheS†-gateisclearlyanP-gatewith$\phi=-\pi/2$: $$S=\begin{bmatrix}1&0\\0&e^{\frac{i\pi}{2}}\end{bmatrix},\quadS^\dagger=\begin{bmatrix}1&0\\0&e^{-\frac{i\pi}{2}}\end{bmatrix}$$Thename"$\sqrt{Z}$-gate"isduetothefactthattwosuccessivelyappliedS-gateshasthesameeffectasoneZ-gate: $$SS|q\rangle=Z|q\rangle$$Thisnotationiscommonthroughoutquantumcomputing. ToaddanS-gateinQiskit: qc=QuantumCircuit(1) qc.s(0)#ApplyS-gatetoqubit0 qc.sdg(0)#ApplySdg-gatetoqubit0 qc.draw() 6.3TheT-gateTheT-gateisaverycommonlyusedgate,itisanP-gatewith$\phi=\pi/4$: $$T=\begin{bmatrix}1&0\\0&e^{\frac{i\pi}{4}}\end{bmatrix},\quadT^\dagger=\begin{bmatrix}1&0\\0&e^{-\frac{i\pi}{4}}\end{bmatrix}$$AswiththeS-gate,theT-gateissometimesalsoknownasthe$\sqrt[4]{Z}$-gate. InQiskit: qc=QuantumCircuit(1) qc.t(0)#ApplyT-gatetoqubit0 qc.tdg(0)#ApplyTdg-gatetoqubit0 qc.draw() Youcanusethewidgetbelowtoplayaroundwithallthegatesintroducedinthischaptersofar: #Runthecodeinthiscelltoseethewidget fromqiskit_textbook.widgetsimportgate_demo gate_demo() 7.TheU-gateAswesawearlier,theI,Z,S&T-gateswereallspecialcasesofthemoregeneralP-gate.Inthesameway,theU-gateisthemostgeneralofallsingle-qubitquantumgates.Itisaparametrisedgateoftheform: $$ U(\theta,\phi,\lambda)=\begin{bmatrix}\cos(\frac{\theta}{2})&-e^{i\lambda}\sin(\frac{\theta}{2})\\ e^{i\phi}\sin(\frac{\theta}{2})&e^{i(\phi+\lambda)}\cos(\frac{\theta}{2}) \end{bmatrix} $$Everygateinthischaptercouldbespecifiedas$U(\theta,\phi,\lambda)$,butitisunusualtoseethisinacircuitdiagram,possiblyduetothedifficultyinreadingthis. Asanexample,weseesomespecificcasesoftheU-gateinwhichitisequivalenttotheH-gateandP-gaterespectively. $$ \begin{aligned} U(\tfrac{\pi}{2},0,\pi)=\tfrac{1}{\sqrt{2}}\begin{bmatrix}1&1\\ 1&-1 \end{bmatrix}=H &\quad& U(0,0,\lambda)=\begin{bmatrix}1&0\\ 0&e^{i\lambda}\\ \end{bmatrix}=P \end{aligned} $$ #Let'shaveU-gatetransforma|0>to|+>state qc=QuantumCircuit(1) qc.u(pi/2,0,pi,0) qc.draw() #Let'sseetheresult qc.save_statevector() qobj=assemble(qc) state=sim.run(qobj).result().get_statevector() plot_bloch_multivector(state) Itshouldbeobviousfromthisthatthereareaninfinitenumberofpossiblegates,andthatthisalsoincludesRxandRy-gates,althoughtheyarenotmentionedhere.ItmustalsobenotedthatthereisnothingspecialabouttheZ-basis,exceptthatithasbeenselectedasthestandardcomputationalbasis.QiskitalsoprovidestheXequivalentoftheSandSdg-gatei.e.theSX-gateandSXdg-gaterespectively.Thesegatesdoaquarter-turnwithrespecttotheX-axisaroundtheBlocksphereandareaspecialcaseoftheRx-gate. BeforerunningonrealIBMquantumhardware,allsingle-qubitoperationsarecompileddownto$I$,$X$,$SX$and$R_{z}$.Forthisreasontheyaresometimescalledthephysicalgates. 8.AdditionalresourcesYoucanfindacommunity-createdcheat-sheetwithsomeofthecommonquantumgates,andtheirpropertieshere. importqiskit.tools.jupyter %qiskit_version_table VersionInformationQiskitSoftwareVersionQiskit0.27.0Terra0.17.4Aer0.8.2Ignis0.6.0Aqua0.9.2IBMQProvider0.14.0SysteminformationPython3.8.10|packagedbyconda-forge|(default,May112021,07:01:05) [GCC9.3.0]OSLinuxCPUs8Memory(Gb)31.40896224975586TueJul1302:37:132021UTC 〈RepresentingQubitStates TheCaseforQuantum〉