Qubit basis states - Quantum Inspire

Single-qubit computational basis states ... The two orthogonal z-basis states of a qubit are defined as: ... When we talk about the qubit basis states we implicitly ... SelectapageKnowledgebasemenu Introductiontoquantumcomputing ThebasicsofQuantumComputing Whatisaqubit? Superpositionandentanglement Whatisaquantumalgorithm? Helloquantumworld Accounts Quickguide Introductionquickguide Workingwiththeeditor Executingyouralgorithm Displayinganddownloadingyourresults Creatinganewproject Managingyourprojects Managingyouraccount Qubitregister Binaryregister Rotationoperators Advanceduserguide Introductionadvancedguide SoftwareDevelopmentKit Low-levelAPI cQASM:AQuantumProgrammingLanguage cQASM:Qubitgateoperations cQASM:SingleGateMultiple-Qubits cQASM:Qubitinitializationandmeasurement cQASM:Displaycommands LibKet Codeexamples Codeexample:Deutsch-Jozsaalgorithm Codeexample:Quantumfulladder Codeexample:Grover'salgorithm Codeexample:Repetitioncode Codeexample:Quantumclassification Codeexample:SAT Hardwarebackendtopics Hardwarebackends Spin-2:Operationalspecifics Starmon-5:Operationalspecifics Emulatorbackendtopics Emulatorbackends cQASM:Binarycontrolledgates cQASM:Displaycommands cQASM:Errormodels Optimizationofsimulations Resettingqubits cQASMinstructions Qubits Prep_z Prep_y Prep_x Pauli-Xgate Pauli-Ygate Pauli-Zgate Hadamardgate Identitygate RxGate RyGate RzGate 90-degreerotations SGate SdaggerGate TGate TdaggerGate CNOTGate CZGate SwapGate CRgate CRkgate ToffoliGate Measure_z Measure_y Measure_x Measure_all Display Display_binary Binarycontrolledgates Not QubitbasisstatesSingle-qubitcomputationalbasisstatesThetwoorthogonalz-basisstatesofaqubitaredefinedas: ∣0⟩\vert0\rangle∣0⟩ ∣1⟩\vert1\rangle∣1⟩ Whenwetalkaboutthequbitbasisstatesweimplicitlyrefertothez-basisstatesasthecomputationalbasisstates. Thetwoorthogonalx-basisstatesare: ∣+⟩=∣0⟩+∣1⟩2\vert+\rangle=\frac{\vert0\rangle+\vert1\rangle}{\sqrt{2}}∣+⟩=2​∣0⟩+∣1⟩​ ∣−⟩=∣0⟩−∣1⟩2\vert-\rangle=\frac{\vert0\rangle-\vert1\rangle}{\sqrt{2}}∣−⟩=2​∣0⟩−∣1⟩​ Thetwoorthogonaly-basisstatesare: ∣R⟩=∣0⟩+ı∣1⟩2\vertR\rangle=\frac{\vert0\rangle+\imath\vert1\rangle}{\sqrt{2}}∣R⟩=2​∣0⟩+ı∣1⟩​ ∣L⟩=∣0⟩−ı∣1⟩2\vertL\rangle=\frac{\vert0\rangle-\imath\vert1\rangle}{\sqrt{2}}∣L⟩=2​∣0⟩−ı∣1⟩​ ThebasisstatesarelocatedatoppositepointsontheBlochsphere: Blochspherecourtesyofhttp://www.laborsciencenetwork.com Multi-qubitcomputationalbasisstatesAsingle-qubithastwocomputationalbasisstates.Inthez-basistheseare∣0⟩\vert0\rangle∣0⟩and∣1⟩\vert1\rangle∣1⟩.Atwo-qubitsystemhas4computationalbasisstatesdenotedas∣00⟩\vert00\rangle∣00⟩,∣01⟩\vert01\rangle∣01⟩,∣10⟩\vert10\rangle∣10⟩,∣11⟩\vert11\rangle∣11⟩. Amulti-qubitsystemofNqubitshas2N2^{N}2Ncomputationalbasisstatesdenotedas∣00...00⟩\vert00...00\rangle∣00...00⟩,∣00⋯01⟩\vert00\cdots01\rangle∣00⋯01⟩,∣00⋯10⟩\vert00\cdots10\rangle∣00⋯10⟩...∣11⋯11⟩\vert11\cdots11\rangle∣11⋯11⟩. ProbabilityamplitudesAssociatedwitheachcomputationalbasisstateisaprobabilityamplitudeαi\alpha_{i}αi​,whichisacomplexnumber. Asanexample,asystemofthreequbitsisdescribedbytheexpression: ∣Ψ⟩=α0∣000⟩+α1∣001⟩+α2∣010⟩+⋯+α7∣111⟩\lvert\Psi\rangle=\alpha_{0}\lvert000\rangle+\alpha_{1}\lvert001\rangle+\alpha_{2}\lvert010\rangle+\cdots+\alpha_{7}\lvert111\rangle∣Ψ⟩=α0​∣000⟩+α1​∣001⟩+α2​∣010⟩+⋯+α7​∣111⟩ whereαi\alpha_{i}αi​aretheprobabilityamplitudesassociatedtothecomputationalbasisstates. InitializationandmeasurementbasesBydefault,allqubitsareinitializedinthe∣0⟩|0\rangle∣0⟩stateinthez-basis. StateinitializationinaspecificbasiscanbedoneexplicitlywiththecQASMinstructionsprep_z,prep_yandprep_x,whichpreparequbitsinthe∣0⟩\vert0\rangle∣0⟩,∣R⟩\vertR\rangle∣R⟩and∣+⟩\vert+\rangle∣+⟩statesrespectively. Bydefault,qubitsaremeasuredwiththemeasureormeasure_allinstructioninthez-basis. QubitmeasurementinaspecificbasiscanbedoneexplicitlywiththecQASMinstructionsmeasure_x,measure_yandmeasure_z. Declaredstates Whenaqubitisinthe∣0⟩\vert0\rangle∣0⟩state(∣1⟩\vert1\rangle∣1⟩state),ameasurementinthez-basiswillresultin0(1) Whenaqubitisinthe∣R⟩\vertR\rangle∣R⟩state(∣L⟩\vertL\rangle∣L⟩state),ameasurementinthey-basiswillresultin0(1) Whenaqubitisinthe∣+⟩\vert+\rangle∣+⟩state(∣−⟩\vert-\rangle∣−⟩state),ameasurementinthex-basiswillresultin0(1) Notes∣R⟩\vertR\rangle∣R⟩and∣L⟩\vertL\rangle∣L⟩standforRightandLeft.Othernotationsthatareoftenusedforthesestatesare∣ı⟩\vert\imath\rangle∣ı⟩and∣−ı⟩\vert-\imath\rangle∣−ı⟩.