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Jack D. Hidary Quantum Computing: An Applied Approach Second Edition Quantum Computing: An Applied Approach Jack D. Hidary Quantum Computing: An Applied Approach Second Edition Jack D. Hidary Palo Alto, CA, USA ISBN 978-3-030-83273-5 ISBN 978-3-030-83274-2 (eBook ) https://doi.org/10.1007/978-3-030-83274-2 1st edition: © Jack D. Hidary under exclusive license to Springer Nature Switzerland AG 2019 2nd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. 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This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Contents PrefacetotheSecondEdition xiii PrefacetotheFirstEdition xv Acknowledgements xix NavigatingthisBook xxi I Foundations 1 Superposition,EntanglementandReversibility 3 1.1 SuperpositionandEntanglement 4 1.2 TheBornRule 5 1.3 Schrödinger’sEquation 8 1.4 ThePhysicsofComputation 11 2 ABriefHistoryofQuantumComputing 15 2.1 EarlyDevelopmentsandAlgorithms 16 2.2 ShorandGrover 18 2.3 DefiningaQuantumComputer 19 3 Qubits,OperatorsandMeasurement 23 3.1 QuantumOperators 28 UnaryOperators 28 BinaryOperators 32 TernaryOperators 34 v vi Contents 3.2 ComparisonwithClassicalGates 36 3.3 UniversalityofQuantumOperators 37 3.4 Gottesman-KnillandSolovay-Kitaev 37 3.5 TheBlochSphere 38 3.6 TheMeasurementPostulate 39 3.7 Computation-in-Place 41 4 ComplexityTheory 43 4.1 Problemsvs. Algorithms 43 4.2 TimeComplexity 44 4.3 ComplexityClasses 46 4.4 QuantumComputingandtheChurch-TuringThesis 49 II Hardware and Applications 5 BuildingaQuantumComputer 53 5.1 AssessingaQuantumComputer 54 5.2 NeutralAtoms 55 5.3 NMR 56 5.4 NVCenter-in-Diamond 57 5.5 Photonics 58 5.6 SpinQubits 60 5.7 SuperconductingQubits 61 5.8 TopologicalQuantumComputation 63 5.9 TrappedIon 64 5.10 Summary 65 Contents vii 6 DevelopmentLibrariesforQuantumComputerProgramming 67 6.1 QuantumComputersandQCSimulators 68 6.2 Cirq 70 6.3 Qiskit 72 6.4 Forest 75 6.5 QuantumDevelopmentKit 77 6.6 DevLibrariesSummary 80 UsingtheLibraries 81 OtherDevelopmentLibraries 81 6.7 AdditionalQuantumPrograms 82 BellStates 82 GateswithParameters 84 7 Teleportation,SuperdenseCodingandBell’sInequality 87 7.1 QuantumTeleportation 87 7.2 SuperdenseCoding 90 7.3 Code for Quantum Teleportation and Superdense Com- munication 91 7.4 BellInequalityTest 94 8 TheCanon: CodeWalkthroughs 101 8.1 TheDeutsch-JozsaAlgorithm 103 8.2 TheBernstein-VaziraniAlgorithm 110 8.3 Simon’sProblem 113 8.4 QuantumFourierTransform 114 8.5 Shor’sAlgorithm 117 RSACryptography 117 ThePeriodofaFunction 119 PeriodofaFunctionasanInputtoaFactorizationAlgo- rithm 121 viii Contents 8.6 Grover’sSearchAlgorithm 135 Grover’sAlgorithminQiskit 140 3-QubitGrover’sAlgo 141 9 QuantumComputingMethods 143 9.1 VariationalQuantumEigensolver 143 VQEwithNoise 148 MoreSophisticatedAnsatzes 150 9.2 QuantumChemistry 151 9.3 QuantumApproximateOptimizationAlgorithm(QAOA) 156 ExampleImplementationofQAOA 159 9.4 MachineLearningonQuantumProcessors 167 9.5 QuantumPhaseEstimation 174 ImplementionofQPE 177 9.6 SolvingLinearSystems 180 DescriptionoftheHHLAlgorithm 182 ExampleImplementationoftheHHLAlgorithm 184 9.7 QuantumRandomNumberGenerator 192 9.8 QuantumWalks 194 ImplementationofaQuantumWalk 196 9.9 UnificationFrameworkforQuantumAlgorithms(QSVT) 202 9.10 Dequantization 203 9.11 Summary 205 10 ApplicationsandQuantumSupremacy 207 10.1 Applications 207 QuantumSimulationandChemistry 207 SamplingfromProbabilityDistributions 208 LinearAlgebraSpeedupwithQuantumComputers 208 Optimization 208 TensorNetworks 208 Contents ix 10.2 QuantumSupremacy 208 RandomCircuitSampling 209 OtherProblemsforDemonstratingQuantumSupremacy 214 Quantum Advantage and Beyond Classical Computation 214 10.3 QuantumErrorCorrection 215 ContextandImportance 215 ImportantPreliminaries 216 MotivatingExample: TheRepetitionCode 217 TheStabilizerFormalism 219 10.4 DoingPhysicswithQuantumComputers 224 III Toolkit 11 MathematicalToolsforQuantumComputingI 227 11.1 IntroductionandSelf-Test 227 11.2 LinearAlgebra 229 Vectors 229 IntroductiontoDiracNotation 230 BasicVectorOperations 231 TheNormofaVector 234 TheDotProduct 237 11.3 TheComplexNumbersandtheInnerProduct 240 ComplexNumbers 240 TheInnerProductasaRefinementoftheDotProduct 242 ThePolarCoordinateRepresentationofaComplexNum- ber 246 11.4 AFirstLookatMatrices 254 BasicMatrixOperations 254 TheIdentityMatrix 261 Transpose,ConjugateandTrace 263 MatrixExponentiation 270 11.5 TheOuterProductandtheTensorProduct 271 TheOuterProductasaWayofBuildingMatrices 271 x Contents TheTensorProduct 273 11.6 SetTheory 276 TheBasicsofSetTheory 276 TheCartesianProduct 279 RelationsandFunctions 280 ImportantPropertiesofFunctions 286 11.7 TheDefinitionofaLinearTransformation 291 11.8 HowtoBuildaVectorSpaceFromScratch 293 Groups 294 Rings 300 Fields 305 TheDefinitionofaVectorSpace 308 Subspaces 311 11.9 Span,LinearIndependence,BasesandDimension 313 Span 313 LinearIndependence 315 BasesandDimension 317 OrthonormalBases 320 12 MathematicalToolsforQuantumComputingII 323 12.1 LinearTransformationsasMatrices 323 12.2 MatricesasOperators 328 AnIntroductiontotheDeterminant 328 TheGeometryoftheDeterminant 332 MatrixInversion 334 12.3 EigenvectorsandEigenvalues 341 ChangeofBasis 344 12.4 FurtherInvestigationofInnerProducts 346 TheKroneckerDeltaFunctionasanInnerProduct 349 12.5 HermitianOperators 350 WhyWeCan’tMeasurewithComplexNumbers 350 HermitianOperatorsHaveRealEigenvalues 352