Principles of Superconducting Quantum Computers

365 Pages·2022·26.213 MB·English

by  Daniel D. Stancil

PrinciplesofSuperconductingQuantumComputers
Principles of Superconducting Quantum
Computers
Daniel D. Stancil
NorthCarolinaStateUniversity
Raleigh,NorthCarolina
Gregory T. Byrd
NorthCarolinaStateUniversity
Raleigh,NorthCarolina
Thiseditionfirstpublished2022
©2022byJohnWiley&Sons,Inc.
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or
transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,
exceptaspermittedbylaw.Adviceonhowtoobtainpermissiontoreusematerialfromthistitleisavailableat
hhttp://www.wiley.com/go/permissions.
TherightofDanielD.StancilandGregoryT.Byrdtobeidentifiedastheauthorsofthisworkhasbeen
assertedinaccordancewithlaw.
RegisteredOffice
JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030,USA
EditorialOffice
111RiverStreet,Hoboken,NJ07030,USA
Fordetailsofourglobaleditorialoffices,customerservices,andmoreinformationaboutWileyproductsvisit
usathwww.wiley.com.
Wileyalsopublishesitsbooksinavarietyofelectronicformatsandbyprint-on-demand.Somecontentthat
appearsinstandardprintversionsofthisbookmaynotbeavailableinotherformats.
LimitofLiability/DisclaimerofWarranty
Thecontentsofthisworkareintendedtofurthergeneralscientificresearch,understanding,anddiscussion
onlyandarenotintendedandshouldnotberelieduponasrecommendingorpromotingscientificmethod,
diagnosis,ortreatmentbyphysiciansforanyparticularpatient.Inviewofongoingresearch,equipment
modifications,changesingovernmentalregulations,andtheconstantflowofinformationrelatingtotheuse
ofmedicines,equipment,anddevices,thereaderisurgedtoreviewandevaluatetheinformationprovidedin
thepackageinsertorinstructionsforeachmedicine,equipment,ordevicefor,amongotherthings,any
changesintheinstructionsorindicationofusageandforaddedwarningsandprecautions.Whilethe
publisherandauthorshaveusedtheirbesteffortsinpreparingthiswork,theymakenorepresentationsor
warrantieswithrespecttotheaccuracyorcompletenessofthecontentsofthisworkandspecificallydisclaim
allwarranties,includingwithoutlimitationanyimpliedwarrantiesofmerchantabilityorfitnessfora
particularpurpose.Nowarrantymaybecreatedorextendedbysalesrepresentatives,writtensalesmaterials
orpromotionalstatementsforthiswork.Thefactthatanorganization,website,orproductisreferredtoin
thisworkasacitationand/orpotentialsourceoffurtherinformationdoesnotmeanthatthepublisherand
authorsendorsetheinformationorservicestheorganization,website,orproductmayprovideor
recommendationsitmaymake.Thisworkissoldwiththeunderstandingthatthepublisherisnotengaged
inrenderingprofessionalservices.Theadviceandstrategiescontainedhereinmaynotbesuitableforyour
situation.Youshouldconsultwithaspecialistwhereappropriate.Further,readersshouldbeawarethat
websiteslistedinthisworkmayhavechangedordisappearedbetweenwhenthisworkwaswrittenand
whenitisread.Neitherthepublishernorauthorsshallbeliableforanylossofprofitoranyother
commercialdamages,includingbutnotlimitedtospecial,incidental,consequential,orotherdamages.
AcataloguerecordforthisbookisavailablefromtheLibraryofCongress
HardbackISBN:9781119750727;ePubISBN:9781119750741;ePDFISBN:9781119750734;
ObookISBN:9781119750758
Coverimage:©Adaptrographics/GettyImages
CoverdesignbyWiley
Setin9.5/12.5ptSTIXTwoTextbyIntegraSoftwareServicesPvt.Ltd,Pondicherry,India
10 9 8 7 6 5 4 3 2 1
DedicatedtothepioneersofthefirstQuantumRevolution,whopavedtheway.
vii
Contents
ListofFigures xiii
ListofTables xxv
Preface xxvii
Acknowledgments xxix
AbouttheCompanionWebsite xxxi
1 Qubits,Gates,andCircuits 1
1.1 BitsandQubits 1
1.1.1 CircuitsinSpacevs.CircuitsinTime 1
1.1.2 Superposition 1
1.1.3 NoCloning 3
1.1.4 Reversibility 3
1.1.5 Entanglement 3
1.2 Single-QubitStates 4
1.3 MeasurementandtheBornRule 5
1.4 UnitaryOperationsandSingle-QubitGates 6
1.5 Two-QubitGates 8
1.5.1 Two-QubitStates 8
1.5.2 MatrixRepresentationofTwo-QubitGates 9
1.5.3 Controlled-NOT 11
1.6 BellState 12
1.7 NoCloning,Revisited 13
1.8 Example:Deutsch’sProblem 15
1.9 KeyCharacteristicsofQuantumComputing 18
1.10 QuantumComputingSystems 18
1.11 Exercises 22
2 PhysicsofSingleQubitGates 25
2.1 RequirementsforaQuantumComputer 25
2.2 SingleQubitGates 25
2.2.1 Rotations 25
2.2.2 TwoStateSystems 33
2.2.3 CreatingRotations:RabiOscillations 38
2.3 QuantumStateTomography 42
2.4 ExpectationValuesandthePauliOperators 44
viii Contents
2.5 DensityMatrix 45
2.6 Exercises 48
3 P√hysicsofTwoQubitGates 51
3.1 iSWAPGate 51
3.2 CoupledTunableQubits 53
3.3 CrossResonanceScheme 55
3.4 OtherControlledGates 57
3.5 Two-QubitStatesandtheDensityMatrix 59
3.6 Exercises 62
4 SuperconductingQuantumComputerSystems 63
4.1 TransmissionLines 63
4.1.1 GeneralTransmissionLineEquations 63
4.1.2 LosslessTransmissionLines 65
4.1.3 TransmissionLineswithLoss 67
4.2 TerminatedLosslessLine 71
4.2.1 ReflectionCoefficient 71
4.2.2 Power(FlowofEnergy)andReturnLoss 72
4.2.3 StandingWaveRatio(SWR) 73
4.2.4 ImpedanceasaFunctionofPosition 74
4.2.5 QuarterWaveTransformer 76
4.2.6 Coaxial,Microstrip,andCoplanarLines 77
4.3 SParameters 80
4.3.1 LosslessCondition 81
4.3.2 Reciprocity 81
4.4 Transmission(ABCD)Matrices 81
4.5 Attenuators 85
4.6 CirculatorsandIsolators 87
4.7 PowerDividers/Combiners 89
4.8 Mixers 92
4.9 Low-PassFilters 95
4.10 Noise 97
4.10.1 ThermalNoise 97
4.10.2 EquivalentNoiseTemperature 99
4.10.3 NoiseFactorandNoiseFigure 100
4.10.4 AttenuatorsandNoise 101
4.10.5 NoiseinCascadedSystems 103
4.11 LowNoiseAmplifiers 104
4.12 Exercises 105
5 Resonators:ClassicalTreatment 107
5.1 ParallelLumpedElementResonator 107
5.2 CapacitiveCouplingtoaParallelLumped-ElementResonator 109
5.3 TransmissionLineResonator 111
5.4 CapacitiveCouplingtoaTransmissionLineResonator 113
5.5 Capacitively-CoupledLosslessResonators 117
Contents ix
5.6 ClassicalModelofQubitReadout 120
5.7 Exercises 124
6 Resonators:QuantumTreatment 127
6.1 LagrangianMechanics 127
6.1.1 Hamilton’sPrinciple 127
6.1.2 CalculusofVariations 128
6.1.3 LagrangianEquationofMotion 129
6.2 HamiltonianMechanics 130
6.3 HarmonicOscillators 131
6.3.1 ClassicalHarmonicOscillator 131
6.3.2 QuantumMechanicalHarmonicOscillator 133
6.3.3 RaisingandLoweringOperators 135
6.3.4 CanaHarmonicOscillatorBeUsedasaQubit? 137
6.4 CircuitQuantumElectrodynamics 138
6.4.1 ClassicalLCResonantCircuit 138
6.4.2 QuantizationoftheLCCircuit 139
6.4.3 CircuitElectrodynamicApproachforGeneralCircuits 140
6.4.4 CircuitModelforTransmissionLineResonator 141
6.4.5 QuantizingaTransmissionLineResonator 144
6.4.6 QuantizedCoupledLCResonantCircuits 144
6.4.7 Schrödinger,Heisenberg,andInteractionPictures 147
6.4.8 ResonantCircuitsandQubits 150
6.4.9 TheDispersiveRegime 153
6.5 Exercises 156
7 TheoryofSuperconductivity 159
7.1 BosonsandFermions 159
7.2 BlochTheorem 161
7.3 FreeElectronModelforMetals 163
7.3.1 DiscreteStatesinFiniteSamples 163
7.3.2 Phonons 166
7.3.3 DebyeModel 167
7.3.4 Electron–PhononScatteringandElectricalConductivity 168
7.3.5 PerfectConductorvs.Superconductor 170
7.4 Bardeen,Cooper,andSchriefferTheoryofSuperconductivity 172
7.4.1 CooperPairModel 172
7.4.2 DielectricFunction 175
7.4.3 Jellium 176
7.4.4 ScatteringAmplitudeandAttractiveElectron–ElectronInteraction 179
7.4.5 InterpretationofAttractiveInteraction 180
7.4.6 SuperconductorHamiltonian 181
7.4.7 SuperconductingGroundState 182
7.5 ElectrodynamicsofSuperconductors 185
7.5.1 CooperPairsandtheMacroscopicWaveFunction 185
7.5.2 PotentialFunctions 186
7.5.3 LondonEquations 187
7.5.4 LondonGauge 189
x Contents
7.5.5 PenetrationDepth 190
7.5.6 FluxQuantization 191
7.6 ChapterSummary 192
7.7 Exercises 193
8 JosephsonJunctions 195
8.1 Tunneling 195
8.1.1 ReflectionfromaBarrier 196
8.1.2 FiniteThicknessBarrier 198
8.2 JosephsonJunctions 200
8.2.1 CurrentandVoltageRelations 200
8.2.2 JosephsonJunctionHamiltonian 203
8.2.3 QuantizedJosephsonJunctionAnalysis 205
8.3 SuperconductingQuantumInterferenceDevices(SQUIDs) 207
8.4 JosephsonJunctionParametricAmplifiers 208
8.5 Exercises 209
9 ErrorsandErrorMitigation 211
9.1 NISQProcessors 211
9.2 Decoherence 212
9.3 StatePreparationandMeasurementErrors 214
9.4 CharacterizingGateErrors 215
9.5 StateLeakageandSuppressionUsingPulseShaping 219
9.6 Zero-NoiseExtrapolation 220
9.7 OptimizedControlUsingDeepLearning 223
9.8 Exercises 225
10 QuantumErrorCorrection 227
10.1 ReviewofClassicalErrorCorrection 227
10.1.1 ErrorDetection 228
10.1.2 ErrorCorrection:RepetitionCode 228
10.1.3 HammingCode 229
10.2 QuantumErrors 230
10.3 DetectingandCorrectingQuantumErrors 232
10.3.1 BitFlip 232
10.3.2 PhaseFlip 234
10.3.3 CorrectingBitandPhaseFlips:Shor’s9-QubitCode 235
10.3.4 ArbitraryRotations 236
10.4 StabilizerCodes 238
10.4.1 Stabilizers 238
10.4.2 StabilizersforErrorCorrection 239
10.5 OperatingonLogicalQubits 242
10.6 ErrorThresholds 243
10.6.1 ConcatenationofErrorCodes 243
10.6.2 ThresholdTheorem 244
10.7 SurfaceCodes 245
10.7.1 Stabilizers 246
Contents xi
10.7.2 ErrorDetectionandCorrection 247
10.7.3 LogicalX andZOperators 250
10.7.4 MultipleQubits:LatticeSurgery 253
10.7.5 CNOT 257
10.7.6 Single-QubitGates 258
10.8 SummaryandFurtherReading 259
10.9 Exercises 261
11 QuantumLogic:EfficientImplementationofClassicalComputations 263
11.1 ReversibleLogic 264
11.1.1 ReversibleLogicGates 264
11.1.2 ReversibleLogicCircuits 266
11.2 QuantumLogicCircuits 268
11.2.1 EntanglementandUncomputing 269
11.2.2 Multi-QubitGates 270
11.2.3 QubitTopology 270
11.3 EfficientArithmeticCircuits:Adder 272
11.3.1 QuantumRipple-CarryAdder 273
11.3.2 In-PlaceRipple-CarryAdder 275
11.3.3 Carry-LookaheadAdder 277
11.3.4 AdderComparison 281
11.4 PhaseLogic 283
11.4.1 Controlled-𝑍andControlled-PhaseGates 283
11.4.2 SelectivePhaseChange 285
11.4.3 PhaseLogicGates 287
11.5 SummaryandFurtherReading 288
11.6 Exercises 289
12 SomeQuantumAlgorithms 291
12.1 ComputationalComplexity 291
12.1.1 QuantumProgramRun-Time 292
12.1.2 ClassicalComplexityClasses 292
12.1.3 QuantumComplexity 293
12.2 Grover’sSearchAlgorithm 294
12.2.1 GroverIteration 294
12.2.2 QuantumImplementation 296
12.2.3 Generalizations 299
12.3 QuantumFourierTransform 299
12.3.1 DiscreteFourierTransform 300
12.3.2 InverseDiscreteFourierTransform 300
12.3.3 QuantumImplementationoftheDFT 301
12.3.4 EncodingQuantumStates 302
12.3.5 QuantumImplementation 304
12.3.6 ComputationalComplexity 306
12.4 QuantumPhaseEstimation 307
12.4.1 QuantumImplementation 307
12.4.2 ComputationalComplexityandOtherIssues 308
12.5 Shor’sAlgorithm 309