Quantum Computing: A Gentle Introduction

389 Pages·2011·2.4 MB·English

by  Rieffel E.G., Polak W.H.

QUANTUM COMPUTING
ScientificandEngineeringComputation
WilliamGroppandEwingLusk,editors;JanuszKowalik,foundingeditor
Acompletelistofthebooksinthisseriescanbefoundatthebackofthisbook.
QUANTUM COMPUTING
AGentleIntroduction
EleanorRieffelandWolfgangPolak
TheMITPress
Cambridge,Massachusetts
London,England
©2011MassachusettsInstituteofTechnology
Allrightsreserved.Nopartofthisbookmaybereproducedinanyformbyanyelectronicormechanicalmeans(including
photocopying,recording,orinformationstorageandretrieval)withoutpermissioninwritingfromthepublisher.
Forinformationaboutspecialquantitydiscounts,[email protected]
ThisbookwassetinSyntaxandTimesRomanbyWestchesterBookGroup.PrintedandboundintheUnitedStatesof
America.
LibraryofCongressCataloging-in-PublicationData
Rieffel,Eleanor,1965–
Quantumcomputing:agentleintroduction/EleanorRieffelandWolfgangPolak.
p. cm.—(Scientificandengineeringcomputation)
Includesbibliographicalreferencesandindex.
ISBN978-0-262-01506-6(hardcover: alk. paper) 1. Quantumcomputers. 2. Quantumtheory. I.Polak, Wolfgang,
1950– II.Title.
QA76.889.R54 2011
004.1—dc22
2010022682
10 9 8 7 6 5 4 3 2 1
Contents
Preface xi
1 Introduction 1
I QUANTUMBUILDINGBLOCKS 7
2 Single-QubitQuantumSystems 9
2.1 TheQuantumMechanicsofPhotonPolarization 9
2.1.1 ASimpleExperiment 10
2.1.2 AQuantumExplanation 11
2.2 SingleQuantumBits 13
2.3 Single-QubitMeasurement 16
2.4 AQuantumKeyDistributionProtocol 18
2.5 TheStateSpaceofaSingle-QubitSystem 21
2.5.1 RelativePhasesversusGlobalPhases 21
2.5.2 GeometricViewsoftheStateSpaceofaSingleQubit 23
2.5.3 CommentsonGeneralQuantumStateSpaces 25
2.6 References 25
2.7 Exercises 26
3 Multiple-QubitSystems 31
3.1 QuantumStateSpaces 32
3.1.1 DirectSumsofVectorSpaces 32
3.1.2 TensorProductsofVectorSpaces 33
3.1.3 TheStateSpaceofann-QubitSystem 34
3.2 EntangledStates 38
3.3 BasicsofMulti-QubitMeasurement 41
3.4 QuantumKeyDistributionUsingEntangledStates 43
3.5 References 44
3.6 Exercises 44
4 MeasurementofMultiple-QubitStates 47
4.1 Dirac’sBra/KetNotationforLinearTransformations 47
4.2 ProjectionOperatorsforMeasurement 49
vi Contents
4.3 HermitianOperatorFormalismforMeasurement 53
4.3.1 TheMeasurementPostulate 55
4.4 EPRParadoxandBell’sTheorem 60
4.4.1 SetupforBell’sTheorem 62
4.4.2 WhatQuantumMechanicsPredicts 62
4.4.3 SpecialCaseofBell’sTheorem:WhatAnyLocalHiddenVariableTheoryPredicts 63
4.4.4 Bell’sInequality 64
4.5 References 65
4.6 Exercises 66
5 QuantumStateTransformations 71
5.1 UnitaryTransformations 72
5.1.1 ImpossibleTransformations:TheNo-CloningPrinciple 73
5.2 SomeSimpleQuantumGates 74
5.2.1 ThePauliTransformations 75
5.2.2 TheHadamardTransformation 76
5.2.3 Multiple-QubitTransformationsfromSingle-QubitTransformations 76
5.2.4 TheControlled-NOTandOtherSinglyControlledGates 77
5.3 ApplicationsofSimpleGates 80
5.3.1 DenseCoding 81
5.3.2 QuantumTeleportation 82
5.4 RealizingUnitaryTransformationsasQuantumCircuits 84
5.4.1 DecompositionofSingle-QubitTransformations 84
5.4.2 Singly-ControlledSingle-QubitTransformations 86
5.4.3 Multiply-ControlledSingle-QubitTransformations 87
5.4.4 GeneralUnitaryTransformations 89
5.5 AUniversallyApproximatingSetofGates 91
5.6 TheStandardCircuitModel 93
5.7 References 93
5.8 Exercises 94
6 QuantumVersionsofClassicalComputations 99
6.1 FromReversibleClassicalComputationstoQuantumComputations 99
6.1.1 ReversibleandQuantumVersionsofSimpleClassicalGates 101
6.2 ReversibleImplementationsofClassicalCircuits 103
6.2.1 ANaiveReversibleImplementation 103
6.2.2 AGeneralConstruction 106
6.3 ALanguageforQuantumImplementations 110
6.3.1 TheBasics 111
6.3.2 Functions 112
6.4 SomeExampleProgramsforArithmeticOperations 115
6.4.1 EfficientImplementationofAND 115
6.4.2 EfficientImplementationofMultiply-ControlledSingle-QubitTransformations 116
6.4.3 In-PlaceAddition 117
6.4.4 ModularAddition 117
6.4.5 ModularMultiplication 118
6.4.6 ModularExponentiation 119
Contents vii
6.5 References 120
6.6 Exercises 121
II QUANTUMALGORITHMS 123
7 IntroductiontoQuantumAlgorithms 125
7.1 ComputingwithSuperpositions 126
7.1.1 TheWalsh-HadamardTransformation 126
7.1.2 QuantumParallelism 128
7.2 NotionsofComplexity 130
7.2.1 QueryComplexity 131
7.2.2 CommunicationComplexity 132
7.3 ASimpleQuantumAlgorithm 132
7.3.1 Deutsch’sProblem 133
7.4 QuantumSubroutines 134
7.4.1 TheImportanceofUnentanglingTemporaryQubitsinQuantumSubroutines 134
7.4.2 PhaseChangeforaSubsetofBasisVectors 135
7.4.3 State-DependentPhaseShifts 138
7.4.4 State-DependentSingle-QubitAmplitudeShifts 139
7.5 AFewSimpleQuantumAlgorithms 140
7.5.1 Deutsch-JozsaProblem 140
7.5.2 Bernstein-VaziraniProblem 141
7.5.3 Simon’sProblem 144
7.5.4 DistributedComputation 145
7.6 CommentsonQuantumParallelism 146
7.7 MachineModelsandComplexityClasses 148
7.7.1 ComplexityClasses 149
7.7.2 Complexity:KnownResults 150
7.8 QuantumFourierTransformations 153
7.8.1 TheClassicalFourierTransform 153
7.8.2 TheQuantumFourierTransform 155
7.8.3 AQuantumCircuitforFastFourierTransform 156
7.9 References 158
7.10 Exercises 159
8 Shor’sAlgorithm 163
8.1 ClassicalReductiontoPeriod-Finding 164
8.2 Shor’sFactoringAlgorithm 164
8.2.1 TheQuantumCore 165
8.2.2 ClassicalExtractionofthePeriodfromtheMeasuredValue 166
8.3 ExampleIllustratingShor’sAlgorithm 167
8.4 TheEfficiencyofShor’sAlgorithm 169
8.5 OmittingtheInternalMeasurement 170
8.6 Generalizations 171
8.6.1 TheDiscreteLogarithmProblem 172
8.6.2 HiddenSubgroupProblems 172
viii Contents
8.7 References 175
8.8 Exercises 176
9 Grover’sAlgorithmandGeneralizations 177
9.1 Grover’sAlgorithm 178
9.1.1 Outline 178
9.1.2 Setup 178
9.1.3 TheIterationStep 180
9.1.4 HowManyIterations? 181
9.2 AmplitudeAmplification 183
9.2.1 TheGeometryofAmplitudeAmplification 185
9.3 OptimalityofGrover’sAlgorithm 188
9.3.1 ReductiontoThreeInequalities 189
9.3.2 ProofsoftheThreeInequalities 191
9.4 DerandomizationofGrover’sAlgorithmandAmplitudeAmplification 193
9.4.1 Approach1:ModifyingEachStep 194
9.4.2 Approach2:ModifyingOnlytheLastStep 194
9.5 UnknownNumberofSolutions 196
9.5.1 VaryingtheNumberofIterations 197
9.5.2 QuantumCounting 198
9.6 PracticalImplicationsofGrover’sAlgorithmandAmplitudeAmplification 199
9.7 References 200
9.8 Exercises 201
III ENTANGLEDSUBSYSTEMSANDROBUSTQUANTUMCOMPUTATION 203
10 QuantumSubsystemsandPropertiesofEntangledStates 205
10.1 QuantumSubsystemsandMixedStates 206
10.1.1 DensityOperators 207
10.1.2 PropertiesofDensityOperators 213
10.1.3 TheGeometryofSingle-QubitMixedStates 215
10.1.4 VonNeumannEntropy 216
10.2 ClassifyingEntangledStates 218
10.2.1 BipartiteQuantumSystems 218
10.2.2 ClassifyingBipartitePureStatesuptoLOCCEquivalence 222
10.2.3 QuantifyingEntanglementinBipartiteMixedStates 224
10.2.4 MultipartiteEntanglement 225
10.3 DensityOperatorFormalismforMeasurement 229
10.3.1 MeasurementofDensityOperators 230
10.4 TransformationsofQuantumSubsystemsandDecoherence 232
10.4.1 Superoperators 233
10.4.2 OperatorSumDecomposition 234
10.4.3 ARelationBetweenQuantumStateTransformationsandMeasurements 238
10.4.4 Decoherence 239
10.5 References 240
10.6 Exercises 240
Contents ix
11 QuantumErrorCorrection 245
11.1 ThreeSimpleExamplesofQuantumErrorCorrectingCodes 246
11.1.1 AQuantumCodeThatCorrectsSingleBit-FlipErrors 246
11.1.2 ACodeforSingle-QubitPhase-FlipErrors 251
11.1.3 ACodeforAllSingle-QubitErrors 252
11.2 FrameworkforQuantumErrorCorrectingCodes 253
11.2.1 ClassicalErrorCorrectingCodes 254
11.2.2 QuantumErrorCorrectingCodes 257
11.2.3 CorrectableSetsofErrorsforClassicalCodes 258
11.2.4 CorrectableSetsofErrorsforQuantumCodes 259
11.2.5 CorrectingErrorsUsingClassicalCodes 261
11.2.6 DiagnosingandCorrectingErrorsUsingQuantumCodes 264
11.2.7 QuantumErrorCorrectionacrossMultipleBlocks 268
11.2.8 ComputingonEncodedQuantumStates 268
11.2.9 SuperpositionsandMixturesofCorrectableErrorsAreCorrectable 269
11.2.10 TheClassicalIndependentErrorModel 270
11.2.11 QuantumIndependentErrorModels 271
11.3 CSSCodes 274
11.3.1 DualClassicalCodes 274
11.3.2 ConstructionofCSSCodesfromClassicalCodesSatisfyingaDualityCondition 275
11.3.3 TheSteaneCode 278
11.4 StabilizerCodes 280
11.4.1 BinaryObservablesforQuantumErrorCorrection 280
11.4.2 PauliObservablesforQuantumErrorCorrection 282
11.4.3 DiagnosingandCorrectingErrors 283
11.4.4 ComputingonEncodedStabilizerStates 285
11.5 CSSCodesasStabilizerCodes 289
11.6 References 290
11.7 Exercises 291
12 FaultToleranceandRobustQuantumComputing 293
12.1 SettingtheStageforRobustQuantumComputation 294
12.2 Fault-TolerantComputationUsingSteane’sCode 297
12.2.1 TheProblemwithSyndromeComputation 297
12.2.2 Fault-TolerantSyndromeExtractionandErrorCorrection 298
12.2.3 Fault-TolerantGatesforSteane’sCode 300
12.2.4 Fault-TolerantMeasurement 303
12.2.5 Fault-TolerantStatePreparationof|π(cid:2)/4(cid:2) 304
12.3 RobustQuantumComputation 305
12.3.1 ConcatenatedCoding 306
12.3.2 AThresholdTheorem 308
12.4 References 310
12.5 Exercises 310
13 FurtherTopicsinQuantumInformationProcessing 311
13.1 FurtherQuantumAlgorithms 311
13.2 LimitationsofQuantumComputing 313