Quantum Algorithms via Linear Algebra: A Primer

207 Pages·2014·1.43 MB·English

by  Richard J. Lipton

QUANTUMALGORITHMSVIALINEARALGEBRA

QUANTUMALGORITHMSVIALINEARALGEBRA
APrimer
RichardJ.Lipton
KennethW.Regan
TheMITPress
Cambridge,Massachusetts
London,England
(cid:2)c 2014MassachusettsInstituteofTechnology
Allrightsreserved.Nopartofthisbookmaybereproducedinanyformorbyanyelectronic
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ThisbookwassetinTimesRomanandMathtimePro2bytheauthors,andwasprintedandbound
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LibraryofCongressCataloging-in-PublicationData
Lipton,RichardJ.,1946–
Quantumalgorithmsvialinearalgebra:aprimer/RichardJ.LiptonandKennethW.Regan.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-262-02839-4(hardcover:alk.paper)
1.Quantumcomputers.2.Computeralgorithms.3.Algebra,Linear.I.Regan,KennethW.,1959–
II.Title
QA76.889.L572014
005.1–dc23
2014016946
10987654321
Wededicatethisbooktoallthosewhohelpedcreateandnourishthebeautiful
areaofquantumalgorithms,andtoourfamilieswhohelpedcreateand
nourishus.
RJLandKWR

Contents
Preface xi
Acknowledgements xiii
1 Introduction 1
1.1 TheModel 2
1.2 TheSpaceandtheStates 3
1.3 TheOperations 5
1.4 WhereIstheInput? 6
1.5 WhatExactlyIstheOutput? 7
1.6 SummaryandNotes 8
2 NumbersandStrings 9
2.1 AsymptoticNotation 11
2.2 Problems 12
2.3 SummaryandNotes 13
3 BasicLinearAlgebra 15
3.1 HilbertSpaces 16
3.2 ProductsandTensorProducts 16
3.3 Matrices 17
3.4 ComplexSpacesandInnerProducts 19
3.5 Matrices,Graphs,andSumsOverPaths 20
3.6 Problems 23
3.7 SummaryandNotes 26
4 BooleanFunctions,QuantumBits,andFeasibility 27
4.1 FeasibleBooleanFunctions 28
4.2 AnExample 30
4.3 QuantumRepresentationofBooleanArguments 33
4.4 QuantumFeasibility 35
4.5 Problems 38
4.6 SummaryandNotes 40
5 SpecialMatrices 41
5.1 HadamardMatrices 41
5.2 FourierMatrices 42
5.3 ReversibleComputationandPermutationMatrices 43
5.4 FeasibleDiagonalMatrices 44
5.5 Reflections 45
5.6 Problems 46
viii Contents
5.7 SummaryandNotes 49
6 Tricks 51
6.1 StartVectors 51
6.2 ControllingandCopyingBaseStates 52
6.3 TheCopy-UncomputeTrick 54
6.4 SuperpositionTricks 55
6.5 FlippingaSwitch 56
6.6 MeasurementTricks 58
6.7 PartialTransforms 59
6.8 Problems 60
6.9 SummaryandNotes 62
7 Phil’sAlgorithm 63
7.1 TheAlgorithm 63
7.2 TheAnalysis 63
7.3 AnExample 64
7.4 ATwo-QubitExample 64
7.5 PhilMeasuresUp 66
7.6 QuantumMazesversusCircuitsversusMatrices 69
7.7 Problems 71
7.8 SummaryandNotes 74
8 Deutsch’sAlgorithm 77
8.1 TheAlgorithm 77
8.2 TheAnalysis 78
8.3 SuperdenseCodingandTeleportation 82
8.4 Problems 86
8.5 SummaryandNotes 87
9 TheDeutsch-JozsaAlgorithm 89
9.1 TheAlgorithm 89
9.2 TheAnalysis 90
9.3 Problems 92
9.4 SummaryandNotes 92
10 Simon’sAlgorithm 93
10.1 TheAlgorithm 93
10.2 TheAnalysis 94
Contents ix
10.3 Problems 95
10.4 SummaryandNotes 96
11 Shor’sAlgorithm 97
11.1 Strategy 97
11.2 GoodNumbers 98
11.3 QuantumPartoftheAlgorithm 99
11.4 AnalysisoftheQuantumPart 100
11.5 ProbabilityofaGoodNumber 102
11.6 UsingaGoodNumber 105
11.7 ContinuedFractions 106
11.8 Problems 107
11.9 SummaryandNotes 108
12 FactoringIntegers 109
12.1 SomeBasicNumberTheory 109
12.2 PeriodsGivetheOrder 110
12.3 Factoring 110
12.4 Problems 112
12.5 SummaryandNotes 113
13 Grover’sAlgorithm 115
13.1 TwoVectors 115
13.2 TheAlgorithm 117
13.3 TheAnalysis 117
13.4 TheGeneralCase,withkUnknown 118
13.5 GroverApproximateCounting 119
13.5.1 TheAlgorithm 122
13.5.2 TheAnalysis 122
13.6 Problems 126
13.7 SummaryandNotes 128
14 QuantumWalks 129
14.1 ClassicalRandomWalks 129
14.2 RandomWalksandMatrices 130
14.3 AnEncodingNicety 132
14.4 DefiningQuantumWalks 133
14.5 InterferenceandDiffusion 134