Quantum Computation and Quantum Information

698 Pages·2011·10.22 MB·English

Quantum Computation and Quantum Information
10th Anniversary Edition
Michael A. Nielsen & Isaac L. Chuang
CAMBRIDGE UNIVERSITY PRESS
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Informationonthistitle:www.cambridge.org/9781107002173
C M.NielsenandI.Chuang2010
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Firstpublished2000
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10thAnniversaryeditionpublished2010
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Contents
IntroductiontotheTenthAnniversaryEdition pagexvii
AfterwordtotheTenthAnniversaryEdition xix
Preface xxi
Acknowledgements xxvii
Nomenclatureandnotation xxix
PartI Fundamentalconcepts 1
1 Introductionandoverview 1
1.1 Globalperspectives 1
1.1.1 Historyofquantumcomputationandquantum
information 2
1.1.2 Futuredirections 12
1.2 Quantumbits 13
1.2.1 Multiplequbits 16
1.3 Quantumcomputation 17
1.3.1 Singlequbitgates 17
1.3.2 Multiplequbitgates 20
1.3.3 Measurementsinbasesotherthanthecomputationalbasis 22
1.3.4 Quantumcircuits 22
1.3.5 Qubitcopyingcircuit? 24
1.3.6 Example:Bellstates 25
1.3.7 Example:quantumteleportation 26
1.4 Quantumalgorithms 28
1.4.1 Classicalcomputationsonaquantumcomputer 29
1.4.2 Quantumparallelism 30
1.4.3 Deutsch’salgorithm 32
1.4.4 TheDeutsch–Jozsaalgorithm 34
1.4.5 Quantumalgorithmssummarized 36
1.5 Experimentalquantuminformationprocessing 42
1.5.1 TheStern–Gerlachexperiment 43
1.5.2 Prospectsforpracticalquantuminformationprocessing 46
1.6 Quantuminformation 50
1.6.1 Quantuminformationtheory:exampleproblems 52
1.6.2 Quantuminformationinawidercontext 58
x Contents
2 Introductiontoquantummechanics 60
2.1 Linearalgebra 61
2.1.1 Basesandlinearindependence 62
2.1.2 Linearoperatorsandmatrices 63
2.1.3 ThePaulimatrices 65
2.1.4 Innerproducts 65
2.1.5 Eigenvectorsandeigenvalues 68
2.1.6 AdjointsandHermitianoperators 69
2.1.7 Tensorproducts 71
2.1.8 Operatorfunctions 75
2.1.9 Thecommutatorandanti-commutator 76
2.1.10 Thepolarandsingularvaluedecompositions 78
2.2 Thepostulatesofquantummechanics 80
2.2.1 Statespace 80
2.2.2 Evolution 81
2.2.3 Quantummeasurement 84
2.2.4 Distinguishingquantumstates 86
2.2.5 Projectivemeasurements 87
2.2.6 POVMmeasurements 90
2.2.7 Phase 93
2.2.8 Compositesystems 93
2.2.9 Quantummechanics:aglobalview 96
2.3 Application:superdensecoding 97
2.4 Thedensityoperator 98
2.4.1 Ensemblesofquantumstates 99
2.4.2 Generalpropertiesofthedensityoperator 101
2.4.3 Thereduceddensityoperator 105
2.5 TheSchmidtdecompositionandpurifications 109
2.6 EPRandtheBellinequality 111
3 Introductiontocomputerscience 120
3.1 Modelsforcomputation 122
3.1.1 Turingmachines 122
3.1.2 Circuits 129
3.2 Theanalysisofcomputationalproblems 135
3.2.1 Howtoquantifycomputationalresources 136
3.2.2 Computationalcomplexity 138
3.2.3 DecisionproblemsandthecomplexityclassesPandNP 141
3.2.4 Aplethoraofcomplexityclasses 150
3.2.5 Energyandcomputation 153
3.3 Perspectivesoncomputerscience 161
PartII Quantumcomputation 171
4 Quantumcircuits 171
4.1 Quantumalgorithms 172
4.2 Singlequbitoperations 174
Contents xi
4.3 Controlledoperations 177
4.4 Measurement 185
4.5 Universalquantumgates 188
4.5.1 Two-levelunitarygatesareuniversal 189
4.5.2 SinglequbitandCNOTgatesareuniversal 191
4.5.3 Adiscretesetofuniversaloperations 194
4.5.4 Approximatingarbitraryunitarygatesisgenericallyhard 198
4.5.5 Quantumcomputationalcomplexity 200
4.6 Summaryofthequantumcircuitmodelofcomputation 202
4.7 Simulationofquantumsystems 204
4.7.1 Simulationinaction 204
4.7.2 Thequantumsimulationalgorithm 206
4.7.3 Anillustrativeexample 209
4.7.4 Perspectivesonquantumsimulation 211
5 ThequantumFouriertransformanditsapplications 216
5.1 ThequantumFouriertransform 217
5.2 Phaseestimation 221
5.2.1 Performanceandrequirements 223
5.3 Applications:order-findingandfactoring 226
5.3.1 Application:order-finding 226
5.3.2 Application:factoring 232
5.4 GeneralapplicationsofthequantumFourier
transform 234
5.4.1 Period-finding 236
5.4.2 Discretelogarithms 238
5.4.3 Thehiddensubgroupproblem 240
5.4.4 Otherquantumalgorithms? 242
6 Quantumsearchalgorithms 248
6.1 Thequantumsearchalgorithm 248
6.1.1 Theoracle 248
6.1.2 Theprocedure 250
6.1.3 Geometricvisualization 252
6.1.4 Performance 253
6.2 Quantumsearchasaquantumsimulation 255
6.3 Quantumcounting 261
6.4 SpeedingupthesolutionofNP-completeproblems 263
6.5 Quantumsearchofanunstructureddatabase 265
6.6 Optimalityofthesearchalgorithm 269
6.7 Blackboxalgorithmlimits 271
7 Quantumcomputers:physicalrealization 277
7.1 Guidingprinciples 277
7.2 Conditionsforquantumcomputation 279
7.2.1 Representationofquantuminformation 279
7.2.2 Performanceofunitarytransformations 281
xii Contents
7.2.3 Preparationoffiducialinitialstates 281
7.2.4 Measurementofoutputresult 282
7.3 Harmonicoscillatorquantumcomputer 283
7.3.1 Physicalapparatus 283
7.3.2 TheHamiltonian 284
7.3.3 Quantumcomputation 286
7.3.4 Drawbacks 286
7.4 Opticalphotonquantumcomputer 287
7.4.1 Physicalapparatus 287
7.4.2 Quantumcomputation 290
7.4.3 Drawbacks 296
7.5 Opticalcavityquantumelectrodynamics 297
7.5.1 Physicalapparatus 298
7.5.2 TheHamiltonian 300
7.5.3 Single-photonsingle-atomabsorptionand
refraction 303
7.5.4 Quantumcomputation 306
7.6 Iontraps 309
7.6.1 Physicalapparatus 309
7.6.2 TheHamiltonian 317
7.6.3 Quantumcomputation 319
7.6.4 Experiment 321
7.7 Nuclearmagneticresonance 324
7.7.1 Physicalapparatus 325
7.7.2 TheHamiltonian 326
7.7.3 Quantumcomputation 331
7.7.4 Experiment 336
7.8 Otherimplementationschemes 343
PartIII Quantuminformation 353
8 Quantumnoiseandquantumoperations 353
8.1 ClassicalnoiseandMarkovprocesses 354
8.2 Quantumoperations 356
8.2.1 Overview 356
8.2.2 Environmentsandquantumoperations 357
8.2.3 Operator-sumrepresentation 360
8.2.4 Axiomaticapproachtoquantumoperations 366
8.3 Examplesofquantumnoiseandquantumoperations 373
8.3.1 Traceandpartialtrace 374
8.3.2 Geometricpictureofsinglequbitquantum
operations 374
8.3.3 Bitflipandphaseflipchannels 376
8.3.4 Depolarizingchannel 378
8.3.5 Amplitudedamping 380
8.3.6 Phasedamping 383
Contents xiii
8.4 Applicationsofquantumoperations 386
8.4.1 Masterequations 386
8.4.2 Quantumprocesstomography 389
8.5 Limitationsofthequantumoperationsformalism 394
9 Distancemeasuresforquantuminformation 399
9.1 Distancemeasuresforclassicalinformation 399
9.2 Howclosearetwoquantumstates? 403
9.2.1 Tracedistance 403
9.2.2 Fidelity 409
9.2.3 Relationshipsbetweendistancemeasures 415
9.3 Howwelldoesaquantumchannelpreserveinformation? 416
10 Quantumerror-correction 425
10.1 Introduction 426
10.1.1 Thethreequbitbitflipcode 427
10.1.2 Threequbitphaseflipcode 430
10.2 TheShorcode 432
10.3 Theoryofquantumerror-correction 435
10.3.1 Discretizationoftheerrors 438
10.3.2 Independenterrormodels 441
10.3.3 Degeneratecodes 444
10.3.4 ThequantumHammingbound 444
10.4 Constructingquantumcodes 445
10.4.1 Classicallinearcodes 445
10.4.2 Calderbank–Shor–Steanecodes 450
10.5 Stabilizercodes 453
10.5.1 Thestabilizerformalism 454
10.5.2 Unitarygatesandthestabilizerformalism 459
10.5.3 Measurementinthestabilizerformalism 463
10.5.4 TheGottesman–Knilltheorem 464
10.5.5 Stabilizercodeconstructions 464
10.5.6 Examples 467
10.5.7 Standardformforastabilizercode 470
10.5.8 Quantumcircuitsforencoding,decoding,and
correction 472
10.6 Fault-tolerantquantumcomputation 474
10.6.1 Fault-tolerance:thebigpicture 475
10.6.2 Fault-tolerantquantumlogic 482
10.6.3 Fault-tolerantmeasurement 489
10.6.4 Elementsofresilientquantumcomputation 493
11 Entropyandinformation 500
11.1 Shannonentropy 500
11.2 Basicpropertiesofentropy 502
11.2.1 Thebinaryentropy 502
11.2.2 Therelativeentropy 504
xiv Contents
11.2.3 Conditionalentropyandmutualinformation 505
11.2.4 Thedataprocessinginequality 509
11.3 VonNeumannentropy 510
11.3.1 Quantumrelativeentropy 511
11.3.2 Basicpropertiesofentropy 513
11.3.3 Measurementsandentropy 514
11.3.4 Subadditivity 515
11.3.5 Concavityoftheentropy 516
11.3.6 Theentropyofamixtureofquantumstates 518
11.4 Strongsubadditivity 519
11.4.1 Proofofstrongsubadditivity 519
11.4.2 Strongsubadditivity:elementaryapplications 522
12 Quantuminformationtheory 528
12.1 Distinguishingquantumstatesandtheaccessibleinformation 529
12.1.1 TheHolevobound 531
12.1.2 ExampleapplicationsoftheHolevobound 534
12.2 Datacompression 536
12.2.1 Shannon’snoiselesschannelcodingtheorem 537
12.2.2 Schumacher’squantumnoiselesschannelcodingtheorem 542
12.3 Classicalinformationovernoisyquantumchannels 546
12.3.1 Communicationovernoisyclassicalchannels 548
12.3.2 Communicationovernoisyquantumchannels 554
12.4 Quantuminformationovernoisyquantumchannels 561
12.4.1 EntropyexchangeandthequantumFanoinequality 561
12.4.2 Thequantumdataprocessinginequality 564
12.4.3 QuantumSingletonbound 568
12.4.4 Quantumerror-correction,refrigerationandMaxwell’sdemon 569
12.5 Entanglementasaphysicalresource 571
12.5.1 Transformingbi-partitepurestateentanglement 573
12.5.2 Entanglementdistillationanddilution 578
12.5.3 Entanglementdistillationandquantumerror-correction 580
12.6 Quantumcryptography 582
12.6.1 Privatekeycryptography 582
12.6.2 Privacyamplificationandinformationreconciliation 584
12.6.3 Quantumkeydistribution 586
12.6.4 Privacyandcoherentinformation 592
12.6.5 Thesecurityofquantumkeydistribution 593
Appendices 608
Appendix1: Notesonbasicprobabilitytheory 608
Appendix2: Grouptheory 610
A2.1 Basicdefinitions 610
A2.1.1 Generators 611
A2.1.2 Cyclicgroups 611
A2.1.3 Cosets 612