內容簡介
本書分三大部分。
第1部分為總論和量子物理學基礎,還包括少量計算機科學基礎知識。量子物理學基礎部分主要介紹學習量子計算與量子信息所必需的量子力學基礎知識,這一部分採用了側重於數學框架和公理化體系的講述方法,從而更便於為非物理專業的讀者所理解。
第2部分講述量子計算,包括量子算法和實現量子算法的一些物理系統的基礎內容。這部分首先介紹了實現普適量子計算所需要的基本邏輯門——單量子比特門與條件非門(第4章)。之後較詳細地介紹了現有的兩個主要量子算法問題,即質因數分解問題(第5章)和搜尋問題(第6章)。第7章講物理實現,介紹了幾個主要的實現條件非門的物理系統;這些物理系統雖然目前尚未實現大規模量子計算,但是大多數已經實現或基本實現了普適量子計算所需要的基本邏輯門。
第3部分為量子資訊理論,主要介紹量子糾錯碼和量子資訊理論的數學框架。這裡包括了非常重要的量子密碼的基礎內容,即理想條件下的安全性證明(第12.6節)。第3部分未包含物理實現內容。
前言
量子信息處理可以帶來許多意想不到的具有特殊優勢的結果。量子算法可以有效地進行大數質因數分解。在量子算法背景下,很多經典保密通信協定的安全性受到威脅,然而量子保密通信可以抵抗來自包括量子計算機在內的任何針對通道的攻擊。由於存在諸多特殊套用,量子計算與量子信息科學近年來得到蓬勃發展。
對於這一相對年輕但又具有廣闊發展前景的學科,優秀的系統化的教材比較缺乏。本書是本領域公認的最權威的系統化教材之一,也幾乎是本學科研究人員的必備基本材料,有學者曾將之稱為量子計算與量子信息學科教材中的“聖經”。
本書的特點是全面包含量子計算和量子信息的核心內容,且系統性強,結構清晰,深入淺出。這使其很適合作為相關專業的本科生和研究生教材,也適合於用作本領域研究人員的基礎參考資料;對於那些準備從其他研究領域轉行投入本領域研究的具有物理學、信息科學或數學等專業背景的研究人員,本書也是一本非常合適的入門書籍。
如同任何其他書籍一樣,本書的內容也不可能面面俱到。本書幾乎未涉及連續變數量子信息處理的有關內容。這方面,目前有多篇綜述論文和一些專門教材可供參考。另外,儘管同多數同類教材比較起來,本書已經較深入地介紹了量子密碼內容,但是相比於其重要性,量子密碼方面的內容量還是有些偏少,對這方面感興趣的讀者可參閱相關專著。
王向斌
清華大學物理系
2015年9月
圖書目錄
PartI
Fundamentalconcepts1
1Introductionandoverview1
1.1Globalperspectives1
1.1.1Historyofquantumcomputationandquantuminformation2
1.1.2Futuredirections12
1.2Quantumbits13
1.2.1Multiplequbits16
1.3Quantumcomputation17
1.3.1Singlequbitgates17
1.3.2Multiplequbitgates20
1.3.3Measurementsinbasesotherthanthecomputationalbasis22
1.3.4Quantumcircuits22
1.3.5Qubitcopyingcircuit?24
1.3.6Example:Bellstates25
1.3.7Example:quantumteleportation26
1.4Quantumalgorithms28
1.4.1Classicalcomputationsonaquantumcomputer29
1.4.2Quantumparallelism30
1.4.3Deutsch’salgorithm32
1.4.4TheDeutsch–Jozsaalgorithm34
1.4.5Quantumalgorithmssummarized36
1.5Experimentalquantuminformationprocessing42
1.5.1TheStern–Gerlachexperiment43
1.5.2Prospectsforpracticalquantuminformationprocessing46
1.6Quantuminformation50
1.6.1Quantuminformationtheory:exampleproblems52
1.6.2Quantuminformationinawidercontext58
2Introductiontoquantummechanics60
2.1Linearalgebra61
2.1.1Basesandlinearindependence62
2.1.2Linearoperatorsandmatrices63
2.1.3ThePaulimatrices65
2.1.4Innerproducts65
2.1.5Eigenvectorsandeigenvalues68
2.1.6AdjointsandHermitianoperators69
2.1.7Tensorproducts71
2.1.8Operatorfunctions75
2.1.9Thecommutatorandanti-commutator76
2.1.10Thepolarandsingularvaluedecompositions78
2.2Thepostulatesofquantummechanics80
2.2.1Statespace80
2.2.2Evolution81
2.2.3Quantummeasurement84
2.2.4Distinguishingquantumstates86
2.2.5Projectivemeasurements87
2.2.6POVMmeasurements90
2.2.7Phase93
2.2.8Compositesystems93
2.2.9Quantummechanics:aglobalview96
2.3Application:superdensecoding97
2.4Thedensityoperator98
2.4.1Ensemblesofquantumstates99
2.4.2Generalpropertiesofthedensityoperator101
2.4.3Thereduceddensityoperator105
2.5TheSchmidtdecompositionandpurifications109
2.6EPRandtheBellinequality111
3Introductiontocomputerscience120
3.1Modelsforcomputation122
3.1.1Turingmachines122
3.1.2Circuits129
3.2Theanalysisofcomputationalproblems135
3.2.1Howtoquantifycomputationalresources136
3.2.2Computationalcomplexity138
3.2.3DecisionproblemsandthecomplexityclassesPandNP141
3.2.4Aplethoraofcomplexityclasses150
3.2.5Energyandcomputation153
3.3Perspectivesoncomputerscience161
PartIIQuantumcomputation171
4Quantumcircuits171
4.1Quantumalgorithms172
4.2Singlequbitoperations174
4.3Controlledoperations177
4.4Measurement185
4.5Universalquantumgates188
4.5.1Two-levelunitarygatesareuniversal189
4.5.2SinglequbitandCNOTgatesareuniversal191
4.5.3Adiscretesetofuniversaloperations194
4.5.4Approximatingarbitraryunitarygatesisgenericallyhard198
4.5.5Quantumcomputationalcomplexity200
4.6Summaryofthequantumcircuitmodelofcomputation202
4.7Simulationofquantumsystems204
4.7.1Simulationinaction204
4.7.2Thequantumsimulationalgorithm206
4.7.3Anillustrativeexample209
4.7.4Perspectivesonquantumsimulation211
5ThequantumFouriertransformanditsapplications216
5.1ThequantumFouriertransform217
5.2Phaseestimation221
5.2.1Performanceandrequirements223
5.3Applications:order-findingandfactoring226
5.3.1Application:order-finding226
5.3.2Application:factoring232
5.4GeneralapplicationsofthequantumFouriertransform234
5.4.1Period-finding236
5.4.2Discretelogarithms238
5.4.3Thehiddensubgroupproblem240
5.4.4Otherquantumalgorithms?242
6Quantumsearchalgorithms248
6.1Thequantumsearchalgorithm248
6.1.1Theoracle248
6.1.2Theprocedure250
6.1.3Geometricvisualization252
6.1.4Performance253
6.2Quantumsearchasaquantumsimulation255
6.3Quantumcounting261
6.4SpeedingupthesolutionofNP-completeproblems263
6.5Quantumsearchofanunstructureddatabase265
6.6Optimalityofthesearchalgorithm269
6.7Blackboxalgorithmlimits271
7Quantumcomputers:physicalrealization277
7.1Guidingprinciples277
7.2Conditionsforquantumcomputation279
7.2.1Representationofquantuminformation279
7.2.2Performanceofunitarytransformations281
7.2.3Preparationoffiducialinitialstates281
7.2.4Measurementofoutputresult282
7.3Harmonicoscillatorquantumcomputer283
7.3.1Physicalapparatus283
7.3.2TheHamiltonian284
7.3.3Quantumcomputation286
7.3.4Drawbacks286
7.4Opticalphotonquantumcomputer287
7.4.1Physicalapparatus287
7.4.2Quantumcomputation290
7.4.3Drawbacks296
7.5Opticalcavityquantumelectrodynamics297
7.5.1Physicalapparatus298
7.5.2TheHamiltonian300
7.5.3Single-photonsingle-atomabsorptionandrefraction303
7.5.4Quantumcomputation306
7.6Iontraps309
7.6.1Physicalapparatus309
7.6.2TheHamiltonian317
7.6.3Quantumcomputation319
7.6.4Experiment321
7.7Nuclearmagneticresonance324
7.7.1Physicalapparatus325
7.7.2TheHamiltonian326
7.7.3Quantumcomputation331
7.7.4Experiment336
7.8Otherimplementationschemes343
PartIIIQuantuminformation353
8Quantumnoiseandquantumoperations353
8.1ClassicalnoiseandMarkovprocesses354
8.2Quantumoperations356
8.2.1Overview356
8.2.2Environmentsandquantumoperations357
8.2.3Operator-sumrepresentation360
8.2.4Axiomaticapproachtoquantumoperations366
8.3Examplesofquantumnoiseandquantumoperations373
8.3.1Traceandpartialtrace374
8.3.2Geometricpictureofsinglequbitquantumoperations374
8.3.3Bitflipandphaseflipchannels376
8.3.4Depolarizingchannel378
8.3.5Amplitudedamping380
8.3.6Phasedamping383
8.4Applicationsofquantumoperations386
8.4.1Masterequations386
8.4.2Quantumprocesstomography389
8.5Limitationsofthequantumoperationsformalism394
9Distancemeasuresforquantuminformation399
9.1Distancemeasuresforclassicalinformation399
9.2Howclosearetwoquantumstates?403
9.2.1Tracedistance403
9.2.2Fidelity409
9.2.3Relationshipsbetweendistancemeasures415
9.3Howwelldoesaquantumchannelpreserveinformation?416
10Quantumerror-correction425
10.1Introduction426
10.1.1Thethreequbitbitflipcode427
10.1.2Threequbitphaseflipcode430
10.2TheShorcode432
10.3Theoryofquantumerror-correction435
10.3.1Discretizationoftheerrors438
10.3.2Independenterrormodels441
10.3.3Degeneratecodes444
10.3.4ThequantumHammingbound444
10.4Constructingquantumcodes445
10.4.1Classicallinearcodes445
10.4.2Calderbank–Shor–Steanecodes450
10.5Stabilizercodes453
10.5.1Thestabilizerformalism454
10.5.2Unitarygatesandthestabilizerformalism459
10.5.3Measurementinthestabilizerformalism463
10.5.4TheGottesman–Knilltheorem464
10.5.5Stabilizercodeconstructions464
10.5.6Examples467
10.5.7Standardformforastabilizercode470
10.5.8Quantumcircuitsforencoding,decoding,andcorrection472
10.6Fault-tolerantquantumcomputation474
10.6.1Fault-tolerance:thebigpicture475
10.6.2Fault-tolerantquantumlogic482
10.6.3Fault-tolerantmeasurement489
10.6.4Elementsofresilientquantumcomputation493
11Entropyandinformation500
11.1Shannonentropy500
11.2Basicpropertiesofentropy502
11.2.1Thebinaryentropy502
11.2.2Therelativeentropy504
11.2.3Conditionalentropyandmutualinformation505
11.2.4Thedataprocessinginequality509
11.3VonNeumannentropy510
11.3.1Quantumrelativeentropy511
11.3.2Basicpropertiesofentropy513
11.3.3Measurementsandentropy514
11.3.4Subadditivity515
11.3.5Concavityoftheentropy516
11.3.6Theentropyofamixtureofquantumstates518
11.4Strongsubadditivity519
11.4.1Proofofstrongsubadditivity519
11.4.2Strongsubadditivity:elementaryapplications522
12Quantuminformationtheory528
12.1Distinguishingquantumstatesandtheaccessibleinformation529
12.1.1TheHolevobound531
12.1.2ExampleapplicationsoftheHolevobound534
12.2Datacompression536
12.2.1Shannon’snoiselesschannelcodingtheorem537
12.2.2Schumacher’squantumnoiselesschannelcodingtheorem542
12.3Classicalinformationovernoisyquantumchannels546
12.3.1Communicationovernoisyclassicalchannels548
12.3.2Communicationovernoisyquantumchannels554
12.4Quantuminformationovernoisyquantumchannels561
12.4.1EntropyexchangeandthequantumFanoinequality561
12.4.2Thequantumdataprocessinginequality564
12.4.3QuantumSingletonbound568
12.4.4Quantumerror-correction,refrigerationandMaxwell’sdemon569
12.5Entanglementasaphysicalresource571
12.5.1Transformingbi-partitepurestateentanglement573
12.5.2Entanglementdistillationanddilution578
12.5.3Entanglementdistillationandquantumerror-correction580
12.6Quantumcryptography582
12.6.1Privatekeycryptography582
12.6.2Privacyamplificationandinformationreconciliation584
12.6.3Quantumkeydistribution586
12.6.4Privacyandcoherentinformation592
12.6.5Thesecurityofquantumkeydistribution593
Appendices608
Appendix1:Notesonbasicprobabilitytheory608
Appendix2:Grouptheory610
A2.1Basicdefinitions610
A2.1.1Generators611
A2.1.2Cyclicgroups611
A2.1.3Cosets612
A2.2Representations612
A2.2.1Equivalenceandreducibility612
A2.2.2Orthogonality613
A2.2.3Theregularrepresentation614
A2.3Fouriertransforms615
Appendix3:TheSolovay--Kitaevtheorem617
Appendix4:Numbertheory625
A4.1Fundamentals625
A4.2ModulararithmeticandEuclid’salgorithm626
A4.3Reductionoffactoringtoorder-finding633
A4.4Continuedfractions635
Appendix5:PublickeycryptographyandtheRSAcryptosystem640
Appendix6:ProofofLieb’stheorem645
Bibliography649
Index
665