Quantum Spin Glasses, Annealing and Computation PDF
Preview Quantum Spin Glasses, Annealing and Computation
Cambridge University Press 978-1-107-11319-0 — Quantum Spin Glasses, Annealing and Computation Bikas K. Chakrabarti , Jun-ichi Inoue , Ryo Tamura , Shu Tanaka Frontmatter More Information Quantum Spin Glasses, Annealing and Computation Quantumannealingisanew generationtoolofinformationtechnology,whichhelpsinsolving combinatorial optimization problems with high precision, based on the concepts of quantum statisticalphysics. Thisbookfocusesontherecentdevelopmentsinquantumspinglasses,quantumannealing andquantumcomputations. Itoffersadetaileddiscussiononquantumstatisticalphysicsofspin glasses and its application in solving combinatorial optimization problems. Separate chapters onsimulatedannealing,quantumdynamicsandclassicalspinmodelsareprovidedforenhanced understanding. Notes on adiabatic quantum computers and quenching dynamics make it apt for the readers. This text will be useful for the students of quantum computation, quantum information,statisticalphysicsandcomputerscience. ShuTanakaisAssistantProfessoratWasedaInstituteforAdvancedStudy,WasedaUniversity, Tokyo,Japan. HeisalsoPRESTOResearcher,JST.Hehasworkedinvariousfieldsofthetheory ofstatisticalphysics,materialsscience,andquantuminformationtheory. Hehasstudiedboth offundamentalandapplicationsidesofquantumannealing. Hehasalsoeditedthreebooksof KinkiUniversitySeriesonQuantumComputing. Hegotthe9thAwardfortheEncouragement ofYoungPhysicistsfromthePhysicalSocietyofJapan(2015). RyoTamuraisResearcheratInternationalCenterforMaterialsNanoarchitectonics(MANA) inNationalInstituteforMaterialsScience(NIMS),Ibaraki,Japan. Hehasworkedinthefield ofmaterialsscience,inparticular,hisresearchinterestsfocusonmagneticmaterialsincluding randomspinsystemsandmaterialsinformatics. Bikas K. Chakrabarti is Professor of Theoretical Condensed Matter Physics at Saha Institute of Nuclear Physics (SINP), Kolkata, India. His research interests include physics of fracture, quantum glasses and the interdisciplinary sciences of optimisation, brain modelling, and econophysics. Professor Chakrabarti is a recipient of the S. S. Bhatnagar Award (1997).HehasalsoreceivedtheOutstandingRefereeAwardoftheAmericanPhysicalSociety (2010). HeisfellowofIndianNationalScienceAcademy&IndianAcademyofSciences. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11319-0 — Quantum Spin Glasses, Annealing and Computation Bikas K. Chakrabarti , Jun-ichi Inoue , Ryo Tamura , Shu Tanaka Frontmatter More Information Quantum Spin Glasses, Annealing and Computation Shu Tanaka Ryo Tamura Bikas K. Chakrabarti © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11319-0 — Quantum Spin Glasses, Annealing and Computation Bikas K. Chakrabarti , Jun-ichi Inoue , Ryo Tamura , Shu Tanaka Frontmatter More Information UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,vic3207,Australia 4843/24,2ndFloor,AnsariRoad,Daryaganj,Delhi-110002,India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learningandresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle: www.cambridge.org/9781107113190 (cid:13)c Authors2017 Thispublicationisincopyright. Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2017 PrintedinIndia AcataloguerecordforthispublicationisavailablefromtheBritishLibrary ISBN978-1-107-11319-0Hardback Additionalresourcesforthispublicationatwww.cambridge.org/9781107113190 CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11319-0 — Quantum Spin Glasses, Annealing and Computation Bikas K. Chakrabarti , Jun-ichi Inoue , Ryo Tamura , Shu Tanaka Frontmatter More Information ThisbookisdedicatedtothememoryofProf. Jun-ichiInoue © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11319-0 — Quantum Spin Glasses, Annealing and Computation Bikas K. Chakrabarti , Jun-ichi Inoue , Ryo Tamura , Shu Tanaka Frontmatter More Information The book contains three important notes contributed by Eliahu Cohen,UmaDivakaran,SudipMukherjee,AtanuRajakandBoaz Tamir. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11319-0 — Quantum Spin Glasses, Annealing and Computation Bikas K. Chakrabarti , Jun-ichi Inoue , Ryo Tamura , Shu Tanaka Frontmatter More Information Contents Figures xiii Tables xxxi Preface xxxiii 1 Introduction References 3 Part One Quantum Spin Glass, Annealing and Computation 2 ClassicalSpinModels: FromFerromagneticSpinSystemstoSpinGlasses 2.1 IsingModel 8 2.1.1 Phasetransitionandcriticalphenomena 8 2.1.2 Mean-fieldapproximationandinfinite-rangemodel 10 2.1.3 Isingmodelonfinitedimensionallattices 15 2.2 BasicConceptofClassicalSpinGlass 22 2.2.1 Frustrationandrandomness 22 2.2.2 Spinglassorderparameter 25 2.2.3 Replicamethod 26 2.3 Sherrington–KirkpatrickModel 26 2.3.1 FreeenergyoftheSherrington–Kirkpatrickmodel 27 2.3.2 Replicasymmetricsolution 32 2.3.3 Replicasymmetrybreaking 36 2.4 Edwards–AndersonModelontheFiniteDimensionalLattices 42 2.A SeveralSpotlightsonClassicalModels 45 2.A.1 Pottsmodel 46 2.A.2 XYmodel 48 2.A.3 Heisenbergmodel 50 References 51 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11319-0 — Quantum Spin Glasses, Annealing and Computation Bikas K. Chakrabarti , Jun-ichi Inoue , Ryo Tamura , Shu Tanaka Frontmatter More Information viii Contents 3 SimulatedAnnealing 3.1 RelationBetweenRandomIsingModelsandCombinatorialOptimization Problems 60 3.2 RepresentativeCombinatorialOptimizationProblems 64 3.2.1 Computationalcomplexitytheory 64 3.2.2 Classificationofcombinatorialoptimizationproblems 66 3.2.3 Examplesofcombinatorialoptimizationproblems 67 3.3 ConventionalComputationMethodsIncludingSimulatedAnnealing 73 3.3.1 MonteCarlomethods 73 3.3.2 Mean-fieldmethods 77 3.3.3 Precedingtheoreticalstudiesonsimulatedannealing 79 3.4 ConvergenceTheoremforSimulatedAnnealing 84 3.4.1 InhomogeneousMarkovchain 84 3.4.2 Ergodicity 85 3.4.3 Proofoftheconvergencetheorem 86 References 97 4 QuantumSpinGlass 4.1 FerromagneticIsingModelinaTransverseField 98 4.1.1 Semiclassicalanalysisandasimplemean-fieldapproximation oftheferromagneticIsingmodelinatransversefield 100 4.1.2 Phasetransitioninthefinite-dimensionalferromagnetic Isingmodelsinatransversefield 103 4.1.3 Hushimi-Temperely–Curie–Weissmodelinatransversefield 110 4.2 IntroductionofQuantumSpinGlass 115 4.3 Sherrington-KirkpatrickModelinaTransverseField 117 4.3.1 Mean-fieldcalculations 118 4.3.2 Simulationresults 129 4.4 Edwards–AndersonModelinaTransverseField 137 4.5 ExistenceofReplicaSymmetricSpinGlassPhaseinthe RandomQuantumIsingModel 142 References 146 5 QuantumDynamics 5.1 Landau–ZenerTransition 151 5.2 Kibble–ZurekMechanism 158 5.2.1 BasicconceptoftheKibble–Zurekmechanism 159 5.2.2 Kibble–ZurekmechanismintheIsingchainwithrandom ferromagneticinteractions 162 References 167 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11319-0 — Quantum Spin Glasses, Annealing and Computation Bikas K. Chakrabarti , Jun-ichi Inoue , Ryo Tamura , Shu Tanaka Frontmatter More Information Contents ix 6 QuantumAnnealing 6.1 GeneralConceptofQuantumAnnealing 170 6.2 QuantumAnnealingbytheSchro¨dingerEquation 174 6.3 QuantumAnnealingbythePath-integralMonteCarloMethod 180 6.4 QuantumAnnealingbytheGreen’sFunctionMonteCarloMethod 185 6.5 QuantumAnnealingbytheDensityMatrixRenormalizationGroup 191 6.6 QuantumAnnealingbytheMean-fieldApproximation 195 6.7 QuantumFieldResponsetoMagneticMaterials 201 6.8 QuantumAdiabaticEvolution 203 6.8.1 Adiabaticquantumcomputation 204 6.8.2 Quantumadiabatictheorem 205 6.8.3 Quantumadiabaticapproximation 215 6.9 ConvergenceTheoremforQuantumAnnealing 222 6.9.1 ConvergencetheoremforquantumannealingbytheSchro¨dinger equation 222 6.9.2 Correspondencebetweenclassicalandquantumsystems 226 6.9.3 Convergencetheoremforquantumannealingbythe path-integralMonteCarlomethod 231 6.9.4 Convergencetheoremforquantumannealingbythe Green’sfunctionMonteCarlomethod 242 References 248 Part Two Additional Notes 7 NotesonAdiabaticQuantumComputers 7.1 Introduction 255 7.2 TheCircuitModel 256 7.3 AdiabaticComputation 259 7.3.1 Simulatedannealingandadiabaticcomputation 266 7.3.2 Differentpathsfromaninitialtoafinaleigenstate 267 7.3.3 Imaginarytimeandsimulations 268 7.3.4 Complexityanduniversality 268 7.3.5 Additionalmethods 270 7.4 QuantumAnnealing 270 7.4.1 Relationbetweensimulatedannealing,quantumannealing,and adiabaticcomputation 272 7.5 D-WaveComputer 274 7.5.1 IstheD-Waveaquantumcomputer? Thefutureofquantum annealers 276 7.5.2 Universality 276 7.5.3 CoherencetimeoftheSQUIDs 277 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11319-0 — Quantum Spin Glasses, Annealing and Computation Bikas K. Chakrabarti , Jun-ichi Inoue , Ryo Tamura , Shu Tanaka Frontmatter More Information x Contents 7.5.4 Scalability 278 7.5.5 Speedup 278 7.5.6 “Quantumness” 279 7.5.7 Doesthecomputerexhibitentanglement? 281 7.5.8 Openquestionsandpossiblefutureroutes 281 References 282 8 QuantumInformationandQuenchingDynamics 8.1 QuantumInformationTheoreticMeasures: IndicatorofaQCP 290 8.1.1 Quantumfidelity 290 8.1.2 Loschmidtecho 292 8.1.3 Entanglement 295 8.2 DynamicsinQuantumManybodySystems 296 8.3 FidelitySusceptibilityandLoschmidtEchoforGenericPaths 296 8.3.1 Modelandphasediagram 297 8.3.2 Fidelitysusceptibility 299 8.3.3 Loschmidtecho 304 8.4 LoschmidtEcho: EffectofGaplessPhase 307 8.4.1 Model,phasediagramandanisotropicquantum criticalpoint(AQCP) 307 8.4.2 LoschmidtechofortheKitaevmodel 310 8.5 EffectofDoubleLocalQuenchesonLoschmidtEchoandEntanglement Entropy 313 8.5.1 Model 314 8.5.2 Semiclassicaltheoryofquasiparticles 315 8.5.3 Loschmidtechoandentanglemententropyforacriticalchain 316 8.5.4 Entanglemententropy: Ferromagneticphase 321 8.5.5 QP1 andQP2: Acomparativestudy 323 8.6 TopologyinCondensedMatterSystems 324 8.6.1 Majoranafermionsandcondensedmattersystems 326 8.6.2 Kitaevchain: Modelandtopologicalphases 327 8.7 One-dimensionalGeneralizedClusterModel 333 8.8 SuddenQuench: DynamicsofaMajoranaEdgeState 335 8.8.1 Quenchingschemesandresults 337 8.9 SlowQuench: DynamicsofaMajoranaEdgeState 340 8.9.1 Model,phasediagramandenergyspectrum 341 8.9.2 QuenchingdynamicsofaMajoranaedgestate 343 8.10 SummaryandConclusion 347 References 350 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-11319-0 — Quantum Spin Glasses, Annealing and Computation Bikas K. Chakrabarti , Jun-ichi Inoue , Ryo Tamura , Shu Tanaka Frontmatter More Information Contents xi 9 ABriefHistoricalNoteontheStudiesofQuantumSpinGlasses,Annealing andComputation 9.1 Introduction 359 9.2 AShortHistoryoftheDevelopment 360 9.3 ReviewofSomeRecentDiscussionPapersandArxivPreprints 367 9.4 ApplicationsofQA 374 9.5 SummaryandDiscussions 375 References 379 Index 383 © in this web service Cambridge University Press www.cambridge.org
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