Introduction to Topological Quantum Computation

225 Pages·2012·2.783 MB·English

by  Jiannis K. Pachos

IntroductiontoTopologicalQuantumComputation
Combiningphysics,mathematicsandcomputerscience,topologicalquantumcomputation
isarapidlyexpandingresearchareafocusedontheexplorationofquantumevolutionsthat
areimmunetoerrors.Inthisbook,theauthorpresentsavarietyofdifferenttopicsdevel-
opedtogetherforthefirsttime,forminganexcellentintroductiontotopologicalquantum
computation.
Themakingsoftopologicalsystems,theirpropertiesandtheircomputationalpowerare
presented in a pedagogical way. Relevant calculations are fully explained, and numerous
workedexamplesandexercisessupportandaidunderstanding.Specialemphasisisgiven
tothemotivationandphysicalintuitionbehindeverymathematicalconcept.
Demystifyingdifficulttopicsbyusingaccessiblelanguage,thisbookhasbroadappeal
andisidealforgraduatestudentsandresearchersfromvariousdisciplineswhowanttoget
intothisnewandexcitingresearchfield.
JiannisK.Pachos is a Reader in the School of Physics and Astronomy at the University of
Leeds,UK.Heworksonavarietyofresearchtopics,rangingfromquantumfieldtheoryto
quantumoptics.DrPachosisaUniversityResearchFellowoftheRoyalSociety.
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Introduction to Topological
Quantum Computation
JIANNIS K. PACHOS
UniversityofLeeds,UK
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107005044
(cid:2)c J.K.Pachos2012
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-1-107-00504-4Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
ToAlmutandSevi

Contents
Acknowledgements pagexi
PartI Preliminaries 1
1 Introduction 3
1.1 Particleexchangeandquantumphysics 3
1.2 Anyonsandtopologicalsystems 4
1.3 Quantumcomputationwithanyons 5
1.4 Abelianandnon-Abeliananyonicstatistics 6
1.5 Whatareanyonicsystems? 8
1.5.1 Two-dimensionalwavefunctionsandquasiparticles 8
1.5.2 Symmetry,degeneracyandquantumcorrelations 10
Summary 11
Exercises 12
2 Geometricandtopologicalphases 13
2.1 Quantumphasesfromgaugefields 14
2.1.1 Chargedparticleinamagneticfield 14
2.1.2 TheAharonov–Bohmeffect 15
2.1.3 AnyonsandAharonov–Bohmeffect 16
2.2 Geometricphasesandholonomies 17
2.2.1 Spin-1/2particleinamagneticfield 18
2.2.2 Non-Abeliangeometricphases 21
2.2.3 Propertiesofgeometricevolutions 23
2.2.4 Anyonsandgeometricphases 26
2.3 ExampleI:IntegerquantumHalleffect 27
2.3.1 Wavefunctionofachargedparticleinamagneticfield 27
2.3.2 CurrentbehaviourandHallconductivity 29
2.3.3 Laughlin’sthoughtexperimentandgeometricphases 32
Summary 36
Exercises 36
3 Quantumcomputation 38
3.1 Qubitsandtheirmanipulations 39
3.1.1 Quantumbits 39
viii Contents
(cid:2)
3.1.2 Decoherenceandmixedstates 40
3.1.3 Quantumgatesandprojectors 41
3.2 Quantumcircuitmodel 43
3.2.1 Quantumalgorithmanduniversality 44
3.2.2 Computationalcomplexity 45
3.3 Othercomputationalmodels 46
3.3.1 One-wayquantumcomputation 47
3.3.2 Adiabaticquantumcomputation 49
3.3.3 Holonomicquantumcomputation 51
Summary 53
Exercises 53
4 Computationalpowerofanyons 55
4.1 Anyonsandtheirproperties 56
4.1.1 Particletypes 56
4.1.2 Fusionrulesofanyons 57
4.1.3 AnyonicHilbertspace 58
4.1.4 Exchangepropertiesofanyons 61
4.1.5 Pentagonandhexagonidentities 62
4.1.6 Spinandstatistics 64
4.2 Anyonicquantumcomputation 65
4.2.1 Anyonicsetting 66
4.2.2 Stabilityofanyoniccomputation 67
4.3 ExampleI:Isinganyons 68
4.3.1 Themodelanditsproperties 68
4.3.2 FandRmatrices 70
4.4 ExampleII:Fibonaccianyons 73
Summary 75
Exercises 75
PartII Topologicalmodels 77
5 Quantumdoublemodels 79
5.1 Errorcorrection 80
5.1.1 Quantumerrorcorrectingcodes 80
5.1.2 Stabilisercodes 81
5.2 Quantumdoublemodels 83
5.2.1 Thetoriccode 83
5.2.2 GeneralD(G)quantumdoublemodels 91
5.3 ExampleI:Abelianquantumdoublemodels 94
5.4 ExampleII:Thenon-AbelianD(S )model 96
3
5.5 Quantumdoublesasquantummemories 98
5.5.1 Non-Abelianinformationencodingandmanipulation 98