Quantum computer science: an introduction

237 Pages·2007·1.105 MB·English

by  N. David Mermin

Quantum Computer Science
An Introduction
Inthe1990sitwasrealizedthatquantumphysicshassome
spectacularapplicationsincomputerscience.Thisbookisaconcise
introductiontoquantumcomputation,developingthebasicelements
ofthisnewbranchofcomputationaltheorywithoutassumingany
backgroundinphysics.Itbeginswithanovelintroductiontothe
quantumtheoryfromacomputer-scienceperspective.Itillustrates
thequantum-computationalapproachwithseveralelementary
examplesofquantumspeed-up,beforemovingtothemajor
applications:Shor’sfactoringalgorithm,Grover’ssearchalgorithm,
andquantumerrorcorrection.
Thebookisintendedprimarilyforcomputerscientistswhoknow
nothingaboutquantumtheorybutwouldliketolearntheelementsof
quantumcomputationeitheroutofcuriosityaboutthisnew
paradigm,orasabasisforfurtherworkinthesubject.Itwillalsobe
ofinteresttophysicistswhowanttolearnthetheoryofquantum
computation,andtophysicistsandphilosophersofscienceinterested
inquantumfoundationalissues.Itevolvedduringsixyearsofteaching
thesubjecttoundergraduatesandgraduatestudentsincomputer
science,mathematics,engineering,andphysics,atCornellUniversity.
N.DAVIDMERMINisHoraceWhiteProfessorofPhysicsEmeritusat
CornellUniversity.HehasreceivedtheLilienfeldPrizeofthe
AmericanPhysicalSocietyandtheKlopstegawardoftheAmerican
AssociationofPhysicsTeachers.HeisamemberoftheU.S.National
AcademyofSciencesandtheAmericanAcademyofArtsand
Sciences.ProfessorMerminhaswrittenonquantumfoundational
issuesforseveraldecades,andisknownfortheclarityandwitofhis
scientificwritings.AmonghisotherbooksareSolidStatePhysics
(withN.W.Ashcroft,ThomsonLearning1976),BoojumsalltheWay
Through(CambridgeUniversityPress1990),andIt’saboutTime:
UnderstandingEinstein’sRelativity(PrincetonUniversityPress2005).
“Thisisoneofthefinestbooksintherapidlygrowingfieldofquan-
tum information. Almost every page contains a unique insight or a
novelinterpretation.DavidMerminhasonceagaindemonstratedhis
legendarypedagogicalskillstoproduceaclassic.”
LovGrover,BellLabs
“Mermin’s book will be a standard for instruction and reference for
yearstocome.Hehascarefullyselected,fromthemountainofknowl-
edge accumulated in the last 20 years of research in quantum infor-
mationtheory,amanageable,coherentsubsetthatconstitutesacom-
pleteundergraduatecourse.Whileselective,itisinnosense“watered
down”; Mermin moves unflinchingly through difficult arguments in
theShoralgorithm,andinquantumerrorcorrectiontheory,providing
invaluablediagrams,cleararguments,and,whennecessary,extensive
appendices to get the students successfully through to the end. The
book is suffused with Mermin’s unique knowledge of the history of
modern physics, and has some of the most captivating writing to be
foundinacollegetextbook.”
DavidDiVincenzo,IBMT.J.WatsonResearchCenter
“Mermin’sbookisagentleintroductiontoquantumcomputationespe-
ciallyaimedatanaudienceofcomputerscientistsandmathematicians.
Itcoversthebasicsofthefield,explainingthematerialclearlyandcon-
taininglotsofexamples.Merminhasalwaysbeenanentertainingand
comprehensiblewriter,andcontinuestobeinthisbook.Iexpectitto
becomethedefinitiveintroductiontothismaterialfornon-physicists.”
PeterShor,MassachusettsInstituteofTechnology
“Textbookwritersusuallystriveforastreamlinedexposition,smooth-
ingouttheinfelicitiesofthoughtandnotationthatplagueanyfield’s
earlydevelopment.Fortunately,DavidMerministoopassionateand
acuteanobserveroftheculturalsideofsciencetofallintothisbland-
ness.Insteadofomittinginfelicities,heexplainsandcondemnsthem,
atthesametimeusinghisexperienceofhavingtaughtthecoursemany
timestonipnascentmisunderstandingsinthebud.Hecelebratesthe
field’s mongrel origin in a shotgun wedding between classical com-
puterscientists,whothoughttheyknewthelawsofinformation,and
quantumphysicists,whothoughtinformationwasnottheirjob.Dif-
ferences remain: we hear, for example, why physicists love the Dirac
notationandmathematicianshateit.Worked-outexamplesandexer-
cisesfamiliarizestudentswiththenecessaryalgebraicmanipulations,
while Mermin’s lucid prose and gentle humor cajole them toward a
sound intuition for what it all means, not an easy task for a subject
superficiallysocounterintuitive.”
CharlesBennett,IBMT.J.WatsonResearchCenter
Quantum Computer Science
An Introduction
N. David Mermin
CornellUniversity

Inmemoryofmybrother,JoelMermin
Youwouldhaveenjoyedit.
CAMBRIDGEUNIVERSITY PRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521876582
© N. D. Mermin 2007
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-34258-5 eBook (NetLibrary)
ISBN-10 0-511-34258-6 eBook (NetLibrary)
ISBN-13 978-0-521-87658-2 hardback
ISBN-10 0-521-87658-3 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
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Contents
Preface pagexi
Anoteonreferences xv
1 CbitsandQbits 1
1.1 Whatisaquantumcomputer? 1
1.2 Cbitsandtheirstates 3
1.3 ReversibleoperationsonCbits 8
1.4 ManipulatingoperationsonCbits 11
1.5 Qbitsandtheirstates 17
1.6 ReversibleoperationsonQbits 19
1.7 Circuitdiagrams 21
1.8 MeasurementgatesandtheBornrule 23
1.9 ThegeneralizedBornrule 28
1.10 Measurementgatesandstatepreparation 30
1.11 Constructingarbitrary1-and2-Qbitstates 32
1.12 Summary:QbitsversusCbits 34
2 Generalfeaturesandsomesimpleexamples 36
2.1 Thegeneralcomputationalprocess 36
2.2 Deutsch’sproblem 41
2.3 WhyadditionalQbitsneedn’tmessthingsup 46
2.4 TheBernstein–Vaziraniproblem 50
2.5 Simon’sproblem 54
2.6 ConstructingToffoligates 58
3 BreakingRSAencryption 63
3.1 Periodfinding,factoring,andcryptography 63
3.2 Number-theoreticpreliminaries 64
3.3 RSAencryption 66
3.4 Quantumperiodfinding:preliminaryremarks 68
3.5 ThequantumFouriertransform 71
3.6 Eliminatingthe2-Qbitgates 76
3.7 Findingtheperiod 79
viii CONTENTS
3.8 Calculatingtheperiodicfunction 83
3.9 Theunimportanceofsmallphaseerrors 84
3.10 Periodfindingandfactoring 86
4 Searchingwithaquantumcomputer 88
4.1 Thenatureofthesearch 88
4.2 TheGroveriteration 89
4.3 HowtoconstructW 94
4.4 Generalizationtoseveralspecialnumbers 96
4.5 Searchingforoneoutoffouritems 98
5 Quantumerrorcorrection 99
5.1 Themiracleofquantumerrorcorrection 99
5.2 Asimplifiedexample 100
5.3 Thephysicsoferrorgeneration 109
5.4 Diagnosingerrorsyndromes 113
5.5 The5-Qbiterror-correctingcode 117
5.6 The7-Qbiterror-correctingcode 121
5.7 Operationson7-Qbitcodewords 124
5.8 A7-Qbitencodingcircuit 127
5.9 A5-Qbitencodingcircuit 128
6 ProtocolsthatusejustafewQbits 136
6.1 Bellstates 136
6.2 Quantumcryptography 137
6.3 Bitcommitment 143
6.4 Quantumdensecoding 146
6.5 Teleportation 149
6.6 TheGHZpuzzle 154
Appendices 159
A. Vectorspaces:basicpropertiesandDiracnotation 159
B. Structureofthegeneral1-Qbitunitarytransformation 168
C. Structureofthegeneral1-Qbitstate 173
D. Spookyactionatadistance 175
E. ConsistencyofthegeneralizedBornrule 181
F. OtheraspectsofDeutsch’sproblem 183
G. TheprobabilityofsuccessinSimon’sproblem 187
H. OnewaytomakeacNOTgate 189
I. Alittleelementarygrouptheory 193
J. Somesimplenumbertheory 195
K. Periodfindingandcontinuedfractions 197
L. Betterestimatesofsuccessinperiodfinding 201
CONTENTS ix
M. Factoringandperiodfinding 203
N. Shor’s9-Qbiterror-correctingcode 207
O. Acircuit-diagrammatictreatmentofthe7-Qbitcode 210
P. Onbitcommitment 216
Index 218