Table Of ContentCHAPMAN & HALL/CRC APPLIED MATHEMATICS
AND NONLINEAR SCIENCE SERIES
Mathematics of
Quantum Computation
and Quantum Technology
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CHAPMAN & HALL/CRC APPLIED MATHEMATICS
AND NONLINEAR SCIENCE SERIES
Series Editors Goong Chen andThomas J. Bridges
Published Titles
Computing with hp-ADAPTIVE FINITE ELEMENTS, Volume 1, One and Two
Dimensional Elliptic and Maxwell Problems, Leszek Demkowicz
Computing with hp-ADAPTIVE FINITE ELEMENTS, Volume 2, Frontiers:
Three Dimensional Elliptic and Maxwell Problems with Applications,
Leszek Demkowicz, Jason Kurtz, David Pardo, Maciej Paszyn´ski,
Waldemar Rachowicz, and Adam Zdunek
CRC Standard Curves and Surfaces with Mathematica®: Second Edition,
David H. von Seggern
Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations
in Mechanics and Physics, Victor A. Galaktionov and Sergey R. Svirshchevskii
Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications,
Victor A. Galaktionov
Introduction to Fuzzy Systems, Guanrong Chen and Trung Tat Pham
Introduction to non-Kerr Law Optical Solitons, Anjan Biswas and Swapan Konar
Introduction to Partial Differential Equations with MATLAB®,Matthew P. Coleman
Introduction to Quantum Control and Dynamics, Domenico D’Alessandro
Mathematical Methods in Physics and Engineering with Mathematica,
Ferdinand F. Cap
Mathematics of Quantum Computation and Quantum Technology, Goong Chen,
Louis Kauffman, and Samuel J. Lomonaco
Optimal Estimation of Dynamic Systems, John L. Crassidis and John L. Junkins
Quantum Computing Devices: Principles, Designs, and Analysis, Goong Chen,
David A. Church, Berthold-Georg Englert, Carsten Henkel, Bernd Rohwedder,
Marlan O. Scully, and M. Suhail Zubairy
Stochastic Partial Differential Equations, Pao-Liu Chow
Forthcoming Titles
Mathematical Theory of Quantum Computation, Goong Chen and Zijian Diao
Mixed Boundary Value Problems, Dean G. Duffy
Multi-Resolution Methods for Modeling and Control of Dynamical Systems,
John L. Junkins and Puneet Singla
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CHAPMAN & HALL/CRC APPLIED MATHEMATICS
AND NONLINEAR SCIENCE SERIES
Mathematics of
Quantum Computation
and Quantum Technology
Edited by
Goong Chen
Louis Kauffman
Samuel J. Lomonaco
Boca Raton London New York
Chapman & Hall/CRC is an imprint of the
Taylor & Francis Group, an informa business
C8997_FM.indd 3 8/16/07 3:15:28 PM
Chapman & Hall/CRC
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International Standard Book Number‑13: 978‑1‑58488‑899‑4 (Hardcover)
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Library of Congress Cataloging‑in‑Publication Data
Mathematics of quantum computation and quantum technology / editors, Goong
Chen, Louis Kauffman, Samuel J. Lomonaco.
p. cm. ‑‑ (Chapman & Hall/CRC applied mathematics and nonlinear
science series ; no. 14)
Includes bibliographical references and index.
ISBN 978‑1‑58488‑899‑4 (hardback : alk. paper)
1. Quantum computers‑‑Mathematics. 2. Quantum theory‑‑Mathematics. I.
Chen, Goong, 1950‑ II. Kauffman, Louis H., 1945‑ III. Lomonaco, Samuel J. IV.
Title. V. Series.
QA76.889.M383 2007
004.1‑‑dc22 2007024955
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
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Preface
Quantum computing is a vast and fascinating interdisciplinaryproject of the
21stcentury.Researchanddevelopmentinthismonumentalenterpriseinvolve
justabouteveryfieldofscienceandengineering. Inthisvolume,wefocuson
twoimportantdisciplines–mathematicsandphysics. Thechoiceismadewith
goodreasons:
(1) Mathematicsandmathematicianshaveplayedmajorrolesinthedevel-
opment of quantum computation. In the middle 1990’s Peter Shor’s
quantumfactoring algorithm generatedtremendousenthusiasm for the
pushtobuildthequantumcomputer. Shor’salgorithmwasthefirstex-
ampleshowingthat,inprinciple,quantumcomputerscouldout-perform
classical digital computers on problems of significance. This opened
wide the field for discovery of new quantum algorithms. This search
hasgoneonintandemwiththesearchfornewprinciplesandtechniques
that will make the new computers practical and actual. Mathematics
participatesinthisprocessatalllevels.
(2) Physicistsandengineersaretheprincipalplayersinthedesignandevo-
lutionofthequantumcomputer. Thetechnologicalaspectofbuildinga
scalablequantumcomputerwithina reasonabletime horizoniscrucial
forthefieldtoremainviable.Recentadvancesinhardwaredevelopmen-
tare extremelyencouraging. Such quantumtechnologyisthe material
cornerstone of quantum computing. The impact of this research goes
well beyond quantum computing, reaching nanotechnology, chemical
physics,condensedmatterphysicsandthefundamentalnatureofmatter
atthequantummechanicallevel.
Items(1)and(2),mathematicsandphysics,areinextricablylinked. Inthis
field of research we see mathematicians and physicists not only working to-
gether, but building common language and techniques that move back and
v
vi
forth across the disciplines. It has been remarkable to watch, over the last
few years, the extraordinaryclarity of articulationof basic quantumphysical
principlesthatis nowavailablebothforphysicistsandmathematicians. This
isfarbeyonda matterofsimpleexposition. New pointsofviewonquantum
theoryareemergingfromthesestudies, andaspectsofquantumtheory(such
asnon-localityandteleportation),previouslythoughttobemattersofphiloso-
phy,arenowunderstoodtobeattheverybasisofquantuminformationtheory
andthepracticeofquantumcomputation.
Based on the firm understanding that mathematics and physics are equal
partnersin the continuingdiscoveryof quantumcomputing,thethreeeditors
ofthisbookorganizedanNSFconferenceentitled“MathematicsofQuantum
Computation and Quantum Technology”, held at Texas A&M University in
November2005. Duringthe 3-day conference, many centraltopics were re-
portedandexamined,andvividdiscussionsensued.Thefundingorganizations
wereNSF,IMA(InstituteofMathematicsandItsApplications,Minneapolis,
U.S.A.)andtheTexasA&MUniversity.WeareespeciallygratefultoDr.Hen-
ry Warchall of NSF for providingthe largestshare of participantsupportvia
NSFGrantDMS0531131.
Thepresentvolumecontainsmaterialsmuchbroaderanddeeperthanwhat
were presentedatthe conference,dueto the generoustime frameforthe au-
thors to prepare their manuscripts. This is evidenced in the large number of
chapters,sixteenoftheminall,aswellasoversixhundredofpagesofpapers.
Morespecifically,thisvolumeconsistsoffourparts:
Part I: Quantum Computing—quantum algorithms and hidden subgroups,
quantumsearch,algorithmiccomplexityandquantumsimulation;
PartII:QuantumTechnology—mathtools,quantumwavefunctions,SQUID-
s,opticalquantumcomputing;
Part III: Quantum Information—quantumerror correction, quantum cryp-
tography,quantumentanglementandcommunication;and
Part IV:QuantumTopology,CategoricalAlgebra andLogic—knottheory,
category,algebraandlogic.
APanelReporttoNSFcontainingrecommendationsforfederalfundingon
themathematicalresearchonquantumcomputingisalsoattachedatthevery
endasanAppendix.
This book was written collectively by the authors of its many and diverse
chapters. Weareindebtedtothemfortheirinvaluablecontributions. Wealso
wish to thank the reviewers (several of them are not coauthors of any book
chapters)fortheirhelpfulreportsandcomments.
Ms. RobinCampbellhasdonethehighqualityeditorialworkinprocessing
and compiling the book chapters. Mr. Bob Stern of the Taylor and Francis
vii
Group has expedited the book publication in every way. Working together
withthemonthisbookprojectwasindeedagreatpleasure.
GoongChen
LouisKauffman
SamuelJ.Lomonaco
viii
Contributors
SamsonAbramsky Sydney,NewSouthWales
OxfordUniversityComputing Australia
Laboratory and
WolfsonBuilding DepartmentofPhysics
Parks,Road TheUniversityofQueensland
Oxford,OX13QD Brisbane,Queensland
England Australia
GraemeAhokas C.Bracher
DepartmentofComputerScience PhysicsDepartment
UniversityofCalgary BrynMawrCollege
Calgary,Alberta BrynMawr,PA 19010
Canada
and HowardE.Brandt
Institute for Quantum Information U.S.ArmyResearchLaboratory
Science Adelphi,MD 20783-1197
UniversityofCalgary
Calgary,Alberta GoongChen
Canada DepartmentofMathematics
andInstituteforQuantumStudies
SalahA.Aly TexasA&MUniversity
DepartmentofComputerScience CollegeStation,TX 77843-3368
TexasA&MUniversity
CollegeStation,TX77843-3112 RichardCleve
DavidR.CheritonSchoolof
SyedM.Assad ComputerScienceand
DepartmentofPhysics InstituteforQuantumComputing
NationalUniversityofSingapore UniversityofWaterloo
2ScienceDrive3 Waterloo,Ontario,Canada
Singapore117542 and
Singapore Perimeter Institute for Theoretical
Physics
DominicW.Berry Waterloo,Ontario,Canada
CentreforQuantumComputer
Technology
MacquarieUniversity
ix
BobCoecke AndreasKlappenecker
OxfordUniversityComputing DepartmentofComputerScience
Laboratory TexasA&MUniversity
WolfsonBuilding CollegeStation,TX 77843-3112
ParksRoad
OxfordOX13QD M.Kleber
UK Physik-DepartmentT30
TechnischeUniversita¨tMu¨nchen
LeonCohen James-Franck-Straße
CityUniversityofNewYork 85747Garching
695ParkAvenue Germany
NewYork,NY 10021
T.Kramer
ZijianDiao PhysicsDepartment
DepartmentofMathematics HarvardUniversity
OhioUniversity-Eastern OneOxfordStreet
St. Clairsville,OH43950 Cambridge,MA02138
JonathanP.Dowling HwangLee
HearneInstituteforTheoretical HearneInstituteforTheoretical
Physics Physics
DepartmentofPhysics DepartmentofPhysicsand
andAstronomy Astronomy
LouisianaStateUniversity LouisianaStateUniversity
BatonRouge,LA 70803-4001 BatonRouge,LA 70803-4001
Berthold-GeorgEnglert Fu-liLi
DepartmentofPhysics DepartmentofAppliedPhysics
NationalUniversityofSingapore Xi’anJiaotongUniversity
2ScienceDrive3 Xi’an710049
Singapore117542 China
Singapore
SamuelJ.Lomonaco
LouisH.Kauffman DepartmentofComputerScience
DepartmentofMathematics, andComputerEngineering
StatisticsandComputerScience UniversityofMaryland
University of Illinois at Chicago BaltimoreCounty(UMBC)
(UIC) Baltimore,MD 21250
Chicago,IL 60607
