Mathematics of Quantum Computation and Quantum Technology PDF
Preview Mathematics of Quantum Computation and Quantum Technology
CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES Mathematics of Quantum Computation and Quantum Technology C8997_FM.indd 1 8/16/07 3:15:28 PM 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 C8997_FM.indd 2 8/16/07 3:15:28 PM 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 Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑1‑58488‑899‑4 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse‑ quences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. 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 http://www.crcpress.com TC&8F99_7L_OFCM_.Ain_dMda s 4ter.indd 1 78//1186//0077 63::4145::2228 APMM 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
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