Introduction to Quantum Information Science PDF
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Graduate Texts in Physics Masahito Hayashi Satoshi Ishizaka Akinori Kawachi Gen Kimura Tomohiro Ogawa Introduction to Quantum Information Science Graduate Texts in Physics For furthervolumes: http://www.springer.com/series/8431 Graduate Texts in Physics Graduate Texts in Physics publishes core learning/teaching material for graduate- andadvanced-levelundergraduatecoursesontopicsofcurrentandemergingfields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, defi- nitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didacticstyle,comprehensivenessandcoverageoffundamentalmaterialalsomake them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field. Series editors Professor William T.Rhodes Department of Computer andElectrical Engineering andComputer Science, ImagingScience and TechnologyCenter Florida Atlantic University 777GladesRoadSE, Room 456 Boca Raton,FL, 33431,USA [email protected] Professor H.Eugene Stanley Center forPolymer Studies Department ofPhysics Boston University 590Commonwealth Avenue,Room 204B, Boston,MA, 02215, USA [email protected] Professor Richard Needs CavendishLaboratory JJ ThomsonAvenue, Cambridge CB3 0HE,UK [email protected] Professor MartinStutzmann Technische Universität München Am Coulombwall, Garching, 85747,Germany [email protected] Professor Susan Scott Department of QuantumScience Australian National University Canberra, ACT,0200, Australia [email protected] Masahito Hayashi Satoshi Ishizaka • Akinori Kawachi Gen Kimura • Tomohiro Ogawa Introduction to Quantum Information Science 123 MasahitoHayashi GenKimura Graduate School ofMathematics College ofSystems Engineering NagoyaUniversity andScience Nagoya Shibaura Instituteof Technology Japan Saitama Japan Satoshi Ishizaka Graduate School ofIntegrated Arts Tomohiro Ogawa andSciences Graduate School ofInformation Systems HiroshimaUniversity Universityof Electro-Communications Higashi-Hiroshima Tokyo Japan Japan AkinoriKawachi Department of Mathematical andComputingSciences TokyoInstituteof Technology Tokyo Japan LicenseforEnglisheditiononly! ISSN 1868-4513 ISSN 1868-4521 (electronic) ISBN 978-3-662-43501-4 ISBN 978-3-662-43502-1 (eBook) DOI 10.1007/978-3-662-43502-1 Springer Heidelberg NewYork Dordrecht London LibraryofCongressControlNumber:2014942016 (cid:2)Springer-VerlagBerlinHeidelberg2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Original Japanese edition published by Kyoritsu shuppan, Tokyo, 2012 Preface Have you heard of quantum teleportation, quantum cryptography, or quantum computation? These seem science fiction, but are truly most-advanced scientific topics that are growing involving physics, information science, and mathematics. This area, called quantum information science, is information science based on ‘‘quantum theory,’’ which is a fundamental theory of physics in the microscopic world. This area has the potential to produce fascinating technology for teleportation, unconditionally secure cryptography, and ultrahigh-speed computer. Unfortu- nately, although this is an emerging area, non-experts, especially undergraduate students, have no sufficient opportunities to glance at this topic. Consideringthiscircumstance,professorsinquantuminformationsciencehave brought out this textbook, which explains the fundamentals of quantum informa- tion science. This book requires only first-year calculus, first-year linear algebra, andelementaryprobabilitytheoryasbackgroundknowledge,anddoesnotrequire any knowledge of quantum theory and information science so that undergraduate students can read this textbook independently. Before the publication of the ori- ginalJapaneseversion,confrontedwiththeproblemthatthereisnoundergraduate course for quantum information science in Japan, the authors organized Winter School of Quantum Information Science in the seminar house in Tohoku Uni- versity in Japan in 2009, 2010, and 2011. The authors published the following original Japanese version based on the lecture materials and the participants’ responses in the above winter school. Title: Introduction to Quantum Information Science Japanese title: ryoushi jouhou kagaku nyuumon Publisher: Kyoritsu shuppan Year: May, 2012 Number of pages: 377 vii viii Preface Precise description of quantum information science requires various background knowledgeoftherelatedfields.Fortunately,sincethefiveauthorsofthisbookare from different backgrounds, this requirement is satisfied. After finishing the first manuscript, the authors adjusted the relation between chapters. Finally, Hayashi coordinated the whole organization. The organization of this book and the responsible persons for the respective chaptersaregivenasfollows.First,inChap. 1,Hayashiintroducesanoverviewof quantum information science and describes the details of this textbook as its coordinator. Please read Sect. 1.5 ‘‘Organization of this book’’ before reading the contents of this book. Next, in Chap. 2, using vectors and matrices, Kimura explains the simplified formulation of quantum theory as an expert on foundation ofquantumtheorysothatabeginnercaneasilyunderstandit.InChap. 3,Kawachi describes the foundations of quantum computation and quantum circuit as an expert of quantum computation. In Chap. 4, Kawachi treats quantum algorithms, which are algorithms for quantum computer. For example, Shor’s algorithm is treatedasaquantumalgorithmthatsolvesfactorizationproblembyusingquantum computer.InChap. 5,Kimuraexplainstheadvancedstructureofquantumtheory, which is necessary to learn quantum information science. In Chap. 6, Ogawa introduces various information quantities for quantum system as an expert on quantuminformationtheory.InChap. 7,Ishizakatreatsquantumentanglementas an expert on quantum entanglement and statistical physics. In Chap. 8, Ogawa explains quantum channel coding. In Chap. 9, Hayashi treats quantum error cor- recting code and quantum cryptography as an expert on quantum information theory and quantum cryptography. This book is organized so that Chaps. 2–4 can be read with elementary calculus for matrices and inner products of complex vectors.Thelatter chaptersrequire advanced knowledgefor linearalgebra,which are summarized in Appendix A by Kimura, Hayashi, and Ogawa. Sincethisbookcoversvariousfieldsinquantuminformation,itcanbeusedasa textforalecturecourseoraseminar.Especially,sinceitcontainsmanyexercises withsolutionsinAppendixB,italsocanbeusedforanexercisecourse.Further,it also treats a recent development in quantum information science. Hence, the readercaninvestigatemoreadvancedtopicsbyusingthereferencesafterfinishing this book. The authors hope that readers develop interest in quantum information science. Finally, the authors express their gratitude to Graduate School of Information Sciences,TohokuUniversity,Grant-in-AidforScientificResearchonPriorityArea ‘Deepening and Expansion of Statistical Mechanical Informatics (DEX-SMI)’, Global COE: Computationism as a Foundation for the Sciences (CompView inshort)ofTokyoInst.ofTech.,andtheorganizingcommitteemembersofWinter SchoolofQuantumInformationScienceforsupportingtheorganizationofWinter Preface ix School of Quantum Information Science. The authors are also grateful to Mr. Hideaki Hosaka, Dr. Shojun Nakayama, Mr. Yueri Wakakuwa in Tokyo University, Mr. Ryota Nakayama in Tohoku University, Dr. Yutaka Shikano in InstituteforMolecularScience,Dr.MasakazuYoshidainDoshisyaUniversity,and Mr. Seiji Mizukami, Mr. Yohei Kawaguchi, Mr. Ryo Yaguchi, Dr. Wataru KumagaiinNagoyaUniversity for making valuable comments for thisbook. The authors are thankful for Mr. Hideo Kotobuki and Ms Yoko Nakagawa for the publication of the original Japanese version. Finally, the authors express their appreciationtoDr.ClausE.AscheronofSpringerScience+BusinessMediaforhis encouragement and patience during the preparation ofthe manuscript. Masahito Hayashi Satoshi Ishizaka Akinori Kawachi Gen Kimura Tomohiro Ogawa Contents 1 Invitation to Quantum Information Science . . . . . . . . . . . . . . . . . 1 1.1 From Classical Information Science to Quantum Information Science. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Further Expansion of Quantum Information Science . . . . . . . . . 4 1.3 Feedback from Quantum Information Science to Physics. . . . . . 6 1.4 Toward Realization of Quantum Information Processing . . . . . . 8 1.5 Organization of This Book. . . . . . . . . . . . . . . . . . . . . . . . . . . 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Quantum Mechanics for Qubit Systems . . . . . . . . . . . . . . . . . . . . 13 2.1 Preliminary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Preparation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.1 Conceptual Preparation: Physical System, State, Measurement of Physical Quantity . . . . . . . . . . . . . . . . 15 2.2.2 Notational Preparation: Dirac Notation . . . . . . . . . . . . . 16 2.3 Qubit Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 A Qubit System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 Time Evolution in Qubit System. . . . . . . . . . . . . . . . . . 24 2.3.3 Composition of Qubit Systems: n-Qubit Systems . . . . . . 28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Foundations on Quantum Computing. . . . . . . . . . . . . . . . . . . . . . 37 3.1 What is Computation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Mathematical Notation for Information Science . . . . . . . . . . . . 40 3.3 Classical Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Quantum Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 Quantum Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Deutsch–Jozsa Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 xi
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