Introduction to Quantum Information Science

338 Pages·2015·4.49 MB·English

by  Masahito Hayashi

Graduate Texts in Physics
Masahito Hayashi
Satoshi Ishizaka
Akinori Kawachi
Gen Kimura
Tomohiro Ogawa
Introduction
to Quantum
Information
Science
Graduate Texts in Physics
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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
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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
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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