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quantum computation and quantum information.pdf
Cover
Half-title
Series-title
Title
Copyright
Dedication
Contents
Introduction to the Tenth Anniversary Edition
Afterword to the Tenth Anniversary Edition
Preface
Structure of the book
How to use this book
Acknowledgements
Nomenclature and notation
Linear algebra and quantum mechanics
Information theory and probability
Miscellanea
Frequently used quantum gates and circuit symbols
I Fundamental concepts
1 Introduction and overview
1.1 Global perspectives
1.1.1 History of quantum computation and quantum information
Any algorithmic process can be simulated efficiently using a Turing machine
Any algorithmic process can be simulated efficiently using a probabilistic Turing machine
1.1.2 Future directions
1.2 Quantum bits
1.2.1 Multiple qubits
1.3 Quantum computation
1.3.1 Single qubit gates
1.3.2 Multiple qubit gates
1.3.3 Measurements in bases other than the computational basis
1.3.4 Quantum circuits
1.3.5 Qubit copying circuit?
1.3.6 Example: Bell states
1.3.7 Example: quantum teleportation
1.4 Quantum algorithms
1.4.1 Classical computations on a quantum computer
1.4.2 Quantum parallelism
1.4.3 Deutsch’s algorithm
1.4.4 The Deutsch–Jozsa algorithm
1.4.5 Quantum algorithms summarized
Quantum algorithms based upon the Fourier transform
Quantum search algorithms
Quantum simulation
The power of quantum computation
1.5 Experimental quantum information processing
1.5.1 The Stern–Gerlach experiment
1.5.2 Prospects for practical quantum information processing
1.6 Quantum information
1.6.1 Quantum information theory: example problems
Classical information through quantum channels
Quantum information through quantum channels
Quantum distinguishability
1.6.2 Quantum information in a wider context
History and further reading
2 Introduction to quantum mechanics
2.1 Linear algebra
2.1.1 Bases and linear independence
2.1.2 Linear operators and matrices
2.1.3 The Pauli matrices
2.1.4 Inner products
2.1.5 Eigenvectors and eigenvalues
2.1.6 Adjoints and Hermitian operators
2.1.7 Tensor products
2.1.8 Operator functions
2.1.9 The commutator and anti-commutator
2.1.10 The polar and singular value decompositions
2.2 The postulates of quantum mechanics
2.2.1 State space
2.2.2 Evolution
2.2.3 Quantum measurement
2.2.4 Distinguishing quantum states
2.2.5 Projective measurements
2.2.6 POVM measurements
2.2.7 Phase
2.2.8 Composite systems
2.2.9 Quantum mechanics: a global view
2.3 Application: superdense coding
2.4 The density operator
2.4.1 Ensembles of quantum states
2.4.2 General properties of the density operator
2.4.3 The reduced density operator
2.5 The Schmidt decomposition and purifications
2.6 EPR and the Bell inequality
History and further reading
3 Introduction to computer science
3.1 Models for computation
3.1.1 Turing machines
3.1.2 Circuits
3.2 The analysis of computational problems
3.2.1 How to quantify computational resources
Asymptotic notation: examples
3.2.2 Computational complexity
3.2.3 Decision problems and the complexity classes P and NP
3.3 Perspectives on computer science
History and further reading
II Quantum computation
4 Quantum circuits
4.1 Quantum algorithms
4.2 Single qubit operations
4.3 Controlled operations
4.4 Measurement
4.5 Universal quantum gates
4.5.1 Two-level unitary gates are universal
4.5.2 Single qubit and CNOT gates are universal
4.5.3 A discrete set of universal operations
Approximating unitary operators
Universality of Hadamard + phase…
4.5.4 Approximating arbitrary unitary gates is generically hard
4.5.5 Quantum computational complexity
4.6 Summary of the quantum circuit model of computation
4.7 Simulation of quantum systems
4.7.1 Simulation in action
4.7.2 The quantum simulation algorithm
4.7.3 An illustrative example
4.7.4 Perspectives on quantum simulation
History and further reading
5 The quantum Fourier transform and its applications
5.1 The quantum Fourier transform
5.2 Phase estimation
5.2.1 Performance and requirements
5.3 Applications: order-finding and factoring
5.3.1 Application: order-finding
The continued fraction expansion
Performance
5.3.2 Application: factoring
5.4 General applications of the quantum Fourier transform
5.4.1 Period-finding
5.4.2 Discrete logarithms
5.4.3 The hidden subgroup problem
5.4.4 Other quantum algorithms?
History and further reading
6 Quantum search algorithms
6.1 The quantum search algorithm
6.1.1 The oracle
6.1.2 The procedure
6.1.3 Geometric visualization
6.1.4 Performance
6.2 Quantum search as a quantum simulation
6.3 Quantum counting
6.4 Speeding up the solution of NP-complete problems
6.5 Quantum search of an unstructured database
6.6 Optimality of the search algorithm
6.7 Black box algorithm limits
History and further reading
7 Quantum computers: physical realization
7.1 Guiding principles
7.2 Conditions for quantum computation
7.2.1 Representation of quantum information
7.2.2 Performance of unitary transformations
7.2.3 Preparation of fiducial initial states
7.2.4 Measurement of output result
7.3 Harmonic oscillator quantum computer
7.3.1 Physical apparatus
7.3.2 The Hamiltonian
7.3.3 Quantum computation
7.3.4 Drawbacks
Harmonic oscillator quantum computer
7.4 Optical photon quantum computer
7.4.1 Physical apparatus
7.4.2 Quantum computation
7.4.3 Drawbacks
Optical photon quantum computer
7.5 Optical cavity quantum electrodynamics
7.5.1 Physical apparatus
Fabry–Perot cavity
Two-level atoms
7.5.2 The Hamiltonian
7.5.3 Single-photon single-atom absorption and refraction
7.5.4 Quantum computation
Optical cavity quantum electrodynamics
7.6 Ion traps
7.6.1 Physical apparatus
Trap geometry and lasers
Atomic structure
7.6.2 The Hamiltonian
7.6.3 Quantum computation
Single qubit operations
Controlled phase-flip gate
Swap gate
Controlled-NOT gate
Ion trap quantum computer
7.7 Nuclear magnetic resonance
7.7.1 Physical apparatus
7.7.2 The Hamiltonian
Single spin dynamics
Spin–spin couplings
Thermal equilibrium
Magnetization readout
Decoherence
7.7.3 Quantum computation
Refocusing
Controlled-NOT gate
Temporal, spatial, and logical labeling
Ensemble readout of quantum algorithm results
7.7.4 Experiment
State tomography
Quantum logic gates
Quantum algorithms
Drawbacks
7.8 Other implementation schemes
History and further reading
III Quantum information
8 Quantum noise and quantum operations
8.1 Classical noise and Markov processes
8.2 Quantum operations
8.2.1 Overview
8.2.2 Environments and quantum operations
8.2.3 Operator-sum representation
Physical interpretation of the operator-sum representation
Measurements and the operator-sum representation
System–environment models for any operator-sum representation
8.2.4 Axiomatic approach to quantum operations
Freedom in the operator-sum representation
8.3 Examples of quantum noise and quantum operations
8.3.1 Trace and partial trace
8.3.2 Geometric picture of single qubit quantum operations
8.3.3 Bit flip and phase flip channels
8.3.4 Depolarizing channel
8.3.5 Amplitude damping
8.3.6 Phase damping
8.4 Applications of quantum operations
8.4.1 Master equations
8.4.2 Quantum process tomography
8.5 Limitations of the quantum operations formalism
History and further reading
9 Distance measures for quantum information
9.1 Distance measures for classical information
9.2 How close are two quantum states?
9.2.1 Trace distance
9.2.2 Fidelity
9.2.3 Relationships between distance measures
9.3 How well does a quantum channel preserve information?
Quantum sources of information and the entanglement fidelity
History and further reading
10 Quantum error-correction
10.1 Introduction
10.1.1 The three qubit bit flip code
Improving the error analysis
10.1.2 Three qubit phase flip code
10.2 The Shor code
10.3 Theory of quantum error-correction
10.3.1 Discretization of the errors
10.3.2 Independent error models
10.3.3 Degenerate codes
10.3.4 The quantum Hamming bound
10.4 Constructing quantum codes
10.4.1 Classical linear codes
10.4.2 Calderbank–Shor–Steane codes
The Steane code
10.5 Stabilizer codes
10.5.1 The stabilizer formalism
10.5.2 Unitary gates and the stabilizer formalism
10.5.3 Measurement in the stabilizer formalism
10.5.4 The Gottesman–Knill theorem
10.5.5 Stabilizer code constructions
10.5.6 Examples
The three qubit bit flip code
The nine qubit Shor code
The five qubit code
CSS codes and the seven qubit code
10.5.7 Standard form for a stabilizer code
10.5.8 Quantum circuits for encoding, decoding, and correction
10.6 Fault-tolerant quantum computation
10.6.1 Fault-tolerance: the big picture
Fundamental issues
Fault-tolerant operations: definitions
Example: fault-tolerant controlled-NOT
Concatenated codes and the threshold theorem
10.6.2 Fault-tolerant quantum logic
Normalizer operations
Fault-tolerant π/8 gate
10.6.3 Fault-tolerant measurement
Measurement of stabilizer generators
10.6.4 Elements of resilient quantum computation
History and further reading
11 Entropy and information
11.1 Shannon entropy
11.2 Basic properties of entropy
11.2.1 The binary entropy
11.2.2 The relative entropy
11.2.3 Conditional entropy and mutual information
11.2.4 The data processing inequality
11.3 Von Neumann entropy
11.3.1 Quantum relative entropy
11.3.2 Basic properties of entropy
11.3.3 Measurements and entropy
11.3.4 Subadditivity
11.3.5 Concavity of the entropy
11.3.6 The entropy of a mixture of quantum states
11.4 Strong subadditivity
11.4.1 Proof of strong subadditivity
11.4.2 Strong subadditivity: elementary applications
History and further reading
12 Quantum information theory
12.1 Distinguishing quantum states and the accessible information
12.1.1 The Holevo bound
12.2 Data compression
12.2.1 Shannon’s noiseless channel coding theorem
12.2.2 Schumacher’s quantum noiseless channel coding theorem
12.3 Classical information over noisy quantum channels
12.3.1 Communication over noisy classical channels
Random coding for the binary symmetric channel
Shannon’s noisy channel coding theorem
12.3.2 Communication over noisy quantum channels
Random coding
Proof of the upper bound
Examples
12.4 Quantum information over noisy quantum channels
12.4.1 Entropy exchange and the quantum Fano inequality
12.4.2 The quantum data processing inequality
12.4.3 Quantum Singleton bound
12.4.4 Quantum error-correction, refrigeration and Maxwell’s demon
12.5 Entanglement as a physical resource
12.5.1 Transforming bi-partite pure state entanglement
12.5.2 Entanglement distillation and dilution
12.5.3 Entanglement distillation and quantum error-correction
12.6 Quantum cryptography
12.6.1 Private key cryptography
12.6.2 Privacy amplification and information reconciliation
CSS code privacy amplification & information reconciliation
12.6.3 Quantum key distribution
The BB84 protocol
The B92 protocol
12.6.4 Privacy and coherent information
12.6.5 The security of quantum key distribution
Requirements for a secure QKD protocol
Random sampling can upper-bound eavesdropping
The modified Lo–Chau protocol
A quantum error-correction protocol
Reduction to BB84
History and further reading
Appendix 1: Notes on basic probability theory
History and further reading
Appendix 2: Group theory
A2.1 Basic definitions
A2.1.1 Generators
A2.1.2 Cyclic groups
A2.1.3 Cosets
A2.2 Representations
A2.2.1 Equivalence and reducibility
A2.2.2 Orthogonality
A2.2.3 The regular representation
A2.3 Fourier transforms
History and further reading
Appendix 3: The Solovay–Kitaev theorem
History and further reading
Appendix 4: Number theory
A4.1 Fundamentals
A4.2 Modular arithmetic and Euclid’s algorithm
A4.3 Reduction of factoring to order-finding
A4.4 Continued fractions
History and further reading
Appendix 5: Public key cryptography and the RSA cryptosystem
History and further reading
Appendix 6: Proof of Lieb’s theorem
History and further reading
Bibliography
Index
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Quantum Computation and Quantum Information
10th Anniversary Edition
One of the most cited books in physics of all time, Quantum Computation and Quantum
Information remains the best textbook in this exciting field of science. This 10th
Anniversary Edition includes a new Introduction and Afterword from the authors
setting the work in context.
This comprehensive textbook describes such remarkable effects as fast quantum
algorithms, quantum teleportation, quantum cryptography, and quantum
error-correction. Quantum mechanics and computer science are introduced, before
moving on to describe what a quantum computer is, how it can be used to solve problems
faster than “classical” computers, and its real-world implementation. It concludes with
an in-depth treatment of quantum information.
Containing a wealth of figures and exercises, this well-known textbook is ideal for
courses on the subject, and will interest beginning graduate students and researchers in
physics, computer science, mathematics, and electrical engineering.
MICHAEL NIELSEN was educated at the University of Queensland, and as a Fulbright
Scholar at the University of New Mexico. He worked at Los Alamos National
Laboratory, as the Richard Chace Tolman Fellow at Caltech, was Foundation Professor
of Quantum Information Science and a Federation Fellow at the University of
Queensland, and a Senior Faculty Member at the Perimeter Institute for Theoretical
Physics. He left Perimeter Institute to write a book about open science and now lives in
Toronto.
ISAAC CHUANG is a Professor at the Massachusetts Institute of Technology, jointly
appointed in Electrical Engineering & Computer Science, and in Physics. He leads the
quanta research group at the Center for Ultracold Atoms, in the MIT Research
Laboratory of Electronics, which seeks to understand and create information technology
and intelligence from the fundamental building blocks of physical systems, atoms, and
molecules.
In praise of the book 10 years after publication
Ten years after its initial publication, “Mike and Ike” (as it’s affectionately called) remains the quantum
computing textbook to which all others are compared. No other book in the field matches its scope:
from experimental implementation to complexity classes, from the philosophical justifications for the
Church-Turing Thesis to the nitty-gritty of bra/ket manipulation. A dog-eared copy sits on my desk;
the section on trace distance and fidelity alone has been worth many times the price of the book to me.
Scott Aaronson, Massachusetts Institute of Technology
Quantum information processing has become a huge interdisciplinary field at the intersection of both,
theoretical and experimental quantum physics, computer science, mathematics, quantum engineering
and, more recently, even quantum metrology. The book by Michael Nielsen and Isaac Chuang was
seminal in many ways: it paved the way for a broader, yet deep understanding of the underlying
science, it introduced a common language now widely used by a growing community and it became
the standard book in the field for a whole decade. In spite of the fast progress in the field, even after
10 years the book provides the basic introduction into the field for students and scholars alike and
the 10th anniversary edition will remain a bestseller for a long time to come. The foundations of
quantum computation and quantum information processing are excellently laid out in this book and
it also provides an overview over some experimental techniques that have become the testing ground
for quantum information processing during the last decade. In view of the rapid progress of the field
the book will continue to be extremely valuable for all entering this highly interdisciplinary research
area and it will always provide the reference for those who grew up with it. This is an excellent book,
well written, highly commendable, and in fact imperative for everybody in the field.
Rainer Blatt, Universtit¨at Innsbruck
My well-perused copy of Nielsen and Chuang is, as always, close at hand as I write this. It appears
that the material that Mike and Ike chose to cover, which was a lot, has turned out to be a large portion
of what will become the eternal verities of this still-young field. When another researcher asks me to
give her a clear explanation of some important point of quantum information science, I breathe a sigh
of relief when I recall that it is in this book – my job is easy, I just send her there.
David DiVincenzo, IBM T. J. Watson Research Center
If there is anything you want to know, or remind yourself, about quantum information science, then
look no further than this comprehensive compendium by Ike and Mike. Whether you are an expert, a
student or a casual reader, tap into this treasure chest of useful and well presented information.
Artur Ekert, Mathematical Institute, University of Oxford
Nearly every child who has read Harry Potter believes that if you just say the right thing or do the
right thing, you can coerce matter to do something fantastic. But what adult would believe it? Until
quantum computation and quantum information came along in the early 1990s, nearly none. The
quantum computer is the Philosopher’s Stone of our century, and Nielsen and Chuang is our basic
book of incantations. Ten years have passed since its publication, and it is as basic to the field as it
ever was. Matter will do wonderful things if asked to, but we must first understand its language. No
book written since (there was no before) does the job of teaching the language of quantum theory’s
possibilities like Nielsen and Chuang’s.
Chris Fuchs, Perimeter Institute for Theoretical Physics
Nielsen and Chuang is the bible of the quantum information field. It appeared 10 years ago, yet even
though the field has changed enormously in these 10 years - the book still covers most of the important
concepts of the field.
Lov Grover, Bell Labs
Quantum Computation and Quantum Information, commonly referred to as “Mike and Ike,” continues
to be a most valuable resource for background information on quantum information processing. As a
mathematically-impaired experimentalist, I particularly appreciate the fact that armed with a modest
background in quantum mechanics, it is possible to pick up at any point in the book and readily grasp
the basic ideas being discussed. To me, it is still “the” book on the subject.
David Wineland, National Institute of Standards and Technology, Boulder, Colorado
Endorsements for the original publication
Chuang and Nielsen have produced the first comprehensive study of quantum computation. To
develop a robust understanding of this subject one must integrate many ideas whose origins are
variously within physics, computer science, or mathematics. Until this text, putting together the
essential material, much less mastering it, has been a challenge. Our Universe has intrinsic capa-
bilities and limitations on the processing of information. What these are will ultimately determine
the course of technology and shape our efforts to find a fundamental physical theory. This book is
an excellent way for any scientist or graduate student – in any of the related fields – to enter the
discussion.
Michael Freedman, Fields Medalist, Microsoft
Nielsen and Chuang’s new text is remarkably thorough and up-to-date, covering many aspects
of this rapidly evolving field from a physics perspective, complementing the computer science
perspective of Gruska’s 1999 text. The authors have succeeded in producing a self-contained book
accessible to anyone with a good undergraduate grounding in math, computer science or physical
sciences. An independent student could spend an enjoyable year reading this book and emerge ready
to tackle the current literature and do serious research. To streamline the exposition, footnotes have
been gathered into short but lively History and Further Reading sections at the end of each chapter.
Charles H Bennett, IBM
This is an excellent book. The field is already too big to cover completely in one book, but Nielsen
and Chuang have made a good selection of topics, and explain the topics they have chosen very
well.
Peter Shor, Massachusetts Institute of Technology
Quantum Computation and Quantum Information
10th Anniversary Edition
Michael A. Nielsen & Isaac L. Chuang
C A M B R I D G E U N I V E R S I T Y P R E S S
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
S˜ao Paulo, Delhi, Dubai, Tokyo, Mexico City
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/9781107002173
C M. Nielsen and I. Chuang 2010
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2000
Reprinted 2002, 2003, 2004, 2007, 2009
10th Anniversary edition published 2010
Printed in the United Kingdom at the University Press, Cambridge
A catalog record for this publication is available from the British Library
ISBN 978-1-107-00217-3 Hardback
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party internet websites referred to in
this publication, and does not guarantee that any content on such websites is,
or will remain, accurate or appropriate.
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