Quantum Computation and Quantum Information

Front Cover
Cambridge University Press, Oct 23, 2000 - Computers - 676 pages
In this first comprehensive introduction to the main ideas and techniques of quantum computation and information, Michael Nielsen and Isaac Chuang ask the question: What are the ultimate physical limits to computation and communication? They detail such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography and quantum error correction. A wealth of accompanying figures and exercises illustrate and develop the material in more depth. They describe what a quantum computer is, how it can be used to solve problems faster than familiar "classical" computers, and the real-world implementation of quantum computers. Their book concludes with an explanation of how quantum states can be used to perform remarkable feats of communication, and of how it is possible to protect quantum states against the effects of noise.
 

Selected pages

Contents

Fundamental concepts
1
111 History of quantum computation and quantum information
2
112 Future directions
12
12 Quantum bits
13
121 Multiple qubits
16
13 Quantum computation
17
132 Multiple qubit gates
20
133 Measurements in bases other than the computational basis
22
754 Quantum computation
306
76 Ion traps
309
762 The Hamiltonian
317
763 Quantum computation
319
764 Experiment
321
77 Nuclear magnetic resonance
324
771 Physical apparatus
325
772 The Hamiltonian
326

135 Qubit copying circuit?
24
Bell states
25
quantum teleportation
26
14 Quantum algorithms
28
141 Classical computations on a quantum computer
29
142 Quantum parallelism
30
143 Deutschs algorithm
32
144 The DeutschJozsa algorithm
34
145 Quantum algorithms summarized
36
15 Experimental quantum information processing
42
151 The SternGerlach experiment
43
152 Prospects for practical quantum information processing
46
16 Quantum information
50
example problems
52
162 Quantum information in a wider context
58
Introduction to quantum mechanics
60
21 Linear algebra
61
211 Bases and linear independence
62
212 Linear operators and matrices
63
213 The Pauli matrices
65
215 Eigenvectors and eigenvalues
68
216 Adjoints and Hermitian operators
69
217 Tensor products
71
218 Operator functions
75
219 The commutator and anticommutator
76
2110 The polar and singular value decompositions
78
22 The postulates of quantum mechanics
80
222 Evolution
81
223 Quantum measurement
84
224 Distinguishing quantum states
86
225 Projective measurements
87
226 POVM measurements
90
227 Phase
93
a global view
96
superdense coding
97
24 The density operator
98
241 Ensembles of quantum states
99
242 General properties of the density operator
101
243 The reduced density operator
105
25 The Schmidt decomposition and purifications
109
26 EPR and the Bell inequality
111
Introduction to computer science
120
31 Models for computation
122
312 Circuits
129
32 The analysis of computational problems
135
321 How to quantify computational resources
136
322 Computational complexity
138
323 Decision problems and the complexity classes P and NP
141
324 A plethora of complexity classes
150
325 Energy and computation
153
33 Perspectives on computer science
161
Quantum computation
171
41 Quantum algorithms
172
42 Single qubit operations
174
43 Controlled operations
177
44 Measurement
185
45 Universal quantum gates
188
451 Twolevel unitary gates are universal
189
452 Single qubit and CNOT gates are universal
191
453 A discrete set of universal operations
194
454 Approximating arbitrary unitary gates is generically hard
198
455 Quantum computational complexity
200
46 Summary of the quantum circuit model of computation
202
47 Simulation of quantum systems
204
472 The quantum simulation algorithm
206
473 An illustrative example
209
474 Perspectives on quantum simulation
211
The quantum Fourier transform and its applications
216
51 The quantum Fourier transform
217
52 Phase estimation
221
521 Performance and requirements
223
orderfinding and factoring
226
factoring
232
54 General applications of the quantum Fourier transform
234
541 Periodfinding
236
542 Discrete logarithms
238
543 The hidden subgroup problem
240
544 Other quantum algorithms?
242
Quantum search algorithms
248
612 The procedure
250
613 Geometric visualization
252
614 Performance
253
62 Quantum search as a quantum simulation
255
63 Quantum counting
261
64 Speeding up the solution of NPcomplete problems
263
65 Quantum search of an unstructured database
265
66 Optimality of the search algorithm
269
67 Black box algorithm limits
271
Quantum computers physical realization
277
72 Conditions for quantum computation
279
722 Performance of unitary transformations
281
724 Measurement of output result
282
73 Harmonic oscillator quantum computer
283
732 The Hamiltonian
284
733 Quantum computation
286
74 Optical photon quantum computer
287
742 Quantum computation
290
743 Drawbacks
296
75 Optical cavity quantum electrodynamics
297
751 Physical apparatus
298
752 The Hamiltonian
300
753 Singlephoton singleatom absorption and refraction
303
773 Quantum computation
331
774 Experiment
336
78 Other implementation schemes
343
Quantum information
353
81 Classical noise and Markov processes
354
82 Quantum operations
356
822 Environments and quantum operations
357
823 Operatorsum representation
360
824 Axiomatic approach to quantum operations
366
83 Examples of quantum noise and quantum operations
373
831 Trace and partial trace
374
833 Bit flip and phase flip channels
376
834 Depolarizing channel
378
835 Amplitude damping
380
836 Phase damping
383
84 Applications of quantum operations
386
842 Quantum process tomography
389
85 Limitations of the quantum operations formalism
394
Distance measures for quantum information
399
92 How close are two quantum states?
403
922 Fidelity
409
923 Relationships between distance measures
415
93 How well does a quantum channel preserve information?
416
Quantum errorcorrection
425
101 Introduction
426
1011 The three qubit bit flip code
427
1012 Three qubit phase flip code
430
102 The Shor code
432
103 Theory of quantum errorcorrection
435
1031 Diseretization of the errors
438
1032 Independent error models
441
1033 Degenerate codes
444
104 Constructing quantum codes
445
1042 CalderbankShorSteane codes
450
105 Stabilizer codes
453
1051 The stabilizer formalism
454
1052 Unitary gates and the stabilizer formalism
459
1053 Measurement in the stabilizer formalism
463
1054 The GottesmanKnill theorem
464
1056 Examples
467
1057 Standard form for a stabilizer code
470
1058 Quantum circuits for encoding decoding and correction
472
106 Faulttolerant quantum computation
474
the big picture
475
1062 Faulttolerant quantum logic
482
1063 Faulttolerant measurement
489
1064 Elements of resilient quantum computation
493
Entropy and information
500
112 Basic properties of entropy
502
1122 The relative entropy
504
1123 Conditional entropy and mutual information
505
1124 The data processing inequality
509
113 Von Neumann entropy
510
1131 Quantum relative entropy
511
1132 Basic properties of entropy
513
1133 Measurements and entropy
514
1134 Subadditivity
515
1135 Concavity of the entropy
516
1136 The entropy of a mixture of quantum states
518
114 Strong subadditivity
519
elementary applications
522
Quantum information theory
528
121 Distinguishing quantum states and the accessible information
529
1211 The Holevo bound
531
1212 Example applications of the Holevo bound
534
122 Data compression
536
1221 Shannons noiseless channel coding theorem
537
1222 Schumachers quantum noiseless channel coding theorem
542
123 Classical information over noisy quantum channels
546
1231 Communication over noisy classical channels
548
1232 Communication over noisy quantum channels
554
124 Quantum information over noisy quantum channels
561
1242 The quantum data processing inequality
564
1243 Quantum Singleton bound
568
1244 Quantum errorcorrection refrigeration and Maxwells demon
569
125 Entanglement as a physical resource
571
1251 Transforming bipartite pure state entanglement
573
1252 Entanglement distillation and dilution
578
1253 Entanglement distillation and quantum errorcorrection
580
126 Quantum cryptography
582
1262 Privacy amplification and information reconciliation
584
1263 Quantum key distribution
586
1264 Privacy and coherent information
592
1265 The security of quantum key distribution
593
Notes on basic probability theory
608
Group theory
610
A211 Generators
611
A213 Cosets
612
A222 Orthogonality
613
A223 The regular representation
614
A23 Fourier transforms
615
The SolovayKitaev theorem
617
Number theory
625
A42 Modular arithmetic and Euclids algorithm
626
A43 Reduction of factoring to orderfinding
633
A44 Continued fractions
635
Public key cryptography and the RSA cryptosystem
640
Proof of Liebs theorem
645
Bibliography
649
Index
665
Copyright

Other editions - View all

Quantum Computation and Quantum Information: 10th Anniversary Edition
Michael A. Nielsen,Isaac L. Chuang
Limited preview - 2010
Quantum Computation and Quantum Information
Michael A. Nielsen,Isaac L. Chuang
No preview available - 2000
Quantum Computation and Quantum Information
Michael A. Nielsen,Isaac L. Chuang
No preview available - 2000

Common terms and phrases

Alice and Bob ancilla applied arbitrary atom bit flip Bloch sphere C₁ chapter classical computer classical information CNOT computation and quantum computational basis construction controlled-NOT gate defined density matrix density operator described efficiently eigenstates eigenvalues encoded entanglement entropy equation error error-correcting codes example Exercise factor fault-tolerant fidelity Figure finite function gives Hadamard gate Hamiltonian implemented inequality input integer interaction known linear model of computation noise notation obtain operation elements oracle order-finding output Pauli perform phase estimation phase flip physical polynomial possible POVM probability problem procedure proof properties protocol prove quantum algorithms quantum circuit quantum codes quantum computation quantum error-correction quantum Fourier transform quantum information processing quantum information theory quantum mechanics quantum operation quantum search algorithm quantum system result Section Show simulation single qubit solve spin subadditivity Suppose Toffoli gate trace distance Turing machine unitary matrix unitary operator unitary transform vector space

References to this book

About the author (2000)

Dr. Michael Nielsen was born in Brisbane, Australia in 1974, and was educated at the University of Queensland, obtaining postgraduate degrees in mathematics and physics, before being awarded his PhD in physics at the University of New Mexico in 1998. He is currently the Tolman Postdoctoral Fellow and a Fulbright Scholar at the California Institute of Technology Dr. Isaac Chuang is a native of Louisville, KY. He received his doctorate in electrical engineering from Stanford University in 1997, where he was a Hertz Foundation Fellow, and holds two bachelors degrees and one masters degree in physics and electrical engineering from the Massachusetts Institute of Technology. He serves as a consulting professor at Stanford University. He joined IBM Research in 1998. In November 1999 he was named one of the top 100 young innovators of 1999.

Bibliographic information

TitleQuantum Computation and Quantum Information
Issue 2 of Cambridge Series on Information and the Natural Sciences
AuthorsMichael A. Nielsen, Isaac L. Chuang
Editionillustrated, reprint, revised
PublisherCambridge University Press, 2000
ISBN0521635039, 9780521635035
Length676 pages
Subjects ›  › 

Computers / Computer Science
Computers / General
Science / Physics / General
Science / Physics / Mathematical & Computational
Science / Physics / Quantum Theory
Technology & Engineering / Electronics / General
  
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