Classical and quantum computation:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Providence, R.I.
American Mathematical Society
2002
|
| Series: | Graduate studies in mathematics
47 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index. |
| Physical Description: | XIII, 257 S. graph. Darst. |
| ISBN: | 082182161X 0821832298 9780821832295 |
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MARC
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| 100 | 1 | |a Kitaev, Aleksej Ju. |d 1963- |e Verfasser |0 (DE-588)173560156 |4 aut | |
| 240 | 1 | 0 | |a Klassičeskie i kvantovye vyčislenija |
| 245 | 1 | 0 | |a Classical and quantum computation |c A.Yu. Kitaev, A.H. Shen, M.N. Vyalyi ; [translated from the Russian by Lester J. Senechal] |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 2002 | |
| 300 | |a XIII, 257 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate studies in mathematics |v 47 | |
| 500 | |a Includes bibliographical references and index. | ||
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Record in the Search Index
| _version_ | 1805071207751483392 |
|---|---|
| adam_text |
Contents
Foreword
vii
Notation
xi
Introduction
1
Part
1.
Classical Computation
9
1.
Turing machines
9
1.1.
Definition of a Turing machine
10
1.2.
Computable functions and decidable predicates
11
1.3.
Turing's thesis and universal machines
12
1.4.
Complexity classes
14
2.
Boolean circuits
17
2.1.
Definitions. Complete bases
17
2.2.
Circuits versus Turing machines
20
2.3.
Basic algorithms. Depth, space and width
23
3.
The class NP: Reducibility and completeness
27
3.1.
Nondeterministic Turing machines
27
3.2.
Reducibility and NP-completeness
30
4.
Probabilistic algorithms and the class BPP
36
4.1.
Definitions. Amplification of probability
36
4.2.
Primality testing
38
4.3.
BPP and circuit complexity
42
iii
iv Contents
5.
The hierarchy of complexity classes
44
5.1.
Games machines play
44
5.2.
The class PSPACE
48
Part
2.
Quantum Computation
53
6.
Definitions and notation
54
6.1.
The tensor product
54
6.2.
Linear algebra in Dirac's notation
55
6.3.
Quantum gates and circuits
58
7.
Correspondence between classical and quantum computation
60
8.
Bases for quantum circuits
65
8.1.
Exact realization
65
8.2.
Approximate realization
71
8.3.
Efficient approximation over a complete basis
75
9.
Definition of Quantum Computation. Examples
82
9.1.
Computation by quantum circuits
82
9.2.
Quantum search: Grover's algorithm
83
9.3.
A universal quantum circuit
88
9.4.
Quantum algorithms and the class BQP
89
10.
Quantum probability
92
10.1.
Probability for state vectors
92
10.2.
Mixed states (density matrices)
94
10.3.
Distance functions for density matrices
98
11.
Physically realizable transformations of density matrices
100
11.1.
Physically realizable
superoperators:
characterization
100
11.2.
Calculation of the probability for quantum computation
102
11.3.
Decoherence
102
11.4.
Measurements
105
11.5.
The
superoperator
norm
108
12.
Measuring operators
112
12.1.
Definition and examples
112
12.2.
General properties
114
12.3.
Garbage removal and composition of measurements
115
13.
Quantum algorithms for Abelian groups
116
Contents
v
13.1.
The problem of hidden subgroup in
(Ь-і)к\
Simon's
algorithm
117
13.2.
Factoring and finding the period for raising to a power
119
13.3.
Reduction of factoring to period finding
120
13.4.
Quantum algorithm for finding the period: the basic idea
122
13.5.
The phase estimation procedure
125
13.6.
Discussion of the algorithm
130
13.7.
Parallelized version of phase estimation. Applications
131
13.8.
The hidden subgroup problem for
Ћк
135
14.
The quantum analogue of NP: the class BQNP
138
14.1.
Modification of classical definitions
138
14.2.
Quantum definition by analogy
139
14.3.
Complete problems
141
14.4.
Local Hamiltonian is BQNP-complete
144
14.5.
The place of BQNP among other complexity classes
150
15.
Classical and quantum codes
151
15.1.
Classical codes
153
15.2.
Examples of classical codes
154
15.3.
Linear codes
155
15.4.
Error models for quantum codes
156
15.5.
Definition of quantum error correction
158
15.6.
Shor's code
161
15.7.
The
Pauli
operators and symplectic transformations
163
15.8.
Symplectic (stabilizer) codes
167
15.9.
Toric code
170
15.10.
Error correction for symplectic codes
172
15.11.
Anyons (an example based on the toric code)
173
Part
3.
Solutions
177
51. Problems of Section
1 177
52. Problems of Section
2 183
53.
Problems of Section
3 195
55. Problems of Section
5 202
56. Problems of Section
6 203
vi
Contents
57.
Problems
of Section
7 204
58. Problems of Section
8 204
59. Problems of Section
9 216
510. Problems of Section
10 221
511.
Problems of Section
11 224
5
12.
Problems of Section
12 230
513. Problems of Section
13 230
S15. Problems of Section
15 234
Appendix A. Elementary Number Theory
237
A.I. Modular arithmetic and rings
237
A.
2.
Greatest common divisor and unique factorization
239
A.3. Chinese remainder theorem
241
A.
4.
The structure of finite Abelian groups
243
A.5. The structure of the group
(Z/çZ)*
245
A.
6.
Euclid's algorithm
247
A.
7.
Continued fractions
248
Bibliography
251
Index
255 |
| any_adam_object | 1 |
| author | Kitaev, Aleksej Ju. 1963- Šen, Aleksandr Ch. 1958- Vjalyj, Michail N. 1961- |
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| language | English Russian |
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| spelling | Kitaev, Aleksej Ju. 1963- Verfasser (DE-588)173560156 aut Klassičeskie i kvantovye vyčislenija Classical and quantum computation A.Yu. Kitaev, A.H. Shen, M.N. Vyalyi ; [translated from the Russian by Lester J. Senechal] Providence, R.I. American Mathematical Society 2002 XIII, 257 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 47 Includes bibliographical references and index. Quantencomputer (DE-588)4533372-5 gnd rswk-swf Berechnungskomplexität (DE-588)4134751-1 gnd rswk-swf Quantencomputer (DE-588)4533372-5 s Berechnungskomplexität (DE-588)4134751-1 s DE-604 Šen, Aleksandr Ch. 1958- Verfasser (DE-588)130558648 aut Vjalyj, Michail N. 1961- Verfasser (DE-588)173560199 aut Graduate studies in mathematics 47 (DE-604)BV009739289 47 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009848433&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kitaev, Aleksej Ju. 1963- Šen, Aleksandr Ch. 1958- Vjalyj, Michail N. 1961- Classical and quantum computation Graduate studies in mathematics Quantencomputer (DE-588)4533372-5 gnd Berechnungskomplexität (DE-588)4134751-1 gnd |
| subject_GND | (DE-588)4533372-5 (DE-588)4134751-1 |
| title | Classical and quantum computation |
| title_alt | Klassičeskie i kvantovye vyčislenija |
| title_auth | Classical and quantum computation |
| title_exact_search | Classical and quantum computation |
| title_full | Classical and quantum computation A.Yu. Kitaev, A.H. Shen, M.N. Vyalyi ; [translated from the Russian by Lester J. Senechal] |
| title_fullStr | Classical and quantum computation A.Yu. Kitaev, A.H. Shen, M.N. Vyalyi ; [translated from the Russian by Lester J. Senechal] |
| title_full_unstemmed | Classical and quantum computation A.Yu. Kitaev, A.H. Shen, M.N. Vyalyi ; [translated from the Russian by Lester J. Senechal] |
| title_short | Classical and quantum computation |
| title_sort | classical and quantum computation |
| topic | Quantencomputer (DE-588)4533372-5 gnd Berechnungskomplexität (DE-588)4134751-1 gnd |
| topic_facet | Quantencomputer Berechnungskomplexität |
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