An introduction to quantum computing:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2007
|
| Edition: | 1. published |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XI, 274 Seiten Diagramme |
| ISBN: | 9780198570004 9780198570493 0198570007 019857049X |
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| 245 | 1 | 0 | |a An introduction to quantum computing |c Philip Kaye, Raymond Laflamme and Michele Mosca |
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| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2007 | |
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Contents
Preface x
Acknowledgements
xi
1
INTRODUCTION AND BACKGROUND
1
1.1
Overview
1
1.2
Computers and the Strong Church-Turing Thesis
2
1.3
The Circuit Model of Computation
6
1.4
A Linear Algebra Formulation of the Circuit Model
8
1.5
Reversible Computation
12
1.6
A Preview of Quantum Physics
15
1.7
Quantum Physics and Computation
19
2
LINEAR ALGEBRA AND THE DIRAC NOTATION
21
2.1
The Dirac Notation and Hubert Spaces
21
2.2
Dual Vectors
23
2.3
Operators
27
2.4
The Spectral Theorem
30
2.5
Functions of Operators
32
2.6
Tensor Products
33
2.7
The Schmidt Decomposition Theorem
35
2.8
Some Comments on the Dirac Notation
37
3
QUBITS AND THE FRAMEWORK OF QUANTUM
MECHANICS
38
3.1
The State of a Quantum System
38
3.2
Time-Evolution of a Closed System
43
3.3
Composite Systems
45
3.4
Measurement
48
vi
CONTENTS
3.5
Mixed States and General Quantum Operations
53
3.5.1
Mixed States
53
3.5.2
Partial Trace
56
3.5.3
General Quantum Operations
59
4
A QUANTUM MODEL OF COMPUTATION
61
4.1
The Quantum Circuit Model
61
4.2
Quantum Gates
63
4.2.1
1-Qubit Gates
63
4.2.2
Controlled-i/ Gates
66
4.3
Universal Sets of Quantum Gates
68
4.4
Efficiency of Approximating Unitary Transformations
71
4.5
Implementing Measurements with Quantum Circuits
73
5
SUPERDENSE
CODING AND QUANTUM
TELEPORTATION
78
5.1
Superdense
Coding
79
5.2
Quantum
Teleportation
80
5.3
An Application of Quantum
Teleportation
82
6
INTRODUCTORY QUANTUM ALGORITHMS
86
6.1
Probabilistic Versus Quantum Algorithms
86
6.2
Phase Kick-Back
91
6.3
The
Deutsch
Algorithm
94
6.4
The
Deutsch-
Jozsa Algorithm
99
6.5
Simon's Algorithm
103
7
ALGORITHMS WITH SUPERPOLYNOMIAL
SPEED-UP
110
7.1
Quantum Phase Estimation and the Quantum Fourier Trans¬
form
110
7.1.1
Error Analysis for Estimating Arbitrary Phases
117
7.1.2
Periodic States
120
7.1.3
GCD, LCM, the Extended Euclidean Algorithm
124
7.2
Eigenvalue Estimation
125
CONTENTS
vii
7.3
Finding-Orders
130
7.3.1
The Order-Finding Problem
130
7.3.2
Some Mathematical Preliminaries
131
7.3.3
The Eigenvalue Estimation Approach to Order Find¬
ing
134
7.3.4
Shor's Approach to Order Finding
139
7.4
Finding Discrete Logarithms
142
7.5
Hidden Subgroups
146
7.5.1
More on Quantum Fourier Transforms
147
7.5.2
Algorithm for the Finite Abelian Hidden Subgroup
Problem
149
7.6
Related Algorithms and Techniques
151
8
ALGORITHMS BASED ON AMPLITUDE
AMPLIFICATION
152
8.1
Grover's Quantum Search Algorithm
152
8.2
Amplitude Amplification
163
8.3
Quantum Amplitude Estimation and Quantum Counting
170
8.4
Searching Without Knowing the Success Probability
175
8.5
Related Algorithms and Techniques
178
9
QUANTUM COMPUTATIONAL COMPLEXITY THEORY
AND LOWER BOUNDS
179
9.1
Computational Complexity
180
9.1.1
Language Recognition Problems and Complexity
Classes
181
9.2
The Black-Box Model
185
9.2.1
State Distinguishability
187
9.3
Lower Bounds for Searching in the Black-Box Model: Hybrid
Method
188
9.4
General Black-Box Lower Bounds
191
9.5
Polynomial Method
193
9.5.1
Applications to Lower Bounds
194
9.5.2
Examples of Polynomial Method Lower Bounds
196
viii CONTENTS
9.6
Block Sensitivity
197
9.6.1
Examples of Block Sensitivity Lower Bounds
197
9.7
Adversary Methods
198
9.7.1
Examples of Adversary Lower Bounds
200
9.7.2
Generalizations
203
10
QUANTUM ERROR CORRECTION
204
10.1
Classical Error Correction
204
10.1.1
The Error Model
205
10.1.2
Encoding
206
10.1.3
Error Recovery
207
10.2
The Classical Three-Bit Code
207
10.3
Fault Tolerance
211
10.4
Quantum Error Correction
212
10.4.1
Error Models for Quantum Computing
213
10.4.2
Encoding
216
10.4.3
Error Recovery
217
10.5
Three- and Nine-Qubit Quantum Codes
223
10.5.1
The Three-Qubit Code for Bit-Flip Errors
223
10.5.2
The Three-Qubit Code for Phase-Flip Errors
225
10.5.3
Quantum Error Correction Without Decoding
226
10.5.4
The Nine-Qubit Shor Code
230
10.6
Fault-Tolerant Quantum Computation
234
10.6.1
Concatenation of Codes and the Threshold Theorem
237
APPENDIX A
241
A.I Tools for Analysing Probabilistic Algorithms
241
A.
2
Solving the Discrete Logarithm Problem When the Order of
α
Is Composite
243
A.3 How Many Random Samples Are Needed to Generate
a Group?
245
A.4 Finding
r
Given | for Random
к
247
A.
5
Adversary Method Lemma
248
CONTENTS ¡x
Α.
6
Black-Boxes for Group Computations
250
A.
7
Computing Schmidt Decompositions
253
A.
8
General Measurements
255
A.9 Optimal Distinguishing of Two States
258
A.
9.1
A Simple Procedure
258
A.9.2 Optimality of This Simple Procedure
258
Bibliography
260
Index
270 |
| adam_txt |
Contents
Preface x
Acknowledgements
xi
1
INTRODUCTION AND BACKGROUND
1
1.1
Overview
1
1.2
Computers and the Strong Church-Turing Thesis
2
1.3
The Circuit Model of Computation
6
1.4
A Linear Algebra Formulation of the Circuit Model
8
1.5
Reversible Computation
12
1.6
A Preview of Quantum Physics
15
1.7
Quantum Physics and Computation
19
2
LINEAR ALGEBRA AND THE DIRAC NOTATION
21
2.1
The Dirac Notation and Hubert Spaces
21
2.2
Dual Vectors
23
2.3
Operators
27
2.4
The Spectral Theorem
30
2.5
Functions of Operators
32
2.6
Tensor Products
33
2.7
The Schmidt Decomposition Theorem
35
2.8
Some Comments on the Dirac Notation
37
3
QUBITS AND THE FRAMEWORK OF QUANTUM
MECHANICS
38
3.1
The State of a Quantum System
38
3.2
Time-Evolution of a Closed System
43
3.3
Composite Systems
45
3.4
Measurement
48
vi
CONTENTS
3.5
Mixed States and General Quantum Operations
53
3.5.1
Mixed States
53
3.5.2
Partial Trace
56
3.5.3
General Quantum Operations
59
4
A QUANTUM MODEL OF COMPUTATION
61
4.1
The Quantum Circuit Model
61
4.2
Quantum Gates
63
4.2.1
1-Qubit Gates
63
4.2.2
Controlled-i/ Gates
66
4.3
Universal Sets of Quantum Gates
68
4.4
Efficiency of Approximating Unitary Transformations
71
4.5
Implementing Measurements with Quantum Circuits
73
5
SUPERDENSE
CODING AND QUANTUM
TELEPORTATION
78
5.1
Superdense
Coding
79
5.2
Quantum
Teleportation
80
5.3
An Application of Quantum
Teleportation
82
6
INTRODUCTORY QUANTUM ALGORITHMS
86
6.1
Probabilistic Versus Quantum Algorithms
86
6.2
Phase Kick-Back
91
6.3
The
Deutsch
Algorithm
94
6.4
The
Deutsch-
Jozsa Algorithm
99
6.5
Simon's Algorithm
103
7
ALGORITHMS WITH SUPERPOLYNOMIAL
SPEED-UP
110
7.1
Quantum Phase Estimation and the Quantum Fourier Trans¬
form
110
7.1.1
Error Analysis for Estimating Arbitrary Phases
117
7.1.2
Periodic States
120
7.1.3
GCD, LCM, the Extended Euclidean Algorithm
124
7.2
Eigenvalue Estimation
125
CONTENTS
vii
7.3
Finding-Orders
130
7.3.1
The Order-Finding Problem
130
7.3.2
Some Mathematical Preliminaries
131
7.3.3
The Eigenvalue Estimation Approach to Order Find¬
ing
134
7.3.4
Shor's Approach to Order Finding
139
7.4
Finding Discrete Logarithms
142
7.5
Hidden Subgroups
146
7.5.1
More on Quantum Fourier Transforms
147
7.5.2
Algorithm for the Finite Abelian Hidden Subgroup
Problem
149
7.6
Related Algorithms and Techniques
151
8
ALGORITHMS BASED ON AMPLITUDE
AMPLIFICATION
152
8.1
Grover's Quantum Search Algorithm
152
8.2
Amplitude Amplification
163
8.3
Quantum Amplitude Estimation and Quantum Counting
170
8.4
Searching Without Knowing the Success Probability
175
8.5
Related Algorithms and Techniques
178
9
QUANTUM COMPUTATIONAL COMPLEXITY THEORY
AND LOWER BOUNDS
179
9.1
Computational Complexity
180
9.1.1
Language Recognition Problems and Complexity
Classes
181
9.2
The Black-Box Model
185
9.2.1
State Distinguishability
187
9.3
Lower Bounds for Searching in the Black-Box Model: Hybrid
Method
188
9.4
General Black-Box Lower Bounds
191
9.5
Polynomial Method
193
9.5.1
Applications to Lower Bounds
194
9.5.2
Examples of Polynomial Method Lower Bounds
196
viii CONTENTS
9.6
Block Sensitivity
197
9.6.1
Examples of Block Sensitivity Lower Bounds
197
9.7
Adversary Methods
198
9.7.1
Examples of Adversary Lower Bounds
200
9.7.2
Generalizations
203
10
QUANTUM ERROR CORRECTION
204
10.1
Classical Error Correction
204
10.1.1
The Error Model
205
10.1.2
Encoding
206
10.1.3
Error Recovery
207
10.2
The Classical Three-Bit Code
207
10.3
Fault Tolerance
211
10.4
Quantum Error Correction
212
10.4.1
Error Models for Quantum Computing
213
10.4.2
Encoding
216
10.4.3
Error Recovery
217
10.5
Three- and Nine-Qubit Quantum Codes
223
10.5.1
The Three-Qubit Code for Bit-Flip Errors
223
10.5.2
The Three-Qubit Code for Phase-Flip Errors
225
10.5.3
Quantum Error Correction Without Decoding
226
10.5.4
The Nine-Qubit Shor Code
230
10.6
Fault-Tolerant Quantum Computation
234
10.6.1
Concatenation of Codes and the Threshold Theorem
237
APPENDIX A
241
A.I Tools for Analysing Probabilistic Algorithms
241
A.
2
Solving the Discrete Logarithm Problem When the Order of
α
Is Composite
243
A.3 How Many Random Samples Are Needed to Generate
a Group?
245
A.4 Finding
r
Given | for Random
к
247
A.
5
Adversary Method Lemma
248
CONTENTS ¡x
Α.
6
Black-Boxes for Group Computations
250
A.
7
Computing Schmidt Decompositions
253
A.
8
General Measurements
255
A.9 Optimal Distinguishing of Two States
258
A.
9.1
A Simple Procedure
258
A.9.2 Optimality of This Simple Procedure
258
Bibliography
260
Index
270 |
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| index_date | 2024-07-02T15:31:34Z |
| indexdate | 2024-11-06T11:02:59Z |
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| publisher | Oxford Univ. Press |
| record_format | marc |
| spelling | Kaye, Philip Verfasser aut An introduction to quantum computing Philip Kaye, Raymond Laflamme and Michele Mosca 1. published Oxford [u.a.] Oxford Univ. Press 2007 XI, 274 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Kwantumcomputers gtt Quantum computers Quantencomputer (DE-588)4533372-5 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Quantencomputer (DE-588)4533372-5 s DE-604 Laflamme, Raymond 1960- Verfasser (DE-588)1213833248 aut Mosca, Michele Verfasser (DE-588)1044264276 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014962015&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kaye, Philip Laflamme, Raymond 1960- Mosca, Michele An introduction to quantum computing Kwantumcomputers gtt Quantum computers Quantencomputer (DE-588)4533372-5 gnd |
| subject_GND | (DE-588)4533372-5 (DE-588)4151278-9 |
| title | An introduction to quantum computing |
| title_auth | An introduction to quantum computing |
| title_exact_search | An introduction to quantum computing |
| title_exact_search_txtP | An introduction to quantum computing |
| title_full | An introduction to quantum computing Philip Kaye, Raymond Laflamme and Michele Mosca |
| title_fullStr | An introduction to quantum computing Philip Kaye, Raymond Laflamme and Michele Mosca |
| title_full_unstemmed | An introduction to quantum computing Philip Kaye, Raymond Laflamme and Michele Mosca |
| title_short | An introduction to quantum computing |
| title_sort | an introduction to quantum computing |
| topic | Kwantumcomputers gtt Quantum computers Quantencomputer (DE-588)4533372-5 gnd |
| topic_facet | Kwantumcomputers Quantum computers Quantencomputer Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014962015&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kayephilip anintroductiontoquantumcomputing AT laflammeraymond anintroductiontoquantumcomputing AT moscamichele anintroductiontoquantumcomputing |