Quantum computing:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2004
|
| Edition: | 2. ed. |
| Series: | Natural computing series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 214 S. graph. Darst. |
| ISBN: | 3540407049 |
Staff View
MARC
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| 245 | 1 | 0 | |a Quantum computing |c Mika Hirvensalo |
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Record in the Search Index
| _version_ | 1804130290282528768 |
|---|---|
| adam_text | Contents
1. Introduction 1
1.1 A Brief History of Quantum Computation 1
1.2 Classical Physics 2
1.3 Probabilistic Systems 4
1.4 Quantum Mechanics 7
2. Quantum Information 13
2.1 Quantum Bits 13
2.2 Quantum Registers 17
2.3 No Cloning Theorem 20
2.4 Observation 22
2.5 Quantum Teleportation 24
2.6 Superdense Coding 27
2.7 Exercises 28
3. Devices for Computation 29
3.1 Uniform Computation 29
3.1.1 Turing Machines 29
3.1.2 Probabilistic Turing Machines 32
3.1.3 Multitape Turing Machines 36
3.1.4 Quantum Turing Machines 37
3.2 Circuits 41
3.2.1 Boolean Circuits 41
3.2.2 Reversible Circuits 43
3.2.3 Quantum Circuits 45
4. Fast Factorization 49
4.1 Quantum Fourier Transform 49
4.1.1 General Framework 49
4.1.2 Hadamard Walsh Transform 51
4.1.3 Quantum Fourier Transform in Zn 52
4.1.4 Complexity Remarks 57
4.2 Shor s Algorithm for Factoring Numbers 58
4.2.1 From Periods to Factoring 58
XII Contents
4.2.2 Orders of the Elements in Zn 60
4.2.3 Finding the Period 63
4.3 The Correctness Probability 65
4.3.1 The Easy Case 65
4.3.2 The General Case 66
4.3.3 The Complexity of Shor s Factorization Algorithm .... 69
4.4 Excercises 70
5. Finding the Hidden Subgroup 73
5.1 Generalized Simon s Algorithm 74
5.1.1 Preliminaries 74
5.1.2 The Algorithms 75
5.2 Examples 79
5.2.1 Finding the Order 79
5.2.2 Discrete Logarithm 79
5.2.3 Simon s Original Problem 80
5.3 Exercises 80
6. Grover s Search Algorithm 83
6.1 Search Problems 83
6.1.1 Satisfiability Problem 83
6.1.2 Probabilistic Search 84
6.1.3 Quantum Search with One Query 86
6.2 Grover s Amplification Method 89
6.2.1 Quantum Operators for Grover s Search Algorithm ... 89
6.2.2 Amplitude Amplification 90
6.2.3 Analysis of Amplification Method 94
6.3 Utilizing Grover s Search Method 97
6.3.1 Searching with Unknown Number of Solutions 97
7. Complexity Lower Bounds for Quantum Circuits 101
7.1 General Idea 101
7.2 Polynomial Representations 102
7.2.1 Preliminaries 102
7.2.2 Bounds for the Representation Degrees 106
7.3 Quantum Circuit Lower Bound 108
7.3.1 General Lower Bound 108
7.3.2 Some Examples Ill
8. Appendix A: Quantum Physics 113
8.1 A Brief History of Quantum Theory 113
8.2 Mathematical Framework for Quantum Theory 115
8.2.1 Hilbert Spaces 117
8.2.2 Operators 118
8.2.3 Spectral Representation of Self Adjoint Operators .... 122
Contents XIII
8.2.4 Spectral Representation of Unitary Operators 125
8.3 Quantum States as Hilbert Space Vectors 129
8.3.1 Quantum Time Evolution 130
8.3.2 Observables 131
8.3.3 The Uncertainty Principles 134
8.4 Quantum States as Operators 139
8.4.1 Density Matrices 139
8.4.2 Observables and Mixed States 142
8.4.3 Subsystem States 147
8.4.4 More on Time Evolution 154
8.4.5 Representation Theorems 156
8.4.6 Jozsa s Theorem of Cloning and Deleting 163
8.5 Exercises 166
9. Appendix B: Mathematical Background 167
9.1 Group Theory 167
9.1.1 Preliminaries 167
9.1.2 Subgroups, Cosets 168
9.1.3 Factor Groups 169
9.1.4 Group Z; 171
9.1.5 Group Morphisms 173
9.1.6 Direct Product 175
9.2 Fourier Transforms 176
9.2.1 Characters of Abelian Groups 176
9.2.2 Orthogonality of the Characters 178
9.2.3 Discrete Fourier Transform 180
9.2.4 The Inverse Fourier Transform 182
9.2.5 Fourier Transform and Periodicity 183
9.3 Linear Algebra 183
9.3.1 Preliminaries 183
9.3.2 Inner Product 186
9.4 Number Theory 189
9.4.1 Euclid s Algorithm 189
9.4.2 Continued Fractions 190
9.5 Shannon Entropy and Information 197
9.5.1 Entropy 197
9.5.2 Information 201
9.5.3 The Holevo Bound 203
9.6 Exercises 204
References 205
Index 211
|
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| discipline | Informatik |
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| id | DE-604.BV017491735 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:18:41Z |
| institution | BVB |
| isbn | 3540407049 |
| language | English |
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| physical | XIII, 214 S. graph. Darst. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Springer |
| record_format | marc |
| series2 | Natural computing series |
| spelling | Hirvensalo, Mika 1972- Verfasser (DE-588)1146315651 aut Quantum computing Mika Hirvensalo 2. ed. Berlin [u.a.] Springer 2004 XIII, 214 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Natural computing series Quantencomputer Quantum computers Quantencomputer (DE-588)4533372-5 gnd rswk-swf Quantencomputer (DE-588)4533372-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010537619&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hirvensalo, Mika 1972- Quantum computing Quantencomputer Quantum computers Quantencomputer (DE-588)4533372-5 gnd |
| subject_GND | (DE-588)4533372-5 |
| title | Quantum computing |
| title_auth | Quantum computing |
| title_exact_search | Quantum computing |
| title_full | Quantum computing Mika Hirvensalo |
| title_fullStr | Quantum computing Mika Hirvensalo |
| title_full_unstemmed | Quantum computing Mika Hirvensalo |
| title_short | Quantum computing |
| title_sort | quantum computing |
| topic | Quantencomputer Quantum computers Quantencomputer (DE-588)4533372-5 gnd |
| topic_facet | Quantencomputer Quantum computers |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010537619&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT hirvensalomika quantumcomputing |