This book provides a comprehensive overview of the mathematical aspects of quantum computing. It will be useful for graduate students and researchers interested in quantum computing from different areas of physics, mathematics, informatics and computer science. The lecture notes in this volume are written in a self-contained style, and hence are accessible for graduate students and researchers with even less background in the topics.
Author(s): Mikio Nakahara, Robabeh Rahimi, Akira SaiToh
Series: Kinki University Series on Quantum Computing
Publisher: World Scientific Pub Co (
Year: 2008
Language: English
Pages: 239
—Poster Summaries—......Page 16
Preface......Page 6
LIST OF PARTICIPANTS......Page 14
1. Introduction......Page 18
2.1. Notation and conventions......Page 19
2.2. Axioms of quantum mechanics......Page 20
2.3. Simple example......Page 22
2.4. Multipartite system, tensor product and entangled state......Page 23
2.5. Mixed states and density matrices......Page 24
2.6. Negativity......Page 26
2.7. Partial trace and purification......Page 28
3.1. One qubit......Page 29
3.2. Bloch sphere......Page 30
3.3. Multi-qubit systems and entangled states......Page 31
4.1. Introduction......Page 32
4.2.1. Simple quantum gates......Page 33
4.2.2. Walsh-Hadamard transformation......Page 35
4.2.3. SWAP gate and Fredkin gate......Page 36
4.3. No-cloning theorem......Page 37
4.4. Quantum teleportation......Page 38
4.5. Universal quantum gates......Page 39
4.6. Quantum parallelism and entanglement......Page 40
5.1. Deutsch algorithm......Page 42
5.2. Deutsch-Jozsa algorithm......Page 44
6. Decoherence......Page 46
6.1.1. Quantum operations and Kraus operators......Page 47
6.1.2. Operator-sum representation and noisy quantum channel......Page 50
6.1.3. Completely positive maps......Page 51
6.2. Measurements as quantum operations......Page 52
6.2.2. POVM......Page 53
6.3.1. Bit- flip channel......Page 54
6.3.2. Phase-flip channel......Page 56
7.2. Three-qubit bit-flip code: the simplest example......Page 58
7.2.3. Transmission......Page 59
7.2.4. Error syndrome dectection and correction......Page 60
7.2.5. Decoding......Page 62
7.2.7. Continuous rotations......Page 63
8.1. DiVincenzo criteria......Page 64
Acknowledgements......Page 68
References......Page 69
1. Introduction......Page 72
2. Braid Groups......Page 74
3. Knots De.ned by Braids......Page 81
4. Topological Quantum Computing......Page 86
5. Anyon Model......Page 89
6. Fibonacci Anyons......Page 93
Appendix A. Fundamental group......Page 97
References......Page 105
1. Introduction......Page 108
2.1. State space......Page 109
2.2. Evolution......Page 111
2.3. POVMs, projective measurement and observables......Page 112
2.4. Composite systems......Page 114
3. Entanglement and Separability......Page 115
4.1. Local operations and classical communication......Page 120
4.2. Entanglement measures......Page 122
4.3. Uniqueness of measures, order on states......Page 124
4.5. Multipartite entanglement......Page 126
5. Conclusions......Page 128
References......Page 129
1. Introduction......Page 132
2.2. Berry phase in quantum mechanics......Page 133
2.3. Wilczek-Zee holonomy in quantum mechanics......Page 134
2.4.1. Berry phase......Page 136
2.4.2. Λ-type system......Page 138
3. Holonomic Quantum Computer......Page 139
4.1. Geometrical setting......Page 140
4.2. The isoholonomic problem......Page 143
4.3. The solution: horizontal extremal curve......Page 145
5.1. Equivalence class......Page 147
5.2. U(1) holonomy......Page 148
5.3. U(k) holonomy......Page 150
6.1. Hadamard gate......Page 151
7. Discussions......Page 152
7.2. Implementation......Page 153
References......Page 154
1. Introduction......Page 156
2. A Brief Review of Classical Game Theory......Page 158
3.2. Density matrices......Page 160
3.3. Unitary transformation......Page 161
3.4. Measurement......Page 162
3.5. Correlations in quantum mechanical systems......Page 163
4.1. Coin flip game......Page 164
4.2. Eisert’s model - Prisoners’ Dilemma......Page 165
4.3. Present status in quantum game theory......Page 168
5. Study of a Discoordination Game - Samaritan’s Dilemma - Using Eisert’s Model: E.ects of Shared Correlation on the Game Dynamics......Page 170
5.1. Quantum operations and quantum correlations......Page 172
5.2. Quantum operations and classical correlations......Page 175
5.3. Classical operations and classical correlations......Page 178
5.4. Classical operations and quantum correlations......Page 179
5.5. Classical Bob versus quantum Alice......Page 180
6. Decoherence and Quantum Version of Classical Games......Page 181
7. Entanglement and Reproducibility of Multiparty Classical Games in Quantum Mechanical Settings......Page 186
7.1. Reproducibility criterion to play games in quantum mechanical settings......Page 187
7.2. Entangled states and strong reproducibility criterion......Page 189
8. Conclusion......Page 194
References......Page 195
1. Introduction......Page 198
2.1. Introduction to classical error-correcting codes......Page 199
2.2. Linear codes and parity-check matrices......Page 200
2.3. Minimum distance and [n, k, d] codes......Page 201
2.4. Encoding......Page 203
2.5. Decoding......Page 204
2.6. Bit error-rate and block error-rate......Page 207
3.1. Parity-check measurements and stabilizer codes......Page 208
4.1. Twisted condition......Page 210
4.2. Characterization of code space......Page 212
4.3. Correctable error......Page 213
4.4. 7-Qubit code (quantum Hamming code)......Page 214
5.1. LDPC codes and sum-product decoding......Page 216
5.2. Application of the sum-product algorithm for the error-correction of CSS codes......Page 217
5.3. Classical and quantum quasi-cyclic LDPC codes......Page 218
5.4. Twisted condition for quasi-cyclic LDPC codes......Page 219
5.5. CSS QC-LDPC codes from right-shifted matrices......Page 220
5.7. Regular LDPC codes......Page 221
5.8. Construction of CSS LDPC codes......Page 223
5.9. Performance of error-correction and minimum distance......Page 226
6. Postscript......Page 227
References......Page 228
1. Introduction......Page 230
References......Page 231
1. Introduction......Page 232
References......Page 233
1. Introduction......Page 234
References......Page 235
Numerical Computation of Time-Dependent Multipartite Nonclassical Correlation A. SaiToh, R. Rahimi, M. Nakahara, M. Kitagawa......Page 236
References......Page 237
On Classical No-Cloning Theorem Under Liouville Dynamics and Distances T. Yamano, O. Iguchi......Page 238
References......Page 239
