Quantum information and computation is a rapidly expanding and cross-disciplinary subject. This book gives a self-contained introduction to the field for physicists, mathematicians and computer scientists who want to know more about this exciting subject. After a step-by-step introduction to the quantum bit (qubit) and its main properties, the author presents the necessary background in quantum mechanics. The core of the subject, quantum computation, is illustrated by a detailed treatment of three quantum algorithms: Deutsch, Grover and Shor. The final chapters are devoted to the physical implementation of quantum computers, including the most recent aspects, such as superconducting qubits and quantum dots, and to a short account of quantum information. Written at a level suitable for undergraduates in physical sciences, no previous knowledge of quantum mechanics is assumed, and only elementary notions of physics are required. The book includes many short exercises, with solutions available to instructors through
[email protected].
Author(s): Michel Le Bellac
Year: 2006
Language: English
Pages: 179
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Foreword......Page 9
Preface......Page 11
1 Introduction......Page 13
2.1 The polarization of light......Page 17
2.2 Photon polarization......Page 20
2.3 Mathematical formulation: the qubit......Page 23
2.4 Principles of quantum mechanics......Page 29
2.6.2 The (lamda, mu) polarizer......Page 40
2.6.3 Circular polarization and the rotation operator......Page 41
2.6.4 An optimal strategy for Eve?......Page 42
2.6.5 Heisenberg inequalities......Page 43
2.7 Further reading......Page 44
3.1 The Bloch sphere, spin 1/2......Page 45
3.2 Dynamical evolution......Page 49
3.3 Manipulating qubits: Rabi oscillations......Page 52
3.4 Principles of NMR and MRI......Page 56
3.5.2 Rabi oscillations away from resonance......Page 59
3.6 Further reading......Page 60
4.1 Two-qubit states......Page 61
4.2 The state operator (or density operator)......Page 66
4.3 The quantum no-cloning theorem......Page 69
4.4 Decoherence......Page 70
4.5 The Bell inequalities......Page 75
4.6.1 Basis independence of the tensor product......Page 79
4.6.3 The state operator for a qubit and the Bloch vector......Page 80
4.6.4 The SWAP operator......Page 81
4.6.5 The Schmidt purification theorem......Page 82
4.6.6 A model for phase damping......Page 83
4.7 Further reading......Page 84
5.1 General remarks......Page 87
5.2 Reversible calculation......Page 89
5.3 Quantum logic gates......Page 93
5.4 The Deutsch algorithm......Page 96
5.5 Generalization to n + m qubits......Page 98
5.6 The Grover search algorithm......Page 100
5.7 The quantum Fourier transform......Page 103
5.8 The period of a function......Page 106
5.9 Classical algorithms and quantum algorithms......Page 113
5.10.1 Justification of the circuits of Fig. 5.4......Page 115
5.10.2 The Deutsch–Josza algorithm......Page 116
5.11 Further reading......Page 117
6 Physical realizations......Page 119
6.1 NMR as a quantum computer......Page 120
6.2 Trapped ions......Page 126
6.3 Superconducting qubits......Page 134
6.4 Quantum dots......Page 143
6.5.1 Off-resonance Rabi oscillations......Page 145
6.5.3 Construction of a cZ gate using trapped ions......Page 146
6.5.6 Josephson current......Page 148
6.5.7 Charge qubits......Page 149
6.6 Further reading......Page 150
7.1 Teleportation......Page 153
7.2 Shannon entropy......Page 156
7.3 von Neumann entropy......Page 159
7.4 Quantum error correction......Page 166
7.5.2 Shannon entropy versus von Neumann entropy......Page 169
7.6 Further reading......Page 170
References......Page 173
Index......Page 177
