A Short Introduction to Quantum Information and Quantum Computation

0.8 MB·English

by  Michel Le Bellac

Half-title Page: i
Title Page: iii
Copyright Page: iv
Contents Page: v
Foreword Page: vii
Preface Page: ix
1 Introduction Page: 1
2 What is a qubit? Page: 5
2.1 The polarization of light Page: 5
2.2 Photon polarization Page: 8
2.3 Mathematical formulation: the qubit Page: 11
2.4 Principles of quantum mechanics Page: 17
2.6 Exercises Page: 28
2.6.1 Determination of the polarization of a light wave Page: 28
2.6.2 The (lamda, mu) polarizer Page: 28
2.6.3 Circular polarization and the rotation operator Page: 29
2.6.4 An optimal strategy for Eve? Page: 30
2.6.5 Heisenberg inequalities Page: 31
2.7 Further reading Page: 32
3 Manipulating qubits Page: 33
3.1 The Bloch sphere, spin 1/2 Page: 33
3.2 Dynamical evolution Page: 37
3.3 Manipulating qubits: Rabi oscillations Page: 40
3.4 Principles of NMR and MRI Page: 44
3.5 Exercises Page: 47
3.5.1 Rotation operator for spin 1/2 Page: 47
3.5.2 Rabi oscillations away from resonance Page: 47
3.6 Further reading Page: 48
4 Quantum correlations Page: 49
4.1 Two-qubit states Page: 49
4.2 The state operator (or density operator) Page: 54
4.3 The quantum no-cloning theorem Page: 57
4.4 Decoherence Page: 58
4.5 The Bell inequalities Page: 63
4.6 Exercises Page: 67
4.6.1 Basis independence of the tensor product Page: 67
4.6.2 Properties of the state operator Page: 68
4.6.3 The state operator for a qubit and the Bloch vector Page: 68
4.6.4 The SWAP operator Page: 69
4.6.5 The Schmidt purification theorem Page: 70
4.6.6 A model for phase damping Page: 71
4.6.7 Amplitude damping channel Page: 72
4.6.8 Invariance of the state (4.35) under rotation Page: 72
4.7 Further reading Page: 72
5 Introduction to quantum computing Page: 75
5.1 General remarks Page: 75
5.2 Reversible calculation Page: 77
5.3 Quantum logic gates Page: 81
5.4 The Deutsch algorithm Page: 84
5.5 Generalization to n + m qubits Page: 86
5.6 The Grover search algorithm Page: 88
5.7 The quantum Fourier transform Page: 91
5.8 The period of a function Page: 94
5.9 Classical algorithms and quantum algorithms Page: 101
5.10 Exercises Page: 103
5.10.1 Justification of the circuits of Fig. 5.4 Page: 103
5.10.2 The Deutsch–Josza algorithm Page: 104
5.10.4 Example of finding y Page: 105
5.11 Further reading Page: 105
6 Physical realizations Page: 107
6.1 NMR as a quantum computer Page: 108
6.2 Trapped ions Page: 114
6.3 Superconducting qubits Page: 122
6.4 Quantum dots Page: 131
6.5 Exercises Page: 133
6.5.1 Off-resonance Rabi oscillations Page: 133
6.5.2 Commutation relations between the a and a Page: 134
6.5.3 Construction of a cZ gate using trapped ions Page: 134
6.5.4 Vibrational normal modes of two ions in a trap Page: 136
6.5.5 Meissner effect and flux quantization Page: 136
6.5.6 Josephson current Page: 136
6.5.7 Charge qubits Page: 137
6.6 Further reading Page: 138
7 Quantum information Page: 141
7.1 Teleportation Page: 141
7.2 Shannon entropy Page: 144
7.3 von Neumann entropy Page: 147
7.4 Quantum error correction Page: 154
7.5 Exercises Page: 157
7.5.1 Superdense coding Page: 157
7.5.2 Shannon entropy versus von Neumann entropy Page: 157
7.5.3 Information gain of Eve Page: 158
7.5.4 Symmetry of the fidelity Page: 158
7.5.5 Quantum error correcting code Page: 158
7.6 Further reading Page: 158
References Page: 161
Index Page: 165