Table Of Content

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

Description:

Quantum information and computation is a rapidly expanding and cross-disciplinary subject. This book, first published in 2006, 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].