Quantum Computation, Complexity, and Many-Body Physics

Introduction......Page 12
Simulations of Physics with Quantum Computers......Page 20
The Conventional Model of Quantum Computation......Page 22
Hamiltonian Evolutions......Page 28
Controlled Operations......Page 29
Deterministic Quantum Algorithms......Page 31
One-Ancilla Qubit Measurement Processes......Page 32
Quantum Algorithms and Quantum Simulations......Page 33
Simulations of Fermionic Systems......Page 36
Simulations of Anyonic Systems......Page 41
Simulations of Bosonic Systems......Page 42
Applications: The 2D fermionic Hubbard model......Page 48
Quantum Algorithms: Efficiency and Errors......Page 50
Liquid-State NMR Quantum Information Processor......Page 53
Applications: The Fano-Anderson Model......Page 57
Experimental Protocol and Results......Page 62
Summary......Page 65
Quantum Entanglement as an Observer-Dependent Concept......Page 70
Quantum Entanglement......Page 74
Separability and von Neuman Entropy......Page 75
Mixed-State Entanglement and the Concurrence......Page 77
Measures of Quantum Entanglement......Page 78
Generalized Entanglement: Definition......Page 79
Generalized Entanglement and Lie Algebras......Page 80
Generalized Entanglement and Mixed States......Page 84
Two-Spin Systems......Page 85
N-Spin Systems......Page 89
Fermionic Systems......Page 90
Summary......Page 92
Generalized Entanglement as a Resource in Quantum Information......Page 94
Quantum Cryptography......Page 95
Quantum Teleportation......Page 97
Quantum Entanglement and Quantum Computation......Page 98
Efficient Classical Simulations of Quantum Physics......Page 102
Higher Order Correlations......Page 106
Generalized Entanglement and Quantum Computation......Page 108
Efficient Initial State Preparation......Page 109
Summary......Page 111
Generalized Entanglement and Many-Body Physics......Page 112
Entanglement and Quantum Phase Transitions......Page 113
Lipkin-Meshkov-Glick Model......Page 115
Anisotropic XY Model in a Transverse Magnetic Field......Page 121
General Mean-Field Hamiltonians......Page 131
Diagonalization Procedure......Page 133
Example: Fermionic Systems......Page 138
Summary......Page 139
Conclusions......Page 140
Future Directions......Page 144
Discrete Fourier Transforms......Page 146
Discrete Fourier Transform and Propagation of Errors......Page 148
The Adjoint Representation......Page 150
Separability, Generalized Unentanglement, and Local Purities......Page 154
Approximations of the exponential matrix......Page 158
Scaling of the Method......Page 162
Efficient Classical Evaluation of High-Order Correlation Functions......Page 164
Classical Limit in the LMG Model......Page 170