Entanglement, quantum phase transitions and quantum algorithms

Introduction......Page 19
Majorization along parameter and renormalization group flows......Page 29
Global, monotonous and fine-grained entanglement loss......Page 31
Majorization along parameter flows in (1+1)-dimensional quantum systems......Page 32
Quantum Heisenberg spin chain with a boundary......Page 36
Quantum XY spin chain with a boundary......Page 37
Critical quantum XX spin chain with a boundary......Page 40
Conclusions of Chapter 1......Page 42
Single-copy entanglement in (1+1)-dimensional quantum systems......Page 43
Operational definition of the single-copy entanglement......Page 44
Exact conformal field theoretical computation......Page 46
Exact computation in quasi-free fermionic quantum spin chains......Page 47
Single-copy entanglement away from criticality......Page 51
Conclusions of Chapter 2......Page 52
Entanglement entropy in the Lipkin-Meshkov-Glick model......Page 55
The Lipkin-Meshkov-Glick model......Page 56
Entanglement within different regimes......Page 57
Analytical study of the isotropic case......Page 58
Numerical study of the anisotropic case......Page 60
Comparison to quantum spin chains......Page 61
Conclusions of Chapter 3......Page 64
Entanglement entropy in quantum algorithms......Page 65
The factoring quantum algorithm......Page 67
Analytical results......Page 69
The adiabatic quantum algorithm......Page 70
Exact Cover......Page 71
Numerical results up to 20 qubits......Page 73
Entanglement in adiabatic quantum searching algorithms......Page 81
Analytical results......Page 83
Conclusions of Chapter 4......Page 88
Classical simulation of quantum algorithms using matrix product states......Page 91
The matrix product state ansatz......Page 92
Computing dynamics......Page 96
Classical simulation of an adiabatic quantum algorithm solving Exact Cover......Page 101
Discretization of the continuous time evolution in unitary gates......Page 102
Numerical results of a simulation with matrix product states......Page 103
Conclusions of Chapter 5......Page 109
Majorization arrow in quantum algorithm design......Page 111
Applying majorization theory to quantum algorithms......Page 112
Majorization in quantum phase-estimation algorithms......Page 113
The quantum phase-estimation algorithm......Page 114
Analytical results......Page 115
Natural majorization and comparison with quantum searching......Page 122
The quantum hidden affine function determination algorithm......Page 123
Majorization in adiabatic quantum searching algorithms......Page 125
Numerical results......Page 126
Majorization in a quantum walk algorithm with exponential speed-up......Page 130
The exponentially fast quantum walk algorithm......Page 131
Numerical results......Page 133
Conclusions of Chapter 6......Page 135
General conclusions and outlook......Page 139
Majorization......Page 141
Some notions about conformal field theory......Page 143
Some notions about classical complexity theory......Page 147