1 Quantum Approach to Classical Thermodynamics and Optimization
R.D. Somma and G. Ortiz
1.1 Introduction
1.2 Quantum Annealing Strategies for Optimization
1.2.1 Classical-to-Quantum Mapping
1.2.2 Adiabatic State Preparation: Rates of Convergence
1.2.3 Quantum Monte Carlo Methods for Quantum Annealing
1.2.4 Quantum Quadratic Speedup
1.2.5 Efficient Quantum Gibbs State Preparation of One-Dimensional Systems
1.3 Quantum Algorithms for Numerical Integration with No Importance Sampling
1.4 Conclusions
References
2 Non-equilibrium Dynamics of Quantum Systems: Order Parameter Evolution, Defect Generation, and Qubit Transfer
S. Mondal, D. Sen, and K. Sengupta
2.1 Introduction
2.2 Quench Dynamics
2.2.1 Ultracold Atoms in an Optical Lattice
2.2.2 Infinite Range Ising Model in a Transverse Field
2.3 Non-adiabatic Dynamics
2.3.1 Quenching Across a Critical Surface
2.3.2 Non-linear Quenching Across a Critical Point
2.3.3 Experimental Realizations
2.4 Quantum Communication
2.5 Discussion
References
3 Defect Production Due to Quenching Through a Multicritical Point and Along a Gapless Line
U. Divakaran, V. Mukherjee, A. Dutta, and D. Sen
3.1 Introduction
3.2 A Spin Model: Transverse and Anisotropic Quenching
3.3 Quenching Through a Multicritical Point
3.4 Quenching Along a Gapless Line
3.5 Conclusions
References
4 Adiabatic Perturbation Theory: From Landau--Zener Problem to Quenching Through a Quantum Critical Point
C. De Grandi and A. Polkovnikov
4.1 Introduction
4.2 Adiabatic Perturbation Theory
4.2.1 Application to the Landau--Zener Problem
4.3 Adiabatic Dynamics in Gapless Systems withQuasiparticle Excitations
4.4 Adiabatic Dynamics Near a Quantum Critical Point
4.4.1 Scaling Analysis
4.4.2 Examples
4.5 Sudden Quenches Near Quantum Critical Points
4.6 Effect of the Quasiparticle Statistics in the Finite Temperature Quenches
4.7 Conclusions
References
5 Quench Dynamics of Quantum and Classical Ising Chains: From the Viewpoint of the Kibble--Zurek Mechanism
S. Suzuki
5.1 Introduction
5.2 Kibble--Zurek Mechanism
5.3 Quench in the Pure Ising Chain
5.3.1 Classical Quench
5.3.2 Quantum Quench
5.4 Quench in the Random Ising Chain
5.4.1 Classical Quench
5.4.2 Quantum Quench
5.5 Conclusion
5.5.1 Summary of the Results
5.5.2 Discussion and Future Problems
References
6 Quantum Phase Transition in the Spin Boson Model
S. Florens, D. Venturelli, and R. Narayanan
6.1 Introduction
6.2 The Spin Boson Model
6.2.1 The Model
6.2.2 The Phases of the SBM
6.3 The SBM Using the Majorana Representation
6.4 Perturbative Renormalization Group in the Localized Regime
6.5 Analyzing the RG Flow
6.5.1 The RG Equations for the Ohmic Case (s=1)
6.5.2 The RG Equations for the Super-ohmic Case (s>1)
6.5.3 The RG Equations for the Sub-ohmic Case (0<s<1)
6.6 Mapping to a Long-Ranged Ising Model
6.7 A Special Identity
6.8 Shiba's Relation for the Sub-ohmic Spin Boson Model
6.9 On the Possible Breakdown of Quantum to Classical Mapping
References
7 Influence of Local Moment Fluctuations on the Mott Transition
C. Janani, S. Florens, T. Gupta, and R. Narayanan
7.1 Introduction
7.2 The Heisenberg Model
7.2.1 Single-Site Representation of the Heisenberg Model
7.2.2 The Auxiliary Fermion Representation
7.3 The Hubbard--Heisenberg Model
7.4 Numerical Results
7.4.1 Numerical Results for J=0
7.4.2 Numerical Results for Non-zero J
7.5 Conclusions, Future Outlook, and Open Problems
References
8 Signatures of Quantum Phase Transitions via Quantum Information Theoretic Measures
I. Bose and A. Tribedi
8.1 Introduction
8.2 Entanglement and Fidelity Measures Probing QPTs
8.3 QPTs in Model Spin Systems
8.4 Signatures of QPTs in a Spin Ladder
8.5 Concluding Remarks
References
9 How Entangled Is a Many-Electron State?
V. Subrahmanyam
9.1 Introduction
9.2 Spin States
9.3 Many-Electron States
References
10 Roles of Quantum Fluctuation in Frustrated Systems -- Order by Disorder and Reentrant Phase Transition
S. Tanaka, M. Hirano, and S. Miyashita
10.1 Introduction
10.2 Triangular Lattice
10.3 Reentrant Phase Transition
10.3.1 Classical Dynamics
10.3.2 Quantum Dynamics
10.4 Conclusion
References
11 Exploring Ground States of Quantum Spin Glasses by Quantum Monte Carlo Method
A.K. Chandra, A. Das, J. Inoue, and B.K. Chakrabarti
11.1 Introduction
11.2 LRIAF Without Disorder
11.3 Finite Temperature Quantum Monte Carlo Simulation
11.3.1 Suzuki--Trotter Mapping and Simulation
11.3.2 Simulation Results
11.4 LRIAF with SK Disorder: `Liquid' Phase of the SK Spin Glasses Gets Frozen
11.5 Ground State of a Quantum Spin Glass Using a Zero-Temperature Quantum Monte Carlo
11.6 Summary
References
12 Phase Transition in a Quantum Ising Modelwith Long-Range Interaction
A. Ganguli and S. Dasgupta
12.1 Introduction
12.2 Classical Ising Model
12.3 Quantum Ising Model
12.4 A Quantum Ising Model with Long-Range Antiferromagnetic Interaction
12.5 Perturbative Treatment of H
12.6 Phase Transition Properties
12.7 Effect of Disorder
References
13 Length Scale-Dependent Superconductor--Insulator Quantum Phase Transitions in One Dimension: Renormalization Group Theory of Mesoscopic SQUIDs Array
S. Sarkar
13.1 Introduction
13.2 Renormalization Group Study for the Quantum Dissipation Phase in Mesoscopic Lumped SQUIDs
13.3 Quantum Field-Theoretical Study of Model Hamiltonianof the System and Derivation of Tunneling Resistanceat the Quantum Critical Point
13.4 Conclusions
References
14 Quantum-Mechanical Variant of the Thouless--Anderson--Palmer Equation for Error-Correcting Codes
J. Inoue, Y. Saika, and M. Okada
14.1 Introduction
14.2 The Model System and the Generic Properties
14.3 The Bayesian Approach
14.3.1 Decoding for Classical System
14.3.2 Decoding for Quantum System
14.4 Meanfield Decoding via the TAP-Like Equation
14.4.1 Numerical Results
14.5 Concluding Remark
References
15 Probabilistic Model of Fault Detection in Quantum Circuits
A. Banerjee and A. Pathak
15.1 Introduction
15.2 Methodology for Detection of Fault
15.2.1 Specific Examples
15.2.2 Time Complexity of the Fault Detection Algorithm
15.3 Conclusions
References
Subject Index