CONTENTS -- Preface -- Quantum Annealing: From Viewpoints of Statistical Physics, Condensed Matter Physics, and Computational Physics Shu Tanaka and Ryo Tamura -- 1. Introduction -- 2. Ising Model -- 2.1. Magnetic Systems -- 2.2. Nuclear Magnetic Resonance -- 3. Implementation Methods of Quantum Annealing -- 3.1. Monte Carlo Method -- 3.2. Deterministic Method Based on Mean-Field Approximation -- 3.3. Real-Time Dynamics -- 3.4. Experiments -- 4. Optimization Problems -- 4.1. Traveling Salesman Problem -- 4.1.1. Monte Carlo Method -- 4.1.2. Quantum Annealing -- 4.1.3. Comparison with Simulated Annealing and Quantum Annealing -- 4.2. Clustering Problem -- 5. Relationship between Quantum Annealing and Statistical Physics -- 5.1. Kibble-Zurek Mechanism -- 5.1.1. Efficiency of Simulated Annealing and Quantum Annealing -- 5.1.2. Simulated Annealing for Random Ferromagnetic Ising Chain -- 5.1.3. Quantum Annealing for Random Ferromagnetic Ising Chain -- 5.1.4. Comparison between Simulated and Quantum Annealing Methods -- 5.2. Frustration Effects for Simulated Annealing and Quantum Annealing -- 5.2.1. Thermal Fluctuation and Quantum Fluctuation Effect of Geometrical Frustrated Systems -- 5.2.2. Non-Monotonic Behavior of Correlation Function in Decorated Bond System -- 6. Conclusion -- Acknowledgement -- References -- Spin Glass: A Bridge between Quantum Computation and Statistical Mechanics Masayuki Ohzeki -- 1. Introduction: Statistical Mechanics and Quantum Mechanics -- 2. Training: Statistical Mechanics -- 2.1. Student's misreading point: Probability is... -- 2.2. Probability describes... a certain behavior -- 2.3. Large deviation property -- 2.4. Mean-field analysis -- 2.5. Phase transition -- 2.6. Spin glasses -- 2.7. Gauge theory -- 3. Quantum Error Correction: Surface Code -- 3.1. Error model -- 3.2. Surface code.

3.2.1. Check operators and error syndrome -- 3.2.2. Probability of error chains -- 3.3. Analyses on accuracy thresholds for surface code -- 3.3.1. Duality analysis: Simple case -- 3.3.2. Duality analysis: Spin glass -- 3.3.3. Duality analysis with real-space renormalization -- 3.3.4. Other cases -- 3.3.5. Depolarizing channel -- 4. Quantum Annealing and Beyond -- 4.1. Quantum adiabatic computation: Short review -- 4.2. Novel type of quantum annealing -- 4.2.1. Classical quantum mapping -- 4.2.2. Jarzynski equality -- 4.2.3. Quantum Jarzynski annealing -- 4.2.4. Problems in measurement of answer -- 4.3. Non-adiabatic quantum computation -- 4.3.1. Jarzynski equality for quantum system -- 4.3.2. Performance of non-adiabatic quantum annealing -- 4.4. Analyses on non-adiabatic quantum annealing -- 4.4.1. Gauge transformation for quantum spin systems -- 4.4.2. Relationship between two different paths of NQA -- 4.4.3. Exact relations involving inverse statistics -- 5. Summary -- References -- Second Law-like Inequalities with Quantum Relative Entropy: An Introduction Takahiro Sagawa -- 1. Introduction -- 2. Quantum States and Dynamics -- 2.1. Quantum States and Observables -- 2.2. Quantum Dynamics -- 2.2.1. Unitary Evolution -- 2.2.2. Completely Positive Maps -- 2.2.3. Kraus Representation -- 2.2.4. Indirect Measurement Model -- 2.2.5. Heisenberg Picture -- 3. Quantum Relative Entropy -- 3.1. Von Neumann Entropy -- 3.2. Quantum Relative Entropy and Its Positivity -- 3.3. Monotonicity of the Quantum Relative Entropy -- 4. Quantum Mutual Information and Related Quantities -- 4.1. Quantum Mutual Information -- 4.2. Holevo's -quantity -- 4.3. QC-mutual Information (Groenewold-Ozawa Information) -- 5. Second Law of Thermodynamics -- 5.1. Thermodynamic Entropy and the von Neumann Entropy -- 5.2. From the Positivity of the Quantum Relative Entropy.

5.3. From the Quantum Fluctuation Theorem -- 5.4. From the Monotonicity of the Quantum Relative Entropy -- 5.4.1. Time-independent Control -- 5.4.2. Time-dependent Control -- 6. Second Law with Quantum Feedback Control -- 6.1. From the Positivity of the Quantum Relative Entropy -- 6.2. From the Monotonicity of the Quantum Relative Entropy -- 7. Concluding Remarks -- Appendix A. Short Summary of the Linear Algebra -- Appendix B. Proof of the Monotonicity of the Quantum Relative Entropy -- Acknowledgments -- References.