Learning Transitions in Classical Ising Models and Deformed Toric Codes.

Area of Science:

• Statistical Mechanics
• Quantum Information Theory
• Bayesian Inference

Background:

• Conditional probability distributions model Bayesian inference for unknown classical states.
• The classical Ising model is a fundamental system in statistical mechanics.

Purpose of the Study:

• To demonstrate a learning transition in the 2D classical Ising model.
• To investigate the implications of this transition for quantum systems and memory.
• To identify new critical points and phase transitions.

Main Methods:

• Utilizing replica field theory and renormalization group techniques.
• Employing tensor network and Monte Carlo simulations to map phase diagrams.
• Analyzing conditional correlation functions for long-distance behavior.

Main Results:

• A learning transition was identified in the 2D Ising model, extending from high temperatures to the critical state.
• A novel tricritical point was discovered at the intersection of the learning transition and the thermal Ising transition.
• The model accurately describes weak measurements on quantum Hamiltonians, showing robustness of quantum memory near phase transitions.

Conclusions:

• Learning can induce critical states in both classical and quantum systems.
• The discovered tricritical point enhances the robustness of topological quantum memory against weak measurements.
• The analytical and computational methods are generalizable to broader studies of learning effects.