Deterministic flows of order-parameters in stochastic processes of quantum Monte Carlo method

TL;DR

This paper derives deterministic flow equations for order parameters in quantum Monte Carlo processes, linking microscopic stochastic dynamics to macroscopic behavior, and validates the approach through simulations and potential applications in image restoration.

Contribution

It introduces a method to analytically derive macroscopic deterministic equations from microscopic quantum Monte Carlo dynamics, including validation and extensions.

Findings

Derived explicit flow equations for order parameters in quantum spin systems.

Validated static approximation through finite-size system simulations.

Applied the approach to Bayesian image restoration, revealing non-monotonic dynamics.

Abstract

In terms of the stochastic process of quantum-mechanical version of Markov chain Monte Carlo method (the MCMC), we analytically derive macroscopically deterministic flow equations of order parameters such as spontaneous magnetization in infinite-range ($d(=\infty)$-dimensional) quantum spin systems. By means of the Trotter decomposition, we consider the transition probability of Glauber-type dynamics of microscopic states for the corresponding $(d+1)$-dimensional classical system. Under the static approximation, differential equations with respect to macroscopic order parameters are explicitly obtained from the master equation that describes the microscopic-law. In the steady state, we show that the equations are identical to the saddle point equations for the equilibrium state of the same system. The equation for the dynamical Ising model is recovered in the classical limit. We also check the validity of the static approximation by making use of computer simulations for finite size systems and discuss several possible extensions of our approach to disordered spin systems for statistical-mechanical informatics. Especially, we shall use our procedure to evaluate the decoding process of Bayesian image restoration. With the assistance of the concept of dynamical replica theory (the DRT), we derive the zero-temperature flow equation of image restoration measure showing some `non-monotonic' behaviour in its time evolution.