Adiabatic approximation for the imaginary-time Schroedinger equation and its application to simulated annealing

TL;DR

This paper develops an adiabatic approximation for the imaginary-time Schrödinger equation and applies it to analyze the convergence of simulated annealing in classical Ising models, providing new asymptotic formulas.

Contribution

It introduces a novel adiabatic approximation for the imaginary-time Schrödinger equation and applies it to improve understanding of simulated annealing convergence.

Findings

Derived an adiabatic condition with two inequalities.

Obtained an asymptotic formula for ground state reaching probability.

Amended existing convergence theory for simulated annealing.

Abstract

We formulate an adiabatic approximation for the imaginary-time Schroedinger equation. The obtained adiabatic condition consists of two inequalities, one of which coincides with the conventional adiabatic condition for the real-time Schroedinger equation, but the other does not. We apply this adiabatic approximation to the analysis of Markovian dynamics of the classical Ising model, which can be formulated as the imaginary-time Schr\"odinger equation, to obtain an asymptotic formula for the probability that the system reaches the ground state in the limit of a long annealing time in simulated annealing. Using this form, we amend the theory of Somma, Batista, and Ortiz for a convergence condition for simulated annealing.