Convergence condition of simulated quantum annealing for closed and open systems

TL;DR

This paper derives a general convergence condition for simulated quantum annealing to reach thermal equilibrium in both closed and open systems, linking classical stochastic processes with quantum dynamics.

Contribution

It introduces a unified convergence condition for simulated quantum annealing applicable to both closed and open systems, connecting classical and quantum annealing theories.

Findings

Convergence condition derived using imaginary-time Schrödinger equation.

Qualitative agreement with rigorous conditions for closed systems.

Highlights a non-trivial link between classical and quantum annealing convergence.

Abstract

Simulated quantum annealing is a generic classical protocol to simulate some aspects of quantum annealing and is sometimes regarded as a classical alternative to quantum annealing in finding the ground state of a classical Ising model. We derive a generic condition for simulated quantum annealing to converge to thermal equilibrium at a given, typically low, temperature. Both closed and open systems are treated. We rewrite the classical master equation for simulated quantum annealing into an imaginary-time Schr\"odinger equation, to which we apply the imaginary-time variant of asymptotic adiabatic condition to deduce the convergence condition. The result agrees qualitatively with a rigorous convergence condition of simulated quantum annealing for closed systems, which was derived from the theory of inhomogeneous Markov process. Also observed is qualitative agreement with a rigorous convergence condition of quantum annealing for closed systems under the real-time Schr\"odinger dynamics. This coincidence of convergence conditions for classical stochastic processes for simulated quantum annealing and the real-time quantum dynamics for quantum annealing is highly non-trivial and calls for further scrutiny.