Convergence condition of simulated quantum annealing for closed and open systems

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