  Simulated quantum annealing based on the path-integral Monte Carlo is one of
the most common tools to simulate quantum annealing on classical hardware.
Nevertheless, it is in principle highly non-trivial whether or not this
classical algorithm can correctly reproduce the quantum dynamics of quantum
annealing, particularly in the diabatic regime. We study this problem
numerically through the generalized Kibble-Zurek mechanism of defect
distribution in the simplest ferromagnetic one-dimensional transverse-field
Ising model with and without coupling to the environment. We find that,in the
absence of coupling to the environment, simulated quantum annealing correctly
describes the annealing-time dependence of the average number of defects, but a
detailed analysis of the defect distribution shows clear deviations from the
theoretical prediction. When the system is open (coupled to the environment),
the average number of defects does not follow the theoretical prediction but is
qualitatively compatible with the numerical result by the infinite
time-evolving block decimation combined with the quasi-adiabatic propagator
path integral, which is valid in a very short time region. The distribution of
defects in the open system turns out to be not far from the theoretical
prediction. It is surprising that the classical stochastic dynamics of
simulated quantum annealing ostensibly reproduce some aspects of the quantum
dynamics. However, a serious problem is that it is hard to predict for which
physical quantities in which system it is reliable. Those results suggest the
necessity to exert a good amount of caution in using simulated quantum
annealing to study the detailed quantitative aspects of the dynamics of quantum
annealing.
