  Quantum computers use quantum resources to carry out computational tasks and
may outperform classical computers in solving certain computational problems.
Special-purpose quantum computers such as quantum annealers employ quantum
adiabatic theorem to solve combinatorial optimization problems. In this paper,
we compare classical annealings such as simulated annealing and quantum
annealings that are done by the D-Wave machines both theoretically and
numerically. We show that if the classical and quantum annealing are
characterized by equivalent Ising models, then solving an optimization problem,
i.e., finding the minimal energy of each Ising model, by the two annealing
procedures, are mathematically identical. For quantum annealing, we also derive
the probability lower-bound on successfully solving an optimization problem by
measuring the system at the end of the annealing procedure. Moreover, we
present the Markov chain Monte Carlo (MCMC) method to realize quantum annealing
by classical computers and investigate its statistical properties. In the
numerical section, we discuss the discrepancies between the MCMC based
annealing approaches and the quantum annealing approach in solving optimization
problems.
