  Convergence conditions for quantum annealing are derived for optimization
problems represented by the Ising model of a general form. Quantum fluctuations
are introduced as a transverse field and/or transverse ferromagnetic
interactions, and the time evolution follows the real-time Schrodinger
equation. It is shown that the system stays arbitrarily close to the
instantaneous ground state, finally reaching the target optimal state, if the
strength of quantum fluctuations decreases sufficiently slowly, in particular
inversely proportionally to the power of time in the asymptotic region. This is
the same condition as the other implementations of quantum annealing, quantum
Monte Carlo and Green's function Monte Carlo simulations, in spite of the
essential difference in the type of dynamics. The method of analysis is an
application of the adiabatic theorem in conjunction with an estimate of a lower
bound of the energy gap based on the recently proposed idea of Somma et. al.
for the analysis of classical simulated annealing using a classical-quantum
correspondence.
