  The protocol of quantum annealing is applied to an optimization problem with
a one-dimensional continuous degree of freedom, a variant of the problem
proposed by Shinomoto and Kabashima. The energy landscape has a number of local
minima, and the classical approach of simulated annealing is predicted to have
a logarithmically slow convergence to the global minimum. We show by extensive
numerical analyses that quantum annealing yields a power law convergence, thus
an exponential improvement over simulated annealing. The power is larger, and
thus the convergence is faster, than a prediction by an existing
phenomenological theory for this problem. Performance of simulated annealing is
shown to be enhanced by introducing quasi-global searches across energy
barriers, leading to a power-law convergence but with a smaller power than in
the quantum case and thus a slower convergence classically even with
quasi-global search processes. We also reveal how diabatic quantum dynamics,
quantum tunneling in particular, steers the systems toward the global minimum
by a meticulous choice of annealing schedule. This latter result explicitly
contrasts the role of tunneling in quantum annealing against the classical
counterpart of stochastic optimization by simulated annealing.
