  We study the performance of quantum annealing for systems with ground-state
degeneracy by directly solving the Schr\"odinger equation for small systems and
quantum Monte Carlo simulations for larger systems. The results indicate that
naive quantum annealing using a transverse field may not be well suited to
identify all degenerate ground-state configurations, although the value of the
ground-state energy is often efficiently estimated. An introduction of quantum
transitions to all states with equal weights is shown to greatly improve the
situation but with a sacrifice in the annealing time. We also clarify the
relation between the spin configurations in the degenerate ground states and
the probabilities that those states are obtained by quantum annealing. The
strengths and weaknesses of quantum annealing for problems with degenerate
ground states are discussed in comparison with classical simulated annealing.
