  It is not currently known if quantum Turing machines can efficiently simulate
probabilistic computations in the space-bounded case. In this paper we show
that space-bounded quantum Turing machines can efficiently simulate a limited
class of random processes: random walks on undirected graphs. By means of such
simulations, it is demonstrated that the undirected graph connectivity problem
for regular graphs can be solved by one-sided error quantum Turing machines
that run in logspace and halt absolutely. It follows that symmetric logspace is
contained in the quantum analogue of randomized logspace.
