Quantum simulations of classical random walks and undirected graph connectivity

There are a number of questions in quantum complexity that have been resolved in the time-bounded setting, but remain open in the space-bounded setting. For example, it is not currently known if space-bounded probabilistic computations can be simulated by space-bounded quantum machines without allowing measurements during the computation, while it is known that an analogous statement holds in the time-bounded case. A more general question asks if measurements during a quantum computation can allow for more space-efficient solutions to certain problems. 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-without relying on measurements during the computation. 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 require a single measurement at the end of their computations. It follows that symmetric logspace is contained in the quantum analogue of randomized logspace, i.e., SL⊆QR_HL