  We define a class of stochastic processes based on evolutions and
measurements of quantum systems, and consider the complexity of predicting
their long-term behavior. It is shown that a very general class of decision
problems regarding these stochastic processes can be efficiently solved
classically in the space-bounded case. The following corollaries are implied by
our main result: (1) Any space O(s) uniform family of quantum circuits acting
on s qubits and consisting of unitary gates and measurement gates defined in a
typical way by matrices of algebraic numbers can be simulated by an unbounded
error space O(s) ordinary (i.e., fair-coin flipping) probabilistic Turing
machine, and hence by space O(s) uniform classical (deterministic) circuits of
depth O(s^2) and size 2^(O(s)). The quantum circuits are not required to
operate with bounded error and may have depth exponential in s. (2) Any
(unbounded error) quantum Turing machine running in space s, having arbitrary
algebraic transition amplitudes, allowing unrestricted measurements during its
computation, and having no restrictions on running time can be simulated by an
unbounded error space O(s) ordinary probabilistic Turing machine, and hence
deterministically in space O(s^2).
