  We introduce a class of noisy quantum cellular automata on a qubit lattice
that includes all classical Markov chains, as well as maps where quantum
coherence between sites is allowed to build up over time. We apply such a
construction to the problem of excitation transfer through 1-d lattices, and
compare the performance of classical and quantum dynamics with equal local
transition probabilities. Our discrete approach has the merits of stripping
down the complications of the open system dynamics, of clearly isolating
coherent effects, of allowing for an exact treatment of conditional dynamics,
all while capturing a rich variety of dynamical behaviours.
