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We introduce a class of noisy quantum cellular automata on a qubit latticethat includes all classical Markov chains, as well as maps where quantumcoherence between sites is allowed to build up over time. We apply such aconstruction to the problem of excitation transfer through 1-d lattices, andcompare the performance of classical and quantum dynamics with equal localtransition probabilities. Our discrete approach has the merits of strippingdown the complications of the open system dynamics, of clearly isolatingcoherent effects, of allowing for an exact treatment of conditional dynamics,all while capturing a rich variety of dynamical behaviours.

Noisy Quantum Cellular Automata for Quantum vs Classical Excitation Transfer