  We consider a class of noisy, one-dimensional quantum cellular automata that
allow one to shift from unitary dynamics to completely positive maps, and
investigate the notion of reversibility in such a setting. To this aim, we
associate an approximate reverse automaton to each noisy automaton, and assess
its effect, and we define an irreversibility time based on the distance from
the maximally mixed state, which is shown to be the only attractor of the
automaton map in the presence of dephasing.
