  Probabilistic cellular automata provide a simple framework for the
exploration of classical nonequilibrium processes. Recently, quantum cellular
automata have been proposed that rely on the propagation of a one-dimensional
quantum state along a fictitious discrete time dimension via the sequential
application of quantum gates. The resulting $(1+1)$-dimensional space-time
structure makes these automata special cases of feed-forward quantum neural
networks. Here we show how asynchronism -- introduced via non-commuting gates
-- impacts on the collective nonequilibrium behavior of quantum cellular
automata. We illustrate this through a simple model, whose synchronous version
implements a contact process and features a nonequilibrium phase transition in
the directed percolation universality class. Non-commuting quantum gates lead
to an "asynchronism transition", i.e. a sudden qualitative change in the phase
transition behavior once a certain degree of asynchronicity is surpassed. Our
results show how quantum effects may lead to abrupt changes of non-equilibrium
dynamics, which may be relevant for understanding the role of quantum
correlations in neural networks.
