  Motivated by recent progress in the experimental development of quantum
simulators based on Rydberg atoms, we introduce and investigate the dynamics of
a class of $(1+1)$-dimensional quantum cellular automata. These non-equilibrium
many-body models, which are quantum generalisations of the Domany-Kinzel
cellular automaton, possess two key features: they display stationary behaviour
and non-equilibrium phase transitions despite being isolated systems. Moreover,
they permit the controlled introduction of local quantum correlations, which
allows for the impact of quantumness on the dynamics and phase transition to be
assessed. We show that projected entangled pair state tensor networks permit a
natural and efficient representation of the cellular automaton. Here, the
degree of quantumness and complexity of the dynamics is reflected in the
difficulty of contracting the tensor network.
