  Central to the field of quantum machine learning is the design of quantum
perceptrons and neural network architectures. A key question in this regard is
the impact of quantum effects on the way in which such models process
information. Here, we approach this question by establishing a connection
between $(1+1)D$ quantum cellular automata, which implement a discrete
nonequilibrium quantum many-body dynamics through the successive application of
local quantum gates, and recurrent quantum neural networks, which process
information by feeding it through perceptrons interconnecting adjacent layers.
This relation allows the processing of information in quantum neural networks
to be studied in terms of the properties of their equivalent cellular automaton
dynamics. We exploit this by constructing a class of quantum gates
(perceptrons) that allow for the introduction of quantum effects, such as those
associated with a coherent Hamiltonian evolution, and establish a rigorous link
to continuous-time Lindblad dynamics. We further analyse the universal
properties of a specific quantum cellular automaton, and identify a change of
critical behavior when quantum effects are varied, demonstrating that they can
indeed affect the collective dynamical behavior underlying the processing of
information in large-scale neural networks.
