Probabilistic cellular automata provide a simple framework for exploring 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 recurrent quantum neural networks which can implement broad classes of classical nonequilibrium processes. Here, we present a general prescription by which these models can be extended into genuinely quantum nonequilibrium models via the systematic inclusion of asynchronism. This is illustrated for the classical contact process, where the resulting model is closely linked to the quantum contact process (QCP), developed in the framework of open quantum systems. Studying the mean-field behavior of the model, we find evidence of an “asynchronism transition,” i.e., a sudden qualitative change in the phase transition behavior once a certain degree of asynchronicity is surpassed, a phenomenon we link to observations in the QCP.

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• Artificial neural networks
• Cellular automata
• Phase transitions
• Quantum many-body systems
• Tensor network methods