A map on finitely many fermionic modes represents a unitary evolution if and only if it preserves canonical anticommutation relations (CARs). Preliminarily general features of nonlinear fermionic cellular automata (FCAs) that can be extended to the case of cellular automata on infinite graphs are analyzed. We use the preservation of CARs for the classification of FCAs on Cayley graphs of finite groups in two simple but paradigmatic case studies. The physical properties of the solutions are finally discussed. One of the examples allows us to classify all scalar FCAs on the $\mathbb{Z}$ integers.

• Received 23 July 2018

DOI:https://doi.org/10.1103/PhysRevA.98.052337