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A family of reversible deterministic cellular automata, including the rules54 and 201 of [Bobenko et al., Commun. Math. Phys. 158, 127 (1993)] as well astheir kinetically constrained quantum (unitary) or stochastic deformations, isshown to correspond to integrable Floquet circuit models with localinteractions round-a-face. Using inhomogeneous solutions of the star-trianglerelation with a one or two dimensional spectral parameter, changing theirfunctional form depending on the orientation, we provide an explicitconstruction of the transfer matrix and establish its conservation law andinvolutivity properties. Integrability is independently demonstrated bynumerically exploring the spectral statistics via the Berry-Tabor conjecture.Curiously, we find that the deformed rule 54 model generically possesses noother local conserved quantities besides the net soliton current.

Reversible Cellular Automata as Integrable Interactions Round-a-Face: Deterministic, Stochastic, and Quantized