Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps

We consider one dimensional block cellular automata, where the local update rules are given by Yang-Baxter maps, which set theoretical solutions of equations. show that such systems superintegrable: they possess an exponentially large conserved charges, charge densities propagate ballistically on chain. For these quantities we observe a complete absence "operator spreading". In addition, models can also have other charges only additively. discuss concrete up to dimensions $N\le 4$, and give rise rich physical behaviour, including non-trivial scattering particles coexistence ballistic diffusive transport. find classical versions "dual unitary gates" if maps non-degenerate. consequences dual unitarity, family gates obtained non-integrable quantum mechanical deformation maps.