Integrable spin chains and cellular automata with medium range interaction

We study integrable spin chains and quantum and classical cellular automata with interaction range $\ell\ge 3$. This is a family of integrable models for which there was no general theory so far. We develop an algebraic framework for such models, generalizing known methods from nearest neighbor interacting chains. This leads to a new integrability condition for medium range Hamiltonians, which can be used to classify such models. A partial classification is performed in specific cases, including $U(1)$-symmetric three site interacting models, and Hamiltonians that are relevant for interaction-round-a-face models. We find a number of models which appear to be new. As an application we consider quantum brickwork circuits of various types, including those that can accommodate the classical elementary cellular automata on light cone lattices. In this family we find that the so-called Rule150 and Rule105 models are Yang-Baxter integrable with three site interactions. We present integrable quantum deformations of these models, and derive a set of local conserved charges for them. For the famous Rule54 model we find that it does not belong to the family of integrable three site models, but we can not exclude Yang-Baxter integrability with longer interaction ranges.