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We study integrable spin chains and quantum and classical cellular automatawith interaction range $\ell\ge 3$. This is a family of integrable models forwhich there was no general theory so far. We develop an algebraic framework forsuch models, generalizing known methods from nearest neighbor interactingchains. This leads to a new integrability condition for medium rangeHamiltonians, which can be used to classify such models. A partialclassification is performed in specific cases, including $U(1)$-symmetric threesite interacting models, and Hamiltonians that are relevant forinteraction-round-a-face models. We find a number of models which appear to benew. As an application we consider quantum brickwork circuits of various types,including those that can accommodate the classical elementary cellular automataon light cone lattices. In this family we find that the so-called Rule150 andRule105 models are Yang-Baxter integrable with three site interactions. Wepresent integrable quantum deformations of these models, and derive a set oflocal conserved charges for them. For the famous Rule54 model we find that itdoes not belong to the family of integrable three site models, but we can notexclude Yang-Baxter integrability with longer interaction ranges.

Integrable spin chains and cellular automata with medium range interaction