Cellular automata and spin chains: a medium range connection

A cellular automaton is an extremely simple physical system: it has both a discrete time and space dimension. While they were initially developed as a new paradigm for quantum computation, it was soon realized that cellular automata were able to display very interesting behaviours like integrability and large-scale diffusive transport. Indeed rule 54, an elementary cellular automaton, is widely considered the simplest physical system to display integrability, making it the perfect toy model to study various properties of classical and quantum integrability and how they connect. Since a cellular automaton is a fully discrete system it is natural to ask what is the corresponding model in the continuum, if it exists. Recently a new algebraic framework was developed that made it possible to generate classes of integrable quantum spin chains with medium range interactions while also building their corresponding discretization, in the form of quantum cellular automata. In this work we will study what it means for an system to be integrable, both in the classical and quantum realm; then we will describe the structure behind cellular automata and we will show the connection between integrable spin chains and quantum cellular automata. Finally we will try to extend this framework to a more complex system, the class of Restricted Solid-on-Solid models.