# A review of Quantum Cellular Automata

## TL;DR

This review explores quantum cellular automata (QCAs) as models for quantum computation, topological phases, quantum field theories, and their mathematical structures, highlighting recent advances and diverse applications.

## Contribution

It provides a comprehensive overview of QCAs, including their applications in physics and new theoretical insights into their structure and classification.

## Key findings

- QCAs serve as models for topological phases of matter.
- They can discretize quantum field theories and exhibit interacting continuum limits.
- Mathematical frameworks like tensor networks and index theory advance QCA classification.

## Abstract

Discretizing spacetime is often a natural step towards modelling physical systems. For quantum systems, if we also demand a strict bound on the speed of information propagation, we get quantum cellular automata (QCAs). These originally arose as an alternative paradigm for quantum computation, though more recently they have found application in understanding topological phases of matter and have been proposed as models of periodically driven (Floquet) quantum systems, where QCA methods were used to classify their phases. QCAs have also been used as a natural discretization of quantum field theory, and some interesting examples of QCAs have been introduced that become interacting quantum field theories in the continuum limit. This review discusses all of these applications, as well as some other interesting results on the structure of quantum cellular automata, including the tensor-network unitary approach, the index theory and higher dimensional classifications of QCAs.