# Simple derivation of the Weyl and Dirac quantum cellular automata

## TL;DR

This paper derives the unique two-dimensional quantum cellular automata on a body-centred cubic lattice that approximate Weyl and Dirac equations in the continuum limit, highlighting their structure and connection to lattice geometry.

## Contribution

It provides a simple derivation of the only two automata satisfying key symmetries and unitarity, linking transition matrices to lattice vertices and extending to four dimensions.

## Key findings

- Automata reduce to Weyl and Dirac equations in the continuum limit.
- Transition matrices correspond to vertices of the lattice's primitive cell.
- Two families of automata in four dimensions relate to the Dirac equation.

## Abstract

We consider quantum cellular automata on a body-centred cubic lattice and provide a simple derivation of the only two homogenous, local, isotropic, and unitary two-dimensional automata [G. M. D'Ariano and P. Perinotti, Physical Review A 90, 062106 (2014)]. Our derivation relies on the notion of Gram matrix and emphasises the link between the transition matrices that characterise the automata and the body-centred cubic lattice: The transition matrices essentially are the matrix representation of the vertices of the lattice's primitive cell. As expected, the dynamics of these two automata reduce to the Weyl equation in the limit of small wave vectors and continuous time. We also briefly examine the four-dimensional case where we find two one-parameter families of automata that reduce to the Dirac equation in a suitable limit.