Scalar fermionic cellular automata on finite Cayley graphs

TL;DR

This paper classifies fermionic cellular automata on finite Cayley graphs by analyzing unitary maps that preserve anti-commutation relations, discussing their physical properties and potential extensions to infinite graphs.

Contribution

It provides a classification framework for fermionic cellular automata on finite Cayley graphs based on anti-commutation preservation, with insights into physical properties and infinite graph extensions.

Findings

Classification of fermionic automata on finite Cayley graphs

Analysis of physical properties of solutions

Discussion on extension to infinite graphs

Abstract

A map on finitely many fermionic modes represents a unitary evolution if and only if it preserves canonical anti-commutation relations. We use this condition for the classification of fermionic cellu- lar automata (FCA) on Cayley graphs of finite groups in two simple but paradigmatic case studies. The physical properties of the solutions are discussed. Finally, features of the solutions that can be extended to the case of cellular automata on infinite graphs are analyzed.