The Group Structure of Quantum Cellular Automata

TL;DR

This paper investigates the algebraic structure of quantum cellular automata (QCA), demonstrating their abelian nature modulo circuits and introducing a method to define a consistent group of QCA without reliance on families, especially in translation invariant cases.

Contribution

It introduces a general method for ancilla removal in QCA, proving their abelian group structure, and constructs a coherent family framework for QCA that is independent of system size.

Findings

QCA form an abelian group modulo circuits for most control spaces.

A method for constructing coherent families of QCA is developed.

Translation invariant QCA in three dimensions are shown to be coherent.

Abstract

We consider the group structure of quantum cellular automata (QCA) modulo circuits and show that it is abelian even without assuming the presence of ancillas, at least for most reasonable choices of control space; this is a corollary of a general method of ancilla removal. Further, we show how to define a group of QCA that is well-defined without needing to use families, by showing how to construct a coherent family containing an arbitrary finite QCA; the coherent family consists of QCA on progressively finer systems of qudits where any two members are related by a shallow quantum circuit. This construction applied to translation invariant QCA shows that all translation invariant QCA in three dimensions and all translation invariant Clifford QCA in any dimension are coherent.