Remarks on the structure of Clifford quantum cellular automata

TL;DR

This paper analyzes the structure of reversible Clifford quantum cellular automata, showing they are induced by symplectic cellular automata and characterizing their local rules, especially in one dimension where they are generated by elementary operations.

Contribution

It provides a detailed characterization of Clifford quantum cellular automata, linking them to symplectic automata and identifying their fundamental generating operations in 1D.

Findings

Clifford QCA are induced by symplectic cellular automata in phase space.

Local rules are reflection invariant up to a global shift.

All 1D Clifford QCA are generated by a few elementary operations.

Abstract

We report here on the structure of reversible quantum cellular automata with the additional restriction that these are also Clifford operations. This means that tensor products of Weyl operators (projective representation of a finite abelian symplectic group) are mapped to multiples of tensor products of Weyl operators. Therefore Clifford quantum cellular automata are induced by symplectic cellular automata in phase space. We characterize these symplectic cellular automata and find that all possible local rules must be, up to some global shift, reflection invariant with respect to the origin. In the one dimensional case we also find that all 1D Clifford quantum cellular automata are generated by a few elementary operations.