On the structure of Clifford quantum cellular automata

TL;DR

This paper characterizes the structure of one-dimensional Clifford quantum cellular automata, showing they are generated by elementary operations and can prepare all translationally invariant stabilizer states from product states.

Contribution

It provides a complete characterization of 1D Clifford QCA, linking symplectic automata to local rules and stabilizer state preparation.

Findings

All local rules are reflection invariant up to a global shift.

Every translationally invariant stabilizer state can be prepared by a single Clifford QCA step.

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

Abstract

We study reversible quantum cellular automata with the restriction that these are also Clifford operations. This means that tensor products of Pauli operators (or discrete Weyl operators) are mapped to tensor products of Pauli 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 every uniquely determined and translationally invariant stabilizer state can be prepared from a product state by a single Clifford cellular automaton timestep, thereby characterizing these class of stabilizer states, and we show that all 1D Clifford quantum cellular automata are generated by a few elementary operations. We also show that the correspondence between translationally invariant stabilizer states and translationally invariant Clifford operations holds for periodic boundary conditions.