A QCA for every SPT

TL;DR

This paper constructs quantum cellular automata (QCA) that disentangle symmetry protected topological phases in various dimensions, generalizing known models and relating Clifford QCA to classification theorems.

Contribution

It introduces a conjectured generalization of QCA for higher-dimensional SPT phases based on Stiefel--Whitney classes, expanding the understanding of QCA in topological phases.

Findings

Constructed QCA for 3D Walker--Wang model

Conjectured higher-dimensional QCA for SPT phases

Identified Clifford QCA in 4m+1 dimensions with specific circuit properties

Abstract

In three dimensions, there is a nontrivial quantum cellular automaton (QCA) which disentangles the three-fermion Walker--Wang model, a model whose action depends on Stiefel--Whitney classes of the spacetime manifold. Here we present a conjectured generalization to higher dimensions. For an arbitrary symmetry protected topological phase of time reversal whose action depends on Stiefel--Whitney classes, we construct a corresponding QCA that we conjecture disentangles that phase. Some of our QCA are Clifford, and we relate these to a classification theorem of Clifford QCA. We identify Clifford QCA in $4m+1$ dimensions, for which we find a low-depth circuit description using non-Clifford gates but not with Clifford gates.