An index for quantum cellular automata on fusion spin chains

TL;DR

This paper generalizes the GNVW index for 1D quantum cellular automata to fusion spin chains using the Jones index, providing a complete invariant for certain automata groups.

Contribution

It introduces a new index for QCA on fusion spin chains based on subfactor theory, extending previous concepts to more general spin chain models.

Findings

The index is a complete invariant for QCA on fusion spin chains from the $ extbf{Fib}$ category.

The generalized index applies to boundary operators of 2D topological codes.

The approach links quantum automata classification to subfactor theory.

Abstract

Interpreting the GNVW index for 1D quantum cellular automata (QCA) in terms of the Jones index for subfactors leads to a generalization of the index defined for QCA on more general abstract spin chains. These include fusion spin chains, which arise as the local operators invariant under a global (categorical/MPO) symmetry, and as the boundary operators of 2D topological codes. We show that for the fusion spin chains built from the fusion category $\mathbf{Fib}$, the index is a complete invariant for the group of QCA modulo finite depth circuits.