Non-Invertible Symmetries on Tensor-Product Hilbert Spaces and Quantum Cellular Automata

TL;DR

This paper explores how fusion category symmetries can be realized on tensor-product Hilbert spaces with quantum cellular automata, establishing theoretical constraints and providing explicit lattice models.

Contribution

It systematically analyzes QCA-refined realizations of fusion categories, proving index determination and constructing models for weakly integral categories.

Findings

QCA and symmetry indices are determined by categorical data

Constructed lattice models for any weakly integral fusion category

Computed QCA indices matching theoretical predictions

Abstract

We investigate realizations of (1+1)-dimensional fusion category symmetries on tensor-product Hilbert spaces, allowing for mixing with quantum cellular automata (QCAs). It was argued recently that any such realizable symmetry must be weakly integral. We develop a systematic analysis of QCA-refined realizations of fusion categories and prove two statements. First, we show that, under certain physical assumptions on defects, any QCA-refined realization has QCA and symmetry-operator indices determined by the categorical data, up to the freedom of redefining the symmetry operators. Second, we construct a lattice model that provides a QCA-refined realization for any weakly integral fusion category symmetry on a tensor product Hilbert space. We also compute indices of the QCAs in our lattice model and show agreement with the first result. As an application of the general construction, we give an explicit QCA-refined realization of general Tambara-Yamagami categorical symmetries.