# Classification of Matrix-Product Unitaries with Symmetries

## TL;DR

This paper classifies matrix-product unitaries with symmetries using indices and cohomology, introduces symmetry-protected indices for refined classification, and explores implications for Floquet SPT phases.

## Contribution

It provides a complete classification of symmetric MPUs using indices and cohomology, and introduces symmetry-protected indices for detailed characterization.

## Key findings

- MPUs classified by (chiral) index and cohomology class of symmetry group
- Symmetry-protected indices quantify transport imbalance and are experimentally measurable
- Framework enables systematic construction of 2D Floquet SPT phases beyond standard classification

## Abstract

We prove that matrix-product unitaries (MPUs) with on-site unitary symmetries are completely classified by the (chiral) index and the cohomology class of the symmetry group $G$, provided that we can add trivial and symmetric ancillas with arbitrary on-site representations of $G$. If the representations in both system and ancillas are fixed to be the same, we can define symmetry-protected indices (SPIs) which quantify the imbalance in the transport associated to each group element and greatly refines the classification. These SPIs are stable against disorder and measurable in interferometric experiments. Our results lead to a systematic construction of two-dimensional Floquet symmetry-protected topological (SPT) phases beyond the standard classification, and thus shed new light on understanding nonequilibrium phases of quantum matter.