# Matrix Product Unitaries: Structure, Symmetries, and Topological Invariants

## TL;DR

This paper develops the structure theory of Matrix Product Unitary operators, proving their relation to Quantum Cellular Automata, establishing a canonical form, and classifying them under various symmetries, with implications for quantum system evolution.

## Contribution

It introduces a canonical form for MPUs, proves an Index Theorem for their classification, and analyzes symmetry effects on MPU equivalence classes.

## Key findings

- All MPUs have a strict causal cone, making them QCAs.
- A canonical form relates different MPU representations.
- Symmetry considerations classify MPU invariants.

## Abstract

Matrix Product Vectors form the appropriate framework to study and classify one-dimensional quantum systems. In this work, we develop the structure theory of Matrix Product Unitary operators (MPUs) which appear e.g. in the description of time evolutions of one-dimensional systems. We prove that all MPUs have a strict causal cone, making them Quantum Cellular Automata (QCAs), and derive a canonical form for MPUs which relates different MPU representations of the same unitary through a local gauge. We use this canonical form to prove an Index Theorem for MPUs which gives the precise conditions under which two MPUs are adiabatically connected, providing an alternative derivation to that of [Commun. Math. Phys. 310, 419 (2012), arXiv:0910.3675] for QCAs. We also discuss the effect of symmetries on the MPU classification. In particular, we characterize the tensors corresponding to MPU that are invariant under conjugation, time reversal, or transposition. In the first case, we give a full characterization of all equivalence classes. Finally, we give several examples of MPU possessing different symmetries.