A Simple and General Equation for Matrix Product Unitary Generation

TL;DR

This work resolves the practical problem of determining when a local tensor $M$ generates a Matrix Product Unitary (MPU) of length $N$ by proving the $N$-unitarity theorem: $M$ generates an MPU of size $N$ if and only if $\operatorname{Tr}(\mathbb{E}_M^N) = \operatorname{Tr}(\mathbb{E}_T^N) = 1$, where $T = MM^{\dagger}$ and $\mathbb{E}_M$, $\mathbb{E}_T$ are the transfer matrices. The theorem provides a simple, unified criterion to classify uniform MPUs and establishes that MPUs existing for all system sizes are locality-preserving, with $\mathbb{E}_T$ possessing a single nonzero eigenvalue equal to $1$ and hence a rank-1 long-time behavior. Through explicit examples, including a locality-preserving translation MPU and a nonlocality-preserving MPU that works only for odd sizes, the paper clarifies how the transfer-matrix structure governs causal cones and unitary dynamics. The results open avenues for extending the framework to higher dimensions and to symmetry-enriched or nonuniform MPUs, offering a practical tool for analyzing dynamical quantum phases and Floquet systems.

Abstract

Matrix Product Unitaries (MPUs) have emerged as essential tools for representing locality-preserving 1D unitary operators, with direct applications to quantum cellular automata and quantum phases of matter. A key challenge in the study of MPUs is determining when a given local tensor generates an MPU, a task previously addressed through fixed-point conditions and canonical forms, which can be cumbersome to evaluate for an arbitrary tensor. In this work, we establish a simple and efficient necessary and sufficient condition for a tensor $M$ to generate an MPU of size $N$, given by $\operatorname{Tr}(\mathbb{E}_M^N) = \operatorname{Tr}(\mathbb{E}_T^N) = 1$, where $\mathbb{E}_M$ and $\mathbb{E}_T$ are the transfer matrices of $M$ and $T = MM^\dagger$. This condition provides a unified framework for characterizing all uniform MPUs and significantly simplifies their evaluation. Furthermore, we show that locality preservation naturally arises when the MPU is generated for all system sizes. Our results offer new insights into the structure of MPUs, highlighting connections between unitary evolution, transfer matrices, and locality-preserving behavior, with potential extensions to higher-dimensions.

Theorems & Definitions (10)

• Theorem 1: $MPU^{(N)}$-generation theorem
• proof
• Definition 1
• Lemma 1
• Corollary 1
• proof
• Lemma 2
• proof
• Theorem 2
• proof