A Simple and General Equation for Matrix Product Unitary Generation

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.