Matrix-product unitaries: Beyond quantum cellular automata

TL;DR

This paper explores the properties of matrix-product unitaries (MPU), especially those with arbitrary boundaries, revealing their ability to alter correlations and phases, and establishing new links to entanglable states.

Contribution

It introduces a theory for MPU with uniform bulk and arbitrary boundary, analyzing non-QCA structures and their relation to entanglable states.

Findings

MPU with open boundaries can change correlation lengths

A subclass of MPU with a direct-sum form maximally violates QCA properties

MPU formed by site-dependent tensors relate to locally maximally entanglable states

Abstract

Matrix-product unitaries (MPU) are 1D tensor networks describing time evolution and unitary symmetries of quantum systems, while their action on states by construction preserves the entanglement area law. MPU which are formed by a single repeated tensor are known to coincide with 1D quantum cellular automata (QCA), i.e., unitaries with an exact light cone. However, this correspondence breaks down for MPU with open boundary conditions, even if the resulting operator is translation-invariant. Such unitaries can turn short- to long-range correlations and thus alter the underlying phase of matter. Here we make the first steps towards a theory of MPU with uniform bulk but arbitrary boundary. In particular, we study the structure of a subclass with a direct-sum form which maximally violates the QCA property. We also consider the general case of MPU formed by site-dependent (nonuniform) tensors and show a correspondence between MPU and locally maximally entanglable states.