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Matrix-product unitaries: Beyond quantum cellular automata

Georgios Styliaris, Rahul Trivedi, David Pérez-García, J. Ignacio Cirac

TL;DR

This work extends the matrix-product unitary (MPU) framework beyond the standard uniform-bulk, periodic-boundary regime by studying MPU with a uniform bulk and arbitrary boundary, revealing TI MPUs that nevertheless violate the 1D quantum cellular automata (QCA) light-cone property. It develops a recursive, canonical-form formalism for nonuniform MPU, derives necessary and sufficient unitarity conditions, and analyzes sequential-generation limitations, showing that not all MPU admit shallow sequential circuits. The authors introduce semi-simple MPU, show their block-structure is incompatible with nontrivial QCA behavior (except in trivial product cases), and connect MPU to locally maximally entanglable (LME) states through Choi–Jamiołkowski vectorization, leading to a LU-classification result for qubit LME unitaries as phase unitaries. Overall, the paper lays foundational steps toward a broader theory of MPU beyond QCA, including classifications, boundary effects, and links to LME theory with potential implications for circuit complexity and phase structure in 1D quantum systems.

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.

Matrix-product unitaries: Beyond quantum cellular automata

TL;DR

This work extends the matrix-product unitary (MPU) framework beyond the standard uniform-bulk, periodic-boundary regime by studying MPU with a uniform bulk and arbitrary boundary, revealing TI MPUs that nevertheless violate the 1D quantum cellular automata (QCA) light-cone property. It develops a recursive, canonical-form formalism for nonuniform MPU, derives necessary and sufficient unitarity conditions, and analyzes sequential-generation limitations, showing that not all MPU admit shallow sequential circuits. The authors introduce semi-simple MPU, show their block-structure is incompatible with nontrivial QCA behavior (except in trivial product cases), and connect MPU to locally maximally entanglable (LME) states through Choi–Jamiołkowski vectorization, leading to a LU-classification result for qubit LME unitaries as phase unitaries. Overall, the paper lays foundational steps toward a broader theory of MPU beyond QCA, including classifications, boundary effects, and links to LME theory with potential implications for circuit complexity and phase structure in 1D quantum systems.

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.
Paper Structure (17 sections, 18 theorems, 96 equations)

This paper contains 17 sections, 18 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Motivating example
  4. Nonuniform MPU
  5. Structure
  6. When does a sequence of tensors generate a unitary?
  7. Can MPU be sequentially generated?
  8. MPU with uniform bulk and arbitrary boundary
  9. Semi-simple MPU
  10. Structure of semi-simple MPU
  11. MPU and locally maximally-entanglable states
  12. Multipartite unitaries and LME states
  13. Classification of qubit LME states and their unitaries
  14. Summary and Outlook
  15. Proofs of \ref{['sec:inhomo_structure']}
  16. ...and 2 more sections

Key Result

Lemma 3

A sequence of tensors $A_{1}, \dots, A_{N}$ generates a MPU if and only if

Theorems & Definitions (44)

  • Example 1: Multi-control $Z$-gate
  • Definition 2: Nonuniform MPU
  • Lemma 3
  • proof
  • Corollary 4
  • proof
  • Definition 5: Nonuniform canonical form
  • Proposition 5
  • proof
  • Corollary 6
  • ...and 34 more