Matrix-product unitaries: Beyond quantum cellular automata

Georgios Styliaris; Rahul Trivedi; David Perez-Garcia; J. Ignacio Cirac

doi:10.22331/q-2025-02-25-1645

Matrix-product unitaries: Beyond quantum cellular automata

Georgios Styliaris^1,2, Rahul Trivedi^1,2,3, David Perez-Garcia^4,5, and J. Ignacio Cirac^1,2

¹Max Planck Institute of Quantum Optics, Hans-Kopfermann-Str. 1, Garching 85748, Germany
²Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany
³Electrical and Computer Engineering, University of Washington, Seattle, Washington 98195, USA
⁴Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain
⁵Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), 28049 Madrid, Spain

Published:	2025-02-25, volume 9, page 1645
Editor:	Thomas Elliott
Eprint:	arXiv:2406.10195v3
Doi:	https://doi.org/10.22331/q-2025-02-25-1645
Citation:	Quantum 9, 1645 (2025).

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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.

Featured image: A unitary matrix-product operator. This work explores their structure and correlations.

Popular summary

Quantum systems evolve according to the laws of quantum mechanics. One way to describe these evolutions for spins arranged on a chain is through matrix-product unitaries (MPU). These are transformations that preserve a constraint known as the entanglement area law, meaning they map quantum states with limited entanglement to similarly structured states. Traditionally, when MPU follow strict translation-invariant rules — meaning the same local rule is applied everywhere — they behave like quantum cellular automata, ensuring information propagates only between nearest neighbors in a controlled fashion. In this work, we show that when boundaries are introduced or the local rules vary across the system, this picture breaks down. These generalized MPU can generate long-range correlations from initially short-range ones, altering quantum phases of matter. Our results lay the groundwork for understanding these more complex quantum evolutions, revealing connections between MPU and multipartite entanglement structures.

► BibTeX data

@article{Styliaris2025matrixproduct,
  doi = {10.22331/q-2025-02-25-1645},
  url = {https://doi.org/10.22331/q-2025-02-25-1645},
  title = {Matrix-product unitaries: {B}eyond quantum cellular automata},
  author = {Styliaris, Georgios and Trivedi, Rahul and Perez-Garcia, David and Cirac, J. Ignacio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1645},
  month = feb,
  year = {2025}
}

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Cited by

[1] Erickson Tjoa and J. Ignacio Cirac, "Continuous matrix-product operators for quantum fields", Physical Review Research 8 2, L022013 (2026).

[2] Georgios Styliaris, Rahul Trivedi, and J. Ignacio Cirac, "Quantum Circuits for Matrix-Product Unitaries", Physical Review Letters 135 26, 260602 (2025).

[3] Faidon Andreadakis and Paolo Zanardi, "Exact link between nonlocal nonstabilizerness and operator entanglement", Physical Review A 113 1, L010404 (2026).

[4] Sujeet K. Shukla, "A simple and general equation for matrix product unitary generation", Journal of Mathematical Physics 66 10, 101901 (2025).

[5] Augustin Vanrietvelde, Octave Mestoudjian, and Pablo Arrighi, "Causal Decompositions of 1D Quantum Cellular Automata", arXiv:2506.22219, (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 21:05:03) and SAO/NASA ADS (last updated successfully 2026-08-12 21:05:05). The list may be incomplete as not all publishers provide suitable and complete citation data.

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