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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.<\/jats:p>","DOI":"10.22331\/q-2025-02-25-1645","type":"journal-article","created":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T17:07:54Z","timestamp":1740503274000},"page":"1645","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Matrix-product unitaries: Beyond quantum cellular automata"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6809-8505","authenticated-orcid":false,"given":"Georgios","family":"Styliaris","sequence":"first","affiliation":[{"name":"Max Planck Institute of Quantum Optics, Hans-Kopfermann-Str. 1, Garching 85748, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 M\u00fcnchen, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5621-7255","authenticated-orcid":false,"given":"Rahul","family":"Trivedi","sequence":"additional","affiliation":[{"name":"Max Planck Institute of Quantum Optics, Hans-Kopfermann-Str. 1, Garching 85748, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 M\u00fcnchen, Germany"},{"name":"Electrical and Computer Engineering, University of Washington, Seattle, Washington 98195, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2990-791X","authenticated-orcid":false,"given":"David","family":"Perez-Garcia","sequence":"additional","affiliation":[{"name":"Departamento de An\u00e1lisis Matem\u00e1tico, Universidad Complutense de Madrid, 28040 Madrid, Spain"},{"name":"Instituto de Ciencias Matem\u00e1ticas (CSIC-UAM-UC3M-UCM), 28049 Madrid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3359-1743","authenticated-orcid":false,"given":"J. 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