{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T23:57:18Z","timestamp":1790726238685,"version":"4.1.0"},"reference-count":54,"publisher":"American Physical Society (APS)","issue":"10","license":[{"start":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T00:00:00Z","timestamp":1773187200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000288","name":"Royal Society","doi-asserted-by":"publisher","award":["201101"],"award-info":[{"award-number":["201101"]}],"id":[{"id":"10.13039\/501100000288","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000275","name":"Leverhulme Trust","doi-asserted-by":"publisher","award":["ECF-2022-324"],"award-info":[{"award-number":["ECF-2022-324"]}],"id":[{"id":"10.13039\/501100000275","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009708","name":"Novo Nordisk Fonden","doi-asserted-by":"publisher","award":["NNF22OC0071934"],"award-info":[{"award-number":["NNF22OC0071934"]}],"id":[{"id":"10.13039\/501100009708","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009708","name":"Novo Nordisk Fonden","doi-asserted-by":"publisher","award":["NNF20OC0059939"],"award-info":[{"award-number":["NNF20OC0059939"]}],"id":[{"id":"10.13039\/501100009708","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. B"],"accepted":{"date-parts":[[2025,12,30]]},"abstract":"<jats:p>\n                    Random permutation circuits were recently introduced as minimal models for local many-body dynamics that can be interpreted both as classical and quantum. Standard dynamical complexity indicators such as damage spreading and out-of-time-order correlators (OTOCs), show that these systems exhibit sensitivity to initial conditions in the classical setting and operator scrambling in the quantum setting. Here, we address their quantum chaoticity\u2014a stricter property\u2014by studying the time evolution of local operator entanglement (LOE). We show that the behavior of LOE in random permutation circuits depends on the dimension of the local configuration space\n                    <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <a:mi>q<\/a:mi>\n                    <\/a:math>\n                    . When\n                    <b:math xmlns:b=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <b:mrow>\n                        <b:mi>q<\/b:mi>\n                        <b:mo>=<\/b:mo>\n                        <b:mn>2<\/b:mn>\n                      <\/b:mrow>\n                    <\/b:math>\n                    , i.e., the circuits act on qubits, random permutations are Clifford and the LOE of any local operator is bounded by a constant, indicating that they are not truly chaotic. On the other hand, when the dimension of the local configuration space exceeds two, the LOE grows linearly in time. We prove this in the limit of large\n                    <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <c:mi>q<\/c:mi>\n                    <\/c:math>\n                    and present numerical evidence that a three-dimensional local configuration space is sufficient for a linear growth of LOE. Our findings highlight that quantum chaos can be produced by essentially classical dynamics. Moreover, we show that LOE can be defined also in the classical realm and put it forward as a universal indicator chaos, both quantum and classical.\n                  <\/jats:p>","DOI":"10.1103\/ql9x-6kzj","type":"journal-article","created":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T21:01:20Z","timestamp":1767128480000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Random permutation circuits beyond qubits are quantum chaotic"],"prefix":"10.1103","volume":"113","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5560-6876","authenticated-orcid":true,"given":"Bruno","family":"Bertini","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03angcq70","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Birmingham"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6791-8594","authenticated-orcid":true,"given":"Katja","family":"Klobas","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/03angcq70","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Birmingham"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7865-8609","authenticated-orcid":true,"given":"Pavel","family":"Kos","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01vekys64","id-type":"ROR","asserted-by":"publisher"}],"name":"Max-Planck-Institut f\u00fcr Quantenoptik"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8832-0927","authenticated-orcid":true,"given":"Daniel","family":"Malz","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/035b05819","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Copenhagen"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2026,3,11]]},"reference":[{"key":"ql9x-6kzjCc1R1","volume-title":"Entanglement in Spin Chains: From Theory to Quantum Technology Applications","author":"A.\u00a0C. 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Nielsen","year":"2000","type":"book"},{"key":"ql9x-6kzjCc46R1","volume-title":"Quantum Signatures of Chaos","author":"F. Haake","year":"2019","type":"book"},{"key":"ql9x-6kzjCc47R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.8.021062","type":"journal-article"},{"key":"ql9x-6kzjCc48R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/ql9x-6kzj for (i) a proof of Eq.   (9); (ii) further details on the permutation-averaged multireplica gates; (iii) the discussion of dynamics of permutation gates with added phases; (iv) the perturbative large local-Hilbert-space expansion; (v) additional details on the numerical methods to produce the data in the main text.","type":"other"},{"key":"ql9x-6kzjCc49R1","doi-asserted-by":"publisher","DOI":"10.2307\/2300300","type":"journal-article"},{"key":"ql9x-6kzjCc50R1","doi-asserted-by":"publisher","DOI":"10.2307\/2312585","type":"journal-article"},{"key":"ql9x-6kzjCc51R1","unstructured":"The form of |\u03c81\u232a depends on the specific operator O under consideration. For definiteness, in the following we take it to be real and have diagonal elements equal to \u00b11. Our treatment, however, can be easily carried out for any initial operator in diagonal form.","type":"other"},{"key":"ql9x-6kzjCc52R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.10.031066","type":"journal-article"},{"key":"ql9x-6kzjCc53R1","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300001917","type":"journal-article"},{"key":"ql9x-6kzjCc55R1","volume-title":"Pseudorandomness Rroperties of Random Reversible Circuits","author":"W. Gay","year":"2025","type":"book"},{"key":"ql9x-6kzjCc57R1","volume-title":"Ergodic Problems of Classical Mechanics","author":"V. Arnold","year":"1989","type":"book"},{"key":"ql9x-6kzjCc58R1","volume-title":"Geometrical Methods in the Theory of Ordinary Differential Equations","author":"V. Arnold","year":"2012","type":"book"},{"key":"ql9x-6kzjCc59R1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511803260","type":"book","volume-title":"Chaos in Dynamical Systems","author":"E. 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Cornfeld","year":"2012","type":"book"}],"container-title":["Physical Review B"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/ql9x-6kzj","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/ql9x-6kzj\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T23:02:23Z","timestamp":1790722943000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/ql9x-6kzj"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,11]]},"references-count":54,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2026,3]]}},"URL":"https:\/\/doi.org\/10.1103\/ql9x-6kzj","relation":{"has-preprint":[{"id-type":"doi","id":"10.48550\/arXiv.2508.10890","asserted-by":"subject"}]},"ISSN":["2469-9950","2469-9969"],"issn-type":[{"value":"2469-9950","type":"print"},{"value":"2469-9969","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,11]]},"assertion":[{"value":"2025-08-25","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"Published by the American Physical Society. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2026","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/","name":"creative_commons_license","label":"Creative Commons license","group":{"name":"licenses","label":"Licenses"}},{"value":"\u00a92026 The authors","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"L100302"}}