{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:28:41Z","timestamp":1772252921056,"version":"3.50.1"},"reference-count":48,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2018,12,19]],"date-time":"2018-12-19T00:00:00Z","timestamp":1545177600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-15-IDEX-03"],"award-info":[{"award-number":["ANR-15-IDEX-03"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere     S 3    . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold     M 3    . More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group     P S L ( 2 , Z )     correspond to d-fold     M 3    - coverings over the trefoil knot. In this paper, we also investigate quantum information on a few \u2018universal\u2019 knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and     M 3    \u2019s obtained from Dehn fillings are explored.<\/jats:p>","DOI":"10.3390\/sym10120773","type":"journal-article","created":{"date-parts":[[2018,12,19]],"date-time":"2018-12-19T12:12:44Z","timestamp":1545221564000},"page":"773","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Universal Quantum Computing and Three-Manifolds"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5739-546X","authenticated-orcid":false,"given":"Michel","family":"Planat","sequence":"first","affiliation":[{"name":"Institut FEMTO-ST CNRS UMR 6174, Universit\u00e9 de Bourgogne\/Franche-Comt\u00e9, 15 B Avenue des Montboucons, F-25044 Besan\u00e7on, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6953-8202","authenticated-orcid":false,"given":"Raymond","family":"Aschheim","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]},{"given":"Marcelo M.","family":"Amaral","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]},{"given":"Klee","family":"Irwin","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Thurston, W.P. 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