{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:10:39Z","timestamp":1760145039144,"version":"build-2065373602"},"reference-count":90,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,12]],"date-time":"2024-06-12T00:00:00Z","timestamp":1718150400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) algebras (GAs) in quantum computing as originally presented in [C. Cafaro and S. Mancini, Adv. Appl. Clifford Algebras 21, 493 (2011)]. Our focus is on testing the usefulness of geometric algebras (GAs) techniques in two quantum computing applications. First, making use of the geometric algebra of a relativistic configuration space (namely multiparticle spacetime algebra or MSTA), we offer an explicit algebraic characterization of one- and two-qubit quantum states together with a MSTA description of one- and two-qubit quantum computational gates. In this first application, we devote special attention to the concept of entanglement, focusing on entangled quantum states and two-qubit entangling quantum gates. Second, exploiting the previously mentioned MSTA characterization together with the GA depiction of the Lie algebras SO3;R and SU2;C depending on the rotor group Spin+3,0 formalism, we focus our attention to the concept of universality in quantum computing by reevaluating Boykin\u2019s proof on the identification of a suitable set of universal quantum gates. At the end of our mathematical exploration, we arrive at two main conclusions. Firstly, the MSTA perspective leads to a powerful conceptual unification between quantum states and quantum operators. More specifically, the complex qubit space and the complex space of unitary operators acting on them merge in a single multivectorial real space. Secondly, the GA viewpoint on rotations based on the rotor group Spin+3,0 carries both conceptual and computational advantages compared to conventional vectorial and matricial methods.<\/jats:p>","DOI":"10.3390\/sym16060734","type":"journal-article","created":{"date-parts":[[2024,6,12]],"date-time":"2024-06-12T06:47:30Z","timestamp":1718174850000},"page":"734","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4913-7046","authenticated-orcid":false,"given":"Carlo","family":"Cafaro","sequence":"first","affiliation":[{"name":"Department of Nanoscale Science and Engineering, University at Albany-SUNY, Albany, NY 12222, USA"},{"name":"Department of Mathematics and Physics, SUNY Polytechnic Institute, Utica, NY 13502, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Newshaw","family":"Bahreyni","sequence":"additional","affiliation":[{"name":"Physics and Astronomy Department, Pomona College, Claremont, CA 91711, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Leonardo","family":"Rossetti","sequence":"additional","affiliation":[{"name":"Department of Nanoscale Science and Engineering, University at Albany-SUNY, Albany, NY 12222, USA"},{"name":"School of Science and Technology, Physics Division, University of Camerino, I-62032 Camerino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,12]]},"reference":[{"key":"ref_1","unstructured":"Hestenes, D. 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