{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T18:33:44Z","timestamp":1763663624886,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2017,7,11]],"date-time":"2017-07-11T00:00:00Z","timestamp":1499731200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000181","name":"AFOSR","doi-asserted-by":"publisher","award":["FA9550-12-1-0046"],"award-info":[{"award-number":["FA9550-12-1-0046"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Gottesman\u2013Knill theorem established that stabilizer states and Clifford operations can be efficiently simulated classically. For qudits with odd dimension three and greater, stabilizer states and Clifford operations have been found to correspond to positive discrete Wigner functions and dynamics. We present a discrete Wigner function-based simulation algorithm for odd-d qudits that has the same time and space complexity as the Aaronson\u2013Gottesman algorithm for qubits. We show that the efficiency of both algorithms is due to harmonic evolution in the symplectic structure of discrete phase space. The differences between the Wigner function algorithm for odd-d and the Aaronson\u2013Gottesman algorithm for qubits are likely due only to the fact that the Weyl\u2013Heisenberg group is not in     S U ( d )     for     d = 2     and that qubits exhibit state-independent contextuality. This may provide a guide for extending the discrete Wigner function approach to qubits.<\/jats:p>","DOI":"10.3390\/e19070353","type":"journal-article","created":{"date-parts":[[2017,7,11]],"date-time":"2017-07-11T11:13:11Z","timestamp":1499771591000},"page":"353","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Discrete Wigner Function Derivation of the Aaronson\u2013Gottesman Tableau Algorithm"],"prefix":"10.3390","volume":"19","author":[{"given":"Lucas","family":"Kocia","sequence":"first","affiliation":[{"name":"Department of Physics, Tufts University, Medford, MA 02155, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yifei","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Physics, Tufts University, Medford, MA 02155, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Love","sequence":"additional","affiliation":[{"name":"Department of Physics, Tufts University, Medford, MA 02155, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,11]]},"reference":[{"key":"ref_1","unstructured":"Gottesman, D. (arXiv, 1998). The Heisenberg Representation of Quantum Computers, arXiv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"052328","DOI":"10.1103\/PhysRevA.70.052328","article-title":"Improved simulation of stabilizer circuits","volume":"70","author":"Aaronson","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Gottesman, D. (1999). Fault-tolerant quantum computation with higher-dimensional systems. Quantum Computing and Quantum Communications, Springer.","DOI":"10.1007\/3-540-49208-9_27"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"230503","DOI":"10.1103\/PhysRevLett.109.230503","article-title":"Positive Wigner functions render classical simulation of quantum computation efficient","volume":"109","author":"Mari","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1038\/nature13460","article-title":"Contextuality supplies the `magic\u2019 for quantum computation","volume":"510","author":"Howard","year":"2014","journal-title":"Nature"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0003-4916(87)90176-X","article-title":"A Wigner-function formulation of finite-state quantum mechanics","volume":"176","author":"Wootters","year":"1987","journal-title":"Ann. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"122107","DOI":"10.1063\/1.2393152","article-title":"Hudson\u2019s theorem for finite-dimensional quantum systems","volume":"47","author":"Gross","year":"2006","journal-title":"J. Math. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"113011","DOI":"10.1088\/1367-2630\/14\/11\/113011","article-title":"Negative quasi-probability as a resource for quantum computation","volume":"14","author":"Veitch","year":"2012","journal-title":"New J. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"013037","DOI":"10.1088\/1367-2630\/15\/1\/013037","article-title":"Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation","volume":"15","author":"Veitch","year":"2013","journal-title":"New J. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kocia, L., and Love, P. (arXiv, 2016). Semiclassical Formulation of Gottesman\u2013Knill and Universal Quantum Computation, arXiv.","DOI":"10.1103\/PhysRevA.96.032331"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Koh, D.E., Penney, M.D., and Spekkens, R.W. (arXiv, 2017). Computing quopit Clifford circuit amplitudes by the sum-over-paths technique, arXiv.","DOI":"10.26421\/QIC17.13-14-1"},{"key":"ref_12","unstructured":"De Beaudrap, N. (arXiv, 2011). A linearized stabilizer formalism for systems of finite dimension, arXiv."},{"key":"ref_13","unstructured":"Yoder, T.J. (2017, July 06). A Generalization of the Stabilizer Formalism for Simulating Arbitrary Quantum Circuits. Available online: https:\/\/pdfs.semanticscholar.org\/b200\/efe1709d07ffc1b5b7bd90e61c09e2729bdf.pdf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"022334","DOI":"10.1103\/PhysRevA.73.022334","article-title":"Fast simulation of stabilizer circuits using a graph-state representation","volume":"73","author":"Anders","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_15","unstructured":"Bengtsson, I., and Zyczkowski, K. (arXiv, 2017). On discrete structures in finite Hilbert spaces, arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1103\/RevModPhys.65.803","article-title":"Hidden variables and the two theorems of John Bell","volume":"65","author":"Mermin","year":"1993","journal-title":"Rev. Mod. Phys."},{"key":"ref_17","unstructured":"Raussendorf, R., Browne, D.E., Delfosse, N., Okay, C., and Bermejo-Vega, J. (arXiv, 2015). Contextuality as a resource for qubit quantum computation, arXiv."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Kocia, L., and Love, P. (arXiv, 2017). Discrete Wigner Formalism for Qubits and the Non-Contextuality of Clifford Operations on Qubit Stabilizer States, arXiv.","DOI":"10.1103\/PhysRevA.96.062134"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/7\/353\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:42:16Z","timestamp":1760208136000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/7\/353"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,11]]},"references-count":18,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2017,7]]}},"alternative-id":["e19070353"],"URL":"https:\/\/doi.org\/10.3390\/e19070353","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2017,7,11]]}}}