{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T03:51:01Z","timestamp":1774065061421,"version":"3.50.1"},"reference-count":0,"publisher":"Rinton Press","issue":"15&16","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["QIC"],"published-print":{"date-parts":[[2016,11]]},"abstract":"<jats:p>Unitary k-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices. Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design. We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected. Our proof strategy works for any distribution of unitaries satisfying a property we call Pauli 2-mixing and proceeds without the use of heavy mathematical machinery. We also show that the Clifford group does not form a 4-design, thus characterizing how well random Clifford elements approximate Haar-random unitaries. Additionally, we show that the generalized Clifford group for qudits is not a 3-design unless the dimension of the qudit is a power of 2.<\/jats:p>","DOI":"10.26421\/qic16.15-16-8","type":"journal-article","created":{"date-parts":[[2021,2,25]],"date-time":"2021-02-25T02:02:10Z","timestamp":1614218530000},"page":"1379-1400","source":"Crossref","is-referenced-by-count":88,"title":["The Clifford group forms a unitary 3-design"],"prefix":"10.26421","volume":"16","author":[{"given":"Zak","family":"Webb","sequence":"first","affiliation":[]}],"member":"10955","published-online":{"date-parts":[[2016,11]]},"container-title":["Quantum Information and Computation"],"original-title":[],"deposited":{"date-parts":[[2021,2,25]],"date-time":"2021-02-25T02:02:18Z","timestamp":1614218538000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.rintonpress.com\/journals\/doi\/QIC16.15-16-8.html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11]]},"references-count":0,"journal-issue":{"issue":"15&16","published-online":{"date-parts":[[2016,11]]},"published-print":{"date-parts":[[2016,11]]}},"URL":"https:\/\/doi.org\/10.26421\/qic16.15-16-8","relation":{},"ISSN":["1533-7146","1533-7146"],"issn-type":[{"value":"1533-7146","type":"print"},{"value":"1533-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,11]]}}}