{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,13]],"date-time":"2024-06-13T16:36:28Z","timestamp":1718296588305},"reference-count":0,"publisher":"Rinton Press","issue":"1&2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["QIC"],"published-print":{"date-parts":[[2010,1]]},"abstract":"Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set of gates, the so called ${\\mathcal{C}}_k$ hierarchy gates, introduced by Gottesman and Chuang (Nature, 402, 390-392). ${\\mathcal{C}}_k$ gates are a generalization of Clifford operators, but our knowledge of these sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev. A 77, 042313) raise the question of the relation between ${\\mathcal{C}}_k$ hierarchy and the set of semi-Clifford and generalized semi-Clifford operators. They conjecture that any ${\\mathcal{C}}_k$ gate is a generalized semi-Clifford operator. In this paper, we prove this conjecture for $k=3$. Using the techniques that we develop, we obtain more insight on how to characterize ${\\mathcal{C}}_3$ gates. Indeed, the more we understand ${\\mathcal{C}}_3$, the more intuition we have on ${\\mathcal{C}}_k$, $k\\geq 4$, and then we have a way of attacking the conjecture for larger $k$.<\/jats:p>","DOI":"10.26421\/qic10.1-2-4","type":"journal-article","created":{"date-parts":[[2021,3,7]],"date-time":"2021-03-07T20:19:02Z","timestamp":1615148342000},"page":"41-59","source":"Crossref","is-referenced-by-count":3,"title":["C_3, semi-Clifford and genralized semi-Clifford operations"],"prefix":"10.26421","volume":"10","author":[{"given":"S.","family":"Beigi","sequence":"first","affiliation":[]},{"given":"P.W.","family":"Shor","sequence":"additional","affiliation":[]}],"member":"10955","published-online":{"date-parts":[[2010,1]]},"container-title":["Quantum Information and Computation"],"original-title":[],"deposited":{"date-parts":[[2021,3,7]],"date-time":"2021-03-07T20:19:08Z","timestamp":1615148348000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.rintonpress.com\/journals\/doi\/QIC10.1-2-4.html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":0,"journal-issue":{"issue":"1&2","published-online":{"date-parts":[[2010,1]]},"published-print":{"date-parts":[[2010,1]]}},"URL":"https:\/\/doi.org\/10.26421\/qic10.1-2-4","relation":{},"ISSN":["1533-7146","1533-7146"],"issn-type":[{"value":"1533-7146","type":"print"},{"value":"1533-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}