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C_3, semi-Clifford and genralized semi-Clifford operations

Beigi, Salman and Shor, Peter W. (2010) C_3, semi-Clifford and genralized semi-Clifford operations. Quantum Information and Computation, 10 (1-2). pp. 41-59. ISSN 1533-7146.

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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 C_k hierarchy gates, introduced by Gottesman and Chuang (Nature, 402, 390-392). 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 C_k hierarchy and the set of semi-Clifford and generalized semi-Clifford operators. They conjecture that any 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 C_3 gates. Indeed, the more we understand C_3, the more intuition we have on C_k, k ≥ 4, and then we have a way of attacking the conjecture for larger k.

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Additional Information:© 2010 Rinton Press. Received February 16, 2009. Revised August 27, 2009. Communicated by: R Cleve & B Terhal. Authors are thankful to Carlos Mochon, Daniel Gottesman, and Bei Zeng for providing the counterexample of Conjecture 1. They are also grateful to unknown referees for their comments which improved the presentation of the results. SB has been supported in part by NSF under Grant No. PHY-0803371 and by NSA/ARO under Grant No. W911NF-09-1-0442. PWS was supported in part by the NSF under grant number CCF-0829421.
Funding AgencyGrant Number
Army Research Office (ARO)W911NF-09-1-0442
Subject Keywords:Clifford group, Semi-Clifford operations, C_k hierarchy, Fault-tolerant quantum computation
Issue or Number:1-2
Record Number:CaltechAUTHORS:20100201-135157391
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:17368
Deposited By: Tony Diaz
Deposited On:08 Feb 2010 20:24
Last Modified:03 Oct 2019 01:26

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