C[subscript 3], semi-clifford and generalized semi-clifford operations

Abstract

Fault-tolerant quantum computation is a basic problem in quantum computation, andteleportation is one of the main techniques in this theory. Using teleportation on stabi-lizer codes, the most well-known quantum codes, Pauli gates and Clifford operators canbe applied fault-tolerantly. Indeed, this technique can be generalized for an extended setof gates, the so called Ck hierarchy gates, introduced by Gottesman and Chuang (Nature,402, 390-392). Ck gates are a generalization of Clifford operators, but our knowledge ofthese 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 Ck hierarchy and the set ofsemi-Clifford and generalized semi-Clifford operators. They conjecture that any Ck gateis 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 Ckgates. Indeed, the more we understand Ck, the more intuition we have on Ck, k ≥ 4, and then we have a way of attacking the conjecture for larger k.