C3, Semi-Clifford and Generalized Semi-Clifford Operations

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 $k$.