  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 $k$.
