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Teleportation is a crucial element in fault-tolerant quantum computation anda complete understanding of its capacity is very important for the practicalimplementation of optimal fault-tolerant architectures. It is known thatstabilizer codes support a natural set of gates that can be more easilyimplemented by teleportation than any other gates. These gates belong to the socalled $\mathcal{C}_k$ hierarchy introduced by Gottesman and Chuang (Nature\textbf{402}, 390). Moreover, a subset of $\mathcal{C}_k$ gates, calledsemi-Clifford operations, can be implemented by an even simpler architecturethan the traditional teleportation setup (Phys. Rev. \textbf{A62}, 052316).However, the precise set of gates in $\mathcal{C}_k$ remains unknown, even fora fixed number of qubits $n$, which prevents us from knowing exactly whatteleportation is capable of. In this paper we study the structure of$\mathcal{C}_k$ in terms of semi-Clifford operations, which send by conjugationat least one maximal abelian subgroup of the $n$-qubit Pauli group into anotherone. We show that for $n=1,2$, all the $\mathcal{C}_k$ gates are semi-Clifford,which is also true for $\{n=3,k=3\}$. However, this is no longer true for$\{n>2,k>3\}$. To measure the capability of this teleportation primitive, weintroduce a quantity called `teleportation depth', which characterizes how manyteleportation steps are necessary, on average, to implement a given gate. Wecalculate upper bounds for teleportation depth by decomposing gates into bothsemi-Clifford $\mathcal{C}_k$ gates and those $\mathcal{C}_k$ gates beyondsemi-Clifford operations, and compare their efficiency.

Semi-Clifford operations, structure of $\mathcal{C}_k$ hierarchy, and gate complexity for fault-tolerant quantum computation