Semi-Clifford operations, structure of Cₖ hierarchy, and gate complexity for fault-tolerant quantum computation

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Teleportation is a crucial element in fault-tolerant quantum computation and a complete understanding of its capacity is very important for the practical implementation of optimal fault-tolerant architectures. It is known that stabilizer codes support a natural set of gates that can be more easily implemented by teleportation than any other gates. These gates belong to the so called Cₖ hierarchy introduced by Gottesman and Chuang (Nature 402, 390). Moreover, a subset of Cₖ gates, called semi-Clifford operations, can be implemented by an even simpler architecture than the traditional teleportation setup (Phys. Rev. A62, 052316). However, the precise set of gates in Cₖ remains unknown, even for a fixed number of qubits n, which prevents us from knowing exactly what teleportation is capable of. In this paper we study the structure of Cₖ in terms of semi-Clifford operations, which send by conjugation at least one maximal abelian subgroup of the n-qubit Pauli group into another one. We show that for n=1,2, all the Cₖ 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, we introduce a quantity called `teleportation depth', which characterizes how many teleportation steps are necessary, on average, to implement a given gate. We calculate upper bounds for teleportation depth by decomposing gates into both semi-Clifford Cₖ gates and those Cₖ gates beyond semi-Clifford operations, and compare their efficiency.