Abstract
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 hierarchy introduced by Gottesman and Chuang [Nature (London) 402, 390 (1999)]. Moreover, a subset of gates, called semi-Clifford operations, can be implemented by an even simpler architecture than the traditional teleportation setup [X. Zhou, D. W. Leung, and I. L. Chuang, Phys. Rev. A 62, 052316 (2000)]. However, the precise set of gates in remains unknown, even for a fixed number of qubits , which prevents us from knowing exactly what teleportation is capable of. In this paper we study the structure of in terms of semi-Clifford operations, which send by conjugation at least one maximal Abelian subgroup of the -qubit Pauli group into another one. We show that for , all the gates are semi-Clifford, which is also true for . However, this is no longer true for . 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 gates and those gates beyond semi-Clifford operations, and compare their efficiency.
3 More- Received 15 January 2008
DOI:https://doi.org/10.1103/PhysRevA.77.042313
©2008 American Physical Society