Physics – Quantum Physics
Scientific paper
2011-12-01
Physics
Quantum Physics
Extended to a more general formalism using right quasigroups. Also minor changes in Sec. V A
Scientific paper
A nonlocal bipartite unitary gate can sometimes be implemented using prior entanglement and only one round of classical communication in which the two parties send messages to each other simultaneously. This cuts the classical communication time by a half compared to the usual protocols, which require back-and-forth classical communication. We introduce such a "fast" protocol that can implement a class of controlled unitaries exactly, where the controlled operators form a subset of a projective representation of a finite group, which may be Abelian or non-Abelian. We also introduce a modified version of the protocol for the approximate implementation of controlled unitaries. This protocol uses the algebraic structure of right quasigroups, which are generalizations of quasigroups, the latter being equivalent to Latin squares. We then show that by using enough entanglement, this fast protocol can implement all controlled unitaries approximately. In doing so we have effectively discussed an approximation of the special unitary group SU(d) by some right quasigroup. The entanglement cost of our protocols is compared with other fast unitary protocols in the literature. The cost is quite small when the form of the unitary is relatively simple.
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