Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces
Abstract
The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e. a pair of subgroups B and N that generate G, is such that intersection H = B \cap N is normal in G, the group W = N/H is a Coxeter group and two extra axioms are satisfied by the double cosets acting on B. The BN-pair used in this decomposition relies on the swap and match gates already introduced for classically simulating quantum circuits (R Jozsa and A Miyake, Preprint arXiv:0804.4050 [quant-ph]). The two- and three-qubit steps are related to the configuration with 27 lines on a smooth cubic surface.
Cite
@article{arxiv.0811.2109,
title = {Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces},
author = {Michel Planat and Patrick Solé},
journal= {arXiv preprint arXiv:0811.2109},
year = {2009}
}
Comments
7 pages, version to appear in Journal of Physics A: Mathematical and Theoretical (fast track communications)