Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces

Details Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces

2008-11-13

arxiv.org/abs/0811.2109v2

Journal of Physics A Mathematical and Theoretical 42 (2009) 042003

Physics

Quantum Physics

7 pages, version to appear in Journal of Physics A: Mathematical and Theoretical (fast track communications)

Scientific paper

10.1088/1751-8113/42/4/042003

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.

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