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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 ∩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 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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Contributor : Michel Planat <>
Submitted on : Wednesday, December 3, 2008 - 6:47:20 PM
Last modification on : Saturday, November 14, 2020 - 7:06:03 PM
Long-term archiving on: : Wednesday, September 22, 2010 - 10:50:09 AM


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  • HAL Id : hal-00338632, version 2
  • ARXIV : 0811.2109


Michel Planat, Patrick Solé. Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2009, 42, pp.042003. ⟨hal-00338632v2⟩



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