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

Michel Planat (1) , Patrick Solé (2)

1 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)
2 I3S - Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis

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.

Mots clés: quantum computing; Coxeter groups; Clifford groups; BN-pairs; smooth cubic surfaces

Domaines: Physique Quantique [quant-ph]; Théorie des groupes [math.GR]

Michel Planat, Patrick Solé. Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces. 2008. ⟨hal-00338632v1⟩