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

TL;DR

This paper explores the geometric and algebraic structure of Clifford quantum gates, representing them as BN-pairs and connecting their properties to smooth cubic surfaces, providing new insights into quantum gate symmetries.

Contribution

It introduces a novel BN-pair decomposition of Clifford groups and links quantum gate configurations to algebraic geometry of cubic surfaces.

Findings

Clifford groups can be expressed as BN-pairs with specific subgroup properties.

The structure relates to Coxeter groups and double cosets acting on B.

Connections are made between quantum gate configurations 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.