Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates

TL;DR

This paper explores how specific entangling Clifford gates can generate representations of complex Coxeter and Weyl groups, including W(E8), revealing deep connections between quantum entanglement, group theory, and quantum gate structures.

Contribution

It introduces a novel link between entangling Clifford gates and the realization of Weyl/Coxeter groups like W(E8) through unitary representations, expanding understanding of quantum symmetries.

Findings

Representation of W(D5) and W(F4) from Clifford gates

W(E8) representation via three-qubit Clifford group and Toffoli gate

Connections between entangled states and complex reflection groups

Abstract

Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R, S) of entangling Clifford quantum gates, acting on two qubits. From them, a natural unitary representation of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also reflected into the splitting of the n-qubit Clifford group Cn into dipoles C$\pm$n . The union of the three-qubit real Clifford group C+ 3 and the Toffoli gate ensures a orthogonal representation of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex reflection groups, BN pairs, unitary group designs and entangled states of the GHZ family.