Burau representation, Squier's form, and non-Abelian anyons

Abstract

We study a phase-tunable coherent device that reproduces the matrix-valued braid statistics associated with non-Abelian anyons without assuming pre-existing anyonic quasiparticles. The construction uses the reduced Burau representation, made unitary by Squier's Hermitian form, to implement two noncommuting braid generators on a two-dimensional encoded space. A controlled-order circuit applies the generators in both orders, and optimal single-use discrimination gives a closed-form witness for their order dependence. Together with the braid relation, this certifies the implemented non-Abelian braid action. It does not by itself demonstrate the existence of anyonic quasiparticles.

arXiv comment13 pages, 2 figures; GitHub repository at https://github.com/sashakolpakov/burau-switch