Universal topological quantum computing via double-braiding in SU(2) Witten-Chern-Simons theory

TL;DR

This paper proves that in the $SU(2)$ Witten-Chern-Simons topological quantum field theory, double-braiding of certain anyons is sufficient for universal quantum computation, strengthening previous results about braiding universality.

Contribution

It demonstrates that double-braiding alone achieves universality in topological quantum computing within the $SU(2)$ Witten-Chern-Simons framework for a single qubit.

Findings

Double-braiding of $SU(2)$ anyons is universal for quantum computing.

Strengthens previous braiding universality results by showing double-braiding suffices.

Applicable for the case of one qubit in topological quantum computation.

Abstract

We study the problem of universality in the anyon model described by the $SU(2)$ Witten-Chern-Simons theory at level $k$. A classic theorem of Freedman-Larsen-Wang states that for $k \geq 3, \ k \neq 4$, braiding of the anyons of topological charge $1/2$ is universal for topological quantum computing. For the case of one qubit, we prove a stronger result that double-braiding of such anyons alone is already universal.