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

TL;DR

This work addresses universality of the SU(2)_k anyon model for topological quantum computing, focusing on the one-qubit space $V^{\tau\tau\tau}_{\tau}$ with $\tau=\tfrac{1}{2}$. It leverages explicit $F$- and $R$-symbol data for SU(2)_k and a normalized representation $\tilde{\rho}_k$ of the braid group to analyze gate sets arising from braiding. The key contribution is proving that the double-braiding gates, i.e., the images of $\sigma_1^2$ and $\sigma_2^2$, generate a dense subgroup of $U(V^{\tau\tau\tau}_{\tau})$ (indeed of $SU(V^{\tau\tau\tau}_{\tau})$ after normalization) for all $k\ge 3$, $k\neq 4,8$. This strengthens prior universality results by showing a reduced and potentially more robust gate set suffices, with implications for experimental control and extensions to broader anyon models.

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.

Figures (10)

• Figure 1: (Left) A counterclockwise braiding of two anyons; (Right) A counterclockwise double-braiding of two anyons.
• Figure 2: A basis of $V_{c}^{a_1a_2\cdots a_n}$ corresponding to a binary tree
• Figure 3: (Left) the braid diagram $\sigma_i$; (Right) the braid diagram $\sigma_i^{-1}$.
• Figure 4: (Left) A splitting-tree basis for $V_{\tau}^{\tau\tau\tau}$; (Right) A splitting-tree basis for $V_{0}^{\tau\tau\tau\tau}$, where $\tau$ is the anyon of type $\frac{1}{2}$. $a,b = 0,1$.
• Figure 5: The action of $\sigma_1$ on $V_{\tau}^{\tau\tau\tau}$.

Theorems & Definitions (7)

• Definition 3.1
• Theorem 3.2: freedman2002two
• Lemma 3.3: kitaev1997quantum
• proof
• Theorem 3.4: Theorem 7, conway1976trigonometric
• Theorem 3.5
• proof