  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.
