Universal gates via fusion and measurement operations on $\mathrm{SU}{\left(2\right)}_4$ anyons

We examine a class of operations for topological quantum computation based on fusing and measuring topological charges for systems with $\mathrm{SU}{\left(2\right)}_4$ or $k=4$ Jones-Kauffman anyons. We show that such operations augment the braiding operations, which, by themselves, are not computationally universal. This augmentation results in a computationally universal gate set through the generation of an exact, topologically protected irrational phase gate and an approximate, topologically protected controlled-$Z$ gate.