  We examine a class of operations for topological quantum computation based on
fusing and measuring topological charges for systems with SU$(2)_4$ 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.

Quantum Physics;Mesoscale and Nanoscale Physics;Strongly Correlated Electrons