  Quantum gates built out of braid group elements form the building blocks of
topological quantum computation. They have been extensively studied in
$SU(2)_k$ quantum group theories, a rich source of examples of non-Abelian
anyons such as the Ising ($k=2$), Fibonacci ($k=3$) and Jones-Kauffman ($k=4$)
anyons. We show that the fusion spaces of these anyonic systems can be
precisely mapped to the product state zero modes of certain Nicolai-like
supersymmetric spin chains. As a result, we can realize the braid group on the
product state zero modes of these supersymmetric systems. These operators kill
all the other states in the Hilbert space, thus preventing the occurrence of
errors while processing information, making them suitable for quantum
computing.
