  String-net condensation can give rise to non-Abelian anyons whereas loop
condensation usually gives rise to Abelian anyons. It has been proposed that
generalized quantum loop gases with non-orthogonal inner products can produce
non-Abelian anyons. We detail an exact mapping between the string-net and the
generalized loop models and explain how the non-orthogonal products arise. We
also introduce a loop model of double-stranded nets where quantum loops with an
orthogonal inner product and local interactions supports non-Abelian Fibonacci
anyons. Finally we emphasize the origin of the sign problem in such systems and
its consequences on the complexity of their ground state wave functions.
