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

String-nets, single and double-stranded quantum loop gases for non-Abelian anyons