We describe the hashing technique for obtaining a fast approximation of a target quantum gate in the unitary group SU(2) represented by a product of the elements of a universal basis. The hashing exploits the structure of the icosahedral group (or other finite subgroups of SU(2)) and its pseudogroup approximations to reduce the search within a small number of elements. One of the main advantages of the pseudogroup hashing is the possibility of iterating to obtain more accurate representations of the targets in the spirit of the renormalization group approach. We describe the iterative pseudogroup hashing algorithm using the universal basis given by the braidings of Fibonacci anyons. An analysis of the efficiency of the iterations based on the random matrix theory indicates that the runtime and braid length scale poly-logarithmically with the final error, comparing favorably to the Solovay-Kitaev algorithm.
