  We describe the hashing technique to obtain 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 to iterate 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.
The analysis of the efficiency of the iterations based on the random matrix
theory indicates that the runtime and the braid length scale
poly-logarithmically with the final error, comparing favorably to the
Solovay-Kitaev algorithm.
