  We study an efficient algorithm to hash any single qubit gate (or unitary
matrix) into a braid of Fibonacci anyons represented by a product of
icosahedral group elements. By representing the group elements by braid
segments of different lengths, we introduce a series of pseudo-groups. Joining
these braid segments in a renormalization group fashion, we obtain a Gaussian
unitary ensemble of random-matrix representations of braids. With braids of
length O[log(1/epsilon)], we can approximate all SU(2) matrices to an average
error epsilon with a cost of O[log(1/epsilon)] in time. The algorithm is
applicable to generic quantum compiling.
