Topological Quantum Hashing with the Icosahedral Group

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 renor- malization group fashion, we obtain a Gaussian unitary ensemble of random-matrix representations of braids. With braids of length O(log2(1/ε)), we can approximate all SU(2) matrices to an average error ε with a cost of O(log(1/ε)) in time. The algorithm is applicable to generic quantum compiling.