Quantum hashing with the icosahedral group

TL;DR

This paper presents an efficient quantum compiling algorithm that approximates any single qubit gate using braids of Fibonacci anyons, leveraging icosahedral symmetry to achieve logarithmic complexity in accuracy.

Contribution

It introduces a novel method combining icosahedral group elements and pseudo-groups to efficiently approximate SU(2) matrices with logarithmic cost.

Findings

Achieves approximation of SU(2) matrices with error epsilon in O(log(1/epsilon)) time.

Uses a renormalization group approach to construct braids from pseudo-groups.

Demonstrates applicability to generic quantum compiling tasks.

Abstract

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.