Topological Quantum Gate Construction by Iterative Pseudogroup Hashing

TL;DR

This paper introduces an iterative pseudogroup hashing method for efficiently approximating quantum gates in SU(2), leveraging group structures and iterative refinement to outperform traditional algorithms like Solovay-Kitaev.

Contribution

It presents a novel iterative hashing algorithm using pseudogroups and Fibonacci anyon braidings for faster quantum gate approximation.

Findings

Runtime scales poly-logarithmically with error

Braid length scales poly-logarithmically with error

Outperforms Solovay-Kitaev in efficiency

Abstract

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.