On Topological Quantum Computing With Mapping Class Group Representations

TL;DR

This paper explores topological quantum computing using mapping class group representations, showing that for abelian anyons the gates are classically simulatable, while Fibonacci anyons can produce non-Clifford gates, indicating potential for universal quantum computation.

Contribution

It demonstrates the nature of gates generated by mapping class group representations for abelian and Fibonacci anyons, highlighting their computational capabilities.

Findings

Abelian anyons lead to normalizer gates, classically simulatable.

Fibonacci anyons produce non-Clifford gates, enabling universal quantum computation.

Leakage into non-computational subspaces is likely unavoidable for universality.

Abstract

We propose an encoding for topological quantum computation utilizing quantum representations of mapping class groups. Leakage into a non-computational subspace seems to be unavoidable for universality in general. We are interested in the possible gate sets which can emerge in this setting. As a first step, we prove that for abelian anyons, all gates from these mapping class group representations are normalizer gates. Results of Van den Nest then allow us to conclude that for abelian anyons this quantum computing scheme can be simulated efficiently on a classical computer. With an eye toward more general anyon models we additionally show that for Fibonnaci anyons, quantum representations of mapping class groups give rise to gates which are not generalized Clifford gates.