Measuring Chern-Simons level $k$ by braiding $SU(2)_k$ anyons

TL;DR

This paper proposes a method to determine the Chern-Simons level $k$ in unknown materials by analyzing braiding operations of $SU(2)_k$ anyons, which could aid in topological quantum computing.

Contribution

It introduces a novel experimental approach to measure the Chern-Simons level $k$ using braiding rules and pair annihilation probabilities of anyons.

Findings

Probabilities of anyon pair annihilation depend on $k$

Turnaround operations enable $k$ measurement

Method applicable to unknown materials

Abstract

Chern-Simons theory in application to the quantum computing is actively developing at the present. However, most discussed are the questions of using materials with known parameters and building corresponding quantum gates and algorithms. In this paper we discuss opposite problem of finding Chern-Simons level $k$ in the unknown material. For this purpose, we use the previously derived braiding rules for Chern-Simons $SU(2)_k$ anyons. Using certain operations (turnarounds) on three anyons, one can measure probabilities of annihilation of pairs of anyons, which depend on the parameter of the theory. Therefore, Chern-Simons level $k$ can be found from such an experiment. It is implied that anyons additionally possess certain properties which are required for topological quantum computations.