Compatibility of Braiding and Fusion on Wire Networks

TL;DR

This paper explores how braiding and fusion of particles on quantum wire networks relate, revealing generalized equations and solutions beyond traditional planar anyons, with implications for fault-tolerant quantum computation.

Contribution

It introduces compatibility conditions between braiding and fusion on graphs, leading to generalized hexagon equations and new braid solutions beyond planar anyons.

Findings

Derived generalized hexagon equations for graph braiding and fusion.

Identified solutions including traditional planar anyons and novel braid actions.

Applied framework to Abelian, Fibonacci, and Ising fusion rules.

Abstract

Exchanging particles on graphs, or more concretely on networks of quantum wires, has been proposed as a means to perform fault tolerant quantum computation. This was inspired by braiding of anyons in planar systems. However, exchanges on a graph are not governed by the usual braid group but instead by a graph braid group. By imposing compatibility of graph braiding with fusion of topological charges, we obtain generalized hexagon equations. We find the usual planar anyons solutions but also more general braid actions. We illustrate this with Abelian, Fibonacci and Ising fusion rules.