Aaron Conlon

Compatibility of Braiding and Fusion on Wire Networks

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 the braiding of anyons in planar systems. For anyons in the plane, there is an algebraic framework given by unitary braided tensor categories. This naturally incorporates the fusion of topological charge and braid statistics. The braiding statistics are constrained through the hexagon equations, which enforce the compatibility of braiding and fusion. However, the exchange statistics of particles on a graph are not governed by the usual braid group but instead by a graph braid group, which depends on the graph.

We propose here an algebraic model for anyon like excitations on graphs, analogous to its planar counterpart. By imposing compatibility of graph braiding with fusion of topological charges, we obtain generalised hexagon equations. We find the usual planar anyon solutions but also more general braid actions.

Talk recording: https://mediasite.bris.ac.uk/Mediasite/Play/23ccc81076c648f4878045244be77ee31d