Extending the planar theory of anyons to quantum wire networks

TL;DR

This paper extends the theory of anyons to quantum wire networks, revealing how network connectivity influences braiding and fusion models, with implications for topological quantum computing.

Contribution

It establishes graph-braided anyon fusion models for general wire networks, linking network topology to braiding properties and exploring new solutions in low-rank fusion rings.

Findings

Triconnected networks replicate planar anyon braiding.

Modular biconnected networks support independent braiding modules.

Numerous new solutions to polygon equations found in low-rank models.

Abstract

The braiding of the worldlines of particles restricted to move on a network (graph) is governed by the graph braid group, which can be strikingly different from the standard braid group known from two-dimensional physics. It has been recently shown that imposing the compatibility of graph braiding with anyon fusion for anyons exchanging at a single wire junction leads to new types of anyon models with the braiding exchange operators stemming from solutions of certain generalised hexagon equations. In this work, we establish these graph-braided anyon fusion models for general wire networks. We show that the character of braiding strongly depends on the graph-theoretic connectivity of the given network. In particular, we prove that triconnected networks yield the same braiding exchange operators as the planar anyon models. In contrast, modular biconnected networks support independent braiding exchange operators in different modules. Consequently, such modular networks may lead to more efficient topological quantum computer circuits. Finally, we conjecture that the graph-braided anyon fusion models will possess the (generalised) coherence property where certain polygon equations determine the braiding exchange operators for an arbitrary number of anyons. We also extensively study solutions to these polygon equations for chosen low-rank multiplicity-free fusion rings, including the Ising theory, quantum double of Z2, and Tambara-Yamagami models. We find numerous solutions that do not appear in the planar theory of anyons.