Extending the planar theory of anyons to quantum wire networks

TL;DR

This work constructs a universal framework for anyons living on quantum wire networks by enforcing compatibility between graph braiding and fusion through generalized polygon equations (P-/Q-hexagons). It demonstrates how network connectivity shapes braiding statistics, proving that triconnected networks recover planar anyon exchange while modular, biconnected networks can host independent, potentially richer gate sets. The authors develop detailed solutions for various fusion rings (Ising, D$( ext{Z}_2)$, TY models, among others) on building-block graphs (circles, trijunctions, lollipops, and Θ-graphs), revealing coherence and rigidity properties, as well as new, graph-specific exchange possibilities. They also discuss implications for topological quantum computing, showing that certain network architectures can reduce circuit depth by leveraging independent junctions to realize a larger repertoire of topological gates. The work culminates in a Theta-graph result that connects graph braiding to planar coherence and conjectures a general coherence property for graph-braided anyon models across arbitrary particle numbers.

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.

Figures (41)

• Figure 1: The simple braid $\sigma_1^{(1,2)}$ and the associated $R$-symbol. The superscript in $\sigma_1^{(1,2)}$ refers to the edge assignment the particles are sent to under the graph braid.
• Figure 2: The $P$- and $Q$-symbols associated with the simple braids $\sigma_2^{(1,1,2)}$ and $\sigma_2^{(2,1,2)}$ on a trijunction.
• Figure 3: The fusion vertex of anyons $b$ and $c$ can be pulled through the entire braid $\sigma_1^{(1_a,2_c)}\sigma_2^{(2_c,1_a,2_b)}$ so that the resulting process is just a simple braid of an anyon $f$ in the fusion channel of $b$ and $c$ with anyon $a$, i.e. $\sigma_1^{(1_a,2_{b\times c})}$. The diagram on the furthest right expresses the $\sigma_{1}^{(1_{a},2_{b\times c})}$ graph braid.
• Figure 4: The hexagon diagram which we call the $Q$-hexagon. It is derived from the identity $\sigma_1^{(1_a,2_c)}\sigma_2^{(2_c,1_a,2_b)}=\sigma_1^{(1_a,2_{b\times c})}$ shown in Figure \ref{['fig:fusioncommutes']} and applied in the bottom-left corner of the hexagon. The hexagon diagram provides a set of hexagon Equations \ref{['eq:Qhexagon']} which allow one to express the $Q$-symbols via the $R$- and $F$-symbols.
• Figure 5: a) The disks associated with the left-fused basis during the first steps of the $\sigma_3^{(2,1,1,2)}$-exchange. b) The disks associated with the pairwise-fused basis during the first steps of the $\sigma_3^{(2,1,1,2)}$-exchange. Note that the disk bounding anyons $c$ and $d$ is intersected by anyons $a$ and $b$ during the exchange. Consequently, the charge of $c\times d$ may not be conserved during the exchange. c) The disks associated with the left-fused basis during the first steps of the $\sigma_3^{(1,1,1,2)}$-exchange. d) The disks associated with the pairwise-fused basis during the first steps of the $\sigma_3^{(1,1,1,2)}$-exchange. In the panels c) and d) there are no disk intersections, so the appropriate charges must always be conserved in both bases. This results with the consistency equation from Figure \ref{['fig::XReduction']}.