## Focused Research Workshop: Graph Braiding

24 Sep 2021, by Sponsored events in13 – 17 September 2021

**Heilbronn Institute for Mathematical Research, University of Bristol, UK**

**Representations of Graph Braid Group in Topological Quantum Computing **

**Heilbronn Focused Research Workshop**

A topological quantum computer performs its tasks using quasiparticles called anyons that are intrinsically robust against different types of noise and decoherence. Braiding (an exchange) of anyons transforms a state of the corresponding quantum system by a unitary operator which is a topological quantum gate. A robust realisation of controlled braiding of anyons is one of the major challenges in this field. Recently developed experimental and theoretical proposals address this challenge by exploring the possibility of braiding of anyons on junctions of **one-dimensional wire networks** **(graphs)**. Mathematically, such a topological quantum computing scheme realises a unitary representation of a graph braid group which is a group generated by exchanges of point-like anyons restricted to move on a given graph (see Fig. 1).

In the literature, anyons on planar networks are often assumed to behave as particles on the plane in terms of their braiding. However, recent studies have shown that the graph braid group is in fact larger than the planar group and as such may permit more general anyonic statistics [T. Maciazek, A. Sawicki (2019), Comm. Math. Phys. 371 (3), T. Maciazek, B. H. An (2020) Phys. Rev. B 102 (20), 201407]. Hence, such platforms are particularly promising in terms of their potential applications. The key next step is to find physically realisable representations of graph braid groups. This problem requires a new set of mathematical tools which may lead to developing a new facet of representation theory of braid groups. We will approach this problem by extending the

mathematical machinery of topological quantum field theory and modular tensor categories which, in the 2D setting, provides a set of so-called fusion rules whose solutions classify physically realisable representations of braid groups. In particular, applying this approach to braid groups for particles on graphs (networks of 1D wires) requires the use of **higher categories**.

Our team brings together experts in three fields of mathematics and physics that are key to the project: braid group theory, category theory and physics of quantum nanowires.

**Organisers**:

Thomasz Maciazek (Bristol, UK)

Nick Jones (Oxford, UK)**Participants**:

Aaron Conlon (Maynooth, Ireland)

Nick Jones (Oxford, UK)

Tomasz Maciazek (Bristol, UK)

Paul Martin (Leeds, UK)

Jonathan Robbins (Bristol, UK)

Joost Slingerland (Maynooth, Ireland)

Gert Vercleyen (Maynooth, Ireland)

**By Invitation Only**