Unitary Braid Group Representations Research Articles

Non-Abelian Anyons on Graphs from Presentations of Graph Braid Groups

The aim of this paper is to analyse algorithms for constructing presentations of graph braid groups from the point of view of anyonic quantum statistics on graphs. In the first part of this paper, we provide a comprehensive review of an algorithm for constructing so-called minimal Morse presentations of graph braid groups that relies on discrete Morse theory. Next, we introduce the notion of a physical presentation of a graph braid group as a presentation whose generators have a direct interpretation as particle exchanges. We show how to derive a physical presentation of a graph braid group from its minimal Morse presentation. In the second part of the paper, we study unitary representations of graph braid groups that are constructed from their presentations. We point out that algebraic objects called moduli spaces of flat bundles encode all unitary representations of graph braid groups. For $2$-connected graphs, we conclude the stabilisation of moduli spaces of flat bundles over graph configuration spaces for large numbers of particles. Moreover, we set out a framework for studying locally abelian anyons on graphs whose non-abelian properties are only encoded in non-abelian topological phases assigned to cycles of the considered graph.

Majorana Fermions and representations of the braid group

In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this result with the Ivanov braiding associated with Majorana operators. The paper generalizes observations of Kauffman and Lomonaco and of Mo-Lin Ge to show that certain strings of Majorana operators give rise to extraspecial 2-groups and to braiding representations of the Ivanov type.

Braiding, Majorana fermions, Fibonacci particles and topological quantum computing

This paper is an introduction to relationships between topology, quantum computing, and the properties of Fermions. In particular, we study the remarkable unitary braid group representations associated with Majorana fermions.

Braiding and Majorana fermions

In this paper, we study unitary braid group representations associated with Majorana fermions. Majorana fermions are represented by Majorana operators, elements of a Clifford algebra. The paper proves a general result about braid group representations associated with Clifford algebras and compares this result with the Ivanov braiding associated with Majorana operators and with other braiding representations associated with Majorana fermions such as the Fibonacci model for universal topological quantum computing.

On the geometry of some unitary Riemann surface braid group representations and Laughlin-type wave functions

In this note we construct the simplest unitary Riemann surface braid group representations geometrically by means of stable holomorphic vector bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail also with the help of noncommutative geometric tools, and an application of Fourier–Mukai–Nahm techniques is also given, explaining the emergence of an intriguing Riemann surface braid group duality.

A new set of generators and a physical interpretation for the $$SU (3)$$ finite subgroup $$D(9,1,1;2,1,1)$$

After 100 years of effort, the classification of all the finite subgroups of SU(3) is yet incomplete. The most recently updated list can be found in P.O. Ludl, J. Phys. A: Math. Theor. 44 255204 (2011), where the structure of the series (C) and (D) of SU(3)-subgroups is studied. We provide a minimal set of generators for one of these groups which has order 162. These generators appear up to phase as the image of an irreducible unitary braid group representation issued from the Jones-Kauffman version of SU(2) Chern-Simons theory at level 4. In light of these new generators, we study the structure of the group in detail and recover the fact that it is isomorphic to the semidirect product Z_9 \times Z_3 \rtimes S_3 with respect to conjugation.

GENERALIZED YANG–BAXTER EQUATIONS AND BRAIDING QUANTUM GATES

Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N > 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.

Abstract error groups via Jones unitary braid group representations at q = i

In this paper, we classify a type of abstract groups by the central products of dihedral groups and quaternion groups. We recognize them as abstract error groups which are often not isomorphic to the Pauli groups in the literature. We show the corresponding nice error bases equivalent to the Pauli error bases modulo phase factors. The extension of these abstract groups by the symmetric group are finite images of the Jones unitary representations (or modulo a phase factor) of the braid group at q = i or r = 4. We hope this work can finally lead to new families of quantum error correction codes via the representation theory of the braid group.

THE FIBONACCI MODEL AND THE TEMPERLEY-LIEB ALGEBRA

We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85th birthday.

Unitary braid representations with finite image

We characterize unitary representations of braid groups $B_n$ of degree linear in $n$ and finite images of such representations of degree exponential in $n$.

Unitary braid group representations

A large class of braid group representations are proven unitary which means a Hermitian bilinear form left invariant by the representation under consideration exists. This generalizes and improves known results on the Burau and Gassner representations. The elementary methods are naturally suggested by the study of monodromy invariant Hermitian forms. Applications to conformal quantum field theory in two dimensions are mentioned.