Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Braiding non-abelian anyons with representations in Clifford algebra

Joel Wiklund

Published Jan 1, 2018• 1 citation
OpenAlex

Abstract

In this thesis we will talk about why a two-dimensional quantum system gives rise to a particle statistics which neither obey Fermi-Dirac nor Bose-Einstein statistics, but instead ranges continuous ...

Related papers

Braid group statistics and exchange matrices of non-abelian anyons : with representations in Clifford algebra

Erik Sönnerlind, Gustav Brage

When leaving classical physics and entering the realm of quantum physics, there are many new concepts being introduced. One of the most fundamental ideas in quantum mechanics is that particles no longer have exact known positions, but instead expected values and prob- abilities. This leads to the phenomena of truly identical particles, since they no longer can be distinguished simply by their positions. An important property differentiating different kinds of particles is how a system behaves when two such identical particles are exchanged. Historically, this divided particles into bosons and fermions, corresponding to symmetry and antisymmetry under an exchange. However, in two dimensions a new type of particle appears. These particles are called anyons, and behave differently when particles are exchanged. Anyons can be further divided into abelian and non-abelian anyons, of which this thesis will focus on the latter. The ex- changes can then be represented by the fundamental group of the configuration space of the particles, and in two dimensions this fundamental group is the braid group. Using rotors from a Clifford algebra and studying excitations of Majorana fermions, this thesis will show a way to calculate the exchange matrices of non-abelian anyons, and their corresponding eigenvalues. Furthermore, suggestions on a generalization of this framework along with areas where it can be applied are given.

•Jan 1, 2018•0 citations

Braiding of Majorana fermions, topological quantum computation and the nature of Clifford algebras

Louis Hirsch Kauffman

Biochemistry & Analytical Biochemistry•Nov 4, 2016•0 citations

Verifying non-Abelian statistics by numerical braiding Majorana fermions

Qiu-Bo Cheng, Jing He, Su-Peng Kou

arXiv:1412.2195•Physics Letters A•Nov 26, 2015•11 citations

Non-Abelian Anyons: Theoretical Promise, Experimental Gap — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon braiding provides non-abelian statistics for topologically protected quantum computation, but the search results are mostly pedagogical videos and one unrelated ATLAS squark paper — no primary experimental data or new mathematical result is presented. | MATH: Anyons obey fractional statistics: braiding operator \( B \) acts on degenerate ground-state manifold with unitary representation \( \rho(B) \), where \( \rho(B_1)\rho(B_2) \neq \rho(B_2)\rho(B_1) \) for non-abelian anyons. Exchange phase \( \theta \) generalizes Fermi (\( \theta=\pi \)) and Bose (\( \theta=0 \)) to \( \theta = 2\pi s \) with \( s \) fractional (e.g., \( s=1/3, 1/5 \)). Fibonacci anyon fusion rules: \( \tau \times \tau = 1 + \tau \), with quantum dimension \( d_\tau = \varphi = (1+\sqrt{5})/2 \approx 1.618 \). | CONNECTION: **Strong.** The Fibonacci anyon quantum dimension is exactly the golden ratio \( \varphi = 1.618 \). Braid group \( B_n \) representations relate to Temperley–Lieb algebra with l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 20, 2026•0 citations

Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation

Viktor Qvarfordt

As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored. We use the framework of modular tensor categories to show how the statistical repulsion of non-abelian anyons depends on the exchange symmetry. The Fibonacci anyon model is studied, for which explicit results are obtained. We also show how Fibonacci anyons can be used to implement topological quantum computation, providing topologically stable quantum information encoded in the state of non-abelian anyons that can be manipulated via the non-abelian exchange symmetry.

•Jan 1, 2017•3 citations

Quasiparticle operators with non-Abelian braiding statistics

D. C. Cabra, Enrique F. Moreno, G. L. Rossini

arXiv:hep-th/9803047•Physics Letters B•Oct 1, 1998•1 citation
When leaving classical physics and entering the realm of quantum physics, there are many new concepts being introduced. One of the most fundamental ideas in quantum mechanics is that particles no longer have exact known positions, but instead expected values and prob- abilities. This leads to the phenomena of truly identical particles, since they no longer can be distinguished simply by their positions. An important property differentiating different kinds of particles is how a system behaves when two such identical particles are exchanged. Historically, this divided particles into bosons and fermions, corresponding to symmetry and antisymmetry under an exchange. However, in two dimensions a new type of particle appears. These particles are called anyons, and behave differently when particles are exchanged. Anyons can be further divided into abelian and non-abelian anyons, of which this thesis will focus on the latter. The ex- changes can then be represented by the fundamental group of the configuration space of the particles, and in two dimensions this fundamental group is the braid group. Using rotors from a Clifford algebra and studying excitations of Majorana fermions, this thesis will show a way to calculate the exchange matrices of non-abelian anyons, and their corresponding eigenvalues. Furthermore, suggestions on a generalization of this framework along with areas where it can be applied are given.
FINDING: Anyon braiding provides non-abelian statistics for topologically protected quantum computation, but the search results are mostly pedagogical videos and one unrelated ATLAS squark paper — no primary experimental data or new mathematical result is presented. | MATH: Anyons obey fractional statistics: braiding operator B B B acts on degenerate ground-state manifold with unitary representation ρ(B) \rho(B) ρ(B), where ρ(B1)ρ(B2)≠ρ(B2)ρ(B1) \rho(B_1)\rho(B_2) \neq \rho(B_2)\rho(B_1) ρ(B1​)ρ(B2​)=ρ(B2​)ρ(B1​) for non-abelian anyons. Exchange phase θ \theta θ generalizes Fermi (θ=π \theta=\pi θ=π) and Bose (θ=0 \theta=0 θ=0) to θ=2πs \theta = 2\pi s θ=2πs with s s s fractional (e.g., s=1/3,1/5 s=1/3, 1/5 s=1/3,1/5). Fibonacci anyon fusion rules: τ×τ=1+τ \tau \times \tau = 1 + \tau τ×τ=1+τ, with quantum dimension dτ=φ=(1+5)/2≈1.618 d_\tau = \varphi = (1+\sqrt{5})/2 \approx 1.618 dτ​=φ=(1+5​)/2≈1.618. | CONNECTION: **Strong.** The Fibonacci anyon quantum dimension is exactly the golden ratio φ=1.618 \varphi = 1.618 φ=1.618. Braid group Bn B_n Bn​ representations relate to Temperley–Lieb algebra with l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored. We use the framework of modular tensor categories to show how the statistical repulsion of non-abelian anyons depends on the exchange symmetry. The Fibonacci anyon model is studied, for which explicit results are obtained. We also show how Fibonacci anyons can be used to implement topological quantum computation, providing topologically stable quantum information encoded in the state of non-abelian anyons that can be manipulated via the non-abelian exchange symmetry.