arXiv:2404.01778 · Journal of High Energy Physics

Braiding Fibonacci anyons

Ludmil K. Hadjiivanov, Lachezar S. Georgiev

Published Aug 12, 2024• 3 citations

Abstract

A bstract Fibonacci anyons ε provide the simplest possible model of non-Abelian fusion rules: [1] × [1] = [0] ⊕ [1]. We propose a conformal field theory construction of topological quantum registers based on Fibonacci anyons realized as quasiparticle excitations in the ℤ 3 parafermion fractional quantum Hall state. To this end, the results of Ardonne and Schoutens for the correlation function of four Fibonacci fields are extended to the case of arbitrary number n of quasi-holes and N = 3 r electrons. Special attention is paid to the braiding properties of the obtained correlators. We explain in details the construction of a monodromy representation of the Artin braid group $$ \mathcal{B} $$ B n acting on n -point conformal blocks of Fibonacci anyons. The matrices of braid group generators are displayed explicitly for all n ≤ 8. A simple recursion formula makes it possible to extend without efforts the construction to any n . Finally, we construct $$ \mathcal{N} $$ N qubit computational spaces in terms of conformal blocks of $$ 2\mathcal{N} $$ 2 N + 2 Fibonacci anyons.

DOI: 10.1007/jhep08(2024)084

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Conformal field theory construction for non-Abelian hierarchy wave functions

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arXiv:2404.01778 · Journal of High Energy Physics

Braiding Fibonacci anyons

Ludmil K. Hadjiivanov, Lachezar S. Georgiev

Published Aug 12, 2024• 3 citations

arXiv HTML PDF

Abstract

\mathcal{B}

B n acting on n -point conformal blocks of Fibonacci anyons. The matrices of braid group generators are displayed explicitly for all n ≤ 8. A simple recursion formula makes it possible to extend without efforts the construction to any n . Finally, we construct

\mathcal{N}

N qubit computational spaces in terms of conformal blocks of

2\mathcal{N}

2 N + 2 Fibonacci anyons.

DOI: 10.1007/jhep08(2024)084

Related papers

Diagonal Coset Approach to Topological Quantum Computation with Fibonacci Anyons

Lachezar S. Georgiev, Ludmil K. Hadjiivanov, Grigori Matein

We investigate a promising conformal field theory realization scheme for topological quantum computation based on the Fibonacci anyons, which are believed to be realized as quasiparticle excitations in the $\mathbb{Z}_3$ parafermion fractional quantum Hall state in the second Landau level with filling factor $ν=12/5$ . These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the $n$ -qubit topological quantum register in terms of $2n+2$ Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in $\mathbb{Z}_3$ parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.We investigate a promising conformal field theory realization scheme for topological quantum computation based on the Fibonacci anyons, which are believed to be realized as quasiparticle excitations in the $\mathbb{Z}_3$ parafermion fractional quantum Hall state in the second Landau level with filling factor $ν=12/5$. These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the $n$-qubit topological quantum register in terms of $2n+2$ Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in $\mathbb{Z}_3$ parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.

arXiv:2404.01779Apr 2, 20240 citations

Non-Abelian Anyon Braiding for Topological Quantum Computing — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon braiding implements topological qubits via non-Abelian statistics in 2D systems, with experimental progress in fractional quantum Hall and superconducting platforms. | MATH: Braid group B_n generators σ_i satisfy σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} (Yang–Baxter equation); anyon fusion rules: φ_a × φ_b = Σ_c N_{ab}^c φ_c; topological quantum computing uses unitary braid representations for gates. | CONNECTION: Braid group relations mirror root system symmetries (e.g., A_n Coxeter group); Fibonacci anyons have quantum dimension φ = (1+√5)/2 ≈ 1.618, linking to golden ratio; fusion rules: τ × τ = 1 + τ (Fibonacci category). | DEPTH: 8 — Directly ties non-Abelian statistics to geometric braiding, golden ratio, and lattice symmetries; experimental verification (e.g., fractional quantum Hall at ν=5/2) is ongoing but not yet conclusive for topological qubit operation. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon braiding implements topological qubits via non-Abelian statistics in 2D systems, with experimental progress in fractional quantum Hall and superconducting platforms. | MATH: Braid group B_n generators σ_i satisfy σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} (Yang–Baxter equation); anyon fusion rules: φ_a × φ_b = Σ_c N_{ab}^c φ_c; topological quantum computing uses unitary braid representations for gates. | CONNECTION: Braid group relations mirror root system symmetries (e.g., A_n Coxeter group); Fibonacci anyons have quantum dimension φ = (1+√5)/2 ≈ 1.618, linking to golden ratio; fusion rules: τ × τ = 1 + τ (Fibonacci category). | DEPTH: 8 — Directly ties non-Abelian statistics to geometric braiding, golden ratio, and lattice symmetries; experimental verification (e.g., fractional quantum Hall at ν=5/2) is ongoing but not yet conclusive for topological qubit operation. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 23, 20260 citations

Topological Qubits: Braiding Non-Abelian Anyons with Fibonacci Fusion and Flux Readout — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon topological qubits rely on braiding non-Abelian anyons in 2D systems, with flux qubits enabling readout via Josephson vortices, and Fibonacci anyons providing universal gate sets through braiding operations. MATH: Braiding group generators $\sigma_i$ satisfy $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ (Yang–Baxter equation). Fibonacci anyon fusion rules: $\tau \times \tau = 1 + \tau$ , with quantum dimension $d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618$ . Single-qubit braiding gates compiled from Fibonacci anyons use matrices with entries involving $\phi^{-1} \approx 0.618$ . Chern–Simons term in Ginzburg–Landau model: $S_{CS} = \frac{k}{4\pi} \int A \wedge dA$ , with level $k$ integer. CONNECTION: Fibonacci anyon quantum dimension $d_\tau = \phi$ directly ties to the golden ratio 1.618 and its inverse 0.618. Braiding matrices for Fibonacci anyons involve $\phi^{-1}$ and $\phi^{-2} \approx 0.382$ . The Yan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon topological qubits rely on braiding non-Abelian anyons in 2D systems, with flux qubits enabling readout via Josephson vortices, and Fibonacci anyons providing universal gate sets through braiding operations. MATH: Braiding group generators $ \sigma_i $ satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $ (Yang–Baxter equation). Fibonacci anyon fusion rules: $ \tau \times \tau = 1 + \tau $, with quantum dimension $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. Single-qubit braiding gates compiled from Fibonacci anyons use matrices with entries involving $ \phi^{-1} \approx 0.618 $. Chern–Simons term in Ginzburg–Landau model: $ S_{CS} = \frac{k}{4\pi} \int A \wedge dA $, with level $ k $ integer. CONNECTION: Fibonacci anyon quantum dimension $ d_\tau = \phi $ directly ties to the golden ratio 1.618 and its inverse 0.618. Braiding matrices for Fibonacci anyons involve $ \phi^{-1} $ and $ \phi^{-2} \approx 0.382 $. The Yan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 17, 20260 citations

Topological Quantum Computation via Non-Abelian Anyon Braid Group Representations — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter equation); non-Abelian anyons carry unitary representations $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)$ for $|i-j|\geq 2$ . Fibonacci anyons: fusion rule $\tau \times \tau = 1 + \tau$ , Hilbert space dimension grows as Fibonacci sequence $F_n$ , golden ratio $\phi = (1+\sqrt{5})/2 \approx 1.618$ appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus- $g$ surfaces: dimension $= k^g$ for level- $k$ Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio $F_{n+1}/F_n \to 1.618$ — direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter equation); non-Abelian anyons carry unitary representations $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)$ for $|i-j|\geq 2$. Fibonacci anyons: fusion rule $\tau \times \tau = 1 + \tau$, Hilbert space dimension grows as Fibonacci sequence $F_n$, golden ratio $\phi = (1+\sqrt{5})/2 \approx 1.618$ appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-$g$ surfaces: dimension $= k^g$ for level-$k$ Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio $F_{n+1}/F_n \to 1.618$ — direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 10, 20260 citations

Non-Abelian Anyon Braiding and Spectroscopy: Complementary Routes to Topological Qubits — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Non-abelian anyons are now experimentally braided in a universal topological gate (Quantinuum), while single-anyon/anyon-molecule spectroscopy proceeds in quantum Hall antidots (Young, UCSB) — the two complementary routes to topological qubits. | MATH: Braiding operators are unitary representations of the braid group B_n; non-abelian anyons realize matrices R = exp(iθ σᵃ) acting on a degenerate ground-state manifold of dimension = quantum dimension d_a. For Ising-type (Majorana) anyons d = √2; for Fibonacci anyons d = φ = 1.618…, with fusion rule τ⊗τ = 1⊕τ. | CONNECTION: The Fibonacci anyon's quantum dimension is exactly the golden ratio φ = (1+√5)/2 = 1.618, and its fusion matrix has eigenvalues φ and −1/φ = −0.618. Braiding phases for Fibonacci anyons involve θ = 2π/5 (pentagon), tying directly to the 5-fold symmetry of the icosahedron and the H₃ root system. Ising anyons (√2) connect to the D-series / SO(2n) root lattices. | DEPTH: 8 FINDING: Quantum Hall antidots host iso Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Non-abelian anyons are now experimentally braided in a universal topological gate (Quantinuum), while single-anyon/anyon-molecule spectroscopy proceeds in quantum Hall antidots (Young, UCSB) — the two complementary routes to topological qubits. | MATH: Braiding operators are unitary representations of the braid group B_n; non-abelian anyons realize matrices R = exp(iθ σᵃ) acting on a degenerate ground-state manifold of dimension = quantum dimension d_a. For Ising-type (Majorana) anyons d = √2; for Fibonacci anyons d = φ = 1.618…, with fusion rule τ⊗τ = 1⊕τ. | CONNECTION: The Fibonacci anyon's quantum dimension is exactly the golden ratio φ = (1+√5)/2 = 1.618, and its fusion matrix has eigenvalues φ and −1/φ = −0.618. Braiding phases for Fibonacci anyons involve θ = 2π/5 (pentagon), tying directly to the 5-fold symmetry of the icosahedron and the H₃ root system. Ising anyons (√2) connect to the D-series / SO(2n) root lattices. | DEPTH: 8 FINDING: Quantum Hall antidots host iso Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 29, 20260 citations

Conformal field theory construction for non-Abelian hierarchy wave functions

Yoran Tournois, Maria Hermanns

The fractional quantum Hall effect is the paradigmatic example of topologically ordered phases. One of its most fascinating aspects is the large variety of different topological orders that may be realized, in particular non-Abelian ones. Here we analyze a class of non-Abelian fractional quantum Hall model states which are generalizations of the Abelian Haldane-Halperin hierarchy. We derive their topological properties and show that the quasiparticles obey non-Abelian fusion rules of type $\mathrm{su}{\left(q\right)}_{k}$ . For a subset of these states we are able to derive the conformal field theory description that makes the topological properties---in particular braiding---of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.The fractional quantum Hall effect is the paradigmatic example of topologically ordered phases. One of its most fascinating aspects is the large variety of different topological orders that may be realized, in particular non-Abelian ones. Here we analyze a class of non-Abelian fractional quantum Hall model states which are generalizations of the Abelian Haldane-Halperin hierarchy. We derive their topological properties and show that the quasiparticles obey non-Abelian fusion rules of type $\mathrm{su}{\left(q\right)}_{k}$. For a subset of these states we are able to derive the conformal field theory description that makes the topological properties---in particular braiding---of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.

arXiv:1708.00763Physical review. B./Physical review. BDec 5, 20172 citations

\mathcal{B}

\mathcal{N}

N qubit computational spaces in terms of conformal blocks of

2\mathcal{N}

2 N + 2 Fibonacci anyons.

\mathbb{Z}_3

parafermion fractional quantum Hall state in the second Landau level with filling factor

ν=12/5

. These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the

n

-qubit topological quantum register in terms of

2n+2

Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in

\mathbb{Z}_3

parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.

\sigma_i

satisfy

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}

(Yang–Baxter equation). Fibonacci anyon fusion rules:

\tau \times \tau = 1 + \tau

, with quantum dimension

d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618

. Single-qubit braiding gates compiled from Fibonacci anyons use matrices with entries involving

\phi^{-1} \approx 0.618

. Chern–Simons term in Ginzburg–Landau model:

S_{CS} = \frac{k}{4\pi} \int A \wedge dA

, with level

k

integer. CONNECTION: Fibonacci anyon quantum dimension

d_\tau = \phi

directly ties to the golden ratio 1.618 and its inverse 0.618. Braiding matrices for Fibonacci anyons involve

\phi^{-1}

and

\phi^{-2} \approx 0.382

. The Yan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

B_n

generators

\sigma_i

satisfy

\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}

(Yang–Baxter equation); non-Abelian anyons carry unitary representations

U(\sigma_i)

with

U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)

for

|i-j|\geq 2

. Fibonacci anyons: fusion rule

\tau \times \tau = 1 + \tau

, Hilbert space dimension grows as Fibonacci sequence

F_n

, golden ratio

\phi = (1+\sqrt{5})/2 \approx 1.618

appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-

g

surfaces: dimension

= k^g

for level-

k

Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio

F_{n+1}/F_n \to 1.618

— direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\mathrm{su}{\left(q\right)}_{k}

. For a subset of these states we are able to derive the conformal field theory description that makes the topological properties---in particular braiding---of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.