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

Andrew Stewart Caldin

Published Oct 10, 2026• 0 citations

Abstract

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.com

DOI: 10.5281/zenodo.23274589

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

Andrew Stewart Caldin

Published Oct 10, 2026• 0 citations

Publisher

Abstract

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.com

DOI: 10.5281/zenodo.23274589

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Non-Abelian Anyons and Braid Group Representations for Topological Quantum Computation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | 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 braid matrices $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})$ ; Fibonacci anyons have fusion rule $\tau \times \tau = 1 + \tau$ , yielding quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ (golden ratio); topological protection arises from ground-state degeneracy on genus- $g$ surfaces: $\dim \mathcal{H} = 4^g$ for toric code, or $2^g$ for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio $\phi = 1.618$ , and its inverse $\phi^{-1} = 0.618$ appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | 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 braid matrices $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})$; Fibonacci anyons have fusion rule $\tau \times \tau = 1 + \tau$, yielding quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ (golden ratio); topological protection arises from ground-state degeneracy on genus-$g$ surfaces: $\dim \mathcal{H} = 4^g$ for toric code, or $2^g$ for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio $\phi = 1.618$, and its inverse $\phi^{-1} = 0.618$ appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Topological Quantum Computing with Non-Abelian Anyons and Braid Group Operations — E8 Intelligence Research

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FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) to encode qubits in braiding operations, immune to local decoherence. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ ; anyon fusion rules: $\sigma \times \sigma = 1 + \psi$ (Ising model); Majorana operators $\gamma_i$ with $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$ ; topological qubit encoded in parity $i\gamma_1\gamma_2 = \pm 1$ . | CONNECTION: Braid group is related to the Fibonacci sequence via the Burau representation; the golden ratio $\phi = 1.618$ appears in the quantum dimension of Fibonacci anyons: $d_\tau = \phi$ . The Ising anyon quantum dimension is $\sqrt{2}$ , linking to $\sqrt{2} \approx 1.414$ , close to 1.4142 (no direct golden ratio). Crystallographic symmetry: Anyon braiding corresponds to the mapping class group of a punctured surface, related to root systems of Lie alge Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) to encode qubits in braiding operations, immune to local decoherence. | MATH: Braid group $ B_n $ generators $ \sigma_i $ satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $; anyon fusion rules: $ \sigma \times \sigma = 1 + \psi $ (Ising model); Majorana operators $ \gamma_i $ with $ \{\gamma_i, \gamma_j\} = 2\delta_{ij} $; topological qubit encoded in parity $ i\gamma_1\gamma_2 = \pm 1 $. | CONNECTION: Braid group is related to the Fibonacci sequence via the Burau representation; the golden ratio $ \phi = 1.618 $ appears in the quantum dimension of Fibonacci anyons: $ d_\tau = \phi $. The Ising anyon quantum dimension is $ \sqrt{2} $, linking to $ \sqrt{2} \approx 1.414 $, close to 1.4142 (no direct golden ratio). Crystallographic symmetry: Anyon braiding corresponds to the mapping class group of a punctured surface, related to root systems of Lie alge Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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

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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

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Andrew Stewart Caldin

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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

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

Andrew Stewart Caldin

FINDING: Anyon braiding for topological qubits relies on non-Abelian statistics in 2D systems, with quantum gates implemented via worldline braids in spacetime. 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). Unitary representations yield quantum gates. Fusion rules: for Fibonacci anyons, τ × τ = 1 + τ, with quantum dimension d_τ = φ = (1+√5)/2 ≈ 1.618. Braiding matrices involve golden ratio: e.g., F-matrix entries are φ^{-1/2} and φ^{-3/2}. CONNECTION: Fibonacci anyon quantum dimension is exactly the golden ratio φ = 1.618. Braiding phases involve φ^{-1} ≈ 0.618 and φ^{-2} ≈ 0.382. The Yang-Baxter equation is a 2D solvability condition linked to root systems of Lie algebras (e.g., A_n, B_n, C_n, D_n). Anyon statistics arise from 2D topology, not 3D; this mirrors the exceptional role of 2D in crystallographic symmetry (17 wallpaper groups, 5 Bravais lattices in 2D). DEPTH: 8 — Directly ties topological quantum computation to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon braiding for topological qubits relies on non-Abelian statistics in 2D systems, with quantum gates implemented via worldline braids in spacetime. 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). Unitary representations yield quantum gates. Fusion rules: for Fibonacci anyons, τ × τ = 1 + τ, with quantum dimension d_τ = φ = (1+√5)/2 ≈ 1.618. Braiding matrices involve golden ratio: e.g., F-matrix entries are φ^{-1/2} and φ^{-3/2}. CONNECTION: Fibonacci anyon quantum dimension is exactly the golden ratio φ = 1.618. Braiding phases involve φ^{-1} ≈ 0.618 and φ^{-2} ≈ 0.382. The Yang-Baxter equation is a 2D solvability condition linked to root systems of Lie algebras (e.g., A_n, B_n, C_n, D_n). Anyon statistics arise from 2D topology, not 3D; this mirrors the exceptional role of 2D in crystallographic symmetry (17 wallpaper groups, 5 Bravais lattices in 2D). DEPTH: 8 — Directly ties topological quantum computation to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 28, 20260 citations

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | 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 braid matrices

U(\sigma_i)

with

U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})

; Fibonacci anyons have fusion rule

\tau \times \tau = 1 + \tau

, yielding quantum dimension

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

(golden ratio); topological protection arises from ground-state degeneracy on genus-

g

surfaces:

\dim \mathcal{H} = 4^g

for toric code, or

2^g

for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio

\phi = 1.618

, and its inverse

\phi^{-1} = 0.618

appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) to encode qubits in braiding operations, immune to local decoherence. | MATH: Braid group

B_n

generators

\sigma_i

satisfy

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

; anyon fusion rules:

\sigma \times \sigma = 1 + \psi

(Ising model); Majorana operators

\gamma_i

with

\{\gamma_i, \gamma_j\} = 2\delta_{ij}

; topological qubit encoded in parity

i\gamma_1\gamma_2 = \pm 1

. | CONNECTION: Braid group is related to the Fibonacci sequence via the Burau representation; the golden ratio

\phi = 1.618

appears in the quantum dimension of Fibonacci anyons:

d_\tau = \phi

. The Ising anyon quantum dimension is

\sqrt{2}

, linking to

\sqrt{2} \approx 1.414

, close to 1.4142 (no direct golden ratio). Crystallographic symmetry: Anyon braiding corresponds to the mapping class group of a punctured surface, related to root systems of Lie alge Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group

B_n

generators satisfy

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

; Majorana operators

\gamma_i

satisfy

\{\gamma_i, \gamma_j\} = 2\delta_{ij}

; fusion rules for Ising anyons:

\sigma \times \sigma = 1 + \psi

\sigma \times \psi = \sigma

\psi \times \psi = 1

. | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at

q = e^{i\pi/5}

, where the Jones polynomial yields

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

; Fibonacci anyons have fusion rules

\tau \times \tau = 1 + \tau

, directly encoding the golden ratio. | DEPTH: 9 — Non-Abelian statistics and braiding are foundational to topological quantum computation, with direct geometric harmony in Fibonacci anyon models. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\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