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

Andrew Stewart Caldin

Published Aug 17, 2026• 0 citations

Abstract

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.com

DOI: 10.5281/zenodo.21974164

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

Andrew Stewart Caldin

Published Aug 17, 2026• 0 citations

Publisher

Abstract

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.com

DOI: 10.5281/zenodo.21974164

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

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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

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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

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

B_n

generators

\sigma_i

satisfy

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

and

\sigma_i \sigma_j = \sigma_j \sigma_i

for

|i-j|>1

. Anyon fusion rules:

\phi \times \phi = 1 + \phi

(Fibonacci anyon model). Quantum dimension

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

. Topological qubit encoded in fusion space dimension

F_{abc}^{d}

(F-matrix) with entries involving golden ratio. Majorana zero modes obey

\gamma_i^2 = 1

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

. | CONNECTION: Fibonacci anyon quantum dimension

d_\phi = 1.618

directly links to golden ratio. Braiding matrices yield entries proportional to

\phi^{-1} \approx 0.618

and

\phi^{-2} \approx 0.382

. The non-Abelian statistics corresp Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com