Non-Abelian Anyons and Fibonacci Braiding in Topological Quantum Computing — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 21, 2026• 0 citations

Abstract

FINDING: Topological quantum computing uses anyons with non-Abelian braiding statistics, enabling fault-tolerant qubits via topological phases of matter. | MATH: Braid group representations; Fibonacci anyon fusion rules: τ × τ = 1 + τ (golden ratio φ = (1+√5)/2 ≈ 1.618); quantum dimension d_τ = φ; S-matrix element S_{ττ} = 1/φ; non-Abelian Berry phase; Kitaev's toric code Hamiltonian: H = -J Σ_v A_v - J Σ_p B_p, with A_v = Π σ^x_i, B_p = Π σ^z_i. | CONNECTION: Fibonacci anyon braiding directly yields golden ratio φ = 1.618 and its inverse 0.618 in fusion rules and quantum dimensions; the non-Abelian statistics correspond to braid group B_n representations related to root systems (A_n, D_n, E_6, E_7, E_8) and crystallographic Coxeter groups; the toric code lives on a square lattice with Z_2 gauge symmetry, linking to lattice structures and base-2 (not base-60) but anyon models extend to SU(2)_k with k=3 giving φ. | DEPTH: 9 — This is a profound unification of topology, quantum mechanics Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21467096

Related papers

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

Andrew Stewart Caldin

Aug 23, 20260 citations

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

Andrew Stewart Caldin

Jul 28, 20260 citations

Topological Quantum Computing: Non-Abelian Anyons, Braiding, and Majorana Qubits — E8 Intelligence Research

Andrew Stewart Caldin

Aug 17, 20260 citations

Fibonacci Anyons and the Golden Ratio in Topological Quantum Computation — E8 Intelligence Research

Andrew Stewart Caldin

Jul 25, 20260 citations

Fibonacci Anyons Realize Universal Topological Quantum Computation via Golden Ratio Braiding — E8 Intelligence Research

Andrew Stewart Caldin

Oct 18, 20260 citations

Microsoft's Majorana 1 Chip: Topological Qubits via Non-Abelian Anyon Braiding — E8 Intelligence Research

Andrew Stewart Caldin

Aug 14, 20260 citations

Non-Abelian Anyons and Fibonacci Braiding in Topological Quantum Computing — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 21, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.21467096

Related papers

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

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

Topological Quantum Computing: Non-Abelian Anyons, Braiding, and Majorana Qubits — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological quantum computing uses anyons with non-Abelian braiding statistics for fault-tolerant qubits, with Microsoft's Majorana 1 chip claiming topological qubit realization. | MATH: Braiding group $B_n$ generators satisfy $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ ; anyon fusion rules $a \times b = \sum_c N_{ab}^c c$ ; Fibonacci anyon quantum dimension $d = \phi = (1+\sqrt{5})/2 \approx 1.618$ ; Majorana zero modes obey $\gamma_i^\dagger = \gamma_i, \{\gamma_i, \gamma_j\} = 2\delta_{ij}$ . | CONNECTION: Fibonacci anyon dimension $d = 1.618$ is the golden ratio; braid group representations relate to Temperley-Lieb algebras with parameter $\tau = 1/d^2 = 0.382$ ; anyon fusion rules mirror SU(2)_k modular tensor categories with $q = e^{2\pi i/(k+2)}$ , linking to root systems $A_n, D_n, E_6, E_7, E_8$ . | DEPTH: 8 — Directly ties golden ratio, modular tensor categories, and non-Abelian statistics to fault-tolerant quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological quantum computing uses anyons with non-Abelian braiding statistics for fault-tolerant qubits, with Microsoft's Majorana 1 chip claiming topological qubit realization. | MATH: Braiding group $ B_n $ generators satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $; anyon fusion rules $ a \times b = \sum_c N_{ab}^c c $; Fibonacci anyon quantum dimension $ d = \phi = (1+\sqrt{5})/2 \approx 1.618 $; Majorana zero modes obey $ \gamma_i^\dagger = \gamma_i, \{\gamma_i, \gamma_j\} = 2\delta_{ij} $. | CONNECTION: Fibonacci anyon dimension $ d = 1.618 $ is the golden ratio; braid group representations relate to Temperley-Lieb algebras with parameter $ \tau = 1/d^2 = 0.382 $; anyon fusion rules mirror SU(2)_k modular tensor categories with $ q = e^{2\pi i/(k+2)} $, linking to root systems $ A_n, D_n, E_6, E_7, E_8 $. | DEPTH: 8 — Directly ties golden ratio, modular tensor categories, and non-Abelian statistics to fault-tolerant quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 17, 20260 citations

Fibonacci Anyons and the Golden Ratio in Topological Quantum Computation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fibonacci anyons realize non-Abelian statistics with fusion rules governed by the golden ratio, enabling topological quantum computation. MATH: Fusion rules: τ × τ = 1 + τ, where τ is the Fibonacci anyon. Quantum dimension d_τ = φ = (1+√5)/2 ≈ 1.618. The number of fusion channels for n anyons grows as F_{n+1} (Fibonacci numbers). Braiding matrices yield non-Abelian representations of the braid group B_n. CONNECTION: The golden ratio φ appears directly as the quantum dimension, linking topological phases to the same ratio found in Penrose tilings, quasicrystals, and 5-fold crystallographic symmetry. The Fibonacci sequence governs fusion state counting, echoing growth patterns in nature and base-60 harmonic ratios (φ ≈ 1.618, 1/φ ≈ 0.618). DEPTH: 9 — Direct embedding of φ into fundamental quantum statistics; bridges topology, number theory, and condensed matter physics; provides a concrete path to fault-tolerant quantum computing via non-Abelian anyons. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Fibonacci anyons realize non-Abelian statistics with fusion rules governed by the golden ratio, enabling topological quantum computation. MATH: Fusion rules: τ × τ = 1 + τ, where τ is the Fibonacci anyon. Quantum dimension d_τ = φ = (1+√5)/2 ≈ 1.618. The number of fusion channels for n anyons grows as F_{n+1} (Fibonacci numbers). Braiding matrices yield non-Abelian representations of the braid group B_n. CONNECTION: The golden ratio φ appears directly as the quantum dimension, linking topological phases to the same ratio found in Penrose tilings, quasicrystals, and 5-fold crystallographic symmetry. The Fibonacci sequence governs fusion state counting, echoing growth patterns in nature and base-60 harmonic ratios (φ ≈ 1.618, 1/φ ≈ 0.618). DEPTH: 9 — Direct embedding of φ into fundamental quantum statistics; bridges topology, number theory, and condensed matter physics; provides a concrete path to fault-tolerant quantum computing via non-Abelian anyons. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 25, 20260 citations

Fibonacci Anyons Realize Universal Topological Quantum Computation via Golden Ratio Braiding — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fibonacci anyons carry quantum dimension φ = (1+√5)/2 ≈ 1.618, enabling universal topological quantum computation via braiding — now experimentally realized in non-Abelian anyon systems (Quantinuum, 2025). | MATH: Quantum dimension d = φ satisfies d² = d + 1 (golden ratio identity). Braid group Bₙ on Fibonacci anyons yields dense unitary representations on the fusion space — universality proven (Freedman–Kitaev–Wang). Single-qubit braiding gates compiled via algebraic decomposition of SU(2) into Fibonacci braid words (arXiv:2008.03542). | CONNECTION: φ = 1.618 = 2cos(π/5) — directly tied to pentagonal symmetry, the root system of the icosahedral group H₃, and the 5-fold crystallographic (quasicrystal) forbidden symmetry. The Fibonacci fusion rules (τ × τ = 1 + τ) mirror the golden ratio recurrence Fₙ₊₁ = Fₙ + Fₙ₋₁. | DEPTH: 9 — This is not numerology; φ emerges as the unique algebraic solution to the consistency equations (pentagon/hexagon identities) for a non-Abelian anyon m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Fibonacci anyons carry quantum dimension φ = (1+√5)/2 ≈ 1.618, enabling universal topological quantum computation via braiding — now experimentally realized in non-Abelian anyon systems (Quantinuum, 2025). | MATH: Quantum dimension d = φ satisfies d² = d + 1 (golden ratio identity). Braid group Bₙ on Fibonacci anyons yields dense unitary representations on the fusion space — universality proven (Freedman–Kitaev–Wang). Single-qubit braiding gates compiled via algebraic decomposition of SU(2) into Fibonacci braid words (arXiv:2008.03542). | CONNECTION: φ = 1.618 = 2cos(π/5) — directly tied to pentagonal symmetry, the root system of the icosahedral group H₃, and the 5-fold crystallographic (quasicrystal) forbidden symmetry. The Fibonacci fusion rules (τ × τ = 1 + τ) mirror the golden ratio recurrence Fₙ₊₁ = Fₙ + Fₙ₋₁. | DEPTH: 9 — This is not numerology; φ emerges as the unique algebraic solution to the consistency equations (pentagon/hexagon identities) for a non-Abelian anyon m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 18, 20260 citations

Microsoft's Majorana 1 Chip: Topological Qubits via Non-Abelian Anyon Braiding — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Microsoft's Majorana 1 chip claims topological qubits using non-Abelian anyons, enabling error-resistant quantum computation via braiding statistics. | MATH: Non-Abelian braid group representations; anyon fusion rules (e.g., Fibonacci anyons: τ × τ = 1 + τ, quantum dimension φ = (1+√5)/2 ≈ 1.618); topological quantum field theory (TQFT) with Chern-Simons action S = (k/4π) ∫ Tr(A ∧ dA + (2/3)A ∧ A ∧ A). | CONNECTION: Fibonacci anyon quantum dimension φ = 1.618 (golden ratio); braid group B₃ yields unitary matrices with entries involving φ; base-60 not directly present, but anyon statistics link to SU(2)ₖ Chern-Simons with level k = 3 for Fibonacci anyons. | DEPTH: 8 — Directly ties golden ratio to quantum error correction; topological protection reduces decoherence; crystallographic symmetries (e.g., Kitaev's honeycomb model) underlie Majorana zero modes. FINDING: Topological quantum matter (Fan Zhang) explains edge states and bulk-boundary correspondence via topological invar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Microsoft's Majorana 1 chip claims topological qubits using non-Abelian anyons, enabling error-resistant quantum computation via braiding statistics. | MATH: Non-Abelian braid group representations; anyon fusion rules (e.g., Fibonacci anyons: τ × τ = 1 + τ, quantum dimension φ = (1+√5)/2 ≈ 1.618); topological quantum field theory (TQFT) with Chern-Simons action S = (k/4π) ∫ Tr(A ∧ dA + (2/3)A ∧ A ∧ A). | CONNECTION: Fibonacci anyon quantum dimension φ = 1.618 (golden ratio); braid group B₃ yields unitary matrices with entries involving φ; base-60 not directly present, but anyon statistics link to SU(2)ₖ Chern-Simons with level k = 3 for Fibonacci anyons. | DEPTH: 8 — Directly ties golden ratio to quantum error correction; topological protection reduces decoherence; crystallographic symmetries (e.g., Kitaev's honeycomb model) underlie Majorana zero modes. FINDING: Topological quantum matter (Fan Zhang) explains edge states and bulk-boundary correspondence via topological invar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 14, 20260 citations

B_n

generators satisfy

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

; anyon fusion rules

a \times b = \sum_c N_{ab}^c c

; Fibonacci anyon quantum dimension

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

; Majorana zero modes obey

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

. | CONNECTION: Fibonacci anyon dimension

d = 1.618

is the golden ratio; braid group representations relate to Temperley-Lieb algebras with parameter

\tau = 1/d^2 = 0.382

; anyon fusion rules mirror SU(2)_k modular tensor categories with

q = e^{2\pi i/(k+2)}

, linking to root systems

A_n, D_n, E_6, E_7, E_8

. | DEPTH: 8 — Directly ties golden ratio, modular tensor categories, and non-Abelian statistics to fault-tolerant quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com