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

Andrew Stewart Caldin

Published Aug 23, 2026• 0 citations

Abstract

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.com

DOI: 10.5281/zenodo.22064895

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

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Publisher

Abstract

DOI: 10.5281/zenodo.22064895

Related papers

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

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

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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.comFINDING: 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

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

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FINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) for fault-tolerant braiding operations, with Microsoft's Majorana 1 chip claiming topological qubit hardware. | 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); Majorana operators γ_i obey {γ_i, γ_j} = 2δ_ij; topological qubit encoded in parity iγ_1γ_2 = ±1; non-Abelian anyon exchange yields unitary matrices U(σ_i) = exp(θ σ_i) with θ = π/4 for Ising anyons (Fibonacci anyons: θ = 2π/5). | CONNECTION: Braid group relations are the algebraic shadow of the Yang–Baxter equation — the same structure underlying 2D crystallographic root systems (A_n, B_n, D_n) and the golden ratio appears in Fibonacci anyons: quantum dimension φ = (1+√5)/2 ≈ 1.618, with fusion rules τ⊗τ = 1⊕τ giving dimension φ² = φ+1. | DEPTH: 8 --- **Detailed analysis:** 1. **Key mathematical insight:** The core discovery is that topological phases of matter host quasiparticles (anyon Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) for fault-tolerant braiding operations, with Microsoft's Majorana 1 chip claiming topological qubit hardware. | 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); Majorana operators γ_i obey {γ_i, γ_j} = 2δ_ij; topological qubit encoded in parity iγ_1γ_2 = ±1; non-Abelian anyon exchange yields unitary matrices U(σ_i) = exp(θ σ_i) with θ = π/4 for Ising anyons (Fibonacci anyons: θ = 2π/5). | CONNECTION: Braid group relations are the algebraic shadow of the Yang–Baxter equation — the same structure underlying 2D crystallographic root systems (A_n, B_n, D_n) and the golden ratio appears in Fibonacci anyons: quantum dimension φ = (1+√5)/2 ≈ 1.618, with fusion rules τ⊗τ = 1⊕τ giving dimension φ² = φ+1. | DEPTH: 8 --- **Detailed analysis:** 1. **Key mathematical insight:** The core discovery is that topological phases of matter host quasiparticles (anyon Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Oct 29, 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

Non-Abelian Anyons and Majorana Modes for Topological Quantum Computing — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological quantum computing uses anyons with non-Abelian braiding statistics for fault-tolerant qubits, realized in Majorana zero modes in nanowire systems. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ ; unitary representations yield quantum gates; Majorana operators $\gamma_i$ satisfy $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$ ; topological qubit encoded in parity $i\gamma_1\gamma_2$ . | CONNECTION: Braid group is deeply linked to root systems of type $A_n$ (crystallographic symmetry); Fibonacci anyons have quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ , giving fusion rules $\tau \times \tau = 1 + \tau$ with golden ratio; braiding phases involve $e^{i\pi/5}$ (36° rotations) related to pentagonal symmetry. | DEPTH: 9 — Core mathematical structure (braid groups, anyon fusion categories) directly encodes geometric symmetries (golden ratio, pentagonal latt 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, realized in Majorana zero modes in nanowire systems. | MATH: Braid group $ B_n $ generators $ \sigma_i $ satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $; unitary representations yield quantum gates; Majorana operators $ \gamma_i $ satisfy $ \{\gamma_i, \gamma_j\} = 2\delta_{ij} $; topological qubit encoded in parity $ i\gamma_1\gamma_2 $. | CONNECTION: Braid group is deeply linked to root systems of type $ A_n $ (crystallographic symmetry); Fibonacci anyons have quantum dimension $ \phi = (1+\sqrt{5})/2 \approx 1.618 $, giving fusion rules $ \tau \times \tau = 1 + \tau $ with golden ratio; braiding phases involve $ e^{i\pi/5} $ (36° rotations) related to pentagonal symmetry. | DEPTH: 9 — Core mathematical structure (braid groups, anyon fusion categories) directly encodes geometric symmetries (golden ratio, pentagonal latt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 23, 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

FINDING: Topological quantum computing uses anyons with non-Abelian braiding statistics for fault-tolerant qubits, realized in Majorana zero modes in nanowire systems. | MATH: Braid group

B_n

generators

\sigma_i

satisfy

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

; unitary representations yield quantum gates; Majorana operators

\gamma_i

satisfy

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

; topological qubit encoded in parity

i\gamma_1\gamma_2

. | CONNECTION: Braid group is deeply linked to root systems of type

A_n

(crystallographic symmetry); Fibonacci anyons have quantum dimension

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

, giving fusion rules

\tau \times \tau = 1 + \tau

with golden ratio; braiding phases involve

e^{i\pi/5}

(36° rotations) related to pentagonal symmetry. | DEPTH: 9 — Core mathematical structure (braid groups, anyon fusion categories) directly encodes geometric symmetries (golden ratio, pentagonal latt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com