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

Andrew Stewart Caldin

Published Jul 28, 2026• 0 citations
Publisher

Abstract

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.com

DOI
10.5281/zenodo.21637394

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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.com

•Aug 23, 2026•0 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.com

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

Andrew Stewart Caldin

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.com

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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.com

•Aug 17, 2026•0 citations
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.com
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
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
FINDING: Anyons are 2D quasiparticles whose worldlines braid in spacetime; quantum gates are implemented by topological braiding, making qubits immune to local decoherence. | 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) and σ_i σ_j = σ_j σ_i for |i−j|≥2. Fibonacci anyons: fusion rule τ⊗τ = 1 ⊕ τ, quantum dimension d_τ = φ = (1+√5)/2 ≈ 1.618. Ising anyons: d_σ = √2. | CONNECTION: Fibonacci anyon quantum dimension is exactly the golden ratio φ = 1.618; braiding matrices generate SU(2) representations with entries in ℤ[φ], linking topological phases to the same φ that appears in phyllotaxis, Penrose tilings, and 5-fold quasicrystalline symmetry. | DEPTH: 8 FINDING: Topological qubits encode information non-locally in the fusion/braiding space of anyons, so logical operations depend only on topology (isotopy class), not on continuous path. | MATH: Hilbert space dimension for n Fibonacci anyons = Fibonacci number F_{n-1}; unitary gates fr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.com
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.com
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 σi \sigma_i σi​ satisfy σiσi+1σi=σi+1σiσi+1 \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} σi​σi+1​σi​=σi+1​σi​σi+1​ (Yang–Baxter equation). Fibonacci anyon fusion rules: τ×τ=1+τ \tau \times \tau = 1 + \tau τ×τ=1+τ, with quantum dimension dτ=ϕ=(1+5)/2≈1.618 d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 dτ​=ϕ=(1+5​)/2≈1.618. Single-qubit braiding gates compiled from Fibonacci anyons use matrices with entries involving ϕ−1≈0.618 \phi^{-1} \approx 0.618 ϕ−1≈0.618. Chern–Simons term in Ginzburg–Landau model: SCS=k4π∫A∧dA S_{CS} = \frac{k}{4\pi} \int A \wedge dA SCS​=4πk​∫A∧dA, with level k k k integer. CONNECTION: Fibonacci anyon quantum dimension dτ=ϕ d_\tau = \phi dτ​=ϕ directly ties to the golden ratio 1.618 and its inverse 0.618. Braiding matrices for Fibonacci anyons involve ϕ−1 \phi^{-1} ϕ−1 and ϕ−2≈0.382 \phi^{-2} \approx 0.382 ϕ−2≈0.382. The Yan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com