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Non-Abelian Anyon Fusion Mirrors Braid Groups for Fault-Tolerant Quantum Computing — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 20, 2026• 0 citations
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Abstract

FINDING: Non-Abelian anyon fusion algebras exhibit self-similarity that mirrors the braid group structure, enabling fault-tolerant topological quantum computation via braiding operations. MATH: - Braid group \( B_n \) 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 \( |i-j|>1 \). - Fusion algebra: \( \phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c \), where \( N_{ab}^c \) are non-negative integers (fusion coefficients). For non-Abelian anyons, \( \sum_c N_{ab}^c > 1 \) for some \( a,b \). - Fibonacci anyon model: single nontrivial particle \( \tau \) with fusion rule \( \tau \times \tau = 1 + \tau \), leading to quantum dimension \( d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 \). - Braiding matrices yield phases \( e^{i\theta} \) with \( \theta = 2\pi/5, 4\pi/5 \) (related to golden angle \( 2\pi/\phi^2 \approx 137.5^\circ \)). CONNECTION: - Fibonacci anyon quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.21449818

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FINDING: Non-Abelian anyon fusion algebras exhibit self-similarity under braid group representations, enabling topological quantum computation via braiding operations that encode quantum gates. MATH: Fusion algebra for anyon types a, b, c: a×b=∑cNabcc a \times b = \sum_c N_{ab}^c c a×b=∑c​Nabc​c, where Nabc N_{ab}^c Nabc​ are non-negative integers (fusion coefficients). Braid group Bn B_n Bn​ 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). For Fibonacci anyons: fusion rule 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. Braid matrices yield R R R-matrix eigenvalues: e±iπ/5 e^{\pm i\pi/5} e±iπ/5, e±i3π/5 e^{\pm i3\pi/5} e±i3π/5. CONNECTION: Fibonacci anyon quantum dimension dτ=ϕ=1.618 d_\tau = \phi = 1.618 dτ​=ϕ=1.618 (golden ratio). Fusion matrix eigenvalues involve ϕ−1=0.618 \phi^{-1} = 0.618 ϕ−1=0.618. Braid group representations correspond to SU(2)3 SU(2)_3 SU(2)3​ Chern-Simons theory, linked to E8 E_8 E8​ root system 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) and braid group representations to encode fault-tolerant qubits, with Microsoft's Majorana 1 chip claiming hardware realization. | MATH: Braid group Bn B_n Bn​ 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​ and σiσj=σjσi \sigma_i \sigma_j = \sigma_j \sigma_i σi​σj​=σj​σi​ for ∣i−j∣>1 |i-j|>1 ∣i−j∣>1. Anyonic exchange yields unitary matrices U(σi) U(\sigma_i) U(σi​) with non-commuting phases eiθ e^{i\theta} eiθ, where θ \theta θ depends on topological charge (e.g., Ising anyons: θ=π/8 \theta = \pi/8 θ=π/8 for σ \sigma σ particles). Fibonacci anyons: dimension d=ϕ=(1+5)/2≈1.618 d = \phi = (1+\sqrt{5})/2 \approx 1.618 d=ϕ=(1+5​)/2≈1.618, with fusion rules τ×τ=1+τ \tau \times \tau = 1 + \tau τ×τ=1+τ. Majorana zero modes: γi†=γi \gamma_i^\dagger = \gamma_i γi†​=γi​, {γi,γj}=2δij \{\gamma_i, \gamma_j\} = 2\delta_{ij} {γi​,γj​}=2δij​, parity P=ik∏γi P = i^k \prod \gamma_i P=ik∏γi​. | CONNECTION: Fibonacci anyon dimension d=ϕ=1.618 d = \phi = 1.618 d=ϕ=1.618 — direct golden ratio e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling topological quantum computation via braiding operations. MATH: - Braid group Bn B_n Bn​ 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​ and σiσj=σjσi \sigma_i \sigma_j = \sigma_j \sigma_i σi​σj​=σj​σi​ for ∣i−j∣>1 |i-j|>1 ∣i−j∣>1. - Hecke algebra relations: (σi−q1/2)(σi+q−1/2)=0 (\sigma_i - q^{1/2})(\sigma_i + q^{-1/2}) = 0 (σi​−q1/2)(σi​+q−1/2)=0, with q=e2πi/k q = e^{2\pi i/k} q=e2πi/k for SU(2)_k anyons. - Fusion rules: ϕa×ϕb=∑cNabcϕc \phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c ϕa​×ϕb​=∑c​Nabc​ϕc​, where Nabc N_{ab}^c Nabc​ are non-negative integers (fusion coefficients). - Fibonacci anyon fusion: 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. CONNECTION: - Fibonacci anyon quantum dimension dτ=ϕ=1.618 d_\tau = \phi = 1.618 dτ​=ϕ=1.618 (golden ratio). - Braid matrix eigenvalues for Fibonacci anyons: e±3πi/5 e^{\pm 3\pi i/5} e±3πi/5, related to golden ratio via \( 2\cos(3\pi/5) 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 BnB_nBn​ generators σi\sigma_iσi​ 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); non-Abelian anyons carry unitary braid matrices U(σi)U(\sigma_i)U(σi​) with U(σi)U(σi+1)U(σi)=U(σi+1)U(σi)U(σi+1)U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})U(σi​)U(σi+1​)U(σi​)=U(σi+1​)U(σi​)U(σi+1​); Fibonacci anyons have fusion rule τ×τ=1+τ\tau \times \tau = 1 + \tauτ×τ=1+τ, yielding quantum dimension ϕ=(1+5)/2≈1.618\phi = (1+\sqrt{5})/2 \approx 1.618ϕ=(1+5​)/2≈1.618 (golden ratio); topological protection arises from ground-state degeneracy on genus-ggg surfaces: dim⁡H=4g\dim \mathcal{H} = 4^gdimH=4g for toric code, or 2g2^g2g for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio ϕ=1.618\phi = 1.618ϕ=1.618, and its inverse ϕ−1=0.618\phi^{-1} = 0.618ϕ−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) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group Bn B_n Bn​ generators 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​; Majorana operators γi \gamma_i γi​ satisfy {γi,γj}=2δij \{\gamma_i, \gamma_j\} = 2\delta_{ij} {γi​,γj​}=2δij​; fusion rules for Ising anyons: σ×σ=1+ψ \sigma \times \sigma = 1 + \psi σ×σ=1+ψ, σ×ψ=σ \sigma \times \psi = \sigma σ×ψ=σ, ψ×ψ=1 \psi \times \psi = 1 ψ×ψ=1. | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at q=eiπ/5 q = e^{i\pi/5} q=eiπ/5, where the Jones polynomial yields τ=(1+5)/2≈1.618 \tau = (1+\sqrt{5})/2 \approx 1.618 τ=(1+5​)/2≈1.618; Fibonacci anyons have fusion rules τ×τ=1+τ \tau \times \tau = 1 + \tau τ×τ=1+τ, 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