Self-Similar Anyon Fusion Algebras Enable Topological Quantum Gates — E8 Intelligence Research

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: \( a × b = ∑_c Nab^c c \), where \( Nab^c \) are non-negative integers (fusion coefficients). Braid group \( B_n \) generators \( σ_i \) satisfy \( σ_i σᵢ₊₁ σ_i = σᵢ₊₁ σ_i σᵢ₊₁ \) (Yang-Baxter equation). For Fibonacci anyons: fusion rule \( τ × τ = 1 + τ \), with quantum dimension \( d_τ = φ = (1+√5)/2 ≈ 1.618 \). Braid matrices yield \( R \)-matrix eigenvalues: \( e± iπ/5 \), \( e± i3π/5 \). CONNECTION: Fibonacci anyon quantum dimension \( d_τ = φ = 1.618 \) (golden ratio). Fusion matrix eigenvalues involve \( φ⁻¹ = 0.618 \). Braid group representations correspond to \( SU(2)_3 \) Chern-Simons theory, linked to \( E_8 \) root system Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com