Non-Abelian Anyons and Braid Group Representations for Topological Quantum Computation — E8 Intelligence Research

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group \\(B_n\\) generators \\(\\sigma_i\\) satisfy \\(\\sigma_i\\sigma_{i+1}\\sigma_i = \\sigma_{i+1}\\sigma_i\\sigma_{i+1}\\) (Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices \\(U(\\sigma_i)\\) with \\(U(\\sigma_i)U(\\sigma_{i+1})U(\\sigma_i) = U(\\sigma_{i+1})U(\\sigma_i)U(\\sigma_{i+1})\\); Fibonacci anyons have fusion rule \\(\\tau \\times \\tau = 1 + \\tau\\), yielding quantum dimension \\(\\phi = (1+\\sqrt{5})/2 \\approx 1.618\\) (golden ratio); topological protection arises from ground-state degeneracy on genus-\\(g\\) surfaces: \\(\\dim \\mathcal{H} = 4^g\\) for toric code, or \\(2^g\\) for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio \\(\\phi = 1.618\\), and its inverse \\(\\phi^{-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