Golden Ratio Braid Gates: Fibonacci Anyons for Topological Quantum Computation — E8 Intelligence Research

FINDING: Fibonacci anyons realize braid group representations with matrices whose entries are powers of the golden ratio, yielding dense unitary gates for topological quantum computation. | MATH: Fibonacci anyon fusion rule: τ ⊗ τ = 1 ⊕ τ (where τ is the non-Abelian anyon). The braid matrices for two τ's are 2×2 unitaries parameterized by φ = (1+√5)/2 ≈ 1.618. Specifically, the R-matrix and F-matrix (fusion/splitting) involve φ⁻¹ = φ−1 ≈ 0.618 and φ⁻² ≈ 0.382. The braid group B₃ acts irreducibly on the 2-dimensional Hilbert space spanned by {|1⟩, |τ⟩} in the fusion channel. The quantum dimension of τ is d_τ = φ, satisfying d_τ² = d_τ + 1. The Fibonacci anyon model is equivalent to the level-2 SU(2) Wess-Zumino-Witten conformal field theory, with q = eiπ/5, giving q-dimension [2]_q = φ. | CONNECTION: The golden ratio φ and its inverse φ⁻¹ = 0.618, φ⁻² = 0.382 appear directly as matrix elements and quantum dimensions. The braid group Bₙ maps to the Temperley-Lieb algebra w Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com