Golden Ratio Braids: Universal Quantum Gates via Fibonacci Anyons — E8 Intelligence Research

Abstract

FINDING: Fibonacci anyon braiding realizes a dense, universal gate set via the Fibonacci representation of the braid group, with braid matrices whose eigenvalues are powers of the golden ratio. | MATH: The Fibonacci anyon Hilbert space dimension grows as \(F_{n}\) (Fibonacci numbers). The braid group generators \(\sigma_i\) act as \(2\times2\) matrices (for 3 anyons) with eigenvalues \(e^{\pm 4\pi i/5}\) and \(e^{\pm 2\pi i/5}\) — roots of unity tied to the 5th cyclotomic field \(\mathbb{Q}(\zeta_5)\). The quantum dimension is \(\phi = (1+\sqrt{5})/2 = 1.618...\), and the fusion rules satisfy \(\tau \times \tau = 1 + \tau\), giving the golden ratio as the largest eigenvalue of the fusion matrix \(N_\tau = \begin{pmatrix}0&1\\1&1\end{pmatrix}\). The braid matrices are \(R\)-matrices of the Yang–Baxter equation, with entries involving \(\phi^{-1} = 0.618...\) and \(\phi^{-2} = 0.382...\). | CONNECTION: Direct geometric harmony: the 5-fold rotational symmetry (C5) of the braid group repre Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com