Topological Quantum Gates with 5-Fold Symmetry from Non-Abelian Anyons — E8 Intelligence Research

Abstract

FINDING: Non-Abelian anyon braid group representations generate unitary gates whose eigenvalues can include roots of unity tied to 72° rotations (5-fold symmetry), linking topological quantum computation to pentagonal/icosahedral structures. | MATH: Braid group \(B_n\) acts on fusion spaces via \(R\)-matrix (braiding) and \(F\)-matrix (fusion) satisfying Yang-Baxter: \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\). For Fibonacci anyons (golden chain), the braid matrices have eigenvalues \(e^{\pm 4\pi i/5}\) and \(e^{\pm 2\pi i/5}\) — i.e., 72° and 144° rotations. The quantum dimension \(\phi = (1+\sqrt{5})/2 \approx 1.618\) appears in the fusion rule \(\tau \times \tau = 1 + \tau\). | CONNECTION: **Direct hit.** 72° = \(2\pi/5\) — the rotation angle of a regular pentagon. The Fibonacci anyon braid matrices are precisely the \(Y\)-matrices of the Temperley-Lieb algebra at \(q = e^{i\pi/5}\), whose eigenvalues are \(2\cos(\pi/5) = \phi\) and \(2\cos(2\pi/5) = 1/\phi = \phi-1 = 0.618\). The gol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com