Non-Abelian Anyon Braiding for Topological Qubits via Braid Group Representations — E8 Intelligence Research

FINDING: Anyon braiding for topological qubits relies on non-Abelian statistics in 2D systems, with braid group representations encoding quantum information. MATH: Braid group \( B_n \) generators \( \sigma_i \) satisfy \( \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \) and \( \sigma_i \sigma_j = \sigma_j \sigma_i \) for \( |i-j|>1 \). Anyon fusion rules: \( a \times b = \sum_c N_{ab}^c c \). Quantum dimension \( d_a \) satisfies \( d_a d_b = \sum_c N_{ab}^c d_c \). Fibonacci anyons: \( \tau \times \tau = 1 + \tau \), with \( d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 \). Braiding matrices are unitary representations of \( B_n \). CONNECTION: Fibonacci anyon quantum dimension \( d_\tau = \phi = 1.618 \) — the golden ratio. This is a direct geometric harmony constant. The braid group is related to the mapping class group of punctured surfaces, linking to modular tensor categories and the Jones polynomial (knot invariants). The anyon fusion algebra mirrors the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com