Topological Qubits via Non-Abelian Anyons and Braid Groups — E8 Intelligence Research

FINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) and braid group representations to encode fault-tolerant qubits, with Microsoft's Majorana 1 chip claiming hardware realization. | 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 \( |i-j|>1 \). Anyonic exchange yields unitary matrices \( U(\sigma_i) \) with non-commuting phases \( e^{i\theta} \), where \( \theta \) depends on topological charge (e.g., Ising anyons: \( \theta = \pi/8 \) for \( \sigma \) particles). Fibonacci anyons: dimension \( d = \phi = (1+\sqrt{5})/2 \approx 1.618 \), with fusion rules \( \tau \times \tau = 1 + \tau \). Majorana zero modes: \( \gamma_i^\dagger = \gamma_i \), \( \{\gamma_i, \gamma_j\} = 2\delta_{ij} \), parity \( P = i^k \prod \gamma_i \). | CONNECTION: Fibonacci anyon dimension \( d = \phi = 1.618 \) — direct golden ratio e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com