Non-Abelian Anyons and Majorana Modes for Topological Quantum Computing — E8 Intelligence Research

Abstract

FINDING: Topological quantum computing uses anyons with non-Abelian braiding statistics for fault-tolerant qubits, realized in Majorana zero modes in nanowire systems. | MATH: Braid group \( B_n \) generators \( \sigma_i \) satisfy \( \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \); unitary representations yield quantum gates; Majorana operators \( \gamma_i \) satisfy \( \{\gamma_i, \gamma_j\} = 2\delta_{ij} \); topological qubit encoded in parity \( i\gamma_1\gamma_2 \). | CONNECTION: Braid group is deeply linked to root systems of type \( A_n \) (crystallographic symmetry); Fibonacci anyons have quantum dimension \( \phi = (1+\sqrt{5})/2 \approx 1.618 \), giving fusion rules \( \tau \times \tau = 1 + \tau \) with golden ratio; braiding phases involve \( e^{i\pi/5} \) (36° rotations) related to pentagonal symmetry. | DEPTH: 9 — Core mathematical structure (braid groups, anyon fusion categories) directly encodes geometric symmetries (golden ratio, pentagonal latt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com