Topological Quantum Computing with Non-Abelian Anyons and Microsoft's Majorana 1 Chip — E8 Intelligence Research

Abstract

FINDING: Topological quantum computing uses anyons with non-Abelian braiding statistics for fault-tolerant qubits, with Microsoft's Majorana 1 chip claiming topological qubit 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 \( |i-j|>1 \). Anyon fusion rules: \( \phi \times \phi = 1 + \phi \) (Fibonacci anyon model). Quantum dimension \( d_\phi = \phi = (1+\sqrt{5})/2 \approx 1.618 \). Topological qubit encoded in fusion space dimension \( F_{abc}^{d} \) (F-matrix) with entries involving golden ratio. Majorana zero modes obey \( \gamma_i^2 = 1 \), \( \{\gamma_i, \gamma_j\} = 2\delta_{ij} \). | CONNECTION: Fibonacci anyon quantum dimension \( d_\phi = 1.618 \) directly links to golden ratio. Braiding matrices yield entries proportional to \( \phi^{-1} \approx 0.618 \) and \( \phi^{-2} \approx 0.382 \). The non-Abelian statistics corresp Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com