Non-Abelian Majorana Modes Enable Topologically Protected Quantum Gates — E8 Intelligence Research

Abstract

FINDING: Majorana zero modes (MZMs) in topological superconductors obey non-Abelian braid statistics, with exponential suppression of coupling between spatially separated modes enabling topologically protected quantum gates. | MATH: The Kitaev chain Hamiltonian \( H = -t\sum c^\dagger_i c_{i+1} + \Delta\sum c_i c_{i+1} + \text{h.c.} - \mu\sum c^\dagger_i c_i \) yields unpaired Majorana operators \( \gamma_1, \gamma_2 \) with \( \gamma_i^\dagger = \gamma_i \), \( \{\gamma_i,\gamma_j\} = 2\delta_{ij} \). Braiding exchanges \( \gamma_i \to \gamma_j \) via unitary \( U_{ij} = \exp(\frac{\pi}{4}\gamma_i\gamma_j) \), giving non-Abelian representation of the braid group \( B_n \) (e.g., \( U_{12}U_{23} \neq U_{23}U_{12} \)). Splitting energy \( \Delta E \propto e^{-L/\xi} \) where \( \xi = \hbar v_F/\Delta \) is the coherence length — exponential suppression with wire length \( L \). | CONNECTION: The braid group \( B_n \) is generated by transpositions \( \sigma_i \) satisfying \( \sigma_i\s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com