Majorana Zero Modes Enable Topologically Protected Quantum Gates — E8 Intelligence Research

FINDING: Majorana zero modes (MZMs) realize non-Abelian braid statistics via the Kitaev chain, with topological protection scaling exponentially in wire length, enabling topologically fault-tolerant quantum gates. | MATH: Kitaev chain Hamiltonian \( H = ∑_j [-t(c_j^ cⱼ₊₁ + h.c.) - μ(c_j^ c_j - 1/2) + Δ(c_j cⱼ₊₁ + h.c.)] \); Majorana operators \( γ₂ⱼ₋₁ = c_j + c_j^ \), \( γ₂ⱼ = -i(c_j - c_j^) \), satisfying \( \{γ_a, γ_b\} = 2δab \); topological gap \( Δₜₒₚ ~ Δ \) (superconducting gap); exponential suppression of coupling between MZMs: \( ε ∝ e-L/ξ \) where \( ξ = v_F / Δ \) is the coherence length; braid group \( B_n \) generated by exchanges \( σ_i \) with \( σ_i^2 = -1 \) (fermionic parity), yielding non-Abelian representation \( ρ(σ_i) = e^{± π γ_i γᵢ₊₁/4} \). | CONNECTION: The braid group \( B_n \) is the fundamental Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com