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Topological Protection and Non-Abelian Braiding of Majorana Zero Modes — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 29, 2026• 0 citations
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Abstract

FINDING: Majorana zero modes (MZMs) realize non-Abelian braid statistics via the Ising anyon model, with topological protection scaling exponentially in wire length. | MATH: Braid group \( B_n \) generators \( \sigma_i \) act on a 2D degenerate ground space (for \( 2n \) MZMs) via \( \gamma_i \to \gamma_j = U^\dagger \gamma_i U \), where \( U = \exp(\frac{\pi}{4}\gamma_i\gamma_j) \). The non-Abelian exchange yields \( \sigma_i^2 = -1 \) (fermion parity flip), and the topological gap \( \Delta_{\rm top} \propto e^{-L/\xi} \) (exponential suppression of splitting with wire length \( L \), coherence length \( \xi \)). Kitaev chain Hamiltonian: \( H = -\mu\sum c_i^\dagger c_i - t\sum(c_{i+1}^\dagger c_i + {\rm h.c.}) + \Delta\sum(c_{i+1}^\dagger c_i^\dagger + {\rm h.c.}) \), with MZM operators \( \gamma_1 = c_1 + c_1^\dagger \), \( \gamma_2 = -i(c_N - c_N^\dagger) \) at \( \mu=0, t=\Delta \). | CONNECTION: The braid group \( B_n \) is the fundamental group of configuration space of \( n \) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22154694

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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(cj†cj+1+h.c.)−μ(cj†cj−1/2)+Δ(cjcj+1+h.c.)] H = \sum_j [-t(c_j^\dagger c_{j+1} + h.c.) - \mu(c_j^\dagger c_j - 1/2) + \Delta(c_j c_{j+1} + h.c.)] H=∑j​[−t(cj†​cj+1​+h.c.)−μ(cj†​cj​−1/2)+Δ(cj​cj+1​+h.c.)]; Majorana operators γ2j−1=cj+cj† \gamma_{2j-1} = c_j + c_j^\dagger γ2j−1​=cj​+cj†​, γ2j=−i(cj−cj†) \gamma_{2j} = -i(c_j - c_j^\dagger) γ2j​=−i(cj​−cj†​), satisfying {γa,γb}=2δab \{\gamma_a, \gamma_b\} = 2\delta_{ab} {γa​,γb​}=2δab​; topological gap Δtop∼Δ \Delta_{\text{top}} \sim \Delta Δtop​∼Δ (superconducting gap); exponential suppression of coupling between MZMs: ϵ∝e−L/ξ \epsilon \propto e^{-L/\xi} ϵ∝e−L/ξ where ξ=ℏvF/Δ \xi = \hbar v_F / \Delta ξ=ℏvF​/Δ is the coherence length; braid group Bn B_n Bn​ generated by exchanges σi \sigma_i σi​ with σi2=−1 \sigma_i^2 = -1 σi2​=−1 (fermionic parity), yielding non-Abelian representation ρ(σi)=e±πγiγi+1/4 \rho(\sigma_i) = e^{\pm \pi \gamma_i \gamma_{i+1}/4} ρ(σi​)=e±πγi​γi+1​/4. | CONNECTION: The braid group Bn B_n Bn​ is the fundamental Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Majorana zero modes (MZMs) in topological superconductors yield non-Abelian anyonic statistics, enabling topologically protected quantum error correction via ground-state degeneracy in the Kitaev chain. | MATH: Kitaev chain Hamiltonian H=∑j[−t(cj†cj+1+h.c.)+Δ(cjcj+1+h.c.)−μ(cj†cj−1/2)] H = \sum_j [-t(c^\dagger_j c_{j+1} + h.c.) + \Delta(c_j c_{j+1} + h.c.) - \mu(c^\dagger_j c_j - 1/2)] H=∑j​[−t(cj†​cj+1​+h.c.)+Δ(cj​cj+1​+h.c.)−μ(cj†​cj​−1/2)]. In the topological phase (∣μ∣<2t|\mu| < 2t∣μ∣<2t), unpaired Majorana operators γ1,γ2\gamma_1, \gamma_2γ1​,γ2​ satisfy γi†=γi\gamma_i^\dagger = \gamma_iγi†​=γi​, {γi,γj}=2δij\{\gamma_i,\gamma_j\} = 2\delta_{ij}{γi​,γj​}=2δij​, and the nonlocal fermion f=(γ1+iγ2)/2f = (\gamma_1 + i\gamma_2)/2f=(γ1​+iγ2​)/2 gives a two-fold ground-state degeneracy (qubit). Braiding MZMs implements the Clifford group; the fusion rules follow the Ising anyon model: σ×σ=1+ψ\sigma \times \sigma = 1 + \psiσ×σ=1+ψ, with quantum dimension dσ=2d_\sigma = \sqrt{2}dσ​=2​. | CONNECTION: The Ising anyon fusion rule σ×σ=1+ψ\sigma \times \sigma = 1 + \psiσ×σ=1+ψ maps directly to the golden-ratio-adjacent structure of the Fibonacci anyon model (though MZMs are Ising, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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∑ci†ci+1+Δ∑cici+1+h.c.−μ∑ci†ci 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 H=−t∑ci†​ci+1​+Δ∑ci​ci+1​+h.c.−μ∑ci†​ci​ yields unpaired Majorana operators γ1,γ2 \gamma_1, \gamma_2 γ1​,γ2​ with γi†=γi \gamma_i^\dagger = \gamma_i γi†​=γi​, {γi,γj}=2δij \{\gamma_i,\gamma_j\} = 2\delta_{ij} {γi​,γj​}=2δij​. Braiding exchanges γi→γj \gamma_i \to \gamma_j γi​→γj​ via unitary Uij=exp⁡(π4γiγj) U_{ij} = \exp(\frac{\pi}{4}\gamma_i\gamma_j) Uij​=exp(4π​γi​γj​), giving non-Abelian representation of the braid group Bn B_n Bn​ (e.g., U12U23≠U23U12 U_{12}U_{23} \neq U_{23}U_{12} U12​U23​=U23​U12​). Splitting energy ΔE∝e−L/ξ \Delta E \propto e^{-L/\xi} ΔE∝e−L/ξ where ξ=ℏvF/Δ \xi = \hbar v_F/\Delta ξ=ℏvF​/Δ is the coherence length — exponential suppression with wire length L L L. | CONNECTION: The braid group Bn B_n Bn​ is generated by transpositions σi \sigma_i σi​ satisfying \( \sigma_i\s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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FINDING: Topological quantum computing leverages non-Abelian anyons and braid group representations for inherently fault-tolerant quantum gates, with Microsoft's Majorana 1 chip as a physical milestone. | MATH: Braid group BnB_nBn​ generators σi\sigma_iσi​ satisfy σiσi+1σi=σi+1σiσi+1\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}σi​σi+1​σi​=σi+1​σi​σi+1​ and σiσj=σjσi\sigma_i\sigma_j = \sigma_j\sigma_iσi​σj​=σj​σi​ for ∣i−j∣≥2|i-j|\geq 2∣i−j∣≥2. Non-Abelian anyons carry unitary representations U(σi)U(\sigma_i)U(σi​) with U(σi)2≠IU(\sigma_i)^2 \neq IU(σi​)2=I (unlike fermions/bosons). Majorana zero modes obey γi†=γi\gamma_i^\dagger = \gamma_iγi†​=γi​, {γi,γj}=2δij\{\gamma_i,\gamma_j\} = 2\delta_{ij}{γi​,γj​}=2δij​, and a pair encodes a qubit via parity iγ1γ2=±1i\gamma_1\gamma_2 = \pm 1iγ1​γ2​=±1. Topological protection arises from the gap ΔE∼e−L/ξ\Delta E \sim e^{-L/\xi}ΔE∼e−L/ξ (exponential suppression with wire length LLL vs coherence length ξ\xiξ). Fibonacci anyons (from SU(2)3_33​ Chern-Simons theory) have quantum dimension ϕ=(1+5)/2≈1.618\phi = (1+\sqrt{5})/2 \approx 1.618ϕ=(1+5​)/2≈1.618, giving dense braiding — universal quantum co Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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