Topological Quantum Error Correction via Majorana Zero Modes in Kitaev Chains — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 4, 2026• 0 citations

Abstract

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 = \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)] $. In the topological phase ($|\mu| < 2t$), unpaired Majorana operators $\gamma_1, \gamma_2$ satisfy $\gamma_i^\dagger = \gamma_i$, $\{\gamma_i,\gamma_j\} = 2\delta_{ij}$, and the nonlocal fermion $f = (\gamma_1 + i\gamma_2)/2$ gives a two-fold ground-state degeneracy (qubit). Braiding MZMs implements the Clifford group; the fusion rules follow the Ising anyon model: $\sigma \times \sigma = 1 + \psi$, with quantum dimension $d_\sigma = \sqrt{2}$. | CONNECTION: The Ising anyon fusion rule $\sigma \times \sigma = 1 + \psi$ 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

DOI: 10.5281/zenodo.23131531

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Topological Quantum Error Correction via Majorana Zero Modes in Kitaev Chains — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 4, 2026• 0 citations

Publisher

Abstract

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 = \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)] ). In the topological phase ((|\mu| < 2t)), unpaired Majorana operators (\gamma_1, \gamma_2) satisfy (\gamma_i^\dagger = \gamma_i), ({\gamma_i,\gamma_j} = 2\delta_{ij}), and the nonlocal fermion (f = (\gamma_1 + i\gamma_2)/2) gives a two-fold ground-state degeneracy (qubit). Braiding MZMs implements the Clifford group; the fusion rules follow the Ising anyon model: (\sigma \times \sigma = 1 + \psi), with quantum dimension (d_\sigma = \sqrt{2}). | CONNECTION: The Ising anyon fusion rule (\sigma \times \sigma = 1 + \psi) 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

DOI: 10.5281/zenodo.23131531

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Majorana 1: Topological Qubits via Non-Abelian Braiding, Evidence Contested — E8 Intelligence Research

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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)

\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

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

. Anyonic exchange yields unitary matrices

U(\sigma_i)

with non-commuting phases

e^{i\theta}

, where

\theta

depends on topological charge (e.g., Ising anyons:

\theta = \pi/8

for

\sigma

particles). Fibonacci anyons: dimension

d = \phi = (1+\sqrt{5})/2 \approx 1.618

, with fusion rules

\tau \times \tau = 1 + \tau

. Majorana zero modes:

\gamma_i^\dagger = \gamma_i

\{\gamma_i, \gamma_j\} = 2\delta_{ij}

, parity

P = i^k \prod \gamma_i

. | CONNECTION: Fibonacci anyon dimension

d = \phi = 1.618

— direct golden ratio e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\gamma_i^\dagger = \gamma_i

\{\gamma_i, \gamma_j\} = 2\delta_{ij}

. Braid group

B_n

generators

\sigma_i

act on ground state degeneracy

2^{n/2}

. Topological qubit encoded in parity

P = i\gamma_1\gamma_2

. | CONNECTION: Braiding matrices correspond to Fibonacci anyons with golden ratio

\phi = 1.618

fusion rules:

\tau \times \tau = 1 + \tau

, dimension

d_\tau = \phi

. Lattice structure: Kitaev honeycomb model with

\mathbb{Z}_2

gauge fields. | DEPTH: 8 — if verified, enables inherently error-corrected qubits; mathematical structure tied to modular tensor categories and quantum groups at roots of unity (e.g.,

q = e^{i\pi/5}

). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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