Topological Qubits via Anyons and Braid Groups: Majorana 1 Hardware — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 9, 2026• 0 citations

Abstract

FINDING: Topological quantum computing leverages anyonic quasiparticles and braid group representations to encode information non-locally, with Microsoft's Majorana 1 chip claiming topological qubit hardware based on Majorana zero modes. | 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 $. Fibonacci anyons: fusion rule $ \tau \otimes \tau = 1 \oplus \tau $, quantum dimension $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. Non-Abelian statistics: unitary braid matrices $ U(\sigma_i) $ with entries involving $ e^{i\pi/5} $, $ \phi^{-1} $, $ \phi^{-1/2} $. Majorana zero modes: $ \gamma_i^\dagger = \gamma_i $, $ \{\gamma_i, \gamma_j\} = 2\delta_{ij} $, parity $ P = i^k \prod \gamma_i $. Topological protection: energy gap $ \Delta $ vs. error rate $ \sim e^{-\Delta/k_B T} $. | CONNECTION: Fibonacci anyons directly enc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23255404

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Topological Qubits via Anyons and Braid Groups: Majorana 1 Hardware — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 9, 2026• 0 citations

Publisher

Abstract

FINDING: Topological quantum computing leverages anyonic quasiparticles and braid group representations to encode information non-locally, with Microsoft's Majorana 1 chip claiming topological qubit hardware based on Majorana zero modes. | 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 ). Fibonacci anyons: fusion rule ( \tau \otimes \tau = 1 \oplus \tau ), quantum dimension ( d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 ). Non-Abelian statistics: unitary braid matrices ( U(\sigma_i) ) with entries involving ( e^{i\pi/5} ), ( \phi^{-1} ), ( \phi^{-1/2} ). Majorana zero modes: ( \gamma_i^\dagger = \gamma_i ), ( {\gamma_i, \gamma_j} = 2\delta_{ij} ), parity ( P = i^k \prod \gamma_i ). Topological protection: energy gap ( \Delta ) vs. error rate ( \sim e^{-\Delta/k_B T} ). | CONNECTION: Fibonacci anyons directly enc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23255404

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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

B_n

generators

\sigma_i

satisfy

\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}

(Yang–Baxter equation). - Non-Abelian anyon fusion rules:

\sigma \times \sigma = 1 + \psi

(Ising anyons) or Fibonacci anyons:

\tau \times \tau = 1 + \tau

. - Topological quantum number:

\nu = \frac{1}{2\pi}\oint \mathbf{A}\cdot d\mathbf{l}

(Chern number for Majorana modes). - Majorana condition:

\gamma^\dagger = \gamma

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

. - Braid matrices yield phases

e^{i\theta}

with

\theta = \pi/8

for Ising (Clifford) and

\theta = 2\pi/5

for Fibonacci (universal). CONNECTION: - Fibonacci anyons: golden ratio

\phi = 1.618

appears in fusion dimension

d_\tau = \phi

, an 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|\geq 2

. Non-Abelian anyons carry unitary representations

U(\sigma_i)

with

U(\sigma_i)^2 \neq I

(unlike fermions/bosons). Majorana zero modes obey

\gamma_i^\dagger = \gamma_i

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

, and a pair encodes a qubit via parity

i\gamma_1\gamma_2 = \pm 1

. Topological protection arises from the gap

\Delta E \sim e^{-L/\xi}

(exponential suppression with wire length

L

vs coherence length

\xi

). Fibonacci anyons (from SU(2)

_3

Chern-Simons theory) have quantum dimension

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

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