Topological Quantum Computing via Non-Abelian Anyon Braiding and Majorana Modes — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 22, 2026• 0 citations

Abstract

FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group $ B_n $ generators satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $; Majorana operators $ \gamma_i $ satisfy $ \{\gamma_i, \gamma_j\} = 2\delta_{ij} $; fusion rules for Ising anyons: $ \sigma \times \sigma = 1 + \psi $, $ \sigma \times \psi = \sigma $, $ \psi \times \psi = 1 $. | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at $ q = e^{i\pi/5} $, where the Jones polynomial yields $ \tau = (1+\sqrt{5})/2 \approx 1.618 $; Fibonacci anyons have fusion rules $ \tau \times \tau = 1 + \tau $, directly encoding the golden ratio. | DEPTH: 9 — Non-Abelian statistics and braiding are foundational to topological quantum computation, with direct geometric harmony in Fibonacci anyon models. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21484231

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Topological Quantum Computing via Non-Abelian Anyon Braiding and Majorana Modes — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 22, 2026• 0 citations

Publisher

Abstract

FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group ( B_n ) generators satisfy ( \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} ); Majorana operators ( \gamma_i ) satisfy ( {\gamma_i, \gamma_j} = 2\delta_{ij} ); fusion rules for Ising anyons: ( \sigma \times \sigma = 1 + \psi ), ( \sigma \times \psi = \sigma ), ( \psi \times \psi = 1 ). | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at ( q = e^{i\pi/5} ), where the Jones polynomial yields ( \tau = (1+\sqrt{5})/2 \approx 1.618 ); Fibonacci anyons have fusion rules ( \tau \times \tau = 1 + \tau ), directly encoding the golden ratio. | DEPTH: 9 — Non-Abelian statistics and braiding are foundational to topological quantum computation, with direct geometric harmony in Fibonacci anyon models. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21484231

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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.comFINDING: 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

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FINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) for fault-tolerant braid-based computation, with Microsoft's Majorana 1 chip claiming hardware realization. MATH: - Braid group $B_n$ 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.comFINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) for fault-tolerant braid-based computation, with Microsoft's Majorana 1 chip claiming hardware realization. MATH: - Braid group $ 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

Aug 31, 20260 citations

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

B_n

generators satisfy

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

; anyon fusion rules

a \times b = \sum_c N_{ab}^c c

; Fibonacci anyon quantum dimension

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

; Majorana zero modes obey

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

. | CONNECTION: Fibonacci anyon dimension

d = 1.618

is the golden ratio; braid group representations relate to Temperley-Lieb algebras with parameter

\tau = 1/d^2 = 0.382

; anyon fusion rules mirror SU(2)_k modular tensor categories with

q = e^{2\pi i/(k+2)}

, linking to root systems

A_n, D_n, E_6, E_7, E_8

. | DEPTH: 8 — Directly ties golden ratio, modular tensor categories, and non-Abelian statistics to fault-tolerant quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) to encode qubits in braiding operations, immune to local decoherence. | MATH: Braid group

B_n

generators

\sigma_i

satisfy

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

; anyon fusion rules:

\sigma \times \sigma = 1 + \psi

(Ising model); Majorana operators

\gamma_i

with

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

; topological qubit encoded in parity

i\gamma_1\gamma_2 = \pm 1

. | CONNECTION: Braid group is related to the Fibonacci sequence via the Burau representation; the golden ratio

\phi = 1.618

appears in the quantum dimension of Fibonacci anyons:

d_\tau = \phi

. The Ising anyon quantum dimension is

\sqrt{2}

, linking to

\sqrt{2} \approx 1.414

, close to 1.4142 (no direct golden ratio). Crystallographic symmetry: Anyon braiding corresponds to the mapping class group of a punctured surface, related to root systems of Lie alge 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

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