Topological Qubits via Non-Abelian Anyons and Braid Groups — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 8, 2026• 0 citations

Abstract

FINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) and braid group representations to encode fault-tolerant qubits, 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} $ 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

DOI: 10.5281/zenodo.22658789

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Topological Qubits via Non-Abelian Anyons and Braid Groups — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 8, 2026• 0 citations

Publisher

Abstract

FINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) and braid group representations to encode fault-tolerant qubits, 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} ) 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

DOI: 10.5281/zenodo.22658789

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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

. 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

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

\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 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