Topological Quantum Gates with 5-Fold Symmetry from Non-Abelian Anyons — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 23, 2026• 0 citations

Abstract

FINDING: Non-Abelian anyon braid group representations generate unitary gates whose eigenvalues can include roots of unity tied to 72° rotations (5-fold symmetry), linking topological quantum computation to pentagonal/icosahedral structures. | MATH: Braid group $B_n$ acts on fusion spaces via $R$-matrix (braiding) and $F$-matrix (fusion) satisfying Yang-Baxter: $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$. For Fibonacci anyons (golden chain), the braid matrices have eigenvalues $e^{\pm 4\pi i/5}$ and $e^{\pm 2\pi i/5}$ — i.e., 72° and 144° rotations. The quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ appears in the fusion rule $\tau \times \tau = 1 + \tau$. | CONNECTION: **Direct hit.** 72° = $2\pi/5$ — the rotation angle of a regular pentagon. The Fibonacci anyon braid matrices are precisely the $Y$-matrices of the Temperley-Lieb algebra at $q = e^{i\pi/5}$, whose eigenvalues are $2\cos(\pi/5) = \phi$ and $2\cos(2\pi/5) = 1/\phi = \phi-1 = 0.618$. The gol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22065108

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Topological Quantum Gates with 5-Fold Symmetry from Non-Abelian Anyons — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 23, 2026• 0 citations

Publisher

Abstract

FINDING: Non-Abelian anyon braid group representations generate unitary gates whose eigenvalues can include roots of unity tied to 72° rotations (5-fold symmetry), linking topological quantum computation to pentagonal/icosahedral structures. | MATH: Braid group (B_n) acts on fusion spaces via (R)-matrix (braiding) and (F)-matrix (fusion) satisfying Yang-Baxter: (R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}). For Fibonacci anyons (golden chain), the braid matrices have eigenvalues (e^{\pm 4\pi i/5}) and (e^{\pm 2\pi i/5}) — i.e., 72° and 144° rotations. The quantum dimension (\phi = (1+\sqrt{5})/2 \approx 1.618) appears in the fusion rule (\tau \times \tau = 1 + \tau). | CONNECTION: Direct hit. 72° = (2\pi/5) — the rotation angle of a regular pentagon. The Fibonacci anyon braid matrices are precisely the (Y)-matrices of the Temperley-Lieb algebra at (q = e^{i\pi/5}), whose eigenvalues are (2\cos(\pi/5) = \phi) and (2\cos(2\pi/5) = 1/\phi = \phi-1 = 0.618). The gol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.22065108

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Self-Similar Anyon Fusion Algebras Enable Topological Quantum Gates — E8 Intelligence Research

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FINDING: Non-Abelian anyon fusion algebras exhibit self-similarity under braid group representations, enabling topological quantum computation via braiding operations that encode quantum gates. MATH: Fusion algebra for anyon types a, b, c: $a \times b = \sum_c N_{ab}^c c$ , where $N_{ab}^c$ are non-negative integers (fusion coefficients). 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). For Fibonacci anyons: fusion rule $\tau \times \tau = 1 + \tau$ , with quantum dimension $d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618$ . Braid matrices yield $R$ -matrix eigenvalues: $e^{\pm i\pi/5}$ , $e^{\pm i3\pi/5}$ . CONNECTION: Fibonacci anyon quantum dimension $d_\tau = \phi = 1.618$ (golden ratio). Fusion matrix eigenvalues involve $\phi^{-1} = 0.618$ . Braid group representations correspond to $SU(2)_3$ Chern-Simons theory, linked to $E_8$ root system Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Non-Abelian anyon fusion algebras exhibit self-similarity under braid group representations, enabling topological quantum computation via braiding operations that encode quantum gates. MATH: Fusion algebra for anyon types a, b, c: $ a \times b = \sum_c N_{ab}^c c $, where $ N_{ab}^c $ are non-negative integers (fusion coefficients). 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). For Fibonacci anyons: fusion rule $ \tau \times \tau = 1 + \tau $, with quantum dimension $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. Braid matrices yield $ R $-matrix eigenvalues: $ e^{\pm i\pi/5} $, $ e^{\pm i3\pi/5} $. CONNECTION: Fibonacci anyon quantum dimension $ d_\tau = \phi = 1.618 $ (golden ratio). Fusion matrix eigenvalues involve $ \phi^{-1} = 0.618 $. Braid group representations correspond to $ SU(2)_3 $ Chern-Simons theory, linked to $ E_8 $ root system Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Non-Abelian Anyon Braiding for Topological Quantum Gates — E8 Intelligence Research

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Andrew Stewart Caldin

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Oct 5, 20260 citations

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

, where

N_{ab}^c

are non-negative integers (fusion coefficients). 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). For Fibonacci anyons: fusion rule

\tau \times \tau = 1 + \tau

, with quantum dimension

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

. Braid matrices yield

R

-matrix eigenvalues:

e^{\pm i\pi/5}

e^{\pm i3\pi/5}

. CONNECTION: Fibonacci anyon quantum dimension

d_\tau = \phi = 1.618

(golden ratio). Fusion matrix eigenvalues involve

\phi^{-1} = 0.618

. Braid group representations correspond to

SU(2)_3

Chern-Simons theory, linked to

E_8

root system 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 anyons carry unitary representations

U(\sigma_i)

with

U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)

for

|i-j|\geq 2

. Fibonacci anyons: fusion rule

\tau \times \tau = 1 + \tau

, Hilbert space dimension grows as Fibonacci sequence

F_n

, golden ratio

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

appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-

g

surfaces: dimension

= k^g

for level-

k

Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio

F_{n+1}/F_n \to 1.618

— direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | 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 anyons carry unitary braid matrices

U(\sigma_i)

with

U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})

; Fibonacci anyons have fusion rule

\tau \times \tau = 1 + \tau

, yielding quantum dimension

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

(golden ratio); topological protection arises from ground-state degeneracy on genus-

g

surfaces:

\dim \mathcal{H} = 4^g

for toric code, or

2^g

for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio

\phi = 1.618

, and its inverse

\phi^{-1} = 0.618

appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th 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