Topological Quantum Computing: Anyons, Braid Groups, and Majorana Milestones — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 5, 2026• 0 citations

Abstract

FINDING: Topological quantum computing leverages non-Abelian anyons and braid-group representations for fault-tolerant qubits, with Microsoft's Majorana 1 chip as a hardware milestone. | MATH: Braid group B_n generators σ_i satisfy σ_iσ_{i+1}σ_i = σ_{i+1}σ_iσ_{i+1} and σ_iσ_j = σ_jσ_i for |i−j|≥2. Anyonic exchange yields unitary matrices U(σ_i) with eigenvalues e^{±iπ/4} (Ising anyons) or e^{±2πi/5} (Fibonacci anyons). Fibonacci anyons: single qubit encoded in 3 anyons, fusion space dimension = Fibonacci numbers F_n; braid matrices approximate SU(2) densely. Majorana zero modes: γ† = γ, {γ_i, γ_j} = 2δ_ij, parity operator P = iγ_1γ_2 with eigenvalues ±1. Topological protection: energy gap Δ → error rate ~ e^{−L/ξ} (exponential suppression with wire length L, coherence length ξ). | CONNECTION: Fibonacci anyons directly invoke the golden ratio φ = (1+√5)/2 = 1.618 — fusion multiplicities follow Fibonacci sequence, and braid group representations relate to Temperley-Lieb algebra at q = e^ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23152507

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Published Oct 5, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.23152507

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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 σ_i satisfy σ_iσ_{i+1}σ_i = σ_{i+1}σ_iσ_{i+1} and σ_iσ_j = σ_jσ_i for |i−j|≥2; Fibonacci anyons yield fusion space dimension φ (golden ratio) via F-matrix with entries involving φ^{-1/2}; Majorana modes obey γ_i†=γ_i, {γ_i,γ_j}=2δ_ij; topological charge conservation enforces fusion rules; Jones polynomial evaluated at q=e^{2πi/5} (or e^{2πi/k} for SU(2)_k Chern-Simons) encodes braid invariants. | CONNECTION: Fibonacci anyon model has quantum dimension d = φ = (1+√5)/2 ≈ 1.618, with F-matrix elements ∝ 1/φ ≈ 0.618; braid group representations correspond to Temperley-Lieb algebra with loop value δ = φ+φ^{-1} = √5 ≈ 2.236; SU(2)_k level k=3 gives φ; crystallographic root system A_2 (hexagonal lattice) underlies SU(3) anyons; base-60 appears in modular data S,T 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 σ_i satisfy σ_iσ_{i+1}σ_i = σ_{i+1}σ_iσ_{i+1} and σ_iσ_j = σ_jσ_i for |i−j|≥2; Fibonacci anyons yield fusion space dimension φ (golden ratio) via F-matrix with entries involving φ^{-1/2}; Majorana modes obey γ_i†=γ_i, {γ_i,γ_j}=2δ_ij; topological charge conservation enforces fusion rules; Jones polynomial evaluated at q=e^{2πi/5} (or e^{2πi/k} for SU(2)_k Chern-Simons) encodes braid invariants. | CONNECTION: Fibonacci anyon model has quantum dimension d = φ = (1+√5)/2 ≈ 1.618, with F-matrix elements ∝ 1/φ ≈ 0.618; braid group representations correspond to Temperley-Lieb algebra with loop value δ = φ+φ^{-1} = √5 ≈ 2.236; SU(2)_k level k=3 gives φ; crystallographic root system A_2 (hexagonal lattice) underlies SU(3) anyons; base-60 appears in modular data S,T 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 and braid group representations for inherently fault-tolerant quantum gates, with Microsoft's Majorana 1 chip as a physical milestone. | 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 $|i-j|\geq 2$ . 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.comFINDING: Topological quantum computing leverages non-Abelian anyons and braid group representations for inherently fault-tolerant quantum gates, with Microsoft's Majorana 1 chip as a physical milestone. | 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 $|i-j|\geq 2$. 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

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Sep 8, 20260 citations

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

. 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