Topological Quantum Computing: Braid Groups and Majorana 1 Milestone — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 27, 2026• 0 citations

Abstract

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.com

DOI: 10.5281/zenodo.22122204

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Topological Quantum Computing: Braid Groups and Majorana 1 Milestone — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 27, 2026• 0 citations

Publisher

Abstract

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.com

DOI: 10.5281/zenodo.22122204

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Sep 10, 20260 citations

Topological Qubits via Non-Abelian Anyons: Braid Group and Majorana Modes — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) for fault-tolerant braiding operations, with Microsoft's Majorana 1 chip claiming topological qubit hardware. | MATH: Braid group B_n generators σ_i satisfy σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} and σ_i σ_j = σ_j σ_i for |i−j|≥2; non-Abelian anyons yield unitary representations U(σ_i) with U(σ_i)U(σ_{i+1})U(σ_i) ≠ U(σ_{i+1})U(σ_i)U(σ_{i+1}); Majorana zero modes obey γ_i† = γ_i, {γ_i, γ_j} = 2δ_ij; topological qubit encoded in parity (−1)^{n} = i^{2n} γ_1 γ_2 … γ_{2n}; Fibonacci anyons have quantum dimension φ = (1+√5)/2 ≈ 1.618, fusion rules τ×τ = 1 + τ. | CONNECTION: Fibonacci anyons directly encode the golden ratio φ in their quantum dimension — braiding statistics are governed by the golden ratio; the braid group B_n is isomorphic to the mapping class group of the punctured disk, whose moduli space relates to the modular group PSL(2,ℤ) (base-60 heritage); Majorana zero modes on a 1D nanowire fo 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 braiding operations, with Microsoft's Majorana 1 chip claiming topological qubit hardware. | MATH: Braid group B_n generators σ_i satisfy σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} and σ_i σ_j = σ_j σ_i for |i−j|≥2; non-Abelian anyons yield unitary representations U(σ_i) with U(σ_i)U(σ_{i+1})U(σ_i) ≠ U(σ_{i+1})U(σ_i)U(σ_{i+1}); Majorana zero modes obey γ_i† = γ_i, {γ_i, γ_j} = 2δ_ij; topological qubit encoded in parity (−1)^{n} = i^{2n} γ_1 γ_2 … γ_{2n}; Fibonacci anyons have quantum dimension φ = (1+√5)/2 ≈ 1.618, fusion rules τ×τ = 1 + τ. | CONNECTION: Fibonacci anyons directly encode the golden ratio φ in their quantum dimension — braiding statistics are governed by the golden ratio; the braid group B_n is isomorphic to the mapping class group of the punctured disk, whose moduli space relates to the modular group PSL(2,ℤ) (base-60 heritage); Majorana zero modes on a 1D nanowire fo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 4, 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|>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}

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

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