Topological Qubits via Non-Abelian Anyons: Majorana Braiding and Microsoft's Majorana 1 — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 24, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.22075664

Related papers

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

Andrew Stewart Caldin

Sep 4, 20260 citations

Topological Qubits via Non-Abelian Anyons: Braiding for Error-Resistant Quantum Computing — E8 Intelligence Research

Andrew Stewart Caldin

Sep 24, 20260 citations

Topological Quantum Computing: Braiding Anyons and Majorana Zero Modes — E8 Intelligence Research

Andrew Stewart Caldin

Aug 24, 20260 citations

Topological Quantum Computing: Braid Groups, Majorana Modes, and Microsoft's Majorana 1 Chip — E8 Intelligence Research

Andrew Stewart Caldin

Sep 10, 20260 citations

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

Andrew Stewart Caldin

Sep 8, 20260 citations

Majorana 1: Topological Qubits via Non-Abelian Braiding, Evidence Contested — E8 Intelligence Research

Andrew Stewart Caldin

Sep 24, 20260 citations

Topological Qubits via Non-Abelian Anyons: Majorana Braiding and Microsoft's Majorana 1 — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 24, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.22075664

Related papers

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

Topological Qubits via Non-Abelian Anyons: Braiding for Error-Resistant Quantum Computing — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological quantum computing leverages anyonic braiding statistics — non-Abelian exchange phases — to encode error-resistant qubits, with Microsoft's Majorana 1 chip claiming topological qubit hardware based on Majorana zero modes. | 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 matrices 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}) — the Fibonacci anyon gives U(σ_i) acting on a 2D Hilbert space with golden-ratio dimension d = φ = (1+√5)/2 ≈ 1.618. Majorana zero modes satisfy γ_i† = γ_i, {γ_i, γ_j} = 2δ_ij, and a pair encodes a qubit with parity p = iγ_1γ_2 = ±1. | CONNECTION: The Fibonacci anyon's fusion space dimension grows as Fibonacci numbers F_n, whose ratio F_{n+1}/F_n → φ = 1.618. The braid group's relation to the Temperley–Lieb algebra yields Jones polynomials evaluated at q = e^{iπ/5} (or q = e^{2πi/5}), linking to the golden ratio and the 5-fol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological quantum computing leverages anyonic braiding statistics — non-Abelian exchange phases — to encode error-resistant qubits, with Microsoft's Majorana 1 chip claiming topological qubit hardware based on Majorana zero modes. | 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 matrices 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}) — the Fibonacci anyon gives U(σ_i) acting on a 2D Hilbert space with golden-ratio dimension d = φ = (1+√5)/2 ≈ 1.618. Majorana zero modes satisfy γ_i† = γ_i, {γ_i, γ_j} = 2δ_ij, and a pair encodes a qubit with parity p = iγ_1γ_2 = ±1. | CONNECTION: The Fibonacci anyon's fusion space dimension grows as Fibonacci numbers F_n, whose ratio F_{n+1}/F_n → φ = 1.618. The braid group's relation to the Temperley–Lieb algebra yields Jones polynomials evaluated at q = e^{iπ/5} (or q = e^{2πi/5}), linking to the golden ratio and the 5-fol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 24, 20260 citations

Topological Quantum Computing: Braiding Anyons and Majorana Zero 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 hardware realization. | 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; unitary braid matrices yield Fibonacci anyon fusion rules: τ × τ = 1 + τ (golden chain), with quantum dimension φ = (1+√5)/2 ≈ 1.618; Majorana zero modes obey γ_i† = γ_i, {γ_i, γ_j} = 2δ_ij; topological degeneracy scales as 2^(N/2) for N Majoranas. | CONNECTION: **Direct golden ratio link** — Fibonacci anyons have quantum dimension φ = 1.618, and braid matrices approximate single-qubit rotations with error ~1/φ^2 ≈ 0.382 (the golden angle complement). The fusion space dimension grows as Fibonacci numbers F_n, whose ratio F_{n+1}/F_n → φ. Braid group B_3 is isomorphic to the modular group PSL(2,Z), linking to elliptic curves and base-60 (Babylonian sexagesimal) angular systems v 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 hardware realization. | 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; unitary braid matrices yield Fibonacci anyon fusion rules: τ × τ = 1 + τ (golden chain), with quantum dimension φ = (1+√5)/2 ≈ 1.618; Majorana zero modes obey γ_i† = γ_i, {γ_i, γ_j} = 2δ_ij; topological degeneracy scales as 2^(N/2) for N Majoranas. | CONNECTION: **Direct golden ratio link** — Fibonacci anyons have quantum dimension φ = 1.618, and braid matrices approximate single-qubit rotations with error ~1/φ^2 ≈ 0.382 (the golden angle complement). The fusion space dimension grows as Fibonacci numbers F_n, whose ratio F_{n+1}/F_n → φ. Braid group B_3 is isomorphic to the modular group PSL(2,Z), linking to elliptic curves and base-60 (Babylonian sexagesimal) angular systems v Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 24, 20260 citations

Topological Quantum Computing: Braid Groups, Majorana Modes, and Microsoft's Majorana 1 Chip — E8 Intelligence Research

Andrew Stewart Caldin

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

Sep 10, 20260 citations

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

Andrew Stewart Caldin

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.comFINDING: 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

Sep 8, 20260 citations

Majorana 1: Topological Qubits via Non-Abelian Braiding, Evidence Contested — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Microsoft's Majorana 1 chip claims topological qubits via Majorana zero modes, but peer-reviewed evidence remains contested; core math is non-Abelian anyon braiding. | MATH: Topological qubit = ground-state degeneracy of a 1D p-wave superconductor (Kitaev chain): H = -μ Σ n_j - t Σ (c†_j c_{j+1} + h.c.) + Δ Σ (c_j c_{j+1} + h.c.). Majorana operators γ_j = c_j + c†_j, γ_{j+2} = i(c_{j+1} - c†_{j+1}); braiding yields unitary transformations on degenerate space — non-Abelian statistics encoded in the braid group B_n, with Fibonacci anyons giving dense quantum gates. | CONNECTION: Majorana zero modes are tied to Ising anyons — their fusion rules (σ × σ = 1 + ψ) mirror the golden ratio structure: quantum dimension d_σ = √2, but Fibonacci anyons (d_τ = φ = 1.618) are the universal gate set. The braid group B_n is generated by exchanges σ_i satisfying σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} — a relation isomorphic to the Yang-Baxter equation, which itself appears in 2D crystallographic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Microsoft's Majorana 1 chip claims topological qubits via Majorana zero modes, but peer-reviewed evidence remains contested; core math is non-Abelian anyon braiding. | MATH: Topological qubit = ground-state degeneracy of a 1D p-wave superconductor (Kitaev chain): H = -μ Σ n_j - t Σ (c†_j c_{j+1} + h.c.) + Δ Σ (c_j c_{j+1} + h.c.). Majorana operators γ_j = c_j + c†_j, γ_{j+2} = i(c_{j+1} - c†_{j+1}); braiding yields unitary transformations on degenerate space — non-Abelian statistics encoded in the braid group B_n, with Fibonacci anyons giving dense quantum gates. | CONNECTION: Majorana zero modes are tied to Ising anyons — their fusion rules (σ × σ = 1 + ψ) mirror the golden ratio structure: quantum dimension d_σ = √2, but Fibonacci anyons (d_τ = φ = 1.618) are the universal gate set. The braid group B_n is generated by exchanges σ_i satisfying σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} — a relation isomorphic to the Yang-Baxter equation, which itself appears in 2D crystallographic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 24, 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