Topological Quantum Computing: Braid Groups, Majorana Modes, and Microsoft's Majorana 1 Chip — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: 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.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212150683
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