Fibonacci Anyons Enable Universal Quantum Computation via Icosahedral Symmetry — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 23, 2026• 0 citations

Abstract

FINDING: Fibonacci anyons realize non-Abelian braiding statistics governed by the Fibonacci fusion rule τ × τ = 1 + τ, enabling universal topological quantum computation through the D5 (icosahedral) point group symmetry. MATH: - Fusion rule: τ ⊗ τ = 1 ⊕ τ, where τ is the non-Abelian anyon with quantum dimension φ = (1+√5)/2 ≈ 1.618. - Braiding matrices satisfy the Yang-Baxter equation and are representations of the braid group B₃, with entries involving φ and its inverse φ⁻¹ = φ - 1 ≈ 0.618. - The Hilbert space dimension for n anyons grows as F₂ₙ (Fibonacci numbers): dim(Hₙ) = F₂ₙ₊₁, where F₀=0, F₁=1. - The D5 point group (order 10) is the symmetry group of the pentagon, isomorphic to the dihedral group D₅, and its irreducible representations include a 2D representation with character values involving 2cos(π/5)=φ and 2cos(2π/5)=φ⁻¹. CONNECTION: - Golden ratio φ = 1.618 appears as the quantum dimension of the τ anyon and as the scaling factor in the braid group representatio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21503638

Related papers

Fibonacci Anyons: Universal Quantum Computation via Golden Ratio Braiding — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fibonacci anyons carry quantum dimension φ = (1+√5)/2 ≈ 1.618, enabling universal topological quantum computation via braiding alone; recent Quantinuum experiment claims first non-abelian anyon braiding gate. | MATH: Quantum dimension of Fibonacci anyon: d = φ = 2cos(π/5) = (1+√5)/2 ≈ 1.618; satisfies d² = d + 1. Fusion rule: τ ⊗ τ = 1 ⊕ τ (where τ is the anyon). Braid matrices generate a dense subset of SU(2) — specifically, the Yang-Baxter R-matrix eigenvalues are {e^{±4πi/5}, e^{±2πi/5}} — yielding universal gates. Ising anyons: d = √2, fusion τ⊗τ = 1 ⊕ ψ (non-universal, only Clifford gates). | CONNECTION: φ = 2cos(π/5) ties directly to pentagonal symmetry (crystallographic point group 5, root system of H₂ — the icosahedral/dodecahedral symmetry group). The golden ratio appears as the quantum dimension because the Fibonacci fusion category is the simplest non-abelian category with a self-dual object whose fusion matrix has spectral radius φ. The braid group B₃ maps onto the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 18, 20260 citations

Fibonacci Anyons Realize Universal Topological Quantum Computation via Golden Ratio Braiding — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fibonacci anyons carry quantum dimension φ = (1+√5)/2 ≈ 1.618, enabling universal topological quantum computation via braiding — now experimentally realized in non-Abelian anyon systems (Quantinuum, 2025). | MATH: Quantum dimension d = φ satisfies d² = d + 1 (golden ratio identity). Braid group Bₙ on Fibonacci anyons yields dense unitary representations on the fusion space — universality proven (Freedman–Kitaev–Wang). Single-qubit braiding gates compiled via algebraic decomposition of SU(2) into Fibonacci braid words (arXiv:2008.03542). | CONNECTION: φ = 1.618 = 2cos(π/5) — directly tied to pentagonal symmetry, the root system of the icosahedral group H₃, and the 5-fold crystallographic (quasicrystal) forbidden symmetry. The Fibonacci fusion rules (τ × τ = 1 + τ) mirror the golden ratio recurrence Fₙ₊₁ = Fₙ + Fₙ₋₁. | DEPTH: 9 — This is not numerology; φ emerges as the unique algebraic solution to the consistency equations (pentagon/hexagon identities) for a non-Abelian anyon m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 18, 20260 citations

Fibonacci Anyons: Universal Quantum Gates via Non-Abelian Braiding — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological quantum computation via non-Abelian anyon braiding, specifically Fibonacci anyons, encodes qubits in degenerate state spaces where braid operations generate the Fibonacci fusion rules and yield universal quantum gates. | MATH: Fibonacci anyon fusion rule: τ ⊗ τ = 1 ⊕ τ (golden anyon). Dimension of Hilbert space for n anyons grows as Fibonacci numbers: dim(H_n) = F_{n-1} (with F_0=0, F_1=1). Braid generators σ_i satisfy Yang-Baxter: 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. Single-qubit braiding gates compiled via brute-force optimization (arXiv:2008.03542) — the key constant is the golden ratio φ = (1+√5)/2 ≈ 1.618, which appears as the quantum dimension d_τ = φ. | CONNECTION: Direct geometric harmony: Fibonacci anyons have quantum dimension exactly φ = 1.618 (and its inverse 0.618 = 1/φ). The fusion space dimension sequence (1,1,2,3,5,8,...) is the Fibonacci sequence — the same ratio governing pentagonal symmetry (5-fold crystallogr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 5, 20260 citations

Fibonacci Anyons: Universal Topological Quantum Computation via Braiding — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fibonacci anyons encode quantum information via braiding, with quantum dimension φ = (1+√5)/2 ≈ 1.618, enabling universal topological quantum computation; recent Quantinuum hardware demonstrates non-Abelian anyon braiding gates. | MATH: Fibonacci anyon fusion rule: τ ⊗ τ = 1 ⊕ τ (single non-trivial particle type τ); quantum dimension d_τ satisfies d_τ² = d_τ + 1 → d_τ = φ = (1+√5)/2 ≈ 1.6180339887; related constants: 1/φ = φ−1 = 0.6180339887, φ² = φ+1 = 2.6180339887, φ⁻² = 2−φ = 0.3819660113; braid group representations yield unitary matrices with entries in ℚ(√5); single-qubit gate compilation (arXiv:2008.03542) uses Fibonacci anyon braid words approximating Clifford+T-like gates to precision ε with O(log(1/ε)) overhead. | CONNECTION: The quantum dimension φ is the golden ratio itself — the same constant governing pentagonal symmetry (diagonal/edge ratio of regular pentagon), 5-fold crystallographic (forbidden in periodic lattices but allowed in quasicrystals), and the Fibona Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 18, 20260 citations

Fibonacci Anyons: Golden Ratio Quantum Dimension Enables Universal Topological Computation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fibonacci anyons realize the golden ratio φ as the quantum dimension, enabling universal topological quantum computation via braiding; Ising anyons (with √2 quantum dimension) are non-universal for computation. | MATH: Fibonacci anyon fusion rule: τ × τ = 1 + τ (where τ is the non-Abelian anyon, 1 is vacuum). Quantum dimension d_τ satisfies d_τ² = 1 + d_τ → d_τ = φ = (1+√5)/2 ≈ 1.618. Braid matrices yield unitary representations of the braid group B_n with entries involving φ⁻¹ = φ−1 ≈ 0.618 and φ⁻² ≈ 0.382. Ising anyon fusion: σ × σ = 1 + ψ (ψ = fermion), quantum dimension d_σ = √2; braiding yields Clifford gates only (not universal). Fibonacci anyons are dense in SU(2) (or SU(N) for higher rank), hence universal. | CONNECTION: The golden ratio φ appears directly as the quantum dimension — a topological invariant counting fusion-space growth. The Fibonacci fusion tree is isomorphic to the Fibonacci sequence: number of fusion channels for n anyons = F_n (Fibonacci numbers), wh Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 20, 20260 citations

Fibonacci Anyons at Level k=3 Enable Universal Quantum Computation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The braiding of SU(2)_k anyons at level k=3 yields Fibonacci anyons, whose braid group representations are dense in SU(2) — enabling universal quantum computation via non-Abelian statistics. | MATH: Fusion rule: [1]×[1] = [0]+[1] (Fibonacci). Level k=3 ⇒ quantum dimension φ = (1+√5)/2 ≈ 1.618. Braid matrices: σ₁ = e^{i4π/5}·[[1,0],[0,e^{iπ/5}]] (acting on 2-dim Fibonacci space), with σ₁σ₂σ₁ = σ₂σ₁σ₂ (Yang-Baxter). Chern-Simons action: S = (k/4π)∫Tr(A∧dA + (2/3)A∧A∧A). Wilson loop expectation: ⟨W_R⟩ = S_{R0}/S_{00} with S-matrix entries involving sin(π(2j+1)(2j'+1)/(k+2)). | CONNECTION: **Direct golden ratio emergence**: quantum dimension φ = 2cos(π/5) = 1.618. The level k=3 corresponds to q = e^{iπ/5}, whose 5th roots of unity structure links to pentagonal symmetry (crystallographic point group 5). The S-matrix for SU(2)_3 contains entries proportional to sin(π/5), sin(2π/5) — ratios yielding φ and 1/φ. The Fibonacci fusion graph is the golden-ratio tiling (Penrose-like) in 1D Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 19, 20260 citations