Golden Gates: Solovay-Kitaev, Fibonacci Anyons, and the Golden Ratio — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The Solovay-Kitaev theorem guarantees efficient approximation of any single-qubit unitary using finite gate sets; its optimal implementations ("golden gates") are deeply tied to algebraic number theory and, for Fibonacci anyons, to the golden ratio. | MATH: The theorem states: For any finite set S ⊂ SU(2) generating a dense subgroup, any U ∈ SU(2) can be ε-approximated in O(logc(1/ε)) gates, c ≈ 3.97 (improved to c=1 for certain sets). Exact synthesis for Fibonacci anyons uses the braid group B3 representation where the fusion space dimension grows as Fn (Fibonacci numbers). The key constant: the golden ratio φ = (1+√5)/2 = 1.618..., with its inverse φ−1 = 0.618..., and the related φ−2 = 0.382.... The Fibonacci anyon braid matrices have entries in ℤ[φ], the ring of integers of ℚ(√5). | CONNECTION: Direct and explicit. The Fibonac
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026golden,
title = {{Golden Gates: Solovay-Kitaev, Fibonacci Anyons, and the Golden Ratio — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23464948},
url = {https://zenodo.org/records/23464948},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Gates: Solovay-Kitaev, Fibonacci Anyons, and the Golden Ratio — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23464948
UR - https://zenodo.org/records/23464948
AB - FINDING: The Solovay-Kitaev theorem guarantees efficient approximation of any single-qubit unitary using finite gate sets; its optimal implementations ("golden gates") are deeply tied to algebraic number theory and, for Fibonacci anyons, to the golden ratio. | MATH: The theorem states: For any finite set S ⊂ SU(2) generating a dense subgroup, any U ∈ SU(2) can be ε-approximated in O(log^c(1/ε)) gates, c ≈ 3.97 (improved to c=1 for certain sets). Exact synthesis for Fibonacci anyons uses the braid group B_3 representation where the fusion space dimension grows as F_n (Fibonacci numbers). The key constant: the golden ratio φ = (1+√5)/2 = 1.618..., with its inverse φ^−1 = 0.618..., and the related φ^−2 = 0.382.... The Fibonacci anyon braid matrices have entries in ℤ[φ], the ring of integers of ℚ(√5). | CONNECTION: Direct and explicit. The Fibonac Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -