Quaternion Algebras and McKay Correspondence for Single-Qubit Gate Compilation — E8 Intelligence Research

FINDING: Quaternion algebras over number fields enable explicit compilation of single-qubit gates into Clifford+T or other universal gate sets via the McKay correspondence, linking E8 root system to quantum circuit synthesis. MATH: - Quaternion algebra: \( \mathbb{H}(a,b) = \langle i,j \mid i^2 = a, j^2 = b, ij = -ji \rangle \) over number field \( \mathbb{Q}(\sqrt{5}) \) or cyclotomic fields. - McKay correspondence: For finite subgroups of SU(2) (binary polyhedral groups), the extended Dynkin diagram of E8 appears for the binary icosahedral group of order 120. - Gate synthesis: Single-qubit unitaries approximated to precision \(\epsilon\) require \(O(\log(1/\epsilon))\) gates using quaternion norm equations \(N(\alpha) = 2^k\) for \(\alpha \in \mathbb{Z}[\omega]\) (Eisenstein integers) or \(\mathbb{Z}[i]\). - Constants: Golden ratio \(\phi = (1+\sqrt{5})/2 \approx 1.618\) appears in the icosahedral group's rotation angles; quaternion units satisfy \(i^2 = j^2 = k^2 = ijk = -1 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com