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: \ (H (a, b) = i, j i² = a, j² = b, ij = -ji \) over number field \ (Q (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 \ (\) require \ (O ( (1/) ) \) gates using quaternion norm equations \ (N () = 2ᵏ\) for \ (Z\) (Eisenstein integers) or \ (Zi\). - Constants: Golden ratio \ (= (1+5) /2 1. 618\) appears in the icosahedral group's rotation angles; quaternion units satisfy \ (i² = j² = k² = ijk = -1 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com