Approximating Quantum Gates using Quaternion Shells: Covering Exponents and Arithmetic Tails

Abstract

Efficient single-qubit compilation seeks short words in a universal gate set that approximate arbitrary elements of (\mathrm{SU}(2)). Via (\mathrm{SU}(2)\cong S^3), this becomes a covering problem on the three-sphere. We study the classical (p=5) Lubotzky--Phillips--Sarnak quaternionic construction and its norm shells [ P_k={x/5^k\in S^3:x\in\mathbb Z^4,\ |x|^2=5^{2k}}. ] Let (K(T)) denote the covering exponent of the associated gate set. The known deterministic range is (4/3\le K(T)\le2). We show that any positive cap-kernel argument using only the Deligne--LPS square-root spectral estimate cannot improve the exponent (2). Consequently, any deterministic improvement requires arithmetic information beyond the standard positivity framework. We also record the sharp conditional benchmark: the twisted Linnik conjecture of Browning, Kumaraswamy, and Steiner implies (K(T)=4/3), matching Harman's lower bound. In addition, we prove that a shell covering estimate (ρ(P_k)\le C5^{-αk}) implies (K(T)\le4/(3α)), making (α>2/3) the threshold for improving the unconditional bound. Finally, exact enumeration of (P_1,P_2,P_3,P_4) and Haar-random experiments show that typical shell errors follow the geometric (N^{-2/3}) scale expected for well-distributed points on (S^3), while the upper tail remains substantially larger. The results indicate that typical approximation is governed by geometry, whereas the deterministic covering exponent is governed by rare arithmetic holes.