Typical-Case Gate Approximation and Arithmetic Obstructions in Quaternionic Single-Qubit Compilation

Abstract

Fault-tolerant quantum computation requires compiling arbitrary one-qubit unitaries into short words over a fixed gate library. For arithmetic libraries such as the $p=5$ Lubotzky--Phillips--Sarnak, or Clifford$+V$, gate set, this problem is governed by quaternionic lattice points on $S^3\cong \SU(2)$. We study the complete norm shells \[ P_k=\{x/5^k\in S^3:x\in\ZZ^4,\ |x|^2=5^{2k}\} \] and the associated projective gate set $T\subset PSU(2)$. The main worst-case quantity is Sarnak's covering exponent $K(T)$, for which the classical range is $4/3\le K(T)\le 2$. We show that any positive localized cap-kernel certificate using only the Deligne--LPS square-root spectral estimate reaches only the volume-squared threshold $|V_T(t)|\gg \hat{I}¼(B(\epsilon ){)}^{-2}$, hence only exponent $2$. Thus any unconditional improvement requires arithmetic cancellation in localized off-diagonal counting. We also record the conditional benchmark that twisted Linnik gives $K(T)=4/3$, matching Harman's obstruction. We prove the shell-to-gate implication $\ddot{I}(P_k)\le C{5}^{-\hat{I}\pm k}\Rightarrow K(T)\le 4/(3\hat{I}\pm )$, so $\hat{I}\pm >2/3$ is exactly the threshold for improving the unconditional bound. Finally, exact enumeration of $P_1,P_2,P_3,P_4$ and Haar-random tests show median trace-defect error at the optimal $N^{-2/3}$ scale, while high quantiles remain separated. This gives a sharp distinction between strong typical-case performance and rare arithmetic holes controlling worst-case synthesis.