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

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 $S^3\cong \text{SU}(2)$. We study the complete norm shells $$P_k=\{x\mathrm{/}{5}^k\in S^3:x\in {\text{ZZ}}^4,\mathrm{\mid }x{\mathrm{\mid }}^2=5^{2k}\}$$ 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 $4\mathrm{/}3\le K(T)\le 2$. We show that any positive localized cap-kernel certificate using only the Deligne–LPS square-root spectral estimate reaches only the volume-squared threshold $\mathrm{\mid }{V}_T(t)\mathrm{\mid }\gg \mu (B(\epsilon ){)}^{-2}$, 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 $K(T)=4\mathrm{/}3$, matching Harman’s obstruction. We prove the shell-to-gate implication $\rho (P_k)\le C{5}^{-\alpha k}\Rightarrow K(T)\le 4\mathrm{/}(3\alpha )$, so $\alpha >2\mathrm{/}3$ 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 $N^{-2\mathrm{/}3}$ scale, while high quantiles remain separated. This gives a sharp distinction between strong typical-case performance and rare arithmetic holes controlling worst-case synthesis.