Deep Holes in the Clifford Hierarchy

We determine the covering radius of the topological closure of the single-qubit Clifford hierarchy in $\text{SU}(2)\cong S^3$. This closure is a union of $18$ great circles — the Clifford–Pauli circles — and we prove that its covering radius is $\arccos \sqrt{5\mathrm{/}6}$. The extremal points, which we call \emph{deep holes}, form a single orbit of size $192$ under left and right multiplication by Clifford gates, and are described in closed form. Equivalently, the minimum over one-qubit unitaries of the all-level Clifford fidelity is $5\mathrm{/}6$. The proof rests on two structures attached to the configuration of $18$ planes in ${\mathbb{R}}^4$: their centered rank-two projectors form an orthonormal basis of the irreducible $\text{SO}(4)$-module ${\text{Sym}}_0(4)$, and the projection profile of a unit quaternion is exactly its image under the double cover $\text{SU}(2)\to \text{SO}(3)$. These reduce the covering problem to a minimax statement for the ${\mathrm{\ell }}^{\mathrm{\infty }}$-norm on $\text{SO}(3)$ which we solve exactly, classifying its equality cases.