Deep Holes in the Clifford Hierarchy

Abstract

We determine the covering radius of the topological closure of the single-qubit Clifford hierarchy in $\SU(2)\cong S^3$. This closure is a union of $18$ great circles --- the Clifford--Pauli circles --- and we prove that its covering radius is $\arccos \sqrt{5/6}$. 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 $5/6$. 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 $\SO(4)$-module $\Sym_0(4)$, and the projection profile of a unit quaternion is exactly its image under the double cover $\SU(2)\to\SO(3)$. These reduce the covering problem to a minimax statement for the ${\ell }^{\infty }$-norm on $\SO(3)$ which we solve exactly, classifying its equality cases.