Quadratic-Defect Completions of Spherical $2$-Design Orbits

Abstract

We study how spherical $2$-designs arising from finite group orbits can be completed to spherical $4$-designs by adjoining further orbits, allowing weights in the general theory. For an irreducible real orthogonal $G$-module $W$ with $\mathbb{D}={\mathrm{End}}_G(W)\in \{\mathbb{R},\mathbb{C},\mathbb{H}\}$, we consider the multiplicity-two representation $W\oplus W$ and retain the failure of the $2$-design equation $M^{\ast }M=\frac{1}{2}{I}_2$ as a quadratic defect. When the invariant quartics are determined by the Hermitian Gram matrix, the fourth-moment problem reduces to a mean and covariance condition on these defects. This yields a sharp lower bound for the total weight of the correction orbits; at equality, their normalized defects form a weighted spherical $2$-design in the associated defect space. The quartic condition holds for the multiqubit Clifford groups in every dimension $r\ge 1$, giving an unbounded-dimensional family with a fixed three-dimensional defect space; among equality cases using the minimum number of correction orbits, the defect geometry is always a regular tetrahedron. As a complementary unweighted example, we construct a $378$-point $W(E_6)$-invariant spherical $4$-design in $S^{11}$ and prove that it is sharp among unweighted invariant completions containing a spherical $2$-design orbit.

19 pages; includes verification code