Analysis of the sine-kernel Gaussian operator A_G

Determine the classical value and the semidefinite-program (quantum) value, and thus the integrality gap, of the Gaussian integral operator A_G on functions from R^n to R with kernel A_G(X,Y) = exp(-||X||_2^2/2 - ||Y||_2^2/2) sin(<X, Y>), in order to analyze whether this oscillatory instance constitutes a hard case for the Grothendieck problem.

Background

The paper motivates moving beyond the Davie–Reeds Hermite projection game by introducing oscillations in the dependence on the angle between Gaussian inputs. An operator with sine kernel, A_G, is proposed as a promising candidate that alternates signs across Hermite degrees and could produce a larger integrality gap.

Unlike pure Hermite projection games, A_G includes off-diagonal Hermite terms, making its analysis nontrivial. Establishing its classical and SDP (quantum) values would clarify its potential to yield stronger lower bounds on K_G.

References

The work of König suggests a concrete instance of this type, although its analysis remains an open problem. The candidate instance is an operator on functions $\Rn \to \R$ whose ``entries'' are (more precisely, the kernel function of the operator is)

A_G(X,Y) = e{-\norm{X}_22/2 - \norm{Y}_22/2}\sin \ip{X, Y}\,.

— The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound  (2603.30039 - Jones et al., 31 Mar 2026) in Subsection "What are hard instances for the Grothendieck problem?"