[논문 리뷰] The Grasshopper Problem on the Sphere

David Llamas, Dmitry Chistikov외 3명|arXiv (Cornell University)|2026. 03. 09.

Quantum Mechanics and Applications인용 수 0

한 줄 요약

The paper analyzes the spherical grasshopper problem under three variants, develops a detailed geometric and spectral framework, and uses numerical optimization to characterize optimal lawn shapes and their relation to spherical harmonics.

ABSTRACT

The spherical grasshopper problem is a geometric optimization problem that arises in the context of Bell inequalities and can be interpreted as identifying the best local hidden variable approximation to quantum singlet correlations for measurements along random axes separated by a fixed angle. In a parallel publication [arXiv:2504.20953], we presented numerical solutions for this problem and explained their significance for singlet simulation and testing. In this companion paper, we describe in detail the geometric and computational framework underlying these results. We examine the role of spherical discretization and compare three natural variants of the problem: antipodal complementary lawns, antipodal independent lawns, and non-antipodal complementary lawns. We analyze the geometric structure of the corresponding optimal lawn configurations and interpret it in terms of a spherical harmonics expansion. We also discuss connections to other physical models and to classical problems in geometric probability.

연구 동기 및 목표

• Motivate and formalize the spherical grasshopper problem as a geometric approach to classical simulations of quantum singlet correlations.
• Compare three variants (antipodal complementary, antipodal independent, non-antipodal complementary) to understand how constraints affect optimal lawn shapes.
• Develop and validate a discretization-based numerical framework to find and analyze optimal lawns across jump angles.
• Provide a spectral (spherical harmonics) interpretation of optimal configurations and relate findings to Bell nonlocality tests.

제안 방법

• Define the grasshopper success probability p(theta) with integral formulations over lawns on S^2.
• Represent lawn shapes via density functions mu_k(r) in {0,1} and expand them in spherical harmonics.
• Use the Funk-Hecke formula to show K_theta is diagonal in spherical harmonics and derive p(theta) in spectral form.
• Discretize the sphere with t-design, HEALPix, Goldberg, and Coulomb grids; approximate deltas with a smooth kernel phi.
• Optimize discrete lawns via simulated annealing and refine with boundary annealing and a greedy maximum finder.
• Relate the discrete model to an Ising-like Hamiltonian with long-range interactions.

Figure 1: Study of discretization effects. The difference between the discrete probability ${P}_{h}(\theta)$ of a hemispherical lawn for different spherical grid types and sizes, and the exact continuous grasshopper probability $p_{h}(\theta)=1-\theta/\pi$ (top panel) and the corresponding absolute

실험 결과

연구 질문

• RQ1 What is the maximal grasshopper success probability p(theta) for each of the three setups as a function of the jump angle theta?
• RQ2 What are the geometric structures (cogs, stripes, labyrinths) of the optimal lawns across theta, and how do they relate to spherical harmonics?
• RQ3 How do antipodal constraints influence optimal shapes and the corresponding spectral coefficients?
• RQ4 How do discretization schemes affect the computed optimal lawns and the resulting p(theta)?
• RQ5 How do the spherical-grasshopper results inform bounds on classical simulations of singlet correlations and potential Bell-type inequalities?

주요 결과

• Optimal lawns exhibit cogwheel, stripe, labyrinth-like, and mixed patterns depending on theta in the antipodal complementary setup.
• In antipodal one-lawn setups, the number of cogs is odd and close to the jump-derived mode 2pi/theta, with higher modes appearing as local maxima.
• Spectral representation shows p(theta) as a sum over odd or all spherical-harmonic modes depending on antipodal constraints, via P_ell(cos theta).
• Discretization studies show t-design grids with N=52,978 points provide high accuracy and reduced grid-alignment biases compared to Goldberg grids; HEALPix with up to 303,372 sites is used for other theta ranges.
• Numerical results indicate hemispherical lawns are typically not optimal; optimal LHV models require more complex lawn shapes than Bell’s hemispherical model.
• The work connects optimal classical approximations to quantum singlet correlations with patterns observed in geometric probability and pattern-forming systems.

Figure 3: Centered histograms of potential energies ( 20 ) across the entire grid for $t$ -design (left panel), HEALPix (middle panel), and Goldberg polyhedron grids (right panel) for the representative jump angle $\theta=0.30\pi$ . The histograms for $t$ -designs and HEALPix grids are more regular,