We explain how the maximum energy of the Quantum MaxCut, XY, and EPR Hamiltonians on a graph $G$ are related to the spectral radii of the token graphs of $G$. From numerical study, we conjecture new bounds for these spectral radii based on properties of $G$. We show how these conjectures tighten the analysis of existing algorithms, implying state-of-the-art approximation ratios for all three Hamiltonians. Our conjectures also provide simple combinatorial bounds on the ground state energy of the antiferromagnetic Heisenberg model, which we prove for bipartite graphs.

翻译：我们阐释了图$G$上的量子MaxCut、XY和EPR哈密顿量的最大能量如何与$G$的token图谱半径相关联。通过数值研究，我们基于$G$的性质提出了这些谱半径的新猜想边界。我们展示了这些猜想如何强化现有算法的分析，从而为所有三种哈密顿量给出了当前最优的近似比。我们的猜想还为反铁磁海森堡模型的基态能量提供了简洁的组合边界，并针对二分图证明了该边界。