Conjectured Bounds for 2-Local Hamiltonians via Token Graphs

Abstract

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.

arXiv comment47 pages, 2 figures v2: updated with Basu et al's proof of conjectures and added separation oracle to convex relaxations v3: updated with refutation of Conjecture 7