Conjectured Bounds for 2-Local Hamiltonians via Token Graphs

TL;DR

This paper explores bounds on quantum Hamiltonians using token graphs, proposing conjectures that improve approximation ratios and provide combinatorial bounds, with proofs for bipartite cases.

Contribution

It introduces conjectured bounds for spectral radii of token graphs related to quantum Hamiltonians, enhancing analysis of approximation algorithms.

Findings

Conjectured bounds tighten existing algorithm analyses.

Proven combinatorial bounds for bipartite graphs.

Numerical evidence supports new spectral radius bounds.

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.