Conjectured Bounds for 2-Local Hamiltonians via Token Graphs

2025-06-03

AI Summary

This work establishes an exact correspondence between the maximum energy of 2-local quantum Hamiltonians—Quantum MaxCut, XY, and EPR—and the spectral radius of the token graph of a graph (G). It is the first systematic study to uncover the unified spectral-graph-theoretic nature underlying the extremal energies of these three Hamiltonian classes. The authors formulate a verifiable conjecture on upper bounds for the spectral radius, from which they derive a concise combinatorial upper bound on the ground-state energy of the antiferromagnetic Heisenberg model. Methodologically, the approach integrates spectral graph theory, quantum many-body modeling, numerical validation, and approximation algorithm analysis. Key theoretical contributions include: (i) a rigorous proof of the bound for bipartite graphs; (ii) substantial improvements in approximation ratios for all three problems, achieving the current best-known guarantees; and (iii) a novel graph-structural analytical framework for quantum constraint satisfaction problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Security and Privacy: Applications of cryptography

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.

Problem

• Relate maximum energy of Hamiltonians to token graph spectral radii
• Conjecture new spectral radius bounds based on graph properties
• Provide combinatorial bounds for antiferromagnetic Heisenberg model

Innovation

• Relate Hamiltonians to token graph spectral radii
• Conjecture new bounds from graph properties
• Tighten analysis for improved approximation ratios