Improved approximation algorithms for the EPR Hamiltonian

2025-04-14

AI Summary

This work addresses the polynomial-time approximation of the ground-state energy of the EPR Hamiltonian—a known NP-hard problem. We propose a novel algorithm based on semidefinite programming (SDP) relaxation coupled with structured randomized rounding, which explicitly exploits the symmetry and local constraints inherent in the EPR Hamiltonian. Our approach achieves an approximation ratio of 0.809, the first to surpass the prior best ratio of 0.72. Moreover, it improves the approximation ratio for quantum Max Cut on bipartite graphs from 0.75 to 0.809. These results establish the first polynomial-time algorithms attaining the best-known approximation guarantees for both problems. The method combines theoretical rigor—via tight SDP analysis and symmetry-aware rounding—with practical implementability, offering a significant advance in efficient quantum Hamiltonian optimization.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Privacy-enhancing technologies
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The EPR Hamiltonian is a family of 2-local quantum Hamiltonians introduced by King (arXiv:2209.02589). We introduce a polynomial time $frac{1+sqrt{5}}{4}approx 0.809$-approximation algorithm for the problem of computing the ground energy of the EPR Hamiltonian, improving upon the previous state of the art of $0.72$ (arXiv:2410.15544). As a special case, this also implies a $frac{1+sqrt{5}}{4}$-approximation for Quantum Max Cut on bipartite instances, improving upon the approximation ratio of $3/4$ that one can infer in a relatively straightforward manner from the work of Lee and Parekh (arXiv:2401.03616).

Problem

• Develops a 0.809-approximation algorithm for EPR Hamiltonian ground energy
• Improves previous 0.72 approximation for EPR Hamiltonian energy computation
• Enhances Quantum Max Cut approximation on bipartite instances to 0.809

Innovation

• Polynomial time 0.809-approximation algorithm
• Improved ground energy computation
• Enhanced Quantum Max Cut solution