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Improved approximation algorithms for the EPR Hamiltonian

Nathan Ju, Ansh Nagda

TL;DR

The paper addresses the problem of approximating the ground energy of the EPR Hamiltonian, a 2-local quantum Hamiltonian with ferromagnetic EPR terms. It introduces a polynomial-time LP-based rounding framework that binds the ground energy via the star bound, and designs quantum-rounding constructions to convert LP solutions into high-energy states. The authors achieve a new global approximation ratio of $\frac{1+\sqrt{5}}{4}$ (≈0.809) for the EPR problem, with the same bound extending to bipartite Quantum Max Cut, thereby improving prior results of $0.72$ and $3/4$ in the respective settings. The approach combines half-integrality of the fractional matching LP, cycle-state constructions for odd cycles, and tilted-EPR state roundings, advancing the understanding of approximability for EPR/QMC problems and offering concrete, scalable algorithms for near-optimal energies.

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).

Improved approximation algorithms for the EPR Hamiltonian

TL;DR

The paper addresses the problem of approximating the ground energy of the EPR Hamiltonian, a 2-local quantum Hamiltonian with ferromagnetic EPR terms. It introduces a polynomial-time LP-based rounding framework that binds the ground energy via the star bound, and designs quantum-rounding constructions to convert LP solutions into high-energy states. The authors achieve a new global approximation ratio of (≈0.809) for the EPR problem, with the same bound extending to bipartite Quantum Max Cut, thereby improving prior results of and in the respective settings. The approach combines half-integrality of the fractional matching LP, cycle-state constructions for odd cycles, and tilted-EPR state roundings, advancing the understanding of approximability for EPR/QMC problems and offering concrete, scalable algorithms for near-optimal energies.

Abstract

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

Paper Structure

This paper contains 10 sections, 7 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Quantum Max Cut.
  3. EPR Hamiltonian.
  4. Bipartite QMC.
  5. Algorithm
  6. An upper bound on $\lambda_{\max}(H)$
  7. A $3/4$-approximation in the bipartite case
  8. An improvement in the bipartite case
  9. General Case
  10. Proof of \ref{['lem:cycle-state']}

Key Result

Theorem 1.1

$\alpha^*_{\text{EPR}}\geq \frac{1+\sqrt{5}}{4}\approx 0.809$, i.e., there is a polynomial time $\frac{1+\sqrt{5}}{4}$-approximation algorithm for the EPR problem.

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.1
  • Lemma 2.1
  • Lemma 2.2: edmonds1965maximum
  • Claim 2.3
  • Claim 2.4
  • Claim 2.5
  • Lemma 2.6: lovasz2009matching
  • Lemma 2.7
  • Lemma A.1: Restatement of \ref{['lem:cycle-state']}
  • ...and 1 more