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A 0.8395-approximation algorithm for the EPR problem

Anuj Apte, Eunou Lee, Kunal Marwaha, Ojas Parekh, Lennart Sinjorgo, James Sud

TL;DR

The paper advances approximations for the EPR Hamiltonian on graphs by developing a nonlinear monogamy-of-entanglement bound on star graphs and coupling it with a refined, shallow quantum circuit parameterization. It achieves a new efficient $\alpha$-approximation with $\alpha>0.8395$ by solving a level-2 quantum moment-SOS relaxation and deterministically mapping SDP outputs to circuit angles via a monotone function class. A rigorous analysis decomposes into three edge-cases, producing explicit per-edge bounds and culminating in a provable global approximation ratio exceeding 0.8395. The authors also establish limits showing that current MoE-based SDP approaches cannot substantially surpass this value, and that further progress would require fundamentally new techniques or ansatzes.

Abstract

We give an efficient 0.8395-approximation algorithm for the EPR Hamiltonian. Our improvement comes from a new nonlinear monogamy-of-entanglement bound on star graphs and a refined parameterization of a shallow quantum circuit from previous works. We also prove limitations showing that current methods cannot achieve substantially better approximation ratios, indicating that further progress will require fundamentally new techniques.

A 0.8395-approximation algorithm for the EPR problem

TL;DR

The paper advances approximations for the EPR Hamiltonian on graphs by developing a nonlinear monogamy-of-entanglement bound on star graphs and coupling it with a refined, shallow quantum circuit parameterization. It achieves a new efficient -approximation with by solving a level-2 quantum moment-SOS relaxation and deterministically mapping SDP outputs to circuit angles via a monotone function class. A rigorous analysis decomposes into three edge-cases, producing explicit per-edge bounds and culminating in a provable global approximation ratio exceeding 0.8395. The authors also establish limits showing that current MoE-based SDP approaches cannot substantially surpass this value, and that further progress would require fundamentally new techniques or ansatzes.

Abstract

We give an efficient 0.8395-approximation algorithm for the EPR Hamiltonian. Our improvement comes from a new nonlinear monogamy-of-entanglement bound on star graphs and a refined parameterization of a shallow quantum circuit from previous works. We also prove limitations showing that current methods cannot achieve substantially better approximation ratios, indicating that further progress will require fundamentally new techniques.

Paper Structure

This paper contains 23 sections, 10 theorems, 112 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Algorithm
  3. Techniques
  4. Semidefinite relaxation of the EPR problem
  5. Defining the relaxation
  6. Monogamy of entanglement
  7. Analysis
  8. Case 1:
  9. Case 2:
  10. Case 3:
  11. Limits on approximability
  12. Limitations from analysis
  13. Limitations from ansatz
  14. Omitted proofs
  15. Proof of QMC and EPR bipartite equivalence
  16. ...and 8 more sections

Key Result

Theorem 1

There is an efficient $\alpha$-approximation algorithm for the EPR problem for $\alpha > 0.8395$.

Figures (1)

  • Figure 1: Values of the functions $r_1$, $r_2$ and $r_3$ as defined in \ref{['eqn:case1', 'eqn:case2', 'eqn_r3Def']} respectively.

Theorems & Definitions (30)

  • Theorem 1
  • Lemma 1: king2023
  • Lemma 2: Nonlinear monogamy of entanglement on a star
  • Definition 1
  • Lemma 3
  • Corollary 1
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 20 more