A Refined Algorithm For the EPR model

TL;DR

This paper refines algorithms for the EPR model, improving approximation ratios for regular graphs and ensuring good performance on irregular graphs by optimizing fractional matchings to distribute quantum entanglement.

Contribution

It introduces a refined algorithm using homogeneous/quasi-homogeneous fractional matchings to enhance approximation ratios in the EPR model.

Findings

Approximation ratio for regular graphs improved to about 0.872.

Refinement guarantees good performance on irregular graphs with proper fractional matchings.

Enhanced distribution of quantum entanglement through optimized fractional matchings.

Abstract

The Einstein-Podolsky-Rosen~(EPR) model is an analogous model of the anti-ferromagnetic Heisenberg model or the equivalent quantum maximum-cut problem, proposed by R. King two years ago. Adjacent qubits in the model prefer symmetric EPR/Bell parings rather than the antisymmetric one, in order to maximize the energy. Recently, two groups independently develop specific algorithms for the highest-energy state with approximation ratio $\frac{1+\sqrt{5}}{4}\approx.809$, based on maximum fractional matchings. Here we try to refine one of the two algorithms by devising homogeneous/quasi-homogeneous fractional matchings, with the aim to distribute quantum entanglement as much as possible. For regular graphs $G_d$, we immediately obtain increasing approximation ratios $r_d$ with $r_2=\frac{3+\sqrt{5}}{6}\approx.872$. For irregular graphs, we show such a refinement could still guarantee nice performance if the fractional matchings are chosen properly.