Achieving the r2 approximation ratio with magic graph states

Establish whether variation of magic graph states can achieve the approximation ratio r_2=(3+√5)/6≈0.872 for the Einstein–Podolsky–Rosen model.

Background

The paper studies approximation algorithms for the highest-energy state of the Einstein–Podolsky–Rosen (EPR) model, using magic graph states of the form |χ⟩ constructed from commuting two-qubit gates. The authors first show that arbitrary variation of the parameters defining these states cannot yield an approximation ratio larger than r_2=(3+√5)/6≈0.872, as demonstrated by the graph K_{2,2}.

Despite this limitation, the authors report that brute-force variation produces ratios substantially larger than the baseline r_0≈0.809 on many graphs. They therefore conjecture that the optimal guaranteed ratio r_2 may nevertheless be attainable by varying magic graph states. The conjecture is later confirmed for unweighted regular graphs through the paper’s FED algorithm, but it is not established for arbitrary graphs.