Product-State Approximation Algorithms for the Transverse Field Ising Model

We study classical polynomial-time approximation algorithms for the transverse-field Ising model (TFIM) Hamiltonian, allowing a mixture of ferromagnetic and anti-ferromagnetic interactions between pairs of qbits, alongside transverse field terms with arbitrary non-negative weights. Our main results are a series of approximation algorithms (all approximation ratios with respect to the true quantum optimum): (i) a simple maximum of two product state rounding algorithm achieving an approximation ratio $\gamma\approx 0.71$ , (ii) a strengthened rounding, inspired by the anticommutation property of the two $X_i, Z_iZ_j$ observables achieving ratio $\gamma\approx 0.7860$, and (iii) a further improvement by interpolation achieving ratio $\gamma \approx 0.8156$. We also give an explicit (purely ferromagnetic) TFIM instance on three qbits for which every product state achieves at most $169/180\approx 0.9389$ of the true optimum, yielding an upper bound for all algorithms producing product state approximations, even in the purely ferromagnetic case.