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 $γ\approx 0.71$ , (ii) a strengthened rounding, inspired by the anticommutation property of the two $X_i, Z_iZ_j$ observables achieving ratio $γ\approx 0.7860$, and (iii) a further improvement by interpolation achieving ratio $γ\approx 0.8156$. 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.

翻译：我们研究了横向场伊辛模型哈密顿量的经典多项式时间近似算法，该模型允许在量子比特对之间混合铁磁和反铁磁相互作用，同时包含具有任意非负权重的横向场项。我们的主要结果是一系列近似算法（所有近似比均相对于真实的量子最优值）：(i) 一个简单的两种产品态舍入取最大值算法，达到近似比 $γ\approx 0.71$；(ii) 一种受两个可观测量 $X_i, Z_iZ_j$ 的反交换性质启发的强化舍入方法，达到近似比 $γ\approx 0.7860$；(iii) 通过插值实现的进一步改进，达到近似比 $γ\approx 0.8156$。我们还给出了一个三量子比特的显式（纯铁磁）TFIM 实例，其中每个产品态最多只能达到真实最优值的 $169/180\approx 0.9389$，这为所有生成产品态近似的算法提供了一个上界，即使在纯铁磁情况下也是如此。