Product-State Approximation Algorithms for the Transverse Field Ising Model

We study classical polynomial-time approximation algorithms for the transverse field Ising model (TFIM), allowing a mixture of ferromagnetic and antiferromagnetic interactions between pairs of qubits, alongside transverse field terms with arbitrary non-negative weights. In this work, we first prove a second-order conic inequality based on the anticommutation property of the two competing terms (Ising ZiZj vs. field Xi terms), and we use this inequality to strengthen the basic SDP relaxation of the problem. By producing two competing rounded product state solutions and taking the better of the two we achieve an approximation ratio ? ˜ 0.7860. A further improvement by non-uniform interpolation achieves a ratio ? ˜ 0.82197. Finally, we give an explicit purely antiferromagnetic TFIM instance on three qubits for which every product state achieves at most 169/180 ˜ 0.9389 of the true optimum, yielding an upper bound for all algorithms producing product state approximations, even in the purely antiferromagnetic case.