Product-State Approximation Algorithms for the Transverse Field Ising Model

TL;DR

This paper develops classical polynomial-time algorithms to approximate the ground state energy of the transverse-field Ising model, achieving ratios up to approximately 0.816, and establishes bounds on product state approximations.

Contribution

It introduces new approximation algorithms with improved ratios for the TFIM and provides bounds on the best possible product state approximations.

Findings

Approximation ratios of 0.71, 0.786, and 0.816 achieved.

An explicit example shows product states can achieve at most 0.9389 of the true optimum.

Provides upper bounds for product state approximations in ferromagnetic cases.

Abstract

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.