Statistical-mechanical analysis of compressed sensing for Hamiltonian estimation of Ising spin glass

TL;DR

This paper introduces a compressed sensing-based method for estimating Ising model parameters in Hamiltonian systems, analyzing its typical performance with the replica method and validating through numerical experiments.

Contribution

It proposes a novel $L_1$-norm minimization approach for Hamiltonian estimation and provides a theoretical performance analysis using the replica method.

Findings

The method effectively estimates Ising Hamiltonians from limited data.

Analytical results match numerical experiments.

Performance depends on the problem's sparsity and data quality.

Abstract

Several powerful machines, such as the D-Wave 2000Q, dedicated to solving combinatorial optimization problems through the Ising-model formulation have been developed. To input problems into the machines, the unknown parameters on the Ising model must be determined, and this is necessarily a nontrivial task. It could be beneficial to construct a method to estimate the parameters of the Ising model from several pairs of values of the energy and spin configurations. In the present paper, we propose a simple method employing the $L_1$-norm minimization, which is based on the concept of the compressed sensing. Moreover, we analyze the typical performance of our proposed method of the Hamiltonian estimation by using the replica method. We also compare our analytical results through several numerical experiments using the alternating direction method of multipliers.