Supervised learning in Hamiltonian reconstruction from local measurements on eigenstates

TL;DR

This paper explores using neural networks for Hamiltonian reconstruction from eigenstate measurements, demonstrating efficiency for low-lying states and adapting methods for middle-lying states, advancing quantum inverse problem solutions.

Contribution

It introduces a supervised learning approach with neural networks for Hamiltonian reconstruction, including modifications for ill-posed cases and applications to different eigenstates.

Findings

Neural networks efficiently reconstruct low-lying Hamiltonians.

Modified transfer learning improves reconstruction for middle-lying eigenstates.

Neural networks assist in generating initial points for numerical optimization.

Abstract

Reconstructing a system Hamiltonian through measurements on its eigenstates is an important inverse problem in quantum physics. Recently, it was shown that generic many-body local Hamiltonians can be recovered by local measurements without knowing the values of the correlation functions. In this work, we discuss this problem in more depth for different systems and apply the supervised learning method via neural networks to solve it. For low-lying eigenstates, the inverse problem is well-posed, neural networks turn out to be efficient and scalable even with a shallow network and a small data set. For middle-lying eigenstates, the problem is ill-posed, we present a modified method based on transfer learning accordingly. Neural networks can also efficiently generate appropriate initial points for numerical optimization based on the BFGS method.