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Reconstructing a system Hamiltonian through measurements on its eigenstatesis an important inverse problem in quantum physics. Recently, it was shown thatgeneric many-body local Hamiltonians can be recovered by local measurementswithout knowing the values of the correlation functions. In this work, wediscuss this problem in more depth for different systems and apply thesupervised learning method via neural networks to solve it. For low-lyingeigenstates, the inverse problem is well-posed, neural networks turn out to beefficient and scalable even with a shallow network and a small data set. Formiddle-lying eigenstates, the problem is ill-posed, we present a modifiedmethod based on transfer learning accordingly. Neural networks can alsoefficiently generate appropriate initial points for numerical optimizationbased on the BFGS method.

Supervised learning in Hamiltonian reconstruction from local measurements on eigenstates