  A fundamental problem in quantum many-body physics is that of finding ground
states of local Hamiltonians. A number of recent works gave provably efficient
machine learning (ML) algorithms for learning ground states. Specifically,
[Huang et al. Science 2022], introduced an approach for learning properties of
the ground state of an $n$-qubit gapped local Hamiltonian $H$ from only
$n^{\mathcal{O}(1)}$ data points sampled from Hamiltonians in the same phase of
matter. This was subsequently improved by [Lewis et al. Nature Communications
2024], to $\mathcal{O}(\log n)$ samples when the geometry of the $n$-qubit
system is known. In this work, we introduce two approaches that achieve a
constant sample complexity, independent of system size $n$, for learning ground
state properties. Our first algorithm consists of a simple modification of the
ML model used by Lewis et al. and applies to a property of interest known
beforehand. Our second algorithm, which applies even if a description of the
property is not known, is a deep neural network model. While empirical results
showing the performance of neural networks have been demonstrated, to our
knowledge, this is the first rigorous sample complexity bound on a neural
network model for predicting ground state properties. We also perform numerical
experiments that confirm the improved scaling of our approach compared to
earlier results.
