Predicting Ground State Properties: Constant Sample Complexity and Deep Learning Algorithms

A fundamental problem in quantum many-body physics is that of finding ground states of localHamiltonians. A number of recent works gave provably efficient machine learning (ML) algorithmsfor learning ground states. Specifically, [Huang et al. Science 2022], introduced an approach for learningproperties of the ground state of an $n$-qubit gapped local Hamiltonian $H$ from only $n^{\mathcal{O}(1)}$ datapoints sampled from Hamiltonians in the same phase of matter. This was subsequently improvedby [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, independentof system size $n$, for learning ground state properties. Our first algorithm consists of a simplemodification 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 ofthe property is not known, is a deep neural network model. While empirical results showing theperformance of neural networks have been demonstrated, to our knowledge, this is the first rigoroussample 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.