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 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^{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 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., Nature Communications 2024] and applies to a property of interest known in advance. 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 on systems of up to 45 qubits that confirm the improved scaling of our approach compared to previous works.