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

Paper in proceeding, 2024

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. in [1], introduced an approach for learning properties of the ground state of an n-qubit gapped local Hamiltonian H from only nO(1) data points sampled from Hamiltonians in the same phase of matter. This was subsequently improved by Lewis et al. in [2], 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. 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 [1, 2].