Predicting Ground State Properties: Constant Sample Complexity and Deep Learning Algorithms
Keywords: Deep Learning, Learning Theory, Quantum Many-Body Problems
Abstract: 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.
Primary Area: Machine learning for physical sciences (for example: climate, physics)
Submission Number: 10012
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