Provably scalable learning from quantum many-body data via locality

Machine learning (ML) holds great promise for extracting insights from complex quantum many-body data obtained in quantum experiments, potentially solving certain classically intractable problems. However, addressing large-scale problems still requires significant amounts of data that exceed the capabilities of near-term quantum devices. We propose a scalable ML framework, the Geometrically Local Quantum Kernel (GLQK), that efficiently learns from quantum many-body data by leveraging the exponential decay of correlations prevalent in noncritical systems. For learning an unknown polynomial of quantum expectation values, we rigorously prove that the GLQK substantially improves the sample complexity in the system size n over the existing shadow kernel, by constructing a feature space from local quantum information at the correlation length scale. This improvement is particularly notable when each term of the target polynomial involves few local subsystems. Remarkably, for translationally symmetric data, the GLQK achieves constant sample complexity, independent of n. Numerical experiments on two quantum many-body tasks corroborate these theoretical results, demonstrating the high scalability of the GLQK. Beyond the polynomial setting, we further show that it can efficiently learn certain non-polynomial functions. Our framework establishes new avenues for utilizing experimental data to advance the understanding of quantum many-body physics.