Provable and scalable quantum Gaussian processes for quantum learning
Organizations: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA · Department of Computer Science and Institute of Applied Mathematics, University of British Columbia, Vancouver, V6T 1Z4 B.C., Canada · Stewart Blusson Quantum Matter Institute, Vancouver, V6T 1Z4 B.C., Canada · Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545, USA · Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebasti´an, Spain · Department of Applied Physics, Gipuzkoa School of Engineering, University of the Basque Country (UPV/EHU), Plaza Europa 1, 20018 San Sebasti´an, Spain · IQM Quantum Computers, Georg-Brauchle-Ring 23-25, 80992 Munich, Germany · PhD Programme in Condensed Matter Physics, Nanoscience and Biophysics, Doctoral School, Universidad Aut´onoma de Madrid · Department for Quantum Information and Computation at Kepler (QUICK), Johannes Kepler University, Linz, Austria · Los Alamos National Laboratory, Los Alamos, NM 87545, USA · Quantum Science Center, Oak Ridge, TN 37931, USA
Abstract
Despite rapid recent advances in quantum machine learning, the field is in many ways stuck. Existing approaches can exhibit serious limitations, and we still lack learning frameworks that are simple, interpretable, scalable, and naturally suited to quantum data. To address this, here we introduce quantum Gaussian processes, a Bayesian framework for learning from quantum systems through priors over unknown quantum transformations. We show that, under suitable conditions, unitary quantum stochastic processes define Gaussian processes, thereby enabling regression, classification, and Bayesian optimization directly on quantum data. The key ingredient in this framework is sufficient knowledge of a quantum process's structure and symmetries to define an informative prior through its corresponding quantum kernel, effectively injecting a strong, physics-informed inductive bias into the learning model. We then prove that matchgate, or free-fermionic, evolutions give rise to provable and scalable quantum Gaussian processes, providing the first family in our framework where the unknown unitary acts non-trivially on all qubits. Finally, we demonstrate accurate long-range extrapolation, phase-diagram learning in many-body systems, and sample-efficient Bayesian optimization in a quantum sensing task. Our results identify quantum Gaussian processes as a promising route toward simpler and more structured forms of quantum learning.