Provable and scalable quantum Gaussian processes for quantum learning
Authors: Jonas Jäger, Paolo Braccia, Pablo Bermejo, Manuel G. Algaba, Diego García-Martín, M. Cerezo
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
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.
Bosonic Gaussian unitaries are fundamental building blocks of central continuous-variable quantum technologies such as quantum-optic interferometry and bosonic error-correction schemes. In this work, we present the first time-efficient algorithm for learning bosonic Gaussian unitaries with a rigorous analysis. Our algorithm produces an estimate of the unknown unitary that is accurate to small worst-case error, measured by the physically motivated energy-constrained diamond distance. Its runtime and query complexity scale polynomially with the number of modes, the inverse target accuracy, and natural energy parameters quantifying the allowed input energy and the unitary's output-energy growth. The protocol uses only experimentally friendly photonic resources: coherent and squeezed probes, passive linear optics, and heterodyne/homodyne detection. We then employ an efficient classical post-processing routine that leverages a symplectic regularization step to project matrix estimates onto the symplectic group. In the limit of unbounded input energy, our procedure attains arbitrarily high precision using only 2m+2 queries, where m is the number of modes. To our knowledge, this is the first provably efficient learning algorithm for a multiparameter family of continuous-variable unitaries.
We investigate Gaussian process (GP) bandit optimization with quantum kernels, assuming the mean reward function lies in the reproducing kernel Hilbert space (RKHS) induced by the quantum kernel. This setting is motivated by NISQ-era tasks such as quantum control, state preparation and variational quantum algorithms. While quantum kernels can offer a `quantum advantage' via domain-specific inductive biases, naïvely using full, high-dimensional kernels increases model complexity and information gain, leading to higher cumulative regret and poor learnability. To address this, we propose projected quantum kernels and classical kernel approximation techniques that reduce feature dimensionality while preserving key quantum properties. Using these approximate kernels, we develop misspecified GP bandit algorithms and derive regret bounds that characterize the trade-off between approximation error and information gain. The regret bounds provide principled guidance for selecting the optimal model complexity. Empirically, our methods outperform full quantum kernels in sample efficiency, while substantially reducing computational overhead, enabling scalable GP optimization for quantum-native applications.
Two recent results have reshaped quantum Gaussian processes (QGPs). On the one hand, \citet{lowe2025assessing} rule out the exponential speedups claimed by HHL-based QGP regression in the typical, well-conditioned regime; on the other, an independent line of work shows that highly expressive quantum kernels suffer posterior pathologies that break Bayesian optimization. We show that these seemingly unrelated phenomena are governed by the same quantity: the normalized spectral entropy S(K)/logn of the kernel Gram matrix. We prove a Cauchy--Schwarz tail bound on Nyström approximation error, a finite-sample variance-contraction identity in terms of Bach's degrees of freedom dσ(K), and a characterization of the \emph{target-dependent} optimal entropy via the intrinsic dimension of the target in the kernel eigenbasis. Empirically, the diagnostic is kernel-agnostic: hardware-efficient, matchgate, IQP \emph{and} RBF/Matérn/RFF/deep-kernel families all collapse onto identical S/logn curves on dequantization, ECE, and variance-contraction panels. The NLL sweet spot lives at high entropy for smooth targets and at low entropy for band-limited quantum-data targets. The diagnostic transfers from simulator to IBM Heron hardware with median absolute error 3.2% and mean 5.2% in S/logn across 24 configurations at nq=4, with matchgate and IQP within 5% mean and a single HE configuration returning a 30% outlier that drops to 0.5% on rerun (attributed to calibration drift); the same diagnostic transfers to a second Heron backend (mean error 2.7%) and to a nq=6 scale-up on the original backend (mean error 1.7%). No error mitigation is applied throughout.