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arXiv:2608.19306

Quantum Gaussian processes for prediction of channel observations

Jonas Jäger, Yaroslav Khmelnitskiy, Paolo Braccia, Artur Miroszewski, Diego García-Martín, M. Cerezo, Piotr Czarnik

quant-phcs.LGstat.MLSubmitted Aug 19, 2026• Announced Aug 21, 2026• 0 citations
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Abstract

Given a set of input states, we consider the task of predicting the expectation value of a Pauli observable at the output of an unknown quantum evolution, using only a limited number of measurements. Recently, quantum Gaussian process (QGP) regression was introduced for this task across various classes of unitary evolution. Here, we extend the QGP framework beyond unitary dynamics. In particular, we prove convergence of the channel's outputs to a QGP and derive the associated closed-form kernel under a uniform (Lebesgue measure) prior over quantum channels. The kernel's dimensional factor, however, dictates the required observation precision. While manageable when the channel and observable are restricted to small subsystems, exponential suppression precludes learning when the subsystem grows extensively with the system size. Since the Lebesgue prior is overly broad for many applications, we propose an empirical Bayes heuristic that replaces the dimensional factor with a learnable scale parameter while retaining the kernel's state-overlap correlation structure. In numerical simulations of up to 64 qubits, channel QGP regression with the Lebesgue kernel exhibits a strong inductive bias for local channels, enabling faithful extrapolation. For global 64-qubit channels, the rescaled kernel restores learnability, with predictions improving systematically with the shot budget. Results from a noisy quantum computer further demonstrate the robustness of QGP regression under experimental conditions. Beyond regression, we validate QGPs as Bayesian-optimization surrogates for state preparation under noisy XXZ dynamics.

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Given a set of input states, we consider the task of predicting the expectation value of a Pauli observable at the output of an unknown quantum evolution, using only a limited number of measurements. Recently, quantum Gaussian process (QGP) regression was introduced for this task across various classes of unitary evolution. Here, we extend the QGP framework beyond unitary dynamics. In particular, we prove convergence of the channel's outputs to a QGP and derive the associated closed-form kernel under a uniform (Lebesgue measure) prior over quantum channels. The kernel's dimensional factor, however, dictates the required observation precision. While manageable when the channel and observable are restricted to small subsystems, exponential suppression precludes learning when the subsystem grows extensively with the system size. Since the Lebesgue prior is overly broad for many applications, we propose an empirical Bayes heuristic that replaces the dimensional factor with a learnable scale parameter while retaining the kernel's state-overlap correlation structure. In numerical simulations of up to 64 qubits, channel QGP regression with the Lebesgue kernel exhibits a strong inductive bias for local channels, enabling faithful extrapolation. For global 64-qubit channels, the rescaled kernel restores learnability, with predictions improving systematically with the shot budget. Results from a noisy quantum computer further demonstrate the robustness of QGP regression under experimental conditions. Beyond regression, we validate QGPs as Bayesian-optimization surrogates for state preparation under noisy XXZ dynamics.
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.
Quantum kernel estimation on near-term hardware is shot-budgeted: every entry of the kernel Gram matrix is a Bernoulli expectation that must be sampled with a finite number of circuit executions. Recent work on quantum kernel classification has shown that allocating shots non-uniformly across kernel entries, weighted by their downstream task sensitivity, can reduce the shot budget required to reach a target accuracy. We extend this idea to Gaussian process (GP) regression, a setting whose downstream quantities (full-spectrum posterior variance, log-determinant, marginal likelihood) couple to kernel error more tightly than the sign-only outputs of classification. We derive three closed-form pair-level sensitivities predictive coupling ∣αiαj∣|α_iα_j|∣αi​αj​∣, leave-one-out residual, and marginal-likelihood gradient and plug them into a Neyman-style minimum-variance allocation rule. To prevent catastrophic over-concentration when the warm-up sensitivity estimate is itself noisy, we add a high uniform coverage floor justified by a Frobenius lower bound on the missing-entry perturbation. On four UCI benchmarks and two synthetic RBF + Bernoulli controlled studies, the resulting allocator delivers 101010--21%21\%21% test-RMSE improvement over uniform allocation across the moderate-budget regime. The gain transfers (i) to genuine ZZ and Pauli-Z quantum kernels on quantum-natural data (−13-13−13--15%15\%15% at low budget, p<0.05p<0.05p<0.05 paired) and (ii) to four downstream tasks (Bayesian quadrature, heteroscedastic regression, hyperparameter learning, multi-output Cokriging). On UCI features embedded into a ZZ kernel the gain disappears, consistent with the exponential-concentration regime where shot allocation has nothing to exploit.
Classical artificial neural networks initialized from independent and identically distributed priors converge to Gaussian processes in the limit of a large number of neurons per hidden layer. This correspondence plays an important role in the current understanding of the capabilities of neural networks. Here we prove an analogous result for quantum neural networks. We show that the outputs of certain models based on Haar-random unitary or orthogonal quantum neural networks converge to Gaussian processes in the limit of large Hilbert space dimension d. The derivation of this result is more nuanced than in the classical case due to the role played by the input states, the measurement observable and because the entries of unitary matrices are not independent. We show that the efficiency of predicting measurements at the output of a quantum neural network using Gaussian process regression depends on the number of measured qubits. Furthermore, our theorems imply that the concentration of measure phenomenon in Haar-random quantum neural networks is worse than previously thought, because expectation values and gradients concentrate as O(1/e⁡dd){\mathcal{O}}\left({1}/{\operatorname{e}^{d}\sqrt{d}}\right)O(1/edd​) . The connection between classical neural networks and Gaussian processes is a fundamental result in machine learning. It has now been shown that many quantum neural networks converge to Gaussian processes, enabling their use for regression tasks.
Gaussian processes (GPs) are flexible nonparametric Bayesian priors over functions and are widely used in regression, time-series modelling, and Bayesian optimisation. Yet their performance is often bottlenecked by the choice of covariance kernel, especially when the relevant correlations are unknown or inherently non-stationary. In this work, we put forward a NISQ-compatible framework that uses a quantum device as a generator of local quantum kernels for GP learning. At a high level, shallow, geometry-aware randomly parametrized circuits with area-law entanglement encode inputs into many-qubit states, and few-body Pauli measurements-estimated efficiently via classical shadows-produce quantum feature vectors that are subsequently wrapped into classical scalar kernels. A multiplekernel learning layer then fits a convex mixture of these quantum kernels by maximizing the GP marginal likelihood, so that all heavy optimisation remains on the classical side, while the quantum hardware is used solely for kernel estimation. On synthetic regression tasks, we find that the learned mixture can approximate both stationary and non-stationary covariances, achieving high kernel alignment and accurate predictions. Taken together, these results suggest that local quantum kernels can in principle form a family of flexible GP covariance learners within realistic NISQ resource budgets.
Abstract Gaussian process regression is a well-established Bayesian machine learning method. We propose a new approach to Gaussian process regression using quantum kernels based on parameterized quantum circuits. By employing a hardware-efficient feature map and careful regularization of the Gram matrix, we demonstrate that the variance information of the resulting quantum Gaussian process can be preserved. We also show that quantum Gaussian processes can be used as a surrogate model for Bayesian optimization, a task that critically relies on the variance of the surrogate model. To demonstrate the performance of this quantum Bayesian optimization algorithm, we apply it to the hyperparameter optimization of a machine learning model which performs regression on a real-world dataset. We benchmark the quantum Bayesian optimization against its classical counterpart and show that quantum version can match its performance.
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.