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.
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.
Abstract Quantum computing algorithms have been shown to produce performant quantum kernels for machine-learning classification problems. Here, we examine the performance of quantum kernels for regression problems of practical interest. For an unbiased benchmarking of quantum kernels, it is necessary to construct the most optimal functional form of the classical kernels and the most optimal quantum kernels for each given data set. We develop an algorithm that uses an analog of the Bayesian information criterion to optimize the sequence of quantum gates used to estimate quantum kernels for Gaussian process models. The algorithm increases the complexity of the quantum circuits incrementally, while improving the performance of the resulting kernels, and is shown to yield much higher model accuracy with fewer quantum gates than a fixed quantum circuit ansatz. We demonstrate that quantum kernels thus obtained can be used to build accurate models of global potential energy surfaces (PES) for polyatomic molecules. The average interpolation error of the six-dimensional PES obtained with a random distribution of 2000 energy points is 16 cm−1 for H3O+, 15 cm−1 for H2CO and 88 cm−1 for HNO2. We show that a compositional optimization of classical kernels for Gaussian process regression converges to the same errors. This indicates that quantum kernels can achieve the same, though not better, expressivity as classical kernels for regression problems.
Forecasting in probabilistic time series is a complex endeavor that extends beyond predicting future values to also quantifying the uncertainty inherent in these predictions. Gaussian process regression stands out as a Bayesian machine learning technique adept at addressing this multifaceted challenge. This paper introduces a novel approach that blends the robustness of this Bayesian technique with the nuanced insights provided by the kernel perspective on quantum models, aimed at advancing quantum kernelized probabilistic forecasting. We incorporate a quantum feature map inspired by Ising interactions and demonstrate its effectiveness in capturing the temporal dependencies critical for precise forecasting. The optimization of our model's hyperparameters circumvents the need for computationally intensive gradient descent by employing gradient-free Bayesian optimization. Comparative benchmarks against established classical kernel models are provided, affirming that our quantum-enhanced approach achieves competitive performance.
Probabilistic machine learning models are distinguished by their ability to integrate prior knowledge of noise statistics, smoothness parameters, and training data uncertainty. A common approach involves modeling data with Gaussian processes; however, their computational complexity quickly becomes intractable as the training dataset grows. To address this limitation, we introduce a quantum-assisted algorithm for sparse Gaussian process regression based on the random Fourier feature kernel approximation. We start by encoding the data matrix into a quantum state using a multi-controlled unitary operation, which encodes the classical representation of the random Fourier features matrix used for kernel approximation. We then employ a quantum principal component analysis along with a quantum phase estimation technique to extract the spectral decomposition of the kernel matrix. We apply a conditional rotation operator to the ancillary qubit based on the eigenvalue. We then use Hadamard and swap tests to compute the mean and variance of the posterior Gaussian distribution. We achieve a polynomialorder computational speedup relative to the classical method.
Gaussian process regression (GPR) is a nonparametric supervised learning method, widely valued for its ability to quantify uncertainty. Despite its advantages and broad applications, classical GPR implementations face significant scalability challenges, as they involve matrix operations with a cubic complexity in relation to the dataset size. This computational challenge is further compounded by the demand of optimizing the Gaussian Process (GP) model over its hyperparameters, rendering the total computational cost prohibitive for data intensive problems. To address this issue, we propose a quantum gradient descent algorithm to optimize the GP model. Taking advantage of recent advances in quantum algorithms for linear algebra, our algorithm achieves exponential speedup in computing the gradients of the log marginal likelihood. The entire gradient descent process is integrated into the quantum circuit. Through runtime analysis and error bounds, we demonstrate that our approach significantly improves the scalability of GPR model optimization, making it computationally feasible for large-scale applications.
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.
Abstract Gaussian processes are widely used probabilistic models for regression and classification tasks. Their flexibility and ability to quantify uncertainties make them well-suited for many applications. However, their applicability to large datasets is limited by the cubic computational time complexity. While reduced-rank methods such as Hilbert space approximations reduce this complexity, scaling to very large datasets remains challenging. Quantum algorithms are promising candidates to further accelerate such computations. Quantum principal component analysis has previously been applied to Hilbert space Gaussian process regression to address this scaling limitation. However, its applicability is limited by the requirement for fault-tolerant quantum hardware. In this work, we present a hybrid quantum-classical model for reduced-rank Gaussian process regression, where we apply a quantum principal component analysis algorithm based on parameterized quantum circuits to Hilbert space Gaussian process regression. In noiseless simulations on a one-dimensional test function, the resulting model reaches a regression accuracy comparable to that of the exact Gaussian process and the classical Hilbert space Gaussian process. The framework uses shallow parameterized circuits and does not require fault-tolerant quantum hardware, which makes it a candidate for noisy intermediate-scale quantum devices.