The aim of this paper is to develop novel quantum algorithms for Gaussian process quadrature methods. Gaussian process quadratures are numerical integration methods where Gaussian processes are used as functional priors for the integrands to capture the uncertainty arising from the sparse function evaluations. Quantum computers have emerged as potential replacements for classical computers, offering exponential reductions in the computational complexity of machine learning tasks. In this paper, we combine Gaussian process quadratures and quantum computing by proposing a quantum low-rank Gaussian process quadrature method based on a Hilbert space approximation of the Gaussian process kernel and enhancing the quadrature using a quantum circuit. The method combines the quantum phase estimation algorithm with the quantum principal component analysis technique to extract information up to a desired rank. Then, Hadamard and SWAP tests are implemented to find the expected value and variance that determines the quadrature. We use numerical simulations of a quantum computer to demonstrate the effectiveness of the method. Furthermore, we provide a theoretical complexity analysis that shows a polynomial advantage over classical Gaussian process quadrature methods. The code is available at https://github.com/cagalvisf/Quantum_HSGPQ.
The aim of this paper is to develop novel quantum algorithms for Gaussian process quadrature methods. Gaussian process quadratures are numerical integration methods where Gaussian processes are used as functional priors for the integrands to capture the uncertainty arising from the sparse function evaluations. Quantum computers have emerged as potential replacements for classical computers, offering exponential reductions in the computational complexity of machine learning tasks. In this paper, we combine Gaussian process quadratures and quantum computing by proposing a quantum low-rank Gaussian process quadrature method based on a Hilbert space approximation of the Gaussian process kernel and enhancing the quadrature using a quantum circuit. The method combines the quantum phase estimation algorithm with the quantum principal component analysis technique to extract information up to a desired rank. Then, Hadamard and SWAP tests are implemented to find the expected value and variance that determines the quadrature. We use numerical simulations of a quantum computer to demonstrate the effectiveness of the method. Furthermore, we provide a theoretical complexity analysis that shows a polynomial advantage over classical Gaussian process quadrature methods. The code is available at https://github.com/cagalvisf/Quantum_HSGPQ.
Gaussian processes are probabilistic models that are commonly used as functional priors in machine learning. Due to their probabilistic nature, they can be used to capture prior information on the statistics of noise, smoothness of the functions, and training data uncertainty. However, their computational complexity quickly becomes intractable as the size of the data set grows. We propose a Hilbert-space approximation-based quantum algorithm for Gaussian process regression to overcome this limitation. Our method consists of a combination of classical basis function expansion with quantum computing techniques of quantum principal component analysis, conditional rotations, and Hadamard and tests. The quantum principal component analysis is used to estimate the eigenvalues, while the conditional rotations and the Hadamard and tests are employed to evaluate the posterior mean and variance of the Gaussian process. Our method provides polynomial computational complexity reduction over the classical method. Published by the American Physical Society 2024
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.
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.
Abstract Gaussian processes are well-established Bayesian machine learning algorithms with significant merits, despite a strong limitation: lack of scalability. Clever solutions address this issue by inducing sparsity through low-rank approximations, often based on the Nystrom method. Here, we propose a different method to achieve better scalability and higher accuracy using quantum computing, outperforming classical Bayesian neural networks for large datasets significantly. Unlike other approaches to quantum machine learning, the computationally expensive linear algebra operations are not just replaced with their quantum counterparts. Instead, we start from a recent study that proposed a quantum circuit for implementing quantum Gaussian processes and then we use quantum phase estimation to induce a low-rank approximation analogous to that in classical sparse Gaussian processes. We provide evidence through numerical tests, mathematical error bound estimation, and complexity analysis that the method can address the “curse of dimensionality,” where each additional input parameter no longer leads to an exponential growth of the computational cost. This is also demonstrated by applying the algorithm in a practical setting and using it in the data-driven design of a recently proposed metamaterial. The algorithm, however, requires significant quantum computing hardware improvements before quantum advantage can be achieved.
Gaussian processes (GPs) are important models in supervised machine learning. Training in Gaussian processes refers to selecting the covariance functions and the associated parameters in order to improve the outcome of predictions, the core of which amounts to evaluating the logarithm of the marginal likelihood (LML) of a given model. The LML gives a concrete measure of the quality of prediction that a GP model is expected to achieve. The classical computation of the LML typically carries a polynomial time overhead with respect to the input size. We propose a quantum algorithm that computes the logarithm of the determinant of a Hermitian matrix, which runs in logarithmic time for sparse matrices. This is applied in conjunction with a variant of the quantum linear system algorithm that allows for logarithmic-time computation of the form ${\mathbf{y}}^{T}{A}^{\ensuremath{-}1}\mathbf{y}$, where y is a dense vector and A is the covariance matrix. We hence show that quantum computing can be used to estimate the LML of a GP with exponentially improved efficiency under certain conditions.
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.