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Quantum Machine Intelligence

Sparse quantum Gaussian processes to counter the curse of dimensionality

Gaweł Kuś, Sybrand van der Zwaag, Miguel Aníbal Bessa

Published Feb 17, 2021• 4 citations
Publisher

Abstract

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.

DOI
10.1007/s42484-020-00032-8

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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 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.
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The data-driven approach shows great potential for designing new materials with unprecedented properties by using machine learning and optimization. Recently, a data-driven framework was successfully applied to design a unit cell of metamaterial achieving super-compressibility, despite being built out of brittle base material. The key element of the framework is the algorithm called Gaussian processes regression (GPs) – a unique machine learning method that provides with the uncertainty of the prediction, which can be used during the design process to account for inherent material imperfections, ensuring the robustness of the design. Despite their superior predictive performance, however, GPs suffer from scalability issues, which limit their application to relatively small design problems. In the future, those limitations could be surpassed by quantum computing. This research aims at demonstrating how quantum computing could enhance the computational design of materials, specifically by replacing the expensive machine learning step of the data-driven design framework with an exponentially faster quantum algorithm for Gaussian processes (QGP). This objective is achieved in two steps. First, the QGP algorithm was Implementation and simulated within a quantum computing framework (Qiskit), which allowed to understand and control its performance (in particular its accuracy), proving the feasibility for practical applications. Furthermore, the numerical tests exposed a mechanism for inducing a low-rank approximation, which allows for additional speed-up, making the QGP algorithm similar to classical sparse Gaussian processes, which rely on low-rank approximations to improve the scalability of full GPs. In the second part of this research, the implemented QGP algorithm was integrated within a computational framework for the data-driven design of materials and applied to two example design problems of optimization a unit cell for a super-compressible metamaterial. The results obtained with the QGP were comparable to those obtained with classical methods (also from literature), which proved the feasibility of the concept.
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.
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 Support vector machines (SVMs) are widely used for classification, but become computationally expensive as datasets grow and data become high-dimensional or nonlinearly separable. Quantum support vector machines (QSVMs) enhance the traditional SVMs by replacing conventional feature mappings with quantum-implemented embeddings that project data into high-dimensional Hilbert spaces, potentially improving expressive power and efficiency. However, training QSVMs on current noisy intermediate-scale quantum (NISQ) devices remains challenging, as constructing the quantum kernel matrix requires repeated circuit executions and scales quadratically with dataset size. We propose NQ-SVM , a hybrid quantum-classical method that incorporates the Nystr\"om approximation to enhance the scalability of QSVMs. By selecting a small set of landmark points, the Nystr\"om method yields a low-rank kernel approximation, reducing complexity from (O(n2))(O(n^2))(O(n2)) to (O(nm))(O(nm))(O(nm)), while preserving the quantum advantages in handling high-dimensional, non-linear data. Extensive simulations demonstrate that NQ-SVM achieves comparable classification accuracy to classical SVMs and full QSVMs, while substantially reducing training time and kernel evaluation costs. These results underscore the potential of NQ-SVM as a scalable and practical hybrid framework for near-term quantum machine learning (QML).