Quantum-assisted Gaussian process regression using random Fourier features

2025-07-30

AI Summary

To address the $O(N^3)$ time complexity bottleneck of Gaussian process regression (GPR) on large-scale data—arising from explicit kernel matrix construction and inversion—this paper proposes a quantum-assisted sparse regression framework. First, random Fourier features (RFF) are employed to obtain a low-rank approximation of the kernel function. Second, quantum principal component analysis (qPCA) and quantum phase estimation (QPE) are leveraged to efficiently compute the spectral decomposition of the implicit kernel matrix. Finally, conditional rotations and Hadamard tests enable quantum estimation of the posterior mean and variance. Crucially, this work is the first to deeply integrate RFF with quantum algorithms, eliminating the need to explicitly construct or store the full $N imes N$ kernel matrix. Under statistical fidelity guarantees, the posterior inference complexity is reduced to polynomial scaling—e.g., $O(mathrm{poly}(M,log N))$, where $M ll N$ denotes the RFF dimension. Empirical results demonstrate substantial improvements over classical GPR and existing quantum GPR approaches on large datasets.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

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 polynomial-order computational speedup relative to the classical method.

Problem

• Reducing computational complexity of Gaussian process regression
• Implementing quantum-assisted sparse regression with Fourier features
• Achieving polynomial speedup over classical machine learning methods

Innovation

• Quantum-assisted sparse Gaussian process regression
• Random Fourier feature kernel approximation
• Quantum principal component analysis