# Quantum-assisted Gaussian process regression using random Fourier features

arXiv: https://arxiv.org/abs/2507.22629 · HTML: https://mlsanity.com/paper/2507.22629/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 20 references that are in the map.
- First posted on arXiv 30 Jul 2025.

## Builds on (most cited first, 7 of 7)

- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [Quantum assisted Gaussian process regression](https://mlsanity.com/paper/1512.03929/) (2015)
- [Quantum-Assisted Hilbert-Space Gaussian Process Regression](https://mlsanity.com/paper/2402.00544/) (2024)
- [Provable Quantum Algorithm Advantage for Gaussian Process Quadrature](https://mlsanity.com/paper/2502.14467/) (2025)

## Built on by (most cited first, 1 of 1)

- [Spectral Anatomy of Quantum Gaussian Process Kernels](https://mlsanity.com/paper/2605.30952/) (2026)

## Abstract (arXiv)

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.