Quantum-Assisted Hilbert-Space Gaussian Process Regression

TL;DR

Gaussian process regression provides principled uncertainty estimates but scales cubically with data size, hindering large-scale use. The paper introduces a quantum-assisted Hilbert-space GPR (QA-HSGPR) that combines a classical Hilbert-space kernel approximation with quantum routines (qPCA, Hadamard tests, Swap tests) to compute the posterior mean and variance with reduced dependence on the dataset size. By expressing mean/variance in terms of low-rank factors and eigenpairs, the method achieves a polynomial speedup over classical HS-GPR for low-rank kernels. Numerical simulations on quantum-circuit models demonstrate close agreement with HS-GPR and highlight potential scalability benefits on fault-tolerant quantum hardware.

Abstract

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 the 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 Swap tests. The quantum principal component analysis is used to estimate the eigenvalues while the conditional rotations and the Hadamard and Swap 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.

Figures (5)

• Figure 1: In this figure, qPCA is first employed on the matrix $\rho_{\mathbf{X}^{\top}\mathbf{X}}$ Following this, a conditionally controlled unitary operation is executed based on the eigenvalues register. Finally, we revert the additional $\tau$ qubit register to its original state by executing the corresponding inverse quantum operations to prepare the quantum state $\ket{\psi_1}$.
• Figure 2: Hadamard test circuit to estimate the mean of GPR. Here $\eta=\log_{2}(NM)+\tau+1$ qubits.
• Figure 3: In this figure, we illustrate the application of qPCA on the matrix $\rho_{\mathbf{X}^{\top}\mathbf{X}}$. Initially, qPCA identifies the eigenvalues and eigenvectors of the matrix. We then apply a conditionally controlled unitary on ancilla register based on the eigenvalue register. Finally, the Swap test is employed to estimate the variance of the GPR.
• Figure 4: Mean of the GPR using the squared exponential kernel (gray), the Hilbert space approximation of the kernel with $M = 4$ eigenfunctions (black dashed line), and our reduced rank approximation using a quantum circuit (blue lines) with $N=16$ data points (red cross). The blue lines range over $R = 1,2,3,4$ showing how taking a larger rank increases the accuracy of the estimation.
• Figure 5: Mean and variance of GPR using the squared exponential kernel (gray solid), the Hilbert space approximation of the kernel with $M = 8$ eigenfunctions (black dashed), and our QA-HSGPR (blue line) with data points $N=16$ (red cross). Each point in the blue line represents a simulation. The shaded areas around each approximation line indicate the $95\%$ confidence intervals, providing a visual representation of the uncertainty associated with each method. We can see that our proposed scheme approximates the HSGPR method well with $R=4$.