Provable Quantum Algorithm Advantage for Gaussian Process Quadrature

TL;DR

This work tackles the computational bottleneck of Gaussian process quadrature by introducing a quantum, low-rank approach that leverages a Hilbert space kernel approximation. By encoding data into quantum states and applying quantum phase estimation, quantum PCA, and Hadamard/SWAP tests, the method computes the GPQ mean and variance with a polynomial speedup over the classical O(M^3) inversion regime. Theoretical complexity bounds show a cost of O(poly(log NM)) log(M) κ^2 ε^{-3} (or similar polylog factors under data-loaded settings), while simulations on a classical quantum computer validate the approach against classical GPQ and Hilbert-space quadrature designs. The results indicate a practical path toward quantum-accelerated Bayesian quadrature, with future work focusing on circuit optimization, iterative QPE, and hardware demonstrations on fault-tolerant quantum devices. Overall, the paper establishes a principled framework for integrating quantum algorithms with probabilistic numerical integration in Gaussian process quadrature, potentially enabling scalable uncertainty quantification for large-scale machine learning tasks.

Abstract

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.

Figures (6)

• Figure 1: Circuit diagram to implement the quantum Fourier transform on a quantum computer.
• Figure 2: Circuit diagram to implement the quantum phase estimation on a quantum computer.
• Figure 3: Quantum circuit to prepare the $\ket{\psi_1}$ quantum state. The red gates represent the parts of the algorithm where the qPCA is implemented, meanwhile, the green part envelopes the implementation of the QPE algorithm.
• Figure 4: Quantum circuit to estimate the quadrature variance. The red gates represent the parts of the algorithm where the qPCA is implemented, meanwhile, the green part envelopes the implementation of the QPE algorithm.
• Figure 5: Estimate of the quadrature for the integral $\mathcal{I}_1$. The estimation results deliver the expected value and variance of the Gaussian distributions plotted above that approximates the integral. The dotted black plot corresponds to the GPQ, the dashed red plot corresponds to the HSQ and the solid green plots correspond to the estimates using our QHSQ algorithm with $R = 1,2,3,4$.