Provable Quantum Algorithm Advantage for Gaussian Process Quadrature

TL;DR

This paper introduces a quantum algorithm that significantly accelerates Gaussian process quadrature, a numerical integration technique, by leveraging quantum computing to achieve polynomial speedups over classical methods.

Contribution

The paper presents a novel quantum low-rank Gaussian process quadrature method combining Hilbert space approximation and quantum algorithms, demonstrating polynomial complexity advantages.

Findings

Quantum algorithm achieves polynomial speedup over classical methods.

Numerical simulations validate the effectiveness of the quantum quadrature.

Theoretical analysis confirms complexity advantages.

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.