Provable Quantum Algorithm Advantage for Gaussian Process Quadrature

Cristian A. Galvis-Florez, Ahmad Farooq, Simo Särkkä

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 quadrature 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.

Submission Length: Long submission (more than 12 pages of main content)

Changes Since Last Submission: We included the acknowledgments, author contributions and minor typographical corrections.

Code: https://github.com/cagalvisf/Quantum_HSGPQ

Assigned Action Editor: ~Jake_Snell1

Submission Number: 3210