Provable Quantum Algorithm Advantage for Gaussian Process Quadrature

TMLR Paper3210 Authors

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 for 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 \url{https://anonymous.4open.science/r/Quantum_HS_GP_Quadrature/}

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

Changes Since Last Submission:

Summary of changes with respect to the reviewer's comments:
* A Notation table was added before the introduction.
* Introduction section:
* Intuitive reason for complexity reduction.
* Clarification on the Paper contribution.
* Definition of constants in the complexity.
* Hilbert Space Approximation for Gaussian Process Quadrature Section:
* Text was added to clarify the relationship between the classical and the quantum methods.
* Quantum Algorithms Background section:
* More details about the representation of quantum states and operators were added.
* Notation clarifications and small examples to introduce the quantum notation.
* More steps in the derivations were added to better understand the QFT.
* Contrast between classical and quantum bits to represent information,
* Clarifications on what is a quantum register.
* Quantum Hilbert space low-rank Gaussian process quadrature Section:
* Added text to relate the classical quadrature method and the quantum method.
* Clarifications in the quantum register sizes.
* Clarification on why the state $\mid 0\rangle_{a}$ of the ancilla register vanishes after dot product.
* Explanation of why two different circuits were implemented.
* Numerical Simulations
* Typo corrections.
* Conclusion:
* Insights about the deviation of the quantum simulation with respect to the classical.
* Added text referring to the suitable scenario to implement the algorithm
* Appendix A2:
* Swap gate explanation added.

Assigned Action Editor: ~Jake_Snell1

Submission Number: 3210