Assessing Quantum Advantage for Gaussian Process Regression

TL;DR

This paper critically evaluates quantum algorithms for Gaussian Process Regression, demonstrating that under broad conditions, these algorithms do not offer exponential speedups due to the linear scaling of key matrix properties.

Contribution

The paper provides a rigorous proof that the condition number, sparsity, and Frobenius norm of kernel matrices scale linearly, challenging claims of exponential quantum speedup in Gaussian Process Regression.

Findings

Quantum algorithms do not achieve exponential speedup for Gaussian Process Regression.

Kernel matrix properties scale linearly with data size under general assumptions.

Numerical verification confirms theoretical results for popular kernels.

Abstract

Gaussian Process Regression is a well-known machine learning technique for which several quantum algorithms have been proposed. We show here that in a wide range of scenarios these algorithms show no exponential speedup. We achieve this by rigorously proving that the condition number of a kernel matrix scales at least linearly with the matrix size under general assumptions on the data and kernel. We additionally prove that the sparsity and Frobenius norm of a kernel matrix scale linearly under similar assumptions. The implications for the quantum algorithms runtime are independent of the complexity of loading classical data on a quantum computer and also apply to dequantised algorithms. We supplement our theoretical analysis with numerical verification for popular kernels in machine learning.