Quantum speedup of leverage score sampling and its application

TL;DR

This paper introduces a quantum algorithm that significantly accelerates leverage score sampling and related matrix problems, achieving at least quadratic and potentially exponential speedups, with applications to rigid regression.

Contribution

It presents a novel quantum algorithm for leverage score sampling, providing near-optimal speedups and an improved classical algorithm for rigid regression.

Findings

Quantum leverage score sampling speedup is at least quadratic, possibly exponential.

The quantum algorithm for rigid regression achieves polynomial speedups.

Lower bounds suggest the quantum algorithm is near optimal.

Abstract

Leverage score sampling is crucial to the design of randomized algorithms for large-scale matrix problems, while the computation of leverage scores is a bottleneck of many applications. In this paper, we propose a quantum algorithm to accelerate this useful method. The speedup is at least quadratic and could be exponential for well-conditioned matrices. We also prove some quantum lower bounds, which suggest that our quantum algorithm is close to optimal. As an application, we propose a new quantum algorithm for rigid regression problems with vector solution outputs. It achieves polynomial speedups over the best classical algorithm known. In this process, we give an improved randomized algorithm for rigid regression.