Optimal algorithms for linear algebra by quantum inspiration

TL;DR

This paper introduces a new classical algorithm for linear algebra problems inspired by quantum computing, achieving optimal time complexity related to matrix multiplication, and encourages further exploration of quantum techniques in classical algorithms.

Contribution

It presents a novel classical algorithm for linear algebra with optimal matrix multiplication complexity, inspired by quantum intuition, distinct from previous methods.

Findings

Algorithm runs in time $O(n^{ ext{ω}+ u})$, matching optimal matrix multiplication bounds.

The approach is based on low-discrepancy sequences and perturbation analysis.

It bridges quantum insights with classical algorithm design, opening new research directions.

Abstract

Recent results by Harrow et. al. and by Ta-Shma, suggest that quantum computers may have an exponential advantage in solving a wealth of linear algebraic problems, over classical algorithms. Building on the quantum intuition of these results, we step back into the classical domain, and explore its usefulness in designing classical algorithms. We achieve an algorithm for solving the major linear-algebraic problems in time $O(n^{\omega+\nu})$ for any $\nu>0$, where $\omega$ is the optimal matrix-product constant. Thus our algorithm is optimal w.r.t. matrix multiplication, and comparable to the state-of-the-art algorithm for these problems due to Demmel et. al. Being derived from quantum intuition, our proposed algorithm is completely disjoint from all previous classical algorithms, and builds on a combination of low-discrepancy sequences and perturbation analysis. As such, we hope it motivates further exploration of quantum techniques in this respect, hopefully leading to improvements in our understanding of space complexity and numerical stability of these problems.