An Exponential Separation Between Quantum and Quantum-Inspired Classical Algorithms for Linear Systems

TL;DR

This paper demonstrates the first provable exponential quantum speedup over quantum-inspired classical algorithms for solving well-conditioned sparse linear systems, challenging the prevailing notion that quantum advantages are mostly polynomial.

Contribution

It establishes an exponential separation between quantum and quantum-inspired classical algorithms for linear systems, marking a significant breakthrough in quantum machine learning complexity.

Findings

Quantum algorithms solve certain linear systems exponentially faster.

Quantum-inspired classical algorithms cannot achieve this exponential speedup.

This work challenges the belief that quantum advantages are limited to polynomial speedups.

Abstract

Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL quantum algorithm for solving linear systems and the subsequent quantum recommender systems algorithm by Kerenidis and Prakash. These algorithms were initially believed to be strong candidates for exponential speedups, but a lower bound ruling out similar classical improvements remained absent. In breakthrough work by Tang, it was demonstrated that this lack of progress in classical lower bounds was for good reasons. Concretely, she gave a classical counterpart of the quantum recommender systems algorithm, reducing the quantum advantage to a mere polynomial. Her approach is quite general and was named quantum-inspired classical algorithms. Since then, almost all the initially exponential quantum machine learning speedups have been reduced to polynomial via new quantum-inspired classical algorithms. From the current state-of-affairs, it is unclear whether we can hope for exponential quantum speedups for any natural machine learning task. In this work, we present the first such provable exponential separation between quantum and quantum-inspired classical algorithms for the basic problem of solving a linear system when the input matrix is well-conditioned and has sparse rows and columns.