Quantum Linear System Solvers: A Survey of Algorithms and Applications

TL;DR

This survey reviews quantum algorithms for solving linear systems, analyzing their development, limitations, improvements, and applications in fields like machine learning and physics, highlighting progress towards optimal efficiency.

Contribution

It provides a comprehensive taxonomy of quantum linear system algorithms, analyzing their evolution, limitations, and potential applications in various scientific domains.

Findings

HHL algorithm's limitations and reliance on quantum resources

Post-HHL enhancements improve efficiency and error tolerance

Potential applications in differential equations, quantum machine learning, and physics

Abstract

Solving linear systems of equations plays a fundamental role in numerous computational problems from different fields of science. The widespread use of numerical methods to solve these systems motivates investigating the feasibility of solving linear systems problems using quantum computers. In this work, we provide a survey of the main advances in quantum linear systems algorithms, together with some applications. We summarize and analyze the main ideas behind some of the algorithms for the quantum linear systems problem in the literature. The analysis begins by examining the Harrow-Hassidim-Lloyd (HHL) solver. We note its limitations and reliance on computationally expensive quantum methods, then highlight subsequent research efforts which aimed to address these limitations and optimize runtime efficiency and precision via various paradigms. We focus in particular on the post-HHL enhancements which have paved the way towards optimal lower bounds with respect to error tolerance and condition number. By doing so, we propose a taxonomy that categorizes these studies. Furthermore, by contextualizing these developments within the broader landscape of quantum computing, we explore the foundational work that have inspired and informed their development, as well as subsequent refinements. Finally, we discuss the potential applications of these algorithms in differential equations, quantum machine learning, and many-body physics.