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 serves as motivation for investigating the feasibility of solving linear system problems using quantum computers. This review surveys the main advances in quantum linear system algorithms, together with some applications. The main ideas behind certain algorithms for the quantum linear system problem are summarized and analyzed. The analysis begins by examining the Harrow-Hassidim-Lloyd (HHL) solver. The review first notes its limitations and reliance on computationally expensive quantum methods then highlights subsequent research efforts that have aimed to address these limitations and optimize run-time efficiency and precision via various paradigms. A particular focus is placed on the post-HHL enhancements that have paved the way toward optimal lower bounds with respect to error tolerance and condition number. In the process a taxonomy that categorizes these studies is proposed. Furthermore, by contextualizing these developments within the broader landscape of quantum computing, the review explores the foundational work that has inspired and informed their development, as well as subsequent refinements. Some of the open problems regarding the optimal linear solver efficiency are highlighted. Finally, the potential applications of these algorithms in differential equations, machine learning, and many-body physics are discussed.

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• Quantum algorithms & computation
• Quantum computation
• Quantum information processing