Towards a more feasible quantum speedup for solving linear systems problems with Quantum Simulation

Solving systems of linear equations is one of the main problems in sciences, even being already efficient classically, processing a great amount of data, as is required nowadays, can be troublesome. Quantum solutions are also being considered with this purpose. The  HHL quantum algorithm aims to solve such problems with exponential speedup on the number of variables. In recent years, this algorithm has been established as one of the main algorithms in the recent area of Quantum Machine Learning (QML). However, with great enthusiasm regarding its practical usage and the restrictions that the algorithm requires. One restriction imposed is the efficient preparation of the coefficient matrix of the system, already assumed by the authors. In this work, we will expose these problems and offer an implementation of one of the algorithms in the literature that implements the coefficient matrix efficiently for, if it is sparse, and still gain efficiency.