Efficient simulation of sparse Hamiltonian and application in quantum resolution of linear systems

The solution of systems of linear equations is one of the main problems in several areas of science. The HHL quantum algorithm is an algorithm that aims to solve such problems with exponential speedup. In recent years, this algorithm has established itself as one of the main algorithms in the recent area of Quantum Machine Learning (QML). However, alongside the great enthusiasm for the power of the algorithm, there is also some skepticism about the number of constraints the algorithm requires. A main constraint imposed is the efficient preparation of matrix A, already assumed by the authors. In this work, we expose these problems and offer an implementation of one of the algorithms in the literature that efficiently implements the e^−iAt matrix for cases of sparse and computable A by lines, with little overhead on the execution of the HHL itself. Finally, we present how the error rate ε of the approximation behaves in quantum simulation and how it affects HHL's solutions.