  We present an algorithm for solving systems of linear equations based on the
HHL algorithm with a novel qudits methodology, a generalization of the qubits
with more states, to reduce the number of gates to be applied and the amount of
resources. Based on this idea, we perform a quantum-inspired version on tensor
networks, taking advantage of their ability to perform non-unitary operations
such as projection. The main novelty of this proposal is to perform a
simulation as efficient as possible of the HHL algorithm in order to benchmark
the algorithm steps according to its input parameters and the input matrix.
Finally, we use this algorithm to obtain a solution for the harmonic oscillator
with an external force, the forced damped oscillator and the 2D static heat
equation differential equations.
