Solving Systems of Linear Equations: HHL from a Tensor Networks Perspective

We present a new approach for solving systems of linear equations with tensor networks based on the quantum HHL algorithm. We first develop a novel HHL in the qudits formalism, the generalization of qubits, and then transform its operations into an equivalent classical HHL, taking advantage of the non-unitary operations that they can apply. The main novelty of this proposal is to perform a classical simulation as efficiently as possible of the HHL to benchmark the algorithm steps according to its input parameters and the input matrix. We apply this algorithm to three simulation problems, comparing it with an exact inversion algorithm, and we compare its performance against an implementation of the original HHL simulated in the Qiskit framework, providing both codes. Our results show that our approach can achieve a promising performance in computational efficiency to simulate HHL process without quantum noise, providing a lower bound.