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

TL;DR

This paper introduces a tensor network-based classical simulation of the HHL quantum algorithm for solving linear systems, enabling benchmarking and analysis of its performance and sensitivity.

Contribution

It develops a novel tensor network formalism for HHL, allowing efficient classical simulation and detailed performance benchmarking of the algorithm.

Findings

The classical simulation achieves promising efficiency in benchmarking HHL.

Performance saturation points and maximal values are identified.

The approach provides a higher bound for quantum HHL performance.

Abstract

This work presents a new approach for simulating the HHL linear systems of equations solver algorithm with tensor networks. First, a novel HHL in the qudits formalism, the generalization of qubits, is developed, and then its operations are transformed 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 of the HHL as efficiently as possible to benchmark the algorithm steps according to its input parameters and the input matrix. The algorithm is applied to three classical simple simulation problems, comparing it with an exact inversion algorithm, and its performance is compared against an implementation of the original HHL simulated in the Qiskit framework, providing both codes. It is also applied to study the sensitivity of the HHL algorithm with respect to its hyperparameter values, reporting the existence of saturation points and maximal performance values. The results show that this approach can achieve a promising performance in computational efficiency to simulate the HHL process without quantum noise, providing a higher bound for its performance.