https://doi.org/10.1140/epjp/s13360-025-06383-0
Regular Article
Quantum linear system algorithm for solving an ill-posed quasi-linear elliptic problem by preconditioning operator
1
Mathematics Department, Faculty of Basic Science, Khatam-ol-Anbia (PBU) University, Tehran, Iran
2
Physics Department, West Tehran Branch, Islamic Azad University, Tehran, Iran
3
Quantum Technologies Research Center (QTRC), Science and Research Branch, Islamic Azad University, Tehran, Iran
a ahsalehi.kau@gmail.com, ah.salehi@mail.kntu.ac.ir
Received:
13
January
2025
Accepted:
29
April
2025
Published online:
31
May
2025
The HHL quantum algorithm for solving a well-conditioned linear system of equations provides an exponential speedup over the best classical methods. To be exact, in the quantum algorithm to achieve exponential speedup, the condition number of the matrix can scale at most poly logarithmically with the size of the matrix. This is a very strict condition that greatly limits the class of problems that can achieve exponential speedup. On the other hand, the considered quasi-linear elliptic problem is ill-conditioned. Therefore, discretization methods lead to an unbounded condition number, as discretization is refined and the exponential speedup of the quantum linear system algorithm may be lost. In doing so, in this paper, we propose a preconditioned quantum linear system algorithm to control ill-conditioning and achieve an exponential speedup algorithm for solving the obtained linear system of equations. In this way, three methods, i.e., preconditioned Sobolev space gradient method, WEB-spline finite element method and HHL quantum algorithm, are applied. At the end, the numerical results are given in details to show the efficiency and accuracy of the proposed method.
Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
