Quantum linear system algorithm for solving an ill-posed quasi-linear elliptic problem by preconditioning operator

• Published in: European physical journal plus Vol. 140; no. 5; p. 467
• Main Authors: Salehi Shayegan, Amir Hossein, Dejam, Laya
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 31.05.2025; Springer Nature B.V
• Subjects: Algorithms; Applied and Technical Physics; Atomic; Complex Systems; Condensed Matter Physics; Data compression; Discretization; Eigenvalues; Expected values; Finite element analysis; Finite element method; Fourier transforms; Ill-conditioned problems (mathematics); Linear systems; Mathematical and Computational Physics; Molecular; Operators (mathematics); Optical and Plasma Physics; Physics; Physics and Astronomy; Preconditioning; Regular Article; Sobolev space; Sparsity; Theoretical
• ISSN: 2190-5444; 2190-5444
• DOI: 10.1140/epjp/s13360-025-06383-0

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.