Article
| Title: | A regularization method based on HHL quantum algorithm and Levenberg-Marquardt method (English) |
| Author: | Salehi Shayegan, Amir Hossein |
| Author: | Dejam, Laya |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 71 |
| Issue: | 5 |
| Year: | 2026 |
| Pages: | 651-671 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | To achieve exponential speed-up with the Harrow-Hassidim-Lloyd (HHL) quantum algorithm, the condition number of the matrix can scale at most poly-logarithmically with its size. This quantum algorithm significantly outperforms the best classical methods for solving well-conditioned linear systems of equations. However, well-conditioning is a stringent requirement that severely limits the range of problems benefiting from exponential speed-up. In this paper, we introduce a regularization method that combines the HHL quantum algorithm with the Levenberg-Marquardt method to mitigate ill-conditioning and enable exponential speed-up. Specifically, the Levenberg-Marquardt method acts as a regularization method, transforming an ill-conditioned linear system of equations into a sequence of well-conditioned linear systems of equations. The HHL quantum algorithm is then applied to solve these systems efficiently. This approach effectively addresses the ill-conditioning issue while preserving the exponential speed-up advantage. (English) |
| Keyword: | quantum algorithm |
| Keyword: | regularization method |
| Keyword: | Levenberg-Marquardt method |
| Keyword: | ill-posed problem |
| Keyword: | HHL quantum algorithm |
| MSC: | 47A52 |
| MSC: | 68Q12 |
| DOI: | 10.21136/AM.2026.0193-25 |
| . | |
| Date available: | 2026-10-12T12:58:32Z |
| Last updated: | 2026-10-19 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153797 |
| . | |
| Reference: | [1] Cao, Y., Daskin, A., Frankel, S., Kais, S.: Quantum circuit design for solving linear systems of equations.Molecular Phys. 110 (2012), 1675-1680. 10.1080/00268976.2012.668289 |
| Reference: | [2] Cao, Y., Papageorgiou, A., Petras, I., Traub, J., Kais, S.: Quantum algorithm and circuit design solving the Poisson equation.New J. Phys. 15 (2013), Article ID 013021, 29 pages. Zbl 1451.81163, MR 3045573, 10.1088/1367-2630/15/1/013021 |
| Reference: | [3] Clader, B. D., Jacobs, B. C., Sprouse, C. R.: Preconditioned quantum linear system algorithm.Phys. Rev. Lett. 110 (2013), Article ID 250504. 10.1103/PhysRevLett.110.250504 |
| Reference: | [4] Demmel, J. W.: Applied Numerical Linear Algebra.SIAM, Philadelphia (1997). Zbl 0879.65017, MR 1463942, 10.1137/1.9781611971446 |
| Reference: | [5] Duan, B., Yuan, J., Yu, C.-H., Huang, J., Hsieh, C.-Y.: A survey on HHL algorithm: From theory to application in quantum machine learning.Phys. Lett., A 384 (2020), Article ID 126595, 13 pages. Zbl 1448.81232, MR 4105727, 10.1016/j.physleta.2020.126595 |
| Reference: | [6] Hanke, M.: The regularizing Levenberg-Marquardt scheme is of optimal order.J. Integral Equations Appl. 22 (2010), 259-283. Zbl 1204.65059, MR 2661721, 10.1216/JIE-2010-22-2-259 |
| Reference: | [7] Hansen, P. C.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion.SIAM Monographs on Mathematical Modeling and Computation 4. SIAM, Philadelphia (1998). Zbl 0890.65037, MR 1486577, 10.1137/1.9780898719697 |
| Reference: | [8] Hansen, P. C.: Regularization tools version 4.0 for Matlab 7.3.Numer. Algorithms 46 (2007), 189-194. Zbl 1128.65029, MR 2358249, 10.1007/s11075-007-9136-9 |
| Reference: | [9] Harkins, F., (Ed.): Qiskit textbook: IBM Q HHL code.Available at https://qiskit.org/textbook/ch-applications/hhl_tutorial.html (2023). |
| Reference: | [10] Harrow, A. W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations.Phys. Rev. Lett. 103 (2009), Article ID 150502, 4 pages. MR 2551688, 10.1103/PhysRevLett.103.150502 |
| Reference: | [11] Kelley, C. T.: Iterative Methods for Optimization.Frontiers in Applied Mathematics 18. SIAM, Philadelphia (1999). Zbl 0934.90082, MR 1678201, 10.1137/1.9781611970920 |
| Reference: | [12] Nielsen, M. A., Chuang, I. L.: Quantum Computation and Quantum Information.Cambridge University Press, Cambridge (2000). Zbl 1049.81015, MR 1796805, 10.1017/CBO9780511976667 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
