International Journal of Theoretical Physics

, Volume 58, Issue 8, pp 2632–2640 | Cite as

Quantum Algorithms and Circuits for Linear Equations with Infinite or No Solutions

  • Jin-Min Liang
  • Shu-Qian ShenEmail author
  • Ming Li


Linear equations with infinite or no solutions play key roles in machine learning and optimization. However, the existed quantum algorithms cannot be applied directly for these classes of equations. In this paper, based on the modification of algorithm (Wossnig et al., Phys. Rev. Lett. 120, 050502 2017), we describe a quantum algorithm to compute the minimal norm solution or minimum norm least-squares solution for equations with infinite or no solutions, respectively. It can be shown that the presented algorithm can achieve an exponential speedup over the best classical algorithm. Furthermore, the corresponding quantum circuit is designed on a quantum computer.


Quantum algorithm Linear equations Quantum circuits 



The authors thank Dr. Bo-Jia Duan for her invaluable suggestions. This work is supported by the Natural Science Foundation of Shandong Province (ZR2016AM23) and the Fundamental Research Funds for the Central Universities (18CX02035A).


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Authors and Affiliations

  1. 1.College of ScienceChina University of PetroleumQingdaoPeople’s Republic of China
  2. 2.Max-Planck-Institute for Mathematics in SciencesLeipzigGermany

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