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

Abstract

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.