An Iterative Improvement Method for HHL algorithm for Solving Linear System of Equations
Abstract
We propose an iterative improvement method for the HarrowHassidimLloyd (HHL) algorithm to solve a linear system of equations. This is a quantumclassical hybrid algorithm. The accuracy is essential to solve the linear system of equations. However, the accuracy of the HHL algorithm is limited by the number of quantum bits used to express the eigenvalues of the matrix. Our iterative method improves the accuracy of the HHL solutions, and gives higher accuracy which surpasses the accuracy limited by the number of quantum bits. In practical HHL algorithm, a huge number of measurements is required to obtain good accuracy, even if we provide a sufficient number of quantum bits for the eigenvalue expression, since the solution is statistically processed from the measurements. Our improved iterative method can reduce the number of measurements. Moreover, the sign information for each eigenstate of the solution is lost once the measurement is made, although the sign is significant. Therefore, the naïve iterative method of the HHL algorithm may slow down, especially, when the solution includes wrong signs. In this paper, we propose and evaluate an improved iterative method for the HHL algorithm that is robust against the sign information loss, in terms of the number of iterations and the computational accuracy.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.07744
 Bibcode:
 2021arXiv210807744S
 Keywords:

 Quantum Physics
 EPrint:
 7 pages, 7 figures