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We propose an iterative improvement method for the Harrow-Hassidim-Lloyd(HHL) algorithm to solve a linear system of equations. This is aquantum-classical hybrid algorithm. The accuracy is essential to solve thelinear system of equations. However, the accuracy of the HHL algorithm islimited by the number of quantum bits used to express the eigenvalues of thematrix. Our iterative method improves the accuracy of the HHL solutions, andgives higher accuracy which surpasses the accuracy limited by the number ofquantum bits. In practical HHL algorithm, a huge number of measurements isrequired to obtain good accuracy, even if we provide a sufficient number ofquantum bits for the eigenvalue expression, since the solution is statisticallyprocessed from the measurements. Our improved iterative method can reduce thenumber of measurements. Moreover, the sign information for each eigenstate ofthe solution is lost once the measurement is made, although the sign issignificant. Therefore, the na\"ive iterative method of the HHL algorithm mayslow down, especially, when the solution includes wrong signs. In this paper,we propose and evaluate an improved iterative method for the HHL algorithm thatis robust against the sign information loss, in terms of the number ofiterations and the computational accuracy.

An Iterative Improvement Method for HHL algorithm for Solving Linear System of Equations