Error Analysis of HHL-Based Quantum Matrix Solver with Improved Quantum Phase Estimation

A quantum approach to solving matrix equations arising in computational methods of electromagnetics, such as the method of moments or finite element method, can be constructed using Harrow/Hassidim/Lloyd (HHL) algorithm with the promise of O(log N) quantum storage and number of operations. The method is based on eigenvalue estimation enabled by the Quantum Phase Estimation (QPE). We show that the Hamiltonian simulation time and clock register size in QPE have to be properly adjusted to account for the periodicity of the magnitude as a function of the eigenvalue estimation error. This results in improved behavior of the amplitude in the QPE output. Such adjustments also facilitate the error analysis of the matrix solver with respect to the condition number of the matrix and the admitted error level in the solution.