We reiterate the contribution made by Harrow, Hassidim, and Llyod to the quantum matrix equation solver with the emphasis on the algorithm description and the error analysis derivation details. Moreover, the behavior of the amplitudes of the phase register on the completion of the Quantum Phase Estimation is studied. This study is beneficial for the comprehension of the choice of the phase register size and its interrelation with the Hamiltonian simulation duration in the algorithm setup phase.

翻译：我们重新阐述了Harrow、Hassidim和Lloyd在量子矩阵方程求解器中的贡献，重点介绍了算法描述和误差分析推导细节。此外，还研究了量子相位估计完成后相位寄存器幅值的行为。这项研究有助于理解算法设置阶段相位寄存器大小的选择及其与哈密顿量模拟持续时间的相互关系。