An improved method for quantum matrix multiplication

Following the celebrated quantum algorithm for solving linear equations (so-called HHL algorithm), Childs et al. (SIAM J Comput 46:1920-1950, 2017) provided an approach to solve a linear system of equations with exponentially improved dependence on precision. In this note, we aim to complement such a result for applying a matrix to some quantum state, based upon their Chebyshev polynomial approach. A few examples that motivate this application are included, and we further discuss an application of this improved matrix application algorithm explicitly with an efficient quantum procedure.