  Despite the successful enhancement to the Harrow-Hassidim-Lloyd algorithm by
Childs et al., who introduced the Fourier approach leveraging linear
combinations of unitary operators, our research has identified non-trivial
redundancies within this method. This finding points to a considerable
potential for refinement. In this paper, we propose the quantum Krylov-subspace
method (QKSM), which is a hybrid classical-quantum algorithm, to mitigate such
redundancies. By integrating QKSM as a subroutine, we introduce the quantum
Krylov-subspace method based linear solver that not only reduces computational
redundancy but also enhances efficiency and accuracy. Extensive numerical
experiments, conducted on systems with dimensions up to $2^{10} \times 2^{10}$,
have demonstrated a significant reduction in computational resources and have
led to more precise approximations.
