  We consider the quantum implementations of the two classical iterative
solvers for a system of linear equations, including the Kaczmarz method which
uses a row of coefficient matrix in each iteration step, and the coordinate
descent method which utilizes a column instead. These two methods are widely
applied in big data science due to their very simple iteration schemes. In this
paper we use the block-encoding technique and propose fast quantum
implementations for these two approaches, under the assumption that the quantum
states of each row or each column can be efficiently prepared. The quantum
algorithms achieve exponential speed up at the problem size over the classical
versions, meanwhile their complexity is nearly linear at the number of steps.
