  Arnoldi method and conjugate gradient method are important classical
iteration methods in solving linear systems and estimating eigenvalues. Their
efficiency often affected by the high dimension of the space, where quantum
computer can play a role in. In this work, we establish their corresponding
quantum algorithms. To achieve high efficiency, a new method about linear
combination of quantum states will be proposed. The final complexity of quantum
Arnoldi iteration method is $O(m^{3+\log (m/\epsilon)}(\log n)^2 /\epsilon^4)$
and the final complexity of quantum conjugate gradient iteration method is
$O(m^{1+\log m/\epsilon} (\log n)^2 \kappa/\epsilon)$, where $\epsilon$ is
precision parameter, $m$ is the iteration steps, $n$ is the dimension of space
and $\kappa$ is the condition number of the coefficient matrix of the linear
system the conjugate gradient method works on. Compared with the classical
methods, whose complexity are $O(mn^2+m^2n)$ and $O(mn^2)$ respectively, these
two quantum algorithms provide us more efficient methods to solve linear
systems and to compute eigenvalues and eigenvectors of general matrices.
Different from the work \cite{rebentros}, the complexity here is almost
polynomial in the iteration steps. Also this work is more general than the
iteration method considered in \cite{kerenidis}.
