Quantum Arnoldi and conjugate gradient iteration algorithm

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}.