Computing eigenvalues of diagonalizable matrices in a quantum computer.

Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient quantum algorithms for Hermitian and unitary matrices. However, the general case is far from fully understood. Combining quantum phase estimation, quantum algorithm to solve linear differential equations and quantum singular value estimation, we propose two quantum algorithms to compute the eigenvalues of diagonalizable matrices that only have real eigenvalues and normal matrices. The output of the quantum algorithms is a superposition of the eigenvalues and the corresponding eigenvectors. The complexities are dominated by solving a linear system of ODEs and performing quantum singular value estimation, which usually can be solved efficiently in a quantum computer. In the special case when the matrix $M$ is $s$-sparse, the complexity is $\widetilde{O}(s\rho^2 \kappa^2/\epsilon^2)$ for diagonalizable matrices that only have real eigenvalues, and $\widetilde{O}(s\rho\|M\|_{\max} /\epsilon^2)$ for normal matrices. Here $\rho$ is an upper bound of the eigenvalues, $\kappa$ is the conditioning of the eigenvalue problem, and $\epsilon$ is the precision to approximate the eigenvalues. We also extend the quantum algorithm to diagonalizable matrices with complex eigenvalues under an extra assumption.