Computing eigenvalues of matrices in a quantum computer

Abstract

Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum algorithms to solve linear differential equations, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: matrices that only have real eigenvalues and normal matrices. The quantum algorithms return a quantum state such that the first register stores eigenvalues and the second register stores the corresponding eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system of differential equations. Under certain conditions (e.g. sparse), the complexities are polylog on the size $n$ of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. If $M$ is $s$ sparse, then the complexity is $\widetilde{O}(sn\|M\|_{\max}\kappa^2/\epsilon^2)$, where $\kappa$ is the condition number of the matrix of eigenvectors.