Computing eigenvalues of matrices in a quantum computer
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Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum linear solver, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: normal matrices and matrices whose eigenvalues have non-positive imaginary parts. Similar to quantum phase estimation, diagonalizability indicates that to compute the eigenvalues we do not need to know the eigenvectors in advance. The quantum algorithm returns a quantum state such that the first register stores eigenvalues and the second register stores eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum phase estimation and quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system. In this case, we can only estimate the real part of the eigenvalues. Under certain conditions (e.g. sparse or block-encoding), the complexity is polylog on the size n of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. For s sparse matrix M, the complexity can be O(snM_max/ϵ). Generally, the complexity can reach O(nM_F/ϵ).
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