Quantum algorithm for estimating largest eigenvalues

Scientific computation involving numerical methods relies heavily on the manipulation of large matrices, including solving linear equations and finding eigenvalues and eigenvectors. Quantum algorithms have been developed to advance these computational tasks, and some have been shown to provide substantial speedup, such as factoring a large integer and solving linear equations. In this work, we leverage the techniques used in the Harrow-Hassidim-Llyod (HHL) algorithm for linear systems, the classical power, and the Krylov subsapce method to devise a simple quantum algorithm for estimating the largest eigenvalues in magnitude of a Hermitian matrix. Our quantum algorithm offers significant speedup with respect to the size of a given matrix over classical algorithms that solve the same problem.