Solving generalized eigenvalue problems by ordinary differential equations on a quantum computer
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Abstract
Many eigenvalue problems arising in practice are often of the generalized form Ax=λBx. One particularly important case is symmetric, namely A,B are Hermitian and B is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than the standard one based on quantum phase estimation. Moreover, it works for a wider case than symmetric: B is invertible, B−1A is diagonalizable and all the eigenvalues are real.