Solving generalized eigenvalue problems by ordinary differential equations on a quantum computer

Abstract

Many eigenvalue problems arising in practice are often of the generalized formAx=λBx. One particularly important case is symmetric, namelyA,Bare Hermitian andBis 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:Bis invertible,B−1Ais diagonalizable and all the eigenvalues are real.