Quantum phase estimation for a class of generalized eigenvalue problems

Abstract

Quantum phase estimation provides a path to quantum computation of solutions to Hermitian eigenvalue problems $Hv=\ensuremath{\lambda}v$, such as those occurring in quantum chemistry. It is natural to ask whether the same technique can be applied to generalized eigenvalue problems $Av=\ensuremath{\lambda}Bv$, which arise in many areas of science and engineering. We answer this question affirmatively. A restricted class of generalized eigenvalue problems could be solved as efficiently as standard eigenvalue problems. A paradigmatic example is provided by Sturm-Liouville problems. Another example comes from linear ideal magnetohydrodynamics, where phase estimation could be used to determine the stability of magnetically confined plasmas in fusion reactors.