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Quantum phase estimation for a class of generalized eigenvalue problems

Quantum phase estimation provides a path to quantum computation of solutions to Hermitian eigenvalue problems $Hv = λv$, such as those occurring in quantum chemistry. It is natural to ask whether the same technique can be applied to generalized eigenvalue problems $Av = λB v$, 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.