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