  Quantum subspace diagonalization methods are an exciting new class of
algorithms for solving large\rev{-}scale eigenvalue problems using quantum
computers. Unfortunately, these methods require the solution of an
ill-conditioned generalized eigenvalue problem, with a matrix pair corrupted by
a non-negligible amount of noise that is far above the machine precision.
Despite pessimistic predictions from classical \rev{worst-case} perturbation
theories, these methods can perform reliably well if the generalized eigenvalue
problem is solved using a standard truncation strategy. By leveraging and
advancing classical results in matrix perturbation theory, we provide a
theoretical analysis of this surprising phenomenon, proving that under certain
natural conditions, a quantum subspace diagonalization algorithm can accurately
compute the smallest eigenvalue of a large Hermitian matrix. We give numerical
experiments demonstrating the effectiveness of the theory and providing
practical guidance for the choice of truncation level. Our new results can also
be of independent interest to solving eigenvalue problems outside the context
of quantum computation.
