  Solving eigenvalue problems is crucially important for both classical and
quantum applications. Many well-known numerical eigensolvers have been
developed, including the QR and the power methods for classical computers, as
well as the quantum phase estimation(QPE) method and the variational quantum
eigensolver for quantum computers. In this work, we present an alternative type
of quantum method that uses fixed-point quantum search to solve Type II
eigenvalue problems. It serves as an important complement to the QPE method,
which is a Type I eigensolver. We find that the effectiveness of our method
depends crucially on the appropriate choice of the initial state to guarantee a
sufficiently large overlap with the unknown target eigenstate. We also show
that the quantum oracle of our query-based method can be efficiently
constructed for efficiently-simulated Hamiltonians, which is crucial for
analyzing the total gate complexity. In addition, compared with the QPE method,
our query-based method achieves a quadratic speedup in solving Type II
problems.
