  We provide a modification to the quantum phase estimation algorithm (QPEA)
inspired on classical windowing methods for spectral density estimation. From
this modification we obtain an upper bound in the cost that implies a cubic
improvement with respect to the algorithm's error rate. Numerical evaluation of
the costs also demonstrates an improvement. Moreover, with similar techniques,
we detail an iterative projective measurement method for ground state
preparation that gives an exponential improvement over previous bounds using
QPEA. Numerical tests that confirm the expected scaling behavior are also
obtained. For these numerical tests we have used a Lattice Thirring model as
testing ground. Using well-known perturbation theory results, we also show how
to more appropriately estimate the cost scaling with respect to state error
instead of evolution operator error.
