  We revisit quantum phase estimation algorithms for the purpose of obtaining
the energy levels of many-body Hamiltonians and pay particular attention to the
statistical analysis of their outputs. We introduce the mean phase direction of
the parent distribution associated with eigenstate inputs as a new
post-processing tool. By connecting it with the unknown phase, we find that if
used as its direct estimator, it exceeds the accuracy of the standard majority
rule using one less bit of resolution, making evident that it can also be
inverted to provide unbiased estimation. Moreover, we show how to directly use
this quantity to accurately find the energy levels when the initialized state
is an eigenstate of the simulated propagator during the whole time evolution,
which allows for shallower algorithms. We then use IBM Q hardware to carry out
the digital quantum simulation of three toy models: a two-level system, a
two-spin Ising model and a two-site Hubbard model at half-filling.
Methodologies are provided to implement Trotterization and reduce the
variability of results in noisy intermediate scale quantum computers.
