Optimizing quantum phase estimation for the simulation of Hamiltonian eigenstates

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 its outputs. We introduce the estimation of the first trigonometric moment of the parent distribution associated with eigenstate inputs as a new post-processing direction. By showing how it connects with the unknown phase, we find that if used as its direct estimator it permits to match the accuracy of the standard majority rule with one qubit less, allowing for shallower algorithms. Moreover, we make evident this quantity can also be inverted to provide an unbiased estimator of the phase. 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 make use of Trotterization and reduce the variability of NISQ results.