  One of the ways to encode many-body Hamiltonians on a quantum computer to
obtain their eigen-energies through Quantum Phase Estimation is by means of the
Trotter approximation. There were several ways proposed to assess the quality
of this approximation based on estimating the norm of the difference between
the exact and approximate evolution operators. Here, we would like to explore
how these different error estimates are correlated with each other and whether
they can be good predictors for the true Trotter approximation error in finding
eigenvalues. For a set of small molecular systems we calculated the exact
Trotter approximation errors of the first order Trotter formulas for the ground
state electronic energies. Comparison of these errors with previously used
upper bounds show almost no correlation over the systems and various
Hamiltonian partitionings. On the other hand, building the Trotter
approximation error estimation based on perturbation theory up to a second
order in the time-step for eigenvalues provides estimates with very good
correlations with the Trotter approximation errors. The developed perturbative
estimates can be used for practical time-step and Hamiltonian partitioning
selection protocols, which are paramount for an accurate assessment of
resources needed for the estimation of energy eigenvalues under a target
accuracy.
