  Quantum algorithms for ground-state energy estimation of chemical systems
require a high-quality initial state. However, initial state preparation is
commonly either neglected entirely, or assumed to be solved by a simple product
state like Hartree-Fock. Even if a nontrivial state is prepared, strong
correlations render ground state overlap inadequate for quality assessment. In
this work, we address the initial state preparation problem with an end-to-end
algorithm that prepares and quantifies the quality of initial states,
accomplishing the latter with a new metric -- the energy distribution. To be
able to prepare more complicated initial states, we introduce an implementation
technique for states in the form of a sum of Slater determinants that exhibits
significantly better scaling than all prior approaches. We also propose
low-precision quantum phase estimation (QPE) for further state quality
refinement. The complete algorithm is capable of generating high-quality states
for energy estimation, and is shown in select cases to lower the overall
estimation cost by several orders of magnitude when compared with the best
single product state ansatz. More broadly, the energy distribution picture
suggests that the goal of QPE should be reinterpreted as generating
improvements compared to the energy of the initial state and other classical
estimates, which can still be achieved even if QPE does not project directly
onto the ground state. Finally, we show how the energy distribution can help in
identifying potential quantum advantage.
