  Variational quantum eigensolvers (VQEs) represent a powerful class of hybrid
quantum-classical algorithms for computing molecular energies. Various
numerical issues exist for these methods, however, including barren plateaus
and large numbers of local minima. In this work, we consider Adaptive,
Problem-Tailored (ADAPT)-VQE ans\"atze, and examine how they are impacted by
these local minima. We find that while ADAPT-VQE does not remove local minima,
the gradient-informed, one-operator-at-a-time circuit construction seems to
accomplish two things: First, it provides an initialization strategy that is
dramatically better than random initialization, and which is applicable in
situations where chemical intuition cannot help with initialization, i.e., when
Hartree-Fock is a poor approximation to the ground state. Second, even if an
ADAPT-VQE iteration converges to a local trap at one step, it can still
"burrow" toward the exact solution by adding more operators, which
preferentially deepens the occupied trap. This same mechanism helps highlight a
surprising feature of ADAPT-VQE: It should not suffer optimization problems due
to "barren plateaus". Even if barren plateaus appear in the parameter
landscape, our analysis and simulations reveal that ADAPT-VQE avoids such
regions by design.
