  A well-known method to prepare ground states of fermionic many-body
hamiltonians is adiabatic state preparation, in which an easy to prepare state
is time-evolved towards an approximate ground state under a specific
time-dependent hamiltonian. However, which path to take in the evolution is
often unclear, and a direct linear interpolation, which is the most common
method, may not be optimal. In this work, we explore new types of adiabatic
paths based on the spectral decomposition of the two-body projection of the
residual hamiltonian (the difference between the final and initial
hamiltonian). The decomposition defines a set of hamiltonian terms which may be
adiabatically interpolated in a piecewise or combined fashion. We demonstrate
the usefulness of partially piecewise interpolation through examples involving
Fermi-Hubbard models where, due to symmetries, level crossings occur in direct
(fully combined) interpolation. We show that this specific deviation from a
direct path appropriately breaks the relevant symmetries, thus avoiding level
crossings and enabling an adiabatic passage. On the other hand, we show that a
fully piecewise scheme, which interpolates every hamiltonian term separately,
exhibits a worst-case complexity of $O(L^6/\Delta^3)$ as compared to
$O(L^4/\Delta^3)$ for direct interpolation, in terms of the number of one-body
modes $L$ and the minimal gap $\Delta$ along the path. This suboptimality
result suggests that only those terms which break necessary symmetries should
be taken into account for piecewise interpolation, while the rest is treated
with direct interpolation.
