  Because quantum simulation of molecular systems is expected to provide the
strongest advantage over classical computing methods for systems exhibiting
strong electron correlation, it is critical that the performance of VQEs be
assessed for strongly correlated systems. For classical simulation, strong
correlation often results in symmetry-breaking of the Hartree-Fock reference,
leading to L\"owdin's well-known ``symmetry dilemma'' whereby accuracy in the
energy can be increased by breaking spin or spatial symmetries. Here, we
explore the impact of symmetry breaking on the performance of ADAPT-VQE using
two strongly correlated systems: (i) the ``fermionized" anisotropic Heisenberg
model, where the anisotropy parameter controls the correlation in the system,
and (ii) symmetrically-stretched linear \ce{H4}, where correlation increases
with increasing H-H separation. In both of these cases, increasing the level of
correlation of the system leads to spontaneous symmetry breaking (parity and
$\hat{S}^{2}$, respectively) of the mean-field solutions. We analyze the role
that symmetry breaking in the reference states and orbital mappings of the
fermionic Hamiltonians have on the compactness and performance of ADAPT-VQE. We
observe that improving the energy of the reference states by breaking symmetry
has a deleterious effect on ADAPT-VQE by increasing the length of the ansatz
necessary for energy convergence and exacerbating the problem of ``gradient
troughs".
