  We use the Lipkin-Meshkov-Glick (LMG) model and the valence-space nuclear
shell model to examine the likely performance of variational quantum
eigensolvers in nuclear-structure theory. The LMG model exhibits both a phase
transition and spontaneous symmetry breaking at the mean-field level in one of
the phases, features that characterize collective dynamics in medium-mass and
heavy nuclei. We show that with appropriate modifications, the ADAPT-VQE
algorithm, a particularly flexible and accurate variational approach, is not
troubled by these complications. We treat up to 12 particles and show that the
number of quantum operations needed to approach the ground-state energy scales
linearly with the number of qubits. We find similar scaling when the algorithm
is applied to the nuclear shell model with realistic interactions in the $sd$
and $pf$ shells. Although most of these simulations contain no noise, we use a
noise model from real IBM hardware to show that for the LMG model with four
particles, weak noise has no effect on the efficiency of the algorithm.
