Abstract: Quantum computers promise to allow quasi-exact solutions of quantum many-body problems in chemistry and physics without the exponential scaling that plagues classical methods. Among the many ways of exploiting quantum computers, hybrid algorithms known as Variational Quantum Eigensolvers (VQEs) which are based on the variational principle of quantum mechanics, are being developed particularly intensively. These algorithms allocate optimization of wave functions to classical computers, using their quantum counterparts only to realize the parameterized state that the optimization scheme calls for. The result is fewer quantum operations at the expense of more measurements, leading to the hope that near-term quantum circuits, which are noisy, can implement the procedures without becoming too inaccurate. 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.