By design, the variational quantum eigensolver (VQE) strives to recover the lowest-energy eigenvalue of a given Hamiltonian by preparing quantum states guided by the variational principle. In practice, the prepared quantum state is indirectly assessed by the value of the associated energy. Novel adaptive derivative-assembled pseudo-trotter (ADAPT) ansatz approaches and recent formal advances now establish a clear connection between the theory of quantum chemistry and the quantum state ansatz used to solve the electronic structure problem. Here we benchmark the accuracy of VQE and ADAPT-VQE to calculate the electronic ground states and potential energy curves for a few selected diatomic molecules, namely H2, NaH, and KH. Using numerical simulation, we find both methods provide good estimates of the energy and ground state, but only ADAPT-VQE proves to be robust to particularities in optimization methods. Another relevant finding is that gradient-based optimization is overall more economical and delivers superior performance than analogous simulations carried out with gradient-free optimizers. The results also identify small errors in the prepared state fidelity which show an increasing trend with molecular size.

We study the classical and quantum percolation of spheres in a three-dimensional continuum. Each sphere has an impenetrable hard core of diameter (J, and two spheres are considered to be directly connected if the distance between their centers is less than d. We calculate the critical percolation density as a function of (J/d. In the classical problem this is the density P at which a~ i~finite cluster of connected spheres first forms. In the quantum problem, we stud~ a tight-bmdl~g model where the hopping matrix element between two spheres is nonzero only if they are dlrectly.con~ected. In this case the critical density P q is the density at which the eigenstates of the Hamlltoman first become extended. Our method uses Monte Carlo simulation and finite-size scaling techniques, and for the quantum problem, the concept of quantum connectivity. We find that both Pc and P q exhibit nonmonotonic behavior as a function of (J/d. We also find that for all values of (J/d, P q >Pc> although the ratio of the thresholds decreases with increasing (J/d. We argue that a better understanding of this ratio is obtained by considering the average coordination number. We speculate about the nature of both classical and quantum percol~tion as (J/d approaches 1.

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