  Approximating the ground states of strongly interacting electron systems in
quantum chemistry and condensed matter physics is expected to be one of the
earliest applications of quantum computers. In this paper, we prepare highly
accurate ground states of the Fermi-Hubbard model for small grids up to 6 sites
(12 qubits) by using an interpretable, adaptive variational quantum
eigensolver(VQE) called ADAPT-VQE. In contrast with non-adaptive VQE, this
algorithm builds a system-specific ansatz by adding an optimal gate built from
one-body or two-body fermionic operators at each step. We show this adaptive
method outperforms the non-adaptive counterpart in terms of fewer variational
parameters, short gate depth, and scaling with the system size. The fidelity
and energy of the prepared state appear to improve asymptotically with ansatz
depth. We also demonstrate the application of adaptive variational methods by
preparing excited states and Green functions using a proposed ADAPT-SSVQE
algorithm. Lower depth, asymptotic convergence, noise tolerance of a
variational approach, and a highly controllable, system-specific ansatz make
the adaptive variational methods particularly well-suited for NISQ devices.
