  Hybrid quantum-classical algorithms have been proposed to circumvent noise
limitations in quantum computers. Such algorithms delegate only a calculation
of the expectation value to the quantum computer. Among them, the Variational
Quantum Eigensolver (VQE) has been implemented to study molecules and condensed
matter systems on small size quantum computers. Condensed matter systems
described by the Hubbard model exhibit a rich phase diagram alongside exotic
states of matter. In this manuscript, we try to answer the question: how much
of the underlying physics of a 1D Hubbard chain is described by a
problem-inspired Variational Hamiltonian Ansatz (VHA) in a broad range of
parameter values ? We start by probing how much does the solution increases
fidelity with increasing ansatz complexity. Our findings suggest that even low
fidelity solutions capture energy and number of doubly occupied sites well,
while spin-spin correlations are not well captured even when the solution is of
high fidelity. Our powerful simulation platform allows us to incorporate a
realistic noise model and shows a successful implementation of noise-mitigation
strategies - post-selection and the Richardson extrapolation. Finally, we
compare our results with an experimental realization of the algorithm on IBM
Quantum's ibmq_quito device.
