  The widespread use of the noninteracting ground state as the initial state
for the digital quantum simulation of the Fermi-Hubbard model is largely due to
the scarcity of alternative easy-to-prepare approximations to the exact ground
state in the literature. Exploiting the fact that the spin-$\frac{1}{2}$
Heisenberg model is the effective low-energy theory of the Fermi-Hubbard model
at half-filling in the strongly interacting limit, here we propose a three-step
deterministic quantum routine to prepare an educated guess of the ground state
of the Fermi-Hubbard model through a shallow circuit suitable for near-term
quantum hardware. First, the ground state of the Heisenberg model is
initialized via a hybrid variational method using an ansatz that explores only
the correct symmetry subspace. Second, a general method is devised to convert a
multi-spin-$\frac{1}{2}$ wave function into its fermionic version. Third,
taking inspiration from the Baeriswyl ansatz, a constant-depth single-parameter
layer that adds doublon-holon pairs is applied to this fermionic state.
Numerical simulations on chains and ladders with up to 12 sites confirm the
improvement over the noninteracting ground state of the overlap with the exact
ground state for the intermediate values of the interaction strength at which
quantum simulation is bound to be most relevant.
