  The determination of the ground state of quantum many-body systems via
digital quantum computers rests upon the initialization of a sufficiently
educated guess. This requirement becomes more stringent the greater the system.
Preparing physically-motivated ans\"{a}tze on quantum hardware is therefore
important to achieve quantum advantage in the simulation of correlated
electrons. In this spirit, we introduce the Gutzwiller Wave Function (GWF)
within the context of the digital quantum simulation of the Fermi-Hubbard
model. We present a quantum routine to initialize the GWF that comprises two
parts. In the first, the noninteracting state associated with the $U = 0$ limit
of the model is prepared. In the second, the non-unitary Gutzwiller projection
that selectively removes states with doubly-occupied sites from the wave
function is performed by adding to every lattice site an ancilla qubit, the
measurement of which in the $|0\rangle$ state confirms the projection was made.
Due to its non-deterministic nature, we estimate the success rate of the
algorithm in generating the GWF as a function of the lattice size and the
interaction strength $U/t$. The scaling of the quantum circuit metrics and its
integration in general quantum simulation algorithms are also discussed.
