  Local Hamiltonians of fermionic systems on a lattice can be mapped onto local
qubit Hamiltonians. Maintaining the locality of the operators comes at the
expense of increasing the Hilbert space with auxiliary degrees of freedom. In
order to retrieve the lower-dimensional physical Hilbert space that represents
fermionic degrees of freedom, one must satisfy a set of constraints. In this
work, we introduce quantum circuits that exactly satisfy these stringent
constraints. We demonstrate how maintaining locality allows one to carry out a
Trotterized time-evolution with constant circuit depth per time step. Our
construction is particularly advantageous to simulate the time evolution
operator of fermionic systems in d>1 dimensions. We also discuss how these
families of circuits can be used as variational quantum states, focusing on two
approaches: a first one based on general constant-fermion-number gates, and a
second one based on the Hamiltonian variational ansatz where the eigenstates
are represented by parametrized time-evolution operators. We apply our methods
to the problem of finding the ground state and time-evolved states of the
$t$-$V$ model.
