Local Hamiltonians of fermionic systems on a lattice can be mapped onto local qubit Hamiltonians. Maintaining the lo-cality of the operators comes at the ex-pense of increasing the Hilbert space with auxiliary degrees of freedom. In order to retrieve the lower-dimensional physical Hilbert space that represents fermionic de-grees of freedom, one must satisfy a set of constraints. In this work, we intro-duce quantum circuits that exactly satisfy these stringent constraints. We demon-strate how maintaining locality allows one to carry out a Trotterized time-evolution with constant circuit depth per time step. Our construction is particularly advanta-geous to simulate the time evolution op-erator of fermionic systems in d>1 di-mensions. We also discuss how these families of circuits can be used as vari-ational quantum states, focusing on two approaches: a first one based on gen-eral 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 meth-ods to the problem of finding the ground state and time-evolved states of the t -V model.