  We perform a systematic investigation of variational forms (wave function
Ans\"atze), to determine the ground state energies and properties of
two-dimensional model fermionic systems on triangular lattices (with and
without periodic boundary conditions), using the Variational Quantum
Eigensolver (VQE) algorithm. In particular, we focus on the nature of the
entangler blocks which provide the most efficient convergence to the system
ground state inasmuch as they use the minimal number of gate operations, which
is key for the implementation of this algorithm in NISQ computers. Using the
concurrence measure, the amount of entanglement of the register qubits is
monitored during the entire optimization process, illuminating its role in
determining the efficiency of the convergence. Finally, we investigate the
scaling of the VQE circuit depth as a function of the desired energy accuracy.
We show that the number of gates required to reach a solution within an error
$\varepsilon$ follows the Solovay-Kitaev scaling,
$\mathcal{O}(\log^c(1/\varepsilon))$, with an exponent $c = 1.31
{\rm{\pm}}0.13$.
