  The two-dimensional Hubbard model is studied using the variational quantum
Monte Carlo technique with Gutzwiller-type variational wave functions. In
addition to the simple one-site correlated Gutzwiller wave function, we use a
form with correlation between electrons on nearest- and next-nearest-neighbor
sites. We show that the stochastic gradient approximation method suits very
well for the optimization of the free parameters of the variational wave
functions.
