Generalized Lanczos algorithm for variational quantum Monte Carlo

We show that the standard Lanczos algorithm can be efficiently implemented statistically and self-consistently improved, using the stochastic reconfiguration method, which has been recently introduced to stabilize the Monte Carlo sign problem instability. With this scheme a few Lanczos steps over a given variational wave function are possible even for large size as a particular case of a more general and more accurate technique that allows to obtain lower variational energies. This method has been tested extensively for a strongly correlated model like the $t-J$ model. With the standard Lanczos technique it is possible to compute any kind of correlation functions, with no particular computational effort. By using the fact that the variance $〈H^2〉-〈H{〉}^2$ is zero for an exact eigenstate, we show that the approach to the exact solution with few Lanczos iterations is indeed possible even for $\sim 100$ electrons for reasonably good initial wave functions. The variational stochastic reconfiguration technique presented here allows in general a many-parameter energy optimization of any computable many-body wave function, including for instance generic long-range Jastrow factors and arbitrary site-dependent orbital determinants. This scheme improves further the accuracy of the calculation, especially for long-distance correlation functions.