Bayesian optimization of variational wave functions

Traditional variational wave-functions, such as those of the Jastrow type, or Gutzwiller projected, for instance, are typically derived from analytical solutions or educated guesses based on some physical insight. However, they are limited by the available phase space defined by their functional form and the number of variational parameters. We present an approach based on systematically improving the wave function approximator by representing the wave-function as a Gaussian Process (GP), in which the state is systematically modified by changing the parameters of the kernel. GPs allow one to guess unknown data by means of Bayesian inference. Our protocol proceeds as follows: (i) pick an initial partition of the Hilbert space using, for instance, importance sampling; (ii) use the initial variational guess as the prior; (iii) interpolate using a GP with a pre-defined metric defined by a chosen variational manifold; (iv) optimize the GP by modifying the parameters in the kernel in order to minimize the energy. This process has to be carried out using variational Monte Carlo to estimate the different measures required in the optimization of the energy. We demonstrate the method on the Heisenberg model in one and two dimensions.