Overview of Projective Quantum Monte Carlo Methods for Stoquastic Hamiltonians

Abstract

Projective quantum Monte Carlo algorithms are among the most powerful computational techniques to simulate the ground state properties of quantum many-body systems. In the present thesis, we provide an implementation of the algorithms both in the simple case where no knowledge of the wave function is needed and in the case of importance sampling where we advocate the use of a guiding wave function to guide the simulation. We show that in the latter, the algorithm converges faster with less relative error. In the particular case of the 1D transverse field Ising model, we find the algorithm to be unstable due to the biases introduced by the system size, time step, and auto-correlation. We find in the case of non-symmetrized and symmetrized Trotter approximation of the Green’s function a linear and quadratic dependence of the relative error on the time step, respectively, which is nothing more than the consequence of the bias due to the time step. First, we address the issue of auto-correlation in our data set with the so-called block averaging, which yields uncorrelated data sets. Later, we implement projective quantum Monte Carlo in the continuous time limit, which enables us to eradicate completely the error due to the time step. Finally, we propose a protocol using recurrent neural networks that enables us to formally generalize the idea of generating, through an unsupervised self-learning process, a good approximated guiding wave function.