Simulating quantum annealing via projective quantum Monte Carlo algorithms

In this thesis, we implement projective quantum Monte Carlo (PQMC) methods to simulate quantum annealing on classical computers. A detailed derivation of the algorithm is presented. We show that in the regime where the systematic errors are well controlled, PQMC algorithms are indeed capable of simulating the imaginary-time dynamics of the Schroedinger equation both on continuous space models and discrete basis systems. We also demonstrate that the tunneling time of the PQMC method is quadratically faster than the one of incoherent quantum tunneling. It also shows remarkable stability when applied to frustrated systems compared to the path integral Monte Carlo algorithm, the algorithm mostly chosen to do comparisons with quantum annealers. However, a major drawback of the method comes from the finite number of random walkers needed to implement the simulations. It grows exponentially with the system size when no or poor guiding wave-functions are utilized. Nevertheless, we demonstrated that when good enough guiding wave-functions are used -- in our case we choose artificial neural networks -- the computational complexity seems to go from exponential to polynomial in the system size. We advocate for a search of more efficient guiding wave functions since they determine when and how feasible the PQMC simulations are on classical computers, a question closely related to a provable need or speed-up of a quantum computer.