Neural quantum states from a computational perspective

Abstract

The main problem of computational quantum physics is the the Curse of Dimensionality, that is the exponential growth of the length of the description, needed to specify a many-body/part quantum state, with the increase of the number of parts the system consists of. Advances in the field on Machine Learning brought new perspectives, and, relatively recently, the idea of Neural Quantum States (NQSs), that are the states which can be encoded with a specific type of artificial neural networks, the so-called Restricted Boltzmann Machines (RBMs), was proposed. It was shown that NQSs are able to capture complex quantum states characterized by a high degree of entanglement, while keeping the length of the state description under control. In this thesis we consider the NQS concept from an ’operational’ point of view, by addressing such computational aspects as scaling of time and memory size (the number of parameters needed to specify a NQS) with the size of a quantum model. For benchmarking we use two scalable models, a completely random Hamiltonian and an Ising chain. We demonstrate that the scaling are essentially different for these models which result highlights an interesting link between quantum physics and Machine Learning. We also discuss different strategies which could help to enhance the convergence, while keeping computational cost at a minimum. Especially, we exploit the structure of the Ising Hamiltonian to speed up the training of the network. This thesis has resulted in a Python framework that implements Neural Quantum State (NQS) as a way to approximate ground state energies for discrete quantum spin systems. The framework can implement NQSs for general Hamiltonians and Ising Hamiltonians, and trains the NQS by using a variational quantum Monte Carlo method to approximate the energy ground state of discrete quantum spin systems. We show that our implementation can achieve promising results with sub-exponential growth of computational cost for the Ising model. We propose a heuristic N_λ = K · N^3 for finding a good number of Markov Chain Monte Carlo (MCMC) steps, which makes the sampling independent of the system size. The experiments done in this thesis shows that the implementation manages to approximate the ground state energy of structured Ising Hamiltonians with sub-exponential number of parameters.