Treffer: Neural quantum state methods for simulating quantum many-body systems

Computational methods for the efficient simulation of quantum many-body systems are crucial for the study of condensed matter physics. In this thesis, we investigate numerical properties of neural quantum states (NQS), a machine-learning-inspired variational ansatz based on using an artificial neural network to represent the quantum wave function. This representation can be used to stochastically estimate quantum expectation values, and the NQS ansatz can be trained to approximate ground states as well as real-time dynamics of quantum systems by classical optimization algorithms. First, we investigate the stability of NQS time propagation with the time-dependent variational Monte Carlo method. Using the antiferromagnetic Heisenberg ladder as a benchmark system, we find that stochastic noise inherent to Monte Carlo sampling can be amplified by the variational equation of motion which can cause numerical instabilities. We propose an error diagnostic that can be used to quantify this effect and demonstrate the influence of regularization methods for the equation of motion on the stability of the dynamics. Subsequently, we discuss the importance of symmetries for improving NQS ground state calculations and propose a symmetry-projection scheme for the honeycomb Kitaev model. Furthermore, we present results of a systematic study of the capabilities of NQS based on feed-forward neural networks to represent highly entangled ground states in the Sachdev-Ye-Kitaev model. In this case, we find that this NQS ansatz does not learn a more efficient representation compared to the exponential scaling of the exact quantum states. This observation highlights the importance of further study to determine which properties decide whether a quantum state is amenable to an efficient approximation by neural quantum states. Finally, we present NetKet, an open-source project and software framework for numerical calculations in quantum many-body systems based on the NQS ansatz and variational Monte Carlo. Altogether, our work highlights ...