Scaling and generalization in neural quantum states

Abstract

Because of their relevance to our understanding of quantum materials, the ground states of quantum many-body systems are a central object of study in condensed matter physics. However, the exponential growth of the Hilbert space with physical system size makes it difficult to study these states. Even with sophisticated numerical methods, we are often limited to studying finite-size systems that may not be representative of the thermodynamic limit, where properties of the system coincide with measurements of real materials. Scaling studies are therefore crucial for bridging the gap between what we can compute and what we want to understand. Quantum simulators, which are engineered and programmable quantum systems, offer an alternative approach to understanding quantum many-body systems. While these devices enable the direct preparation of certain quantum states, the need to verify the states prepared on these devices presents another exponentially difficult problem.

The tools of modern deep learning, such as neural networks, have proven to be extraordinarily capable of extracting patterns in complex and high-dimensional data. Crucially, the learned patterns often correctly describe new data that the network was not exposed to during training, a phenomenon known as generalization. Despite belonging to exponentially large Hilbert spaces, quantum many-body ground states are often highly structured. Neural networks, which generalize precisely by learning and exploiting such structure, offer a promising approach to the problems of representing and characterizing such states.

In this thesis, I focus on the use of neural networks to study the ground states of quantum many-body systems. When used in this context, neural networks are referred to as neural quantum states (NQS). Not only are NQS flexible and expressive ansätze, but they can be trained in different ways, depending on the information about the target state that is available. On the one hand, NQS can be trained with a data-driven approach, using measurement data from quantum simulators. We show that, because of their generalization abilities, NQS are poised to maximize the value of limited and imperfect data from experiments on today's quantum devices. On the other hand, NQS can be trained with a Hamiltonian-driven approach, which only requires knowledge of a system's Hamiltonian. Using this approach, we demonstrate how the generalization abilities of certain NQS architectures can be leveraged to enable efficient and accurate large-scale simulations of quantum many-body systems. Finally, we directly investigate generalization in the context of NQS, connecting our results to important observations in the broader machine learning research community. Together, these results demonstrate that the generalization abilities of NQS are not only essential for, but fundamentally linked to, their capacity to enable scalable studies of quantum many-body systems.