Applications of Machine Learning Techniques to the Quantum Many-Body Problem

Abstract

In recent years, the remarkable progress in machine learning has revolutionized various fields, including natural sciences, and in particular the physical sciences. One area where machine learning has shown great promise is in tackling complex problems in quantum many-body physics, a field that seeks to understand the behavior of large ensembles of interacting quantum particles. By leveraging the power and the mathematical principles of machine learning techniques, researchers have unlocked new avenues for exploring fundamental questions in quantum physics, enabling breakthroughs in the understanding of properties of condensed matter systems, quantum materials and chemical systems, and even the behavior of quantum computers.

The numerical approach to the quantum-many body problem and machine learning share interesting similarities as they are both rooted in the realms of applied linear algebra in large dimensional spaces, probability and statistics, and intricate optimization tasks. On one hand, the quantum-many body problem deals with an exponentially growing Hilbert space and the understanding of the complex collective phenomena of interacting particles. Similarly, machine learning addresses complex tasks by processing high-dimensional data (images, sequences of text, etc.), seeking to capture intricate patterns and relationships. Probability and statistics play a vital role as well. In the numerical many-body context they provide a framework to obtain approximate values for expectation values, while in the machine learning context they provide with tools to analyze and infer information from the available data. Moreover, complex optimization tasks arise in both arenas, since both a subset numerical quantum-many body and machine learning techniques rely on the optimization of a cost function. Consequently, it is only natural to deploy machine learning techniques to tackle the numerical approach to the quantum many-body problem.

The deep connection between machine learning and the quantum many-body problem holds immense potential for both fields. Machine learning algorithms can provide a deeper understanding of quantum systems, shedding light on fundamental questions and even enabling the design of new materials with tailored properties. Conversely, the inherent complexity and richness of quantum systems offer unique challenges and opportunities for developing novel machine learning techniques and algorithms. The cross-pollination of ideas between these disciplines promises to fuel innovation and drive fundamental advancements across a wide range of applications.

This thesis dissertation focuses on one of the research directions bringing together the quantum many-body problem and machine learning: the application of machine learning techniques to existing many-body numerical methods. In particular devoting most of our attention to the parametrization of fermionic wave functions using neural networks to study the ground-state properties of electronic systems (Part II of this manuscript) and to the representation of density functionals using neural networks in a supervised learning setting (Part III of this manuscript). The findings that culminated in this dissertation show that indeed machine learning techniques can improve standard numerical many-body methods.

Part I containing Chapters 1 and 2 is a preamble to Part II and Part III, which contain the chapters devoted to the most relevant research projects performed during the duration of the thesis. The aim of Chapter 1 is to review the mathematical principles of the wave-function-based description of quantum interacting particles. Chapter 1 also sets up the notation for the remainder of the manuscript. In particular, it focuses on the construction of the Hilbert space, wave function and operators for systems of indistinguishable particles of fermionic nature, including a comprehensive introduction to the Second Quantization formalism. The electronic structure problem and relevant simplified models are also introduced in this chapter. Chapter 1 ends with a description of the approximate many-body numerical techniques that are the basis for the application of machine learning techniques, including variational approaches like the Variational Monte Carlo, Variational Quantum Eigensolver and Density Matrix Renormalization Group, as well as Density Functional Theory-based techniques. Chapter 2 is devoted to the foundational concepts of the field of machine learning. The chapter begins by providing an extensive description of tasks that can be tackled using machine learning techniques, focusing on many-body applications. Chapter 2 also provides a comprehensive discussion around gradient-based optimization methods, including the properties of the vanilla and natural gradient descents. This discussion also details the impact that the stochastic estimation of gradients has in the classes of solutions that the algorithm converges to. Different classes of neural networks (the machine learning tool by excellence) and the universal approximation theorem make their appearance in Chapter 2. Chapter 2 ends with an introduction to statistical learning and the concept of generalization in supervised machine learning.

Part II (Chapters 3, 4, 5, 6 and 7) is devoted to the parametrization of fermionic wave functions using neural networks (Neural Quantum States) applied to the study of the ground-state properties of electronic systems. In particular these neural network parametrizations are employed in a variational setting, using the Variational Monte Carlo technique to optimize the variational parameters. The variational approach to the ground-state search involves the proposal of a trial wave function, which depends on a set of variational parameters that are determined by an optimization procedure. While neural networks are employed in this context, this technique is not a supervised learning approach, that requires training data. The authors of Ref. [113] propose to label it as a “Self-Generative Reinforcement Learning” within the machine learning taxonomy. Chapter 3 contains a fairly standard introduction to Variational Monte Carlo. Chapter 4 discusses different families of fermionic trial states, including the traditional, and physically motivated ansätze, as well as the more modern Neural Quantum States. This chapter helps putting into perspective the contributions of this dissertation to the field. Chapter 5 studies the use of neural-network correlation factors to correct the amplitudes and signs of a reference Slater determinant. This class of wave functions provides a universal wave function approximator. We apply this state to the ground state properties and phase diagram of spin-polarized lattice fermions in two dimensions. These results are part of the work published in Ref. [271]. Chapter 6 is devoted to he introduction of a new family of neural-network based trial states for fermionic systems. This family consists of Slater determinants in an augmented Hilbert space involving extra “hidden” fermionic degrees of freedom. The projection of Slater determinants in the augmented Hilbert space is parametrized by a neural network, that is optimized together with the Slater determinant itself to find the ground state of the system. This novel class of trial states provides an extremely flexible wave function that is shown to be universal. Furthermore, we show that it can explicitly and efficiently represent other classes of trial states, including linear combinations of combinatorially-many Slater determinants, and backflow wave functions. This class of trial states is applied to the ground-state properties of the Hubbard model in the square lattice, achieving levels of accuracy competitive with state-of-the-art variational methods. Some of these results are part of the work published in Ref. [245]. Chapter 7 describes a procedure based on orbital rotations to adapt the single-particle basis that defines the many-body Hamiltonian to maximize the expressive power of the trial state representing the ground-state wave function. These orbital rotations make the variational procedure invariant under the reparametrization of the problem given by single-particle basis transformations. We demonstrate that the optimization of the single-particle basis as part of the variational procedure yields improvements both in the accuracy of the converged ground state and the dynamics of the optimization itself. This procedure is tested for a collection of variational methods (Variational Monte Carlo with Neural Quantum States, Variational Quantum Eigensolver and Density Matrix Renormalization Group) in several electronic systems, including molecules and lattice models. Some of these results are part of the pre-print version of Ref. [244]. 

Part III (Chapters 8 and 9) is devoted to the use of neural networks to parametrize the Hohenberg-Kohn maps of Density Functional Theory within a supervised statistical learning framework. The Hohenberg-Kohn theorem establishes a one-to-one correspondence between the ground-state electronic density and the associated ground-state wave function. This also implies that the ground state and any other observable of the system are connected via an injective map. Chapter 8, which was the first research project of this thesis, analyzes the viability of the representation of the Hohenberg-Kohn maps using neural networks in a system of spin-polarized lattice fermions in one dimensions using synthetic data. We show that neural networks allow to accurately reconstruct the many-body ground state of the system and observables from the knowledge of the density. These findings are part of the work published in Ref. [242]. Chapter 9 contains the results of the research project which is a follow up to the proof of principle project described in Chapter 8. We show that neural networks can accurately represent the map between approximate ground-state density distributions and experimental band gaps of real semiconducting materials. The relevance of these findings is rooted in the observation that for modern electronic, optoelectronic, and photovoltaic applications, the reliable estimation of band gaps of solids is essential. However, the direct computation using traditional numerical approaches is an extremely resource-intensive task. These findings are part of the work published in Ref. [243].