The study of quantum many-body systems remains a central challenge in condensed matter physics due to the exponential growth of the Hilbert space with system size. Traditional numerical approaches, such as exact diagonalization, quantum Monte Carlo, and tensor networks, often struggle with respective computational limitations, particularly in two and three-dimensional systems with strong correlations. In recent years, deep learning has emerged as a powerful alternative, offering new perspectives on quantum state representation and simulation. This thesis explores the large-scale simulation of deep neural quantum states (NQSs), leveraging artificial neural networks to approximate quantum many-body wave functions efficiently.
We begin by introducing the foundational concepts of NQS and their potential in overcoming the curse of dimensionality in quantum many-body problems. Variational Monte Carlo (VMC) is employed as the primary optimization framework, allowing the ground-state properties of complex Hamiltonians to be explored stochastically. The stochastic reconfiguration (SR) and minimum-norm SR (MinSR) are then introduced for accurate and efficient optimization of deep NQSs. We further investigate various neural network architectures, including feed-forward neural networks, restricted Boltzmann machines, convolutional neural networks, and transformer-based wave functions, assessing their expressivity and efficiency in encoding quantum correlations in strongly correlated quantum matters. The symmetry projection of NQSs starting from the group theory is introduced in detail, and particular attention is given to symmetry-preserving network architectures.
With the help of VMC, MinSR, and symmetry projection, the NQS achieves outstanding accuracy and outperforms existing numerical methods in several benchmark models of spin systems. We then show its application in quantum spin liquids (QSLs), the study of which relies heavily on the advances of computational methods. The NQS provides an accurate estimation of phase diagrams and energy gaps of several QSL candidates in the square J1-J2 model, the triangular J1-J2 model, and the Shastry-Sutherland model, leading to a deeper understanding of the emergence of QSL in frustrated magnets.
In fermion systems, fermionic mean-field wavefunctions are introduced to combine with the NQS for efficient expression of fermion sign structures. We then discuss the application of fermionic NQSs in the Fermi-Hubbard model, which reflects the mechanism of the Mott-insulator transition and probably high-temperature superconductivity. The fermionic NQS successfully reproduces the insulator transition and the spin density wave predicted by other numerical methods and further observes the possible superconducting order in the Hubbard model.
These findings highlight the immense potential of deep NQSs in quantum many-body physics. By combining large-scale neural networks with variational quantum algorithms, we make the NQS a powerful tool in solving quantum many-body systems, paving the way for future advancements in the computational study of quantum materials.