Capturing the correlation emerging between constituents of many-body systems accurately is one of the key challenges for the appropriate description of various systems whose properties are underpinned by quantum mechanical fundamentals. This thesis discusses novel tools and techniques for the (classical) modelling of quantum many-body wavefunctions with the ultimate goal to introduce a universal framework for finding accurate representations from which system properties can be extracted efficiently. It is outlined how synergies with standard machine learning approaches can be exploited to enable an automated inference of the most relevant intrinsic characteristics through rigorous Bayesian regression techniques. Based on the probabilistic framework forming the foundation of the introduced ansatz, coined the Gaussian Process State, different compression techniques are explored to extract numerically feasible representations of relevant target states within stochastic schemes. By following intuitively motivated design principles, the resulting model carries a high degree of interpretability and offers an easily applicable tool for the numerical study of quantum systems, including ones which are notoriously difficult to simulate due to a strong intrinsic correlation. The practical applicability of the Gaussian Process States framework is demonstrated within several benchmark applications, in particular, ground state approximations for prototypical quantum lattice models, Fermi-Hubbard models and $J_1-J_2$ models, as well as simple ab-initio quantum chemical systems.
Capturing the correlation emerging between constituents of many-body systems accurately is one of the key challenges for the appropriate description of various systems whose properties are underpinned by quantum mechanical fundamentals. This thesis discusses novel tools and techniques for the (classical) modelling of quantum many-body wavefunctions with the ultimate goal to introduce a universal framework for finding accurate representations from which system properties can be extracted efficiently. It is outlined how synergies with standard machine learning approaches can be exploited to enable an automated inference of the most relevant intrinsic characteristics through rigorous Bayesian regression techniques. Based on the probabilistic framework forming the foundation of the introduced ansatz, coined the Gaussian Process State, different compression techniques are explored to extract numerically feasible representations of relevant target states within stochastic schemes. By following intuitively motivated design principles, the resulting model carries a high degree of interpretability and offers an easily applicable tool for the numerical study of quantum systems, including ones which are notoriously difficult to simulate due to a strong intrinsic correlation. The practical applicability of the Gaussian Process States framework is demonstrated within several benchmark applications, in particular, ground state approximations for prototypical quantum lattice models, Fermi-Hubbard models and J1−J2 models, as well as simple ab-initio quantum chemical systems.
The recently introduced Gaussian Process State (GPS) provides a highly flexible, compact, and physically insightful representation of quantum many-body states based on ideas from the zoo of machine learning approaches. In this work, we give a comprehensive description of how such a state can be learned from given samples of a potentially unknown target state and show how regression approaches based on Bayesian inference can be used to compress a target state into a highly compact and accurate GPS representation. By application of a type II maximum likelihood method based on relevance vector machines, we are able to extract many-body configurations from the underlying Hilbert space, which are particularly relevant for the description of the target state, as support points to define the GPS. Together with an introduced optimization scheme for the hyperparameters of the model characterizing the weighting of modeled correlation features, this makes it possible to easily extract physical characteristics of the state such as the relative importance of particular correlation properties. We apply the Bayesian learning scheme to the problem of modeling ground states of small Fermi-Hubbard chains and show that the found solutions represent a systematically improvable trade-off between sparsity and accuracy of the model. Moreover, we show how the learned hyperparameters and the extracted relevant configurations, characterizing the correlation of the wave function, depend on the interaction strength of the Hubbard model and the target accuracy of the representation.
The quantum many-electron problem is not just at the heart of condensed matter phenomena, but also essential for first-principles simulation of chemical phenomena. Strong correlation in chemical systems are prevalent and present a formidable challenge in the simulation of these systems, while predictive phenomena in this domain often also requires a demanding level of accuracy to inform chemical behavior. Efficient representations of the many-electron states of chemical systems are therefore also being inspired by machine learning principles to provide an alternative to established approaches. In this chapter, we review recent progress in this endeavor for quantum chemical problems represented in second quantization, and the particular challenges present in this field. In particular, we focus on the application of Gaussian Process States emerging from efficient representations of the many-body wavefunction with rigorous Bayesian modeling frameworks, allowing for the unification of multiple paradigms under a common umbrella. We show how such models (and other representations derived from machine learning) can be used as novel tools to compute ab initio chemical properties, while in turn also informing the design of machine learning models to extract correlation patterns in classical data.
Despite rapid recent advances in quantum machine learning, the field is in many ways stuck. Existing approaches can exhibit serious limitations, and we still lack learning frameworks that are simple, interpretable, scalable, and naturally suited to quantum data. To address this, here we introduce quantum Gaussian processes, a Bayesian framework for learning from quantum systems through priors over unknown quantum transformations. We show that, under suitable conditions, unitary quantum stochastic processes define Gaussian processes, thereby enabling regression, classification, and Bayesian optimization directly on quantum data. The key ingredient in this framework is sufficient knowledge of a quantum process's structure and symmetries to define an informative prior through its corresponding quantum kernel, effectively injecting a strong, physics-informed inductive bias into the learning model. We then prove that matchgate, or free-fermionic, evolutions give rise to provable and scalable quantum Gaussian processes, providing the first family in our framework where the unknown unitary acts non-trivially on all qubits. Finally, we demonstrate accurate long-range extrapolation, phase-diagram learning in many-body systems, and sample-efficient Bayesian optimization in a quantum sensing task. Our results identify quantum Gaussian processes as a promising route toward simpler and more structured forms of quantum learning.
We present a general framework for the efficient simulation of realistic fermionic systems with modern machine-learning-inspired representations of quantum many-body states, towards a universal tool for ab initio electronic structure. These machine-learning-inspired Ans\"atze have recently come to the fore in both a (first-quantized) continuum and discrete Fock space representations, where, however, the inherent scaling of the latter approach for realistic interactions has so far limited practical applications. With application to the ``Gaussian Process State,'' a recently introduced Ansatz inspired by systematically improvable kernel models in machine learning, we discuss different choices to define the representation of the computational Fock space. We show how local representations are particularly suited for stochastic sampling of expectation values, while also indicating a route to overcome the discrepancy in the scaling compared with continuum-formulated models. We are able to show competitive accuracy for systems with up to 64 electrons, including a simplified (yet fully ab initio) model of the Mott transition in three-dimensional hydrogen, indicating a significant improvement over similar approaches, even for moderate numbers of configurational samples.
This thesis illustrates the use of machine learning algorithms and exact numerical methods to study quantum observables for different systems. The first part of this thesis depicts how to construct accurate potential energy surfaces (PESs) using supervised learning algorithms such as Gaussian Process (GP) regression. PESs have a leading part in quantum chemistry since they are used to study chemical reaction dynamics. Constructing the PES from quantum reactive scattering calculations, as the reaction probability, is known as the inverse scattering problem. Here, we illustrate a possible solution to the inverse scattering problem with a two-tiered GP model one GP model interpolates the PES and the second in Bayesian optimization (BO) algorithm. The end result is an accurate PES constructed with a GP with fewer points than with standard methods previously used for PES. BO is an optimization algorithm for black-box functions that use GP regression as an approximation of the interrogative function. We applied BO to find the optimal parameters of hybrid-density functionals. Quantum observables can differ between phases of matter. GP models with kernel combinations can extrapolate quantum observables such as the polaron dispersion energy between different phases and discover phases of matter. The same algorithm can predict quantum observables where standard numerical techniques lack convergence. In the second half of the dissertation, we studied the evolution of quantum walks in various graphs with Hamiltonians permitting particle number changes. We showed that particle number-changing interactions accelerate quantum walks for any of the graph considered. Quantum simulators to study many-body physics is an active research field. We proposed the use of magnetic atoms trapped in optical lattices to experimentally mimic Bose-Hubbard type models by preparing atoms in different Zeeman states.
This thesis focuses on the development, application, and high-performance implementation of numerical methods for simulating quantum many-body systems. In the first chapter, I provide an overview of several most successful conventional approaches for simulating quantum many-body systems, along with their current limitations. Then, the basic properties of the quantum spin model, which will be the subject of study in the following chapters, and its ground state are briefly summarized. Additionally, I discuss the background and the motivations of the research projects carried out during my PhD research. Chapter 2 introduces two numerical algorithms that are extensively used in this thesis, which are exact diagonalization (ED) and variational Monte Carlo. I outline the process of constructing a Hamiltonian matrix and employing iterative methods for exact diagonalization. Subsequently, I introduce the mechanism of the four important components in the ``standard" variational Monte Carlo that are widely used in the most recent research. In Chapter 3, we propose a symmetry-projected variational ansatz in the form of linear combinations of simple restricted Boltzmann machines. This construction allows one to explore states outside of the original variational manifold and increase the representation power with moderate computational effort. Besides allowing one to restore spatial symmetries, an expansion in terms of Krylov states using a Lanczos recursion offers a solution that can further improve the quantum state accuracy. We illustrate these ideas with an application to the Heisenberg J1−J2 model on the square lattice, a paradigmatic problem under debate in condensed matter physics,and achieve state-of-the-art accuracy in the representation of the ground state. In Chapter 4, we present a variational approach based on a Chebyshev expansion of the spectral function, and each Chebyshev polynomial is represented by a neural network quantum state. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the "Chebyshev wave functions." We present results for the one-dimensional and two-dimensional Heisenberg models on the square lattice and compare them with those obtained by other methods in the literature. In Chapter 5, we describe a method to carry out thermodynamic simulations of quantum many-body models using typicality and variational representations of quantum states. Minimally entangled typical thermal states (METTS) are a construction that allows one to solve for the imaginary time evolution of quantum many-body systems. By using wave functions that are weakly entangled, one can take advantage of efficient representations in the form of matrixproduct states. We generalize these ideas to arbitrary variational wave functions and we focus, as an illustration, on the particular case of restricted Boltzmann machines. We demonstrate the application of the method to quantum spin systems in one and two spatial dimensions. In Chapter 6, we study the properties of the impurity band in heavily doped nonmagnetic semiconductors using the Jacobi-Davidson algorithm and the supervised deep learning method. The disorder averaged inverse participation ratio and Thouless number calculation show us the rich structure inside the impurity band. A convolutional neural network model, which is trained to distinguish the extended/localized phase of the Anderson model with high accuracy, shows us the results in good agreement with the conventional approach. Together, we find that there are three mobility edges in the impurity band for a specific on-site impurity potential, which means the presence of the extended states while filling the impurity band. In the final chapter, Chapter 7, we discuss the implementation of architecture-aware optimizations for spin dynamics simulations on x86 CPUs and Nvidia GPUs. The time cost of evaluating the spin-spin correlation functions in real space increases quadratically as the number of spins N, leading to significant computational effort, making the simulation of large spin systems very challenging. We propose to use a highly optimized general matrix multiply (GEMM) subroutine to calculate the dynamical spin-spin correlation function that can achieve near-optimal hardware utilization. Furthermore, we fuse the element-wise operations in the calculation of S(q,t) into the in-house GEMM kernel to reduce memory latency. On Nvidia GPU, we perform the on-the-fly calculation in the epilogue of the GEMM subroutine to avoid saving intermediate results on global memory, which significantly reduces memory usage and achieves better performance than cuBLAS.--Author's abstract