This thesis is divided into three chapters. In the first chapter we outline a simple and numerically inexpensive approach to describe the spectral features of the single-impurity Anderson model. The method combines aspects of the density matrix embedding theory (DMET) approach with a spectral broadening approach inspired by those used in numerical renormalization group (NRG) methods. At zero temperature for a wide range of U , the spectral function produced by this approach is found to be in good agreement with general expectations as well as more advanced and complex numerical methods such as DMRG-based schemes. The theory developed here is simply transferable to more complex impurity problems The second chapter outlines the density matrix embedding methodology in the context of electronic structure applications. We formulate analytical gradients for energies obtained from DMET focusing on two scenarios: RHF-in-RHF embedding and FCI-in-RHF embedding. The former involves solving the small embedded system at Restricted Hartree-Fock (RHF) level of theory. This serves to check the validity of the formulas by reproducing the RHF results on the full system for energies and gradients. The latter scenario employs full configuration interaction (FCI) as a high level solver for the small embedded system. Our results show that only Hellmann-Feynman terms, which involve derivatives of one- and two-electron terms in the atomic orbital basis, are required to calculate energy gradients in both cases. We applied our methodology to the problem of H 10 ring dissociation where the analytical gradients matched those obtained numerically. The gradient formulation is applicable to geometry optimization of strongly correlated molecules and solids. It can also be used in ab initio molecular dynamics where forces on nuclei are obtained from DMET energy gradients. In the final chapter we focus on the study of finite-temperature equilibrium properties of quantum systems in continuous space. We formulate an expansion of the partition function in continuous-time and use Monte Carlo to sample terms in the resulting infinite series. Such a strategy has been highly successful in quantum lattice models but has found scant application in off-lattice systems. The Monte Carlo estimate of the average energy of quantum particles in continuous space subject to simple model potentials is found to converge with low statistical error to the exact solutions even when very high perturbation order is required. We outline two ways in which the algorithm can be applied to more complex problems. First, by drawing an analogy between the formulation in continuous-time with the discretized, Trotter factorized version of standard path integral Monte Carlo (PIMC). This allows one to use the suite of standard PIMC moves to carry out the position sampling required to obtain the weight of each time configuration. Finally, we propose an alternate route by fitting the many-particle potential with multidimensional Gaussians which provides an analytical form for the position integrals.
This thesis is divided into three chapters. In the first chapter we outline a simple and numerically inexpensive approach to describe the spectral features of the single-impurity Anderson model. The method combines aspects of the density matrix embedding theory (DMET) approach with a spectral broadening approach inspired by those used in numerical renormalization group (NRG) methods. At zero temperature for a wide range of U , the spectral function produced by this approach is found to be in good agreement with general expectations as well as more advanced and complex numerical methods such as DMRG-based schemes. The theory developed here is simply transferable to more complex impurity problems The second chapter outlines the density matrix embedding methodology in the context of electronic structure applications. We formulate analytical gradients for energies obtained from DMET focusing on two scenarios: RHF-in-RHF embedding and FCI-in-RHF embedding. The former involves solving the small embedded system at Restricted Hartree-Fock (RHF) level of theory. This serves to check the validity of the formulas by reproducing the RHF results on the full system for energies and gradients. The latter scenario employs full configuration interaction (FCI) as a high level solver for the small embedded system. Our results show that only Hellmann-Feynman terms, which involve derivatives of one- and two-electron terms in the atomic orbital basis, are required to calculate energy gradients in both cases. We applied our methodology to the problem of H 10 ring dissociation where the analytical gradients matched those obtained numerically. The gradient formulation is applicable to geometry optimization of strongly correlated molecules and solids. It can also be used in ab initio molecular dynamics where forces on nuclei are obtained from DMET energy gradients. In the final chapter we focus on the study of finite-temperature equilibrium properties of quantum systems in continuous space. We formulate an expansion of the partition function in continuous-time and use Monte Carlo to sample terms in the resulting infinite series. Such a strategy has been highly successful in quantum lattice models but has found scant application in off-lattice systems. The Monte Carlo estimate of the average energy of quantum particles in continuous space subject to simple model potentials is found to converge with low statistical error to the exact solutions even when very high perturbation order is required. We outline two ways in which the algorithm can be applied to more complex problems. First, by drawing an analogy between the formulation in continuous-time with the discretized, Trotter factorized version of standard path integral Monte Carlo (PIMC). This allows one to use the suite of standard PIMC moves to carry out the position sampling required to obtain the weight of each time configuration. Finally, we propose an alternate route by fitting the many-particle potential with multidimensional Gaussians which provides an analytical form for the position integrals.
This thesis details four research projects related to zero temperature quantum Monte Carlo. Chapters 2-4 focus on continuum quantum Monte Carlo, and specifically its application to molecular systems; whereas Chapter 5 focuses on quantum Monte Carlo in a discrete space. Chapter 2 focuses on improving upon the single-particle basis functions employed in quantum Monte Carlo calculations for molecular systems. For calculations requiring non-diverging pseudopotentials, a class of functions is introduced that is capable of producing the short- and long-range asymptotic behavior of the exact wavefunction. It is demonstrated that this form of basis function produces superior accuracy and efficiency when compared to the basis sets typically employed in quantum Monte Carlo. Although the basis functions introduced in Chapter 2 are capable of producing superior results, it is necessary that the parameters of the functional form are near-optimal for the full potential of the functions to be realized. Chapter 3 introduces a simple yet general method for constructing basis sets of a desired functional form appropriate for molecular electronic structure calculations. A standard basis set is created for each of the elements from hydrogen to argon. Chapter 4 explores the effect of different aspects of the trial wavefunction on the accuracy of quantum Monte Carlo. By systematically testing the effect of the basis size, orbital quality, and determinant expansion quality, this work offers guidance to quantum Monte Carlo practitioners for achieving results to within chemical accuracy of experiment. In Chapter 5, semistochastic projection, a hybrid of deterministic and stochastic projection, is introduced for finding the dominant eigenvalue and eigenvector of a matrix. This method, like stochastic projection, is applicable to matrices well beyond the size that can be handled by deterministic methods. Semistochastic projection improves over stochastic projection by significantly reducing the computational time required to obtain the eigenvalue within a specified statistical uncertainty. After the semistochastic projection method is introduced, it is applied to determine the ground state energy of the Hamiltonian in a discrete basis. This special case of semistochastic projection, dubbed semistochastic quantum Monte Carlo, is shown to be orders of magnitude more efficient than stochastic quantum Monte Carlo.
The focus of this thesis is set on the calculation of spectral functions for low-dimensional quantum systems. At first the quantum many-body problem is introduced and the related spectral functions of interest are defined. In a next step the applied numerical algorithms are presented: Exact diagonalization & Lanczos algorithm, the numerical renormalization group (NRG) and the density matrix renormalization group (DMRG) in the formulation of matrix product states (MPS). The main focus is set on the calculation of spectral functions with these algorithms. In particular for the DMRG a new algorithm is presented which combines the Lanczos algorithm with matrix product states. The presented algorithms are applied to different problems. The first system denotes a copper crystal with magnetic (cobalt/iron) impurities. These materials are known Kondo-systems, i.e. they show an untypical increase in the resistivity with decreasing temperature. At low temperatures local moments will develop, which will be screened from the surrounding conduction band electrons. The ground state of the system is a long-range strongly correlated many-body state between conduction band electrons and local moments. In the singleparticle impurity spectral function a so-called Kondo resonance at the Fermi energy is formed below the Kondo temperature. In scanning tunneling microscope measurements the spectral function at the surface - the so-called local density of states - of the copper crystals with buried impurities in varying depth was measured. The goal of this study is to verify the existence of long-range Kondo signatures via a comparison of the spectral functions for impurities with different depth below the surface. Therefore in this work numerical simulations are used to calculate the local density of states at the surface. The single impurity Anderson model (SIAM) is applied which shows the Kondo resonance in the impurity spectral function. In order to calculate the local density of states at the surface of the copper crystal the equations of motion of the SIAM and the Dyson equation are used and then solved by a combination of numerical renormalization group and band structure calculations of the copper crystal (linear combination of atomic orbitals). The simulation for impurities in different depth below the surface and the experimental data are in very good agreement and hence support the measurement of a long-range Kondo signature. Furthermore the simulations allow to determine the Kondo temperature of a single impurity. The focus of the second part is set on the one-dimensional Heisenberg model. The spectral function of interest is here the dynamic spin structure factor. The newly developed MPS-Lanczos algorithm within the DMRG is used to calculate the dynamic spin structure factor. The results are compared to analytic results from the Bethe ansatz. In the end the new algorithm is discussed in respect to existing methods within the DMRG to calculate spectral functions.
Quantum chemistry models atoms and molecules using quantum mechanics, with the electronic structure problem at its core. This problem, which determines the energy and properties of electrons in a molecule, is exponentially hard to solve exactly on classical computers. To approximate solutions, effective models introduce layers of simplifications. However, these approximations can fail for strongly correlated systems, necessitating alternative approaches. Quantum computers, capable of processing quantum information natively, offer a promising path to simulate molecular systems beyond classical reach, with potential applications in catalysis, materials science, and photochemistry. Despite rapid advancements in quantum hardware, practical quantum computations remain challenging due to limited resources. This thesis optimizes quantum resource usage in computational chemistry, introducing novel embedding techniques and exploring applications beyond ground-state energies. Chapter 1 establishes the foundation by reviewing quantum chemistry methods, from Hartree-Fock and Density Functional Theory (DFT) to active space approaches. It also provides an introduction to quantum computing for chemistry, focusing on near-term algorithms. The following chapters present novel methods aimed at reducing quantum resource requirements and tackling problems classical methods struggle with. Chapter 2 introduces a technique to reduce the λ-norm of the electronic structure Hamiltonian, a key measure of quantum resource demand. By transforming molecular orbitals, this method enhances sampling efficiency in Noisy Intermediate-Scale Quantum (NISQ) algorithms and reduces runtime in Fault-Tolerant Quantum Computing (FTQC). Localized orbitals outperform canonical ones, and an orbital-optimization scheme is proposed to minimize the λ-norm. Benchmarks on hydrogen chains, FeMoco, and Ruthenium complexes demonstrate its effectiveness. Chapter 3 presents the FragPT2 framework for simulating large systems with multiple interacting fragments. This method embeds each fragment in the mean field of others and applies multi-reference perturbation theory to capture inter-fragment correlations such as dispersion and charge transfer efficiently. The approach reduces computational overhead and enables scalable simulations, as shown in applications to \ce{N2} dimers, aromatic dimers, and butadiene. Chapter 4 focuses on photochemical systems, where multiple electronic states must be treated equally. The state-averaged orbital-optimized variational quantum eigensolver (SAOOVQE) is extended with analytical energy gradients and non-adiabatic couplings, enabling geometry optimizations and simulations of non-adiabatic dynamics on quantum hardware. This is demonstrated by locating the conical intersection in formaldimine, highlighting SAOOVQE’s potential for studying photoisomerization in biomolecules. Building on this, Chapter 5 develops a quantum algorithm to detect conical intersections, where the Born-Oppenheimer approximation breaks down. These intersections are characterized by a discrete Berry phase (0 or π), representing their topological nature. The algorithm extracts this phase as a single bit of information, ensuring resilience to noise. Analytical proof and numerical validation using a toy molecular model illustrate its effectiveness for studying light-perception biochemistry and related photochemical processes. Finally, Chapter 6 explores machine learning density functionals trained on noisy quantum data. While DFT is computationally efficient, its accuracy depends on approximated exchange-correlation functionals. This work proposes a neural network-based approach to learn the exact functional from quantum data, showing that training on noisy outputs from NISQ devices can mitigate sampling errors. Applications to the Fermi-Hubbard model and other systems demonstrate its potential to generalize and provide accurate results under noise. This thesis advances quantum computational chemistry by optimizing quantum resource usage, developing new embedding strategies, and expanding the scope of quantum simulations beyond ground-state properties. These contributions pave the way for practical quantum algorithms that could revolutionize molecular simulations in the near future.
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
In the first part of the Thesis we mostly concentrate on spectral properties of strongly correlated systems and on their equilibrium properties. This is accomplished by the general concept of imaginary-time dynamics which we apply to a number of different problems in which different strengths of this approach emerge. In Chapter 1 we introduce the formalism that allows for a connection between the quantum and the classical worlds. The connection is established by means of the imaginary-time quantum evolution which, under certain circumstances, is shown to be equivalent to a classical stochastic process. It is further shown that exact static and spectral properties of correlated systems can be obtained when this mapping is feasible. The relationship between the imaginary-time dynamics in different frameworks such as the path-integral and the perturbative one is also underlined. In Chapter 2 we present a specific implementation of the general ideas previously presented. In particular we introduced an extension to lattice systems of the Reptation Monte Carlo algorithm [30] which benefits of a sampling scheme based on directed updates. Specific improvements over the existing methodologies consist in the unbiased evaluation of the imaginary-time path integrals for bosons and a systematic scheme to improve over the Fixed-node approximation for fermions. Applications to the Hubbard and the Heisenberg models are presented. In Chapter 3 we demonstrate the application of the imaginary-time dynamics to the exact study of spectral properties. Subject of our attention is a highly anharmonic and correlated quantum crystal such as Helium 4 at zero temperature.[33] Concerning this system, we have obtained the first ab-initio complete phonon dispersion in good agreement with neutron spectroscopy experiments. Moreover, we have also studied the density excitations of solid helium in a region of wave-vectors in between the collective (phonon) and the single-particle regimes, where the presence of residual coherence in the dynamics shows analogies between the highly anharmonic crystal and the superfluid phase. In Chapter 4 we introduce a novel method, based on the imaginary-time dynamics, to obtain unbiased estimates of fermionic properties.[34] By means of this method and of a very accurate variational state, we provide strong evidence for the stability of a saturated ferromagnetic phase in the high-density regime of the two-dimensional infinite-U Hubbard model. By decreasing the electron density, we observe a discontinuous transition to a paramagnetic phase, accompanied by a divergence of the susceptibility on the paramagnetic side. This behavior, resulting from a high degeneracy among different spin sectors, is consistent with an infinite-order phase transition scenario. In Chapter 5 the use of imaginary-time dynamics in the context of finite-temperature response functions is highlighted. As an application, we study an intriguing quantum phase featuring both glassy order and Bose-Einstein condensation. [35] We introduce and validate a model for the role of geometrical frustration in the coexistence of off-diagonal long range order with an amorphous density profile. The exact characterization of the response of the system to an external density perturbation is what allows here to establish the existence of a spin-glass phase. The differences between such a phase and the otherwise insulating "Bose glasses" are further elucidated in the Chapter. In the second part of the Thesis we focus our attention on the dynamics of closed systems out of equilibrium. This is accomplished by both non-stochastic exact methods for the dynamics and the introduction of a novel time-dependent Variational Monte Carlo scheme. In Chapter 6 exact diagonalization schemes and renormalization-based methods for one-dimensional systems are introduced. We identify key phenomenological traits resulting from the many-body correlation in closed systems driven sufficiently away from equilibrium.[31] We provide evidences that the dynamics of interacting lattice bosons away from equilibrium can be trapped into extremely long-lived inhomogeneous metastable states. The slowing down of incoherent density excitations above a threshold energy, much reminiscent of a dynamical arrest on the verge of a glass transition, is identified as the key feature of this phenomenon. In Chapter 7 we present an extension to dynamical properties of the Variational Quantum Monte Carlo method.[32] This is accomplished by introducing a general class of time-dependent variational states which is based on the mapping of the many-body dynamics onto an instantaneous ground-state problem. The application of the method to the experimentally relevant quantum quenches of interacting bosons reveals the accuracy and the reliability of the introduced numerical scheme. We indeed obtain for the first time a consistent variational description of the approach to the equilibrium of local observables and underline the origin of the metastability and glassy behavior previously identified. In the very last part we draw our conclusions and show some possible paths for stimulating future research.
This chapter, like the previous one, describes quantum Monte Carlo methods for the simulation of a system of interacting electrons at nonzero temperature. Here, we target a particular class of problems, characterized by a small number of correlated sites or orbitals, that we call impurity problems. While we could apply the methods of the previous chapter to such impurity models, the algorithms discussed here are more advantageous not only because they work in continuous imaginary time and hence lack the Trotter approximation error inherent to the previous methods, but also because they manipulate determinants of matrices of smaller size, which can be handled more efficiently. At the same time, the continuous-time approach is applicable to broader classes of impurity problems. Driving the development of these continuous-time impurity solvers was the desire to perform more efficient simulations of correlated lattice models within the framework of dynamical mean-field theory. This formalism maps the lattice problem onto an impurity problem, whose parameters are determined by a self-consistency condition. In some cases, the self-consistent solution gives a good approximation of the properties of the original lattice problem. When we combine this dynamical mean-field approximation with the local-density approximation for electronic structure calculations, we obtain a powerful scheme for electronic structure calculations of strongly correlated materials. In this chapter we discuss the most important classes of impurity models, sketch the basics of the dynamical mean-field approximation, and detail the different variants of continuous-time impurity solvers. Quantum impurity models A quantum impurity model describes an atom or molecule embedded in some host with which it exchanges electrons, spin, and energy. This exchange allows the impurity to make transitions between different quantum states, and in the presence of interactions within or between the impurity orbitals, these transitions lead to nontrivial dynamical properties. Quantum impurity models play a prominent role, for example, in the theoretical description of the magnetic and electric properties of dilute metal alloys and in theoretical studies of quantum dots and molecular conductors. These models also appear as an auxiliary problem whose solution yields the dynamical mean-field description of correlated lattice models.