In this thesis, the numerical simulation of strongly-interacting many-body quantum-mechanical systems using matrix product states (MPS) is considered. Compared to classical systems, quantum many-body systems possess an exponentially enlarged number of degrees of freedom, significantly complicating a simulation on a classical computer. Matrix-Product-States are a novel representation of arbitrary quantum many-body states. Using quantum information theory, it is possible to show that Matrix-Product-States provide a polynomial-sized representation of one-dimensional quantum systems, thus allowing an efficient simulation of one-dimensional quantum system on classical computers. Matrix-Product-States form the conceptual framework of the density-matrix renormalization group (DMRG). Based upon this connection, deeper understanding of the density matrix renormalization group can be obtained. After a general introduction in the first chapter of this thesis, the second chapter deals with Matrix-Product-States, focusing on the development of fast and stable algorithms. It is possible to extend the Matrix-Product-States approach to be able to represent arbitrary operators, the so-called Matrix-Product-Operators, which allows a fast and flexible calculation of arbitrary expectation values. To obtain algorithms to efficiently calculate groundstates, the density-matrix renormalization group is reformulated using the Matrix-Product-States framework. Further, time-dependent problems are considered. Two different algorithms are presented, one based on a Trotter decomposition of the time-evolution operator, the other one on Krylov subspaces. Finally, the evaluation of dynamical spectral functions is discussed, and a correction vector-based method is presented. In the following chapters, the methods presented in the second chapter, are applied to a number of different physical problems. The third chapter deals with the existence of chiral phases in isotropic one-dimensional quantum spin systems. A preceding analytical study based on a mean-field approach indicated the possible existence of those phases in an isotopic Heisenberg model with a frustrating zig-zag interaction and a magnetic field. In this thesis, the existence of the chiral phases will be shown numerically by using Matrix-Product-States-based algorithms. A key effect of interacting one-dimensional quantum-mechanical many-body systems is the spin-charge separation. However, up to now only signs of the spin-charge separation have been observed in experiments. In the fourth chapter, we propose an experiment using ultracold atomic gases in optical lattices, which allows a well controlled observation of the spin-charge separation (of different hyperfine states of the ultracold atoms) with current state of the art experimental techniques. Ultracold atoms in optical lattices are well described by (Bose)-Hubbard models. In order to support this proposal, we present numerical results for realistic system parameters. Matrix-Product-States are an excellent tool for the simulation of one-dimensional quantum systems, however, they are not well suited for the simulation of higher dimensional systems. For strongly-correlated systems, for instance cuprates-based high-temperature superconductors, quantum fluctuations play an essential role. Classical mean-field theories neglect any kind of fluctuations, thus they are not suitable to describe strongly-correlated systems. The dynamical mean-field theory (DMFT) fully takes local quantum fluctuations into account but neglects any kind of spatial fluctuations. The many-body problem on the lattice is mapped onto an impurity problem, which needs to be solved self-consistently. In the last chapter of this thesis, Matrix-Product-States-based algorithms are used to solve the impurity problem of the dynamical mean-field theory. We present results for a Hubbard model on a one-dimensional lattice and on a Bethe lattice obtained by the dynamical mean-field and compare them with exact results.
In this thesis, the numerical simulation of strongly-interacting many-body quantum-mechanical systems using matrix product states (MPS) is considered. Compared to classical systems, quantum many-body systems possess an exponentially enlarged number of degrees of freedom, significantly complicating a simulation on a classical computer. Matrix-Product-States are a novel representation of arbitrary quantum many-body states. Using quantum information theory, it is possible to show that Matrix-Product-States provide a polynomial-sized representation of one-dimensional quantum systems, thus allowing an efficient simulation of one-dimensional quantum system on classical computers. Matrix-Product-States form the conceptual framework of the density-matrix renormalization group (DMRG). Based upon this connection, deeper understanding of the density matrix renormalization group can be obtained. After a general introduction in the first chapter of this thesis, the second chapter deals with Matrix-Product-States, focusing on the development of fast and stable algorithms. It is possible to extend the Matrix-Product-States approach to be able to represent arbitrary operators, the so-called Matrix-Product-Operators, which allows a fast and flexible calculation of arbitrary expectation values. To obtain algorithms to efficiently calculate groundstates, the density-matrix renormalization group is reformulated using the Matrix-Product-States framework. Further, time-dependent problems are considered. Two different algorithms are presented, one based on a Trotter decomposition of the time-evolution operator, the other one on Krylov subspaces. Finally, the evaluation of dynamical spectral functions is discussed, and a correction vector-based method is presented. In the following chapters, the methods presented in the second chapter, are applied to a number of different physical problems. The third chapter deals with the existence of chiral phases in isotropic one-dimensional quantum spin systems. A preceding analytical study based on a mean-field approach indicated the possible existence of those phases in an isotopic Heisenberg model with a frustrating zig-zag interaction and a magnetic field. In this thesis, the existence of the chiral phases will be shown numerically by using Matrix-Product-States-based algorithms. A key effect of interacting one-dimensional quantum-mechanical many-body systems is the spin-charge separation. However, up to now only signs of the spin-charge separation have been observed in experiments. In the fourth chapter, we propose an experiment using ultracold atomic gases in optical lattices, which allows a well controlled observation of the spin-charge separation (of different hyperfine states of the ultracold atoms) with current state of the art experimental techniques. Ultracold atoms in optical lattices are well described by (Bose)-Hubbard models. In order to support this proposal, we present numerical results for realistic system parameters. Matrix-Product-States are an excellent tool for the simulation of one-dimensional quantum systems, however, they are not well suited for the simulation of higher dimensional systems. For strongly-correlated systems, for instance cuprates-based high-temperature superconductors, quantum fluctuations play an essential role. Classical mean-field theories neglect any kind of fluctuations, thus they are not suitable to describe strongly-correlated systems. The dynamical mean-field theory (DMFT) fully takes local quantum fluctuations into account but neglects any kind of spatial fluctuations. The many-body problem on the lattice is mapped onto an impurity problem, which needs to be solved self-consistently. In the last chapter of this thesis, Matrix-Product-States-based algorithms are used to solve the impurity problem of the dynamical mean-field theory. We present results for a Hubbard model on a one-dimensional lattice and on a Bethe lattice obtained by the dynamical mean-field and compare them with exact results.
This thesis treats the classical simulation of strongly-interacting many-body quantum-mechanical systems in more than one dimension using matrix product states and the more general tensor product states. Contrary to classical systems, quantum many-body systems possess an exponentially larger number of degrees of freedom, thereby significantly complicating their numerical treatment on a classical computer. For this thesis two different representations of quantum many-body states were employed. The first, the so-called matrix product states (MPS) form the basis for the extremely successful density matrix renormalization group (DMRG) algorithm. While originally conceived for one-dimensional systems, MPS are in principle capable of describing arbitrary quantum many-body states. Using concepts from quantum information theory it is possible to show that MPS provide a representation of one-dimensional quantum systems that scales polynomially in the number of particles, therefore allowing an efficient simulation of one-dimensional systems on a classical computer. One of the key results of this thesis is that MPS representations are indeed efficient enough to describe even large systems in two dimensions, thereby enabling the simulation of such systems using DMRG. As a demonstration of the power of the DMRG algorithm, it is applied to the Heisenberg antiferromagnet with spin S=1/2 on the kagome lattice. This model's ground state has long been under debate, with proposals ranging from static spin configurations to so-called quantum spin liquids, states where quantum fluctuations destroy conventional order and give rise to exotic quantum orders. Using a fully SU(2)-symmetric implementation allowed us to handle the exponential growth of entanglement and to perform a large-scale study of this system, finding the ground state for cylinders of up to 700 sites. Despite employing a one-dimensional algorithm for a two-dimensional system, we were able to compute the spin gap (i.e. the energy gap to the first spinful excitation) and study the ground state properties, such as the decay of correlation functions, the static spin structure factors, and the structure and distribution of the nearest-neighbor spin-spin correlations. Additionally, by applying a new tool from quantum information theory, the topological entanglement entropy, we could also with high confidence demonstrate the ground state of this model to be the elusive gapped Z2 quantum spin liquid with topological order. To complement this study, we also considered the extension of MPS to higher dimensions, known as tensor product states (TPS). We implemented an optimization algorithm exploiting symmetries for this class of states and applied it to the bilinear-biquadratic-bicubic Heisenberg model with spin S=3/2 on the z=3 Bethe lattice. By carefully analyzing the simulation data we were able to determine the presence of both conventional and symmetry-protected topological order in this model, thereby demonstrating the analytically predicted existence of the Haldane phase in higher dimensions within an extended region of the phase diagram. Key properties of this symmetry-protected topological order include a doubling of the levels in the entanglement spectrum and the presence of edge spins, both of which were confirmed in our simulations. This finding simultaneously validated the applicability of the novel TPS algorithms to the search for exotic order.
The Thesis explores the application of Matrix Product States (MPS) in the domain of quantum many-body systems and quantum computing, show casing the versatility and effectiveness of MPS for addressing complex com putational challenges. The opening chapter introduces essential tools for Tensor Networks, addressing the curse of dimensionality and elucidating the main characteristics of Matrix Product States and Matrix Product Op erators. The discussion encompasses also important numerical techniques, including Density Matrix Renormalization Group and Time Evolving Block Decimation. Furthermore, the relation between MPS and quantum circuits is discussed. The subsequent Chapters delve into specific applications of MPS. In the second Chapter, we introduce a Tensor Network ansatz inspired by the backflow transformation for correlated systems. This extension of the MPS representation ensures an area law for entanglement in dimensions one or greater. We employ an optimization scheme combining DMRG and variational Monte Carlo algorithms for efficient ground-state search. Bench marking against spin models demonstrates high accuracy. The third Chapter explores quantum annealing for optimizing complex classical spin Hamil tonians, as the Hopfield model and the binary perceptron. We introduce an efficient Tensor Network representation for the adiabatic time evolution of quantum annealing, enabling scalable classical simulations. The use of MPS in mitigating Trotter errors and mapping to quantum circuits is also explored. In the fourth Chapter we present a novel method for evaluating the amount of nonstabilizerness (also known as quantum magic) contained in an MPS. We overcome the exponentially hard evaluation of this quan tity by employing a simple perfect sampling technique of Pauli string. Our innovative MPS approach enables efficient computation. Benchmarked on magic states and the quantum Ising chain ground state, this method offers easy access to the non-equilibrium dynamics of nonstabilizerness following a quantum quench. The Appendices provide additional insights, including: an introduction to variational Monte Carlo, technical details on MPS simula tions for the quantum annealing and a comprehensive overview on stabilizer formalism and nonstabilizerness.
Experimental advances have made it possible to realize and control quantum many-body systems, allowing the experimental study of non-equilibrium phenomena. Due to the Hilbert space growing exponentially with the number of particles, advanced numerical techniques are usually required to understand many-body effects at the theoretical level. In this thesis, we address the question of how to efficiently simulate the non-equilibrium dynamics of many-body systems using matrix product states (MPS). In particular, we consider quantum systems in interaction with macroscopic environments, as they appear in Anderson impurity problems, quantum thermodynamics, or open quantum systems. We develop a low-entanglement representation of the environment with only short-range interactions, perfectly suitable for simulations with MPS. We further show that an interleaved ordering of tight-binding chains can significantly reduce the creation of entanglement, as opposed to an intuitive implementation of the Hamiltonian's geometry. Our approach allows long-time simulations and an analysis of the environmental dynamics, in which, after a sudden quench, we find clear signatures of many-body effects. We employ our techniques to compute spectral functions of impurity models, with possible applications to dynamical mean-field theory impurity solvers, and to calculate dissipation in the non-equilibrium Anderson model with explicit time-dependence, as relevant for ongoing experiments with oscillating tip atomic force microscopes.
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
The main topic of this thesis is the study of many-body effects in strongly correlated one- or quasi one-dimensional condensed matter systems. These systems are characterized by strong quantum and thermal fluctuations, which make mean-field methods fail and demand for a fully numerical approach. Fortunately, a numerical method exist, which allows to treat unusually large one -dimensional system at very high precision. This method is the density-matrix renormalization group method (DMRG), introduced by Steve White in 1992. Originally limited to the study of static problems, time-dependent DMRG has been developed allowing one to investigate non-equilibrium phenomena in quantum mechanics. In this thesis I present the solution of three conceptionally different problems, which have been addressed using mostly the Krylov-subspace version of the time-dependent DMRG. My findings are directly relevant to recent experiments with ultracold atoms, also carried out at LMU in the group of Prof. Bloch. The first project aims the ultimate goal of atoms in optical lattices, namely, the possibility to act as a quantum simulator of more complicated condensed matter system. The underline idea is to simulate a magnetic model using ultracold bosonic atoms of two different hyperfine states in an optical superlattice. The system, which is captured by a two-species Bose-Hubbard model, realizes in a certain parameter range the physics of a spin-1/2 Heisenberg chain, where the spin exchange constant is given by second order processes. Tuning of the superlattice parameters allows one to controlling the effect of fast first order processes versus the slower second order ones. The analysis is motivated by recent experiments, %\cite{Folling2007,Trotzky2008} where coherent two-particle dynamics with ultracold bosonic atoms in isolated double wells were detected. My project investigates the coherent many-particle dynamics, which takes place after coupling the double well. I provide the theoretical background for the next step, the observation of coherent many-particle dynamics after coupling the double wells. The tunability between the Bose-Hubbard model and the Heisenberg model in this setup could be used to study experimentally the differences in equilibration processes for non-integrable and Bethe ansatz integrable models. It turns out that the relaxation in the Heisenberg model is connected to a phase averaging effect, which is in contrast to the typical scattering driven thermalization in nonintegrable models In the second project I study a many-body generalization of the original Landau-Zener formula. This formula gives the transition probability between the two states of a quantum mechanical two-level system, where the offset between the two levels is varying linearly in time. In a recent experiment this framework has been extended to a many-body system consisting of pairwise tunnel-coupled one-dimensional Bose liquids. It was found that the tunnel coupling between the tubes and the intertube interactions strongly modify the original Landau-Zener picture. After a introduction to the two-level and the three-level Landau-Zener problem I present my own results for the quantum dynamics of the microscopic model and the comparison to the experimental results. I have calculated both Landau-Zener sweeps as well as the time-evolution after sudden quenches of the energy offset. A major finding is that for sufficiently large initial density quenches can be efficiently used to create quasi-thermal states of arbitrary temperatures. The third project is more mathematical and connects the fields of quantum computation and of quantum information. Here, the main purpose is to analyse systematically the effects of decoherence on maximally entangled multi-partite states, which arise typically during quantum computation processes. The bigger the number of entangled qubits the more fragile is its entanglement under the influence decoherence. As starting point I consider first two entangled qubits, whereby one qubit interacts with an arbitrary environment. For this particular case I have derived a factorization law for the disentanglement. Next, I calculate the decrease of entanglement of two , three and four entangled qubits, general W- and general GHZ-state, coupled to a global spin-1/2 bath or several independent spin-1/2 baths , one for each qubit. Although there is no appropriate entanglement measure for three and more qubits, it turns out that this decrease is directly related to the increase of entanglement between the central system and the bath. This implies the formation of a much bigger multipartite entangled network. Thus, using the von Neumann entropy and the Wootters concurrence, I derive a simple upper bound for the bath-induced entanglement breaking power of the initially maximally entangled multi-partite states.
The research work in this thesis is based on strongly interacting quantum lattice systems. The biggest part of research was conducted using state-of-the-art tensor network simulations. Tensor networks provide efficient and highly accurate representations of quantum states when any simply connected patch of the quantum state is slightly entangled. Matrix Product States (MPS) is a tensor network representation for quantum states which is quasi-exact for one-dimensional systems when entanglement entropy of any bipartition of the state follows ``area-law". Projected Entangled Pair States (PEPS) is the extension of MPS for higher dimensional systems, where entanglement entropy area-law is non-trivial. These formalisms and their relevant ground state optimization techniques, Density Matrix Renormalization Group (DMRG) for MPS and Simple Update (SU) with Tensor Renormalization Group (TRG) for PEPS are thoroughly analysed. Entanglement entropy area-law is usually obeyed by the ground state of local Hamiltonians, while generic highly excited states follow a volume-law (they span a finite part of the Hilbert space). Recently, a class of interacting system was shown to undergo a dynamical phase transition (Many Body Localization) where the entropy of every eigenstate follows an area-law. This transition is achieved when the system is highly disordered and the quantum many body state becomes localized similarly to the free particle Anderson localization. The area-law property makes highly excited eigenstates of Many Body Localized (MBL) systems efficiently represented by the MPS ansatz. We develop a highly optimized algorithm (eDMRG) which goes beyond ground state optimization and successfully target eigenstates in any part of the spectrum. This algorithm is used together with analytical calculations, based on local integrability of MBL systems, to identify the universal behaviour of the entanglement spectrum of highly excited eigenstates in MBL systems. In the second part we study interacting bosons in 2D optical lattices. Strongly interacting bosons are simulated using the Bose-Hubbard (BH) model when the interactions are strictly local and the Extended Bose-Hubbard (EBH) model, when additional dipolar interactions are present. The ground states of BH/EBH Hamiltonians in a hexagonal lattice are studied. The phase spectrum includes various insulating and critical phases which are studied in detail using infinite-PEPS ansatz. Additional results on entanglement entropy scaling with respect to the filling of the lattice are presented. Finally, a strongly interacting Harper-Hofstadter Hamiltonian is realized by combining synthetic fields with the BH model. The homogeneity of the system is then broken by a parabolic trapping potential, similar to the ones used in cold atom experiments. Using time-dependent Gutzwiller ansatz (GA) the expansion dynamics of the cloud in large square lattices are studied. In contrast to the expansion dynamics of the BH model, it is found that the synthetic fields generate a self-trapping effect. Using phenomenology and simulations the dynamics are studied in the hard-core and soft-core limit.