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A fundamental longstanding problem in studying spin models is the efficient and accurate numerical simulation of the long-time behavior of larger systems. The exponential growth of the Hilbert space and the entanglement accumulation at long times pose major challenges for current methods. To address these issues, we employ the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) framework to simulate the many-body spin dynamics of the Heisenberg model in various settings, including the Ising and XYZ limits with different interaction ranges and random couplings. Benchmarks with analytical and exact numerical approaches show that ML-MCTDH accurately captures the time evolution of one- and two-body observables in both one- and two-dimensional lattices. A comparison with the discrete truncated Wigner approximation (DTWA) highlights that ML-MCTDH is particularly well-suited for handling anisotropic models and provides more reliable results for two-point observables across all tested cases. The behavior of the corresponding entanglement dynamics is analyzed to reveal the complexity of the quantum states. Our findings indicate that the rate of entanglement growth strongly depends on the interaction range and the presence of disorder. This particular relationship is then used to examine the convergence behavior of ML-MCTDH. Our results indicate that the multilayer structure of ML-MCTDH is a promising numerical framework for handling the dynamics of generic many-body spin systems.

Quantum gases of ultracold atoms are ubiquitous in modern physics as they offer excellent isolation from the environment as well as fine-grained control over their relevant characteristics such as interparticle interactions. Almost arbitrary spatial arrangements of these particles can be realized and manipulated by employing external potentials. This versatility renders ultracold atoms an ideal platform for the simulation of other quantum system as well as promising candidates in the field of quantum information. However, the corresponding theoretical description usually involves complex many-body problems which can rarely be solved analytically, thus rendering the development of powerful numerical approaches crucial. The present thesis employs the family of multi-layer multi-configuration time-dependent Hartree (ML-MCTDH) methods in order to simulate ultracold quantum many-body systems. While this class of ab-initio approaches originates from the description of molecular dynamics in quantum chemistry, it was later applied to a plethora of other problems and extended to capture indistinguishable particles such as ultracold atoms. The strength of this class of algorithms stems from the fact that they employ variationally optimal, time-dependent basis functions in order to obtain a compact representation of the many-body wave function. The construction of hierarchical multi-layer ansätze allows for the treatment of large and complex composite quantum systems. The present thesis focuses on the development of methodological and implementational improvements as well as the application of the method to novel scenarios. Even though ML-MCTDH methods can often yield compact representations of the many-body wave function, they too cannot escape the exponential scaling of computational complexity as the number of particles increases or when strong correlations in the system require numerous basis functions in order to obtain accurate results. In recent years, various different approaches have been proposed to tackle this problem and reduce the numerical effort. Unfortunately, these schemes cannot be easily transferred to ultracold atom setups or are unable to adapt to non-trivial dynamics. Hence, a novel dynamical pruning approach targeting bosonic particles is developed in the scope of the present thesis. The scheme automatically selects the most relevant many-body states and adapts to the time-evolution of the system. The algorithm is benchmarked using two typical scenarios motivated from ultracold atom physics and found to capture the physics accurately while significantly reducing the computational effort in some cases. A particularly fascinating aspect of quantum simulation is the emulation ultrafast processes such as electronic dynamics with slower-moving atomic particles. In light of this strategy, controlled collisions of ultracold confined in moving potential wells may serve as a test bed to unravel the fundamental processes in atom-atom collisions by taking on the role of electrons. Furthermore, similar scenarios have been proposed as a means to generate entanglement and implement quantum gates in the context of quantum computing. Therefore, the second focus of the present dissertation is to investigate the nonequilibrium dynamics of bosonic particles in colliding potential wells which can be realized experimentally using optical tweezers. This study illuminates the main signatures of the dynamics such as entanglement build-up as well as the transport and untrapping of particles. Quantum spin models are relevant in many areas of physics such as quantum information or condensed matter physics and have been realized experimentally using ultracold atoms or in the related field of Rydberg atoms, among others. The theoretical description of these systems is often challenging, when disorder comes into play. Disorder can result in a high level of degeneracy in the low-energy spectrum and the violation of the so-called area law of entanglement entropy which is a fundamental assumption of many numerical approaches, such as those based on matrix product states. The present thesis studies how the ML-MCTDH method can handle such scenarios by computing the ground states of different disordered models and comparing the results with other established numerical approaches. ML-MCTDH is found to yield accurate results even in the presence of strong disorder and should be considered as another promising approach for the investigation of quantum spin systems.

Numerical simulations of quantum spin models are crucial for a profound understanding of many-body phenomena in a variety of research areas in physics. An outstanding problem is the availability of methods to tackle systems that violate area laws of entanglement entropy. Such scenarios cover a wide range of compelling physical situations including disordered quantum spin systems among others. In this paper, we employ a numerical technique referred to as multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) to evaluate the ground state of several disordered spin models. ML-MCTDH has previously been used to study problems of high-dimensional quantum dynamics in molecular and ultracold physics but is here applied to study spin systems. We exploit the inherent flexibility of the method to present results in one and two spatial dimensions and treat challenging setups that incorporate long-range interactions as well as disorder. Our results suggest that the hierarchical multilayering inherent to ML-MCTDH allows to tackle a wide range of quantum many-body problems such as spin dynamics of varying dimensionality.

We employ the multiconfiguration time-dependent Hartree method for bosons in order to investigate the correlated nonequilibrium quantum dynamics of two bosons confined in two colliding and uniformly accelerated Gaussian wells. As the wells approach each other an effective, transient double-well structure is formed. This induces a transient and oscillatory over-barrier transport. We monitor both the amplitude of the intrawell dipole mode in the course of the dynamics as well as the final distribution of the particles between the two wells. For fast collisions we observe an emission process which we attribute to two distinct mechanisms. Energy transfer processes lead to an untrapped fraction of bosons and a resonant enhancement of the deconfinement for certain kinematic configurations can be observed. Despite the comparatively weak interaction strengths employed in this work, we identify strong interparticle correlations by analyzing the corresponding von Neumann entropy.

The investigation of the nonequilibrium quantum dynamics of bosonic many-body systems is very challenging due to the excessively growing Hilbert space and poses a major problem for their theoretical description and simulation. We present a novel dynamical pruning approach in the framework of the multiconfiguration time-dependent Hartree method for bosons (MCTDHB) to tackle this issue by dynamically detecting the most relevant number states of the underlying physical system and modifying the many-body Hamiltonian accordingly. We discuss two different number state selection criteria as well as two different ways to modify the Hamiltonian. Our scheme regularly re-evaluates the number state selection in order to dynamically adapt to the time evolution of the system. To benchmark our methodology, we study the nonequilibrium dynamics of bosonic particles confined either in an optical lattice or in a double-well potential. It is shown that our approach reproduces the unpruned MCTDHB results accurately while yielding a significant reduction of the simulation time. The speedup is particularly pronounced in the case of the optical lattice.

This presentation explores hybrid quantum algorithms, specifically the Variational Quantum Eigensolver (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), as practical approaches for leveraging near-term quantum hardware. After introducing the digital and analogue paradigm of quantum computing, VQE is established as a capable tool for solving eigenvalue problems — here in the context of calculating molecular energies and highlighting some advantages over quantum phase estimation. Next, QAOA is presented as a approach to solving optimization problems where the solution is encoded in the computational basis. This approach is motivated by trotterization of the adiabatic evolution of a quantum state. The merits of this method are demonstrated using the Max-Cut problem as an example. QAOA efficiently produces high-quality approximate solutions with relatively shallow quantum circuits while avoiding the strong occupation of excited states that can occur in adiabatic quantum computing.