Many-body entangled dynamics of closed and open systems for quantum simulators

Abstract

When D-wave was founded in 1999, quantum computing was still an ambitious vision. In the past five years, quantum computing has moved from this vision to a very reachable goal with major companies such as Google, IBM, Intel, and Microsoft investigating quantum computation investing in this technology. Together with a vivid start-up scene including companies such as 1QBit, ionQ, Q-CTRL, QuantumCircuits, and Rigetti, they aim to develop hardware and software for a new generation of computers which can tackle problems intractable on classical computers. This thesis revolves around the simulation of such quantum systems and, thus, it has some direct connections to this intriguing new wave of technology. Our focus is shifted toward the side of quantum simulators, which use quantum systems to investigate a specific research question, e.g., predicting the properties of new materials. The results of such simulations can be thought of as an analog measurement outcome of the final state of the quantum simulator. Quantum annealing is one prominent example where the final state encodes the solution to an optimization problem. In contrast, quantum computers work in the spirit of a classical computer calculating in terms of zeros and ones while allowing superpositions and entanglement to reach a quantum advantage. They favor a formulation in terms of quantum gates, which are adapted from the logical gates used in classical computers. We establish a common understanding of the terminology and questions arising in this area of research in the first part of the thesis. We consider the dynamics of closed quantum systems, which has direct connections to the quantum annealing architectures. We analyze how long-range interactions in the quantum Ising model modify the quantum phase diagram. The quantum critical point for the ferromagnetic model shifts to higher external fields to break up the order in comparison to the nearest neighbor model. In contrast, the antiferromagnetic order breaks at lower external fields when including long-range interactions. In the dynamics, we quench through the quantum critical points evaluated for different values for the long-range interactions and analyze the Kibble-Zurek hypothesis via the defects generated in the ferro- and antiferromagnetic limits. These simulations reveal how the number of defects depends on the quench rate and the strength of the long-range interactions, where we find that the defect density decreases toward the nearest-neighbor limit in the ferromagnetic model. Such results are valuable when considering the balance between long-range interactions and quench rates in adiabatic quantum computing. We then move on to open quantum systems. Closed systems are an incomplete description of a quantum system when it is disturbed from an environment or measurement devices not included in the description of the closed system; such effects lead to decoherence of the system. We focus on large reservoirs described by the Lindblad master equation. The research questions revolve around the thermalization of a many-body quantum system. This study treating open quantum systems can be seen as the analog of the eigenstate thermalization hypothesis vs. generalized Gibbs ensemble handling closed quantum systems with regards to the question of whether or not thermalization occurs. The choice of the operator in the interaction Hamiltonian between the system and the reservoir governs which limit of the phase diagram cannot thermalize by definition in the global multi-channel approach. Furthermore, we find regions in the phase diagram of a small quantum Ising chain protected from thermalization and decoherence. Evidently, other regions are favorable to reach thermalization as quick as possible, which is essential to dissipative state preparation. In addition to the global multi-channel model, we analyze the concept of neighborhood single-channel Lindblad operators, which allows one to install non-trivial long-range order; this is a task where single-site single-channel Lindblad operators fail. The next part treats the numerical methods used in the first chapters of this thesis. We describe the exact diagonalization and tensor network algorithms; both are available in our open source package Open Source Matrix Product States. The algorithms contain a selection of state-of-the-art methods for closed and open quantum system, e.g., matrix product density operators as tensor network type and the time-dependent variational principle as time evolution method. We aim for an upgrade from a stand-alone package to the integration into a science gateway. Science gateways serve as a cloud resource for research software. The results from the research on tensor network methods during this thesis will significantly contribute to an efficient science gateway using the algorithms with optimal scaling and versatile features such as meaningful error bars. The equations to estimate error bars for many observables depend on easily accessible parameters such as the system size, the energy gap to the first excited state, and the variance of the energy. The error bounds of the bond entropy in the quantum Ising model are at the level of 0.01, where most other observables such as spin measurements have a tighter error bound of around 0.001 for 256 sites. The choice of the most efficient algorithm plays a key role in many-body mixed states as we simulate the density matrix instead of a pure state. Providing all three major methods for mixed states within our library, we have the option to pick locally purified tensor networks to calculate thermal states and matrix product density operators for the dynamics governed by the Lindblad equation. Both choices are based on the examples in this thesis. Quantum trajectories are favorable for systems generating a large amount of entanglement and simulations parallelized across a large number of CPU cores, which is on the order of magnitude of the number of trajectories. We close the thesis with a discussion of the results in the context of the present research and highlight how the work presented raises even more intriguing questions. This collection of open questions is considered in the second part of the conclusion. These projects can be divided loosely into additions to the numerical methods enabling novel modeling of quantum systems and simulations predicting the actual behavior of physical systems, where most of them can be solved with the present software.