The numerical description of many-body systems constitutes a formidable challenge, since its complexity grows - a priori - exponentially with the number of microscopic constituents (electrons, spins, qubits). Taking inspiration from statistical physics, quantum information theory, and computer science, we develop, implement, and employ a blend of numerical techniques, often in conjunction with analytical ones, to shed light on their collective physical phenomena. In our daily activity, our algorithmic and conceptual developments include classical and quantum Monte Carlo, exact diagonalization, (functional) renormalization group techniques, and tensor-network based approaches. Complex quantum systems often are partially, or even fully chaotic. Origins of quantum chaos include many-body interactions, the presence of disorder, device imperfections, or combinations of these. The ensuing quantum fluctuations and interference effects manifest themselves in intriguing physical phenomena relevant to a wide class of systems, including (topological) quantum materials, synthetic quantum devices such as superconductor qubit arrays or nano-electronic conductors, and even gravitational systems.
Correlated quantum many-body systems can give rise to remarkable collective behavior that have no classical counterpart such as superconductors, superfluids, or quantum spin liquids. When coming from a materials perspective such states often reflect an intricate interplay of strong correlations, topology, and spin-orbit coupling, which we study using a combination of analytical and numerical approaches exploiting concepts from field theory, statistical physics, and quantum information theory. From a conceptual perspective the study of correlated matter is rich in phenomenology, including the formation of long-range entanglement, fractionalization of quantum numbers and the emergence of gauge theories. Driven open quantum matter is characterized by an interplay of coherent quantum dynamics with external driving, dissipation, and quantum measurement. This scenario emerges in platforms ranging from ultracold atomic gases over light-driven quantum materials to the first quantum computing architectures. What are the universal principles and phenomena governing such systems? We construct novel theoretical and numerical frameworks to understand this question, bringing together concepts from quantum optics, solid state- and quantum field theory.
Computing and communication devices are subject to the laws of Nature. Therefore, a complete theory of computation and communication must be informed by physics. In the 1990s, two break-through results made this abstract principle concrete. First the discovery of quantum algorithms: Methods for solving computational problems that are fundamentally more efficient than any classical approach. Second the realization that quantum communication can be used to establish cryptographic keys whose security rests on the laws of physics, rather than unproven computational assumptions. Quantum information theory studies these applications. More general, the field is interested in making the fundamental differences between the classical and the quantum world mathematically precise. Driven by the significant progress in the realization of quantum hardware, quantum information theory has recently been transformed from a niche subject to one of the most active branches of physics.  In extreme systems, such as in the context of black holes or early universe physics, the classical theory of general relativity breaks down and a quantized version of it is required to describe nature. One of the most promising approaches to quantum gravity is holography, positing that a theory of quantum gravity can be defined as a quantum theory without gravity in one lower dimension. The gravitational dynamics is then in a sense ‘emergent’ from regular quantum physics. In particular, it’s quantum information theoretical concepts such as entanglement that to a large extent carry the information about the emergent dynamical geometry of the quantum gravity theory. In addition to quantum info, random matrix theory and quantum chaos also play an increasingly important role in holography and thus the quest for a better understanding of gravity.