Simulation of strongly correlated systems

Understanding properties of complex systems is of fundamental interest in physics and beyond. Strongly correlated systems constitute important examples of such complex systems, however only a very limited number of special problems are exactly solvable. Although powerful numerical methods such as Quantum Monte Carlo (QMC) or density matrix renormalization group (DMRG) have been developed and successfully applied, their applicability is still limited. Many relevant problems, including high-Tc superconductivity or magnetic properties of solids, are still not well understood. We aim to develop and implement novel methods for the simulation of complex classical and quantum systems based on insights gained in quantum information theory. Our main goal is to propose new variational families and methods to accurately describe ground states, thermal states as well as time evolution of certain classes of complex systems, in such a way that the strength of various proposed methods should be combined. As a first example, we will consider the combination of matrix product states (MPS) -the basis of DMRG- with weighted graph states (WGS) - a class of multipartite entangled states which we recently introduced. The bounded amount of entanglement carried by MPS explains their limited applicability to 2D systems which obey an area law, while WGS have rich entanglement features and include states with an arbitrary large amount of entanglement and may hence yield to better performance and accuracies. We will then move to recently proposed generalizations of MPS, including tensor tree networks, projected entangled pair states (PEPS) and multiscale entanglement renormalization ansatz (MERA), which are believed to be capable of successfully describing higher dimensional or critical systems. We will investigate possible combinations with WGS, their entanglement features as well as the practical performance, improvements and extensions of the variational methods. The application of the developed methods to various complex systems, including simulation of quantum and classical spin models as well as certain types of quantum computations shall be performed. Possibilities to use Monte Carlo sampling techniques within the proposed methods will be analyzed. Furthermore, the design of new types of quantum simulators and quantum algorithms to describe and investigate complex systems, based on ansatzes for classical simulations described above, will be investigated.

Understanding properties of complex systems is of fundamental interest in physics and beyond. Strongly correlated systems constitute important examples of such complex systems, however only a very limited number of special problems are exactly solvable. Although powerful numerical methods such as Quantum Monte Carlo (QMC) or density matrix renormalization group (DMRG) have been developed and successfully applied, their applicability is still limited. Many relevant problems, including high-Tc superconductivity or magnetic properties of solids, are still not well understood. During this research project, we have developed and utilized novel classical simulation techniques for complex quantum systems. Our methods are based on the usage of (concatenated) tensor networks, and (in part) their unification with other approaches such as weighted graph states. This allowed us to develop variational approaches for the simulation of ground states as well as the time evolution of quantum systems, including the classical simulation of certain quantum computations. These methods combine the strength of different Ansatzes, where in particular the limitations on the amount of entanglement that can be properly described with such variational state families can be significantly lifted. Based on this approach, we have developed a method for the treatment of systems with long-ranged interactions. In this approach, not only states but also operators (e.g. the system Hamiltonian) are described by tensor networks, and we studied the optimal construction of such tensor-network operators for 1D and 2D systems. This also allowed us to develop an algorithm for the numerical simulation of infinite 1D systems with long-ranged interactions and broken translational invariance, which we successfully applied for the study of phase diagrams and entropy scaling of systems consisting of polar bosons or Rydberg atoms. These simulation techniques are however not limited to the treatment of ground states or unitary dynamics. In fact, we have adapted and utilized them for the study of entanglement properties of (macroscopic) multipartite quantum states under decoherence. In particular, we have investigated the stability of large-scale entanglement and macroscopic quantum superpositions, where we could show the existence of stable macroscopic quantum superpositions, i.e. states that maintain their distinct quantum features despite the presence of noise and decoherence. In addition, we have put forward a quantum simulator as well as a new quantum algorithm for the treatment of classical complex systems, in particular the approximation of the partition function of classical spin systems. This allows us to determine the quantum complexity of classical problems, where it turns out that the simulation of such classical systems in a certain parameter regime is as hard as simulating arbitrary quantum computations.