abstract: In the first part of the talk, we describe a variational method to approximate the ground state of strongly correlated quantum system in arbitrary spatial geometry based on weighted graph states (WGS). We illustrate the the method for the Ising model, the XY model as well as the Bose-Hubbard model in 2D. Ground state energies and correlations can be obtained with relatively good accuracy, and the phase diagram can be reconstructed. We then show how to obtain a time evolution algorithm, and how to combine the WGS method with the PEPS and tree tensor network (TTN) approaches.
In the second part, we consider the classical simulation of measurement-based quantum computation on graph states. We provide the optimal description of graph states in terms of a TTN, and establish a relation of the optimal TTN with a graph-theoretical measure called rank width. This allows us to identify large classes of graph states on which measurement based quantum computation can be efficiently simulated classically. We also use these results to provide a new method to classically evaluate the partition function of (inhomogeneous) classical spin models. This is done by establishing a relation between the partition function of classical spin models on the one hand, and overlaps between certain stabilizer (or graph) states and product states on the other hand. The interaction pattern is encoded in the graph state, while the details of the interaction as well as the temperature of the system is given by the product state.