Quantum many-body systems are difficult to study as their Hilbert-space dimensions grow exponentially with the physical system size. The non-locality in these systems can be quantified by entanglement measures. One finds that the entanglement in many states of interest (like ground states) is far below the theoretical maximum. Hence, it is possible to work with a reduced set of effective degrees of freedom. This is exploited in simulation techniques based on tensor network states. I will discuss corresponding algorithms employing different types of tensor networks: matrix product states (MPS), matrix product operators (MPO), tree tensor networks (TTN), projected entangled-pair states (PEPS), and the multiscale entanglement renormalization ansatz (MERA). The computation costs are intimately related to the scaling of entanglement entropies in the simulated systems, for which I will give a short overview.
For 1d (or quasi-1d) systems, MPS allow us to study groundstate phase diagrams, dynamic response functions, finite-temperature states, non- equilibrium dynamics, and open quantum systems (decoherence and dissipation).
While there are still open questions, TTN and MPS with open boundary conditions have a well-established mathematical foundation. I will indicate how things become more delicate for tensor networks with loops. Such issues and fundamental theorems on tensor networks will be discussed in a subsequent tutorial by Frank Verstraete.
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