Heidelberg 2015 – scientific programme
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Q: Fachverband Quantenoptik und Photonik
Q 18: Quantum Information: Concepts and Methods III
Q 18.4: Talk
Tuesday, March 24, 2015, 11:45–12:00, K/HS1
Matrix product operators and states: NP-hardness and undecidability — •Martin Kliesch1, David Gross2, and Jens Eisert1 — 1Freie Universität Berlin, 14195 Berlin, Germany — 2Universität Freiburg, 79104 Freiburg, Germany
Tensor network states constitute an important variational set of quantum states for numerical studies of strongly correlated systems in condensed-matter physics, as well as in mathematical physics. This is specifically true for finitely correlated states or matrix-product operators, designed to capture mixed states of one-dimensional quantum systems. It is a well-known open problem to find an efficient algorithm that decides whether a given matrix-product operator actually represents a physical state that in particular has no negative eigenvalues. We address and answer this question by showing that the problem is provably undecidable in the thermodynamic limit and that the bounded version of the problem is NP-hard in the system size. Furthermore, we discuss numerous connections between tensor network methods and (seemingly) different concepts treated before in the literature, such as hidden Markov models and tensor trains.