Journal
PHYSICAL REVIEW LETTERS
Volume 113, Issue 16, Pages -Publisher
AMER PHYSICAL SOC
DOI: 10.1103/PhysRevLett.113.160503
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Funding
- Studienstiftung des Deutschen Volkes
- FQXi
- EU
- ERC
- Excellence Initiative of the German Federal and State Governments [ZUK 43]
- U.S. Army Research Office [W911NF-14-1-0098, W911NF-14-1-0133]
- Freiburg Research Innovation Fund
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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 (nondeterministic-polynomial-time 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.
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