Matrix-product operators and states: NP-hardness and undecidability

Publikationstyp
Journal Article
Date Issued
2014-10-16
Sprache
English
Author(s)
Kliesch, Martin  
Gross, Dietmar  
Eisert, Jens  
TORE-URI
http://hdl.handle.net/11420/14111
Journal
Physical review letters  
Volume
113
Issue
16
Article Number
160503
Citation
Physical Review Letters 113 (16): 160503 (2014-10-16)
Publisher DOI
10.1103/PhysRevLett.113.160503
Scopus ID
2-s2.0-84908053380
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.