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

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.

[1]  Yuri V. Matiyasevich,et al.  Decision problems for semi-Thue systems with a few rules , 2005, Theor. Comput. Sci..

[2]  Leonid Gurvits.  Classical deterministic complexity of Edmonds' Problem and quantum entanglement , 2003, STOC '03.

[3]  Sevag Gharibian,et al.  Strong NP-hardness of the quantum separability problem , 2008, Quantum Inf. Comput..

[4]  M. Fannes,et al.  Finitely correlated states on quantum spin chains , 1992 .

[5]  S. Papson,et al.  “Model” , 1981 .

[6]  Mathukumalli Vidyasagar,et al.  The complete realization problem for hidden Markov models: a survey and some new results , 2011, Math. Control. Signals Syst..

[7]  D. Gross,et al.  All reversible dynamics in maximally nonlocal theories are trivial. , 2009, Physical review letters.

[8]  Michael Zwolak,et al.  Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm. , 2004, Physical review letters.

[9]  Jason Morton,et al.  Undecidability in tensor network states , 2012, 1205.3315.

[10]  Jonathan Barrett.  Information processing in generalized probabilistic theories , 2005 .

[11]  Stephen A. Vavasis,et al.  On the Complexity of Nonnegative Matrix Factorization , 2007, SIAM J. Optim..

[12]  Steven R. White,et al.  Spectral functions in one-dimensional quantum systems at finite temperature using the density matrix renormalization group , 2009 .

[13]  J. Ignacio Cirac,et al.  Purifications of multipartite states: limitations and constructive methods , 2013, 1308.1914.

[14]  U. Schollwoeck.  The density-matrix renormalization group in the age of matrix product states , 2010, 1008.3477.

[15]  Vincent D. Blondel,et al.  Undecidable Problems for Probabilistic Automata of Fixed Dimension , 2003, Theory of Computing Systems.

[16]  Ronald de Wolf,et al.  Nondeterministic Quantum Query and Communication Complexities , 2003, SIAM J. Comput..

[17]  Rekha R. Thomas,et al.  Lifts of Convex Sets and Cone Factorizations , 2011, Math. Oper. Res..

[18]  Eitan M. Gurari,et al.  Introduction to the theory of computation , 1989 .

[19]  R. V. Erickson.  Functions of Markov Chains , 1970 .

[20]  M. G. Nadkarni,et al.  Some Regular and Non-regular Functions of Finite Markov Chains , 1970 .

[21]  Guifré Vidal.  Efficient simulation of one-dimensional quantum many-body systems. , 2004, Physical review letters.

[22]  S. Rommer,et al.  CLASS OF ANSATZ WAVE FUNCTIONS FOR ONE-DIMENSIONAL SPIN SYSTEMS AND THEIR RELATION TO THE DENSITY MATRIX RENORMALIZATION GROUP , 1997 .

[23]  Frank Verstraete,et al.  Matrix product state representations , 2006, Quantum Inf. Comput..

[24]  Mihalis Yannakakis,et al.  Expressing combinatorial optimization problems by linear programs , 1991, STOC '88.

[25]  J Eisert,et al.  Quantum measurement occurrence is undecidable. , 2011, Physical review letters.

[26]  Roman Orus,et al.  A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States , 2013, 1306.2164.

[27]  Mika Hirvensalo,et al.  Improved Undecidability Results on the Emptiness Problem of Probabilistic and Quantum Cut-Point Languages , 2007, SOFSEM.

[28]  F. Verstraete,et al.  Matrix product density operators: simulation of finite-temperature and dissipative systems. , 2004, Physical review letters.

[29]  R. Tourky,et al.  Cones and duality , 2007 .

[30]  J. Eisert,et al.  Area laws for the entanglement entropy - a review , 2008, 0808.3773.

[31]  Emil L. Post.  A variant of a recursively unsolvable problem , 1946 .

[32]  B. M. Fulk.  MATH , 1992 .

[33]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[34]  H. Sebastian Seung,et al.  Learning the parts of objects by non-negative matrix factorization , 1999, Nature.

[35]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[36]  F. Verstraete,et al.  Computational complexity of projected entangled pair states. , 2007, Physical review letters.

[37]  Eugene E. Tyrtyshnikov,et al.  Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions , 2009, SIAM J. Sci. Comput..

[38]  F. Verstraete,et al.  Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems , 2008, 0907.2796.

[39]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[40]  M. W. Shields.  An Introduction to Automata Theory , 1988 .

[41]  Steven R. White,et al.  Minimally entangled typical thermal state algorithms , 2010, 1002.1305.

[42]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[43]  Ivan Oseledets,et al.  Tensor-Train Decomposition , 2011, SIAM J. Sci. Comput..

[44]  J. Kiefer,et al.  Asymptotically Optimum Sequential Inference and Design , 1963 .

[45]  D. Vernon.  Inform , 1995, Encyclopedia of the UN Sustainable Development Goals.

[46]  Zi Cai,et al.  Algebraic versus Exponential Decoherence in Dissipative Many-Particle Systems. , 2013, Physical review letters.

[47]  John N. Tsitsiklis,et al.  When is a Pair of Matrices Mortal? , 1997, Inf. Process. Lett..