Projected Entangled Pair States: Fundamental Analytical and Numerical Limitations.

Matrix product states and projected entangled pair states (PEPS) are powerful analytical and numerical tools to assess quantum many-body systems in one and higher dimensions, respectively. While matrix product states are comprehensively understood, in PEPS fundamental questions, relevant analytically as well as numerically, remain open, such as how to encode symmetries in full generality, or how to stabilize numerical methods using canonical forms. Here, we show that these key problems, as well as a number of related questions, are algorithmically undecidable, that is, they cannot be fully resolved in a systematic way. Our work thereby exposes fundamental limitations to a full and unbiased understanding of quantum many-body systems using PEPS.

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

[2]  Norbert Schuch,et al.  Characterizing Topological Order with Matrix Product Operators , 2014, Annales Henri Poincaré.

[3]  J. Ignacio Cirac,et al.  Approximating Gibbs states of local Hamiltonians efficiently with projected entangled pair states , 2014, 1406.2973.

[4]  Andreas Weichselbaum,et al.  Non-abelian symmetries in tensor networks: A quantum symmetry space approach , 2012, 1202.5664.

[5]  D. Pérez-García,et al.  PEPS as ground states: Degeneracy and topology , 2010, 1001.3807.

[6]  Xiao-Gang Wen,et al.  Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations , 2011, 1106.4752.

[7]  David Perez-Garcia,et al.  Irreducible forms of matrix product states: Theory and applications , 2017, 1708.00029.

[8]  M. B. Hastings,et al.  Solving gapped Hamiltonians locally , 2006 .

[9]  Umesh Vazirani,et al.  Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D , 2016, Communications in Mathematical Physics.

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

[11]  Guoming Wang,et al.  Tensor network non-zero testing , 2014, Quantum Inf. Comput..

[12]  A. H. Werner,et al.  Tensor network representations from the geometry of entangled states , 2018, SciPost Physics.

[13]  M. Wolf,et al.  Undecidability of the spectral gap , 2015, Nature.

[14]  Xiao-Gang Wen,et al.  Symmetry protected topological orders and the group cohomology of their symmetry group , 2011, 1106.4772.

[15]  Guifre Vidal,et al.  Tensor network decompositions in the presence of a global symmetry , 2009, 0907.2994.

[16]  F. Verstraete,et al.  Variational matrix product ansatz for dispersion relations , 2011, 1103.2286.

[17]  Frank Verstraete,et al.  Peps as unique ground states of local hamiltonians , 2007, Quantum Inf. Comput..

[18]  Philippe Corboz,et al.  Variational optimization with infinite projected entangled-pair states , 2016, 1605.03006.

[19]  U. Schollwoeck.  The density-matrix renormalization group , 2004, cond-mat/0409292.

[20]  O. Buerschaper.  Twisted injectivity in projected entangled pair states and the classification of quantum phases , 2013, 1307.7763.

[21]  Xiao-Gang Wen,et al.  Classification of gapped symmetric phases in one-dimensional spin systems , 2010, 1008.3745.

[22]  Robert L. Berger.  The undecidability of the domino problem , 1966 .

[23]  Frank Verstraete,et al.  Gradient methods for variational optimization of projected entangled-pair states , 2016, 1606.09170.

[24]  David Pérez-García,et al.  Order parameter for symmetry-protected phases in one dimension. , 2012, Physical review letters.

[25]  F. Verstraete,et al.  Matrix product states represent ground states faithfully , 2005, cond-mat/0505140.

[26]  Christopher T. Chubb,et al.  Hand-waving and interpretive dance: an introductory course on tensor networks , 2016, 1603.03039.

[27]  M. Troyer,et al.  Implementing global Abelian symmetries in projected entangled-pair state algorithms , 2010, 1010.3595.

[28]  Frank Pollmann,et al.  Symmetry protection of topological phases in one-dimensional quantum spin systems , 2009, 0909.4059.

[29]  F. Barahona.  On the computational complexity of Ising spin glass models , 1982 .

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

[31]  Mikhail N. Vyalyi,et al.  Commutative version of the local Hamiltonian problem and common eigenspace problem , 2005, Quantum Inf. Comput..

[32]  B. Normand,et al.  Tensor renormalization of quantum many-body systems using projected entangled simplex states , 2013, 1307.5696.

[33]  Michael Marien,et al.  Anyons and matrix product operator algebras , 2015, 1511.08090.

[34]  Yuri Gurevich,et al.  The Classical Decision Problem , 1997, Perspectives in Mathematical Logic.

[35]  David Pérez-García,et al.  Classifying quantum phases using matrix product states and projected entangled pair states , 2011 .

[36]  Ying Ran,et al.  Symmetric tensor networks and practical simulation algorithms to sharply identify classes of quantum phases distinguishable by short-range physics , 2015, 1505.03171.

[37]  F. Verstraete,et al.  Criticality, the area law, and the computational power of projected entangled pair states. , 2006, Physical review letters.

[38]  M. Hastings,et al.  An area law for one-dimensional quantum systems , 2007, 0705.2024.

[39]  D. Perez-Garcia,et al.  Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories , 2016, 1606.00608.

[40]  Frank Pollmann,et al.  Detection of symmetry-protected topological phases in one dimension , 2012, 1204.0704.