Publication: Projected entangled pair states: fundamental analytical and numerical limitations
Loading...
Official URL
Full text at PDC
Publication Date
2020-11-20
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Physical Society
Citations
Exportar
Abstract
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 manybody systems using PEPS.
Description
UCM subjects
Unesco subjects
Keywords
Citation
[1] U. Schollwöck, The density-matrix renormalization group,
Rev. Mod. Phys. 77, 259 (2005).
[2] U. Schollwöck, The density-matrix renormalization group
in the age of matrix product states, Ann. Phys. (Amsterdam)
326, 96 (2011).
[3] R. Orus, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (Amsterdam) 349, 117 (2014).
[4] J. C. Bridgeman and C. T. Chubb, Hand-waving and interpretive dance: An introductory course on tensor networks, J. Phys. A 50, 223001 (2017).
[5] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa,
Symmetry protection of topological order in onedimensional quantum spin systems, Phys. Rev. B 85, 075125 (2012).
[6] X. Chen, Z. C. Gu, and X. G. Wen, Classification of gapped
symmetric phases in 1D spin systems, Phys. Rev. B 83,
035107 (2011).
[7] N. Schuch, D. Perez-Garcia, and I. Cirac, Classifying
quantum phases using matrix product states and PEPS,
Phys. Rev. B 84, 165139 (2011).
[8] F. Verstraete and J. I. Cirac, Matrix product states
represent ground states faithfully, Phys. Rev. B 73,
094423 (2006).
[9] M. Hastings, An area law for one dimensional quantum
systems, J. Stat. Mech. P08024 (2007).
[10] I. Arad, Z. Landau, U. Vazirani, and T. Vidick, Rigorous RG algorithms and area laws for low energy eigenstates in 1D,
Commun. Math. Phys. 356, 65 (2017).
[11] J. Haegeman, B. Pirvu, D. J. Weir, J. I. Cirac, T. J. Osborne, H. Verschelde, and F. Verstraete, Variational matrix product ansatz for dispersion relations, Phys. Rev. B 85, 100408(R)(2012).
[12] S. Singh, R. N. C. Pfeifer, and G. Vidal, Tensor network
decompositions in the presence of a global symmetry,
Phys. Rev. A 82, 050301 (2010).
[13] B. Bauer, P. Corboz, R. Orús, and M. Troyer, Implementing
global Abelian symmetries in projected entangled-pair state
algorithms, Phys. Rev. B 83, 125106 (2011).
[14] A. Weichselbaum, Non-Abelian symmetries in tensor
networks: A quantum symmetry space approach, Ann.
Phys. (Amsterdam) 327, 2972 (2012).
[15] J. Haegeman, D. Perez-Garcia, I. Cirac, and N. Schuch, An
Order Parameter for Symmetry-Protected Phases in One
Dimension, Phys. Rev. Lett. 109, 050402 (2012).
[16] F. Pollmann and A. M. Turner, Detection of symmetry
protected topological phases in 1D, Phys. Rev. B 86,
125441 (2012).
[17] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac,Matrix product state representations, Quantum Inf. Comput. 7, 401 (2007).
[18] J. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete,
Matrix product density operators: Renormalization fixed
points and boundary theories, Ann. Phys. (Amsterdam) 378,
100 (2017).
[19] G. De las Cuevas, J. I. Cirac, N. Schuch, and D. Perez-Garcia, Irreducible forms of matrix product states: Theory and
applications, J. Math. Phys. (N.Y.) 58, 121901 (2017).
[20] M. B. Hastings, Solving gapped Hamiltonians locally, Phys. Rev. B 73, 085115 (2006).
[21] A. Molnar, N. Schuch, F. Verstraete, and J. I. Cirac,
Approximating Gibbs states of local Hamiltonians
efficiently with PEPS, Phys. Rev. B 91, 045138 (2015).
[22] P. Corboz, Variational optimization with infinite projected entangled-pair states, Phys. Rev. B 94, 035133 (2016).
[23] L. Vanderstraeten, J. Haegeman, P. Corboz, and F. Verstraete,Gradient methods for variational optimization of projected entangled-pair states, Phys. Rev. B 94, 155123 (2016).
[24] S. Jiang and Y. Ran, Symmetric tensor networks and
practical simulation algorithms to sharply identify classes
of quantum phases distinguishable by short-range physics,
Phys. Rev. B 92, 104414 (2015).
[25] X. Chen, Z.-X. Liu, and X.-G. Wen, 2D symmetry protected
topological orders and their protected gapless edge excitations, Phys. Rev. B 84, 235141 (2011).
[26] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry
protected topological orders and the group cohomology of
their symmetry group, Phys. Rev. B 87, 155114 (2013).
[27] O. Buerschaper, Twisted injectivity in PEPS and the
classification of quantum phases, Ann. Phys. (Amsterdam)
351, 447 (2014).
[28] M. B. Sahinoglu, D. Williamson, N. Bultinck, M.
Marien, J. Haegeman, N. Schuch, and F. Verstraete,
Characterizing topological order with matrix product
operators, arXiv:1409.2150.
[29] N. Bultinck, M. Mariën, D. J. Williamson, M. B. Şahinoğlu,
J. Haegeman, and F. Verstraete, Anyons and matrix product
operator algebras, Ann. Phys. (Amsterdam) 378, 183 (2017).
[30] Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, and T.
Xiang, Tensor Renormalization of Quantum Many-Body
Systems Using Projected Entangled Simplex States,
Phys. Rev. X 4, 011025 (2014).
[31] M. Christandl, A. Lucia, P. Vrana, and A. H. Werner, Tensor network representations from the geometry of entangled
states, SciPost Phys. 9, 042 (2020).
[32] E. Börger, E. Grädel, and Y. Gurevich, The Classical
Decision Problem, Universitext (Springer Berlin Heidelberg,
Berlin, Heidelberg, 2001).
[33] R. Berger, The undecidability of the domino problem, Mem.
Am. Math. Soc. 66, 72 (1966).
[34] D. Perez-Garcia, F. Verstraete, J. I. Cirac, and M. M. Wolf, PEPS as unique ground states of local Hamiltonians,
Quantum Inf. Comput. 8, 0650 (2008).
[35] N. Schuch, I. Cirac, and D. P´erez-García, PEPS as ground
states: Degeneracy and topology, Ann. Phys. (Amsterdam)
325, 2153 (2010).
[36] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevLett.125.210504 for computational-hardness
results and proof of the undecidability of the gap of parent
Hamiltonians, which includes Refs. [37–41].
[37] M. W. P. Savelsbergh and P. van Emde Boas,Bounded
tiling, an alternative to satisfiability?, Report Mathematisch Centrum (Amsterdam, Netherlands). Afdeling
Mathematische Besliskunde en Systeemtheorie, 1984.
[38] F. Barahona, On the computational complexity of Ising spin glass models, J. Phys. A 15, 3241 (1982).
[39] S. Bravyi and M. Vyalyi, Commutative version of the
k-local Hamiltonian problem and common eigenspace
problem, Quantum Inf. Comput. 5, 187 (2005).
[40] F. Verstraete, M. M. Wolf, D. P´erez-García, and J. I. Cirac, Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States, Phys. Rev. Lett. 96, 220601 (2006).
[41] S. Gharibian, Z. Landau, S. W. Shin, and G. Wang, Tensor
network non-zero testing, Quantum Inf. Comput. 15, 885(2015).
[42] T. Cubitt, D. Perez-Garcia, and M. M. Wolf, Undecidability of the spectral gap, Nature (London) 528, 207 (2015).