Entanglement renormalization, quantum error correction, and bulk causality

  • Isaac H. KimEmail author
  • Michael J. Kastoryano
Open Access
Regular Article - Theoretical Physics


Entanglement renormalization can be viewed as an encoding circuit for a family of approximate quantum error correcting codes. The logical information becomes progres-sively more well-protected against erasure errors at larger length scales. In particular, an approximate variant of holographic quantum error correcting code emerges at low energy for critical systems. This implies that two operators that are largely separated in scales behave as if they are spatially separated operators, in the sense that they obey a Lieb-Robinson type locality bound under a time evolution generated by a local Hamiltonian.


Models of Quantum Gravity Renormalization Group 


Open Access

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  1. [1]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    Z. Yang, P. Hayden and X.-L. Qi, Bidirectional holographic codes and sub-AdS locality, JHEP 01 (2016) 175 [arXiv:1510.03784] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    W. Donnelly, B. Michel, D. Marolf and J. Wien, Living on the edge: a toy model for holographic reconstruction of algebras with centers, arXiv:1611.05841 [INSPIRE].
  10. [10]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].
  12. [12]
    R.N.C. Pfeifer, G. Evenbly and G. Vidal, Entanglement renormalization, scale invariance and quantum criticality, Phys. Rev. A 79 (2009) 040301 [arXiv:0810.0580] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28 (1972) 251 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  15. [15]
    B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
  16. [16]
    X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282 [INSPIRE].
  17. [17]
    M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Continuous multiscale entanglement renormalization ansatz as holographic surface-state correspondence, Phys. Rev. Lett. 115 (2015) 171602 [arXiv:1506.01353] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    A. Castro, M.R. Gaberdiel, T. Hartman, A. Maloney and R. Volpato, The gravity dual of the Ising model, Phys. Rev. D 85 (2012) 024032 [arXiv:1111.1987] [INSPIRE].ADSGoogle Scholar
  19. [19]
    A.J. Ferris and D. Poulin, Tensor networks and quantum error correction, Phys. Rev. Lett. 113 (2014) 030501 [arXiv:1312.4578].ADSCrossRefGoogle Scholar
  20. [20]
    F. Pastawski, J. Eisert and H. Wilming, Quantum source-channel codes, arXiv:1611.07528 [INSPIRE].
  21. [21]
    A. Yu. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2 [quant-ph/9707021] [INSPIRE].
  22. [22]
    S.T. Flammia, J. Haah, M.J. Kastoryano and I.H. Kim, Limits on the storage of quantum information in a volume of space, arXiv:1610.06169 [INSPIRE].
  23. [23]
    G. Evenbly and G. Vidal, Algorithms for entanglement renormalization, Phys. Rev. B 79 (2009) 144108 [arXiv:0707.1454] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    V. Giovannetti, S. Montangero and R. Fazio, Quantum multiscale entanglement renormalization ansatz channels, Phys. Rev. Lett. 101 (2008) 180503 [arXiv:0804.0520].ADSCrossRefGoogle Scholar
  25. [25]
    D. Kretschmann, D. Schlingemann and R.F. Werner, The information-disturbance tradeoff and the continuity of Stinesprings representation, IEEE Trans. Informat. Theor. 54 (2008) 1708.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    C. Bény and O. Oreshkov, General conditions for approximate quantum error correction and near-optimal recovery channels, Phys. Rev. Lett. 104 (2010) 120501 [arXiv:0907.5391].CrossRefGoogle Scholar
  27. [27]
    G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    G. Evenbly and G. Vidal, Entanglement renormalization in two spatial dimensions, Phys. Rev. Lett. 102 (2009) 180406 [arXiv:0811.0879] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    F. Pastawski and J. Preskill, Code properties from holographic geometries, arXiv:1612.00017 [INSPIRE].
  30. [30]
    S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New J. Phys. 11 (2009) 043029 [arXiv:0810.1983].ADSCrossRefGoogle Scholar
  31. [31]
    S. Bravyi, D. Poulin and B. Terhal, Tradeoffs for reliable quantum information storage in 2d systems, Phys. Rev. Lett. 104 (2010) 050503 [arXiv:0909.5200].ADSCrossRefzbMATHGoogle Scholar
  32. [32]
    S. Bravyi, M.B. Hastings and F. Verstraete, Lieb-Robinson bounds and the generation of correlations and topological quantum order, Phys. Rev. Lett. 97 (2006) 050401 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, 10th anniversary edition, Cambridge University Press, New York NY U.S.A., (2011).Google Scholar
  35. [35]
    M.M. Wolf, Quantum channels & operations: guided tour,, (2012).

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.IBM T.J. Watson Research CenterYorktown HeightsU.S.A.
  2. 2.NBIA, Niels Bohr InstituteUniversity of CopenhagenCopenhagenDenmark

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