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Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence

A preprint version of the article is available at arXiv.


We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].


  1. A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  2. J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  3. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  4. S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  6. M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  7. M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].

  8. M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. P. Hayden, M. Headrick and A. Maloney, Holographic mutual information is monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].

    ADS  Google Scholar 

  10. A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglementthermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].

    ADS  Article  Google Scholar 

  13. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91 (2003) 147902 [quant-ph/0301063].

  14. F. Verstraete and J.I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, cond-mat/0407066 [INSPIRE].

  15. F. Verstraete, J. Cirac and V. Murg, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57 (2008) 143 [arXiv:0907.2796].

    ADS  Article  Google Scholar 

  16. G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].

    ADS  Article  Google Scholar 

  17. G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [INSPIRE].

    ADS  Article  Google Scholar 

  18. G. Evenbly and G. Vidal, Entanglement renormalization in two spatial dimensions, Phys. Rev. Lett. 102 (2009) 180406 [arXiv:0811.0879] [INSPIRE].

    ADS  Article  Google Scholar 

  19. G. Evenbly and G. Vidal, Algorithms for entanglement renormalization, Phys. Rev. B 79 (2009) 144108 [arXiv:0707.1454].

    ADS  Article  Google Scholar 

  20. G. Evenbly and G. Vidal, Frustrated antiferromagnets with entanglement renormalization: ground state of the spin-1/2 Heisenberg model on a Kagome lattice, Phys. Rev. Lett. 104 (2010) 187203 [arXiv:0904.3383] [INSPIRE].

    ADS  Article  Google Scholar 

  21. L. Susskind and E. Witten, The holographic bound in anti-de Sitter space, hep-th/9805114 [INSPIRE].

  22. B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].

    ADS  Google Scholar 

  23. B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].

  24. B. Yoshida, Information storage capacity of discrete spin systems, Annals Phys. 338 (2013) 134 [arXiv:1111.3275] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  25. J.I. Latorre and G. Sierra, Holographic codes, arXiv:1502.06618 [INSPIRE].

  26. A.J. Ferris and D. Poulin, Tensor networks and quantum error correction, Phys. Rev. Lett. 113 (2014) 030501 [arXiv:1312.4578].

    ADS  Article  Google Scholar 

  27. D. Bacon, S.T. Flammia, A.W. Harrow and J. Shi, Sparse quantum codes from quantum circuits, arXiv:1411.3334.

  28. X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282 [INSPIRE].

  29. W. Helwig, W. Cui, A. Riera, J.I. Latorre and H.-K. Lo, Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A 86 (2012) 052335 [arXiv:1204.2289] [INSPIRE].

    ADS  Article  Google Scholar 

  30. W. Helwig, Absolutely maximally entangled qudit graph states, arXiv:1306.2879.

  31. R. Cleve, D. Gottesman and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett. 83 (1999) 648 [quant-ph/9901025] [INSPIRE].

    ADS  Article  Google Scholar 

  32. J. Weeks, KaleidoTile. A computer program for creating spherical, Euclidean and hyperbolic tilings,

  33. C.H. Papadimitriou and K. Steiglitz, Combinatorial optimization: algorithms and complexity, Courier Corporation, (1998).

  34. A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  35. I.A. Morrison, Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography, JHEP 05 (2014) 053 [arXiv:1403.3426] [INSPIRE].

    ADS  Article  Google Scholar 

  36. D. Kribs, R. Laflamme and D. Poulin, Unified and generalized approach to quantum error correction, Phys. Rev. Lett. 94 (2005) 180501 [quant-ph/0412076].

    ADS  Article  Google Scholar 

  37. D.W. Kribs, R. Laflamme, D. Poulin and M. Lesosky, Operator quantum error correction, Quant. Inf. Comp. 6 (2006) 383 [quant-ph/0504189].

  38. C. Bény, A. Kempf and D. Kribs, Quantum error correction of observables, Phys. Rev. A 76 (2007) 042303 [arXiv:0705.1574].

    ADS  Article  Google Scholar 

  39. C. Bény, A. Kempf and D. Kribs, Generalization of quantum error correction via the Heisenberg picture, Phys. Rev. Lett. 98 (2007) 100502 [quant-ph/0608071].

  40. M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].

    ADS  Article  Google Scholar 

  41. A.C. Wall, Maximin surfaces and the strong subadditivity of the covariant holographic entanglement entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  42. B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  43. D.L. Jafferis and S.J. Suh, The gravity duals of modular hamiltonians, arXiv:1412.8465 [INSPIRE].

  44. D. Gottesman, An introduction to quantum error correction and fault-tolerant quantum computation, arXiv:0904.2557.

  45. E. Mintun, J. Polchinski and V. Rosenhaus, Bulk-boundary duality, gauge invariance and quantum error correction, arXiv:1501.06577 [INSPIRE].

  46. 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].

    ADS  Article  Google Scholar 

  47. F. Pastawski and B. Yoshida, Fault-tolerant logical gates in quantum error-correcting codes, Phys. Rev. A 91 (2015) 012305 [arXiv:1408.1720].

    ADS  MathSciNet  Article  Google Scholar 

  48. A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  49. D. Harlow, Jerusalem lectures on black holes and quantum information, arXiv:1409.1231 [INSPIRE].

  50. J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  51. S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].

  52. T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  53. L. Susskind, Computational complexity and black hole horizons, arXiv:1403.5695 [INSPIRE].

  54. D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  55. D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error, in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, (1997), pg. 176 [quant-ph/9906129] [INSPIRE].

  56. M. Grassl, T. Beth and M. Roetteler, On optimal quantum codes, Int. J. Quant. Inf. 2 (2004) 55 [quant-ph/0312164].

  57. D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].

  58. S. Goldstein, J.L. Lebowitz, R. Tumulka and N. Zanghì, Canonical typicality, Phys. Rev. Lett. 96 (2006) 050403 [cond-mat/0511091].

  59. J. Adler, Bootstrap percolation, Phys. A 171 (1991) 453.

    Google Scholar 

  60. D.A. Levin, Y. Peres and E.L. Wilmer, Markov chains and mixing times, American Mathematical Society, U.S.A. (2008).

    Book  Google Scholar 

  61. M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2000).

    MATH  Google Scholar 

  62. N. Schuch, I. Cirac and D. Perez-Garcia, PEPS as ground states: degeneracy and topology, Ann. Phys. 325 (2010) 2153 [arXiv:1001.3807].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  63. M.B. ¸ahinoğlu et al., Characterizing topological order with matrix product operators, arXiv:1409.2150.

  64. O. Buerschaper, Twisted injectivity in projected entangled pair states and the classification of quantum phases, Annals Phys. 351 (2014) 447 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  65. B. Yoshida and I.L. Chuang, Framework for classifying logical operators in stabilizer codes, Phys. Rev. A 81 (2010) 052302 [arXiv:1002.0085].

    ADS  Article  Google Scholar 

  66. J. Haah and J. Preskill, Logical-operator tradeoff for local quantum codes, Phys. Rev. A 86 (2012) 032308 [arXiv:1011.3529].

    ADS  Article  Google Scholar 

  67. P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

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Correspondence to Fernando Pastawski.

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ArXiv ePrint: 1503.06237

These authors contributed equally to this work (Fernando Pastawski and Beni Yoshida).

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Pastawski, F., Yoshida, B., Harlow, D. et al. Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence. J. High Energ. Phys. 2015, 149 (2015).

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  • AdS-CFT Correspondence
  • Lattice Integrable Models