Advertisement

Toy models of holographic duality between local Hamiltonians

  • Tamara KohlerEmail author
  • Toby Cubitt
Open Access
Regular Article - Theoretical Physics

Abstract

Holographic quantum error correcting codes (HQECC) have been proposed as toy models for the AdS/CFT correspondence, and exhibit many of the features of the duality. HQECC give a mapping of states and observables. However, they do not map local bulk Hamiltonians to local Hamiltonians on the boundary. In this work, we combine HQECC with Hamiltonian simulation theory to construct a bulk-boundary mapping between local Hamiltonians, whilst retaining all the features of the HQECC duality. This allows us to construct a duality between models, encompassing the relationship between bulk and boundary energy scales and time dynamics.

It also allows us to construct a map in the reverse direction: from local boundary Hamiltonians to the corresponding local Hamiltonian in the bulk. Under this boundary-to-bulk mapping, the bulk geometry emerges as an approximate, low-energy, effective theory living in the code-space of an (approximate) HQECC on the boundary. At higher energy scales, this emergent bulk geometry is modified in a way that matches the toy models of black holes proposed previously for HQECC. Moreover, the duality on the level of dynamics shows how these toy-model black holes can form dynamically.

Keywords

AdS-CFT Correspondence Black Holes 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    J.M. Maldacena, Eternal black holes in Anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  2. [2]
    X.D. Ahmed Almheiri and D. Harlow, Bulk locality and quantum error correction in Ads/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  3. [3]
    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].MathSciNetCrossRefzbMATHADSGoogle Scholar
  4. [4]
    B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 46 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
  5. [5]
    B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
  6. [6]
    Z. Yang, P. Hayden and X.L. Qi, Bidirectional holographic codes and sub-AdS locality, JHEP 01 (2016) 175 [arXiv:1510.03784] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  7. [7]
    P. Hayden et al., Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  8. [8]
    A. Bhattacharyya, Z.S. Gao, L.Y. Hung and S.N. liu, Exploring the tensor networks/AdS correspondence, JHEP 08 (2016) 086 [arXiv:1606.00621] [INSPIRE].
  9. [9]
    T.J. Osborne and D.E. Stiegemann, Dynamics for holographic codes, arXiv:1706.08823 [INSPIRE].
  10. [10]
    T. Cubitt, A. Montanaro and S. Piddock, Universal quantum Hamiltonians, Proce. Natl. Acad. Sci. 115 (2018) 9497 [arXiv:1701.05182].MathSciNetCrossRefzbMATHADSGoogle Scholar
  11. [11]
    M.P. Woods and A.M. Alhambra, Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames, arXiv:1902.07725 [INSPIRE].
  12. [12]
    P. Faist et al., Continuous symmetries and approximate quantum error correction, arXiv:1902.07714 [INSPIRE].
  13. [13]
    A.L.M. Headrick, V.E. Hubeny and M. Rangamani, Causality and holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].CrossRefADSGoogle Scholar
  14. [14]
    K.G. Wilson, Renormalization group and critical phenomena. I: renormalization group and the Kadanoff scaling pictre, Phys. Rev. B 4 (1971) 3174.Google Scholar
  15. [15]
    K.G. Wilson, The renormalization group and critical phenomena. II: phase space cell analysis of critical behavior, Phys. Rev. B 4 (1971) 3184.Google Scholar
  16. [16]
    R. Oliveira and B. Terhal, The complexity of quantum spin systems on a two-dimensional square lattice, Quant. Inf. Comput. 8 (2005) 0900 [quant-ph/0504050].
  17. [17]
    G. Gour and N.R. Wallach, All maximally entangled four-qubit states, J. Math. Phys. 51 (2010)112201 [arXiv:1006.0036].
  18. [18]
    E.M. Rains, Quantum codes of minimum distance two, IEEE Trans. Inf. Theor. 45 (1999) 266 [quant-ph/9704043] [INSPIRE].
  19. [19]
    F. Huber, O. Guene and J. Siewert, Absolutely maximally entanged states of seven qubits do not exist, Phys. Rev. Lett. 118 (2017) 200502 [arXiv:1608.06228].CrossRefADSGoogle Scholar
  20. [20]
    W. Helwig, Absolutely maximally entangled qudit graph states, arXiv:1306.2879.
  21. [21]
    S. Bravyi and M. Hastings, On complexity of the quantum Ising model, Comm. Math. Phys. 349 (2017)1 [arXiv:1410.0703] [INSPIRE].
  22. [22]
    J. Beckenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333.Google Scholar
  23. [23]
    S.W. Hawking, Particle creation by black holes, Comm. Math. Phys. 43 (1975) 199.MathSciNetCrossRefzbMATHADSGoogle Scholar
  24. [24]
    P. Hayden and G. Penington, Learning the alpha-bits of black holes, arXiv:1807.06041 [INSPIRE].
  25. [25]
    D. Harlow, The Ryu-Takayanagi formula from quantum error correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  26. [26]
    S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  27. [27]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  28. [28]
    S. Nezami and M. Walter, Multipartite entanglement in stabilizer tensor networks, arXiv:1608.02595 [INSPIRE].
  29. [29]
    A. Felikson and P. Tumarkin, Hyperbolic Coxeter polytopes, http://www.maths.dur.ac.uk/users/anna.felikson/Polytopes/polytopes.html.
  30. [30]
    J. Tits, Groupes et géométries de Coxeter, (1961)Google Scholar
  31. [31]
    M.W. Davis, The geometry and topology of Coxeter groups, Princeton University Press, Princeton U.S.A. (2007).Google Scholar
  32. [32]
    E.B. Vinberg, Hyperbolic reflection groups, Russian Math. Surveys 40 (1985) 31.CrossRefzbMATHADSGoogle Scholar
  33. [33]
    P. Abramenko and K. Brown, Buildings theory and applications, Graduate Texts in Mathematics, Springer, Germany (2008).Google Scholar
  34. [34]
    A.M. Cohen, Finite Coxeter groups, lecture notes (2008).Google Scholar
  35. [35]
  36. [36]
    D. Aharonov and L. Zhou, Hamiltonian sparsification and gap-simulation, arXiv:1804.11084.
  37. [37]
    S. Piddock and A. Montanaro, The complexity of antiferromagnetic interactions and 2d lattices, Quant. Inf. Comput. 17 (2015) 636 [arXiv:1506.04014].MathSciNetGoogle Scholar
  38. [38]
    R. Guglielmetti, CoxiterWeb, https://coxiterweb.rafaelguglielmetti.ch.
  39. [39]
    W. Helwig and W. Cui, Absolutely maximally entangled states: existence and applications, arXiv:1306.2536.
  40. [40]
    D. Goyeneche et al., Absolutely maximally entangled states, combinatorial design and multi-unitary matrices, Phys. Rev. A 92 (2015) 032316 [arXiv:1506.08857] [INSPIRE].
  41. [41]
    W. Helwig et al., Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A 86 (2012)052335 [arXiv:1204.2289] [INSPIRE].
  42. [42]
    R. Cleve, D. Gottesman and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett. 83 (1999) 648 [quant-ph/9901025] [INSPIRE].
  43. [43]
    V. Gheorghiu, Standard form of qudit stabilizer groups, Phys. Lett. A 378 (2014) 505 [arXiv:1101.1519].
  44. [44]
    H. Kurzweil and B. Stellmacher, The theory of finite groups: an introduction, Springer, Germany (2004).Google Scholar
  45. [45]
    D. Gottesman, Stabilizer Codes and Quantum Error Correction, Ph.D. thesis, Caltech, Pasadena, U.S.A. (1997).Google Scholar
  46. [46]
    I. Reed and G. Solomon, Polynomial codes over certain finite fields, J. Soc. Industr. Appl. Math. 8 (1960) 300.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    G. Seroussi and R. Roth, On MDS extensions of generalized Reed-Solomon codes, IEEE Trans. Inf. Theor. 32 (1986) 349.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    M. Grassl and M. Roetteler, Quantum MDS codes over small fields, IEEE Proc. Int. Symp. Inf. Theor. (2015) 1104 [arXiv:1502.05267].
  49. [49]
    T. Kohler and T. Cubitt, Translationally invariant universal classical Hamiltonians, J. Stat. Phys. 176 (2019) 228 [arXiv:1807.01715].MathSciNetCrossRefzbMATHADSGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity College LondonLondonU.K.

Personalised recommendations