Journal of High Energy Physics

, 2018:39 | Cite as

Complexity is simple!

  • William Cottrell
  • Miguel Montero
Open Access
Regular Article - Theoretical Physics


In this note we investigate the role of Lloyd’s computational bound in holographic complexity. Our goal is to translate the assumptions behind Lloyd’s proof into the bulk language. In particular, we discuss the distinction between orthogonalizing and ‘simple’ gates and argue that these notions are useful for diagnosing holographic complexity. We show that large black holes constructed from series circuits necessarily employ simple gates, and thus do not satisfy Lloyd’s assumptions. We also estimate the degree of parallel processing required in this case for elementary gates to orthogonalize. Finally, we show that for small black holes at fixed chemical potential, the orthogonalization condition is satisfied near the phase transition, supporting a possible argument for the Weak Gravity Conjecture first advocated in [1].


AdS-CFT Correspondence Black Holes in String Theory 


Open Access

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  1. [1]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic complexity equals bulk action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].ADSGoogle Scholar
  4. [4]
    N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The string landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A. de la Fuente, P. Saraswat and R. Sundrum, Natural inflation and quantum gravity, Phys. Rev. Lett. 114 (2015) 151303 [arXiv:1412.3457] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    T. Rudelius, On the possibility of large axion moduli spaces, JCAP 04 (2015) 049 [arXiv:1409.5793] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    T. Rudelius, Constraints on axion inflation from the weak gravity conjecture, JCAP 09 (2015) 020 [arXiv:1503.00795] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    M. Montero, A.M. Uranga and I. Valenzuela, Transplanckian axions!?, JHEP 08 (2015) 032 [arXiv:1503.03886] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    J. Brown, W. Cottrell, G. Shiu and P. Soler, Fencing in the swampland: quantum gravity constraints on large field inflation, JHEP 10 (2015) 023 [arXiv:1503.04783] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    T.C. Bachlechner, C. Long and L. McAllister, Planckian axions and the weak gravity conjecture, JHEP 01 (2016) 091 [arXiv:1503.07853] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    A. Hebecker, P. Mangat, F. Rompineve and L.T. Witkowski, Winding out of the swamp: evading the weak gravity conjecture with F-term winding inflation?, Phys. Lett. B 748 (2015) 455 [arXiv:1503.07912] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  12. [12]
    J. Brown, W. Cottrell, G. Shiu and P. Soler, On axionic field ranges, loopholes and the weak gravity conjecture, JHEP 04 (2016) 017 [arXiv:1504.00659] [INSPIRE].ADSMathSciNetGoogle Scholar
  13. [13]
    D. Junghans, Large-field inflation with multiple axions and the weak gravity conjecture, JHEP 02 (2016) 128 [arXiv:1504.03566] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    E. Palti, On natural inflation and moduli stabilisation in string theory, JHEP 10 (2015) 188 [arXiv:1508.00009] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    B. Heidenreich, M. Reece and T. Rudelius, Sharpening the weak gravity conjecture with dimensional reduction, JHEP 02 (2016) 140 [arXiv:1509.06374] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    K. Kooner, S. Parameswaran and I. Zavala, Warping the weak gravity conjecture, Phys. Lett. B 759 (2016) 402 [arXiv:1509.07049] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  17. [17]
    B. Heidenreich, M. Reece and T. Rudelius, Weak gravity strongly constrains large-field axion inflation, JHEP 12 (2015) 108 [arXiv:1506.03447] [INSPIRE].ADSMathSciNetGoogle Scholar
  18. [18]
    L.E. Ibáñez, M. Montero, A. Uranga and I. Valenzuela, Relaxion monodromy and the weak gravity conjecture, JHEP 04 (2016) 020 [arXiv:1512.00025] [INSPIRE].ADSMathSciNetGoogle Scholar
  19. [19]
    M. Montero, G. Shiu and P. Soler, The weak gravity conjecture in three dimensions, JHEP 10 (2016) 159 [arXiv:1606.08438] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    B. Heidenreich, M. Reece and T. Rudelius, Evidence for a sublattice weak gravity conjecture, JHEP 08 (2017) 025 [arXiv:1606.08437] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    A. Hebecker, P. Mangat, S. Theisen and L.T. Witkowski, Can gravitational instantons really constrain axion inflation?, JHEP 02 (2017) 097 [arXiv:1607.06814] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    P. Saraswat, Weak gravity conjecture and effective field theory, Phys. Rev. D 95 (2017) 025013 [arXiv:1608.06951] [INSPIRE].ADSGoogle Scholar
  23. [23]
    A. Herraez and L.E. Ibáñez, An axion-induced SM/MSSM Higgs landscape and the weak gravity conjecture, JHEP 02 (2017) 109 [arXiv:1610.08836] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  24. [24]
    H. Ooguri and C. Vafa, Non-supersymmetric AdS and the swampland, arXiv:1610.01533 [INSPIRE].
  25. [25]
    G. Shiu, P. Soler and W. Cottrell, Weak gravity conjecture and extremal black hole, arXiv:1611.06270 [INSPIRE].
  26. [26]
    A. Hebecker, P. Henkenjohann and L.T. Witkowski, What is the magnetic weak gravity conjecture for axions?, Fortsch. Phys. 65 (2017) 1700011 [arXiv:1701.06553] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    E. Palti, The weak gravity conjecture and scalar fields, JHEP 08 (2017) 034 [arXiv:1705.04328] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    A. Hebecker and P. Soler, The weak gravity conjecture and the axionic black hole paradox, JHEP 09 (2017) 036 [arXiv:1702.06130] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    D. Klaewer and E. Palti, Super-Planckian spatial field variations and quantum gravity, JHEP 01 (2017) 088 [arXiv:1610.00010] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    L.E. Ibáñez, V. Martin-Lozano and I. Valenzuela, Constraining the EW hierarchy from the weak gravity conjecture, arXiv:1707.05811 [INSPIRE].
  31. [31]
    Y. Hamada and G. Shiu, Weak gravity conjecture, multiple point principle and the Standard Model landscape, JHEP 11 (2017) 043 [arXiv:1707.06326] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    M. Montero, A.M. Uranga and I. Valenzuela, A Chern-Simons pandemic, JHEP 07 (2017) 123 [arXiv:1702.06147] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    M. Montero, Are tiny gauge couplings out of the swampland?, JHEP 10 (2017) 208 [arXiv:1708.02249] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    L.E. Ibáñez and M. Montero, A note on the WGC, effective field theory and clockwork within string theory, arXiv:1709.02392 [INSPIRE].
  35. [35]
    D. Lüst and E. Palti, Scalar fields, hierarchical UV/IR mixing and the weak gravity conjecture, arXiv:1709.01790 [INSPIRE].
  36. [36]
    S. Lloyd, Ultimate physical limits to computation, Nature 406 (2000) 1047 [quant-ph/9908043].
  37. [37]
    L. Susskind, The typical-state paradox: diagnosing horizons with complexity, Fortsch. Phys. 64 (2016) 84 [arXiv:1507.02287] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    J. Watrous, Quantum computational complexity, arXiv:0804.3401.
  39. [39]
    R.A. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information: 10th anniversary edition, 10th ed., Cambridge University Press, New York NY U.S.A., (2011).Google Scholar
  41. [41]
    S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Towards complexity for quantum field theory states, arXiv:1707.08582 [INSPIRE].
  42. [42]
    L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    N. Margolus and L.B. Levitin, The maximum speed of dynamical evolution, Physica D 120 (1998) 188 [quant-ph/9710043] [INSPIRE].
  45. [45]
    S.P. Jordan, Fast quantum computation at arbitrarily low energy, Phys. Rev. A 95 (2017) 032305 [arXiv:1701.01175] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    N.A. Sinitsyn, Computing with a single qubit faster than the quantum speed limit, arXiv:1701.05550.
  47. [47]
    M. Dugić and M.M. Ćirković, Quantum information processing: the case of vanishing interaction energy, Phys. Lett. A 302 (2002) 291 [quant-ph/0210186].
  48. [48]
    R.-G. Cai, S.-M. Ruan, S.-J. Wang, R.-Q. Yang and R.-H. Peng, Action growth for AdS black holes, JHEP 09 (2016) 161 [arXiv:1606.08307] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60 (1999) 064018 [hep-th/9902170] [INSPIRE].ADSMathSciNetGoogle Scholar
  50. [50]
    A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Holography, thermodynamics and fluctuations of charged AdS black holes, Phys. Rev. D 60 (1999) 104026 [hep-th/9904197] [INSPIRE].ADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute for Theoretical Physics AmsterdamUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Institute for Theoretical Physics and Center for Extreme Matter and Emergent PhenomenaUtrecht UniversityUtrechtThe Netherlands

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