Black holes, complexity and quantum chaos

Journal of High Energy Physics, Sep 2018

Abstract We study aspects of black holes and quantum chaos through the behavior of computational costs, which are distance notions in the manifold of unitaries of the theory. To this end, we enlarge Nielsen geometric approach to quantum computation and provide metrics for finite temperature/energy scenarios and CFT’s. From the framework, it is clear that costs can grow in two different ways: operator vs ‘simple’ growths. The first type mixes operators associated to different penalties, while the second does not. Important examples of simple growths are those related to symmetry transformations, and we describe the costs of rotations, translations, and boosts. For black holes, this analysis shows how infalling particle costs are controlled by the maximal Lyapunov exponent, and motivates a further bound on the growth of chaos. The analysis also suggests a correspondence between proper energies in the bulk and average ‘local’ scaling dimensions in the boundary. Finally, we describe these complexity features from a dual perspective. Using recent results on SYK we compute a lower bound to the computational cost growth in SYK at infinite temperature. At intermediate times it is controlled by the Lyapunov exponent, while at long times it saturates to a linear growth, as expected from the gravity description.

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Black holes, complexity and quantum chaos

Journal of High Energy Physics September 2018, 2018:43 | Cite as Black holes, complexity and quantum chaos AuthorsAuthors and affiliations Javier M. Magán Open Access Regular Article - Theoretical Physics First Online: 07 September 2018 Received: 26 June 2018 Revised: 28 August 2018 Accepted: 02 September 2018 29 Downloads Abstract We study aspects of black holes and quantum chaos through the behavior of computational costs, which are distance notions in the manifold of unitaries of the theory. To this end, we enlarge Nielsen geometric approach to quantum computation and provide metrics for finite temperature/energy scenarios and CFT’s. From the framework, it is clear that costs can grow in two different ways: operator vs ‘simple’ growths. The first type mixes operators associated to different penalties, while the second does not. Important examples of simple growths are those related to symmetry transformations, and we describe the costs of rotations, translations, and boosts. For black holes, this analysis shows how infalling particle costs are controlled by the maximal Lyapunov exponent, and motivates a further bound on the growth of chaos. The analysis also suggests a correspondence between proper energies in the bulk and average ‘local’ scaling dimensions in the boundary. Finally, we describe these complexity features from a dual perspective. Using recent results on SYK we compute a lower bound to the computational cost growth in SYK at infinite temperature. At intermediate times it is controlled by the Lyapunov exponent, while at long times it saturates to a linear growth, as expected from the gravity description. Keywords Black Holes AdS-CFT Correspondence Random Systems  ArXiv ePrint: 1805.05839 Download to read the full article text 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] 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] D. Harlow, TASI lectures on the emergence of the bulk in AdS/CFT, arXiv:1802.01040 [INSPIRE]. [3] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE]. [4] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar [5] 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 [6] A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar [7] S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar [8] S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar [9] J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar [10] L. Susskind and J. Lindesay, An introduction to black holes, information and the string theory revolution: the holographic universe, World Scientific, Hackensack, U.S.A., (2005) [INSPIRE]. [11] J.L.F. Barbon and J.M. Magan, Chaotic fast scrambling at black holes, Phys. Rev. D 84 (2011) 106012 [arXiv:1105.2581] [INSPIRE]. [12] J.L.F. Barbon and J.M. Magan, Fast scramblers, horizons and expander graphs, JHEP 08 (2012) 016 [arXiv:1204.6435] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar [13] L. Susskind, Why do things fall?, arXiv:1802.01198 [INSPIRE]. [14] M.A. Nielsen, A geometric approach to quantum lower bounds, quant-ph/0502070. [15] M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum computation as geometry, Science 311 (2006) 1133 [quant-ph/0603161]. [16] M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, quant-ph/0701004. [17] L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 24 [Addendum ibid. 64 (2016) 44] [arXiv:1402.5674] [arXiv:1403.5695] [INSPIRE]. [18] S. Aaronson, The complexity of quantum states and transformations: from quantum money to black holes, arXiv:1607.05256 [INSPIRE]. [19] A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE]. [20] P. Caputa and J.M. Magan, Quantum computation as gravity, arXiv:1807.04422 [INSPIRE]. [21] A.R. Brown, L. Susskind and Y. Zhao, Quantum comple (...truncated)


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Javier M. Magán. Black holes, complexity and quantum chaos, Journal of High Energy Physics, 2018, pp. 43, Volume 2018, Issue 9, DOI: 10.1007/JHEP09(2018)043