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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References
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
D. Harlow, TASI lectures on the emergence of the bulk in AdS/CFT, arXiv:1802.01040 [INSPIRE].
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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
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].
J.L.F. Barbon and J.M. Magan, Chaotic fast scrambling at black holes, Phys. Rev. D 84 (2011) 106012 [arXiv:1105.2581] [INSPIRE].
J.L.F. Barbon and J.M. Magan, Fast scramblers, horizons and expander graphs, JHEP 08 (2012) 016 [arXiv:1204.6435] [INSPIRE].
L. Susskind, Why do things fall?, arXiv:1802.01198 [INSPIRE].
M.A. Nielsen, A geometric approach to quantum lower bounds, quant-ph/0502070.
M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum computation as geometry, Science 311 (2006) 1133 [quant-ph/0603161].
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, quant-ph/0701004.
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].
S. Aaronson, The complexity of quantum states and transformations: from quantum money to black holes, arXiv:1607.05256 [INSPIRE].
A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
P. Caputa and J.M. Magan, Quantum computation as gravity, arXiv:1807.04422 [INSPIRE].
A.R. Brown, L. Susskind and Y. Zhao, Quantum complexity and negative curvature, Phys. Rev. D 95 (2017) 045010 [arXiv:1608.02612] [INSPIRE].
D.A. Roberts, D. Stanford and A. Streicher, Operator growth in the SYK model, JHEP 06 (2018) 122 [arXiv:1802.02633] [INSPIRE].
D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Anti-de Sitter space from optimization of path integrals in conformal field theories, Phys. Rev. Lett. 119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].
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].
R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].
S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a definition of complexity for quantum field theory states, Phys. Rev. Lett. 120 (2018) 121602 [arXiv:1707.08582] [INSPIRE].
R.-Q. Yang, Complexity for quantum field theory states and applications to thermofield double states, Phys. Rev. D 97 (2018) 066004 [arXiv:1709.00921] [INSPIRE].
R.-Q. Yang, C. Niu, C.-Y. Zhang and K.-Y. Kim, Comparison of holographic and field theoretic complexities for time dependent thermofield double states, JHEP 02 (2018) 082 [arXiv:1710.00600] [INSPIRE].
R. Khan, C. Krishnan and S. Sharma, Circuit complexity in fermionic field theory, arXiv:1801.07620 [INSPIRE].
K. Hashimoto, N. Iizuka and S. Sugishita, Thoughts on holographic complexity and its basis-dependence, Phys. Rev. D 98 (2018) 046002 [arXiv:1805.04226] [INSPIRE].
M. Guo, J. Hernandez, R.C. Myers and S.-M. Ruan, Circuit complexity for coherent states, arXiv:1807.07677 [INSPIRE].
N. Margolus and L.B. Levitin, The maximum speed of dynamical evolution, Physica D 120 (1998) 188 [quant-ph/9710043] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, http://online.kitp.ucsb.edu/online/entangled15/kitaev/, University of California, Santa Barbara, CA, U.S.A., 7 April 2015.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/, University of California, Santa Barbara, CA, U.S.A., 27 May 2015.
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
E. Witten, An SYK-like model without disorder, arXiv:1610.09758 [INSPIRE].
I.R. Klebanov and G. Tarnopolsky, Uncolored random tensors, melon diagrams and the Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 046004 [arXiv:1611.08915] [INSPIRE].
S. Sachdev, Bekenstein-Hawking entropy and strange metals, Phys. Rev. X 5 (2015) 041025 [arXiv:1506.05111] [INSPIRE].
I. Danshita, M. Hanada and M. Tezuka, Creating and probing the Sachdev-Ye-Kitaev model with ultracold gases: towards experimental studies of quantum gravity, PTEP 2017 (2017) 083I01 [arXiv:1606.02454] [INSPIRE].
L. Benet and H.A. Weidenmuller, Review of the k body embedded ensembles of Gaussian random matrices, J. Phys. A 36 (2003) 3569 [cond-mat/0207656] [INSPIRE].
J.M. Magan, Random free fermions: an analytical example of eigenstate thermalization, Phys. Rev. Lett. 116 (2016) 030401 [arXiv:1508.05339] [INSPIRE].
J.M. Magan, Black holes as random particles: entanglement dynamics in infinite range and matrix models, JHEP 08 (2016) 081 [arXiv:1601.04663] [INSPIRE].
J.M. Magan, Decoherence and microscopic diffusion at the Sachdev-Ye-Kitaev model, Phys. Rev. D 98 (2018) 026015 [arXiv:1612.06765] [INSPIRE].
V.K.B. Kota and N.D. Chavda, Embedded random matrix ensembles from nuclear structure and their recent applications, Int. J. Mod. Phys. E 27 (2018) 1830001 [INSPIRE].
J. Sonner and M. Vielma, Eigenstate thermalization in the Sachdev-Ye-Kitaev model, JHEP 11 (2017) 149 [arXiv:1707.08013] [INSPIRE].
M. Haque and P. McClarty, Eigenstate thermalization scaling in Majorana clusters: from integrable to chaotic SYK models, arXiv:1711.02360 [INSPIRE].
J.M. Magan, De Finetti theorems and entanglement in large-N theories and gravity, Phys. Rev. D 96 (2017) 086002 [arXiv:1705.03048] [INSPIRE].
S. Lloyd and H. Pagels, Complexity as thermodynamic depth, Annals Phys. 188 (1988) 186 [INSPIRE].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
J.M. Magan and S. Vandoren, Entanglement in Fock space of random QFT states, JHEP 07 (2015) 150 [arXiv:1504.01346] [INSPIRE].
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Magán, J.M. Black holes, complexity and quantum chaos. J. High Energ. Phys. 2018, 43 (2018). https://doi.org/10.1007/JHEP09(2018)043
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DOI: https://doi.org/10.1007/JHEP09(2018)043