Advertisement

Chaos and complexity by design

  • Daniel A. RobertsEmail author
  • Beni Yoshida
Open Access
Regular Article - Theoretical Physics

Abstract

We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the “frame poten-tial,” which is minimized by unitary k-designs and measures the 2-norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is out-of-time-order (OTO) four-point correlation functions. We show that the norm squared of a generalization of out-of-time-order 2k-point correlators is proportional to the kth frame potential, providing a quantitative connection between chaos and pseudorandomness. Additionally, we prove that these 2k-point correlators for Pauli operators completely determine the k-fold channel of an ensemble of unitary operators. Finally, we use a counting argument to obtain a lower bound on the quantum circuit complexity in terms of the frame potential. This provides a direct link between chaos, complexity, and randomness.

Keywords

AdS-CFT Correspondence Gauge-gravity correspondence Random Systems Holography and condensed matter physics (AdS/CMT) 

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]
    P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the fast scrambling conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An apologia for firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    F. Dupuis, M. Berta, J. Wullschleger and R. Renner, One-shot decoupling, Comm. Math. Phys. 328 (2014) 251 [arXiv:1012.6044].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    W. Brown and O. Fawzi, Decoupling with random quantum circuits, Comm. Math. Phys. 340 (2015) 867.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    . Chamon, A. Hamma and E.R. Mucciolo, Emergent irreversibility and entanglement spectrum statistics, Phys. Rev. Lett. 112 (2014) 240501 [arXiv:1310.2702].
  11. [11]
    P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    A. Nahum, J. Ruhman, S. Vijay and J. Haah, Quantum entanglement growth under random unitary dynamics, arXiv:1608.06950 [INSPIRE].
  13. [13]
    D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    A. Larkin and Y. Ovchinnikov, Quasiclassical method in the theory of superconductivity, JETP 28 (1969) 1200.ADSGoogle Scholar
  15. [15]
    A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk given at the Fundamental Physics Prize Symposium, November 10 (2014).Google Scholar
  16. [16]
    S.H. Shenker and D. Stanford, Multiple shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    D. Stanford, Many-body chaos at weak coupling, JHEP 10 (2016) 009 [arXiv:1512.07687] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    A.L. Fitzpatrick and J. Kaplan, A quantum correction to chaos, JHEP 05 (2016) 070 [arXiv:1601.06164] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    Y. Gu and X.-L. Qi, Fractional statistics and the butterfly effect, JHEP 08 (2016) 129 [arXiv:1602.06543] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    P. Caputa, T. Numasawa and A. Veliz-Osorio, Out-of-time-ordered correlators and purity in rational conformal field theories, PTEP 2016 (2016) 113B06 [arXiv:1602.06542] [INSPIRE].
  23. [23]
    B. Swingle, G. Bentsen, M. Schleier-Smith and P. Hayden, Measuring the scrambling of quantum information, Phys. Rev. A 94 (2016) 040302 [arXiv:1602.06271] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    E. Perlmutter, Bounding the space of holographic CFTs with chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    M. Blake, Universal charge diffusion and the butterfly effect in holographic theories, Phys. Rev. Lett. 117 (2016) 091601 [arXiv:1603.08510] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    D.A. Roberts and B. Swingle, Lieb-Robinson bound and the butterfly effect in quantum field theories, Phys. Rev. Lett. 117 (2016) 091602 [arXiv:1603.09298] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    M. Blake, Universal diffusion in incoherent black holes, Phys. Rev. D 94 (2016) 086014 [arXiv:1604.01754] [INSPIRE].ADSGoogle Scholar
  28. [28]
    B. Swingle and D. Chowdhury, Slow scrambling in disordered quantum systems, Phys. Rev. B 95 (2017) 060201 [arXiv:1608.03280] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    Y. Huang, Y.-L. Zhang and X. Chen, Out-of-time-ordered correlator in many-body localized systems, arXiv:1608.01091 [INSPIRE].
  30. [30]
    R. Fan, P. Zhang, H. Shen and H. Zhai, Out-of-time-order correlation for many-body localization, arXiv:1608.01914 [INSPIRE].
  31. [31]
    N. Yunger Halpern, Jarzynski-like equality for the out-of-time-ordered correlator, Phys. Rev. A 95 (2017) 012120 [arXiv:1609.00015] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    D.P. DiVincenzo, D.W. Leung and B.M. Terhal, Quantum data hiding, IEEE Trans. Inf. Theory 48 (2002) 580.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    J. Emerson, E. Livine and S. Lloyd, Convergence conditions for random quantum circuits, Phys. Rev. A 72 (2005) 060302.ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    A. Ambainis and J. Emerson, Quantum t-designs: t-wise independence in the quantum world, in the proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity (CCC07), June 13-16, Washington, U.S.A. (2007).Google Scholar
  35. [35]
    D. Gross, K. Audenaert and J. Eisert, Evenly distributed unitaries: on the structure of unitary designs, J. Math. Phys. 48 (2007) 052104 [quant-ph/0611002]
  36. [36]
    C. Dankert, R. Cleve, J. Emerson and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A 80 (2009) 012304.ADSCrossRefGoogle Scholar
  37. [37]
    J. Emerson et al., Pseudo-random unitary operators for quantum information processing, Science 302 (2003) 2098.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    W.G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104 (2010) 250501.ADSCrossRefGoogle Scholar
  39. [39]
    A.W. Harrow and R.A. Low, Random quantum circuits are approximate 2-designs, Comm. Math. Phys. 291 (2009) 257 [arXiv:0802.1919].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    E. Knill et al., Randomized benchmarking of quantum gates, Phys. Rev. A 77 (2008) 012307.ADSCrossRefGoogle Scholar
  41. [41]
    F.G.S.L. Brandao, A.W. Harrow and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, arXiv:1208.0692.
  42. [42]
    R. Kueng and D. Gross, Qubit stabilizer states are complex projective 3-designs, arXiv:1510.02767.
  43. [43]
    Z. Webb, The Clifford group forms a unitary 3-design, arXiv:1510.02769.
  44. [44]
    R.A. Low., Pseudo-randomness and learning in quantum computation, arXiv:1006.5227.
  45. [45]
    A.J. Scott, Optimizing quantum process tomography with unitary 2-designs, J. Phys. A 41 (2008) 055308.ADSMathSciNetzbMATHGoogle Scholar
  46. [46]
    H. Zhu, Multiqubit clifford groups are unitary 3-designs, arXiv:1510.02619.
  47. [47]
    B. Collins and I. Nechita, Random matrix techniques in quantum information theory, J. Math. Phys. 57 (2016) 015215 [arXiv:1509.04689].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    Y. Nakata, C. Hirche, M. Koashi and A. Winter, Efficient unitary designs with nearly time-independent Hamiltonian dynamics, arXiv:1609.07021 [INSPIRE].
  49. [49]
    D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
  50. [50]
    E. Knill, Approximation by quantum circuits, quant-ph/9508006 [INSPIRE].
  51. [51]
    L. Susskind, The Typical-State Paradox: Diagnosing Horizons with Complexity, Fortsch. Phys. 64 (2016) 84 [arXiv:1507.02287] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    A. Kitaev, Ph/CS 219C: quantum computation, course taught at Caltech, California, U.S.A. (2016).Google Scholar
  53. [53]
    Y. Gu, Moments of random matrices and weingarten functions, M.Sc. thesis, Queen’s University, Ontario, Canada (2013).Google Scholar
  54. [54]
    J. Watrous, Theory of quantum information, lecture notes (2015)Google Scholar
  55. [55]
    B. Collins, Moments and cumulants of polynomial random variables on unitarygroups, the Itzykson-Zuber integral, and free probability, Int. Math. Res. Not. 2003 (2003) 953.CrossRefzbMATHGoogle Scholar
  56. [56]
    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].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    A. Roy and A.J. Scott, Unitary designs and codes, Des. Codes Cryptogr. 53 (2009) 13.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    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].
  59. [59]
    A. Kitaev, A simple model of quantum holography, talks given at KITP, April 7 and May 27 Santa Barbara, U.S.A. (2015).Google Scholar
  60. [60]
    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].ADSGoogle Scholar
  61. [61]
    L. Dyson, M. Kleban and L. Susskind, Disturbing implications of a cosmological constant, JHEP 10 (2002) 011 [hep-th/0208013] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    J.L.F. Barbon and E. Rabinovici, Very long time scales and black hole thermal equilibrium, JHEP 11 (2003) 047 [hep-th/0308063] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  63. [63]
    J.L.F. Barbon and E. Rabinovici, Geometry and quantum noise, Fortsch. Phys. 62 (2014) 626 [arXiv:1404.7085] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, quant-ph/0701004.
  65. [65]
    A.R. Brown, L. Susskind and Y. Zhao, Quantum complexity and negative curvature, Phys. Rev. D 95 (2017) 045010 [arXiv:1608.02612] [INSPIRE].ADSGoogle Scholar
  66. [66]
    S. Chapman, H. Marrochio and R.C. Myers, Complexity of formation in holography, to appear.Google Scholar
  67. [67]
    L. Susskind and Y. Zhao, Switchbacks and the bridge to nowhere, arXiv:1408.2823 [INSPIRE].
  68. [68]
    W. Chemissany and T.J. Osborne, Holographic fluctuations and the principle of minimal complexity, JHEP 12 (2016) 055 [arXiv:1605.07768] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  69. [69]
    A.R. Brown and L. Susskind, The second law of quantum complexity, arXiv:1701.01107 [INSPIRE].
  70. [70]
    D. Harlow and P. Hayden, Quantum computation vs. firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
  71. [71]
    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].ADSGoogle Scholar
  75. [75]
    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
  76. [76]
    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
  77. [77]
    T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  79. [79]
    B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
  80. [80]
    J. Maldacena, Spacetime from entanglement, talk give at KITP, August 20, Santa Barbara, U.S.A. (2013).Google Scholar
  81. [81]
    A. Almheiri, X. Dong, and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041].ADSMathSciNetCrossRefGoogle Scholar
  82. [82]
    M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, U.K. (2000).zbMATHGoogle Scholar
  83. [83]
    M. Mozrzymas et al., Local random quantum circuits are approximate polynomial-designs: numerical results, J. Phys. A 46 (2013) 305301 [arXiv:1212.2556].MathSciNetzbMATHGoogle Scholar
  84. [84]
    A. Brown, Wormholes and complexity, talk given at the Perimeter Institute for Theoretical Physics, August 21, Waterloo, Canada (2015).Google Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Center for Theoretical Physics and Department of PhysicsMassachusetts Institute of TechnologyCambridgeU.S.A.
  2. 2.School of Natural Sciences, Institute for Advanced StudyPrincetonU.S.A.
  3. 3.Perimeter Institute for Theoretical PhysicsWaterlooCanada

Personalised recommendations