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.
References
P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the fast scrambling conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An apologia for firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].
F. Dupuis, M. Berta, J. Wullschleger and R. Renner, One-shot decoupling, Comm. Math. Phys. 328 (2014) 251 [arXiv:1012.6044].
W. Brown and O. Fawzi, Decoupling with random quantum circuits, Comm. Math. Phys. 340 (2015) 867.
. Chamon, A. Hamma and E.R. Mucciolo, Emergent irreversibility and entanglement spectrum statistics, Phys. Rev. Lett. 112 (2014) 240501 [arXiv:1310.2702].
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].
A. Nahum, J. Ruhman, S. Vijay and J. Haah, Quantum entanglement growth under random unitary dynamics, arXiv:1608.06950 [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
A. Larkin and Y. Ovchinnikov, Quasiclassical method in the theory of superconductivity, JETP 28 (1969) 1200.
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk given at the Fundamental Physics Prize Symposium, November 10 (2014).
S.H. Shenker and D. Stanford, Multiple shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
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].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
D. Stanford, Many-body chaos at weak coupling, JHEP 10 (2016) 009 [arXiv:1512.07687] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, A quantum correction to chaos, JHEP 05 (2016) 070 [arXiv:1601.06164] [INSPIRE].
Y. Gu and X.-L. Qi, Fractional statistics and the butterfly effect, JHEP 08 (2016) 129 [arXiv:1602.06543] [INSPIRE].
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].
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].
E. Perlmutter, Bounding the space of holographic CFTs with chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].
M. Blake, Universal charge diffusion and the butterfly effect in holographic theories, Phys. Rev. Lett. 117 (2016) 091601 [arXiv:1603.08510] [INSPIRE].
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].
M. Blake, Universal diffusion in incoherent black holes, Phys. Rev. D 94 (2016) 086014 [arXiv:1604.01754] [INSPIRE].
B. Swingle and D. Chowdhury, Slow scrambling in disordered quantum systems, Phys. Rev. B 95 (2017) 060201 [arXiv:1608.03280] [INSPIRE].
Y. Huang, Y.-L. Zhang and X. Chen, Out-of-time-ordered correlator in many-body localized systems, arXiv:1608.01091 [INSPIRE].
R. Fan, P. Zhang, H. Shen and H. Zhai, Out-of-time-order correlation for many-body localization, arXiv:1608.01914 [INSPIRE].
N. Yunger Halpern, Jarzynski-like equality for the out-of-time-ordered correlator, Phys. Rev. A 95 (2017) 012120 [arXiv:1609.00015] [INSPIRE].
D.P. DiVincenzo, D.W. Leung and B.M. Terhal, Quantum data hiding, IEEE Trans. Inf. Theory 48 (2002) 580.
J. Emerson, E. Livine and S. Lloyd, Convergence conditions for random quantum circuits, Phys. Rev. A 72 (2005) 060302.
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 (CCC’07), June 13-16, Washington, U.S.A. (2007).
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]
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.
J. Emerson et al., Pseudo-random unitary operators for quantum information processing, Science 302 (2003) 2098.
W.G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104 (2010) 250501.
A.W. Harrow and R.A. Low, Random quantum circuits are approximate 2-designs, Comm. Math. Phys. 291 (2009) 257 [arXiv:0802.1919].
E. Knill et al., Randomized benchmarking of quantum gates, Phys. Rev. A 77 (2008) 012307.
F.G.S.L. Brandao, A.W. Harrow and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, arXiv:1208.0692.
R. Kueng and D. Gross, Qubit stabilizer states are complex projective 3-designs, arXiv:1510.02767.
Z. Webb, The Clifford group forms a unitary 3-design, arXiv:1510.02769.
R.A. Low., Pseudo-randomness and learning in quantum computation, arXiv:1006.5227.
A.J. Scott, Optimizing quantum process tomography with unitary 2-designs, J. Phys. A 41 (2008) 055308.
H. Zhu, Multiqubit clifford groups are unitary 3-designs, arXiv:1510.02619.
B. Collins and I. Nechita, Random matrix techniques in quantum information theory, J. Math. Phys. 57 (2016) 015215 [arXiv:1509.04689].
Y. Nakata, C. Hirche, M. Koashi and A. Winter, Efficient unitary designs with nearly time-independent Hamiltonian dynamics, arXiv:1609.07021 [INSPIRE].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
E. Knill, Approximation by quantum circuits, quant-ph/9508006 [INSPIRE].
L. Susskind, The Typical-State Paradox: Diagnosing Horizons with Complexity, Fortsch. Phys. 64 (2016) 84 [arXiv:1507.02287] [INSPIRE].
A. Kitaev, Ph/CS 219C: quantum computation, course taught at Caltech, California, U.S.A. (2016).
Y. Gu, Moments of random matrices and weingarten functions, M.Sc. thesis, Queen’s University, Ontario, Canada (2013).
J. Watrous, Theory of quantum information, lecture notes (2015)
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.
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].
A. Roy and A.J. Scott, Unitary designs and codes, Des. Codes Cryptogr. 53 (2009) 13.
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].
A. Kitaev, A simple model of quantum holography, talks given at KITP, April 7 and May 27 Santa Barbara, U.S.A. (2015).
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
L. Dyson, M. Kleban and L. Susskind, Disturbing implications of a cosmological constant, JHEP 10 (2002) 011 [hep-th/0208013] [INSPIRE].
J.L.F. Barbon and E. Rabinovici, Very long time scales and black hole thermal equilibrium, JHEP 11 (2003) 047 [hep-th/0308063] [INSPIRE].
J.L.F. Barbon and E. Rabinovici, Geometry and quantum noise, Fortsch. Phys. 62 (2014) 626 [arXiv:1404.7085] [INSPIRE].
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, quant-ph/0701004.
A.R. Brown, L. Susskind and Y. Zhao, Quantum complexity and negative curvature, Phys. Rev. D 95 (2017) 045010 [arXiv:1608.02612] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Complexity of formation in holography, to appear.
L. Susskind and Y. Zhao, Switchbacks and the bridge to nowhere, arXiv:1408.2823 [INSPIRE].
W. Chemissany and T.J. Osborne, Holographic fluctuations and the principle of minimal complexity, JHEP 12 (2016) 055 [arXiv:1605.07768] [INSPIRE].
A.R. Brown and L. Susskind, The second law of quantum complexity, arXiv:1701.01107 [INSPIRE].
D. Harlow and P. Hayden, Quantum computation vs. firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].
L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].
D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
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].
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].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
J. Maldacena, Spacetime from entanglement, talk give at KITP, August 20, Santa Barbara, U.S.A. (2013).
A. Almheiri, X. Dong, and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041].
M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, U.K. (2000).
M. Mozrzymas et al., Local random quantum circuits are approximate polynomial-designs: numerical results, J. Phys. A 46 (2013) 305301 [arXiv:1212.2556].
A. Brown, Wormholes and complexity, talk given at the Perimeter Institute for Theoretical Physics, August 21, Waterloo, Canada (2015).
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1610.04903
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Roberts, D.A., Yoshida, B. Chaos and complexity by design. J. High Energ. Phys. 2017, 121 (2017). https://doi.org/10.1007/JHEP04(2017)121
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2017)121
Keywords
- AdS-CFT Correspondence
- Gauge-gravity correspondence
- Random Systems
- Holography and condensed matter physics (AdS/CMT)