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Chaos and complexity by design

  • Regular Article - Theoretical Physics
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  • Published: 20 April 2017
  • volume 2017, Article number: 121 (2017)
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Chaos and complexity by design
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  • Daniel A. Roberts1,2 &
  • Beni Yoshida3 

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A preprint version of the article is available at arXiv.

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.

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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.

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Authors and Affiliations

  1. Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, U.S.A.

    Daniel A. Roberts

  2. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, U.S.A.

    Daniel A. Roberts

  3. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada

    Beni Yoshida

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Correspondence to Daniel A. Roberts.

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ArXiv ePrint: 1610.04903

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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

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  • Received: 10 November 2016

  • Revised: 28 January 2017

  • Accepted: 28 March 2017

  • Published: 20 April 2017

  • DOI: https://doi.org/10.1007/JHEP04(2017)121

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Keywords

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

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