Chaos and complexity by design

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


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.


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


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.


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© 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

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