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Title: Chaos and complexity by design

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 also 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. In addition, 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.
Authors:
 [1] ;  [2]
  1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States). Center for Theoretical Physics, Dept. of Physics; Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
  2. Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
Publication Date:
Grant/Contract Number:
SC0012567
Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 4; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Research Org:
Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
Sponsoring Org:
USDOE
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdS-CFT Correspondence; Gauge-gravity correspondence; Random Systems; Holography and condensed matter physics (AdS/CMT)
OSTI Identifier:
1393480

Roberts, Daniel A., and Yoshida, Beni. Chaos and complexity by design. United States: N. p., Web. doi:10.1007/JHEP04(2017)121.
Roberts, Daniel A., & Yoshida, Beni. Chaos and complexity by design. United States. doi:10.1007/JHEP04(2017)121.
Roberts, Daniel A., and Yoshida, Beni. 2017. "Chaos and complexity by design". United States. doi:10.1007/JHEP04(2017)121. https://www.osti.gov/servlets/purl/1393480.
@article{osti_1393480,
title = {Chaos and complexity by design},
author = {Roberts, Daniel A. and Yoshida, Beni},
abstractNote = {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 also 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. In addition, 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.},
doi = {10.1007/JHEP04(2017)121},
journal = {Journal of High Energy Physics (Online)},
number = 4,
volume = 2017,
place = {United States},
year = {2017},
month = {4}
}