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 potential,” which is minimized by unitary kdesigns and measures the 2norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is outoftimeorder (OTO) fourpoint correlation functions. We also show that the norm squared of a generalization of outoftimeorder 2kpoint correlators is proportional to the kth frame potential, providing a quantitative connection between chaos and pseudorandomness. In addition, we prove that these 2kpoint correlators for Pauli operators completely determine the kfold 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]}
 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
 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 10298479
 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; AdSCFT Correspondence; Gaugegravity 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 potential,” which is minimized by unitary kdesigns and measures the 2norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is outoftimeorder (OTO) fourpoint correlation functions. We also show that the norm squared of a generalization of outoftimeorder 2kpoint correlators is proportional to the kth frame potential, providing a quantitative connection between chaos and pseudorandomness. In addition, we prove that these 2kpoint correlators for Pauli operators completely determine the kfold 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}
}