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Chaos, Complexity, and Random Matrices

Cotler, Jordan and Hunter-Jones, Nicholas and Liu, Junyu and Yoshida, Beni (2017) Chaos, Complexity, and Random Matrices. Journal of High Energy Physics, 2017 (11). Art. No. 048. ISSN 1126-6708. doi:10.1007/JHEP11(2017)048.

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Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an O(1) scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce k-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate k-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.

Item Type:Article
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URLURL TypeDescription Paper
Liu, Junyu0000-0003-1669-8039
Additional Information:© 2017 The Author(s). 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. Received: September 15, 2017; Accepted: October 26, 2017; Published: November 9, 2017. We thank Yoni Bentov, Fernando Brandão, Clifford Cheung, Patrick Hayden, Alexei Kitaev, John Preskill, Daniel Ranard, Daniel Roberts, Lukas Schimmer, and Steve Shenker for valuable comments and insights. JC is supported by the Fannie and John Hertz Foundation and the Stanford Graduate Fellowship program. JC and NHJ would like to thank the Perimeter Institute for their hospitality during the completion of part of this work. BY and NHJ acknowledge support from the Simons Foundation through the "It from Qubit" collaboration. NHJ is supported the Institute for Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (NSF Grant PHY-1125565) with support from the Gordon and Betty Moore Foundation (GBMF-2644). JL is supported in part by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
Group:Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
Fannie and John Hertz FoundationUNSPECIFIED
Stanford Graduate FellowshipUNSPECIFIED
Simons FoundationUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Gordon and Betty Moore FoundationGBMF-2644
Department of Energy (DOE)DE-SC0011632
Industry CanadaUNSPECIFIED
Ontario Ministry of Research and InnovationUNSPECIFIED
Subject Keywords:AdS-CFT Correspondence; Black Holes; Matrix Models; Random Systems
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Issue or Number:11
Record Number:CaltechAUTHORS:20171102-135408184
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Official Citation:Cotler, J., Hunter-Jones, N., Liu, J. et al. J. High Energ. Phys. (2017) 2017: 48.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:82905
Deposited By: Bonnie Leung
Deposited On:03 Nov 2017 02:36
Last Modified:15 Nov 2021 19:53

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