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Title: Chaos, complexity, and random matrices

Abstract

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. Here,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.

Authors:
 [1]; ORCiD logo [2];  [2];  [3]
  1. Stanford Univ., Stanford, CA (United States). Stanford Inst. for Theoretical Physics
  2. California Inst. of Technology (CalTech), Pasadena, CA (United States)
  3. Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
Publication Date:
Research Org.:
California Inst. of Technology, Pasadena, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP)
OSTI Identifier:
1501469
Grant/Contract Number:  
SC0011632
Resource 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: 11; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdS-CFT Correspondence; Black Holes; Matrix Models; Random Systems

Citation Formats

Cotler, Jordan, Hunter-Jones, Nicholas, Liu, Junyu, and Yoshida, Beni. Chaos, complexity, and random matrices. United States: N. p., 2017. Web. doi:10.1007/jhep11(2017)048.
Cotler, Jordan, Hunter-Jones, Nicholas, Liu, Junyu, & Yoshida, Beni. Chaos, complexity, and random matrices. United States. doi:10.1007/jhep11(2017)048.
Cotler, Jordan, Hunter-Jones, Nicholas, Liu, Junyu, and Yoshida, Beni. Thu . "Chaos, complexity, and random matrices". United States. doi:10.1007/jhep11(2017)048. https://www.osti.gov/servlets/purl/1501469.
@article{osti_1501469,
title = {Chaos, complexity, and random matrices},
author = {Cotler, Jordan and Hunter-Jones, Nicholas and Liu, Junyu and Yoshida, Beni},
abstractNote = {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. Here,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.},
doi = {10.1007/jhep11(2017)048},
journal = {Journal of High Energy Physics (Online)},
number = 11,
volume = 2017,
place = {United States},
year = {2017},
month = {11}
}

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Works referenced in this record:

Optimizing quantum process tomography with unitary 2 -designs
journal, January 2008


Information in black hole radiation
journal, December 1993


Black holes as mirrors: quantum information in random subsystems
journal, September 2007


Towards the fast scrambling conjecture
journal, April 2013

  • Lashkari, Nima; Stanford, Douglas; Hastings, Matthew
  • Journal of High Energy Physics, Vol. 2013, Issue 4
  • DOI: 10.1007/JHEP04(2013)022

Eternal black holes in anti-de Sitter
journal, April 2003


Chaos and complexity by design
journal, April 2017

  • Roberts, Daniel A.; Yoshida, Beni
  • Journal of High Energy Physics, Vol. 2017, Issue 4
  • DOI: 10.1007/JHEP04(2017)121

Chaos in quantum channels
journal, February 2016

  • Hosur, Pavan; Qi, Xiao-Liang; Roberts, Daniel A.
  • Journal of High Energy Physics, Vol. 2016, Issue 2
  • DOI: 10.1007/JHEP02(2016)004

Random-matrix theories in quantum physics: common concepts
journal, June 1998


Spectral form factor in a random matrix theory
journal, April 1997


Average entropy of a subsystem
journal, August 1993


Holographic duality from random tensor networks
journal, November 2016

  • Hayden, Patrick; Nezami, Sepehr; Qi, Xiao-Liang
  • Journal of High Energy Physics, Vol. 2016, Issue 11
  • DOI: 10.1007/JHEP11(2016)009

Black holes and the butterfly effect
journal, March 2014

  • Shenker, Stephen H.; Stanford, Douglas
  • Journal of High Energy Physics, Vol. 2014, Issue 3
  • DOI: 10.1007/JHEP03(2014)067

Gapless spin-fluid ground state in a random quantum Heisenberg magnet
journal, May 1993


Fast scramblers
journal, October 2008


Stringy effects in scrambling
journal, May 2015

  • Shenker, Stephen H.; Stanford, Douglas
  • Journal of High Energy Physics, Vol. 2015, Issue 5
  • DOI: 10.1007/JHEP05(2015)132

Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order
journal, July 2006


Convergence to equilibrium under a random Hamiltonian
journal, September 2012

  • Brandão, Fernando G. S. L.; Ćwikliński, Piotr; Horodecki, Michał
  • Physical Review E, Vol. 86, Issue 3
  • DOI: 10.1103/PhysRevE.86.031101

Asymptotic behavior of group integrals in the limit of infinite rank
journal, May 1978

  • Weingarten, Don
  • Journal of Mathematical Physics, Vol. 19, Issue 5
  • DOI: 10.1063/1.523807

Local Random Quantum Circuits are Approximate Polynomial-Designs
journal, August 2016

  • Brandão, Fernando G. S. L.; Harrow, Aram W.; Horodecki, Michał
  • Communications in Mathematical Physics, Vol. 346, Issue 2
  • DOI: 10.1007/s00220-016-2706-8

Characteristic Vectors of Bordered Matrices With Infinite Dimensions
journal, November 1955

  • Wigner, Eugene P.
  • The Annals of Mathematics, Vol. 62, Issue 3
  • DOI: 10.2307/1970079

On the late-time behavior of Virasoro blocks and a classification of semiclassical saddles
journal, April 2017

  • Fitzpatrick, A. Liam; Kaplan, Jared
  • Journal of High Energy Physics, Vol. 2017, Issue 4
  • DOI: 10.1007/JHEP04(2017)072

Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws
journal, January 1984


Black holes and random matrices
journal, May 2017

  • Cotler, Jordan S.; Gur-Ari, Guy; Hanada, Masanori
  • Journal of High Energy Physics, Vol. 2017, Issue 5
  • DOI: 10.1007/JHEP05(2017)118

On the eigenvalues of random matrices
journal, January 1994


Random Free Fermions: An Analytical Example of Eigenstate Thermalization
journal, January 2016


Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
journal, October 2010


2D CFT partition functions at late times
journal, August 2017


Phase Transition in the Density of States of Quantum Spin Glasses
journal, December 2014

  • Erdős, László; Schröder, Dominik
  • Mathematical Physics, Analysis and Geometry, Vol. 17, Issue 3-4
  • DOI: 10.1007/s11040-014-9164-3

Power-law out of time order correlation functions in the SYK model
journal, August 2017


Entanglement complexity in quantum many-body dynamics, thermalization, and localization
journal, July 2017


Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory
journal, September 2015


A bound on chaos
journal, August 2016

  • Maldacena, Juan; Shenker, Stephen H.; Stanford, Douglas
  • Journal of High Energy Physics, Vol. 2016, Issue 8
  • DOI: 10.1007/JHEP08(2016)106

The Spectral Form Factor Is Not Self-Averaging
journal, March 1997


From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics
journal, May 2016


On information loss in AdS3/CFT2
journal, May 2016

  • Fitzpatrick, A. Liam; Kaplan, Jared; Li, Daliang
  • Journal of High Energy Physics, Vol. 2016, Issue 5
  • DOI: 10.1007/JHEP05(2016)109

Exact and approximate unitary 2-designs and their application to fidelity estimation
journal, July 2009


Sachdev-Ye-Kitaev model and thermalization on the boundary of many-body localized fermionic symmetry-protected topological states
journal, March 2017


Complexity, action, and black holes
journal, April 2016


Extension of level-spacing universality
journal, July 1997


Decoupling with Random Quantum Circuits
journal, September 2015


Echoes of chaos from string theory black holes
journal, March 2017

  • Balasubramanian, Vijay; Craps, Ben; Czech, Bartlomiej
  • Journal of High Energy Physics, Vol. 2017, Issue 3
  • DOI: 10.1007/JHEP03(2017)154

Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence
journal, June 2015

  • Pastawski, Fernando; Yoshida, Beni; Harlow, Daniel
  • Journal of High Energy Physics, Vol. 2015, Issue 6
  • DOI: 10.1007/JHEP06(2015)149

Notes on black-hole evaporation
journal, August 1976


Statistical Theory of the Energy Levels of Complex Systems. I
journal, January 1962

  • Dyson, Freeman J.
  • Journal of Mathematical Physics, Vol. 3, Issue 1
  • DOI: 10.1063/1.1703773

Particle creation by black holes
journal, August 1975

  • Hawking, S. W.
  • Communications In Mathematical Physics, Vol. 43, Issue 3
  • DOI: 10.1007/BF02345020

Chaos and quantum thermalization
journal, August 1994


Scrambling the spectral form factor: Unitarity constraints and exact results
journal, June 2017


Particle creation by black holes
journal, June 1976

  • Hawking, S. W.
  • Communications in Mathematical Physics, Vol. 46, Issue 2
  • DOI: 10.1007/bf01608497

On the Eigenvalues of Random Matrices
journal, January 1994

  • Diaconis, Persi; Shahshahani, Mehrdad
  • Journal of Applied Probability, Vol. 31
  • DOI: 10.2307/3214948

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    • Cotler, Jordan; Han, Xizhi; Qi, Xiao-Liang
    • Journal of High Energy Physics, Vol. 2019, Issue 7
    • DOI: 10.1007/jhep07(2019)042

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    journal, July 2019

    • Cotler, Jordan; Han, Xizhi; Qi, Xiao-Liang
    • Journal of High Energy Physics, Vol. 2019, Issue 7
    • DOI: 10.1007/jhep07(2019)042

    Disentangling Scrambling and Decoherence via Quantum Teleportation
    journal, January 2019


    Contrasting SYK-like models
    journal, January 2018

    • Krishnan, Chethan; Pavan Kumar, K. V.; Rosa, Dario
    • Journal of High Energy Physics, Vol. 2018, Issue 1
    • DOI: 10.1007/jhep01(2018)064

    Conformal field theory and the web of quantum chaos diagnostics
    journal, January 2020

    • Kudler-Flam, Jonah; Nie, Laimei; Ryu, Shinsei
    • Journal of High Energy Physics, Vol. 2020, Issue 1
    • DOI: 10.1007/jhep01(2020)175

    Spectral form factor and semi-circle law in the time direction
    journal, February 2019


    Chaos and random matrices in supersymmetric SYK
    journal, May 2018

    • Hunter-Jones, Nicholas; Liu, Junyu
    • Journal of High Energy Physics, Vol. 2018, Issue 5
    • DOI: 10.1007/jhep05(2018)202

    Note on global symmetry and SYK model
    journal, May 2019


    A bound on quantum chaos from Random Matrix Theory with Gaussian Unitary Ensemble
    journal, May 2019

    • Choudhury, Sayantan; Mukherjee, Arkaprava
    • Journal of High Energy Physics, Vol. 2019, Issue 5
    • DOI: 10.1007/jhep05(2019)149

    Onset of random matrix behavior in scrambling systems
    journal, July 2018

    • Gharibyan, Hrant; Hanada, Masanori; Shenker, Stephen H.
    • Journal of High Energy Physics, Vol. 2018, Issue 7
    • DOI: 10.1007/jhep07(2018)124

    The Thouless time for mass-deformed SYK
    journal, September 2018

    • Nosaka, Tomoki; Rosa, Dario; Yoon, Junggi
    • Journal of High Energy Physics, Vol. 2018, Issue 9
    • DOI: 10.1007/jhep09(2018)041

    On the replica structure of Sachdev-Ye-Kitaev model
    journal, September 2019

    • Wang, Hanteng; Bagrets, D.; Chudnovskiy, A. L.
    • Journal of High Energy Physics, Vol. 2019, Issue 9
    • DOI: 10.1007/jhep09(2019)057

    On the evolution of operator complexity beyond scrambling
    journal, October 2019

    • Barbón, J. L. F.; Rabinovici, E.; Shir, R.
    • Journal of High Energy Physics, Vol. 2019, Issue 10
    • DOI: 10.1007/jhep10(2019)264

    Eigenstate thermalization in the Sachdev-Ye-Kitaev model
    journal, November 2017


    Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs
    journal, November 2018

    • Jia, Yiyang; Verbaarschot, Jacobus J. M.
    • Journal of High Energy Physics, Vol. 2018, Issue 11
    • DOI: 10.1007/jhep11(2018)031

    Scrambling and complexity in phase space
    journal, June 2019


    Sampling and Scrambling on a Chain of Superconducting Qubits
    journal, August 2018


    Universal level statistics of the out-of-time-ordered operator
    journal, July 2019


    Entanglement features of random Hamiltonian dynamics
    journal, July 2018


    Operator spreading in quantum maps
    journal, March 2019


    Thouless and relaxation time scales in many-body quantum systems
    journal, May 2019

    • Schiulaz, Mauro; Torres-Herrera, E. Jonathan; Santos, Lea F.
    • Physical Review B, Vol. 99, Issue 17
    • DOI: 10.1103/physrevb.99.174313

    Majorana fermion quantum mechanics for higher rank tensors
    journal, October 2019


    Chaos and complexity in quantum mechanics
    journal, January 2020


    Dynamical signatures of quantum chaos and relaxation time scales in a spin-boson system
    journal, July 2019

    • Lerma-Hernández, S.; Villaseñor, D.; Bastarrachea-Magnani, M. A.
    • Physical Review E, Vol. 100, Issue 1
    • DOI: 10.1103/physreve.100.012218

    Level repulsion and dynamics in the finite one-dimensional Anderson model
    journal, August 2019

    • Torres-Herrera, E. Jonathan; Méndez-Bermúdez, J. A.; Santos, Lea F.
    • Physical Review E, Vol. 100, Issue 2
    • DOI: 10.1103/physreve.100.022142

    Fine Grained Chaos in AdS 2 Gravity
    journal, March 2018


    Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators
    journal, September 2018


    Extreme Decoherence and Quantum Chaos
    journal, January 2019


    Experimental Implementation of Efficient Quantum Pseudorandomness on a 12-Spin System
    journal, July 2019


    Measurement-Induced Phase Transitions in the Dynamics of Entanglement
    journal, July 2019


    Revealing Nonperturbative Effects in the SYK Model
    journal, November 2019

    • Aref’eva, I. Ya.; Volovich, I. V.; Khramtsov, M. A.
    • Theoretical and Mathematical Physics, Vol. 201, Issue 2
    • DOI: 10.1134/s0040577919110059

    Work Statistics, Loschmidt Echo and Information Scrambling in Chaotic Quantum Systems
    journal, March 2019


    Almost Markovian processes from closed dynamics
    journal, April 2019