Chaos, complexity, and random matrices

Cotler, Jordan; Hunter-Jones, Nicholas; Liu, Junyu; Yoshida, Beni

doi:10.1007/jhep11(2017)048

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:

Cotler, Jordan ^[1];

^[2]; Liu, Junyu ^[2]; Yoshida, Beni ^[3]

Stanford Univ., Stanford, CA (United States). Stanford Inst. for Theoretical Physics
California Inst. of Technology (CalTech), Pasadena, CA (United States)
Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)

Publication Date:: Thu Nov 09 00:00:00 EST 2017

Research Org.:: California Institute of Technology (CalTech), 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.

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                    Cotler, Jordan, Hunter-Jones, Nicholas, Liu, Junyu, & Yoshida, Beni. Chaos, complexity, and random matrices.  United States.  https://doi.org/10.1007/jhep11(2017)048

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                    Cotler, Jordan, Hunter-Jones, Nicholas, Liu, Junyu, and Yoshida, Beni. Thu .  
"Chaos, complexity, and random matrices".  United States.  https://doi.org/10.1007/jhep11(2017)048.  https://www.osti.gov/servlets/purl/1501469.

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@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         = {Thu Nov 09 00:00:00 EST 2017},

  month        = {Thu Nov 09 00:00:00 EST 2017}

}

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Universität Regensburg
DOI: 10.5283/epub.37336

Eigenstate thermalization in the Sachdev-Ye-Kitaev model
text, January 2017

Sonner, Julian; Vielma, Manuel
arXiv
DOI: 10.48550/arxiv.1707.08013

Chaos and random matrices in supersymmetric SYK
text, January 2017

Hunter-Jones, Nicholas; Liu, Junyu
arXiv
DOI: 10.48550/arxiv.1710.08184

Sampling and scrambling on a chain of superconducting qubits
text, January 2017

Geller, Michael R.
arXiv
DOI: 10.48550/arxiv.1711.11026

Universal level statistics of the out-of-time-ordered operator
text, January 2018

Rozenbaum, Efim B.; Ganeshan, Sriram; Galitski, Victor
arXiv
DOI: 10.48550/arxiv.1801.10591

Entanglement Features of Random Hamiltonian Dynamics
text, January 2018

You, Yi-Zhuang; Gu, Yingfei
arXiv
DOI: 10.48550/arxiv.1803.10425

Experimental Implementation of Efficient Quantum Pseudorandomness on a 12-spin System
text, January 2018

Li, Jun; Luo, Zhihuang; Xin, Tao
arXiv
DOI: 10.48550/arxiv.1807.07419

Operator Spreading in Quantum Maps
text, January 2018

Moudgalya, Sanjay; Devakul, Trithep; von Keyserlingk, C. W.
arXiv
DOI: 10.48550/arxiv.1808.04889

Dynamical signatures of quantum chaos and relaxation timescales in a spin-boson system
text, January 2019

Lerma-Hernández, S.; Villaseñor, D.; Bastarrachea-Magnani, M. A.
arXiv
DOI: 10.48550/arxiv.1905.03253

Revealing nonperturbative effects in the SYK model
text, January 2019

Aref'eva, Irina; Khramtsov, Mikhail; Volovich, Igor
arXiv
DOI: 10.48550/arxiv.1905.04203

Majorana Fermion Quantum Mechanics for Higher Rank Tensors
text, January 2019

Klebanov, Igor R.; Pallegar, Preethi N.; Popov, Fedor K.
arXiv
DOI: 10.48550/arxiv.1905.06264

On The Evolution Of Operator Complexity Beyond Scrambling
text, January 2019

Barbon, J. L. F.; Rabinovici, E.; Shir, R.
arXiv
DOI: 10.48550/arxiv.1907.05393

Conformal field theory and the web of quantum chaos diagnostics
text, January 2019

Kudler-Flam, Jonah; Nie, Laimei; Ryu, Shinsei
arXiv
DOI: 10.48550/arxiv.1910.14575

Spectral properties of three-dimensional Anderson model
text, January 2021

Šuntajs, Jan; Prosen, Tomaž; Vidmar, Lev
arXiv
DOI: 10.48550/arxiv.2103.05680

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Chaos and random matrices in supersymmetric SYK

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We use random matrix theory to explore late-time chaos in supersymmetric quantum mechanical systems. Motivated by the recent study of supersymmetric SYK models and their random matrix classification, we consider the Wishart-Laguerre unitary ensemble and compute the spectral form factors and frame potentials to quantify chaos and randomness. Compared to the Gaussian ensembles, we observe the absence of a dip regime in the form factor and a slower approach to Haar-random dynamics. Here we find agreement between our random matrix analysis and predictions from the supersymmetric SYK model, and discuss the implications for supersymmetric chaotic systems.
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Similar Records

Title: Chaos, complexity, and random matrices

Abstract

Citation Formats

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

Addendum to computational complexity and black hole horizons: Addendum to computational complexity and black hole horizons journal, January 2016

Towards the fast scrambling conjecture journal, April 2013

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

Chaos and complexity by design journal, April 2017

Chaos in quantum channels journal, February 2016

Particle creation by black holes journal, June 1976

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

Black holes and the butterfly effect journal, March 2014

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

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

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

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

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

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

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

Black holes and random matrices journal, May 2017

On the eigenvalues of random matrices journal, January 1994

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

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

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

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

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

Notes on black-hole evaporation journal, August 1976

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

Particle creation by black holes journal, August 1975

Chaos and quantum thermalization journal, August 1994

Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group journal, March 2006

Quantum computation vs. firewalls journal, June 2013

Black holes as mirrors: quantum information in random subsystems text, January 2007

Optimizing quantum process tomography with unitary 2-designs text, January 2007

Fast Scramblers text, January 2008

Towards the fast scrambling conjecture text, January 2011

Stringy effects in scrambling preprint, January 2014

A bound on chaos text, January 2015

Random free fermions: An analytical example of eigenstate thermalization text, January 2015

From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics text, January 2015

Chaos in quantum channels text, January 2015

Complexity, action, and black holes text, January 2015

Holographic duality from random tensor networks text, January 2016

On Information Loss in AdS$_3$/CFT$_2$ preprint, January 2016

On the Late-Time Behavior of Virasoro Blocks and a Classification of Semiclassical Saddles text, January 2016

Chaos and complexity by design text, January 2016

Black Holes and Random Matrices text, January 2016

Echoes of chaos from string theory black holes text, January 2016

The spectral form factor is not self-averaging text, January 1996

Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet text, January 1992

Chaos and Quantum Thermalization text, January 1994

Random Matrix Theories in Quantum Physics: Common Concepts text, January 1997

Lieb-Robinson bounds and the generation of correlations and topological quantum order text, January 2006

An Extension of Level-spacing Universality text, January 1997

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

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

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

Fine Grained Chaos in AdS 2 Gravity journal, March 2018

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

Addendum to computational complexity and black hole horizons: Addendum to computational complexity and black hole horizons
journal, January 2016

Towards the fast scrambling conjecture
journal, April 2013

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

Chaos and complexity by design
journal, April 2017

Chaos in quantum channels
journal, February 2016

Particle creation by black holes
journal, June 1976

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

Black holes and the butterfly effect
journal, March 2014

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

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

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

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

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

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

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

Black holes and random matrices
journal, May 2017

On the eigenvalues of random matrices
journal, January 1994

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

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

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

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

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

Notes on black-hole evaporation
journal, August 1976

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

Particle creation by black holes
journal, August 1975

Chaos and quantum thermalization
journal, August 1994

Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group
journal, March 2006

Quantum computation vs. firewalls
journal, June 2013

Black holes as mirrors: quantum information in random subsystems
text, January 2007

Optimizing quantum process tomography with unitary 2-designs
text, January 2007

Fast Scramblers
text, January 2008

Towards the fast scrambling conjecture
text, January 2011

Stringy effects in scrambling
preprint, January 2014

A bound on chaos
text, January 2015

Random free fermions: An analytical example of eigenstate thermalization
text, January 2015

From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
text, January 2015

Chaos in quantum channels
text, January 2015

Complexity, action, and black holes
text, January 2015

Holographic duality from random tensor networks
text, January 2016

On Information Loss in AdS$_3$/CFT$_2$
preprint, January 2016

On the Late-Time Behavior of Virasoro Blocks and a Classification of Semiclassical Saddles
text, January 2016

Chaos and complexity by design
text, January 2016

Black Holes and Random Matrices
text, January 2016

Echoes of chaos from string theory black holes
text, January 2016

The spectral form factor is not self-averaging
text, January 1996

Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
text, January 1992

Chaos and Quantum Thermalization
text, January 1994

Random Matrix Theories in Quantum Physics: Common Concepts
text, January 1997

Lieb-Robinson bounds and the generation of correlations and topological quantum order
text, January 2006

An Extension of Level-spacing Universality
text, January 1997

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

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

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

Fine Grained Chaos in ${AdS}_{2}$ Gravity
journal, March 2018

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

Extreme Decoherence and Quantum Chaos
journal, January 2019

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