Complexity functionals and complexity growth limits in continuous MERA circuits

Molina Vilaplana, J.; del Campo, Adolfo

doi:10.1007/JHEP08(2018)012

Title: Complexity functionals and complexity growth limits in continuous MERA circuits

Abstract

Using the path integral associated to a cMERA tensor network, we provide an operational definition for the complexity of a cMERA circuit/state which is relevant to investigate the complexity of states in quantum field theory. In this framework, it is possible to explicitly establish the correspondence (Minimal) Complexity = (Least) Action. Remarkably, it is also shown how the cMERA complexity action functional can be seen as the action of a Liouville field theory, thus establishing a connection with two dimensional quantum gravity. Concretely, the Liouville mode is identified with the variational parameter defining the cMERA circuit. The rate of complexity growth along the cMERA renormalization group flow is obtained and shown to saturate limits which are in close resemblance to the fundamental bounds to the speed of evolution in unitary quantum dynamics, known as quantum speed limits. We also show that the complexity of a cMERA circuit measured through these complexity functionals, can be cast in terms of the variationally-optimized amount of left-right entanglement created along the cMERA renormalization flow. Our results suggest that the patterns of entanglement in states of a QFT could determine their dual gravitational descriptions through a principle of least complexity.

Authors:

Molina Vilaplana, J. ^[1]; del Campo, Adolfo ^[2]

Universidad Politencia de Cartagena (Spain)
Univ. of Massachusetts, Boston, MA (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:: Fri Aug 03 00:00:00 EDT 2018

Research Org.:: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1467335

Report Number(s):: LA-UR-18-21985
Journal ID: ISSN 1029-8479

Grant/Contract Number:: AC52-06NA25396

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: 2018; Journal Issue: 8; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Atomic and Nuclear Physics

Citation Formats


                    Molina Vilaplana, J., and del Campo, Adolfo. Complexity functionals and complexity growth limits in continuous MERA circuits.  United States: N. p., 2018. 
Web.  doi:10.1007/JHEP08(2018)012.

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                    Molina Vilaplana, J., & del Campo, Adolfo. Complexity functionals and complexity growth limits in continuous MERA circuits.  United States.  https://doi.org/10.1007/JHEP08(2018)012

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                    Molina Vilaplana, J., and del Campo, Adolfo. Fri .  
"Complexity functionals and complexity growth limits in continuous MERA circuits".  United States.  https://doi.org/10.1007/JHEP08(2018)012.  https://www.osti.gov/servlets/purl/1467335.

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@article{osti_1467335,

  title        = {Complexity functionals and complexity growth limits in continuous MERA circuits},

  author       = {Molina Vilaplana, J. and del Campo, Adolfo},

  abstractNote = {Using the path integral associated to a cMERA tensor network, we provide an operational definition for the complexity of a cMERA circuit/state which is relevant to investigate the complexity of states in quantum field theory. In this framework, it is possible to explicitly establish the correspondence (Minimal) Complexity = (Least) Action. Remarkably, it is also shown how the cMERA complexity action functional can be seen as the action of a Liouville field theory, thus establishing a connection with two dimensional quantum gravity. Concretely, the Liouville mode is identified with the variational parameter defining the cMERA circuit. The rate of complexity growth along the cMERA renormalization group flow is obtained and shown to saturate limits which are in close resemblance to the fundamental bounds to the speed of evolution in unitary quantum dynamics, known as quantum speed limits. We also show that the complexity of a cMERA circuit measured through these complexity functionals, can be cast in terms of the variationally-optimized amount of left-right entanglement created along the cMERA renormalization flow. Our results suggest that the patterns of entanglement in states of a QFT could determine their dual gravitational descriptions through a principle of least complexity.},

  doi          = {10.1007/JHEP08(2018)012},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 8,

  volume       = 2018,

  place        = {United States},

  year         = {Fri Aug 03 00:00:00 EDT 2018},

  month        = {Fri Aug 03 00:00:00 EDT 2018}

}

Copy to clipboard

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Quantum complexity of time evolution with chaotic Hamiltonians
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Title: Complexity functionals and complexity growth limits in continuous MERA circuits

Abstract

Citation Formats

Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence journal, May 2006

Building up spacetime with quantum entanglement journal, June 2010

Cool horizons for entangled black holes journal, August 2013

Gravitational dynamics from entanglement “thermodynamics” journal, April 2014

Anti de Sitter space and holography journal, January 1998

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

Holographic Complexity Equals Bulk Action? journal, May 2016

Complexity, action, and black holes journal, April 2016

Toward a Definition of Complexity for Quantum Field Theory States journal, March 2018

Circuit complexity in quantum field theory journal, October 2017

Entanglement Renormalization for Quantum Fields in Real Space journal, March 2013

Holographic geometry of entanglement renormalization in quantum field theories journal, October 2012

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT journal, November 2017

Quantum geometry of bosonic strings journal, July 1981

Notes on Quantum Liouville Theory and Quantum Gravity journal, January 1990

Comparison of holographic and field theoretic complexities for time dependent thermofield double states journal, February 2018

An energy dispersion estimate journal, January 1992

Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight journal, October 2009

Quantum Speed Limit for Physical Processes journal, January 2013

Ultimate physical limits to computation journal, August 2000

Entanglement is not enough: Entanglement is not enough journal, January 2016

Complexity is simple! journal, February 2018

Riemannian structure on manifolds of quantum states journal, September 1980

Entanglement and correlations in the continuous multi-scale entanglement renormalization ansatz journal, December 2017

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

Holographic fluctuations and the principle of minimal complexity journal, December 2016

Holographic geometry of cMERA for quantum quenches and finite temperature journal, March 2014

Analytic continuation of Liouville theory journal, December 2011

Perturbative expansion around the Gaussian effective action: The background field method journal, November 1997

Information geometry of entanglement renormalization for free quantum fields journal, September 2015

Geometry of quantum evolution journal, October 1990

Evolution of entanglement entropy in one-dimensional systems journal, April 2005

On the time dependence of holographic complexity journal, November 2017

Quantum complexity of time evolution with chaotic Hamiltonians journal, January 2020

Binding complexity and multiparty entanglement journal, February 2019

Time evolution of complexity: a critique of three methods journal, April 2019

Quantum Computation as Gravity text, January 2018

Circuit Complexity for Coherent States text, January 2018

Holographic Spacetimes as Quantum Circuits of Path-Integrations text, January 2018

Time Evolution of Complexity: A Critique of Three Methods text, January 2018

Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence
journal, May 2006

Building up spacetime with quantum entanglement
journal, June 2010

Cool horizons for entangled black holes
journal, August 2013

Gravitational dynamics from entanglement “thermodynamics”
journal, April 2014

Anti de Sitter space and holography
journal, January 1998

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

Holographic Complexity Equals Bulk Action?
journal, May 2016

Complexity, action, and black holes
journal, April 2016

Toward a Definition of Complexity for Quantum Field Theory States
journal, March 2018

Circuit complexity in quantum field theory
journal, October 2017

Entanglement Renormalization for Quantum Fields in Real Space
journal, March 2013

Holographic geometry of entanglement renormalization in quantum field theories
journal, October 2012

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT
journal, November 2017

Quantum geometry of bosonic strings
journal, July 1981

Notes on Quantum Liouville Theory and Quantum Gravity
journal, January 1990

Comparison of holographic and field theoretic complexities for time dependent thermofield double states
journal, February 2018

An energy dispersion estimate
journal, January 1992

Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight
journal, October 2009

Quantum Speed Limit for Physical Processes
journal, January 2013

Ultimate physical limits to computation
journal, August 2000

Entanglement is not enough: Entanglement is not enough
journal, January 2016

Complexity is simple!
journal, February 2018

Riemannian structure on manifolds of quantum states
journal, September 1980

Entanglement and correlations in the continuous multi-scale entanglement renormalization ansatz
journal, December 2017

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

Holographic fluctuations and the principle of minimal complexity
journal, December 2016

Holographic geometry of cMERA for quantum quenches and finite temperature
journal, March 2014

Analytic continuation of Liouville theory
journal, December 2011

Perturbative expansion around the Gaussian effective action: The background field method
journal, November 1997

Information geometry of entanglement renormalization for free quantum fields
journal, September 2015

Geometry of quantum evolution
journal, October 1990

Evolution of entanglement entropy in one-dimensional systems
journal, April 2005

On the time dependence of holographic complexity
journal, November 2017

Quantum complexity of time evolution with chaotic Hamiltonians
journal, January 2020

Binding complexity and multiparty entanglement
journal, February 2019

Time evolution of complexity: a critique of three methods
journal, April 2019

Quantum Computation as Gravity
text, January 2018

Circuit Complexity for Coherent States
text, January 2018

Holographic Spacetimes as Quantum Circuits of Path-Integrations
text, January 2018

Time Evolution of Complexity: A Critique of Three Methods
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