Complexity growth in integrable and chaotic models

Balasubramanian, Vijay; DeCross, Matthew; Kar, Arjun; Li, Yue Cathy; Parrikar, Onkar

doi:10.1007/jhep07(2021)011

Cited by 58 publications

(91 citation statements)

References 71 publications

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“…A number of works have suggested that certain measures of quantum complexity may have a semiclassical bulk dual [5,12,39,40]. Notions of state complexity [41][42][43][44][45][46][47][48][49] or unitary complexity of time evolution [1,[50][51][52][53][54][55][56] have been intriguingly linked to measures of "size" of the bulk universe at the given boundary time, e.g. the volume of the bulk maximal volume slice [12,31,57] anchored at that time, or the gravitational action in the corresponding Wheeler-de Witt diamond [40].…”

Section: K-complexity and The Black Hole Interiormentioning

confidence: 99%

Random matrix theory for complexity growth and black hole interiors

Kar

Lamprou

Rozali

et al. 2022

J. High Energ. Phys.

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We study a precise and computationally tractable notion of operator complexity in holographic quantum theories, including the ensemble dual of Jackiw-Teitelboim gravity and two-dimensional holographic conformal field theories. This is a refined, “microcanonical” version of K-complexity that applies to theories with infinite or continuous spectra (including quantum field theories), and in the holographic theories we study exhibits exponential growth for a scrambling time, followed by linear growth until saturation at a time exponential in the entropy — a behavior that is characteristic of chaos. We show that the linear growth regime implies a universal random matrix description of the operator dynamics after scrambling. Our main tool for establishing this connection is a “complexity renormalization group” framework we develop that allows us to study the effective operator dynamics for different timescales by “integrating out” large K-complexities. In the dual gravity setting, we comment on the empirical match between our version of K-complexity and the maximal volume proposal, and speculate on a connection between the universal random matrix theory dynamics of operator growth after scrambling and the spatial translation symmetry of smooth black hole interiors.

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Section: K-complexity and The Black Hole Interiormentioning

confidence: 99%

Random matrix theory for complexity growth and black hole interiors

Kar

Lamprou

Rozali

et al. 2022

J. High Energ. Phys.

Self Cite

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“…An interesting approach based on the Euler-Arnold formalism to study complexity in chaotic quantum systems was developed in[149,150].…”

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confidence: 99%

Holographic and QFT complexity with angular momentum

Bernamonti

Bigazzi

Billo

et al. 2021

J. High Energ. Phys.

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We study the influence of angular momentum on quantum complexity for CFT states holographically dual to rotating black holes. Using the holographic complexity=action (CA) and complexity=volume (CV) proposals, we study the full time dependence of complexity and the complexity of formation for two dimensional states dual to rotating BTZ. The obtained results and their dependence on angular momentum turn out to be analogous to those of charged states dual to Reissner-Nordström AdS black holes. For CA, our computation carefully accounts for the counterterm in the gravity action, which was not included in previous analysis in the literature. This affects the complexity early time dependence and its effect becomes negligible close to extremality. In the grand canonical ensemble, the CA and CV complexity of formation are linear in the temperature, and diverge with the same structure in the speed of light angular velocity limit. For CA the inclusion of the counterterm is crucial for both effects. We also address the problem of studying holographic complexity for higher dimensional rotating black holes, focusing on the four dimensional Kerr-AdS case. Carefully taking into account all ingredients, we show that the late time limit of the CA growth rate saturates the expected bound, and find the CV complexity of formation of large black holes diverges in the critical angular velocity limit. Our holographic analysis is complemented by the study of circuit complexity in a two dimensional free scalar model for a thermofield double (TFD) state with angular momentum. We show how this can be given a description in terms of non-rotating TFD states introducing mode-by-mode effective temperatures and times. We comment on the similarities and differences of the holographic and QFT complexity results.

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“…We should finally point out again that we did not aim at writing a comprehensive review of the subject, therefore we left out many topics that we felt were too advanced for an introduction, such as the thermodynamics and resource theory aspects of complexity [43,46,151,152], the relation with bulk dynamics (in the sense of reconstructing the Einstein equations in the bulk from the complexity of the boundary) [85], alternative conjectures, most notably the one in [92] (sometimes called CV 2.0), complexity in de Sitter space [52,[153][154][155], the evolution of complexity after a quench [77,156], the relation between complexity and chaos [32,157], other notions of complexity such as the operator complexity [158,159]. We hope that our readers will be encouraged to delve further into this fascinating subject and contribute to its development.…”

Section: Discussionmentioning

confidence: 99%

Quantum Computational Complexity -- From Quantum Information to Black Holes and Back

Chapman,

Policastro

2021

Preprint

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Quantum computational complexity estimates the difficulty of constructing quantum states from elementary operations, a problem of prime importance for quantum computation. Surprisingly, this quantity can also serve to study a completely different physical problem -that of information processing inside black holes. Quantum computational complexity was suggested as a new entry in the holographic dictionary, which extends the connection between geometry and information and resolves the puzzle of why black hole interiors keep growing for a very long time. In this pedagogical review, we present the geometric approach to complexity advocated by Nielsen and show how it can be used to define complexity for generic quantum systems; in particular, we focus on Gaussian states in QFT, both pure and mixed, and on certain classes of CFT states. We then present the conjectured relation to gravitational quantities within the holographic correspondence and discuss several examples in which different versions of the conjectures have been tested. We highlight the relation between complexity, chaos and scrambling in chaotic systems. We conclude with a discussion of open problems and future directions. This article was written for the special issue of EPJ-C Frontiers in Holographic Duality.

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Complexity growth in integrable and chaotic models

Cited by 58 publications

References 71 publications

Random matrix theory for complexity growth and black hole interiors

Random matrix theory for complexity growth and black hole interiors

Holographic and QFT complexity with angular momentum

Quantum Computational Complexity -- From Quantum Information to Black Holes and Back

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