On the evolution of operator complexity beyond scrambling

Barbón, J.L.F.; Rabinovici, Eliezer; Shir, Ruth; Sinha, Ritam

doi:10.1007/jhep10(2019)264

Cited by 144 publications

(188 citation statements)

References 31 publications

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“…Second, because of the relation between operator growth, quantum complexity and the emergence of near horizon symmetries [9][10][11][12]. Finally, due to the broader connection between complexity and operator growth, as discussed from different perspectives in [10,[13][14][15], such as using Nielsen's geometric approach to quantum circuit complexity [16,17] or the recursion method in many-body physics [18]. 1 The main goal of this work is twofold.…”

Section: Jhep05(2020)071mentioning

confidence: 99%

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On operator growth and emergent Poincaré symmetries

Magán

Simón

2020

J. High Energ. Phys.

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We consider operator growth for generic large-N gauge theories at finite temperature. Our analysis is performed in terms of Fourier modes, which do not mix with other operators as time evolves, and whose correlation functions are determined by their two-point functions alone, at leading order in the large-N limit. The algebra of these modes allows for a simple analysis of the operators with whom the initial operator mixes over time, and guarantees the existence of boundary CFT operators closing the bulk Poincaré algebra, describing the experience of infalling observers. We discuss several existing approaches to operator growth, such as number operators, proper energies, the many-body recursion method, quantum circuit complexity, and comment on its relation to classical chaos in black hole dynamics. The analysis evades the bulk vs boundary dichotomy and shows that all such approaches are the same at both sides of the holographic duality, a statement that simply rests on the equality between operator evolution itself. In the way, we show all these approaches have a natural formulation in terms of the Gelfand-Naimark-Segal (GNS) construction, which maps operator evolution to a more conventional quantum state evolution, and provides an extension of the notion of operator growth to QFT.

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Section: Jhep05(2020)071mentioning

confidence: 99%

“…They are convenient to determine the thermal nature of the Rindler modes in |0 M [43]. Indeed, usingâ 14) and the fact that Unruh modes annihilate |0 M (eq. (3.6)), one easily finds…”

Section: Jhep05(2020)071mentioning

confidence: 99%

On operator growth and emergent Poincaré symmetries

Magán

Simón

2020

J. High Energ. Phys.

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“…In [30] the growth of complexity with time of the operator, the so-called K-complexity, is defined introducing the Krylov basis of operators associated with O (see also [43]). The moments of CðtÞ are introduced as…”

Section: The Quantum Length Of Time Evolution and A New View On Cmentioning

confidence: 99%

“…with σ m;n ¼ ðρ m −ρ n Þ 2 ρ n þρ m plus any antisymmetric term. For chaotic Hamiltonians we can use the eigenstate thermalization hypothesis [44,45] that implies that the eigen basis of O and that of the Hamiltonian H are uncorrelated [43]. In this case we can approximate…”

Section: The Quantum Length Of Time Evolution and A New View On Cmentioning

confidence: 99%

Complexity and time

Gómez

2020

Phys. Rev. D

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For any quantum algorithm given by a path in the space of unitary operators we define the computational complexity as the typical computational time associated with the path. This time is defined using a quantum time estimator associated with the path. This quantum time estimator is fully characterized by the Lyapunov generator of the path and the corresponding quantum Fisher information. The computational metric associated with this definition of computational complexity leads to a natural characterization of cost factors on the Lie algebra generators. Operator complexity growth in time is analyzed from this perspective leading to a simple characterization of the Lyapunov exponent in the case of chaotic Hamiltonians. The connection between complexity and entropy is expressed using the relation between quantum Fisher information about quantum time estimation and von Neumann entropy. This relation suggests a natural bound on computational complexity that generalizes the standard time energy quantum uncertainty. The connection between Lyapunov and modular Hamiltonian is briefly discussed. In the case of theories with holographic duals and for those reduced density matrix defined by tracing over a bounded region of the bulk, quantum estimation theory is crucial to estimate quantum mechanically the geometry of the tracing region. It is suggested that the corresponding quantum Fisher information associated with this estimation problem is at the root of the holographic bulk geometry.

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“…Measures of operator complexity have received considerable recent attention in studies of information scrambling in many-body quantum systems [1][2][3][4][5][6][7]. One motivation is the characterization of quantum complexity in holographic systems.…”

Section: Introductionmentioning

confidence: 99%

Momentum/Complexity duality and the black hole interior

Barbón

Martín-García

Sasieta

2020

J. High Energ. Phys.

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We establish a version of the Momentum/Complexity (PC) duality between the rate of operator complexity growth and an appropriately defined radial component of bulk momentum for a test system falling into a black hole. In systems of finite entropy, our map remains valid for arbitrarily late times after scrambling. The asymptotic regime of linear complexity growth is associated to a frozen momentum in the interior of the black hole, measured with respect to a time foliation by extremal codimension-one surfaces which saturate without reaching the singularity. The detailed analysis in this paper uses the Volume-Complexity (VC) prescription and an infalling system consisting of a thin shell of dust, but the final PC duality formula should have a much wider degree of generality.

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On the evolution of operator complexity beyond scrambling

Cited by 144 publications

References 31 publications

On operator growth and emergent Poincaré symmetries

On operator growth and emergent Poincaré symmetries

Complexity and time

Momentum/Complexity duality and the black hole interior

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