Tensor Network States and Geometry

Evenbly, Glen; Vidal, Guifré

doi:10.1007/s10955-011-0237-4

Cited by 262 publications

(307 citation statements)

References 106 publications

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“…A promising starting point for addressing these issues is the MERA tensor network construction of a discrete version of AdS/CFT [40][41][42]. It seems possible that in that fairly controlled setting one could rigorously confirm the quantum error correction structure we have motivated in this paper.…”

Section: Mera As An Error Correcting Code?mentioning

confidence: 83%

Bulk locality and quantum error correction in AdS/CFT

Almheiri

Dong

Harlow

2015

J. High Energ. Phys.

704

1,132

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We point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction. Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction. We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a "quantum secret sharing scheme", and suggest a tensor network calculation that may settle the issue. Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard "operator algebra quantum error correction" of Beny, Kempf, and Kribs. Our proposal gives a precise formulation of the idea of "subregion-subregion" duality in AdS/CFT, and clarifies the limits of its validity.

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Section: Mera As An Error Correcting Code?mentioning

confidence: 83%

Bulk locality and quantum error correction in AdS/CFT

Almheiri

Dong

Harlow

2015

J. High Energ. Phys.

704

1,132

View full text Add to dashboard Cite

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“…This very general bound has been related to the Ryu-Takayanagi formula for entanglement in holographic conformal field theories [50][51][52][53]74], and has also been applied to unitary networks as a heuristic picture for entanglement growth [10].…”

Section: Directed Polymers and Minimal Cutmentioning

confidence: 99%

Quantum Entanglement Growth under Random Unitary Dynamics

et al. 2017

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Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the "entanglement tsunami" in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated KardarParisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like ðtimeÞ 1=3 and are spatially correlated over a distance ∝ ðtimeÞ 2=3 . We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a "minimal cut" picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the "velocity" of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the wellstudied problem of pinning of a membrane or domain wall by disorder.

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“…Tensor networks are a graphical description of wave functionals in quantum many-body systems in terms of networks of quantum entanglement (see e.g. [45,46]). The optimization of tensor network was introduced in [23,24], called tensor network renormalization.…”

Section: Connection To Computational Complexitymentioning

confidence: 99%

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

Caputa

Kundu

Miyaji

et al. 2017

J. High Energ. Phys.

285

402

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Abstract:We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.

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Tensor Network States and Geometry

Cited by 262 publications

References 106 publications

Bulk locality and quantum error correction in AdS/CFT

Bulk locality and quantum error correction in AdS/CFT

Quantum Entanglement Growth under Random Unitary Dynamics

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

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