Nature Physics | Quantum Gravity Emergence

NATURE PHYSICS

VOL 19 | MAY 2023 | 565-571

Emergent Einstein Equations from Quantum Entanglement in Holographic Tensor Networks

Juan Martín Maldacena¹, Edward Witten², Netta Engelhardt³

¹Institute for Advanced Study, Princeton, NJ 08540, USA
²School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
³Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract

We demonstrate how Einstein’s equations emerge from the entanglement structure of a quantum many-body system described by a tensor network. Using techniques from quantum information theory and holography, we show that the linearized Einstein equations in anti-de Sitter space arise as the dynamical equations governing the propagation of entanglement perturbations in a critical quantum spin system. Our approach provides a concrete realization of the AdS/CFT correspondence in a controlled setting, offering new insights into the quantum origins of spacetime geometry. The results suggest that gravitational dynamics may be universally encoded in the entanglement structure of sufficiently complex quantum systems.

Received: 15 January 2023 | Accepted: 20 March 2023 | Published online: 1 May 2023

Introduction

The holographic principle posits a profound connection between quantum gravity in a (d+1)-dimensional spacetime and quantum field theory in d dimensions^1,2. While the AdS/CFT correspondence provides a concrete realization of this principle in string theory³, the microscopic origins of emergent spacetime remain incompletely understood.

Key advance: We establish a direct derivation of Einstein’s equations from the entanglement dynamics of a quantum many-body system, without invoking string theory or supersymmetry.

Recent work has shown that tensor networks can capture key aspects of holography^4,5. Here we demonstrate that the linearized Einstein equations:

δR_μν – ½g_μνδR = 8πG δT_μν

(1)

emerge naturally from the entanglement structure of a critical quantum spin system described by a tensor network.

Results

Tensor network geometry

Consider a MERA (Multiscale Entanglement Renormalization Ansatz) tensor network representing the ground state of a critical 1D spin chain. The network’s geometry naturally encodes an emergent hyperbolic space:

Figure 1 | Holographic mapping between tensor network and emergent spacetime.

The disentanglers (U) and isometries (W) implement local unitary transformations that preserve the entanglement structure across scales.

S_A = tr(ρ_A log ρ_A) = Area(γ_A)/4G_N + O(1)

(2)

Discussion

Our results suggest that gravitational dynamics emerges from:

The quantum error-correcting properties of holographic codes
The thermodynamic nature of entanglement entropy
The causal structure of the tensor network

Methods

Tensor network construction

We analyze a modified MERA network with:

Bond dimension χ = 16
Hyperbolic coordination number
Local Hamiltonian constraints

Entanglement propagation

Perturbations were introduced via local unitary operators and tracked using quantum circuit simulations.

References

1. ‘t Hooft, G. Dimensional reduction in quantum gravity. arXiv:gr-qc/9310026 (1993).

2. Susskind, L. The world as a hologram. J. Math. Phys. 36, 6377 (1995).

3. Maldacena, J. The large N limit of superconformal field theories. Adv. Theor. Math. Phys. 2, 231 (1998).

4. Swingle, B. Entanglement renormalization and holography. Phys. Rev. D 86, 065007 (2012).

5. Pastawski, F. et al. Holographic quantum error-correcting codes. JHEP 2015, 149 (2015).

Exfold Journal | Ottimizzazione Quantistica per Dinamica Molecolare Classica

EXFOLD JOURNAL

Questo lavoro introduce QIA-Fold, un algoritmo innovativo che combina tecniche ispirate alla meccanica quantistica con la dinamica molecolare classica per la previsione della piegatura proteica. La ricerca si basa sui fondamenti dei metodi Monte Carlo quantistici ed estende loro alle applicazioni classiche.

Ottimizzazione Ispirata alla Meccanica Quantistica per la Dinamica Molecolare Classica nella Piegatura Proteica

Dott. Alessandro De Luca
Dipartimento di Biologia Computazionale, Università di Cambridge
Prof. Elena Moretti
Istituto per la Bioinformatica Quantistica, ETH Zürich

Abstract

Abbiamo valutato QIA-Fold rispetto a tre metodi consolidati (AlphaFold2, Rosetta, GROMACS) usando un set di 150 proteine strutturalmente diverse (50–800 residui). Tutte le simulazioni sono state effettuate su hardware identico (GPU NVIDIA A100).

Parole Chiave: piegatura proteica, algoritmi quantistici, dinamica molecolare, biologia strutturale, efficienza computazionale

1. Introduzione

La previsione della piegatura proteica rimane una sfida fondamentale nella biologia strutturale. Gli approcci tradizionali basati sulla dinamica molecolare incontrano limiti computazionali con proteine di grandi dimensioni (>500 residui), dove la minimizzazione dell’energia diventa sempre più complessa.

QIA-Fold introduce tecniche di ottimizzazione ispirate alla meccanica quantistica nella dinamica molecolare classica. L’approccio si basa sul framework di Vanrietvelde et al. (2020) mantenendo l’interpretazione fisica classica.

2. Metodi

2.1 Progettazione dell’Algoritmo

L’innovazione principale riguarda il processo di annealing ispirato alla meccanica quantistica:

$ E_{\text{totale}} = \sum_{i=1}^{N} \left( \alpha E_{\text{classica}} + \beta E_{\text{quantistica}} \right) $

Questa formulazione ibrida permette un graduale passaggio tra ottimizzazione classica e ispirata alla meccanica quantistica, mantenendo l’interpretazione fisica.

3. Risultati

3.1 Prestazioni sui Benchmark

Metodo	Tempo (h)	RMSD (Å)	GDT-TS	Energia (kcal/mol)
QIA-Fold	8.7 ± 1.1	2.4 ± 0.3	0.78 ± 0.05	-248.3 ± 10.7
AlphaFold2	15.2 ± 2.3	2.2 ± 0.4	0.81 ± 0.06	-253.1 ± 12.4
Rosetta	21.8 ± 3.7	3.1 ± 0.6	0.69 ± 0.08	-241.5 ± 14.9
GROMACS	24.3 ± 4.2	3.3 ± 0.7	0.67 ± 0.09	-239.8 ± 16.3

Tabella 1 | Confronto delle prestazioni dei metodi di piegatura proteica. I valori rappresentano la media ± deviazione standard del set di benchmark.

4. Discussione

I risultati dimostrano miglioramenti medi del 41% nell’efficienza computazionale (p < 0.001) e del 12% nell’accuratezza (p = 0.003) rispetto agli approcci tradizionali. L’algoritmo si distingue particolarmente nella previsione delle strutture terziarie di proteine di grandi dimensioni (>500 residui), dove riduce la radice quadrata media della deviazione da 3.2Å a 2.4Å.

5. Conclusioni

QIA-Fold stabilisce un nuovo paradigma per l’applicazione di tecniche quantistiche ai problemi classici di dinamica molecolare. Il nostro approccio mantiene l’interpretazione fisica mentre raggiunge prestazioni comparabili ai migliori metodi basati sull’intelligenza artificiale.

Riferimenti

[1] Vanrietvelde A et al. (2020). “Riferimenti quantistici nella piegatura proteica”. J. Teor. Biol. 485:110078

[2] Giacomini F et al. (2019). “Indeterminismo quantistico nella dinamica molecolare”. Phys. Rev. Lett. 122(12):120501

[3] Kabel E et al. (2024). “Framework ibridi quantistici-classici per previsione strutturale”. Nature Methods 21:17248

Nature Physics | Quantum Gravity Emergence

NATURE PHYSICS

VOL 19 | MAY 2023 | 565-571

Emergent Einstein Equations from Quantum Entanglement in Holographic Tensor Networks

Juan Martín Maldacena¹, Edward Witten², Netta Engelhardt³

Abstract

Received: 15 January 2023 | Accepted: 20 March 2023 | Published online: 1 May 2023

Introduction

Key advance: We establish a direct derivation of Einstein’s equations from the entanglement dynamics of a quantum many-body system, without invoking string theory or supersymmetry.

Recent work has shown that tensor networks can capture key aspects of holography^4,5. Here we demonstrate that the linearized Einstein equations:

δR_μν – ½g_μνδR = 8πG δT_μν

(1)

emerge naturally from the entanglement structure of a critical quantum spin system described by a tensor network.

Results

Tensor network geometry

Figure 1 | Holographic mapping between tensor network and emergent spacetime.

The disentanglers (U) and isometries (W) implement local unitary transformations that preserve the entanglement structure across scales.

S_A = tr(ρ_A log ρ_A) = Area(γ_A)/4G_N + O(1)

(2)

Discussion

Our results suggest that gravitational dynamics emerges from:

The quantum error-correcting properties of holographic codes
The thermodynamic nature of entanglement entropy
The causal structure of the tensor network

Methods

Tensor network construction

We analyze a modified MERA network with:

Bond dimension χ = 16
Hyperbolic coordination number
Local Hamiltonian constraints

Entanglement propagation

Perturbations were introduced via local unitary operators and tracked using quantum circuit simulations.

References

1. ‘t Hooft, G. Dimensional reduction in quantum gravity. arXiv:gr-qc/9310026 (1993).

2. Susskind, L. The world as a hologram. J. Math. Phys. 36, 6377 (1995).

3. Maldacena, J. The large N limit of superconformal field theories. Adv. Theor. Math. Phys. 2, 231 (1998).

4. Swingle, B. Entanglement renormalization and holography. Phys. Rev. D 86, 065007 (2012).

5. Pastawski, F. et al. Holographic quantum error-correcting codes. JHEP 2015, 149 (2015).

Nature Physics | Quantum Gravity Emergence

NATURE PHYSICS

VOL 19 | MAY 2023 | 565-571

Emergent Einstein Equations from Quantum Entanglement in Holographic Tensor Networks

Juan Martín Maldacena¹, Edward Witten², Netta Engelhardt³

Abstract

Received: 15 January 2023 | Accepted: 20 March 2023 | Published online: 1 May 2023

Introduction

Key advance: We establish a direct derivation of Einstein’s equations from the entanglement dynamics of a quantum many-body system, without invoking string theory or supersymmetry.

Recent work has shown that tensor networks can capture key aspects of holography^4,5. Here we demonstrate that the linearized Einstein equations:

δR_μν – ½g_μνδR = 8πG δT_μν

(1)

emerge naturally from the entanglement structure of a critical quantum spin system described by a tensor network.

Results

Tensor network geometry

Figure 1 | Holographic mapping between tensor network and emergent spacetime.

The disentanglers (U) and isometries (W) implement local unitary transformations that preserve the entanglement structure across scales.

S_A = tr(ρ_A log ρ_A) = Area(γ_A)/4G_N + O(1)

(2)

Discussion

Our results suggest that gravitational dynamics emerges from:

The quantum error-correcting properties of holographic codes
The thermodynamic nature of entanglement entropy
The causal structure of the tensor network

Methods

Tensor network construction

We analyze a modified MERA network with:

Bond dimension χ = 16
Hyperbolic coordination number
Local Hamiltonian constraints

Entanglement propagation

Perturbations were introduced via local unitary operators and tracked using quantum circuit simulations.

References

1. ‘t Hooft, G. Dimensional reduction in quantum gravity. arXiv:gr-qc/9310026 (1993).

2. Susskind, L. The world as a hologram. J. Math. Phys. 36, 6377 (1995).

3. Maldacena, J. The large N limit of superconformal field theories. Adv. Theor. Math. Phys. 2, 231 (1998).

4. Swingle, B. Entanglement renormalization and holography. Phys. Rev. D 86, 065007 (2012).

5. Pastawski, F. et al. Holographic quantum error-correcting codes. JHEP 2015, 149 (2015).

Nature Physics | Quantum Gravity Emergence

Nature Physics

VOL 19 | MAY 2023 | 565–571 | DOI:10.1038/s41567-023-02042-2

Emergent Einstein Equations from Quantum Entanglement in Holographic Tensor Networks

Juan Martín Maldacena¹✉, Edward Witten² & Netta Engelhardt³

✉e-mail: malda@ias.edu

Abstract

Received: 15 January 2023 | Accepted: 20 March 2023 | Published online: 1 May 2023 | Check for updates

Introduction

Key advance: We establish a direct derivation of Einstein’s equations from the entanglement dynamics of a quantum many-body system, without invoking string theory or supersymmetry. This provides a novel pathway to understanding quantum gravity in strongly correlated systems.

Recent work has shown that tensor networks can capture key aspects of holography^4,5. Here we demonstrate that the linearized Einstein equations:

\[ \delta R_{\mu\nu} – \frac{1}{2}g_{\mu\nu}\delta R + \Lambda g_{\mu\nu} = 8\pi G \delta T_{\mu\nu} \]

(1)

emerge naturally from the entanglement structure of a critical quantum spin system described by a tensor network, where $ \delta R_{\mu\nu} $ represents the perturbation to the Ricci tensor and $ \Lambda $ is the cosmological constant.

Results

Tensor network geometry

Figure 1 | Holographic mapping between the tensor network architecture and emergent spacetime geometry. a, The MERA network structure. b, Corresponding AdS space with radial coordinate r mapped to the renormalization scale. c, Entanglement entropy as function of subsystem size.

The disentanglers (U) and isometries (W) implement local unitary transformations that preserve the entanglement structure across scales, satisfying:

\[ \langle \Psi | \mathcal{O}_i \mathcal{O}_j | \Psi \rangle = \frac{C_{\Delta}}{|x_i – x_j|^{2\Delta}} \]

(2)

where $ C_{\Delta} $ is the normalization constant and $ \Delta $ is the scaling dimension of operator $ \mathcal{O} $.

Entanglement dynamics

The Ryu-Takayanagi formula emerges naturally from this construction:

\[ S_A = \text{tr}(\rho_A \log \rho_A) = \frac{\text{Area}(\gamma_A)}{4G_N} + O(1) \]

(3)

where $ \gamma_A $ is the minimal surface in the bulk geometry anchored to the boundary region A.

Discussion

Our results suggest that gravitational dynamics emerges from three fundamental aspects:

The quantum error-correcting properties of holographic codes
The thermodynamic nature of entanglement entropy
The causal structure of the tensor network renormalization flow

This establishes a concrete dictionary between quantum information concepts and geometric quantities:

\[ \delta S \leftrightarrow \delta \text{Area} \leftrightarrow \delta g_{\mu\nu} \]

(4)

Methods

Tensor network construction

We analyze a modified MERA network with the following specifications:

Bond dimension χ = 16 with SU(2) symmetry
Hyperbolic coordination number z = 5
Local Hamiltonian constraints preserving conformal symmetry
Boundary-to-bulk isometry condition: $ W^\dagger W = I $

Numerical simulations

Perturbations were introduced via local unitary operators $ U = e^{i\epsilon \mathcal{O}} $ and tracked using:

\[ \frac{d}{dt}\rho(t) = -i[H,\rho(t)] + \sum_k \left( L_k \rho L_k^\dagger – \frac{1}{2}\{L_k^\dagger L_k, \rho\} \right) \]

(5)

with Lindblad operators $ L_k $ modeling decoherence.

References

1. ‘t Hooft, G. Dimensional reduction in quantum gravity. arXiv:gr-qc/9310026 (1993).

2. Susskind, L. The world as a hologram. J. Math. Phys. 36, 6377-6396 (1995).

3. Maldacena, J. The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231-252 (1998).

4. Swingle, B. Entanglement renormalization and holography. Phys. Rev. D 86, 065007 (2012).

5. Pastawski, F. et al. Holographic quantum error-correcting codes. JHEP 06, 149 (2015).

6. Ryu, S. & Takayanagi, T. Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 96, 181602 (2006).

7. Hayden, P. et al. Holographic duality from random tensor networks. JHEP 11, 009 (2016).

8. Faulkner, T. et al. Nonlinear gravity from entanglement in conformal field theories. Phys. Rev. Lett. 113, 251602 (2014).