Machine-learning emergent spacetime from linear response in future tabletop quantum gravity experiments

TL;DR

A novel interpretable neural network model designed to perform precision bulk reconstruction under the AdS/CFT correspondence is introduced and it is confirmed that the machine can let a higher-dimensional gravity metric be automatically emergent as its interpretable weights, using a linear response of the condensed matter system as data, through supervised machine learning.

Abstract

We introduce a novel interpretable Neural Network (NN) model designed to perform precision bulk reconstruction under the AdS/CFT correspondence. According to the correspondence, a specific condensed matter system on a ring is holographically equivalent to a gravitational system on a bulk disk, through which tabletop quantum gravity experiments may be possible as reported in arXiv:2211.13863. The purpose of this paper is to reconstruct a higher-dimensional gravity metric from the condensed matter system data via machine learning using the NN. Our machine reads spatially and temporarily inhomogeneous linear response data of the condensed matter system, and incorporates a novel layer that implements the Runge-Kutta method to achieve better numerical control. We confirm that our machine can let a higher-dimensional gravity metric be automatically emergent as its interpretable weights, using a linear response of the condensed matter system as data, through supervised machine learning. The developed method could serve as a foundation for generic bulk reconstruction, i.e., a practical solution to the AdS/CFT correspondence, and would be implemented in future tabletop quantum gravity experiments.

Figures (10)

• Figure 1: The profile of $\Phi_n(\xi)$ on the BTZ black hole metric is shown, comparing the exact solution with numerical results obtained using the Euler method and the Runge-Kutta method. The parameters chosen are $(k_n, \omega) = (1.00, 1.00)$. The blue line represents results from the Euler method, the orange line represents results from the Runge-Kutta method, and the green dotted line represents the exact solution.
• Figure 2: The profile of $\Phi_n(\xi)$ on the AdS soliton metric is shown, comparing the exact solution with numerical results obtained using the Euler method and the Runge-Kutta method. The parameters chosen are $(k_n, \omega) = (3.00, 3.00)$ and $r_s = 0.10$. The blue line represents results from the Euler method, the orange line represents results from the Runge-Kutta method, and the green dotted line represents the exact solution.
• Figure 3: The model structure.
• Figure 4: The data flow of the Runge-Kutta layer.
• Figure 5: The bulk layer. Here $Z^{1,2}$ is a component of ${\bf Z}$ and the activation ${\bf f}$ is a function of a six dimensional vector ${\bf x}=(x_1,x_2,\ldots,x_6)$ and gives the four dimensional vector $(x_3, x_4+x_5x_1+x_6x_2, x_5,x_6)$.