Holographic Spacetimes as Quantum Circuits of Path-Integrations

TL;DR

The paper proposes that holographic spacetimes emerge from quantum circuits defined by path-integrations on codimension-one surfaces in AdS, extending the surface/state duality to a covariant framework. It connects path-integral complexity to bulk gravity actions: Euclidean cases yield a complexity tied to surface area via corner contributions, while Lorentzian cases reproduce the complexity=action picture on WDW patches for static backgrounds. A key claim is that the time component of the bulk metric arises from the density of unitary gates, linking circuit complexity to spacetime geometry and even gravitational force, with entanglement growth encoded by surface areas. The work also introduces ghost D-brane holography to compute certain entanglement measures and discusses how gravitational dynamics and emergent spacetime may be underpinned by quantum circuits, offering a unifying, covariant bridge between tensor networks and AdS/CFT. It lays groundwork for extending holographic complexity and emergent gravity to time-dependent and non-AdS settings in future research.

Abstract

We propose that holographic spacetimes can be regarded as collections of quantum circuits based on path-integrals. We relate a codimension one surface in a gravity dual to a quantum circuit given by a path-integration on that surface with an appropriate UV cut off. Our proposal naturally generalizes the conjectured duality between the AdS/CFT and tensor networks. This largely strengthens the surface/state duality and also provides a holographic explanation of path-integral optimizations. For static gravity duals, our new framework provides a derivation of the holographic complexity formula given by the gravity action on the WDW patch. We also propose a new formula which relates numbers of quantum gates to surface areas, even including time-like surfaces, as a generalization of the holographic entanglement entropy formula. We argue the time component of the metric in AdS emerges from the density of unitary quantum gates in the dual CFT. Our proposal also provides a heuristic understanding how the gravitational force emerges from quantum circuits.

Figures (12)

• Figure 1: The left picture is a sketch of surface/state correspondence in the context of AdS/CFT MiTa. The right picture explains the new correspondence proposed in the present paper, based on path-integrations in surface/state duality for Euclidean AdS.
• Figure 2: Various constructions of the same state $|\Psi_\Sigma\rangle$ (i.e. the vacuum state in a CFT) from path-integrations on different surfaces $M_\Sigma$ and their gravity duals.
• Figure 3: The path-integral construction of the state $|\Psi_\Sigma\rangle$ when we take $\Sigma$ in Lorentzian AdS. $M_\Sigma$ can be either time-like, null or space-like.
• Figure 4: The sketch of holographic computation of path-integral complexity in a global Euclidean AdS. For simplicity, we choose $\Sigma$ is a codimension two convex surface on a time slice $t=0$. Originally, the state $|\Psi_\Sigma\rangle$ dual to the surface $\Sigma$ is obtained by the path-integration along the Euclidean time with a coarse-grained CFT action as in the left picture. Then we can deform the space $M_\Sigma$ on which we perform the path-integration without changing the quantum state $|\Psi_\Sigma\rangle$ as depicted in the middle picture. The bulk region surrounded by the time slice $t=0$ and $M_{\Sigma}$ is called $N_\Sigma$. During this process we can reduce the normalization of wave function and this normalization is computed by doubling the system namely the inner product $\langle\Psi_\Sigma|\Psi_{\Sigma}\rangle$. In the gravity dual, this inner product is given by the gravity action evaluated on the Euclidean space given by a double copy of $N_\Sigma$, depicted in the right picture.
• Figure 5: The setup of holographic calculation of path-integral complexity in a Poincare Euclidean AdS$_3$.