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.
Highlights
In the original conjecture [12], it was argued that a canonical time slice in an AdS corresponds to a special tensor network called MERA [19, 20]
We propose that holographic spacetimes can be regarded as collections of quantum circuits based on path-integrals
We propose a new framework of holography where each codimension one surface MΣ in the gravity dual is interpreted as a quantum circuit defined by a path-integration on MΣ with a suitable UV cut off, both in Lorentzian and Euclidean signature
Summary
The surface/state duality [44] argues that an arbitrary d dimensional (i.e. codimention two) connected closed surface Σ which is convex and space-like in a d + 2 dimensional gravitational spacetime Nd+2 (either Euclidean or Lorentzian), corresponds to a certain quantum state |ΨΣ in a Hilbert space HN specific to the spacetime Nd+2: Σd ∈ Nd+2 ↔ |ΨΣ ∈ HN. We consider the surface/state duality in the AdS/CFT case and would like to argue that it leads to an interpretation of codimension one surfaces in AdS, called MΣ, as quantum circuits of path-integrals (see the right picture in figure 1). We would like to study how we can construct the state |ΨΣ in a CFT We consider both Euclidean AdS and Lorentzian AdS separately below. There is a definite AdS spacetime for a given boundary and we consider the surface Σ and MΣ in this fixed AdS spacetime
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