Spacetime as a quantum circuit

Abstract

We propose that finite cutoff regions of holographic spacetimes represent quantum circuits that map between boundary states at different times and Wilsonian cutoffs, and that the complexity of those quantum circuits is given by the gravitational action. The optimal circuit minimizes the gravitational action. This is a generalization of both the "complexity equals volume" conjecture to unoptimized circuits, and path integral optimization to finite cutoffs. Using tools from holographic $T\bar T$, we find that surfaces of constant scalar curvature play a special role in optimizing quantum circuits. We also find an interesting connection of our proposal to kinematic space, and discuss possible circuit representations and gate counting interpretations of the gravitational action.

Figures (2)

• Figure 1: We consider a subregion $M$ of Euclidean Poincaré AdS$_3$. We introduce two time-slices $t=t_{i}$ and $t=t_{f}$ corresponding to the field theory ground states $|0\rangle_{z_{i}}$ and $|0\rangle_{z_{f}}$, which are prepared for different values of the radial cutoff. The radial boundary is at finite cutoff, $z=\rho(t)$. Our proposal is that the complexity of the circuit that maps between these ground states with different finite Wilsonian cutoffs is given by the gravitational action on $M$.
• Figure 2: We can parametrize a generic bulk curve $\rho(t)$ by the pairs of boundary points $(t_1(t),t_2(t))$, such that a bulk geodesic connecting these two points is tangent to the bulk curve at $z=\rho(t)$. This way, the profile $\rho(t)$ is encoded as a path in kinematic space, the space of bulk geodesics.