Spacetime as a quantum circuit

TL;DR

This paper proposes a novel interpretation of holographic spacetimes as quantum circuits, where the circuit complexity is given by gravitational action, extending previous complexity conjectures and connecting to kinematic space.

Contribution

It introduces a framework linking finite cutoff holographic regions to quantum circuits and identifies gravitational action as a measure of circuit complexity, generalizing existing complexity conjectures.

Findings

Surfaces of constant scalar curvature optimize quantum circuits.

Gravitational action correlates with circuit complexity.

Connections established to kinematic space and gate counting.

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.