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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.