Complexity, action, and black holes

TL;DR

The paper proposes a holographic complexity conjecture (CA-duality) equating boundary state complexity with the bulk action of the Wheeler-DeWitt patch, improving on CV-duality by using the full spacetime region behind horizons. It derives and tests late-time action growth across neutral, charged, and rotating black holes, showing neutral and small-charge cases saturate a Lloyd-type bound on complexity growth, while large charged-hole cases highlight subtleties due to hair and UV completion. Through shock-wave analyses, static shells, and a tensor-network model, the work demonstrates consistency between CA-duality, black-hole interiors, and minimal circuit descriptions, and clarifies when the conjecture matches or challenges expected bounds. The discussion outlines open questions, regulator issues, and possible stringy corrections, positioning CA-duality as a diagnostic tool and a unifying framework for understanding black-hole interior growth via computational complexity. The results offer a principled link between gravitational action, information processing limits, and the emergent geometry of spacetime in holography, with implications for horizon transparency and the dynamics of quantum information in gravity.

Abstract

Our earlier paper "Complexity Equals Action" conjectured that the quantum computational complexity of a holographic state is given by the classical action of a region in the bulk (the "Wheeler-DeWitt" patch). We provide calculations for the results quoted in that paper, explain how it fits into a broader (tensor) network of ideas, and elaborate on the hypothesis that black holes are the fastest computers in nature.

Figures (20)

• Figure 1: The Penrose diagrams for two-sided eternal black holes (left) and one-sided black holes that form from collapsing shock waves (right). The two-sided black hole is dual to an entangled state of two CFTs that live on the left and right boundaries; the one-sided black hole is dual to a single CFT. The (old) complexity/volume conjecture related the complexity of the entangled CFT state to the volume of the maximal spatial slice anchored at the CFT state. Our (new) complexity/action conjecture relates the complexity of the CFT state to the action of the Wheeler-DeWitt patch.
• Figure 2: An uncharged AdS black hole. When $t_L$ increases, the Wheeler-DeWitt patch gains a slice (in blue) and loses a slice (in red).
• Figure 3: The WDW patch divided into quadrants. In quadrants III and IV the WDW patch intersects the AdS boundary and causes a divergence in the action.
• Figure 4: A charged AdS black hole. When $t_L$ increases, the Wheeler-DeWitt patch gains a slice (in blue) and loses a slice (in red). It is useful to consider separately the pieces above and below $r = r_\textrm{meet}(t_L,t_R)$.
• Figure 5: The phase diagram for charged Reissner-Nordstrom black holes in AdS. Top pane: at fixed $M$, black holes exist only for small enough $Q$. For black holes that are small compared to $\ell_{\text{AdS}}$, the extremal line is $Q=\sqrt{G} M$; for black holes that are large compared to $\ell_{\text{AdS}}$, the extremal line becomes $Q \sim M^{1/3}$. Middle pane: curves of constant chemical potential $\mu$. Small extremal black holes have $\sqrt{G} \mu = 1$; larger extremal black holes have larger $\mu$. Thus for $\sqrt{G} \mu < 1$ the lines of constant $\mu$ end at $M = Q = 0$, and for $\sqrt{G} \mu > 1$ the lines of constant $\mu$ end on the extremal line. Bottom pane: for a given large charged black hole (red star), we may define $M_\mu$ (gray star) as the mass of the extremal black hole with the same chemical potential $\mu$, and $M_Q$ (blue star) as the mass of the extremal black hole with the same charge $Q$.