Complexity is Simple

TL;DR

We investigate how Lloyd's bound on computation translates to holographic complexity, focusing on the distinction between orthogonalizing versus simple gates. The analysis shows large AdS black holes, modeled as serial circuits, must employ simple gates, which invalidates the Lloyd bound for holographic complexity, while near the Hawking-Page transition small black holes can satisfy an orthogonalization condition, offering a potential link to the Weak Gravity Conjecture. By comparing coherence time $\tau_{\text{coh}}$, computation time $\tau_{\text{comp}}$, and Margolus-Levitin time $\tau_{\text{ML}}$, the work identifies regimes where holographic gates are simple ($\tau_{\text{comp}}\ll\tau_{\text{coh}}$) and where parallelization could be required for a bound to hold, but stresses that no universal holographic Lloyd bound is guaranteed. Overall, the paper clarifies the limitations of applying classical computation bounds to holographic complexity and outlines future avenues to refine bounds and their physical implications, including potential connections to the Weak Gravity Conjecture.

Abstract

In this note we investigate the role of Lloyd's computational bound in holographic complexity. Our goal is to translate the assumptions behind Lloyd's proof into the bulk language. In particular, we discuss the distinction between orthogonalizing and `simple' gates and argue that these notions are useful for diagnosing holographic complexity. We show that large black holes constructed from series circuits necessarily employ simple gates, and thus do not satisfy Lloyd's assumptions. We also estimate the degree of parallel processing required in this case for elementary gates to orthogonalize. Finally, we show that for small black holes at fixed chemical potential, the orthogonalization condition is satisfied near the phase transition, supporting a possible argument for the Weak Gravity Conjecture first advocated in Brown et al.

Figures (6)

• Figure 1: Complexity as a function of $\theta$ for the state (\ref{['ss']}). The plot was done with $k_{\text{max}}=19$ and $\epsilon=0.015$. We see that complexity is a fairly discontinuous thing.
• Figure 2: Illustration of Wheeler-DeWitt patch in the neutral AdS-Schwarzschild black hole for a pair of times $(t_{L},t_{R})$. The patch is defined by the intersection of both future and past-directed light rays from both boundary points, and the singularity.
• Figure 3: (a) Plot of $\sigma$ versus $r_{+}/\ell$ for $\tilde{Q} = 10$. (b) The same plot, zoomed in to the small $r_{+}/\ell$ region. One can see that $\sigma$ turns around before ever becoming small.
• Figure 4: $\Sigma$ as a function of the black hole radius $s$ for fixed charge $=1.2, 1$ and $.8$ depicted in blue, orange and green, respectively. Here we have chosen $\ell/\sqrt{G} = 100$.
• Figure 5: A violation of Lloyd's bound at $x=.8$. The assumptions of Lloyd's bound are satisfied at the transition region between stable (blue) and unstable (red) regions.