Universal Time Evolution of Holographic and Quantum Complexity

TL;DR

This paper demonstrates that holographic and quantum complexity exhibit universal spectral behavior characterized by linear growth and saturation, rooted in spectral statistics and chaos theory.

Contribution

It introduces a spectral representation for holographic complexity measures and links their universal evolution to random matrix universality and spectral pole structures.

Findings

Complexity measures show a universal slope-ramp-plateau pattern.

Linear growth of complexity is tied to spectral pole structures.

Late-time saturation results from spectral level repulsion.

Abstract

Holographic complexity, as the bulk dual of quantum complexity, encodes the geometric structure of black hole interiors. Motivated by the complexity=anything proposal, we introduce the spectral representation for generating functions associated with codimension-one and codimension-zero holographic complexity measures. These generating functions exhibit a universal slope-ramp-plateau structure, analogous to the spectral form factor in chaotic quantum systems. In such systems, quantum complexity evolves universally, displaying long-time linear growth followed by saturation at late times. By employing the generating function formalism, we demonstrate that this universal behavior originates from random matrix universality in spectral statistics and from a particular pole structure of the matrix elements of the generating functions in the energy eigenbasis. Using the residue theorem, we prove that the existence of this pole structure is both a necessary and sufficient condition for the linear growth of complexity measures. Furthermore, we show that the late-time saturation plateau arises directly from the spectral level repulsion, a hallmark of quantum chaos.