# Chaos, Complexity, and Random Matrices

## TL;DR

This paper analyzes quantum chaos using Gaussian Unitary Ensemble Hamiltonians, computing correlation functions and frame potentials to understand scrambling and complexity, and introduces the concept of $k$-invariance to describe the transition to random matrix behavior.

## Contribution

It provides an analytical study of chaos and complexity through GUE Hamiltonians and introduces $k$-invariance as a new framework for understanding the onset of random matrix behavior in quantum systems.

## Key findings

- Qualitative prediction of late-time chaotic behavior
- Unphysical early-time behavior including $ ext{O}(1)$ scrambling time
- Introduction of $k$-invariance to characterize dynamics

## Abstract

Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an $\mathcal{O}(1)$ scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce $k$-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate $k$-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.