Chaos, complexity, and random matrices

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 $$ \mathcal{O}(1) $$ 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.