Chaos, complexity, and random matrices

By Jordan S. Cotler, Nicholas Hunter-Jones, Junyu Liu, B. Yoshida. Journal of High Energy Physics, 2017

299 citations recorded by the source corpus.

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 O1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O}(1) $$\end{document} 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.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8d70532e8450f027bb0c18b4c7f27b280a0d001b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. A. Nahum
2. B. Skinner
3. J. Ruhman
4. Nicholas Hunter-Jones
5. Tianci Zhou
6. Phil Saad
7. Matthew Fisher
8. Yaodong Li

In Statistical and Nonlinear Physics

1. J. Ruhman
2. B. Skinner
3. A. Nahum
4. Nicholas Hunter-Jones
5. Phil Saad
6. Tianci Zhou
7. D. Stanford
8. S. Shenker

Citations in the corpus

• Unitary designs from statistical mechanics in random quantum circuits. (2019, background): “Specifically, [25] related the frame potential to both averaged out-of-time-ordered correlators and measures of complexity, and [26] computed the frame potentials for random Hamiltonian evolution.”
• Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity (2019, background): “We also expect that certain out-of-time-ordered correlation functions (OTOCs) in this theory will exhibit a ramp and plateau structure [30].”
• Entanglement Membrane in Chaotic Many-Body Systems (2019, background): “The approach could also be used to study questions motivated by quantum information such as the emergence of the “design” property in random circuits with non-Haar–distributed gates [5, 17, 80–82].”
• Measurement-Induced Phase Transitions in the Dynamics of Entanglement (2018, background): “[91] Jordan Cotler, Nicholas Hunter-Jones, Junyu Liu, and”