Complexity Growth in Integrable and Chaotic Models

01/06/2021
by   Vijay Balasubramanian, et al.
0

We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such "shortcuts" through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O(√(N)), and we find an explicit operator which "fast-forwards" the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O( poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(e^N), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/14/2019

Quantum Complexity of Time Evolution with Chaotic Hamiltonians

We study the quantum complexity of time evolution in large-N chaotic sys...
research
02/28/2022

Bounds on quantum evolution complexity via lattice cryptography

We address the difference between integrable and chaotic motion in quant...
research
05/26/2021

Entropy and complexity unveil the landscape of memes evolution

On the Internet, information circulates fast and widely, and the form of...
research
01/22/2020

Time-invariant degree growth in preferential attachment network models

Preferential attachment drives the evolution of many complex networks. I...
research
01/18/2018

Complexity of Combinations of Qualitative Constraint Satisfaction Problems

The CSP of a first-order theory T is the problem of deciding for a given...
research
06/22/2017

A Minimal Developmental Model Can Increase Evolvability in Soft Robots

Different subsystems of organisms adapt over many time scales, such as r...
research
09/03/2022

Global Population Growth as Socio-Economic Soft Matter System Dynamics Evolution

The report considers the dynamics of the global population as the unique...

Please sign up or login with your details

Forgot password? Click here to reset