Complexity growth of operators in the SYK model and in JT gravity

TL;DR

This paper investigates the growth of operator complexity in the SYK model and JT gravity, revealing exponential-to-linear growth patterns and connections between operator size and complexity, enhancing understanding of quantum chaos and holography.

Contribution

It compares microscopic K-complexity with holographic complexity, demonstrating their similar growth behaviors and linking operator size to complexity in quantum gravity contexts.

Findings

Both K-complexity and CV duality show exponential-to-linear growth.

Operator size saturates at the scrambling time.

Complexity remains a useful measure of operator evolution over time.

Abstract

The concepts of operator size and computational complexity play important roles in the study of quantum chaos and holographic duality because they help characterize the structure of time-evolving Heisenberg operators. It is particularly important to understand how these microscopically defined measures of complexity are related to notions of complexity defined in terms of a dual holographic geometry, such as complexity-volume (CV) duality. Here we study partially entangled thermal states in the Sachdev-Ye-Kitaev (SYK) model and their dual description in terms of operators inserted in the interior of a black hole in Jackiw-Teitelboim (JT) gravity. We compare a microscopic definition of complexity in the SYK model known as K-complexity to calculations using CV duality in JT gravity and find that both quantities show an exponential-to-linear growth behavior. We also calculate the growth of operator size under time evolution and find connections between size and complexity. While the notion of operator size saturates at the scrambling time, our study suggests that complexity, which is well defined in both quantum systems and gravity theories, can serve as a useful measure of operator evolution at both early and late times.