# Time evolution of the complexity in chaotic systems: concrete examples

## TL;DR

This paper explores the time evolution of operator complexity in chaotic systems using the SYK model, demonstrating that bi-invariant Finsler geometry captures key features like linear growth, saturation, and bounds, supporting its relevance in quantum complexity.

## Contribution

It provides a concrete example showing bi-invariant complexity geometry reproduces expected chaotic complexity evolution, highlighting its naturalness in quantum systems.

## Key findings

- Complexity exhibits linear growth and saturation in the SYK model.
- Bi-invariant Finsler geometry captures the complexity evolution features.
- Lloyd's bound is realized within the model.

## Abstract

We investigate the time evolution of the complexity of the operator by the Sachdev-Ye-Kitaev (SYK) model with $N$ Majorana fermions. We follow Nielsen's idea of complexity geometry and geodesics thereof. We show that it is possible that the bi-invariant complexity geometry can exhibit the conjectured time evolution of the complexity in chaotic systems: i) linear growth until $t\sim e^{N}$, ii) saturation and small fluctuations after then. We also show that the Lloyd's bound is realized in this model. Interestingly, these characteristic features appear only if the complexity geometry is the most natural "non-Riemannian" Finsler geometry. This serves as a concrete example showing that the bi-invariant complexity may be a competitive candidate for the complexity in quantum mechanics/field theory (QM/QFT). We provide another argument showing a naturalness of bi-invariant complexity in QM/QFT. That is that the bi-invariance naturally implies the equivalence of the right-invariant complexity and left-invariant complexity, either of which may correspond to the complexity of a given operator. Without bi-invariance, one needs to answer why only right (left) invariant complexity corresponds to the "complexity", instead of only left (right) invariant complexity.