Krylov Complexity

TL;DR

Krylov complexity is a versatile measure of operator growth rooted in quantum chaos, with applications spanning holography and black hole physics, offering insights into chaotic and integrable systems without arbitrary parameters.

Contribution

This review consolidates the theory of Krylov complexity, highlighting its general features, mathematical foundations, and relevance to diverse areas like holography and quantum chaos.

Findings

Krylov complexity effectively probes chaotic dynamics at late times.

It is independent of arbitrary control parameters.

The review establishes theoretical foundations and geometric interpretations.

Abstract

We introduce and review a new complexity measure, called `Krylov complexity', which takes its origins in the field of quantum-chaotic dynamics, serving as a canonical measure of operator growth and spreading. Krylov complexity, underpinned by the Lanczos algorithm, has since evolved into a highly diverse field of its own right, both because of its attractive features as a complexity, whose definition does not depend on arbitrary control parameters, and whose phenomenology serves as a rich and sensitive probe of chaotic dynamics up to exponentially late times, but also because of its relevance to seemingly far-afield subjects such as holographic dualities and the quantum physics of black holes. In this review we give a unified perspective on these topics, emphasizing the robust and most general features of K-complexity, both in chaotic and integrable systems, state and prove theorems on its generic features and describe how it is geometrised in the context of (dual) gravitational dynamics. We hope that this review will serve both as a source of intuition about K-complexity in and of itself, as well as a resource for researchers trying to gain an overview over what is by now a rather large and multi-faceted literature. We also mention and discuss a number of open problems related to K-complexity, underlining its currently very active status as a field of research.