Krylov complexity, path integrals, and instantons

TL;DR

This paper formulates Krylov complexity as a path integral and links its late-time plateau in chaotic systems to quantum instantons, providing new insights into quantum chaos and complexity growth.

Contribution

It introduces a path integral formulation of Krylov complexity and connects its plateau value to quantum instantons in classical chaotic systems.

Findings

Krylov complexity can be expressed as a path integral.

The plateau of Krylov complexity relates to quantum instantons.

Tested ideas in a simple toy model.

Abstract

Krylov complexity has emerged as an important tool in the description of quantum information and, in particular, quantum chaos. Here we formulate Krylov complexity $K(t)$ for quantum mechanical systems as a path integral, and argue that at large times, for classical chaotic systems with at least two minima of the potential, that have a plateau for $K(t)$, the value of the plateau is described by quantum mechanical instantons, as is the case for standard transition amplitudes. We explain and test these ideas in a simple toy model.