Krylov construction and complexity for driven quantum systems

TL;DR

This paper extends the concept of Krylov complexity to time-dependent driven quantum systems, especially Floquet systems like the quantum kicked rotor, analyzing operator growth and complexity growth across different coupling regimes.

Contribution

It introduces a method for constructing Krylov space in driven systems and studies K-complexity and Krylov subspace growth numerically for the quantum kicked rotor.

Findings

K-complexity can be defined for driven systems using a natural Krylov construction.

Krylov subspace dimension growth varies with system coupling strength.

Numerical results show operator complexity growth behavior in kicked quantum systems.

Abstract

Krylov complexity is an important dynamical quantity with relevance to the study of operator growth and quantum chaos, and has recently been much studied for various time-independent systems. We initiate the study of K-complexity in time-dependent (driven) quantum systems. For periodic time-dependent (Floquet) systems, we develop a natural method for doing the Krylov construction and then define (state and operator) K-complexity for such systems. Focusing on kicked systems, in particular the quantum kicked rotor on a torus, we provide a detailed numerical study of the time dependence of Arnoldi coefficients as well as of the K-complexity with the system coupling constant interpolating between the weak and strong coupling regime. We also study the growth of the Krylov subspace dimension as a function of the system coupling constant.