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

Krylov construction and complexity for driven quantum systems