Probing the localization effects in Krylov basis

Krylov complexity (K-complexity) is a measure of quantum state complexity that minimizes wave-function spreading across all the possible bases. It serves as a key indicator of operator growth and quantum chaos. In this work, we use K-complexity and Arnoldi coefficients to investigate diverse localization phenomena in the quantum kicked rotor (QKR). We analyze four distinct localization scenarios-ranging from strong localization effect arising from quantum antiresonance to a weaker form of power-law localization-each one exhibiting distinct K-complexity signatures and Arnoldi coefficient variations. The long-time behavior of K-complexity and the wave-function evolution on Krylov chain can distinguish various types of observed localization in QKR. We show that K-complexity not only captures the degree of localization but also the nature of localization. In particular, the time-averaged K-complexity and scaling of the variance of Arnoldi coefficients with effective Planck's constant can distinguish the localization effects induced by the classical regular phase structures and the dynamical localization arising from quantum interferences. Further, we also show that the Arnoldi coefficients effectively reveal the onset of chaos, even with the quantum dynamics being localized.