Information scrambling in quantum walks: Discrete-time formulation of Krylov complexity

We study information scrambling (a spread of initially localized quantum information into the system's many degree of freedom) in discrete-time quantum walks. We consider out-of-time-ordered correlators (OTOC) and K complexity as a probe of information scrambling. The OTOC for local spin operators in all directions has a light-cone structure which is ``shell-like.'' As the wavefront passes, the OTOC approaches to zero in the long-time limit, showing no signature of scrambling. The introduction of spatial or temporal disorder changes the shape of the light-cone akin to localization of wavefuction. We formulate the K complexity in system with discrete-time evolution, and show that it grows linearly in discrete-time quantum walk. The presence of disorder modifies this growth to sublinear. Our study present interesting case to explore many-body phenomenon in a discrete-time quantum walk using scrambling.