The simplest 2D quantum walk detects chaoticity

Quantum walks are at present an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found for the Bunimovich stadium -- a chaotic billiard -- level statistics described by a Brody distribution with parameter $\delta \simeq 0.1$. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio (PR) $\simeq$ 1150) compared to the rectangular billiard (regular) case, where the average PR $\simeq$ 1500. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.