Robustness of periodicity in Grover walks under a magnetic vector potential

Abstract

We study the effect of magnetic vector potentials on periodic Grover walks on finite graphs. The magnetic vector potential is introduced through the framework of quantum graphs, which induces the Grover walk as a special case. We regard the magnetic vector potential as a perturbation of a periodic Grover walk and investigate the robustness of its periodicity. Our analysis reveals that the response to such perturbations depends on the spectral structure of the underlying graph. In particular, when the graph possesses at least one non-simple eigenvalue, we derive a Hermitian matrix that characterizes the robustness of its periodicity. As a consequence, for initial states orthogonal to the eigenspaces of the unperturbed Grover walk corresponding to the eigenvalues $\pm1$, we show that the perturbed dynamics is asymptotically described by a continuous-time quantum walk generated by this Hermitian matrix.