Flux-fluctuation relation for quantum walks on networks

For classical random walks on networks, the flux-fluctuation relation is an important characteristic that relates the mean flux and strength of fluctuations on the nodes. It was extensively studied before, although equivalent relations for quantum walks on networks remain unknown. In this work, a quantum version of the flux-fluctuation relation is obtained. It is valid for quantum walks on an arbitrary network, subject to the condition that the spectrum of the unitary walk operator is nondegenerate. Remarkably, for phase coherent quantum walks, the quantum flux-fluctuation relation is qualitatively similar to its classical counterpart, and quantum effects are insignificant. We numerically simulate quantum walks on periodic lattices and scale-free and Erdős-Rényi networks, and the results are consistent with the analytical quantum flux-fluctuation relation. We discuss the implications for the quantum walk dynamics if the spectrum of the unitary walk operator violate the nondegeneracy condition.