Extreme events of quantum walks on networks

Extreme events are the ones for which the magnitude of the event is much larger than its typical values. They have been extensively investigated on classical dynamical and stochastic processes on networks. In this work, in analogy with well-studied extreme events of random walks, the extreme events of quantum walks on regular lattices, a scale-free network, and Erdős-Rényi network are studied. Due to unitary quantum evolution, the quantum walk dynamics usually differs from that of the classical random walks, and hence we expect extreme event properties to be different in classical and quantum settings. In contrast to this expectation, we report that extreme events of phase coherent quantum walks are qualitatively similar to that of classical random walks. We show that this counterintuitive behavior originates from the dominance of nodal fluctuations over internode correlations in disordered network structures. In particular, the occurrence probability for extreme events on scale-free and Erdős-Rényi networks displays a decaying power law with the degree of nodes. That is, the extreme event probability is larger for small degree nodes compared to the hubs on the network, in qualitative agreement with corresponding classical random walk results. Further, it is shown that extreme event probability scales with the threshold used to define extreme events.