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We quantitatively differentiate between the spreads of discrete-time quantumand classical random walks on a cyclic graph. Due to the closed nature of anycyclic graph, there is additional "collision"- like interference in the quantumrandom walk along with the usual interference in any such walk on any graph,closed or otherwise. We find that the quantum walker remains localized incomparison to the classical one, even in the absence of disorder, a phenomenonthat is potentially attributable to the additional interference in the quantumcase. This is to be contrasted with the situation on open graphs, where thequantum walker, being effectively denied the collision-like interference,garners a much higher spread than its classical counterpart. We use Shannonentropy of the position probability distribution to quantify spread of thewalker in both quantum and classical cases. We find that for a given number ofvertices on a cyclic graph, the entropy with respect to number of steps for thequantum walker saturates, on average, to a value lower than that for thecorresponding classical one. We also analyze variations of the entropies withrespect to system size, and look at the corresponding asymptotic growth rates.

Interference-induced localization in quantum random walk on clean cyclic graph