Fluctuations of Quantum Random Walks on Circles

Temporal fluctuations in the Hadamard walk on circles are studied. A temporal standard deviation of probability that a quantum random walker is positive at a given site is introduced to manifest striking differences between quantum and classical random walks. An analytical expression of the temporal standard deviation on a circle with odd sites is shown and its asymptotic behavior is considered for large system size. In contrast with classical random walks, the temporal fluctuation of quantum random walks depends on the position and initial conditions, since temporal standard deviation of the classical case is zero for any site. It indicates that the temporal fluctuation of the Hadamard walk can be controlled.