Abstract
We investigate the discrete-time quantum random walks on a line in periodic potential. The probability distribution with periodic potential is more complex compared to the normal quantum walks, and the standard deviation σ has interesting behaviors for different period q and parameter θ. We study the behavior of standard deviation with variation in walk steps, period, and θ. The standard deviation increases approximately linearly with θ and decreases with 1/q for θ ∈ (0, π/4), and increases approximately linearly with 1/q for θ ∈ [π/4, π/2). For θ ∈ (π/4, 3π/4), the transmission is larger than the reflection, with sensibility the diffussion will be accelerated. However, when q = 2, the standard deviations are nearly the same, and when q > 2, the standard deviation will decrease.