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Analytical solutions for quantum walks on 1D chain with different shift operators

In this paper, we study the discrete-time quantum walks on 1D Chain with the moving and swapping shift operators. We derive analytical solutions for the eigenvalues and eigenstates of the evolution operator $\hat{U}$ using the Chebyshev polynomial technique, and calculate the long-time averaged probabilities for the two different shift operators respectively. It is found that the probability distributions for the moving and swapping shift operators display completely different characteristics. For the moving shift operator, the probability distribution exhibits high symmetry where the probabilities at mirror positions are equal. The probabilities are inversely proportional to the system size $N$ and approach to zero as $N\rightarrow \infty$. On the contrary, for the swapping shift operator, the probability distribution is not symmetric, the probability distribution approaches to a power-law stationary distribution as $N\rightarrow \infty$ under certain coin parameter condition. We show that such power-law stationary distribution is determined by the eigenstates of the eigenvalues $\pm1$ and calculate the intrinsic probability for different starting positions. Our findings suggest that the eigenstates corresponding to eigenvalues $\pm1$ play an important role for the swapping shift operator.