Abstract
In this paper, we consider discrete-time quantum walks with moving shift (MS) and flip-flop shift (FF) on two-dimensional lattice Z² and torus π_N²=(Z/N)². Weak limit theorems for the Grover walks on Z² with MS and FF were given by Watabe et al. and Higuchi et al., respectively. The existence of localization of the Grover walks on Z² with MS and FF was shown by Inui et al. and Higuchi et al., respectively. Non-existence of localization of the Fourier walk with MS on Z² was proved by Komatsu and Tate. Here our simple argument gave non-existence of localization of the Fourier walk with both MS and FF. Moreover we calculate eigenvalues and the corresponding eigenvectors of the (k₁,k₂)-space of the Fourier walks on π_N² with MS and FF for some special initial conditions. The probability distributions are also obtained. Finally, we compute amplitudes of the Grover and Fourier walks on π₂².