The Fourier and Grover walks on the two-dimensional lattice and torus

In this paper, we consider discrete-time quantum walks with moving shift (MS) and flip-flop shift (FF) on two-dimensional lattice mathbb{Z}² and torus π_N²=\(mathbb{Z}/N\)². Weak limit theorems for the Grover walks on mathbb{Z}² with MS and FF were given by Watabe et al. and Higuchi et al., respectively. The existence of localization of the Grover walks on mathbb{Z}² 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 mathbb{Z}² 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.