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

Abstract

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 ${\pi }_N^2=(\mathbb{Z}/N{)}^2$. 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 $(k_1,k_2)$-space of the Fourier walks on ${\pi }_N^2$ 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 ${\pi }_2^2$.

13 pages