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 $π_N^2=(\mathbb{Z}/N)^2$ . 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 $π_N^2$ 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 $π_2^2$ .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 $π_N^2=(\mathbb{Z}/N)^2$. 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 $π_N^2$ 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 $π_2^2$.

arXiv:1811.05302Nov 13, 20181 citation

Walk/Zeta Correspondence for quantum and correlated random walks

Norio Konno, Shunya Tamura

In this paper, following the recent paper on Walk/Zeta Correspondence by the first author and his coworkers, we compute the zeta function for the three- and four-state quantum walk and correlated random walk, and the multi-state random walk on the one-dimensional torus by using the Fourier analysis. We deal with also the four-state quantum walk and correlated random walk on the two-dimensional torus. In addition, we introduce a new class of models determined by the generalized Grover matrix bridging the gap between the Grover matrix and the positive-support of the Grover matrix. Finally, we give a generalized version of the Konno-Sato theorem for the new class. As a corollary, we calculate the zeta function for the generalized Grover matrix on the d-dimensional torus.In this paper, following the recent paper on Walk/Zeta Correspondence by the first author and his coworkers, we compute the zeta function for the three- and four-state quantum walk and correlated random walk, and the multi-state random walk on the one-dimensional torus by using the Fourier analysis. We deal with also the four-state quantum walk and correlated random walk on the two-dimensional torus. In addition, we introduce a new class of models determined by the generalized Grover matrix bridging the gap between the Grover matrix and the positive-support of the Grover matrix. Finally, we give a generalized version of the Konno-Sato theorem for the new class. As a corollary, we calculate the zeta function for the generalized Grover matrix on the d-dimensional torus.

arXiv:2109.07664Sep 16, 20211 citation

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_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

π_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

π_2^2