3 years ago

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

Masahiro Asano, Takashi Komatsu, Norio Konno, Akihiro Narimatsu

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 $\pi_2^2$.

Publisher URL: http://arxiv.org/abs/1811.05302

DOI: arXiv:1811.05302v1

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