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arXiv · 2110.00716

Quantum walks driven by quantum coins with two multiple eigenvalues

Norio Konno·Iwao Sato·Etsuo Segawa·Yutaka Shikano
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Abstract

We consider a spectral analysis on the quantum walks on graph $G=(V,E)$ with the local coin operators $\{C_u\}_{u\in V}$ and the flip flop shift. The quantum coin operators have commonly two distinct eigenvalues $\kappa,\kappa'$ and $p=\dim(\ker(\kappa-C_u))$ for any $u\in V$ with $1\leq p\leq \delta(G)$, where $\delta(G)$ is the minimum degrees of $G$. We show that this quantum walk can be decomposed into a cellular automaton on $\ell^2(V;\mathbb{C}^p)$ whose time evolution is described by a self adjoint operator $T$ and its remainder. We obtain how the eigenvalues and its eigenspace of $T$ are lifted up to as those of the original quantum walk. As an application, we express the eigenpolynomial of the Grover walk on $\mathbb{Z}^d$ with the moving shift in the Fourier space.

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Norio Konno, Iwao Sato, Etsuo Segawa, Yutaka Shikano. 2021-10-02. Quantum walks driven by quantum coins with two multiple eigenvalues. https://doi.org/10.1007/s40509-022-00281-1

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