Quantum walks driven by quantum coins with two multiple eigenvalues
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866918244809441280 |
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| author | Konno, Norio Sato, Iwao Segawa, Etsuo Shikano, Yutaka |
| author_facet | Konno, Norio Sato, Iwao Segawa, Etsuo Shikano, Yutaka |
| contents | 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 $κ,κ'$ and $p=\dim(\ker(κ-C_u))$ for any $u\in V$ with $1\leq p\leq δ(G)$, where $δ(G)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_00716 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quantum walks driven by quantum coins with two multiple eigenvalues Konno, Norio Sato, Iwao Segawa, Etsuo Shikano, Yutaka Mathematical Physics Quantum Physics 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 $κ,κ'$ and $p=\dim(\ker(κ-C_u))$ for any $u\in V$ with $1\leq p\leq δ(G)$, where $δ(G)$ 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. |
| title | Quantum walks driven by quantum coins with two multiple eigenvalues |
| topic | Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2110.00716 |