A method of approximation of discrete Schrödinger equation with the normalized Laplacian by discrete-time quantum walk on graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917982993645568 |
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| author | Saito, Kei Segawa, Etsuo |
| author_facet | Saito, Kei Segawa, Etsuo |
| contents | We propose a class of continuous-time quantum walk models on graphs induced by a certain class of discrete-time quantum walk models with the parameter $ε\in [0,1]$. Here the graph treated in this paper can be applied both finite and infinite cases. The induced continuous-time quantum walk is an extended version of the (free) discrete-Schrödinger equation driven by the normalized Laplacian: the element of the weighted Hermitian takes not only a scalar value but also a matrix value depending on the underlying discrete-time quantum walk. We show that each discrete-time quantum walk with an appropriate setting of the parameter $ε$ in the long time limit identifies with its induced continuous-time quantum walk and give the running time for the discrete-time to approximate the induced continuous-time quantum walk with a small error $δ$. We also investigate the detailed spectral information on the induced continuous-time quantum walk. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_13741 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A method of approximation of discrete Schrödinger equation with the normalized Laplacian by discrete-time quantum walk on graphs Saito, Kei Segawa, Etsuo Quantum Physics Mathematical Physics We propose a class of continuous-time quantum walk models on graphs induced by a certain class of discrete-time quantum walk models with the parameter $ε\in [0,1]$. Here the graph treated in this paper can be applied both finite and infinite cases. The induced continuous-time quantum walk is an extended version of the (free) discrete-Schrödinger equation driven by the normalized Laplacian: the element of the weighted Hermitian takes not only a scalar value but also a matrix value depending on the underlying discrete-time quantum walk. We show that each discrete-time quantum walk with an appropriate setting of the parameter $ε$ in the long time limit identifies with its induced continuous-time quantum walk and give the running time for the discrete-time to approximate the induced continuous-time quantum walk with a small error $δ$. We also investigate the detailed spectral information on the induced continuous-time quantum walk. |
| title | A method of approximation of discrete Schrödinger equation with the normalized Laplacian by discrete-time quantum walk on graphs |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2308.13741 |