dorsal/arxiv
View SchemaInvestigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm
| Authors | M. A. Jafarizadeh, S. Salimi, R. Sufiani |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0606241 |
| URL | https://arxiv.org/abs/quant-ph/0606241 |
| DOI | 10.1140/epjb/e2007-00281-5 |
Abstract
In papers\cite{js,jsa}, the amplitudes of continuous-time quantum walk on graphs possessing quantum decomposition (QD graphs) have been calculated by a new method based on spectral distribution associated to their adjacency matrix. Here in this paper, it is shown that the continuous-time quantum walk on any arbitrary graph can be investigated by spectral distribution method, simply by using Krylov subspace-Lanczos algorithm to generate orthonormal bases of Hilbert space of quantum walk isomorphic to orthogonal polynomials. Also new type of graphs possessing generalized quantum decomposition have been introduced, where this is achieved simply by relaxing some of the constrains imposed on QD graphs and it is shown that both in QD and GQD graphs, the unit vectors of strata are identical with the orthonormal basis produced by Lanczos algorithm. Moreover, it is shown that probability amplitude of observing walk at a given vertex is proportional to its coefficient in the corresponding unit vector of its stratum, and it can be written in terms of the amplitude of its stratum. Finally the capability of Lanczos-based algorithm for evaluation of walk on arbitrary graphs (GQD or non-QD types), has been tested by calculating the probability amplitudes of quantum walk on some interesting finite (infinite) graph of GQD type and finite (infinite) path graph of non-GQD type, where the asymptotic behavior of the probability amplitudes at infinite limit of number of vertices, are in agreement with those of central limit theorem of Ref.\cite{nko}.
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"abstract": "In papers\\cite{js,jsa}, the amplitudes of continuous-time quantum walk on\ngraphs possessing quantum decomposition (QD graphs) have been calculated by a\nnew method based on spectral distribution associated to their adjacency matrix.\nHere in this paper, it is shown that the continuous-time quantum walk on any\narbitrary graph can be investigated by spectral distribution method, simply by\nusing Krylov subspace-Lanczos algorithm to generate orthonormal bases of\nHilbert space of quantum walk isomorphic to orthogonal polynomials. Also new\ntype of graphs possessing generalized quantum decomposition have been\nintroduced, where this is achieved simply by relaxing some of the constrains\nimposed on QD graphs and it is shown that both in QD and GQD graphs, the unit\nvectors of strata are identical with the orthonormal basis produced by Lanczos\nalgorithm. Moreover, it is shown that probability amplitude of observing walk\nat a given vertex is proportional to its coefficient in the corresponding unit\nvector of its stratum, and it can be written in terms of the amplitude of its\nstratum. Finally the capability of Lanczos-based algorithm for evaluation of\nwalk on arbitrary graphs (GQD or non-QD types), has been tested by calculating\nthe probability amplitudes of quantum walk on some interesting finite\n(infinite) graph of GQD type and finite (infinite) path graph of non-GQD type,\nwhere the asymptotic behavior of the probability amplitudes at infinite limit\nof number of vertices, are in agreement with those of central limit theorem of\nRef.\\cite{nko}.",
"arxiv_id": "quant-ph/0606241",
"authors": [
"M. A. Jafarizadeh",
"S. Salimi",
"R. Sufiani"
],
"categories": [
"quant-ph"
],
"doi": "10.1140/epjb/e2007-00281-5",
"title": "Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm",
"url": "https://arxiv.org/abs/quant-ph/0606241"
},
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