dorsal/arxiv
View SchemaInvestigation of Continuous-Time Quantum Walk Via Spectral Distribution Associated with Adjacency Matrix
| Authors | M. A. Jafarizadeh, S. Salimi |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0510174 |
| URL | https://arxiv.org/abs/quant-ph/0510174 |
Abstract
Using the spectral distribution associated with the adjacency matrix of graphs, we introduce a new method of calculation of amplitudes of continuous-time quantum walk on some rather important graphs, such as line, cycle graph $C_n$, complete graph $K_n$, graph $G_n$, finite path and some other finite and infinite graphs, where all are connected with orthogonal polynomials such as Hermite, Laguerre, Tchebichef and some other orthogonal polynomials. It is shown that using the spectral distribution, one can obtain the infinite time asymptotic behavior of amplitudes simply by using the method of stationary phase approximation(WKB approximation), where as an example, the method is applied to star, two-dimensional comb lattices, infinite Hermite and Laguerre graphs. Also by using the Gauss quadrature formula one can approximate infinite graphs with finite ones and vice versa, in order to derive large time asymptotic behavior by WKB method. Likewise, using this method, some new graphs are introduced, where their amplitude are proportional to product of amplitudes of some elementary graphs, even though the graphs themselves are not the same as Cartesian product of their elementary graphs. Finally, via calculating mean end to end distance of some infinite graphs at large enough times, it is shown that continuous time quantum walk at different infinite graphs belong to different universality classes which are also different than those of the corresponding classical ones.
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"abstract": "Using the spectral distribution associated with the adjacency matrix of\ngraphs, we introduce a new method of calculation of amplitudes of\ncontinuous-time quantum walk on some rather important graphs, such as line,\ncycle graph $C_n$, complete graph $K_n$, graph $G_n$, finite path and some\nother finite and infinite graphs, where all are connected with orthogonal\npolynomials such as Hermite, Laguerre, Tchebichef and some other orthogonal\npolynomials. It is shown that using the spectral distribution, one can obtain\nthe infinite time asymptotic behavior of amplitudes simply by using the method\nof stationary phase approximation(WKB approximation), where as an example, the\nmethod is applied to star, two-dimensional comb lattices, infinite Hermite and\nLaguerre graphs. Also by using the Gauss quadrature formula one can approximate\ninfinite graphs with finite ones and vice versa, in order to derive large time\nasymptotic behavior by WKB method. Likewise, using this method, some new graphs\nare introduced, where their amplitude are proportional to product of amplitudes\nof some elementary graphs, even though the graphs themselves are not the same\nas Cartesian product of their elementary graphs. Finally, via calculating mean\nend to end distance of some infinite graphs at large enough times, it is shown\nthat continuous time quantum walk at different infinite graphs belong to\ndifferent universality classes which are also different than those of the\ncorresponding classical ones.",
"arxiv_id": "quant-ph/0510174",
"authors": [
"M. A. Jafarizadeh",
"S. Salimi"
],
"categories": [
"quant-ph"
],
"title": "Investigation of Continuous-Time Quantum Walk Via Spectral Distribution Associated with Adjacency Matrix",
"url": "https://arxiv.org/abs/quant-ph/0510174"
},
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