# Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix

## 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 {sub n}, complete graph K {sub n}, graph G {sub n}, finite path and some other finite and infinite graphs, where all are connected with orthogonal polynomials such as Hermite, Laguerre, Tchebichef, and 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 the 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 amplitudes are proportional to the product of amplitudes of some elementary graphs, even though the graphs themselves are not the same as the Cartesian product of their elementary graphs. Finally, by calculating the mean end to end distance of some infinitemore »

- Authors:

- Department of Theoretical Physics and Astrophysics, Tabriz University, Tabriz 51664 (Iran, Islamic Republic of) and Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795 (Iran, Islamic Republic of) and Research Institute for Fundamental Sciences, Tabriz 51664 (Iran, Islamic Republic of). E-mail: jafarizadeh@tabrizu.ac.ir
- Department of Theoretical Physics and Astrophysics, Tabriz University, Tabriz 51664 (Iran, Islamic Republic of) and Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795 (Iran, Islamic Republic of). E-mail: shsalimi@tabrizu.ac.ir

- Publication Date:

- OSTI Identifier:
- 20976753

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics (New York); Journal Volume: 322; Journal Issue: 5; Other Information: DOI: 10.1016/j.aop.2007.01.009; PII: S0003-4916(07)00017-6; Copyright (c) 2007 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AMPLITUDES; DISTRIBUTION; MATRICES; POLYNOMIALS; QUADRATURES; TWO-DIMENSIONAL CALCULATIONS; WKB APPROXIMATION

### Citation Formats

```
Jafarizadeh, M.A., and Salimi, S..
```*Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix*. United States: N. p., 2007.
Web. doi:10.1016/j.aop.2007.01.009.

```
Jafarizadeh, M.A., & Salimi, S..
```*Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix*. United States. doi:10.1016/j.aop.2007.01.009.

```
Jafarizadeh, M.A., and Salimi, S.. Tue .
"Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix". United States.
doi:10.1016/j.aop.2007.01.009.
```

```
@article{osti_20976753,
```

title = {Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix},

author = {Jafarizadeh, M.A. and Salimi, S.},

abstractNote = {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 {sub n}, complete graph K {sub n}, graph G {sub n}, finite path and some other finite and infinite graphs, where all are connected with orthogonal polynomials such as Hermite, Laguerre, Tchebichef, and 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 the 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 amplitudes are proportional to the product of amplitudes of some elementary graphs, even though the graphs themselves are not the same as the Cartesian product of their elementary graphs. Finally, by calculating the 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 from those of the corresponding classical ones.},

doi = {10.1016/j.aop.2007.01.009},

journal = {Annals of Physics (New York)},

number = 5,

volume = 322,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}