JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (8): 84-93.doi: 10.6040/j.issn.1671-9352.0.2023.123

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A class of two-state quantum walk based on Hadamard walk with the same eigenvalues and continuous spectrum

Pingtao LYU(),Caishi WANG*(),Jijun ZHAO   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Received:2023-03-21 Online:2024-08-20 Published:2024-07-31
  • Contact: Caishi WANG E-mail:2328158374@qq.com;wangcs@nwnu.edu.cn

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Abstract:

In this paper, we generalize the Wojcik model to make the whole model contain two parameters ε and ω. Especially, although the extended model adds one parameter ε, the eigenvalue does not change, i.e., it does not depend on the newly added parameter. At the same time, we compare the eigenvalue distribution of the extended model on the unit circle with the region of Hadamard walk continuous spectrum, and obtain relevant conclusions.

Key words: discrete-time quantum walk, eigenvalue, stationary measure, continuous spectrum

CLC Number: 

  • O211.6

Table 1

Parameter dependence of the eigenvalues and the corresponding stationary measures for the extend model $ \beta=\alpha \mathrm{e}^{e i} \mathrm{i}$"

$ \phi$ $ \beta=\alpha \mathrm{e}^{\varepsilon i} \mathrm{i}$
$ \frac{1}{8}\left(\omega=\mathrm{e}^{\frac{\pi \mathrm{i}}{4}}=\frac{\sqrt{2}+\sqrt{2} \mathrm{i}}{2}\right)$ $ \lambda^2=-1, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{3-2 \sqrt{2}}\right)^{|x|} \begin{cases}2-\sqrt{2}, & x \neq 0, \\ 1, & x=0\end{cases}$
$ \frac{1}{6}\left(\omega=\mathrm{e}^{\frac{\pi i}{3}}=\frac{1+\sqrt{3} i}{2}\right)$ $ \lambda^2=\frac{-1+\sqrt{3} i}{2}, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{2-\sqrt{3}}\right)^{|x|} \begin{cases}\frac{3-\sqrt{3}}{2}, & x \neq 0， \\ 1, & x=0\end{cases}$
$ \frac{1}{4}\left(\omega=\mathrm{e}^{\frac{\pi \mathrm{i}}{2}}=\mathrm{i}\right)$ $ \lambda^2=\mathrm{i}, \quad \mu(x)=2|\alpha|^2$
$ \frac{1}{2}\left(\omega=\mathrm{e}^{\pi \mathrm{i}}=-1\right)$ $ \lambda^2=\frac{-4+3 \mathrm{i}}{5}, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{5}\right)^{|x|} \begin{cases}3, & x \neq 0 ，\\ 1, & x=0\end{cases}$
$ \frac{2}{3}\left(\omega=\mathrm{e}^{\frac{4 \pi \mathrm{i}}{3}}=\frac{-1-\sqrt{3} \mathrm{i}}{2}\right)$ $ \lambda^2=\frac{-6-3 \sqrt{3}+(1-2 \sqrt{3}) \mathrm{i}}{8+2 \sqrt{3}}, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{4+\sqrt{3}}\right){ }^{|x|} \begin{cases}\frac{5+\sqrt{3}}{2}, & x \neq 0, \\ 1, & x=0\end{cases}$
$ \frac{3}{4}\left(\omega=\mathrm{e}^{\frac{3 \pi \mathrm{i}}{2}}=-\mathrm{i}\right)$ $ \lambda^2=\frac{-4-3 \mathrm{i}}{5}, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{5}\right)^{|x|} \begin{cases}3, & x \neq 0， \\ 1, & x=0\end{cases}$

Table 2

Parameter dependence of the eigenvalues and the corresponding stationary measures for the extend model $ \beta=-\alpha \mathrm{e}^{\varepsilon i} \mathrm{i}$"

$ \phi$ $ \beta=-\alpha \mathrm{e}^{\varepsilon i} \mathrm{i}$
$ \frac{1}{8}\left(\omega=\mathrm{e}^{\frac{\mathrm{i}}{4}}=\frac{\sqrt{2}+\sqrt{2} \mathrm{i}}{2}\right)$ $ \lambda^2=\frac{-1+2 \sqrt{2} \mathrm{i}}{3}, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{3}\right)^{|x|} \begin{cases}2, & x \neq 0, \\ 1, & x=0\end{cases}$
$ \frac{1}{6}\left(\omega=\mathrm{e}^{\frac{i}{3}}=\frac{1+\sqrt{3} i}{2}\right)$ $ \lambda^2=\frac{-(2+\sqrt{3})+(3+2 \sqrt{3}) \mathrm{i}}{4+2 \sqrt{3}}, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{2+\sqrt{3}}\right)^{|x|} \begin{cases}\frac{3+\sqrt{3}}{2}, & x \neq 0, \\ 1, & x=0\end{cases}$
$ \frac{1}{4}\left(\omega=\mathrm{e}^{\frac{\mathrm{i}}{2}}=\mathrm{i}\right)$ $ \lambda^2=\frac{-4+3 \mathrm{i}}{5}, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{5}\right)^{|x|} \begin{cases}3, & x \neq 0， \\ 1, & x=0\end{cases}$
$ \frac{1}{2}\left(\omega=\mathrm{e}^{\pi \mathrm{i}}=-1\right)$ $ \lambda^2=\frac{-4-3 \mathrm{i}}{5}, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{5}\right)^{|x|} \begin{cases}3, & x \neq 0， \\ 1, & x=0\end{cases}$
$ \frac{2}{3}\left(\omega=e^{\frac{4 \pi i}{3}}=\frac{-1-\sqrt{3} \mathrm{i}}{2}\right)$ $ \lambda^2=\frac{3 \sqrt{3}-6-(1+2 \sqrt{3}) \mathrm{i}}{8-2 \sqrt{3}}, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{4-\sqrt{3}}\right)^{|x|} \begin{cases}\frac{5-\sqrt{3}}{2}, & x \neq 0, \\ 1, & x=0\end{cases}$
$ \frac{3}{4}\left(\omega=e^{\frac{3 \pi i}{2}}=-i\right)$ $ \lambda^2=-\mathrm{i}, \quad \mu(x)=2|\alpha|^2\left(\frac{1}{5}\right)^{|x|} \begin{cases}3, & x \neq 0， \\ 1, & x=0\end{cases}$

Fig.1

Illustrations of the eigenvalues movements of the extend model, when $ \beta=\alpha \mathrm{e}^{\varepsilon i} \mathrm{i}$"

Fig.2

$ \beta=-\alpha \mathrm{e}^{\varepsilon i} \mathrm{i}$"

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