Eigenvalue analysis of three-state quantum walks
with general coin matrices

Quantum walks (QWs) are fundamental models for understanding quantum dynamics. Despite their apparent simplicity, quantum walks exhibit a wide range of intricate quantum features, making them essential tools in various domains, including the study of topological phases [1] and the formulation of models and designs for quantum algorithms [2, 3, 4]. Within the context of QWs, the combination of the coin operator and the shift operator determines how the system evolves. A vital part of studying QWs is looking at the spectral properties of the time evolution operator. This helps us understand the behaviors and features of the walk, thereby enriching our comprehension of the underlying dynamics.

In this paper, our focus is on three-state QWs on a one-dimensional integer lattice, which involves working within an infinite-dimensional Hilbert space. This model is intensively studied in various articles with different contexts [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. It is well-known that the occurrence of localization, a characteristic property of QWs, is equivalent to the existence of eigenvalues of the time evolution operator. Moreover, a quantitative investigation of localization can be conducted by deriving the time-averaged limit distribution using the eigenvalues and eigenvectors [21]. The transfer matrix method stands as a powerful tool for eigenvalue analysis and has been applied to several types of QW models [22, 23, 24], and even for non-unitary QWs [25]. We specifically consider the two-phase model with a finite number of defects. Previous studies [24] have conducted eigenvalue analysis for three-state QWs with a specific class of coin matrices, including Grover matrices, through the transfer matrix method. In contrast, this work extends the result by demonstrating that the method remains applicable and effective even for cases involving a general coin matrix.

Calculating explicit eigenvalues analytically for a general coin matrix remains highly challenging. Therefore, we resort to numerical calculations to determine eigenvalues, utilizing the method of the transfer matrix. For our numerical analysis, we concentrate on one-defect and two-phase Fourier walks. The Fourier walk has been intensively studied by [26, 27, 28] for the space-homogeneous model, where the coin matrices are identical at any position. Notably, it has been proven that the Fourier walk does not exhibit localization for the homogeneous model. However, we numerically demonstrate that the Fourier walk does exhibit localization for both the one-defect and two-phase models.

The remainder of this paper is organized as follows. In Section 2, we define our three-state QWs with a self-loop on the integer lattice. Section 3 introduces the transfer matrix with a general coin matrix and establishes methods for the eigenvalue problem. Theorem 1 constitutes the main theorem, providing a necessary and sufficient condition for the eigenvalue problem. Section 4 is dedicated to the numerical analysis of the Fourier walk. We present figures to visually depict the results of the numerical analysis on the eigenvalue equation and its probability distribution, thereby illustrating the occurrence of localization and the existence of the eigenvalue.

We define a three-state QW on the integer lattice $\mathbb{Z}$. Let us define the Hilbert space $\mathcal{H}$ as

$$\mathcal{H}=\ell^{2}(\mathbb{Z};\mathbb{C}^{3})=\left\{\Psi:\mathbb{Z}\rightarrow\mathbb{C}^{3}\ \middle|\ \sum_{x\in\mathbb{Z}}\|\Psi(x)\|_{\mathbb{C}^{3}}^{2}<\infty\right\}.$$

Here, $\mathbb{C}$ denotes the set of complex numbers. The quantum state $\Psi\in\mathcal{H}$ is expressed as

$$\Psi(x)=\left[\begin{array}[]{ c }\Psi_{1}(x)\\ \Psi_{2}(x)\\ \Psi_{3}(x)\end{array}\right].$$

Let $\{C_{x}\}_{x\in\mathbb{Z}\cup\{\pm\infty\}}$ be a sequence of $3\times 3$ unitary matrices, which is written as below:

$$C_{x}=\left[\begin{array}[]{ c c c }a_{x}^{(1,1)}&a_{x}^{(1,2)}&a_{x}^{(1,3)}\\ a_{x}^{(2,1)}&a_{x}^{(2,2)}&a_{x}^{(2,3)}\\ a_{x}^{(3,1)}&a_{x}^{(3,2)}&a_{x}^{(3,3)}\end{array}\right].$$

Here, we assume that $\left|a_{x}^{(2,2)}\right|\neq 1$. The coin operator is then defined using the coin matrices as:

$$(C\Psi)(x)=C_{x}\Psi(x).$$

Next, we define a shift operator $S$ that moves $\Psi_{1}(x)$ to the left and $\Psi_{3}(x)$ to the right.

$$\ (S\Psi)(x)=\left[\begin{array}[]{ c }\Psi_{1}(x+1)\\ \Psi_{2}(x)\\ \Psi_{3}(x-1)\end{array}\right].$$

Finally, the time evolution of the QW is determined by the following unitary operator

$$U=SC.$$

We impose a condition essential for considering eigenvalue analysis with our method.

$$C_{x}=\begin{cases}C_{\infty},&x\in[x_{+},\infty),\\ C_{x},&x\in(x_{-},x_{+}),\\ C_{-\infty},&x\in(-\infty,x_{-}],\end{cases}$$

where $x_{+}>0,x_{-}<0$.  We say the model with this condition as two-phase QW with finite number of defects. As a remark, we can easily replace $C_{\pm\infty}$ with periodic coin matrices by following the previous research [29]. However we opt not to explore this generalization as it leads to notably more complex notation. For an initial state $\Psi_{0}\in\mathcal{H}\left(\|\Psi_{0}\|_{\mathcal{H}}^{2}=1\right)$, the finding probability of a walker in position $x$ at time $t\in\mathbb{Z}_{\geq 0}$ is defined by

$$\mu_{t}^{(\Psi_{0})}(x)=\left\|\left(U^{t}\Psi_{0}\right)(x)\right\|_{\mathbb{C}^{3}}^{2},$$

where $\mathbb{Z}_{\geq 0}$ is the set of non-negative integers. We say that the $\mathrm{QW}$ exhibits localization if there exists a position $x_{0}\in\mathbb{Z}$ and an initial state $\Psi_{0}\in\mathcal{H}$ satisfying $\lim\sup_{t\rightarrow\infty}\mu_{t}^{(\Psi_{0})}(x_{0})>0$. It is known that the QW exhibits localization if and only if there exists an eigenvalue of $U$, that is, there exists $\lambda\in[0,2\pi)$ and $\Psi\in\mathcal{H}\setminus\{\mathbf{0}\}$ such that

$$U\Psi=e^{i\lambda}\Psi.$$

Let $\sigma_{p}(U)$ hereafter denote the set of eigenvalues of $U$.

The eigenvalue equation $U\Psi=e^{i\lambda}\Psi$ can be rewritten as the following system of equations:

	$\displaystyle e^{i(\lambda-\Delta_{x})}\Psi_{1}(x-1)=A_{x}(\lambda)\Psi_{1}(x)+B_{x}(\lambda)\Psi_{3}(x),$		(1)
	$\displaystyle e^{i(\lambda-\Delta_{x})}\Psi_{3}(x+1)=C_{x}(\lambda)\Psi_{1}(x)+D_{x}(\lambda)\Psi_{3}(x),$		(2)
	$\displaystyle\Psi_{2}(x)=E_{x}(\lambda)\Psi_{1}(x)+F_{x}(\lambda)\Psi_{3}(x).$		(3)

where

		$\displaystyle A_{x}(\lambda)=a_{x}^{(1,1)}+\frac{a_{x}^{(1,2)}a_{x}^{(2,1)}}{e^{i\lambda}-a_{x}^{(2,2)}},\ B_{x}(\lambda)=a_{x}^{(1,3)}+\frac{a_{x}^{(1,2)}a_{x}^{(2,3)}}{e^{i\lambda}-a_{x}^{(2,2)}},$
		$\displaystyle C_{x}(\lambda)=a_{x}^{(3,1)}+\frac{a_{x}^{(3,2)}a_{x}^{(2,1)}}{e^{i\lambda}-a_{x}^{(2,2)}},\ D_{x}(\lambda)=a_{x}^{(3,3)}+\frac{a_{x}^{(3,2)}a_{x}^{(2,3)}}{e^{i\lambda}-a_{x}^{(2,2)}},$
		$\displaystyle E_{x}(\lambda)=\frac{a_{x}^{(2,1)}}{e^{i\lambda}-a_{x}^{(2,2)}},\ F_{x}(\lambda)=\frac{a_{x}^{(2,3)}}{e^{i\lambda}-a_{x}^{(2,2)}}.$

Here, we define transfer matrices $T_{x}(\lambda)$ by

$$T_{x}(\lambda)=\frac{1}{A_{x}(\lambda)}\left[\begin{array}[]{ c c }e^{i\lambda}&-B_{x}(\lambda)\\ C_{x}(\lambda)&-e^{-i\lambda}(B_{x}(\lambda)C_{x}(\lambda)-A_{x}(\lambda)D_{x}(\lambda))\end{array}\right],$$

then equations (1), (2) can be expressed as

$$\left[\begin{array}[]{ c }\Psi_{1}(x)\\ \Psi_{3}(x+1)\end{array}\right]=T_{x}(\lambda)\left[\begin{array}[]{ c }\Psi_{1}(x-1)\\ \Psi_{3}(x)\end{array}\right].$$

From the unitarity of the coin matrix, we can write

$$C_{x}=e^{i\Delta_{x}}\left[\begin{array}[]{ c c c }\overline{\det C_{x}^{(1,1)}}&-\overline{\det C_{x}^{(1,2)}}&\overline{\det C_{x}^{(1,3)}}\\ -\overline{\det C_{x}^{(2,1)}}&\overline{\det C_{x}^{(2,2)}}&-\overline{\det C_{x}^{(2,3)}}\\ \overline{\det C_{x}^{(3,1)}}&-\overline{\det C_{x}^{(3,2)}}&\overline{\det C_{x}^{(3,3)}}\end{array}\right],$$

where $e^{i\Delta_{x}}=\det C_{x}$ and  $C_{x}^{(i,j)}$ denote the $2\times 2$ submatrix obtained by excluding the $i$-th row and $j$-th column of $C_{x}$. Using the above expression, we can simplify each component of the transfer matrix as:

		$\displaystyle A_{x}(\lambda)=\frac{a_{x}^{(1,1)}e^{i\lambda}-e^{i\Delta_{x}}\overline{a_{x}^{(3,3)}}}{e^{i\lambda}-a_{x}^{(2,2)}},\ B_{x}(\lambda)=\frac{a_{x}^{(1,3)}e^{i\lambda}+e^{i\Delta_{x}}\overline{a_{x}^{(3,1)}}}{e^{i\lambda}-a_{x}^{(2,2)}},$
		$\displaystyle C_{x}(\lambda)=\frac{a_{x}^{(3,1)}e^{i\lambda}+e^{i\Delta_{x}}\overline{a_{x}^{(1,3)}}}{e^{i\lambda}-a_{x}^{(2,2)}},\ D_{x}(\lambda)=\frac{a_{x}^{(3,3)}e^{i\lambda}-e^{i\Delta_{x}}\overline{a_{x}^{(1,1)}}}{e^{i\lambda}-a_{x}^{(2,2)}}$

and we can further calculate

$$A_{x}(\lambda)D_{x}(\lambda)-B_{x}(\lambda)C_{x}(\lambda)=\frac{\det C_{x}^{(2,2)}e^{i\lambda}-e^{i\Delta_{x}}}{e^{i\lambda}-a_{x}^{(2,2)}}=-e^{i(\lambda+\Delta_{x})}\frac{e^{-i\lambda}-\overline{a_{x}^{(2,2)}}}{e^{i\lambda}-a_{x}^{(2,2)}}.$$

Hence, we can simplify the expression of the transfer matrix as

$$T_{x}(\lambda)=\frac{1}{a_{x}^{(1,1)}e^{i\lambda}-e^{i\Delta_{x}}\overline{a_{x}^{(3,3)}}}\begin{bmatrix}e^{i\lambda}\left(e^{i\lambda}-a_{x}^{(2,2)}\right)&-a_{x}^{(1,3)}e^{i\lambda}-e^{i\Delta_{x}}\overline{a_{x}^{(3,1)}}\\ a_{x}^{(3,1)}e^{i\lambda}+e^{i\Delta_{x}}\overline{a_{x}^{(1,3)}}&-e^{i\Delta_{x}}\left(e^{-i\lambda}-\overline{a_{x}^{(2,2)}}\right)\end{bmatrix}.$$

Note that, when $A_{x}(\lambda)=0$, we cannot construct a transfer matrix. Therefore, we must treat this case separately. Let us define

$$\Lambda_{0}=\left\{e^{i\lambda}\in\mathbb{C}\mid\exists x\in\mathbb{Z}\cup\{\pm\infty\},\ A_{x}(\lambda)=0\right\}$$

and we consider the case $e^{i\lambda}\notin\Lambda_{0}$ first. For simplification, we may write $T_{x}(\lambda)$ as $T_{x}$ henceforward. Let $e^{i\lambda}\notin\Lambda_{0}$ and $\varphi\in\mathbb{C}^{2}$, we define $\tilde{\Psi}_{\varphi}:\mathbb{Z}\rightarrow\mathbb{C}^{2}$ as follows:

$$\tilde{\Psi}_{\varphi}(x)=\begin{cases}T_{\infty}^{x-x_{+}}T_{+}\varphi,&x_{+}\leq x,\\ T_{x-1}\cdots T_{0}\varphi,&0<x<x_{+},\\ \varphi,&x=0,\\ T_{x}^{-1}\cdots T_{-1}^{-1}\varphi,&x_{-}<x<0,\\ T_{-\infty}^{x-x_{-}}T_{-}\varphi,&x\leq x_{-},\end{cases}$$

(4)

where $T_{+}=T_{x_{+}-1}\cdots T_{0},T_{-}=T_{x_{-}}^{-1}\cdots T_{-1}^{-1}$ and $T_{\pm\infty}=T_{x_{\pm}}$. Let $V_{\lambda}$ be a set of generalized eigenvectors and $W_{\lambda}$ be a set of reduced vectors $\tilde{\Psi}_{\varphi}$ defined by (4):

		$\displaystyle V_{\lambda}=\left\{\Psi:\mathbb{Z}\rightarrow\mathbb{C}^{3}\mid U\Psi=e^{i\lambda}\Psi\right\},$
		$\displaystyle W_{\lambda}=\left\{\tilde{\Psi}_{\varphi}\mid\varphi\in\mathbb{C}^{2}\right\}.$

For $e^{i\lambda}\notin\Lambda_{0}$, we define a bijective map $\iota:V_{\lambda}\rightarrow W_{\lambda}$ by

$$(\iota\Psi)(x)=\left[\begin{array}[]{ c }\Psi_{1}(x-1)\\ \Psi_{3}(x)\end{array}\right].$$

Here, the inverse of $\iota$ is given as

$$\left(\iota^{-1}\tilde{\Psi}\right)(x)=\left[\begin{array}[]{ c }\tilde{\Psi}_{1}(x+1)\\ E_{x}(\lambda)\tilde{\Psi}_{1}(x+1)+F_{x}(\lambda)\tilde{\Psi}_{2}(x)\\ \tilde{\Psi}_{2}(x)\end{array}\right]$$

for $\tilde{\Psi}\in W_{\lambda},\ \tilde{\Psi}(x)=\left[\begin{array}[]{ c c }\tilde{\Psi}_{1}(x)&\tilde{\Psi}_{2}(x)\end{array}\right]^{t}$. Thus, from the definition, $\Psi\in V_{\lambda}$ if and only if there exists $\tilde{\Psi}\in W_{\lambda}$ such that $\Psi=\iota^{-1}\tilde{\Psi}$. We can also see that $\iota^{-1}\tilde{\Psi}\in\mathcal{H}\setminus\{\mathbf{0}\}$ is equivalent to $\tilde{\Psi}\in\ell^{2}\left(\mathbb{Z};\mathbb{C}^{2}\right)\setminus\{\mathbf{0}\}$.

The subsequent property is pivotal in formulating the method for the eigenvalue problem.

Thus, we can convince that the following two eigenvalues  $\zeta_{x}^{+}$, $\zeta_{x}^{-}$ of the transfer matrix satisfies $|\zeta_{x}^{+}||\zeta_{x}^{-}|=1.$

$$\zeta_{x}^{\pm}=\frac{\mathrm{tr}(T_{x})\pm\sqrt{\mathrm{tr}(T_{x})^{2}-4\det T_{x}}}{2}.$$

In this paper, we define the square root of the complex number $a=|a|e^{i\theta},\ \theta\in[0,2\pi)$ as $\sqrt{a}=\sqrt{|a|}e^{i\frac{\theta}{2}}$. Consequently, we designate $\zeta_{x}^{<}$ as one of $\zeta_{x}^{\pm}$ such that $\left|\zeta_{x}^{<}\right|\leq 1$, and $\zeta_{x}^{>}$ such that $\left|\zeta_{x}^{>}\right|\geq 1.$ Additionally, we introduce $\ket{v_{x}^{>}}$ and $\ket{v_{x}^{<}}$ as normalized eigenvector of $T_{x}$ corresponding to $\zeta_{x}^{>}$ and $\zeta_{x}^{<}$, respectively.

Let

$$\Lambda=\left\{e^{i\lambda}\in\mathbb{C}\ \middle|\ \left|\zeta_{\pm\infty}^{>}\right|\neq\left|\zeta_{\pm\infty}^{<}\right|\right\}.$$

With this, we present the following theorem:

Let $T_{+}T_{-}^{-1}\ket{v_{-\infty}^{>}}$ be represented as $\begin{bmatrix}u_{1}&u_{2}\end{bmatrix}$, and let $\ket{v_{\infty}^{<}}$ be given by $\begin{bmatrix}v_{1}&v_{2}\end{bmatrix}$. Define

$$\chi(\lambda)=u_{1}v_{2}-u_{2}v_{1}.$$

Then, we can rewrite Theorem 1 with equation $\chi(\lambda)$.

From Theorem 1, for $e^{i\lambda}\in\sigma_{p}(U)\setminus\Lambda_{0}$, the eigenvector $\Psi\in\ker(U-e^{i\lambda})\setminus\{\mathbf{0}\}$ is given by $\Psi=\iota^{-1}\tilde{\Psi}$, where $\tilde{\Psi}\in W_{\lambda}\setminus\{\mathbf{0}\}$ becomes

$\displaystyle\tilde{\Psi}(x)=\begin{cases}(\zeta_{\infty}^{<})^{x-x_{+}}T_{+}\varphi,&x_{+}\leq x,\\ T_{x-1}\cdots T_{0}\varphi,&0<x<x_{+},\\ \varphi,&x=0,\\ T_{x}^{-1}\cdots T_{-1}^{-1}\varphi,&x_{-}<x<0,\\ (\zeta_{-\infty}^{>})^{x-x_{-}}T_{-}\varphi,&x\leq x_{-},\end{cases}$

with $\varphi\in\ker\left(\left(T_{\infty}-\zeta_{\infty}^{<}\right)T_{+}\right)\cap\ker\left(\left(T_{-\infty}-\zeta_{-\infty}^{>}\right)T_{-}\right)\setminus\{\mathbf{0}\}.$ Furthermore, it is straightforward to verify that $\dim\ker(U-e^{i\lambda})=1$.

If $A_{x}=0$ for some $x\in\mathbb{Z}$, we cannot construct a transfer matrix at $x$. Therefore, we have to deal with this case separately.

As a consequence of the last proposition, we obtain the following prposition.

We get the following corollary, which was also proved in [30].