The spectra of the unitary marix of a 2-tessellable staggered quantum walk on a graph

TL;DR

This paper derives explicit formulas for the spectra of the unitary matrices in 2-tessellable staggered quantum walks and Szegedy walks on graphs, with applications to quantum search algorithms.

Contribution

It provides new spectral formulas for these quantum walk models, enhancing understanding of their dynamics and applications in quantum algorithms.

Findings

Derived spectra formulas for 2-tessellable SQWs

Provided spectra formulas for Szegedy matrices on bipartite graphs

Applied results to quantum search characteristic polynomials

Abstract

Recently, the staggered quantum walk (SQW) on a graph is discussed as a generalization of coined quantum walks on graphs and Szegedy walks. We present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so directly give its spectra. Furthermore, we present a formula for the Szegedy matrix of a bipartite graph by the same method, and so give its spectra. As an application, we present a formula for the characteristic polynomial of the modified Szegedy matrix in the quantum search problem on a graph, and give its spectra.

Equations

$$U_{ef}=\left\{\begin{array}[]{ll}2/d_{t(f)}(=2/d_{o(e)})&\mbox{if $t(f)=o(e)$ and $f\neq e^{-1}$, }\\ 2/d_{t(f)}-1&\mbox{if $f=e^{-1}$, }\\ 0&\mbox{otherwise.}\end{array}\right.$$

$$F^{+}_{ij}=\left\{\begin{array}[]{ll}1&\mbox{if $F_{ij}>0$, }\\ 0&\mbox{otherwise.}\end{array}\right.$$

$$T_{uv}=\left\{\begin{array}[]{ll}1/(\deg{}_{G}u)&\mbox{if $(u,v)\in D(G)$, }\\ 0&\mbox{otherwise.}\end{array}\right.$$

$$\det (\lambda {\mathbf{I}}_{2m}-\mathbf{U})$$

$$=\frac{(\lambda {}^2-1{)}^{m-n}\det ((\lambda {}^2+1)\mathbf{D}-2\lambda \mathbf{A}(G))}{d_{v_1}\cdots d_{v_n}},$$

$$\lambda =\lambda {}_T\pm i\sqrt{1-\lambda {}_T^2},$$

$$\lambda =\frac{\lambda {}_A}{2}\pm i\sqrt{k-1-\lambda {}_A^2/4},$$

$$\sum _{X(e)=x}p(e)=\sum _{Y(e)=y}q(e)=1,\forall x\in X,\forall y\in Y,$$

$$|\varphi {}_x⟩=\sum _{X(e)=x}\sqrt{p(e)}|e⟩and|\psi {}_y⟩=\sum _{Y(e)=y}\sqrt{q(e)}|e⟩.$$

$${\mathbf{R}}_0=2\sum _{x\in X}|\varphi {}_x⟩⟨\varphi {}_x|-{\mathbf{I}}_{ϵ},{\mathbf{R}}_1=2\sum _{y\in Y}|\psi {}_y⟩⟨\psi {}_y|-{\mathbf{I}}_{ϵ}$$

$$\mathbf{W}={\mathbf{R}}_1{\mathbf{R}}_0.$$

$$\mathbf{V}=\sum _x|\psi {}_x^{+}⟩⟨\psi {}_x^{+}|-\sum _y|\psi {}_y^{-}⟩⟨\psi {}_y^{-}|,$$

$$\sum _x|\psi {}_x^{+}⟩⟨\psi {}_x^{+}|+\sum _y|\psi {}_y^{-}⟩⟨\psi {}_y^{-}|=\mathbf{I},$$

$$\mathbf{U}=2\sum _x|\psi {}_x^{+}⟩⟨\psi {}_x^{+}|-\mathbf{I}\cdots (\ast ).$$

$$\mathbf{U}={\mathbf{U}}_1{\mathbf{U}}_0,$$

$${\mathbf{U}}_0=2\sum _{k=1}^m|\alpha {}_k⟩⟨\alpha {}_k|-{\mathbf{I}}_{\nu },{\mathbf{U}}_1=2\sum _{l=1}^n|\beta {}_l⟩⟨\beta {}_l|-{\mathbf{I}}_{\nu },$$

$$|\alpha {}_k⟩=\sum _{k^{\prime }\in \alpha {}_k}{a}_{k{k}^{\prime }}|k^{\prime }⟩(1\le k\le m),|\beta {}_l⟩=\sum _{l^{\prime }\in \beta {}_l}{b}_{l{l}^{\prime }}|l^{\prime }⟩(1\le l\le n).$$

$$V(\alpha {}_1)\bigsqcup \cdots \bigsqcup V(\alpha {}_m)=V(\beta {}_1)\bigsqcup \cdots \bigsqcup V(\beta {}_n)=V(G)$$

$$(E(\alpha {}_1)\bigsqcup \cdots \bigsqcup E(\alpha {}_m))\cup (E(\beta {}_1)\bigsqcup \cdots \bigsqcup E(\beta {}_n))=E(G),$$

$$V(G_2)=V\bigsqcup V^{\prime },$$

$$E_G(u,v)\subset E(G)\iff E_{G_2}(u,v^{\prime })\subset E(G_2)$$

$$\{p(e)|e\in E_{G_2}(u,v^{\prime })\}=\{q(e)|e\in E_{G_2}(u^{\prime },v)\}$$

$$\sum _{V(e)=x}p(e)=\sum _{V^{\prime }(e)=y}q(e)=1,\forall x\in V,\forall y\in V^{\prime }.$$

$$(\mathbf{P}{)}_{u,v}=p_{uv}=\{\begin{array}{ll}{\sum }_{V(e)=u,V^{\prime }(e)=v}p(e)& \text{if u∈V, v∈V′, }\\ {\sum }_{V^{\prime }(e)=u,V(e)=v}q(e)& \text{if u∈V′, v∈V, }\\ 0& \text{otherwise.}\end{array}$$

$$M_2=M\cup \{u^{\prime }\mid u\in M\}.$$

$$V({\overrightarrow{G}}_2)$$

$$A({\overrightarrow{G}}_2)$$

$$p^{\prime }(e)$$

$$q^{\prime }(e)$$

$$({\mathbf{P}}^{\prime }{)}_{u,v}=p_{uv}^{\prime }=\{\begin{array}{ll}{\sum }_{V(e)=u,V^{\prime }(e)=v}{p}^{\prime }(e)& \text{if u∈V, v∈V′, }\\ {\sum }_{V^{\prime }(e)=u,V(e)=v}{q}^{\prime }(e)& \text{if u∈V′, v∈V, }\\ 0& \text{otherwise.}\end{array}$$

$${\psi }_0=\sum _{e\in E(G_2)}\sqrt{p(e)}|e⟩.$$

$$E_M:=E_2\cup F_2.$$

Full text

THE SPECTRA OF THE UNITARY MATRIX OF A 2-TESSELLABLE STAGGERED QUANTUM WALK ON A GRAPH

Norio Konno

Department of Applied Mathematics, Faculty of Engineering, Yokohama National University

Hodogaya, Yokohama 240-8501, Japan

e-mail: konno@ynu.ac.jp, Tel.: +81-45-339-4205, Fax: +81-45-339-4205

Iwao Sato

Oyama National College of Technology

Oyama, Tochigi 323-0806, Japan

e-mail: isato@oyama-ct.ac.jp. Tel.: +81-285-20-2100, Fax: +81-285-20-2880

Etsuo Segawa

Graduate School of Information Sciences, Tohoku University

Sendai 980-8579, Japan

Abstract. Recently, the staggered quantum walk (SQW) on a graph is discussed as a generalization of coined quantum walks on graphs and Szegedy walks. We present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so directly give its spectra. Furthermore, we present a formula for the Szegedy matrix of a bipartite graph by the same method, and so give its spectra. As an application, we present a formula for the characteristic polynomial of the modified Szegedy matrix in the quantum search problem on a graph, and give its spectra.

000 Abbr. title: The transition matrix of a quantum walk on a graph 000 *AMS 2000 subject classifications: * 60F05, 05C50, 15A15, 05C60 000 *PACS: * 03.67.Lx, 05.40.Fb, 02.50.Cw 000 *Keywords: * Quantum walk, Szegedy walk, staggered quantum walk

1 Introduction

As a quantum counterpart of the classical random walk, the quantum walk has recently attracted much attention for various fields. The review and book on quantum walks are Ambainis [3], Kempe [8], Kendon [11], Konno [12], Venegas-Andraca [25], Manouchehri and Wang [15], Portugal [18], examples.

Quantum walks of graphs were studied by many researchers. A discrete-time quantum walk on a line was proposed by Aharonov et al [1]. In [2], a discrete-time quantum walk on a regular graph was proposed. The Grover walk is a discrete-time quantum walk on a graph which originates from the Grover algorithm. The Grover algorithm which was introduced in [7] is a quantum search algorithm that performs quadratically faster than the best classical search algorithm. Using a different quantization procedure, Szegedy [24] proposed a new coinless discrete-time quantum walk, i.e., the Szegedy walk on a bipartite graph and provided a natural definition of quantum hitting time. Also, Szegedy developed quantum walk-based search algorithm, which can detect the presence of a marked vertex at a hitting time that is quadratically smaller than the classical average time on ergodic Markov chains. Portugal [19], [20], [21], defined the staggered quantum walk (SQW) on a graph as a generalization of coined quantum walks on graphs and Szegedy walks. In [19], [20], Portugal studied the relation between SQW and coined quantum walks, Szegedy walks. In [21], Portugal presented some properties of 2-tessellable SQW on graphs by using several results of the graph theory.

Spectra of various quantum walk on a graph were computed by many researchers. Related to graph isomorphism problems, Emms et al. [4] presented spectra of the Grover matrix (the time evolution matrix of the Grover walk) on a graph and those of the positive supports of the Grover matrix and its square. Konno and Sato [13] computed the characteristic polynomials for the Grover matrix and its positive supports of a graph by using determinant expressions for several graph zeta functions, and so directly gave their spectra. Godsil and Guo [6] gave new proofs of the results of Emms et al. [4].

In the quantum search problem, the notion of hitting time in classical Markov chains is generalized to quantum hitting time. Kempe [9] provided two definitions and proved that a quantum walker hits the opposite corner of an $n$-hypercube in time $O(n)$. Krovi and Brun [14] provided a definition of average hitting time that requires a partial measurement of the position of the walker at each step. Kempe and Portugal [10] discussed the relation between hitting times and the walker’s group velocity. Szegedy [24] gave a definition of quantum hitting time that is a natural generalization of the classical definition of hitting time. Magniez et al [16] extended Szegedy’s work to non-symmetric ergodic Markov chains. Recently, Santos and Portugal [23] calculated analytically Szegedy’s hitting time and the probability of finding a set of marked vertices on the complete graph.

The rest of the paper is organized as follows. Section 2 states some definitions and notation on graph theory, and gives the definitions of the Grover walk, the Szegedy walk, the staggered quantum walk (SQW) on a graph and a short review on the quantum search problem on a graph. In Sect. 3, we present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so give its spectra. In Sect. 4, we present a formula for the Szegedy matrix of a bipartite graph, and so give its spectra. In Sect. 6, we present a formula for the modified time evolution matrix of the duplication of the modified digraph which is appeared in the quantum search problem on a graph, and so give its spectra.

2 Definition of several quantum walks on a graph

2.1 Definitions and notation

Graphs treated here are finite. Let $G=(V(G),E(G))$ be a connected graph (possibly multiple edges and loops) with the set $V=V(G)$ of vertices and the set $E=E(G)$ of unoriented edges $uv$ joining two vertices $u$ and $v$. Two vertices $u$ and $v$ of $G$ are adjacent if there exits an edge $e$ joining $u$ and $v$ in $G$. Furthermore, two vertices $u$ and $v$ of $G$ are incident to $e$. The degree $\deg v=\deg{}_{G}\ v$ of a vertex $v$ of $G$ is the number of edges incident to $v$. For a natural number $k$, a graph $G$ is called $k$*-regular * if $\deg{}_{G}\ v=k$ for each vertex $v$ of $G$.

For $uv\in E(G)$, an arc $(u,v)$ is the oriented edge from $u$ to $v$. Set $D(G)=\{(u,v),(v,u)|uv\in E(G)\}$. For $e=(u,v)\in D(G)$, set $u=o(e)$ and $v=t(e)$. Furthermore, let $e^{-1}=(v,u)$ be the inverse of $e=(u,v)$. A path $P=(v_{1},v_{2},\ldots,v_{n+1})$ of length $n$ in $G$ is a sequence of $(n+1)$ vertices such that $v_{i}v_{i+1}\in E(G)$ for $i=1,\ldots,n$. Then $P$ is called a $(v_{1},v_{n+1})$-path. If $e_{i}=v_{i}v_{i+1}(1\leq i\leq n)$, then we write $P=(e_{1},\ldots e_{n})$.

A graph $G$ is called a complete if any two vertices of $G$ are adjacent. We denote the complete graph with $n$ vertices by $K_{n}$. Furthermore, a graph $G$ is called bipartite, denoted by $G=(V_{1},V_{2})$ if there exists a partition $V(G)=V_{1}\cup V_{2}$ of $V(G)$ such that the vertices in $V_{i}$ are mutually nonadjacent for $i=1,2$. The subsets $V_{1},V_{2}$ of $V(G)$ is called the bipartite set or the bipartition of $G$. A bipartite graph $G=(V_{1},V_{2})$ is called complete if any vertex of $V_{1}$ and any vertex of $V_{2}$ are adjacent. If $|V_{1}|=m$ and $|V_{2}|=n$, then we denote the complete bipartite woth bipartition $V_{1},V_{2}$ by $K_{m,n}$.

Next, we define two operations of a graph. Let $G$ be a connected graph. Then a subgraph $H$ of $G$ is called a clique if $H$ is a complete subgraph of $G$. The clique graph $K(G)$ of $G$ has its vertex set the maximal cliques of $G$, with two vertices adjacent whenever they have some vertex of $G$ in common. Furthermore, the line graph $L(G)$ of $G$ has its vertex set the edges of $G$, with two vertices adjacent whenever they have some vertex of $G$ in common.

2.2 The Grover walk on a graph

A discrete-time quantum walk is a quantum analog of the classical random walk on a graph whose state vector is governed by a matrix called the transition matrix. Let $G$ be a connected graph with $n$ vertices and $m$ edges, $V(G)=\{v_{1},\ldots,v_{n}\}$ and $D(G)=\{e_{1},\ldots,e_{m},e^{-1}_{1},\ldots,e^{-1}_{m}\}$. Set $d_{j}=d_{u_{j}}=\deg v_{j}$ for $i=1,\ldots,n$. The transition matrix ${\bf U}={\bf U}(G)=(U_{ef})_{e,f\in D(G)}$ of $G$ is defined by

[TABLE]

The matrix ${\bf U}$ is called the Grover matrix of $G$.

We introduce the positive support $\>{\bf F}^{+}=(F^{+}_{ij})$ of a real matrix ${\bf F}=(F_{ij})$ as follows:

[TABLE]

Let $G$ be a connected graph. If the degree of each vertex of $G$ is not less than 2, i.e., $\delta(G)\geq 2$, then $G$ is called an md2 graph.

The transition matrix of a discrete-time quantum walk in a graph is closely related to the Ihara zeta function of a graph. We stare a relationship between the discrete-time quantum walk and the Ihara zeta function of a graph by Ren et al. [22].

Konno and Sato [13] obtained the following formula of the characteristic polynomial of ${\bf U}$ by using the determinant expression for the second weighted zeta function of a graph.

Let $G$ be a connected graph with $n$ vertices and $m$ edges. Then the $n\times n$ matrix ${\bf T}(G)=(T_{uv})_{u,v\in V(G)}$ is given as follows:

[TABLE]

Note that the matrix ${\bf T}(G)$ is the transition matrix of the simple random walk on $G$.

Theorem 2.1 (Konno and Sato [13])

Let $G$ be a connected graph with $n$ vertices $v_{1},\ldots,v_{n}$ and $m$ edges. Then, for the transition matrix ${\bf U}$ of $G$, we have

[TABLE]

where ${\bf A}(G)$ is the adjacency matrix of $G$, and ${\bf D}=(d_{uv})$ is the diagonal matrix given by $d_{uu}=\deg u\ (u\in V(G))$.

From this Theorem, the spectra of the Grover matrix on a graph is obtained by means of those of ${\bf T}(G)$ (see [4]). Let $Spec({\bf F})$ be the spectra of a square matrix ${\bf F}$ .

Corollary 2.2 (Emms, Hancock, Severini and Wilson [4])

Let $G$ be a connected graph with $n$ vertices and $m$ edges. The transition matrix ${\bf U}$ has $2n$ eigenvalues of the form

[TABLE]

where $\lambda{}_{T}$ is an eigenvalue of the matrix ${\bf T}(G)$. The remaining $2(m-n)$ eigenvalues of ${\bf U}$ are $\pm 1$ with equal multiplicities.

Emms et al. [4] determined the spectra of the transition matrix ${\bf U}$ by examining the elements of the transition matrix of a graph and using the properties of the eigenvector of a matrix. And now, we could explicitly obtain the spectra of the transition matrix ${\bf U}$ from its characteristic polynomial.

Next, we state about the positive support of the transition matrix of a graph.

Emms et al [4] expressed the spectra of the positive support ${\bf U}^{+}$ of the transition matrix of a regular graph $G$ by means of those of the adjacency matrix ${\bf A}(G)$ of $G$.

Theorem 2.3 (Emms, Hancock, Severini and Wilson [4])

Let $G$ be a connected $k$-regular graph with $n$ vertices and $m$ edges, and $\delta(G)\geq 2$. The positive support ${\bf U}^{+}$ has $2n$ eigenvalues of the form

[TABLE]

where $\lambda{}_{A}$ is an eigenvalue of the matrix ${\bf A}(G)$. The remaining $2(m-n)$ eigenvalues of ${\bf U}^{+}$ are $\pm 1$ with equal multiplicities.

Godsil and Guo [6] presented a new proof of Theorem by using linear algebraic technique.

Konno and Sato [13] obtained the following formula of the characteristic polynomial of ${\bf U}^{+}$ by using the determinant expression for the Ihara zeta function of a graph, and directly presented the spectra of the positive support ${\bf U}^{+}$ of the transition matrix of a regular graph $G$.

2.3 The Szegedy quantum walk on a bipartite graph

Let $G=(X\sqcup Y,E)$ be a connected bipartite graph with partite set $X$ and $Y$. Moreover, set $|V(G)|=\nu$, $|E|=|E(G)|=\epsilon$, $|X|=m$ and $|Y|=n$. Then we consider the Hilbert space ${\cal H}=\ell{}^{2}(E)=span\{|e\rangle\mid e\in E\}$. Let $p:E\rightarrow[0,1]$ and $q:E\rightarrow[0,1]$ be the functions such that

[TABLE]

where $X(e)$ and $Y(e)$ are the vertex of $e$ belonging to $X$ and $Y$, respectively.

For each $x\in X$ and $y\in Y$, let

[TABLE]

From these vectors, we construct two $\epsilon\times\epsilon$ matrices ${\bf R}_{0}$ and ${\bf R}_{1}$ as follows:

[TABLE]

Furthermore, we define an $\epsilon\times\epsilon$ matrix ${\bf W}$ as follows:

[TABLE]

Note that two matrices ${\bf R}_{0}$ and ${\bf R}_{1}$ are unitary, and ${\bf R}^{2}_{0}={\bf R}^{2}_{1}={\bf I}_{mn}$.

The quantum walk on $G$ with ${\bf W}$ as a time evolution matrix is called Szegedy walk on $G$, and the matrix ${\bf W}$ is called the Szegedy matrix of $G$.

2.4 The staggered quantum walk on a graph

Let $G$ be a connected graph with $\nu$ vertices and $\epsilon$ edges. Furthermore, let ${\cal H}^{\nu}$ be the Hilbert space generated by the vertices of $G$. We take a standard basis as $\{|u\rangle\;\mid\;u\in V\}$. In general, a unitary and Hermitian operator ${\bf V}$ on ${\cal H}^{\nu}$ can be written by

[TABLE]

where the set of vectors $|\psi{}^{+}_{x}\rangle$ is a normal orthogonal basis of $(+1)$-eigenspace, and the set of vectors $|\psi{}^{-}_{x}\rangle$ is a normal orthogonal basis of $(-1)$-eigenspace. Since

[TABLE]

we obtain

[TABLE]

A unitary and Hermitian matrix ${\bf V}$ in ${\cal H}^{\nu}$ given by (*) is called an orthogonal reflection of $G$ if the set of the orthogonal set of $(+1)$-eigenvectors $\{|\psi{}^{+}_{x}\rangle\}_{x}$ obeying the following properties:

If the $i$-th entry of $|\psi{}^{+}_{x}\rangle$ for a fixed $x$ is nonzero, the $i$-th entry of the other $(+1)$-eigenvectors are zero, that is, if $\langle i|\psi_{x}\rangle\neq 0$, then $\langle i|\psi_{x^{\prime}}\rangle=0$ for any $x^{\prime}\neq x$; 2. 2.

The vector $\sum_{x}|\psi{}^{+}_{x}\rangle$ has no zero entries.

Next, a polygon of a graph $G$ induced by a vector $|\psi\rangle\in{\cal H}^{\nu}$ is a clique. That is, two vertices of $G$ are adjacent if the corresponding entries of $|\psi\rangle$ in the basis associated with $G$ are nonzero. Thus if $\langle u|\psi\rangle\neq 0$ and $\langle v|\psi\rangle\neq 0$, then $u$ is connected to $v$. A vertex belongs to the polygon if and only if it corresponding entry in $|\psi\rangle$ is nonzero. An edge belongs to the polygon if and only if the polygon contains the endpoints of the edge.

A tessellation induced by an orthogonal reflection ${\bf V}$ of $G$ is the union of the polygons induced by the $(+1)$-eigenvectors $\{|\psi{}^{+}_{x}\rangle\}_{x}$ of ${\bf V}$ described in the above. The staggered quantum walk(SQW) on $G$ associated with the Hilbert space ${\cal H}^{\nu}$ is driven by

[TABLE]

where ${\bf U}_{0}$ and ${\bf U}_{1}$ are orthogonal reflections of $G$. The union of the tessellations $\alpha$ and $\beta$ by ${\bf U}_{0}$ and ${\bf U}_{1}$ must cover the edges of $G$. Furthermore, set $\alpha=\{\alpha{}_{1},\ldots,\alpha{}_{m}\}$ and $\beta=\{\beta{}_{1},\ldots,\beta{}_{n}\}$. Then ${\bf U}_{0}$ and ${\bf U}_{1}$ are given as follows:

[TABLE]

where

[TABLE]

A graph $G$ is 2-tessellable if the following conditions holds:

[TABLE]

and

[TABLE]

where ${\bf U}={\bf U}_{1}{\bf U}_{0}$ is the unitary matrix of a SQW on $G$, and $\alpha=\{\alpha{}_{1},\ldots,\alpha{}_{m}\}$ and $\beta=\{\beta{}_{1},\ldots,\beta{}_{n}\}$ are tessellations of ${\bf U}$ corresponding to ${\bf U}_{0}$ and ${\bf U}_{1}$, respectively.

2.5 The quantum search problem on a graph

Let $G=(V,E)$ be a connected non-bipartite graph with $n$ vertices and $\epsilon$ edges which may have multiple edges and self loops. Let $E_{H}(u,v)$ be the subset of the edge set of a graph $H$ connecting between vertices $u$ and $v$. It holds $\sqcup_{u,v,\in V(H)}E_{H}(u,v)=E(H)$, where “$\sqcup$” means the disjoint union. We want to set the quantum search of an element of $M\subset V$ by the Szegedy walk. The Szegedy walk is defined by a bipartite graph. To this end, we construct the duplication of $G$. The duplication $G_{2}$ of $G$ is defined as follows: The duplication graph $G_{2}$ of $G$ is defined as follows.

[TABLE]

where $v^{\prime}$ is the copy of $v\in V$, therefore $V^{\prime}=\{v^{\prime}:v\in V\}$. The edge set of $E(G_{2})$ is denoted by

[TABLE]

The end vertex of $e\in E(G_{2})$ included in $V$ is denoted by $V(e)$, and one included in $V^{\prime}$ is denoted by $V^{\prime}(e)$. We consider two functions $p:E(G_{2})\rightarrow[0,1]$ and $q:E(G_{2})\rightarrow[0,1]$ be the functions such that

[TABLE]

where

[TABLE]

The $2n\times 2n$ stochastic matrix ${\bf P}$ is denoted by

[TABLE]

Let $V(G)=\{v_{1},\ldots,v_{n}\}$ and $M=\{v_{n-m+1},\ldots,v_{n}\}$. Then define the modified digraph $\vec{G}$ from $G$ as follows: The modified digraph $\vec{G}$ with respect to $M$ is obtained from the symmetric digraph $D_{G}$ by converting all arcs leaving from the marked vertices into loops. In the duplication $G_{2}$, the set $M_{2}$ of marked vertices is

[TABLE]

The modified bipartite digraph $\vec{G}_{2}$ is obtained from the symmetric digraph of $G_{2}$ by deleting all arcs leaving from the marked vertices of $G_{2}$, but keeping the incoming arcs to the marked vertices of $G_{2}$ and all other arcs unchanged. Moreover, we add new $2m=2|M|$ arcs $(u,u^{\prime}),(u^{\prime},u)$ for $u\in M$. Then the modified bipartite digraph $\vec{G}_{2}$ is obtained by taking the duplication of $\vec{G}$. More precisely, let $A(G_{2})=D(G_{2})$ be the set of symmetric arcs naturally induced by $E(G_{2})$, then

[TABLE]

Here $M^{\prime}\subset V^{\prime}$ is the copy of $M$. We put the first arcset in RHS by $A_{2}$, and the second one by $N_{2}$. The modified graph $\vec{G}_{2}$ keeps the bipartiteness with $V$ and $V^{\prime}$. Thus once a random walker steps in $M_{2}$, then she will be trapped in $M_{2}$ forever.

We want to induce the Szegedy walk from this absorption picture into $M_{2}$ of $\vec{G}_{2}$. The Szegedy is denoted by non-directed edges of the bipartite graph. So we consider the support of $A(\overrightarrow{G}_{2})$ by $E_{2}:=[A(\overrightarrow{G}_{2})]:=\{[a]\;|\;a\in A(\overrightarrow{G}_{2})\}$. Here $[a]$ is the edge obtained by removing the direction of the arc $a$. Thus $E_{2}=[A_{2}]\sqcup[N_{2}]$, and remark that $[N_{2}]$ describes the set of the matching between $m$ and $m^{\prime}$ for $m\in M$. Taking the following modification to $p$ and $q$, the above absorption picture of a classical walk is preserved by the following random walk as follows. For $e\in E_{2}$,

[TABLE]

The modified $2n\times 2n$ stochastic matrix ${\bf P}^{\prime}$ is given by changing $p$ and $q$ to $p^{\prime}$ and $q^{\prime}$ as follows:

[TABLE]

If there exists a marked element connecting to another marked element in $G$, then such an edge is omitted by the procedure of the deformation to $\vec{G}_{2}$, thus $[A_{2}]\subset E(G_{2})$, on the other hand, otherwise, $[A_{2}]=E(G_{2})$. We set $F_{2}=\{e\in E(G_{2})\;|\;V(e),V^{\prime}(e)\in M_{2}\}$. Now we are considering a quantum search setting without any connected information about marked elements, so we want to set the initial state as a usual way,

[TABLE]

However in the above situation, that is, $F_{2}\neq\emptyset$, since an original edge of $G_{2}$ is omitted, we cannot define this initial state. So we expand the considering edge set

[TABLE]

We re-define $p^{\prime}$ and $q^{\prime}$ whose domain is changed to $E_{M}$: for every $e\in E_{M}$.

[TABLE]

Remark that the above “otherwise” in the definition of $p^{\prime}$ is equivalent to the situation of “$V(e)\in M$ and $V^{\prime}(e)\notin M^{\prime}$” or “$e\in F_{2}$”.

Now we are ready to give the setting of quantum search problem. Remark that $E_{M}=2\epsilon+m$. For each $x\in V$ and $y\in V^{\prime}$, let

[TABLE]

From these unit vectors, we construct two $(2\epsilon+m)\times(2\epsilon+m)$ matrices ${\bf R}^{\prime}_{0}$ and ${\bf R}^{\prime}_{1}$ as follows:

[TABLE]

Furthermore, we define an $(2\epsilon+m)\times(2\epsilon+m)$ matrix ${\bf W}^{\prime}$ as follows:

[TABLE]

Then ${\bf W}^{\prime}$ is the time evolution matrix of the modified Szegedy walk on $\ell^{2}(E_{M})$.

The initial condition of the quantum walk is

[TABLE]

Note that $|\psi(0)\rangle$ is defined using a random walk on $G$ determined by $p$, and it is invariant under the action of ${\bf W}={\bf R}_{1}{\bf R}_{0}$ associated with $G$(see [18]). We assume that $p_{uv^{\prime}}=p_{v^{\prime}u},u,v\in V(G)$ for the stochastic matrix ${\bf P}$. Then ${\bf P}$ is doubly stochastic. Let

[TABLE]

and

[TABLE]

Then the quantum hitting time $H_{P,M}$ of a quantum walk on $G$ is fined as the smallest number of steps $T$ such that

[TABLE]

where $n=|V(G)|$ and $m=|M|$. The quantum hitting time is evaluated by the square of the spectral gap of the $n\times n$ matrix ${\bf P}_{M}$:

[TABLE]

3 Key method

From now on, we will attempt to three cases of the characteristic polynomials of the time evolution; “a 2-tessellable staggered quantum matrix”, “Szegedy matrix ” and “modified Szegedy matrix of quantum search”. To this end, we provide the key lemma.

Theorem 3.1

Let ${\bf A}$ and ${\bf B}$ be $N\times s$ and $N\times t$ complex valued isometry matrices, that is,

[TABLE]

where ${}^{*}{\bf Y}$ is the conjugate and transpose of ${\bf Y}$. Putting ${\bf U}={\bf U}_{B}{\bf U}_{A}$ with ${\bf U}_{B}=(2{\bf B}\ {}^{*}{\bf B}-{\bf I}_{N})$ and ${\bf U}_{A}=(2{\bf A}\ {}^{*}{\bf A}-{\bf I}_{N})$, we have

[TABLE]

Proof: At first, we have

[TABLE]

Therefore once we can show the first equality, then changing the variables by $A\leftrightarrow B$ and $t\leftrightarrow s$, we have the second equality.

Now we will show the first equality.

[TABLE]

If ${\bf A}^{\prime}$ and ${\bf B}^{\prime}$are a $m\times n$ and $n\times m$ matrices, respectively, then we have

[TABLE]

Thus, we have

[TABLE]

Furthermore, we have

[TABLE]

Therefore, it follows that

[TABLE]

$\Box$

We put ${\bf T}_{BA}:={}^{*}{\bf B}{\bf A}$ and ${\bf T}_{AB}:={}^{*}{\bf A}{\bf B}$. Thus ${}^{*}{\bf T}_{BA}={\bf T}_{AB}$.

Lemma 3.2

For any eigenvalue $\lambda_{q}$ of ${\bf T}_{BA}{\bf T}_{AB}$,

[TABLE]

*Proof *. At first, let

[TABLE]

Then we have

[TABLE]

Thus,

[TABLE]

Since $\langle g,{\bf T}_{BA}{\bf T}_{AB}g\rangle\geq 0$ for every $g$, we have $0\leq=\lambda_{q}$ holds. Therefore $\mu\in[0,1]$ $\Box$

Remark 3.3

Let $s\geq t$. Then it holds

[TABLE]

where $\{0\}^{s-t}$ is the multi-set of $s-t$ 0. Thus $0\leq\lambda_{p}\leq 1$ for any $\lambda_{p}\in Spec(T_{AB}T_{BA})$.

Corollary 3.4

For the unitary matrix ${\bf U}={\bf U}_{B}{\bf U}_{A}$, we have

[TABLE]

*Proof *. Let $u=1/\lambda$. Then, by Theorem 3.1, we have

[TABLE]

and so,

[TABLE]

$\Box$

Corollary 3.5

Set $Spec({\bf T}_{BA}{\bf T}_{AB})=\{\lambda_{q,1},\dots,\lambda_{q,t}\}$ with $0\leq\lambda_{q,1}\leq\cdots\leq\lambda_{q,t}\leq 1$. Moreover the two solutions of

[TABLE]

is denoted by $\alpha_{j}^{(\pm)}$. Then $N$ eigenvalues of ${\bf U}$ are described as follows:

$|N-(s+t)|$-multiple eigenvalue: $1$; 2. 2.

$|t-s|$-multiple eigenvalue: $(-1)$; 3. 3.

$2(\mathrm{Min}\{t,N-s\}-\mathrm{Max}\{0,t-s\})$* eigenvalues:*

[TABLE]

Here an expression for $\alpha_{j}^{(\pm)}$ is

[TABLE]

Remark 3.6

It holds

[TABLE]

In particular,

If $t<s$, then $\lambda_{q,1}=\cdots=\lambda_{q,s-t}=0$. 2. 2.

If $N<t+s$, then $\lambda_{q,N-s+1}=\cdots=\lambda_{q,t}=1$.

Corollary 3.7

Set $Spec({\bf T}_{AB}{\bf T}_{BA})=\{\lambda_{p,1},\dots,\lambda_{p,s}\}$ with $0\leq\lambda_{p,1}\leq\cdots\leq\lambda_{p,s}\leq 1$. Moreover the two solutions of

[TABLE]

is denoted by $\beta_{j}^{(\pm)}$. Then $N$ eigenvalues of ${\bf U}$ are described as follows:

$|N-(s+t)|$-multiple eigenvalue: $1$; 2. 2.

$|t-s|$-multiple eigenvalue: $(-1)$; 3. 3.

$2(\mathrm{Min}\{s,N-t\}-\mathrm{Max}\{0,s-t\})$* eigenvalues:*

[TABLE]

Here an expression for $\beta_{j}^{(\pm)}$ is

[TABLE]

Remark 3.8

If $s<t$, then $\lambda_{p,1}=\cdots=\lambda_{p,t-s}=0$. 2. 2.

If $N<t+s$, then $\lambda_{p,N-t+1}=\cdots=\lambda_{p,s}=1$.

Once we show Corollary 3.5, then Corollary 3.7 automatically holds by Theorem 3.1. So in the following we give a proof of Corollary 3.5.

Proof of Corollary 3.5:

By Corollary 3.4, we can rewrite the characteristic polynomial of ${\bf U}$ by

[TABLE]

We put the two solution of $\lambda{}^{2}-2(2\lambda{}_{q,j}-1)\lambda+1=0$ by $\alpha_{j}^{(\pm)}$. Then

[TABLE]

Concerning that RHS is an $N$-th degree polynomial of $\lambda$, we consider the four cases with respect to the signes of $N-(s+t)$ and $s-t$.

$N-(s+t)\geq 0$, $s-t\geq 0$ case:

we directly obtain $(N-s-t)$-multiple eigenvalue $1$, $(s-t)$-multiple eigenvalue $-1$ and $2t$ eigenvalues $\alpha_{q,j}^{(\pm)}$ $(j=1,\dots,t)$. 2. 2.

$N-(s+t)\geq 0$, $s-t<0$ case:

Since $s-t<0$, $(\lambda+1)^{s-t}$ is a negative power term. To cancel down it, $\{(\lambda-\alpha_{j}^{(+)}),\;(\lambda-\alpha_{j}^{(-)})\}_{j=1}^{t}$ must contain $(t-s)$ terms of $(\lambda+1)$. Remark that if $\lambda=-1$, then $\lambda_{q,j}=0$ from the above quadratic equation. So $\lambda_{q,1}=\cdots=\lambda_{q,t-s}=0$. By the above consideration, the characteristic polynomial is expressed by

[TABLE]

Then we obtain $(N-s-t)$-multiple eigenvalue $1$, $(t-s)$-multiple eigenvalue $-1$ and $2s$ eigenvalues $\alpha_{q,j}^{(\pm)}$ $(j=t-s+1,\dots,t)$. 3. 3.

$N-(s+t)<0$, $s-t\geq 0$ case:

Since $N-(s+t)<0$, $(\lambda-1)^{N-(s+t)}$ is a negative power term. To cancel down it, $\{(\lambda-\alpha_{j}^{(+)}),\;(\lambda-\alpha_{j}^{(-)})\}_{j=1}^{t}$ must contain $(s+t)-N$ terms of $(\lambda-1)$. Remark that if $\lambda=1$, then $\lambda_{q,j}=1$ from the above quadratic equation. So $\lambda_{q,N-s+1}=\cdots=\lambda_{q,t}=1$. By the above consideration, the characteristic polynomial is expressed by

[TABLE]

Then we obtain $(s+t-N)$-multiple eigenvalue $1$, $(s-t)$-multiple eigenvalue $-1$ and $2(N-s)$ eigenvalues $\alpha_{q,j}^{(\pm)}$ $(j=1,\dots,N-s)$. 4. 4.

$N-(s+t)<0$, $s-t<0$ case:

Since $N-(s+t)<0$ and $s-t<0$, both $(\lambda-1)^{N-(s+t)}$ and $(\lambda+1)^{s-t}$ are negative power terms. To cancel down it, $\{(\lambda-\alpha_{j}^{(+)}),\;(\lambda-\alpha_{j}^{(-)})\}_{j=1}^{t}$ must contain $(s+t)-N$ terms of $(\lambda-1)$ and $t-s$ terms of $(\lambda+1)$. From the arguments of cases (2) and (3), we have $\lambda_{q,N-s+1}=\cdots=\lambda_{q,t}=1$ and $\lambda_{q,1}=\cdots=\lambda_{q,t-s}=0$. By the above consideration of the characteristic polynomial is expressed by

[TABLE]

Then we obtain $(s+t-N)$-multiple eigenvalue $1$, $(t-s)$-multiple eigenvalue $-1$ and $2(N-t)$ eigenvalues $\alpha_{q,j}^{(\pm)}$ $(j=t-s+1,\dots,N-s)$.

Compiling the four cases, we have the desired conclusion. $\Box$

4 The characteristic polynomial of the unitary matrix of a 2-tessellable staggered quantum matrix

Let $G$ be a connected graph with $\nu$ vertices and $\epsilon$ edges, and let ${\bf U}={\bf U}_{1}{\bf U}_{0}$ be the unitary matrix of a 2-tessellable SQW on $G$ such that both ${\bf U}_{0}$ and ${\bf U}_{1}$ are orthogonal reflections. Furthermore, let $\alpha$ and $\beta$ be tessellations of ${\bf U}$ corresponding to ${\bf U}_{0}$ and ${\bf U}_{1}$, respectively. Set $\alpha=\{\alpha{}_{1},\ldots,\alpha{}_{m}\}$ and $\beta=\{\beta{}_{1},\ldots,\beta{}_{n}\}$. Then we have

[TABLE]

[TABLE]

Now, let $X$ be a finite nonempty set and $S=\{S_{1},\ldots,S_{r}\}$ a family of subsets of $X$. Then the generalized intersection graph $\Omega(S)$ is defined as follows: $V(\Omega(S))=S=\{S_{1},\ldots,S_{r}\}$; $S_{i}$ and $S_{j}$ are joined by $|S_{i}\cap S_{j}|$ edges in $\Omega(S)$.

Peterson [17] gave a necessary and sufficient condition for a graph to be 2-tessellable.

Proposition 4.1 (Perterson)

A graph $G$ is 2-tessellable if and only if $G$ is the line graph of a bipartite graph.

Sketch of proof Let $G$ be a 2-tessellable graph with two tessellations $\alpha$ and $\beta$. Set $S=\alpha\cup\beta$ and $H=\Omega(S)$. Then $H$ is a bipartite multi graph with partite set $\alpha$ and $\beta$. Furthermore, we have $G=L(\Omega(S))$.

Conversely, it is clear that the line graph of a bipartite graph is 2-tessellable. Q.E.D.

By Proposition 4.1, we can rewrite $|\alpha{}_{k}\rangle$ and $|\beta{}_{l}\rangle$. Let $H=\Omega(\alpha\cup\beta)$ be a bipartite graph with bipartition $X=\{x_{1},\ldots,x_{m}\}$, $Y=\{y_{1},\ldots,y_{n}\}$ such that $G=L(H)$. Furthermore, we set $\alpha{}_{k}=N(x_{k})(1\leq k\leq m)$ and $\beta{}_{l}=N(y_{l})(1\leq l\leq n)$, where $N(x)=\{e\in E(H)\mid x\in e\}$. Then we can write

[TABLE]

where $a_{e}(or\ b_{f})$ corresponds to $a_{kk^{\prime}}(or\ b_{ll^{\prime}})$ if $k^{\prime}(or\ l^{\prime})\in V(G)$ corresponds to an edge $e(or\ f)\in E(H)$.

Now, we define an $m\times m$ matrix $\hat{{\bf A}}=(a_{xx^{\prime}})_{x,x^{\prime}\in X}$ as follows:

[TABLE]

Then we obtain the following formula for the unitary matrix of a SQW on a 2-tessellable graph.

Theorem 4.2

Let $G$ be a connected 2-tessellable graph with $\nu$ vertices and $\epsilon$ edges, and let ${\bf U}={\bf U}_{1}{\bf U}_{0}$ be the unitary matrix of a 2-tessellable SQW on $G$ such that both ${\bf U}_{0}$ and ${\bf U}_{1}$ are orthogonal reflections. Furthermore, let $\alpha$ and $\beta$ be tessellations of ${\bf U}$ corresponding to ${\bf U}_{0}$ and ${\bf U}_{1}$, respectively. Set $|\alpha|=m$ and $|\beta|=n$. Then, for the unitary matrix ${\bf U}={\bf U}_{1}{\bf U}_{0}$, we have

[TABLE]

*Proof *. Let $\alpha=\{\alpha{}_{1},\ldots,\alpha{}_{m}\}$ and $\beta=\{\beta{}_{1},\ldots,\beta{}_{n}\}$. Then we have

[TABLE]

[TABLE]

Furthermore, $H=\Omega(\alpha\cup\beta)$ is expressed as follows:

[TABLE]

[TABLE]

[TABLE]

where $N(x)=\{e\in E(H)\mid x\in e\},\ x\in V(H)$ and $d_{1}+\cdots+d_{m}=\overline{d}_{1}+\cdots+\overline{d}_{n}=\nu$.

We consider $\alpha{}_{k}=N(x_{k})(1\leq k\leq m)$ and $\beta{}_{l}=N(y_{l})(1\leq l\leq n)$. By Proposition 4.1, we can write

[TABLE]

[TABLE]

Now, let $x=x_{1}\in X$, $d=d_{x}=\deg x$ and $N(x)=\{e_{1},\ldots,e_{d}\}$. Set $\alpha{}_{x}=\alpha{}_{i}(x=x_{i})$. Then the submatrix of $|\alpha{}_{x}\rangle\langle\alpha{}_{x}|$ corresponding to the $e_{1},\ldots,e_{d}$ rows and the $e_{1},\ldots,e_{d}$ columns is we have

[TABLE]

where $E(H)=\{e_{1},\ldots,e_{d},\ldots\}$. Thus, the submatrix of ${\bf U}_{0}=2\sum^{m}_{i=1}|\alpha{}_{i}\rangle\langle\alpha{}_{i}|-{\bf I}_{\nu}$ corresponding to the $e_{1},\ldots,e_{d}$ rows and the $e_{1},\ldots,e_{d}$ columns is

[TABLE]

Let $y=y_{1}$, $d^{\prime}=d_{y}=\deg y$ and $N(y)=\{f_{1},\ldots,f_{d^{\prime}}\}$. Similarly to ${\bf U}_{0}$, the submatrix of ${\bf U}_{1}=2\sum^{n}_{j=1}|\beta{}_{j}\rangle\langle\beta{}_{j}|-{\bf I}_{\nu}$ corresponding to the $f_{1},\ldots,f_{d^{\prime}}$ rows and the $f_{1},\ldots,f_{d^{\prime}}$ columns is we have

[TABLE]

Now, let ${\bf K}=({\bf K}_{ex})$ e∈E(H);x∈X be the $\nu\times m$ matrix defined as follows:

[TABLE]

Furthermore, we define the $\nu\times n$ matrix ${\bf L}=({\bf L}_{ey})_{e\in E(H);y\in Y}$ as follows:

[TABLE]

Then we have

[TABLE]

Furthermore, since

[TABLE]

we have

[TABLE]

Therefore, by Theorem 3.1, it follows that

[TABLE]

But, we have

[TABLE]

Furthermore, we have

[TABLE]

Thus, for $x,x^{\prime}\in X$,

[TABLE]

Thus, we have

[TABLE]

Hence,

[TABLE]

$\Box$

By Theorem 4.2 and Corollary 3.4, we obtain the following.

Corollary 4.3

Let $G$ be a connected 2-tessellable graph with $\nu$ vertices and $\epsilon$ edges, and let ${\bf U}={\bf U}_{1}{\bf U}_{0}$ be the unitary matrix of a 2-tessellable SQW on $G$ such that both ${\bf U}_{0}$ and ${\bf U}_{1}$ are orthogonal reflections. Furthermore, let $\alpha$ and $\beta$ be tessellations of ${\bf U}$ corresponding to ${\bf U}_{0}$ and ${\bf U}_{1}$, respectively. Set $|\alpha|=m$ and $|\beta|=n$. Then, for the unitary matrix ${\bf U}={\bf U}_{1}{\bf U}_{0}$, we have

[TABLE]

By Corollary 3.7, we obtain the spectrum of ${\bf U}$.

Corollary 4.4

Let $G$ be a connected 2-tessellable graph with $\nu$ vertices and $\epsilon$ edges, and let ${\bf U}={\bf U}_{1}{\bf U}_{0}$ be the unitary matrix of a 2-tessellable SQW on $G$ such that both ${\bf U}_{0}$ and ${\bf U}_{1}$ are orthogonal reflections. Furthermore, let $\alpha$ and $\beta$ be tessellations of ${\bf U}$ corresponding to ${\bf U}_{0}$ and ${\bf U}_{1}$, respectively. Set $|\alpha|=m$ and $|\beta|=n$. Then the spectra of the unitary matrix ${\bf U}={\bf U}_{1}{\bf U}_{0}$ are given as follows: Let $0\leq\lambda_{p,1}\leq\cdots\leq\lambda_{p,m}$ be the eigenvalues of $\hat{{\bf A}}$.

$2(\mathrm{Max}\{n,\nu-m\}-\mathrm{Max}(0,n-m))$* eigenvalues:*

[TABLE] 2. 2.

$|\nu-m-n|$* eigenvalues: 1;* 3. 3.

$|n-m|$* eigenvalues: -1.*

5 The characteristic polynomial of the Szegedy matrix

We present a formula for the characteristic polynomial of the Szegedy matrix of a bipartite graph. Let $G=(X\sqcup Y,E)$ be a connected multi-bipartite graph with partite set $X$ and $Y$. Moreover, set $|V(G)|=\nu$, $|E|=|E(G)|=\epsilon$, $|X|=m$ and $|Y|=n$. Then we consider the Hilbert space ${\cal H}=\ell{}^{2}(E)=span\{|e\rangle\mid e\in E\}$. Let $p:E\rightarrow[0,1]$ and $q:E\rightarrow[0,1]$ be the functions such that

[TABLE]

where $X(e)$ and $Y(e)$ are the vertex of $e$ belonging to $X$ and $Y$, respectively.

Let ${\bf W}={\bf R}_{1}{\bf R}_{0}$ be a Szegedy matrix of $G$, where

[TABLE]

[TABLE]

Then we define an $m\times m$ matrix ${\bf A}_{p}=(a^{(p)}_{xx^{\prime}})_{x,x^{\prime}\in X}$ as follows:

[TABLE]

Note that

[TABLE]

Then the characteristic polynomial of the Szegedy matrix of a bipartite graph is given as follows.

Theorem 5.1

Let $G=(X\sqcup Y,E)$ and ${\bf W}$ be as the above. Then, for the Szegedy matrix ${\bf W}={\bf R}_{1}{\bf R}_{0}$, we have

[TABLE]

*Proof *. Let $X=\{x_{1},\ldots,x_{m}\}$ and $Y=\{y_{1},\ldots,y_{n}\}$. Let $x\in X$ and $y\in Y$. Then, let

[TABLE]

Now, let $x\in X$, $d=d_{x}=\deg x$ and $N(x)=\{e_{1},\ldots,e_{d}\}$. Moreover, set $e_{j}=xy_{k_{j}}$ and $p_{xj}=p(xy_{k_{j}})$ for $j=1,\ldots,d$. Then the submatrix of $|\phi{}_{x}\rangle\langle\phi{}_{x}|$ corresponding to the $e_{1},\ldots,e_{d}$ rows and the $e_{1},\ldots,e_{d}$ columns is

[TABLE]

Thus, the submatrix of ${\bf R}_{0}=2\sum_{x\in X}|\phi{}_{x}\rangle\langle\phi{}_{x}|-{\bf I}_{\epsilon}$ corresponding to the $e_{1},\ldots,e_{d}$ rows and the $e_{1},\ldots,e_{d}$ columns is

[TABLE]

Let $y\in Y$, $d^{\prime}=d_{y}=\deg y$ and $N(y)=\{f_{1},\ldots,f_{d^{\prime}}\}$. Moreover, set $f_{j}=yx_{k_{l}}$ and $q_{yl}=q(yx_{k_{l}})$ for $l=1,\ldots,d^{\prime}$. Similarly to ${\bf R}_{0}$, the submatrix of ${\bf R}_{1}=2\sum_{x\in X}|\psi{}_{y}\rangle\langle\psi{}_{y}|-{\bf I}_{\epsilon}$ corresponding to the $f_{1},\ldots,f_{d^{\prime}}$ rows and the $f_{1},\ldots,f_{d^{\prime}}$ columns is

[TABLE]

Now, let ${\bf K}=({\bf K}_{ex})$ e∈E(G);x∈X be the $\epsilon\times m$ matrix defined as follows:

[TABLE]

Furthermore, we define the $\epsilon\times n$ matrix ${\bf L}=({\bf L}_{ey})_{e\in E(G);y\in Y}$ as follows:

[TABLE]

Moreover, since

[TABLE]

we have

[TABLE]

Thus, by Theorem 3.1, for ${\bf W}={\bf R}_{1}{\bf R}_{0}$ and $|u|<1$,

[TABLE]

But, we have

[TABLE]

Furthermore, we have

[TABLE]

Thus, for $x,x^{\prime}\in X(x\neq x^{\prime})$,

[TABLE]

In the case of $x=x^{\prime}$,

[TABLE]

Therefore, it follows that

[TABLE]

Hence,

[TABLE]

$\Box$

By Theorem 5.1 and Corollary 3.4, we obtain the following.

Corollary 5.2

Let $G=(X\sqcup Y,E)$ and ${\bf W}$ be as the above. Then, for the Szegedy matrix ${\bf W}={\bf R}_{1}{\bf R}_{0}$, we have

[TABLE]

By Theorem 5.1 and Corollary 3.5, we obtain the spectrum of ${\bf W}$, which is consistency with [24].

Corollary 5.3

*Let $G=(X\sqcup Y,E)$ and ${\bf W}$ be as the above. Suppose that $n\geq m$. Then, the spectra of the Szegedy matrix ${\bf W}={\bf R}_{1}{\bf R}_{0}$ are given as follows:

If $G$ is not a tree, then*

$2m$* eigenvalues:*

[TABLE] 2. 2.

$\epsilon-\nu$* eigenvalues: 1;* 3. 3.

$n-m$* eigenvalues: -1.*

If $G$ is a tree, then

$2m-2$* eigenvalues:*

[TABLE] 2. 2.

one eigenvalue: 1; 3. 3.

$n-m$* eigenvalues: -1.*

Similarly, if $n<m$, then the following result holds.

Corollary 5.4

Let $G=(X\sqcup Y,E)$ and ${\bf W}$ be as the above. Suppose that $m\geq n$. Then we define an $n\times n$ matrix ${\bf A}_{q}=(a^{(q)}_{yy^{\prime}})_{y,y^{\prime}\in Y}$ as follows:

[TABLE]

Note that

[TABLE]

Then, the spectra of the Szegedy matrix ${\bf W}={\bf R}_{1}{\bf R}_{0}$ are given as follows: If $G$ is not a tree, then

$2n$* eigenvalues:*

[TABLE] 2. 2.

$\epsilon-\nu$* eigenvalues: 1;* 3. 3.

$m-n$* eigenvalues: -1.*

If $G$ is a tree, then

$2n-2$* eigenvalues:*

[TABLE] 2. 2.

one eigenvalue: 1; 3. 3.

$m-n$* eigenvalues: -1.*

6 An example

Let $G=K_{2,2}$ be the complete bipartite graph with partite set $X=\{a,b\},Y=\{c,d\}$. Then we arrange edges of $G$ as follows:

[TABLE]

Furthermore, we consider the following two functions $p:E\rightarrow[0,1]$ and $q:E\rightarrow[0,1]$ such that

[TABLE]

Now, we have

[TABLE]

Thus, we have

[TABLE]

Therefore, it follows that

[TABLE]

[TABLE]

Hence,

[TABLE]

[TABLE]

where

[TABLE]

Thus,

[TABLE]

But,

[TABLE]

Thus,

[TABLE]

Therefore, it follows that

[TABLE]

Furthermore, since $m=n=2$, we have $mn-m-n=n-m=0$. By Corollary 4.3, the eigenvalues of ${\bf W}$ are

[TABLE]

There are eigenvalues induced from ${\bf A}_{p}$.

7 The characteristic polynomial of the modified time evolution matrix of the duplication of the modified digraph

Let $G$ be a connected graph with $n$ vertices and $\epsilon$ edges which may have multiple edges and self loops , and the duplication graph be $G_{2}$. We set $p,q:E(G_{2})\to[0,1]$ so that $\sum_{V(e)=v}p(e)=\sum_{V^{\prime}(e)=v^{\prime}}q(e)=1$ with

[TABLE]

for any $v\in V$ and $u^{\prime}\in V^{\prime}$. Thus $q$ is determined by $p$. The $2n\times 2n$ stochastic matrix ${\bf P}$ is denoted by

[TABLE]

Furthermore, let $M$ be a set of $m$ marked vertices in $G$, and the modified bipartite graph of the duplication graph $G_{2}$ with the marked element

[TABLE]

be denoted by $\vec{G}_{2}$. Let ${\bf W}^{\prime}={\bf R}^{\prime}_{1}{\bf R}^{\prime}_{0}$ be the modified time evolution matrix of the modified Szegedy walk on $\ell^{2}(\tilde{E}_{2})$. Here $E_{M}=E(G_{2})\cup[N_{2}]$, where $[N_{2}]$ is set of the matching edges between marked elements and its copies, that is $[N_{2}]=\{mm^{\prime}\;|\;m\in M\}$. Thus the cardinality of $\ell^{2}(E_{M})$ is $2\epsilon+m$. Under the setting of ${\bf W^{\prime}}$, we took the modification of $p$ and $q$ as follows. Let $p^{\prime},q^{\prime}:E_{M}\to[0,1]$ be

[TABLE]

[TABLE]

where

[TABLE]

The modified $2n\times 2n$ stochastic matrix ${\bf P}^{\prime}$ is given by changing $p$ and $q$ to $p^{\prime}$ and $q^{\prime}$ as follows:

[TABLE]

The reflection operators ${\bf R}_{0}^{\prime}$ and ${\bf R}_{1}^{\prime}$ are described by $\{\phi^{\prime}_{v}\}_{v\in V}$ and $\{\psi^{\prime}_{u}\}_{u\in V^{\prime}}$ as follows:

[TABLE]

where $\phi^{\prime}_{v}=\sum_{V(e)=v}\sqrt{p^{\prime}(e)}|e\rangle$, $\psi^{\prime}_{v}=\sum_{V^{\prime}(e)=v}\sqrt{q^{\prime}(e)}|e\rangle$. See Sect. 2.5 for more detailed this setting. Let $\{|v\rangle\}_{v\in V}$ be the standard basis of $\mathbb{C}^{n}$, that is, $(|v\rangle)_{u}=1$ if $v=u$, $(|v\rangle)_{u}=0$ otherwise, where $n=|V|$. We define $(2\epsilon+m)\times n$ matrices as follows, where $2\epsilon=|E(G_{2})|$:

[TABLE]

that is,

[TABLE]

[TABLE]

Let $r$ be the number of edges connecting non-marked elements and its copies, that is,

[TABLE]

Let $s$ be the number of edges connecting non-marked elements and copies of marked elements, that is,

[TABLE]

We set $\epsilon{}^{\prime}=r+2s+m$. Remark that if there is no marked element connecting to another marked element in the original graph $G$, then $\epsilon^{\prime}=2\epsilon+m$, on the other hand, if not, $\epsilon^{\prime}<2\epsilon+m$ since such an edge connecting marked element in $G$ is omitted in the procedure making $\vec{G}_{2}$ from $G$. By the definitions of ${\bf R}^{\prime}_{0}$ and ${\bf R}^{\prime}_{1}$, ${\bf K}\ {}^{t}{\bf K}$ is equal to $\sum_{x\in X}|\phi{}^{\prime}_{x}\rangle\langle\phi{}^{\prime}_{x}|$, and ${\bf L}\ {}^{t}{\bf L}$ is equal to $\sum_{y\in Y}|\psi{}^{\prime}_{y}\rangle\langle\psi{}^{\prime}_{y}|$. Thus,

[TABLE]

[TABLE]

Now, we define an $n\times n$ matrix $\hat{{\bf A}}^{\prime}_{p}$ as follows:

[TABLE]

Remark that $q$ is determined by $p$ and so as $p^{\prime}$ and $q^{\prime}$. The $v,u$ element of this symmetric matrix $\hat{{\bf A}}^{\prime}_{p}$ is computed as follows: ${}^{t}{\bf LK}$ is expressed by

[TABLE]

Thus

[TABLE]

which is the summation of a real valued weight over all the path from $u\in V^{\prime}$ to $v\in V$. Therefore

[TABLE]

Since $p^{\prime}(e)=0$, $q^{\prime}(f)=0$ for every “$V(e)\in M$, $V^{\prime}(e)\notin M$” and “$V^{\prime}(f)\in M^{\prime}$, $V(f)\notin M$”,

[TABLE]

Here the summation $Q_{2}$ is over all the $2$-length path in $G_{2}$ from $u\in V$ to $v\in V$ never going into $M$ and $M^{\prime}$.

If the following condition holds, we say $p$ satisfies the detailed balanced condition: there exists $\pi:V\sqcup V^{\prime}$ such that

[TABLE]

for every $e\in E_{M}$ with $V(e)\notin M$ and $V^{\prime}(e)\notin M^{\prime}$, and $\pi(u)=1$ if $u\in M\sqcup M^{\prime}$. A typical setting of $p(e)=1/\mathrm{deg}(V(e))$ and $q(e)=1/\mathrm{deg}(V^{\prime}(e))$ satisfies the detailed balanced condition by $\pi(u)=\mathrm{deg}(u)$ for every $u\in(V\setminus M)\sqcup(V^{\prime}\setminus M^{\prime})$. If the detailed balanced condition holds, Since the values $q(e)$ and $p(f)$, where $(e,f)$ is $(u,v)$-path in $G_{2}$, are equivalent to

[TABLE]

we have

[TABLE]

Then it is expressed by

[TABLE]

Therefore if the detailed balanced condition holds, $\hat{{\bf A}}^{\prime}_{p}$ is unitary equivalent to the square of ${\bf P}_{M}^{\prime}:={\bf P}_{M}\oplus{\bf I}_{m}$, where ${\bf P}_{M}$ is an $(n-m)\times(n-m)$ matrix describing the random walk with the Dirichlet boundary condition at $M$: for $u,v\notin M$,

[TABLE]

Thus

[TABLE]

Now we are in the place to give the following formula for the the modified time evolution matrix of the modified Szegedy walk on $\ell^{2}(E_{M})$.

Theorem 7.1

Let $G$ be a connected graph with $n$ vertices and $\epsilon$ edges which may have multiple edges and self loops. Let ${\bf W}^{\prime}={\bf R}^{\prime}_{1}{\bf R}^{\prime}_{0}$ be the modified time evolution matrix of the modified Szegedy walk on $\ell^{2}(E_{M})$ induced by random walk $p:E(G_{2})\to[0,1]$ and the marked element $M$ with $|M|=m$.

Then, for ${\bf W}^{\prime}$, we have

[TABLE]

In particular, if $p$ satisfies the detailed balanced condition, then

[TABLE]

*Proof *. The subset of edges connecting marked elements and its copies in $E_{M}$ denotes $F_{M}$, that is,

[TABLE]

The cardinality of $F_{M}=2\epsilon+m-\epsilon^{\prime}$. The definitions of $p^{\prime}$ and $q^{\prime}$ give $p^{\prime}(e)=q^{\prime}(e)=0$ for $e\in F_{M}$, which implies $\langle e|\phi^{\prime}_{v}\rangle=\langle e|\psi^{\prime}_{u}\rangle=0$ for any $u,v\in V$. Thus

[TABLE]

for every $e,f\in F_{M}$. Concerning the above, it holds that

[TABLE]

Therefore if $F_{M}\neq\emptyset$, then

[TABLE]

Therefore, if $F_{M}\neq\emptyset$, then at least $|F_{M}|$-multiple eigenvalue $1$ of ${\bf W}^{\prime}$ exists.

From now on we consider the second term of the above RHS. To this end, it is not a loss of generality that we take the assumption that $F_{M}=\emptyset$ putting $2{\bf K}\;{\bf{}^{t}K}-{\bf I}_{\epsilon^{\prime}}={\bf R}^{\prime}_{0}$, $2{\bf L}\;{\bf{}^{t}L}-{\bf I}_{\epsilon^{\prime}}={\bf R}^{\prime}_{1}$ and ${\bf W}^{\prime}={\bf R}^{\prime}_{1}{\bf R}^{\prime}_{0}$. Since

[TABLE]

we have

[TABLE]

Therefore, by Theorem 3.1, it follows that

[TABLE]

But,

[TABLE]

Hence, If $F_{M}=\emptyset$, then

[TABLE]

Therefore if $F_{M}\neq\emptyset$, then

[TABLE]

Concerning the fact that $F_{M}=\emptyset$ if and only if $\epsilon^{\prime}=2\epsilon+m$, then we have obtained the desired conclusion. If the detailed balanced condition holds, $\hat{{\bf A}}^{\prime}_{p}=({\bf D}\oplus{\bf I}_{m}){{\bf P}_{M}^{\prime}}^{2}({\bf D}^{-1}\oplus{\bf I}_{m})$, ${\bf D}$ is an $(n-m)\times(n-m)$ diagonal matrix $\mathrm{diag}[\sqrt{\pi(u)}\;|\;u\notin M]$, that is, $({\bf D}\oplus{\bf I}_{m})|u\rangle=\sqrt{\pi(u)}$ if $u\notin M$, $({\bf D}\oplus{\bf I}_{m})|u\rangle=|u\rangle$ if $u\in M$. $\Box$

By Theorem 7.1 and Corollary 3.4, we have obtain following.

Corollary 7.2

Let $G$ be a connected graph with $n$ vertices and $\epsilon$ edges which may have multiple edges and self loops. Let ${\bf W}^{\prime}={\bf R}^{\prime}_{1}{\bf R}^{\prime}_{0}$ be the modified time evolution matrix of the modified Szegedy walk on $\ell^{2}(E_{M})$ induced by random walk $p:E(G_{2})\to[0,1]$ and the marked element $M$ with $|M|=m$. Then, for the ${\bf W}^{\prime}={\bf R}^{\prime}_{1}{\bf R}^{\prime}_{0}$, we have

[TABLE]

By Theorem 7.1 and Corollary 3.5, we obtain the eigenvalues of ${\bf W}^{\prime}$.

Corollary 7.3

Let $G$ be a connected graph with $n$ vertices and $\epsilon$ edges which may have multiple edges and self loops. Let ${\bf W}^{\prime}={\bf R}^{\prime}_{1}{\bf R}^{\prime}_{0}$ be the modified time evolution matrix of the modified Szegedy walk on $\ell^{2}(E_{M})$ induced by random walk $p:E(G_{2})\to[0,1]$ and the marked element $M$ with $|M|=m$. Then the spectra of the unitary matrix ${\bf W}^{\prime}={\bf R}^{\prime}_{1}{\bf R}^{\prime}_{0}$ are given as follows:

If $2(\epsilon-n)+m\geq 0$, that is, “$G$* is not a tree” or “$m>1$”, then*

1. (a)

$2n$* eigenvalues:*

[TABLE] 2. (b)

$2(\epsilon-n)+m$* eigenvalues: 1.* 2. 2.

otherwise, that is, $G$ is a tree and $m\in\{0,1\}$, then

1. (a)

$2(n-1)$* eigenvalues:*

[TABLE] 2. (b)

$m$-multiple eigenvalue $1$.

*Proof *. Since $\epsilon-n<0$ if and only if $G$ is a tree, thus $2(\epsilon-n)+m<0$ if and only if $G$ is a tree and $m\in\{0,1\}$. By Corollary 7.2,

[TABLE]

holds, where $\alpha_{j}^{(\pm)}$ are the solutions of $\lambda^{2}-2(2\mu-1)\lambda+1=0$ with $\nu\in Spec(\hat{{\bf A}}^{\prime}_{p})$. The second term has $2n=2\epsilon+2$ solutions while the dimension of the total space is now $2\epsilon+m$. But in this situation since $2(\epsilon-n)+m=-2+m<0$, then the power of the first term $(1-\lambda)^{2(\epsilon-n)+m}$ is negative. Thus the second term should includes the $(\lambda-1)^{(2-m)}$ term counteracted by the first term. The result follows. $\Box$

8 An example

Let $G=K_{3}$ be the complete graph with three vertices $v_{1},v_{2},v_{3}$, and ${\bf P}=(p_{uv})_{u,v\in V(G)}$ the following stochastic matrix of $G$:

[TABLE]

Furthermore, let $M=\{v_{3}\}$ be a set of $m$ marked vertices in $G$. Thus we set $E_{M}$ by

[TABLE]

where $e_{1}=\{v_{1},v_{2}^{\prime}\}$, $e_{2}=\{v_{2},v_{1}^{\prime}\}$, $f_{1}=\{v_{1},v_{3}^{\prime}\}$, $f_{2}=\{v_{2},v_{3}^{\prime}\}$, $f_{1}^{\prime}=\{v_{3},v_{2}^{\prime}\}$, $f_{2}^{\prime}=\{v_{3},v_{1}^{\prime}\}$ and $g=\{v_{3},v_{3}^{\prime}\}$. The duplication graph of $G$ is denoted by $G_{2}$. $E_{M}$ is the union of $E(G_{2})$ and $\{g\}$. The modified stochastic matrix ${\bf P}^{\prime}=(p^{\prime}_{uv})_{u,v\in V(G_{2})}$ derived from ${\bf P}$ with the marked element $M=\{v_{3}\}$ is given as follows:

[TABLE]

which means

[TABLE]

Then the dimension of the total state space is

[TABLE]

We put $X=\{v_{1},v_{2},v_{3}\}$ and its copy $X^{\prime}=\{v_{1}^{\prime},v_{2}^{\prime},v_{3}^{\prime}\}$. The $7\times 3$ matrix $K$ is an incidence matrix between $7$ edges $e_{1},e_{2},f_{1},f_{2},f^{\prime}_{1},f^{\prime}_{2},g$ and $X$ as follows:

[TABLE]

Furthermore, the $7\times 3$ matrix $L$ is an incidence matrix between $7$ edges $e_{1},e_{2},f_{1},f_{2},f_{1},f_{2},g$ and $Y$ as follows:

[TABLE]

Thus, we have

[TABLE]

and

[TABLE]

Therefore, it follows that

[TABLE]

and

[TABLE]

Hence,

[TABLE]

Now, we have

[TABLE]

Thus, we have

[TABLE]

Thus,

[TABLE]

Therefore, it follows that

[TABLE]

Furthermore, since $n=3$, we have $\epsilon{}^{\prime}-2n=7-6=1$. By Corollary 6.3, the eigenvalues of ${\bf W}^{\prime}$ are

[TABLE]

Acknowledgments

The first author is partially supported by the Grant-in-Aid for Scientific Research (Chal- lenging Exploratory Research) of Japan Society for the Promotion of Science (Grant No. 15K13443). The second author is partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 15K04985). The third author is partially supported by the Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science (Grant No. 25800088).