# Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### 2.1. Graph Notation

#### 2.2. Definition of the Grover Walk

#### 2.3. Boundary Operators

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

## 3. Definition of the Grover Walk on Graphs with Sinks

## 4. Main Theorem

**Case A:**- ${A}_{0,\sigma}=\varnothing $ and ${G}_{0}$ is a bipartite graph;
**Case B:**- ${A}_{0,\sigma}=\varnothing $ and ${G}_{0}$ is a non-bipartite graph;
**Case C:**- ${A}_{0,\sigma}\ne \varnothing $ and ${G}_{0}\setminus {A}_{0,\sigma}$ is a bipartite graph;
**Case D:**- ${A}_{0,\sigma}\ne \varnothing $ and ${G}_{0}\setminus {A}_{0,\sigma}$ is a non-bipartite graph.

**Theorem**

**1.**

- 1.
- ${lim}_{n\to \infty}{\gamma}_{n}=\gamma $ exists.
- 2.
- The survival probability γ is expressed by$$\gamma =\left|\right|{\Pi}_{\mathcal{T}}{\chi}_{S}{\varphi}_{0}{\left|\right|}^{2}+\left|\right|{\Pi}_{\mathcal{K}}{\chi}_{S}{\varphi}_{0}{\left|\right|}^{2}+\left\{\begin{array}{cc}\left|\right|{\Pi}_{\mathcal{A}}{\chi}_{S}{\varphi}_{0}{\left|\right|}^{2}\hfill & :\phantom{\rule{4.pt}{0ex}}\mathit{Case}\phantom{\rule{4.pt}{0ex}}A\hfill \\ \left|\right|{\Pi}_{\mathcal{B}}{\chi}_{S}{\varphi}_{0}{\left|\right|}^{2}\hfill & :\phantom{\rule{4.pt}{0ex}}\mathit{Case}\phantom{\rule{4.pt}{0ex}}B\hfill \\ \left|\right|{\Pi}_{\mathcal{C}}{\chi}_{S}{\varphi}_{0}{\left|\right|}^{2}\hfill & :\phantom{\rule{4.pt}{0ex}}\mathit{Case}\phantom{\rule{4.pt}{0ex}}C\hfill \\ \left|\right|{\Pi}_{\mathcal{D}}{\chi}_{S}{\varphi}_{0}{\left|\right|}^{2}\hfill & :\phantom{\rule{4.pt}{0ex}}\mathit{Case}\phantom{\rule{4.pt}{0ex}}D\hfill \end{array}\right.$$

**Proof.**

**Corollary**

**1.**

**Remark**

**1.**

## 5. Example

## 6. Relation between Grover Walk with Sinks and Grover Walk with Tails

#### 6.1. Grover Walk on Graphs with Tails

**Theorem**

**2**

**.**Let $\tilde{G}=(\tilde{V},\tilde{A})$ be the graph with infinite tails ${\left\{{\mathbb{P}}_{j}\right\}}_{j=1}^{r}$ induced by ${G}_{0}$ and its boundaries $\delta {V}_{0}$. Assume the initial state ${\psi}_{0}$ is

**Remark**

**2.**

**Remark**

**3.**

#### 6.2. Relation between Grover Walk with Sinks and Grover Walk with Tails

**Lemma**

**2.**

**Proof.**

**Corollary**

**2.**

**Remark**

**4.**

## 7. Centered Generalized Eigenspace of E for the Grover Walk Case

#### 7.1. The Stationary States from the Viewpoint of the Centered Generalized Eigenspace

**Proposition**

**2**

**.**

- 1.
- For any $\lambda \in \mathrm{Spec}\left(E\right)$, it holds that $\left|\lambda \right|\le 1$, i.e.,$${\mathcal{H}}_{s}=\left\{\psi \phantom{\rule{0.277778em}{0ex}}\right|\phantom{\rule{0.277778em}{0ex}}\exists \phantom{\rule{0.277778em}{0ex}}m\in {\mathbb{N},\phantom{\rule{0.277778em}{0ex}}\exists \phantom{\rule{0.277778em}{0ex}}\left|\lambda \right|<1,\phantom{\rule{0.277778em}{0ex}}(U-\lambda )}^{m}\psi )=0\}.$$
- 2.
- Let ${P}_{c}$ be the projection operator on ${\mathcal{H}}_{c}$ along with ${\mathcal{H}}_{s}$; that is, ${P}_{c}E=E{P}_{c}$ and ${P}_{c}^{2}={P}_{c}$. Then, ${P}_{c}$ is the orthogonal projection onto ${\mathcal{H}}_{c}$, i.e., ${P}_{c}={P}_{c}^{\ast}$.
- 3.
- The operator E acts as a unitary operator on ${\mathcal{H}}_{c}$, that is, ${\mathcal{H}}_{c}={\oplus}_{\left|\lambda \right|=1}ker(\lambda -E)$ and ${U}_{G}{\chi}_{S}^{\ast}\phi =\lambda {\chi}_{S}^{\ast}\phi $ for any $\phi \in ker(\lambda -E)$ with $\left|\lambda \right|=1$.

**Corollary**

**3.**

**Proposition**

**3.**

- 1.
- The state ${\chi}_{T}{\psi}_{n}$ in Theorem 2 belongs to ${\mathcal{H}}_{s}$ for any time step $n\in \mathbb{N}$.
- 2.
- The state of QW with sinks, ${\chi}_{S}{\varphi}_{n}$, asymptotically belongs to ${\mathcal{H}}_{c}$ in the long time limit n.

**Proof.**

**Proposition**

**4.**

**Proof.**

#### 7.2. Characterization of Centered Generalized Eigenspace by Graph Notations

**Lemma**

**3**

**.**Assume $\lambda \in \mathrm{Spec}\left(E\right)$ with $\left|\lambda \right|=1$. Then, we have

- 1.
- $\lambda =\pm 1$ if and only if $ker(\lambda -E)=ker(-\lambda -S)\cap ker{d}_{1}$.
- 2.
- $\lambda \ne \pm 1$ if and only if supp$\left(g\right)\subset {V}_{0}\setminus \delta {V}_{0}$ for any $g\in ker((\lambda +{\lambda}^{-1})/2-T)\ne 0$.

**Lemma**

**4.**

**Proof.**

**${A}_{0,\sigma}=\varnothing $ case**:

**${A}_{0,\sigma}\ne \varnothing $ case**: Since $\left({\partial}_{-}^{\ast}f\right)\left(a\right)=f\left(t\left(a\right)\right)=0$ if $a\in {A}_{0,\sigma}$, then f takes the value 0 at the other vertices since $f\left(t\right(a\left)\right)=-f\left(o\right(a\left)\right)$ for any $a\in {A}_{+}$, which implies $ker\left({\partial}_{+}^{\ast}\right)=0$ if ${A}_{0,\sigma}\ne \varnothing $.

**Proposition**

**5.**

**Proof.**

**Definition**

**2.**

**Construction of**${\eta}_{x,y}\in {\mathbb{C}}^{{A}_{0}}$:

- 1.
- $x\in {\Gamma}_{o}$, $y\in {\Gamma}_{o}$ case:If ${G}_{0}$ is a bipartite graph, let us fix an odd length fundamental cycle ${c}_{\ast}=({a}_{0},\cdots ,{a}_{r-1})\in {\Gamma}_{o}$ and pick up another $c\in {\Gamma}_{o}=({b}_{0},\cdots ,{b}_{s-1})$. We set the following walk q and define the function on ${\mathbb{C}}^{{A}_{0}}$; ${\xi}_{q}^{(-)}=:\overline{a}{a}_{{c}_{\ast}-c}$, induced by ${c}_{\ast},c\in {\Gamma}_{o}$:
- (a)
- ${c}_{0}\cap c\ne \varnothing $ case: We set q as the shortest closed walk starting from a vertex ${u}_{0}\in V\left({c}_{0}\right)\cap V\left(c\right)$ and visiting all the vertices of $V\left({c}_{0}\right)$ and $V\left(c\right)$; that is, $q=({a}_{i},\cdots ,{a}_{i+r},{b}_{j},\cdots ,{b}_{s+j})$. Here, $o\left({a}_{i}\right)=o\left({b}_{j}\right)={u}_{0}$ and the suffices are modulus of r and s.
- (b)
- ${c}_{0}\cap c=\varnothing $ case: Let us fix the shortest path between ${c}_{0}$ and c by $p=({p}_{1},\cdots ,{p}_{t})$. Denoting the vertex in $V\left({c}_{\ast}\right)$ connecting to p by ${u}_{0}\in V\left({c}_{\ast}\right)$, we set q by the shortest closed walk q starting from ${u}_{\ast}$ and visiting all the vertices; that is, $q=({a}_{i},\cdots ,{a}_{r+i},{p}_{0}\cdots ,{p}_{t},{b}_{j}\cdots ,{b}_{s+j},{\overline{p}}_{t}\cdots ,{\overline{p}}_{1})$, where $o\left({a}_{i}\right)=t\left({a}_{r+i}\right)=o\left({p}_{1}\right)={u}_{0}$, $t\left({p}_{t}\right)=o\left({b}_{j}\right)=t\left({b}_{s+j}\right)$.

Note that, by the definition of the fundamental cycle, the intersection ${c}_{0}\cap c$ is a path in Case (1). Since ${G}_{0}$ is connected, there is a path connecting ${c}_{\ast}$ to c and we fix such a path for every pair of $({c}_{\ast},c)$ in Case (2). - 2.
- $x\in {A}_{\sigma}$ and $y\in {A}_{\sigma}$ case:If the number of self-loops $|{A}_{\sigma}|\ge 2$, let us fix a self-loop ${a}_{\ast}$ from ${A}_{\sigma}$ and a path between ${a}_{\ast}$ to each $a\in {A}_{\sigma}\setminus \left\{{a}_{\ast}\right\}$. Let us denote the path between ${a}_{\ast}$ and a by $p=({p}_{1},\cdots ,{p}_{t})$. Then, we set the walk from ${a}_{\ast}$ to a by $q=({a}_{\ast},{p}_{1},\cdots ,{p}_{t},a)$ and ${\xi}_{q}^{(-)}=:{\eta}_{{a}_{\ast}-a}$.
- 3.
- $x\in {A}_{\sigma}$ and $y\in {\Gamma}_{o}$ case:If $|{A}_{\sigma}|\ge 1$ and $G\setminus {A}_{\sigma}$ is a non-bipartite graph, let us fix a self-loop ${a}_{\ast}$ and pick up an odd cycle $c=({b}_{1},\cdots ,{b}_{t})\in {\Gamma}_{o}$; if the self-loop $o\left({a}_{\ast}\right)\in V\left(c\right)$, we set the walk starting from ${a}_{\ast}$ visiting all the vertices $V\left(c\right)$ and returning back to ${a}_{\ast}$ by $q=({a}_{\ast},{b}_{1},\cdots ,{b}_{t},{a}_{\ast})$; and, for $o\left({a}_{\ast}\right)\notin V\left(c\right)$, let us fix a path $p=({p}_{1},\cdots ,{p}_{t})$ between $o\left({a}_{\ast}\right)$ and $o\left({b}_{1}\right)$ and set the walk starting from ${a}_{\ast}$ visiting all the vertices $V\left(p\right)\cup V\left(c\right)$ and returning back to ${a}_{\ast}$; $q=({a}_{\ast},{p}_{1},\cdots ,{p}_{t},{b}_{0}\cdots ,{b}_{t},{\overline{p}}_{t},\cdots ,{\overline{p}}_{1},{a}_{\ast})$. Then, we set ${\xi}_{q}^{(-)}=:{\eta}_{{a}_{\ast},c}$.
- 4.
- $x\in {\Gamma}_{o}$ and $y\in {A}_{\sigma}$ case:Let us fix an odd length fundamental cycle ${c}_{\ast}\in {\Gamma}_{o}=({b}_{1},\cdots ,{b}_{s-1})$ and pick up a self-loop $a\in {A}_{\sigma}$. Let us set a short length path p between $o\left(a\right)$ and $o\left({b}_{1}\right)$. Then, we consider the same walk q as in Case (3) and set ${\xi}_{q}^{(-)}=:{\eta}_{{c}_{\ast},a}.$

**Proposition**

**6.**

**Proof.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**

**The setting of graphs:**The original graph G is depicted in the left corner. The sinks ${V}_{s}$ are the white vertices. The subgraph ${G}_{0}$ of G is the black colored graph in the center. The set of boundary vertices $\delta V$ is $\{2,4\}$. The semi-infinite graph $\tilde{G}$ is constructed by connecting the infinite length path to each boundary vertex of ${G}_{0}$.

**Figure 2.**

**The centered eigenspace of the example:**The centered eigenspace to which Grover walk with sinks asymptotically belongs in this example is $\mathcal{T}\oplus \mathcal{K}\oplus \mathcal{C}$. Each weighted sub-digraph represents a function in ${\mathbb{C}}^{{A}_{0}}$; the complex value at each arc is the returned value of the function. Each eigenspace, $\mathcal{T}$, $\mathcal{K}$, and $\mathcal{C}$, is spanned by the functions represented by these weighted sub-digraphs.

**Figure 3.**

**Construction of eigenfunction ${\eta}_{x,y}\in {\mathbb{C}}^{{A}_{0}}$**: Each graph with signs ± represents the function ${\eta}_{x,y}$. The support of ${\eta}_{x,y}$ is included in the arcs of each graphs. The signs are the return values of this function at each arcs. The return values of the inverse arcs are the same as the original arcs. The signs are assigned alternatively along the red colored walks. At each time where the walk runs through an arc, we take the sum of the signs; e.g., in the case for $x\in {A}_{0,\sigma},y\in {\Gamma}_{o}$, the walk runs through the self-loop twice, and then the return value at the self-loops of the function is $1+1=2$.

**Figure 4.**

**Eigenspaces**($\mathcal{A}$–$\mathcal{D}$): This figure shows examples of four graphs for Cases ($\mathit{A}$)–($\mathit{D}$) and their induced eigenspaces of the Grover walk ($\mathcal{A}$–$\mathcal{D}$). The figures at the right corner are the fundamental cycles for each case. The weighted graphs represent bases of each eigenspace. The weights are the return values at each arcs of the bases, where every base takes the value 0 at the dashed arcs.

$\mathit{\rho}$ | $\mathit{\nu}$ | State in ${\mathit{G}}_{0}$ | |
---|---|---|---|

QW with tails in the setting of Theorem 2 [19] | $\ne 0$ | $=0$ | $\in {\mathcal{H}}_{s}$ (for any n) |

QW with sinks | $=0$ | $\ne 0$ | $\in {\mathcal{H}}_{c}$ (asymptotically) |

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**MDPI and ACS Style**

Konno, N.; Segawa, E.; Štefaňák, M. Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs. *Symmetry* **2021**, *13*, 1169.
https://doi.org/10.3390/sym13071169

**AMA Style**

Konno N, Segawa E, Štefaňák M. Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs. *Symmetry*. 2021; 13(7):1169.
https://doi.org/10.3390/sym13071169

**Chicago/Turabian Style**

Konno, Norio, Etsuo Segawa, and Martin Štefaňák. 2021. "Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs" *Symmetry* 13, no. 7: 1169.
https://doi.org/10.3390/sym13071169