**5. Example**

Let us consider a simple example in Figure 1. *G*0 = (*<sup>V</sup>*0, *<sup>A</sup>*0) with *V*0 = {1, 2, 3, 4} and *A*0 = {*<sup>a</sup>*1, *a*2, *a*3, *a*4, *a*1, *a*2, *a*3, *a*4, *b*1, *b*2}, where *a*1 has the origin 1 and the terminus 2; *a*2 has the origin 2 and the terminus 3; *a*3 has the origin 1 and the terminus 4; *a*4 has the origin 1 and the terminus 1; and *b*1 and *b*2 are the self loops on 1 and 3, respectively.

This graph fits into Case C. Thus, let *q* be the closed walk by *q* = (*<sup>a</sup>*1, *a*2, *a*3, *<sup>a</sup>*4) and *q* be the walk between two selfloops by (*b*1, *a*1, *a*2, *b*2). Then, *ξ*(+) *q* , and the functions defined by (6) and Definition 2 are given by

$$\begin{split} \xi\_{q}^{(+)} &= (\delta\_{\mathfrak{a}\_{1}} + \delta\_{\mathfrak{a}\_{2}} + \delta\_{\mathfrak{a}\_{3}} + \delta\_{\mathfrak{a}\_{4}}) - (\delta\_{\mathfrak{a}\_{1}} + \delta\_{\mathfrak{a}\_{2}} + \delta\_{\mathfrak{a}\_{3}} + \delta\_{\mathfrak{a}\_{4}}, \\ \xi\_{q}^{(-)} &= (\delta\_{\mathfrak{a}\_{1}} + \delta\_{\mathfrak{a}\_{1}}) - (\delta\_{\mathfrak{a}\_{2}} + \delta\_{\mathfrak{a}\_{2}}) + (\delta\_{\mathfrak{a}\_{3}} + \delta\_{\mathfrak{a}\_{3}}) - (\delta\_{\mathfrak{a}\_{4}} + \delta\_{\mathfrak{a}\_{4}}), \\ \eta\_{b\_{1} - b\_{2}} &= \delta\_{\mathfrak{b}\_{1}} - (\delta\_{\mathfrak{a}\_{1}} + \delta\_{\mathfrak{a}\_{1}}) + (\delta\_{\mathfrak{a}\_{2}} + \delta\_{\mathfrak{a}\_{2}}) - \delta\_{\mathfrak{b}\_{2}}. \end{split}$$

The matrix representation of the self adjoint operator *T* is expressed by

$$T = \frac{1}{3} \begin{bmatrix} 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \end{bmatrix}.$$

The eigenvector of *T* which has no overlaps to *δV*0 = {2, 4} is easily obtained by

$$f = [1, \ 0, \ -1, \ 0]^\top$$

which satisfies *T f* = (1/3)*f* . Here, the symbol "&" is the transpose. The eigenfunctions lifted up to C*<sup>A</sup>* from *f* is

$$(\varrho\_{\pm})(a) = f(t(a)) - \lambda\_{\pm} f(o(a))$$

by (2), where

$$
\lambda\_{\pm} = \frac{1}{3}(1 \pm i\sqrt{8}) = \varepsilon^{\pm i\theta}, \quad \theta = \arccos\frac{1}{3}.
$$

Then, we have

$$\begin{aligned} \varrho\_{\pm}(a\_1) &= -\lambda\_{\pm}, \; \varrho\_{\pm}(a\_2) = -1, \; \varrho\_{\pm}(a\_3) = \lambda\_{\pm}, \; \varrho\_{\pm}(a\_4) = 1, \\\varrho\_{\pm}(\mathfrak{a}\_1) &= 1, \; \varrho\_{\pm}(\mathfrak{a}\_2) = \lambda\_{\pm}, \; \varrho\_{\pm}(\mathfrak{a}\_3) = -1, \; \varrho\_{\pm}(\mathfrak{a}\_4) = -\lambda\_{\pm}, \\\varrho\_{\pm}(b\_1) &= 1 - \lambda\_{\pm}, \; \varrho\_{\pm}(b\_2) = -1 + \lambda\_{\pm}. \end{aligned}$$

It holds that *Eϕ*± = *λ*±*ϕ*<sup>±</sup>. We obtain

$$\begin{aligned} \mathcal{T} &= \mathbb{C}\mathfrak{q}\_{+} \oplus \mathbb{C}\mathfrak{q}\_{-}, \\ \mathcal{K} &= \mathbb{C}\mathfrak{q}\_{+}^{(+)} \\ \mathcal{C} &= \mathbb{C}\mathfrak{q}\_{(\mathfrak{a}\_{1}, \mathfrak{a}\_{2}, \mathfrak{a}\_{3}, \mathfrak{a}\_{4})}^{(+)} \end{aligned}$$

$$\mathcal{C} = \mathbb{C}\mathfrak{q}\_{(\mathfrak{a}\_{1}, \mathfrak{a}\_{2}, \mathfrak{a}\_{3}, \mathfrak{a}\_{4})}^{(-)} \oplus \mathbb{C}\mathfrak{q}\_{\mathfrak{b}\_{1} - \mathfrak{b}\_{2}}.$$

After the Gram–Schmidt procedure to C, we have

$$\mathcal{C} = \mathbb{C} \mathfrak{x}^{(-)}\_{(a\_1 a\_2, a\_3, a\_4)} \oplus \mathbb{C} (\eta\_{b\_1 - b\_2} + \eta'\_{b\_1 - b\_2}) \dots$$

Here, we denote

$$
\eta\_{b\_1 \cdots b\_2}' = \delta\_{b\_1} - (\delta\_{a\_4} + \delta\_{\mathbb{Z}\_4}) + (\delta\_{a\_3} + \delta\_{\mathbb{Z}\_3}) - \delta\_{b\_2}
$$

(see Figure 2). We express the functions *ϕ*<sup>±</sup>, *ξ*(+) (*<sup>a</sup>*1,*a*2,*a*3,*a*4), *ηb*1−*b*<sup>2</sup> , *<sup>η</sup>b*1−*b*<sup>2</sup> , *ηb*1−*b*<sup>2</sup> + *<sup>η</sup>b*1−*b*<sup>2</sup> by weighted sub-digraphs of *G*0. Then, the time evolution of the asymptotic dynamics of this quantum walk is described by

$$\begin{split} \mathcal{U}^{n} \sim \frac{1}{8} |\boldsymbol{\xi}^{(+)}\_{(a\_{1}a\_{2}a\_{3}a\_{4})} \rangle \langle \boldsymbol{\xi}^{(+)}\_{(a\_{1}a\_{2}a\_{3}a\_{4})} | \\ + (-1)^{n} \Big( \frac{1}{8} |\boldsymbol{\xi}^{(-)}\_{(a\_{1}a\_{2}a\_{3}a\_{4})} \rangle \langle \boldsymbol{\xi}^{(-)}\_{(a\_{1}a\_{2}a\_{3}a\_{4})} | + \frac{1}{16} |\eta\_{b\_{1}-b\_{2}} + \eta\_{b\_{1}-b\_{2}}' \rangle \langle \eta\_{b\_{1}-b\_{2}} + \eta\_{b\_{1}-b\_{2}}' | \Big) \\ + e^{in\theta} \frac{3}{32} |\varrho\_{+} \rangle \langle \boldsymbol{\varrho}\_{+} | + e^{-in\theta} \frac{3}{32} |\varrho\_{-} \rangle \langle \boldsymbol{\varrho}\_{-} |. \end{split} \tag{4}$$

**Figure 2. The centered eigenspace of the example:** The centered eigenspace to which Grover walk with sinks asymptotically belongs in this example is T ⊕K⊕C. Each weighted sub-digraph represents a function in C*A*0 ; the complex value at each arc is the returned value of the function. Each eigenspace, T , K, and C, is spanned by the functions represented by these weighted sub-digraphs.

Finally, for example, if the initial state is *ϕ*0 = *δb*1 , then the survival probability can be computed by

$$\begin{split} \gamma &= ||\Pi\_{\mathcal{T}}\varphi\_{0}||^{2} + ||\Pi\_{\mathcal{K}}\varphi\_{0}||^{2} + ||\Pi\_{\mathcal{C}}\varphi\_{0}||^{2} \\ &= \frac{1}{16} |\langle \eta\_{b\_{1}-b\_{2}} + \eta\_{b\_{1}-b\_{2}}'\varphi\_{0} \rangle|^{2} + \frac{3}{32} |\langle \varphi\_{+}, \rho\_{0} \rangle|^{2} + \frac{3}{32} |\langle \varphi\_{-}, \rho\_{0} \rangle|^{2} \\ &= \frac{1}{16} |2|^{2} + \frac{3}{32} |1 - \lambda\_{+}|^{2} + \frac{3}{32} |1 - \lambda\_{-}|^{2} \\ &= 1/2. \end{split}$$

The second equality derives from the fact that the orthonormalized eigenvectors in the centered generalized eigenspace which have an overlap with the self-loop *b*1 are given by (1/4)(*ηb*1−*b*<sup>2</sup> + *<sup>η</sup>b*1−*b*<sup>2</sup>) and √3/32 *ϕ*<sup>±</sup>.

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

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

Let *G* = (*<sup>V</sup>*, *A*) be a finite and connected graph with the set of sinks *Vs* ⊂ *V*. We introduce the infinite graph *G* ˜ = (*V*˜ , *A*˜) by adding the semi-infinite paths to each vertex of *δV* = {*<sup>v</sup>*1,..., *vr*}, that is,

$$\begin{aligned} \tilde{V} &= (V \backslash V\_s) \cup (\cup\_{j=1}^r V(\mathbb{P}\_j)), \\ \tilde{A} &= \cup\_{j=1}^r A(\mathbb{P}\_j) \cup (A \backslash \{a \in A \mid t(a) \in V\_s \text{ or } o(a) \in V\_s\}). \end{aligned}$$

Here, P*i*<sup>s</sup> are the semi-infinite paths named the tail whose origin vertex is identified with *vi* (*i* = 1, ... ,*<sup>r</sup>*) (see Figure 1). Recall that *G*0 = (*<sup>V</sup>*0, *<sup>A</sup>*0) is the subgraph of *G* eliminating the sinks *Vs*. Recall also that *χS* : C*<sup>A</sup>* → C*A*0 is

$$(\chi\_S \phi)(a) = \phi(a)$$

˜

for all *a* ∈ *A*0. In the same way, we newly introduce *χT* : C*<sup>A</sup>* → C*A*0 by

$$(\chi\_T \phi)(a) = \phi(a).$$

for all *a* ∈ *A*0. The adjoint *<sup>χ</sup>*<sup>∗</sup>*T*: C*A*0 → C*<sup>A</sup>* ˜ is

$$(\chi\_T^\*f)(a) = \begin{cases} f(a) & : a \in A\_{0\prime} \\ 0 & : \text{otherwise} \end{cases}$$

The only difference between *χS* and *χT* is the domain. A matrix representation of *χT* is

$$\chi\_T \cong \begin{bmatrix} I\_{A\_0} \ \vert \; 0 \end{bmatrix}$$

which is a |*<sup>A</sup>*0| × ∞ matrix because |*A*˜ \ *<sup>A</sup>*0| = ∞. The following theorem was proven by [19].

**Theorem 2** ([19])**.** *Let G* ˜ = (*V*˜ , *A*˜) *be the graph with infinite tails* {<sup>P</sup>*j*}*rj*=<sup>1</sup> *induced by G*0 *and its boundaries δV*0*. Assume the initial state ψ*0 *is*

$$\psi\_0(a) = \begin{cases} a\_1 & : a \in A(\mathbb{P}\_1), \text{dist}(o(a), v\_1) > \text{dist}(t(a), v\_1), \\ \vdots \\ a\_r & : a \in A(\mathbb{P}\_r), \text{dist}(o(a), v\_r) > \text{dist}(t(a), v\_r), \\ 0 & : otherwise. \end{cases}$$

*Then,* lim*n*→∞ *ψn*(*a*) =: *ψ*∞(*a*) *exists and ψ*∞(*a*) *is expressed by*

$$
\psi\_{\infty}(a) = \frac{\alpha\_1 + \dots + \alpha\_r}{r} + \mathbf{j}(a).
$$

*Here,* j(·) *is the electric current flow on the electric circuit assigned the resistance value* 1 *at each edge, that is,* j(·) *satisfies the following properties:*

$$\begin{aligned} d\_1 \mathbf{j} &= 0, \mathbf{j}(\overline{a}) = -\mathbf{j}(a) \text{ (Kirchhoff's current law)}\\ \partial\_2^\* \mathbf{j} &= 0 \text{ (Kirchhoff's voltage law)} \end{aligned}$$

*with the boundary conditions*

$$\mathbf{j}(\mathbf{e}\_i) = \mathbf{a}\_i - \frac{\mathbf{a}\_1 + \dots + \mathbf{a}\_r}{r} \tag{5}$$

*for any ei (i* = 1, . . . ,*r) such that <sup>t</sup>*(*ei*) = *vj and o*(*ei*) ∈ *<sup>V</sup>*(<sup>P</sup>*i*)*.*

**Remark 2.** *The stationary state ψ*∞ *satisfies the equation*

$$\psi\_{\infty}(a) = (\mathcal{U}\_G \psi\_{\infty})(a)$$

*for any a* ∈ *A and ψ*∞ ∈ <sup>∞</sup>*, however* ||*ψ*∞||*A*˜ = ∞*.*

$$\textbf{Remark 3.}\text{ }The\text{ }function\text{ }\hat{\mathfrak{g}}\_{\mathfrak{c}}^{(+)} = (1-\mathcal{S})\partial\_{2}\delta\_{\mathfrak{c}}^{(\Gamma)}\text{ also } satisfies\text{ }$$

$$\chi\_T^\* \mathfrak{z}\_c^{(+)}(a) = (\mathcal{U}\_G \chi\_T^\* \mathfrak{z}\_c^{(+)})(a)$$

*and Kirchhoff's current and voltage laws if the internal graph G*0 *is not a tree, while it does not satisfy the boundary condition (5) because the support of this function <sup>χ</sup>*<sup>∗</sup>*Tξ*(+) *c has no overlaps to the tails but is included in the fundamental cycle c in the internal graph G*0*.*

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

Let us consider the Grover walk on *G* with sinks *Vs* and with the initial state *ψ*(*S*) 0 ∈ C*A*. We describe *UG* as the time evolution operator of Grover walk on *G*. The *n*th iteration of this walk following (3) is denoted by *ψ*(*S*) *n* . Let us also consider the Grover walk on *G*˜ with the tails and with the "same" initial state

$$\psi\_0^{(T)}(a) = \begin{cases} \psi\_0^{(S)}(a) & : a \in A\_{0\prime} \\ 0 & : \text{otherwise} \end{cases}$$

Note that the initial state *ψ*(*S*) 0 is different from the one in the setting of Theorem 2. Putting the time evolution operator on *G* ˜ by *UG*˜ , we denote the *n*th iteration of this walk by *ψ*(*T*) *n* = *UG*˜*ψ*(*T*) *<sup>n</sup>*−1. Then, we obtain a simple but important relation between QW with sinks and QW with tails.

**Lemma 2.** *Let the setting of the QW with sinks and QW with tails be as the above. Then, for any time step n, we have*

$$
\chi\_S \psi\_n^{(S)} = \chi\_T \psi\_n^{(T)}.
$$

**Proof.** The initial state of *<sup>χ</sup>Sψ*(*S*) 0 coincides with *<sup>χ</sup>Tψ*(*T*) 0 because of the setting. Note that *χ*∗*J χJ* is the projection operator onto C*A*0 while *<sup>χ</sup>Jχ*<sup>∗</sup>*J* is the identity operator on C*A*0 (*J* ∈ {*<sup>S</sup>*, *<sup>T</sup>*}). Since *ψ*(*S*) *n* (*a*) = 0 for any *a* ∈ *Vs*, we have

$$(1 - \chi\_S^\* \chi\_S) \psi\_n^{(S)} = 0$$

for any *n* ∈ N. Then, putting *<sup>χ</sup>Sψ*(*S*) *n* =: *φ*(*S*) *n* and *<sup>χ</sup>SUGχ*<sup>∗</sup>*S* =: *E*, we have

$$\begin{split} \boldsymbol{\phi}\_{n}^{(S)} &= \chi\_{S} \boldsymbol{\upmu}\_{n}^{(S)} = \chi\_{S} \boldsymbol{\upmu}\_{G} \boldsymbol{\upmu}\_{n-1}^{(S)} \\ &= \chi\_{S} \boldsymbol{\upmu}\_{G} \left( \chi\_{S}^{\*} \boldsymbol{\upchi}\_{S} + (1 - \chi\_{S}^{\*} \boldsymbol{\upchi}\_{S}) \boldsymbol{\upmu}\_{n-1}^{(S)} \right) \\ &= E \boldsymbol{\upphi}\_{n-1}^{(S)} + \left( \chi\_{S} \boldsymbol{\upmu}\_{G} (1 - \chi\_{S}^{\*} \boldsymbol{\upchi}\_{S}) \right) \boldsymbol{\upmu}\_{n-1}^{(S)} \\ &= E \boldsymbol{\upphi}\_{n-1}^{(S)} . \end{split}$$

It is easy to see that *E* = *<sup>χ</sup>SUGχ*<sup>∗</sup>*S* = *<sup>χ</sup>TUG*˜ *<sup>χ</sup>*<sup>∗</sup>*T*. Since the support of the initial state is included in the internal graph, the inflow never comes into the internal graph from the tail for any time *n*, which implies

$$(\chi\_T \mathcal{U}\_{\mathbb{G}}(1 - \chi\_T^\* \chi\_T))\psi\_n^{(T)} = 0.$$

It holds that *E* = *χSU* ˜ *χ*∗*S* = *<sup>χ</sup>TUG*˜ *<sup>χ</sup>*<sup>∗</sup>*T*. Then, putting *φ*(*T*) *n* := *<sup>χ</sup>Tψ*(*T*) *n* , in the same way as *ψ*(*S*) *n* , we have

$$\begin{aligned} \phi\_n^{(T)} &= \chi\_T \mathcal{U}\_\mathbb{G} (\chi\_T^\* \chi\_T + (1 - \chi\_T^\* \chi\_T)) \psi\_{n-1}^{(T)} \\ &= E \phi\_{n-1}^{(T)} . \end{aligned}$$

Therefore, *<sup>χ</sup>Sψ*(*S*) *n* and *<sup>χ</sup>Tψ*(*T*) *n* follow the same recurrence and have the same initial state which means *<sup>χ</sup>Sψ*(*S*) *n* = *<sup>χ</sup>Tψ*(*T*) *n* for any *n* ∈ N.

**Corollary 2.** *Let the initial state for the Grover walk with sinks be φ*0 *with supp*(*φ*0) ⊂ *A*0*. The survival probability γ can be expressed by*

$$\gamma = \left| \left| \phi \right| \right|\_{A}^{2} - \sum\_{n=0}^{\infty} \tau\_{n},$$

*where τn is the outflow of the QW with tails from the internal graph G*0*, i.e.,*

$$\tau\_n = \sum\_{o(a)\in\delta\mathcal{V},\ t(a)\notin A\_0} |\left(\mathcal{U}\_{\complement\mathcal{K}}\chi\_T^\*\phi\_{n-1}^{(T)}\right)(a)|^2$$

**Remark 4.** *The time evolution for φ*(*T*) *n is given by*

$$\phi\_n^{(T)} = E\phi\_{n-1}^{(T)} + \rho\_n$$

*where ρ* = *<sup>χ</sup>TUG*˜*ψ*(*T*) 0 *. In this case, the inflow is ρ* = 0*. On the other hand, in the setting of Theorem 2, ρ is given by a nonzero constant vector.*

˜

Let us now consider a QW with tails with a general initial state Ψ0 ∈ C*<sup>A</sup>* on *G* ˜ . We denote *ν* = *χT*Ψ0 and *ρ* = *<sup>χ</sup>TUG*˜(<sup>1</sup> − *<sup>χ</sup>*<sup>∗</sup>*χ*)<sup>Ψ</sup>0. We summarize the relation between a QW with sinks and a QW for the setting of Theorem 2 in Table 1 from the viewpoint of a QW with tails.

**Table 1.** Relatiion beteween QWs with tails and sinks.


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

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

From the above discussion, we see the importance of the spectral decomposition

$$E = \chi\_S \mathcal{U}\_G \chi\_S^\* = \chi\_T \mathcal{U}\_G \chi\_{T'}^\*$$

to obtain both limit behaviors. The operator *E* is no longer a unitary operator, and, moreover, it is not ensured that it is diagonalizable. The centered generalized eigenspace of *E* is defined by

$$\mathcal{H}\_{\mathfrak{c}} := \{ \psi \in \mathbb{C}^{A\_0} \mid \exists \ m \ge 1 \text{ and } \exists \ |\lambda| = 1 \text{ such that } (E^m - \lambda)\psi = 0 \}$$

Let H*s* be defined by

$$
\mathbb{C}^{\mathcal{A}\_0} = \mathcal{H}\_c \oplus \mathcal{H}\_s.
$$

Here, "⊕" means H*c* and H*s* are complementary spaces, that is, if *uc* + *uv* = 0 for some *uc* ∈ H*c* and *uv* ∈ H*<sup>s</sup>*, then *uc* and *uv* must be *uc* = *uv* = 0. Note that, since *E* is not a normal operator on a vector space H*c* ⊕ H*<sup>s</sup>*, it seems that in general *uc*, *uv* = 0 for *u* ∈ H*c* and H*s* ∈ *N*. However, we can see some important properties of the spectrum of *E* in the following proposition.
