*2.3. Boundary Operators*

Let *G* = (*<sup>V</sup>*, *A*) be the original graph. The set of sinks is denoted by *Vs* ⊂ *V*. The subgraph of *G*; *G*0 = (*<sup>V</sup>*0, *<sup>A</sup>*0), is defined by

$$V\_0 = V \backslash V\_s,\ A\_0 = \{ a \in A \mid t(a), o(a) \notin V\_s \}.$$

The set of self-loops in *G*0 is denoted by *A*0,*σ* ⊂ *A*0 (see Figure 1). The set of the fundamental cycles in *G*0 is denoted by Γ hereafter. The set of boundary vertices of *G*0 is defined by

$$\delta G\_0 = \{ o(a) \mid a \in A, \ o(a) \in V \backslash V\_s, \ t(a) \in V\_s \}.$$

This means that *δG*0 consists of the origins of arcs flowing into the sinks. Under the above settings of graphs, let us now prepare some notations to show our main theorem.

**Figure 1. The setting of graphs:** The original graph *G* is depicted in the left corner. The sinks *Vs* are the white vertices. The subgraph *G*0 of *G* is the black colored graph in the center. The set of boundary vertices *δV* is {2, <sup>4</sup>}. The semi-infinite graph *G*˜ is constructed by connecting the infinite length path to each boundary vertex of *G*0.

**Definition 1.** *Let* deg(*u*) *be the degree of u in the original graph G. Let G*0 = (*<sup>V</sup>*0, *<sup>A</sup>*0) *be the subgraph as above. Then, the boundary operators d*1 : C*A*0 → C*V*0 *and ∂*2 : C<sup>Γ</sup> → C*A*0 *are denoted by*

$$(d\_1\psi)(v) = \frac{1}{\sqrt{\deg(v)}} \sum\_{t(a) = v} \psi(a), \ (\partial\_2 \Psi)(a) = \sum\_{a \in A(c) \subset A\_0} \Psi(c),$$

*respectively, for any ψ* ∈ C*A,* Ψ ∈ C<sup>Γ</sup> *and v* ∈ *V*0*, a* ∈ *A*0*. Here, <sup>A</sup>*(*c*) *is the set of arcs of c* ∈ Γ*.*

The boundary operator *d*1 has the following matrix representation

$$(d\_1)\_{\mathfrak{u},\mathfrak{a}} = \begin{cases} 1/\sqrt{\deg(\mathfrak{u})} & : t(\mathfrak{a}) = \mathfrak{u},\\ 0 & : \text{otherwise}, \end{cases}$$

while the boundary operator *∂*2 has the following matrix representation

$$(\partial\_2)\_{a,c} = \begin{cases} 1 & : a \in A(c), \\ 0 & : \text{otherwise}. \end{cases}$$

Note that deg(*u*) is the degree of *G*; thus, if *u* ∈ *δG*0, then deg(*u*) is greater than the degree in *G*0. The adjoint operators of *d*1 and *∂*2 are defined by

$$
\langle f, d\_1 \psi \rangle\_{V\_0} = \langle d\_1^\* f, \psi \rangle\_{A\_{0'}} \quad \langle \psi, \partial\_2 \Psi \rangle\_{A\_0} = \langle \partial\_2^\* \psi, \Psi \rangle\_{\Gamma}
$$

which imply

$$(d\_1^\*f)(a) = f(t(a)), \ (\partial\_2^\*\psi)(c) = \sum\_{a \in A(c)} \psi(a).$$

Let *S* : C*A*0 → C*A*0 be a unitary operator defined by (*<sup>S</sup>ψ*)(*a*) = *ψ*(*a*). We prove that the composition of *d*1(*<sup>I</sup>* − *S*) ◦ *∂*2 is identically equal to zero as follows.

**Lemma 1.** *Let d*1 *and ∂*2 *be the above. Then, we have*

$$d\_1(I - \mathcal{S})\partial\_2 = 0.$$

**Proof.** For any *c* ∈ Γ, let *δ*(Γ) *c* ∈ C<sup>Γ</sup> be the delta function, i.e.,

$$\delta\_c^{(\Gamma)}(c') = \begin{cases} 1 & : c = c', \\ 0 & : c \neq c'. \end{cases}$$

Then, it is enough to see that *d*1(*<sup>I</sup>* − *<sup>S</sup>*)*∂*2*δ*(Γ) *c* = 0 for any *c* ∈ Γ. Indeed, we find

$$\begin{aligned} d\_1(I - S)\partial\_2\delta\_\epsilon^{(\Gamma)} &= d\_1(\sum\_{a \in A(\epsilon)} \delta\_a^{(A)} - \sum\_{a \in A(\epsilon)} \delta\_a^{(A)}) \\ &= \sum\_{a \in A(\epsilon)} \frac{1}{\sqrt{\deg(t(a))}} \delta\_{t(a)}^{(V)} - \sum\_{a \in A(\epsilon)} \frac{1}{\sqrt{\deg(t(\overline{a}))}} \delta\_{t(\overline{a})}^{(V)} \\ &= 0, \end{aligned}$$

which is the desired conclusion.

> Let us set the function *ξ*(+) *c* induced by *c* ∈ Γ by

$$
\xi\_{\varepsilon}^{(+)} := (I - \mathcal{S}) \partial\_2 \delta\_{\varepsilon}^{(\Gamma)}.
$$

.

In other words, supp(*ξ*(+) *c* ) = *<sup>A</sup>*(*c*) ∪ *<sup>A</sup>*(*c*¯) and

$$(\xi\_{\mathcal{c}}^{(+)})(a) = \begin{cases} 1 & : a \in A(\mathcal{c}), \\ -1 & : \overline{a} \in A(\mathcal{c}), \\ 0 & : \text{otherwise}. \end{cases}$$

The function *ξ*(+) *c* represents the fundamental cycle *c*. Let us introduce *χS* : C*<sup>A</sup>* → C*A*0 by

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

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

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

A matrix representation of *χS* is expressed as follows:

$$\chi\_{\mathcal{S}} \cong [I\_{A\_{\mathbb{Q}}} \mid 0 \mid ],$$

which is a |*<sup>A</sup>*0|×|*A*| matrix. The function *ξ*(+) *c* satisfies the following properties:

**Proposition 1.** *For any fundamental cycle c in G*0 ⊂ *G, we have <sup>χ</sup>*<sup>∗</sup>*Sξ*(+) *c* ∈ ker(1 − *UG*)*.* **Proof.** The following direct computation gives the consequence:

$$\begin{split} (\mathcal{U}\_{\mathcal{G}} \chi\_{\mathcal{G}}^{\*} \mathfrak{z}\_{c}^{(+)})(a) &= - (\chi\_{\mathcal{G}}^{\*} \mathfrak{z}\_{c}^{(+)})(\overline{a}) + \frac{2}{\deg(o(a))} \sum\_{t(b) = o(a)} (\chi\_{\mathcal{G}}^{\*} \mathfrak{z}\_{c}^{(+)})(b) \\ &= (\chi\_{\mathcal{G}}^{\*} \mathfrak{z}\_{c}^{(+)})(a) + \frac{2}{\sqrt{\deg(o(a))}} (d\_{1} \chi\_{\mathcal{G}}^{\*} \mathfrak{z}\_{c}^{(+)})(o(a)) \\ &= (\chi\_{\mathcal{G}}^{\*} \mathfrak{z}\_{c}^{(+)})(a) .\end{split}$$

Here, the first equality derives from the definition of *UG*. In the second equality, since supp(*ξ*(+) *c* ) ⊂ *A*0 ⊂ *A* and the summation of RHS in the first equality is essentially the same as the one over *A*0, we can apply the definition of *d*1 to this. We use Lemma 1 in the last equality.

We set K ⊂ C*A*0 by

$$\mathcal{K} = \text{span}\{\chi\_S \mathfrak{f}\_c^{(+)} \mid \mathcal{c} \in \Gamma \subset \mathcal{G}\_0\}.\tag{1}$$

The self-adjoint operator

$$T := (\chi\_S d\_1) S (\chi\_S d\_1)^\* $$

on C*A*0 is similar to the transition probability operator *P* with the Dirichlet boundary condition on *δV*0; i.e.,

$$P' = D^{-1/2} T D^{1/2} \rho$$

where (*D f*) = deg(*u*)*f*(*u*). Here, the matrix representation of *P* is described by

$$\langle (P')\_{\boldsymbol{u}, \boldsymbol{v}} := \langle \delta\_{\boldsymbol{u}}^{(V\_0)}, P' \delta\_{\boldsymbol{v}}^{(V\_0)} \rangle \boldsymbol{v}\_0 = \begin{cases} 1/\deg(\boldsymbol{u}) & \text{if } \boldsymbol{u} \text{ and } \boldsymbol{v} \text{ are connected,} \\ 0 & \text{: otherwise,} \end{cases}$$

for any *u*, *v* ∈ *V*0. If *T f* = *x f* and *Tg* = *yg* (*x* = *y*), then we find the orthogonality such that

$$\begin{aligned} \langle (1 - e^{i \arccos x} S) d\_1^\* f, (1 - e^{-i \arccos y} S) d\_1^\* g \rangle &= 0, \\ \langle (1 - e^{i \arccos x} S) d\_1^\* f, (1 - e^{i \arccos y} S) d\_1^\* g \rangle &= 0, \\ \langle (1 - e^{i \arccos x} S) d\_1^\* f, (1 - e^{-i \arccos y} S) d\_1^\* g \rangle &= 0. \end{aligned}$$

Then, we set T ⊂ C*A*0 by

$$\mathcal{T} = \bigoplus\_{|\lambda|=1} \{ (1 - \lambda S) d\_1^\* f \mid f \in \ker((\lambda + \lambda^{-1})/2 - T), \ \operatorname{supp}(f) \subset V\_0 \mid \delta V\_0 \}. \tag{2}$$

This is the subspace of C*A*0 lifted up from the eigenfunctions in C*V*0 of the Dirichlet cut random walk *T* by (1 − *<sup>λ</sup>S*)*d*<sup>∗</sup>1 *f* . It is shown that Spec(*E*) ⊂ D, where D is the unit disc {*z* ∈ C | |*z*| ≤ 1} in Proposition 3, and T = ⊕|*λ*|=1, *λ*=±1 ker(*λ* − *<sup>E</sup>*), where *E* := *<sup>χ</sup>SUGχ*<sup>∗</sup>*S* in Lemma 3.

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

Let *G* = (*<sup>V</sup>*, *A*) be a finite and connected graph with sinks *Vs* = {*<sup>v</sup>*1, ... , *vq*} ⊂ *V*. We consider the subgraph *G*0 = (*<sup>V</sup>*0, *<sup>A</sup>*0) as defined in Section 2.3. Assume that *G*0 is connected. For simplicity, in this paper, we consider the initial state of the Grover walk *φ*0 that satisfies the condition supp(*φ*0) ⊂ *A*0. (If we consider general initial state *φ*0 such that supp(*φ*0) ∩ (*A* \ *<sup>A</sup>*0) = ∅, replacing *φ*0 into *φ*0 = *φ*1, we can reproduce the QW with this

initial state after *n* ≥ 1 by our setting.) The time evolution of the Grover walk with sinks *Vs* with such an initial state *φ*0 is defined by

$$\phi\_{\mathcal{U}}(a) = \begin{cases} (\mathcal{U}\_G \phi\_{n-1})(a) & : t(a) \in V \nmid V\_{s\_\star} \\ 0 & : t(a) \in V\_{s\_\star} \end{cases} \tag{3}$$

This means that a quantum walker at a sink falls into a pit trap. We are interested in the survival probability of the Grover walk defined by

$$\gamma := \lim\_{n \to \infty} \sum\_{a \in \mathcal{A}} |\phi\_n(a)|^2.$$

It is the probability that the quantum walker remains in the graph without falling into the sinks forever. Considering the corresponding isotropic random walk with sinks such that 

$$p\_n(v) = \begin{cases} (P p\_{n-1})(v) & : v \in V \backslash V\_{s\prime} \\ 0 & : v \in V\_{s\prime} \end{cases}$$

we find that its survival probability is zero,

$$\gamma^{\mathrm{RW}} := \lim\_{\mathfrak{u} \to \infty} \sum\_{\upsilon \in V} p\_{\mathfrak{u}}(\upsilon) = 0\_{\tau}$$

because the first hitting time of a random walk to an arbitrary vertex for a finite graph is finite. On the other hand, in the case of the Grover walk, the survival probability becomes positive, up to the initial state. In this paper, we clarify a necessary and sufficient condition for *γ* > 0.
