*2.1. Graph Notation*

Let *G* = ( *V*, *A*) be a connected and *symmetric digraph* such that an arc *a* ∈ *A* if and only if its inverse arc *a* ∈ *A*. The *origin and terminal vertices* of *a* ∈ *A* are denoted by *o*(*a*) ∈ *V* and *<sup>t</sup>*(*a*) ∈ *V*, respectively. Assume that *G* has no multiple arcs. If *<sup>t</sup>*(*a*) = *<sup>o</sup>*(*a*), we call such an arc *a* the *self-loop*. In this paper, we regard *a* = *a* for any self-loops. We denote *Aσ* as the set of all self-loops. The *degree* of *v* ∈ *V* is defined by

$$\deg(v) = |\{a \in A \mid t(a) = v\}|.$$

The *support edge* of *a* ∈ *A* \ *Aσ* is denoted by |*a*| with |*a*| = |*a*|. The set of *(non-directed) edges* is

$$E = \{ |a| \mid a \in A \backslash A\_{\sigma} \}.$$

A *walk* in *G* is a sequence of arcs such that *p* = (*<sup>a</sup>*0, *a*1, ... , *ar*−<sup>1</sup>) with *<sup>t</sup>*(*aj*) = *<sup>o</sup>*(*aj*+<sup>1</sup>) for any *j* = 0, ... ,*r* − 2, which may have the same arcs in *p*. The *cycle* in *G* is a subgraph of *G* which is isomorphic to a sequence of arcs (*<sup>a</sup>*0, *a*1, ... , *ar*−<sup>1</sup>) (*r* ≥ 3) satisfying *<sup>t</sup>*(*aj*) = *<sup>o</sup>*(*aj*+<sup>1</sup>) with *aj* = *aj*+<sup>1</sup> for any *j* = 0, ... ,*r* − 1, where the subscript is the modulus of *r*. We identify (*ak*, *ak*+1, ... , *ak*+*r*−<sup>1</sup>) with (*<sup>a</sup>*0, *a*1, ... , *ar*−<sup>1</sup>) for *k* ∈ Z. The *spanning tree* of *G* is a connected subtree of *G* covering all vertices of *G*. A *fundamental cycle* induced by the spanning tree is the cycle in *G* generated by recovering an arc which is outside of the spanning tree to the spanning tree. There are two choices of orientations for each support of the fundamental cycle, but we choose only one of them as the representative. Fixing a spanning tree, we denote the set of fundamental cycles by Γ. Then, the cardinality of Γ is |*E*|−| *V*| + 1 =: *b*1. We call *b*1 the *first Betti number*.

### *2.2. Definition of the Grover Walk*

Let Ω be a discrete set. The vector space whose standard basis is labeled by each element of Ω is denoted by CΩ. The standard basis is denoted by *δ*(Ω) *ω* (*ω* ∈ Ω), i.e.,

$$
\delta^{(\Omega)}\_{\omega}(\omega') = \begin{cases} 1 & : \omega = \omega', \\ 0 & : \text{otherwise}. \end{cases}
$$

Throughout this paper, the inner product is standard, i.e.,

$$
\langle \psi, \Phi \rangle\_{\Omega} = \sum\_{\omega \in \Omega} \bar{\psi}(\omega) \phi(\omega)\_{\omega}
$$

for any *ψ*, *φ* ∈ CΩ, and the norm is defined by

$$||\psi||\_{\Omega} = \sqrt{\langle \Psi, \psi \rangle\_{\Omega}}.$$

For any *ψ* ∈ CΩ, the support of *ψ* is defined by

$$\operatorname{supp}(\psi) := \{ \omega \in \Omega \mid \psi(\omega) \neq 0 \}.$$

For subspaces *M*, *N* ⊂ CΩ, the relation

$$
\mathbb{C}^{\Omega} = M \oplus N\_{\ast}
$$

means that *M* and *N* are complementary spaces in CΩ, i.e., for any *f* ∈ CΩ, *g* ∈ *M* and *h* ∈ *N* are uniquely determined such that *f* = *g* + *h*, which means, if *u* + *u* = 0 for some *u* ∈ Ω and *u* ∈ Ω, then *u* and *u* must be *u* = *u* = 0. Note that *g*, *h*Ω = 0 in general, i.e., *M* and *N* are not necessarily orthogonal subspaces. Especially in this paper, we treat an operator which is a submatrix of a unitary operator, and we are not ensured that it is a normal operator. The vector space describing the whole system of the Grover walk is C*A*. The time evolution operator of the Grover walk on *G* is defined by

$$(\mathcal{U}\_G \psi)(a) = -\psi(\tilde{a}) + \frac{2}{\deg(o(a))} \sum\_{t(b) = o(a)} \psi(b)$$

for any *ψ* ∈ C*<sup>A</sup>* and *a* ∈ *A*. Note that, since *UG* is a unitary operator on C*A*, *UG* preserves the 2 norm, i.e., ||*UGψ*||<sup>2</sup>*A* = ||*ψ*||<sup>2</sup>*A*. Let *ψn* ∈ C*<sup>A</sup>* be the *n*th iteration of the Grover walk *ψn* = *UGψ<sup>n</sup>*−<sup>1</sup> (*n* ≥ 1) with the initial state *ψ*0. Then, the probability distribution at time *n*, *μn* : *V* → [0, 1], can be defined by

$$\mu\_{\mathfrak{n}}(\upsilon) = \sum\_{t(a) = \upsilon} |\psi\_{\mathfrak{n}}(a)|^2$$

if the norm of the initial state is unity. Our interest is the asymptotic behavior of the sequence of probabilities *μn* and also of amplitudes *ψn* on the graph comparing with the behavior of the corresponding random walk.
