Quantum walks on graphs embedded in orientable surfaces

Abstract

A quantum walk model which reflects the $2$-cell embedding on the orientable closed surface of a graph in the dynamics is introduced. We show that the scattering matrix is obtained by finding the faces on the underlying surface which have the overlap to the boundary and the stationary state is obtained by counting two classes of the rooted spanning subgraphs of the dual graph on the underlying embedding.

Figures (12)

• Figure 1: The underlying closed surface of $K_{3,3}$: It is known that the minimum genus of the underlying orientable closed surface for $K_{3,3,}$ is $\gamma(K_{3,3})=1$. On the other hand, since the minimum number of the facial walks is $1$ (see the right bottom of the figure), the maximum number of that is $\gamma_M(K_{3,3,})=2$ by (\ref{['eq:genus']}). Three kinds of rotations are depicted (the first column); we call them $[10,4,4]$-type, $[6,6,6]$-type and $[18]$-type, respectively. The orientable underlying surface and the $2$-cell embedding of $K_{3,3}$ are determined by the rotation (the second column): the genuses of the underlying surfaces induced by the first and second rotations, $[10,4,4]$ and $[6,6,6]$ types, are $1$, but their embedding ways are different because there are $2$-square faces and $1$-decagon faces in the first embedding while there are $3$ hexagon faces in the second embedding (the third column). On the other hand, the third rotation, $[18]$-type, needs the maximal genus $\gamma_M(K_{3,3})=2$ of the underlying orientable surface. The way to assign the vertices on the closed surface is as follows: (i) on each $v\in V$, arrange radially the edges connected to $v$ following the rotation $\rho_v$; (ii) connect the vertices so that on each edge, the rotations of the two end vertices must be opposite direction MT.
• Figure 2: A rotation tailed graph $(G;\delta V;\rho)$ and its blow up graph $G^{BU}$: In the figure of $(G;\delta V;\rho)$, the white vertices are $\delta V$ and each clockwise circle describes the rotation on each vertex. In the figure of $G^{BU}$, the dotted lines are the arcs in $A_{is}$ while the real lines are the arcs in $A_{br}$.
• Figure 3: The names of arcs of $G^{BU}=(V^{BU},A^{BU})$. The island$A_{is}$ is the set of the oriented cycles induced by blowing up the vertices of the original graph. The bridge$A_{br}$ is the set of arcs which is isomorphic to the symmetric arc set of the original graph. The set of $A_{tl}$ is the set of arcs of the tails. The subset $\delta A_{pr}\subset A_{tl}$ called the pier is the set of arcs whose terminal or origin vertices belongs to the internal graph. The quay$\delta A_{qy}$ is the pair of island arcs $(\xi^{out},\xi^{in})$, where $t(\xi^{out})=o(\xi^{in})$ is connected to a tail.
• Figure 4: The internal and external facial walks and the dual graph: The closed oriented cycles correspond to facial walks in $G^{BU}$. The facial walk which passes through the tails, $f_*$, is the external facial walk, while the other facial walks, $f_1^{in},f_2^{in},f_3^{in}$, are the internal facial walk. The right figure illustrating that each facial walk in the left figure corresponds to each vertex and the external facial walk corresponds to the sink, will be used for the discussion in Section \ref{['sect:4.3.3']}.
• Figure 5: The location of tails in each embedding of $K_{3,3}$ for Section \ref{['sect:ex1']}: The bold (green) lines depict the tails in the setting of Section \ref{['sect:ex1']}. The inflow comes from only the tail joining to the vertex $1$. Let us call this tail an incoming tail. For the embedding [4,4,10], the number of tails which is included in the same face as the incoming tail is $N=4$; for the embedding [6,6,6], the number of tails which is included in the same face as the incoming tail is $N=2$; for the embedding [18], the number of tails which is included in the same face as the incoming tail is $N=6$.

Theorems & Definitions (21)

• Theorem 1.1: Duke's interpolation theorem
• Definition 2.1: Facial quantum walk
• Theorem 2.1: Theorem 3.1 in HS
• Lemma 2.1: Key lemma
• proof
• Remark 2.1
• Theorem 3.1
• proof
• Remark 3.1
• Remark 3.2