We introduce a new type of discrete quantum walks, called vertex-face walks, based on orientable embeddings. We first establish a spectral correspondence between the transition matrix U and the vertex-face incidence structure. Using the incidence graph, we derive a formula for the principal logarithm of \(U^2\), and find conditions for its underlying digraph to be an oriented graph. In particular, we show this happens if the vertex-face incidence structure forms a partial geometric design. We also explore properties of vertex-face walks on the covers of a graph. Finally, we study a non-classical behavior of vertex-face walks.

中文翻译：

---

量子在嵌入上行走

我们介绍了一种基于可定向嵌入的新型离散量子行走，称为顶点面行走。我们首先在过渡矩阵U和顶点面入射结构之间建立光谱对应关系。使用入射图，我们得出\（U ^ 2 \）的主要对数的公式，并找到其下有向图成为定向图的条件。特别是，我们显示出如果顶点面入射结构形成局部几何设计会发生这种情况。我们还探索了图的封面上顶点面走动的属性。最后，我们研究了顶点面行走的非经典行为。