标题： 嵌入曲面上图的量子漫步的舒适性 显示英文标题

标题： Comfortability of quantum walks on embedded graphs on surfaces

摘要： 离散时间量子行走的时间演化由图的局部邻接关系决定。 在本文中，首先，我们构建了一个反映嵌入在曲面上的离散时间量子行走模型，使得底层全局几何信息得以体现。 其次，我们考虑了这个量子行走模型的散射问题。 我们得到了由曲面上的面所表征的散射矩阵，并利用散射信息检测了嵌入的可定向性。 在散射问题中的平稳态中，舒适度被定义为限制在内部的平稳态的平方范数。 这表明在嵌入下量子行走者如何存储在内部。 然后我们发现，在某些自然设置下，量子行走者在亏格较小的曲面上会感到更舒适。 我们通过一些有趣的例子来说明我们的结果。 显示英文摘要

摘要： The time evolutions of discrete-time quantum walks on graphs are determined by the local adjacency relations of the graphs. In this paper, first, we construct a discrete-time quantum walk model that reflects the embedding on the surface so that an underlying global geometric information is reflected. Second, we consider the scattering problem of this quantum walk model. We obtain the scattering matrix characterized by the faces on the surface and detect the orientablility of the embedding using scattering information. For the stationary state in the scattering problem, the comfortability is defined as the square norm of the stationary state restricted to the internal. This indicates how a quantum walker is stored in the internal under the embedding. Then we find that a quantum walker feels more comfortable on a surface with small genus in some natural setting. We illustrate our results with some interesting examples.