Transport distance between Grover walks on graphs and coarse Ricci curvature

One direction to investigate the relation between quantum walks and their underlying graphs is to define geometric quantity concerning quantum walks. In order to contribute to this direction, we define a transport distance between Grover walks, which can be seen as a quantum analogue of symmetric random walks. We employ signed optimal transport theory to formulate this distance. Also, we define coarse Ricci curvature induced by Grover walks and investigate its property. It has been found that this coarse Ricci curvature has similar properties to those of coarse Ricci curvature induced by random walks.