We provide a mathematically and conceptually robust notion of quantum superpositions of graphs. We argue that, crucially, quantum superpositions of graphs require node names for their correct alignment, which we demonstrate through a no-signalling argument. Nevertheless, node names are a fiducial construct, serving a similar purpose to the labelling of points through a choice of coordinates in continuous space. Graph renamings, aka isomorphisms, are understood as a change of coordinates on the graph and correspond to a natively discrete analogue of continuous diffeomorphisms. We postulate renaming invariance as a symmetry principle in discrete topology of similar weight to diffeomorphism invariance in the continuous. We explain how to impose renaming invariance at the level of quantum superpositions of graphs, in a way that still allows us to talk about an observable centred at a specific node.

翻译：我们提出了一个在数学和概念上都稳健的图的量子叠加概念。我们认为，关键在于，图的量子叠加需要节点名称以实现正确对齐，这一点通过无信号性论证得以证明。尽管节点名称是一种基准构造，其作用类似于通过坐标选择在连续空间中标记点。图的重命名（即同构）被理解为图上的坐标变换，对应连续微分同胚的本征离散类似物。我们将重命名不变性设定为离散拓扑中的对称性原理，其重要性与连续情形下的微分同胚不变性相当。我们阐释了如何在图的量子叠加层面施加重命名不变性，且这种方式仍允许我们谈论聚焦于特定节点的可观测量。