Demonstration of Topological Data Analysis on a Quantum Processor

Several examples for the explanation of Betti numbers, demonstrating their ability to capture structural information even in the presence of local deformations. Betti numbers are a way to describe the connectivity within a topological space. In simplest terms, the k -th Betti number β k counts the the number of k -dimensional holes in a topological space, for example, - β 0 is the number of connected components; - β 1 is the number of planar holes (1-dimensional holes); - β 2 is the number of two-dimensional voids (2-dimensional holes); - ... Betti numbers are topological invariants. If two Betti numbers are the same for two different spaces then the spaces are homotopy equivalent [1]. To demonstrate Betti numbers more 6 vividly, some examples are shown in Figure 1.