Limitations of Clustering Using Quantum Persistent Homology

Abstract

Different algorithms can be used for clustering purposes with data sets. On of these algorithms, uses topological features extracted from the data set to base the clusters on. The complexity of this algorithm is however exponential in the number of data points. Recently a quantum algorithm was proposed by Lloyd Garnerone and Zanardi with claimed polynomial complexity, hence an exponential improved over classical algorithms. However, we show that this algorithm in general cannot be used to compute these topological features in any dimension but the zeroth. We also give pointers on how to still use the algorithm for clustering purposes.