Scoring Anomalous Vertices Through Quantum Walks

Abstract: With the constant flow of data from vast sources over the past decades, a plethora of advanced analytical techniques have been developed to extract relevant information from different data types ranging from labeled data, quasi-labeled data, and data with no labels known a priori. For data with at best quasi-labels, graphs are a natural representation of these data types and have important applications in many industries and scientific disciplines. Specifically, for unlabeled data, anomaly detection on graphs is a method to determine which data points do not posses the latent characteristics that is present in most other data. There have been a variety of classical methods to compute an anomalous score for the individual vertices of a respected graph, such as checking the local topology of a node,random walks, and complex neural networks. Leveraging the structure of the graph, we propose a first quantum algorithm to calculate the anomaly score of each node by continuously traversing the graph with a uniform starting position of all nodes. The proposed algorithm incorporates well-known characteristics of quantum random walks, and, taking into consideration the NISQ era and subsequent ISQ era, an adjustment to the algorithm is given to mitigate the increasing depth of the circuit. This algorithm is rigorously shown to converge to the expected probability, with respect to the initial condition.