Scoring Anomalous Vertices Through Quantum Walks

TL;DR

This work introduces a quantum algorithm to score anomalous vertices in graphs by running continuous-time quantum walks from a uniform superposition and leveraging an averaging (mixing) framework to converge to a distribution that reveals traversal patterns. It analyzes generators based on the adjacency, Laplacian, and maximal-entropy matrices, and extends to directed graphs via Hermitian adjacency constructions, enabling robust per-node anomaly scoring in both symmetric and asymmetric graphs. To address NISQ limitations, the authors propose modular walk iterations with data-loading techniques to mitigate circuit depth while preserving convergence to the expected distribution. Experimental results on toy graphs demonstrate that adjacency-based CTQW provides balanced exploration and effective outlier discrimination, with classical walks offering a useful baseline. The paper lays groundwork for hardware-aware quantum anomaly detection on graphs and points to future work on real datasets, feature integration, and comparisons across graph families and DTQW versus CTQW frameworks.

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.

Figures (9)

• Figure 1: Anomaly scores for the adjacency, Laplacian, and MEA matrices, and a classical random walk. The node labels are converted to binary. the for the classical walk the damping term has value 0 and there is an error threshold of .05.
• Figure 2: The undirected graph that is randomly generated with weights randomly generated and are within the unit interval. To assist in visualizing the graph, the edges of the vertices are split into three groups, values less than $1/3$ are identified with the blue dashed edge, values greater than $1/3$ and less than $2/3$ are identified by the green dotted edges, and values greater than $2/3$ are identified by the red solid line.
• Figure 3: Eigenvalues and total energy evolution for the adjacency Hamiltonian.
• Figure 4: Eigenvalues and total energy evolution for the Laplacian Hamiltonian.
• Figure 5: Eigenvalues and total energy evolution for the MEA Hamiltonian.

Theorems & Definitions (6)

• Remark 1
• Proposition 1
• Remark 2
• Claim 1
• proof
• Remark 3