Open set anomaly detection (OSAD) is a crucial task that aims to identify abnormal patterns or behaviors in data sets, especially when the anomalies observed during training do not represent all possible classes of anomalies. The recent advances in quantum computing in handling complex data structures and improving machine learning models herald a paradigm shift in anomaly detection methodologies. This study proposes a Quantum Scoring Module (Qsco), embedding quantum variational circuits into neural networks to enhance the model's processing capabilities in handling uncertainty and unlabeled data. Extensive experiments conducted across eight real-world anomaly detection datasets demonstrate our model's superior performance in detecting anomalies across varied settings and reveal that integrating quantum simulators does not result in prohibitive time complexities. Our study validates the feasibility of quantum-enhanced anomaly detection methods in practical applications.