Anomaly detection is used for identifying data that does not obey the `normal' data patterns and it nds diverse applications in many important areas like fraud detection, diagnoses, data cleaning and surveillance. With the advent of quantum computing and the future of quantum computing in the cloud and over the quantum internet, anomaly detection will inevitably play an important role in revealing novel or unusual patterns in communicated quantum data. Machine learning algorithms are playing pivotal roles in classical computing, including in anomaly detection. Two widely-used algorithms include kernel principal component analysis and one-class support vector machine, which we extend to the quantum domain and apply them to quantum data. This is one of the rst instances of a quantum machine learning algorithm applied to quantum data itself. We show that these two quantum algorithms can have an exponential speed-up in the dimensionality of quantum states compared to the analogous classical algorithms. In the realm of pure quantum states, there can also be an exponential speed-up in the number of quantum states. This makes these algorithms potentially applicable to big quantum data applications.