QUACK: Quantum Aligned Centroid Kernel

Quantum computing (QC) seems to show potential for application in machine learning (ML). In particular quantum kernel methods (QKM) exhibit promising properties for use in supervised ML tasks. However, a major disadvantage of kernel methods is their unfavorable quadratic scaling with the number of training samples. Together with the limits imposed by currently available quantum hardware (NISQ devices) with their low qubit coherence times, small number of qubits, and high error rates, the use of QC in ML at an industrially relevant scale is currently impossible. As a small step in improving the potential applications of QKMs, we introduce QUACK, a quantum kernel algorithm whose time complexity scales linear with the number of samples during training, and independent of the number of training samples in the inference stage. In the training process, only the kernel entries for the samples and the centers of the classes are calculated, i.e. the maximum shape of the kernel for n samples and c classes is (n, c). During training, the parameters of the quantum kernel and the positions of the centroids are optimized iteratively. In the inference stage, for every new sample the circuit is only evaluated for every centroid, i.e. c times. We show that the QUACK algorithm nevertheless provides satisfactory results and can perform at a similar level as classical kernel methods with quadratic scaling during training. In addition, our (simulated) algorithm is able to handle high-dimensional datasets such as MNIST with 784 features without any dimensionality reduction.

Semisupervised Anomaly Detection using Support Vector Regression with Quantum Kernel

Anomaly detection (AD) involves identifying observations or events that deviate in some way from the rest of the data. Machine learning techniques have shown success in automating this process by detecting hidden patterns and deviations in large-scale data. The potential of quantum computing for machine learning has been widely recognized, leading to extensive research efforts to develop suitable quantum machine learning (QML) algorithms. In particular, the search for QML algorithms for near-term NISQ devices is in full swing. However, NISQ devices pose additional challenges due to their limited qubit coherence times, low number of qubits, and high error rates. Kernel methods based on quantum kernel estimation have emerged as a promising approach to QML on NISQ devices, offering theoretical guarantees, versatility, and compatibility with NISQ constraints. Especially support vector machines (SVM) utilizing quantum kernel estimation have shown success in various supervised learning tasks. However, in the context of AD, semisupervised learning is of great relevance, and yet there is limited research published in this area. This paper introduces an approach to semisupervised AD based on the reconstruction loss of a support vector regression (SVR) with quantum kernel. This novel model is an alternative to the variational quantum and quantum kernel one-class classifiers, and is compared to a quantum autoencoder as quantum baseline and a SVR with radial-basis-function (RBF) kernel as well as a classical autoencoder as classical baselines. The models are benchmarked extensively on 10 real-world AD data sets and one toy data set, and it is shown that our SVR model with quantum kernel performs better than the SVR with RBF kernel as well as all other models, achieving highest mean AUC over all data sets. In addition, our QSVR outperforms the quantum autoencoder on 9 out of 11 data sets.