The k-nearest neighbors (k-NN) is a basic machine learning (ML) algorithm, and several quantum versions of it, employing different distance metrics, have been presented in the last few years. Although the Euclidean distance is one of the most widely used distance metrics in ML, it has not received much consideration in the development of these quantum variants. In this article, a novel quantum k-NN algorithm based on the Euclidean distance is introduced. Specifically, the algorithm is characterized by a quantum encoding requiring a low number of qubits and a simple quantum circuit not involving oracles, aspects that favor its realization. In addition to the mathematical formulation and some complexity observations, a detailed empirical evaluation with simulations is presented. In particular, the results have shown the correctness of the formulation, a drop in the performance of the algorithm when the number of measurements is limited, the competitiveness with respect to some classical baseline methods in the ideal case, and the possibility of improving the performance by increasing the number of measurements.

Classical shadows constitute a protocol to estimate the expectation values of a collection of M observables acting on O(1) qubits of an unknown n-qubit state with a number of measurements that is independent of n and that grows only logarithmically with M. We propose a local variant of the quantum Wasserstein distance of order 1 of De Palma et al. [IEEE Trans. Inf. Theory 67, 6627–6643 (2021)] and prove that the classical shadow obtained measuring O(log n) copies of the state to be learned constitutes an accurate estimate with respect to the proposed distance. We apply the results to quantum generative adversarial networks, showing that quantum access to the state to be learned can be useful only when some prior information on such state is available.

Support vector machines (SVMs) are widely used machine learning models, with formulations for both classification and regression tasks. In the last years, with the advent of working quantum annealers, hybrid SVM models characterised by quantum training and classical execution have been introduced. These models have demonstrated comparable performance to their classical counterparts. However, they are limited in the training set size due to the restricted connectivity of the current quantum annealers. Hence, to take advantage of large datasets, a strategy is required. In the classical domain, local SVMs, namely, SVMs trained on the data samples selected by a $k$ -nearest neighbors model, have already proven successful. Here, the local application of quantum-trained SVM models is proposed and empirically assessed. In particular, this approach allows overcoming the constraints on the training set size of the quantum-trained models while enhancing their performance. In practice, the Fast Local Kernel Support Vector Machine (FaLK-SVM) method, designed for efficient local SVMs, has been combined with quantum-trained SVM models for binary and multiclass classification. In addition, for comparison, FaLK-SVM has been interfaced for the first time with a classical single-step multiclass SVM model (CS SVM). Concerning the empirical evaluation, D-Wave's quantum annealers and real-world datasets taken from the remote sensing domain have been employed. The results have shown the effectiveness and scalability of the proposed approach, but also its practical applicability in a real-world large-scale scenario.