This paper presents two quantum-enhanced methods for supervised learning, implemented on a superconducting quantum processor. The first method, the quantum variational classifier, uses a variational quantum circuit to classify data by mapping it to a quantum state and finding an optimal separating hyperplane. The second method, the quantum kernel estimator, estimates the kernel function of the quantum feature space directly and implements a conventional support vector machine (SVM). Both methods leverage the large dimensionality of the quantum Hilbert space to achieve a quantum advantage over classical approaches. The quantum variational classifier maps classical data to a quantum state using a feature map circuit, then applies a variational quantum circuit to find an optimal separating hyperplane. The classifier uses a binary measurement to determine the label of a data point. The quantum kernel estimator estimates the kernel function between data points using a quantum circuit and then applies a classical SVM to classify new data points. Both methods are tested on artificial data that can be perfectly classified using the quantum feature map. The quantum variational classifier achieves high classification success rates, even in the presence of noise. The quantum kernel estimator also achieves high classification success rates, with the best results for data sets where the kernel cannot be estimated classically. The paper demonstrates that quantum-enhanced feature spaces can provide a quantum advantage in supervised learning tasks. The methods are implemented on a superconducting quantum processor with five coupled transmons. The quantum variational classifier uses a variational circuit with up to four layers, while the quantum kernel estimator estimates the kernel function directly using a quantum circuit. The results show that both methods can achieve high classification success rates, with the quantum variational classifier achieving up to 100% success for certain data sets. The quantum kernel estimator also achieves high success rates, with the best results for data sets where the kernel cannot be estimated classically. The paper concludes that quantum-enhanced feature spaces can provide a quantum advantage in supervised learning tasks, and that further research is needed to find suitable feature maps for this technique with provable quantum advantages.This paper presents two quantum-enhanced methods for supervised learning, implemented on a superconducting quantum processor. The first method, the quantum variational classifier, uses a variational quantum circuit to classify data by mapping it to a quantum state and finding an optimal separating hyperplane. The second method, the quantum kernel estimator, estimates the kernel function of the quantum feature space directly and implements a conventional support vector machine (SVM). Both methods leverage the large dimensionality of the quantum Hilbert space to achieve a quantum advantage over classical approaches. The quantum variational classifier maps classical data to a quantum state using a feature map circuit, then applies a variational quantum circuit to find an optimal separating hyperplane. The classifier uses a binary measurement to determine the label of a data point. The quantum kernel estimator estimates the kernel function between data points using a quantum circuit and then applies a classical SVM to classify new data points. Both methods are tested on artificial data that can be perfectly classified using the quantum feature map. The quantum variational classifier achieves high classification success rates, even in the presence of noise. The quantum kernel estimator also achieves high classification success rates, with the best results for data sets where the kernel cannot be estimated classically. The paper demonstrates that quantum-enhanced feature spaces can provide a quantum advantage in supervised learning tasks. The methods are implemented on a superconducting quantum processor with five coupled transmons. The quantum variational classifier uses a variational circuit with up to four layers, while the quantum kernel estimator estimates the kernel function directly using a quantum circuit. The results show that both methods can achieve high classification success rates, with the quantum variational classifier achieving up to 100% success for certain data sets. The quantum kernel estimator also achieves high success rates, with the best results for data sets where the kernel cannot be estimated classically. The paper concludes that quantum-enhanced feature spaces can provide a quantum advantage in supervised learning tasks, and that further research is needed to find suitable feature maps for this technique with provable quantum advantages.