Quantum computing has ushered in a new era of exploration in machine learning, offering novel tools such as quantum kernels that promise to trans- form the landscape of data analysis and model development. In this report, we delve into the principles of quantum kernels, explore the SWAP test as a key component, and present a comparative analysis of classical and quantum support vector machine (SVM) classifiers using the Iris dataset.

Kernel methods in classical machine learning leverage kernel functions to map data points into higher-dimensional feature spaces. In quantum ma- chine learning, we extend this concept using quantum feature maps to create quantum kernels. Quantum kernels serve as powerful tools to explore the overlap and relationships between quantum states, providing a unique per- spective on data relationships.

Quantum feature maps, such as angle embeddings, encode classical inputs into quantum states. Angle embeddings use rotation gates (RX, RY , or RZ) to represent features, with the rotation angles determined by classical input 1 features. These quantum states are crucial in creating a quantum kernel that captures the essence of the data relationships.

The SWAP test plays a pivotal role in quantum kernel calculations. It esti- mates the squared inner product between two quantum states, providing a crucial metric for measuring their overlap. The SWAP test is formalized as:

P(|0⟩) = |⟨0...0|S(x′)†S(x)|0...0⟩|^2

The quantum kernel κ(x, x′) is calculated using the squared overlap obtained from the SWAP test:

κ(x, x′) = |⟨ϕ(x′)|ϕ(x)⟩|^2

This quantum kernel represents the squared inner product between the quantum states |ϕ(x′)⟩ and |ϕ(x)⟩, offering a quantitative measure of their similarity.

To optimize qubit usage, we apply the inverse embedding with x′ on the same qubits. The projector onto the initial state |0...0⟩⟨0...0| is then measured, ensuring the quantum kernel remains accurate:

 P(|0⟩) = |⟨0...0|S(x′)†S(x)|0...0⟩|^2

To verify the quantum kernel, we incorporate the SWAP test in the calcula- tion:

       ⟨0...0|S(x′)S(x)†MS(x′)†S(x)|0...0⟩    = ⟨0...0|S(x′)S(x)†|0...0⟩⟨0...0|S(x′)†S(x)|0...0⟩

                                             = |⟨0...0|S(x′)†S(x)|0...0⟩|^2

                                             = |⟨ϕ(x′)|ϕ(x)⟩|^2
                                             
                                             = κ(x, x′)

This equation confirms that the quantum kernel κ(x, x′) is obtained through the SWAP test, providing a robust and efficient means of measuring quantum state overlap.

Moving beyond theoretical considerations, we applied the principles of quan- tum kernels to enhance classifier performance. A classical SVM model and a quantum SVM model were trained on the Iris dataset, which contains 150 data points across three classes with four features each.

One potential reason for this can be that kernels play an important role in the loss function of the SVM classifier. Quantum kernels may perform a more efficient embedding of the data into a high-dimensional feature space, capturing intricate relationships that a classical linear kernel might struggle with.

For the classical SVM model, we utilized the scikit-learn library’s SVM algo- rithm. In contrast, the quantum SVM model leveraged quantum-enhanced features to create a more robust classifier. The quantum model was trained using a hybrid approach, combining classical and quantum processing, allow- ing for improved classification accuracy.

In conclusion, the integration of quantum kernels and the SWAP test provides a powerful framework for measuring the overlap and relationships between quantum states. The theoretical underpinnings of quantum kernels were applied practically in the context of classifier enhancement, demonstrating superior performance of the quantum SVM model on the Iris dataset. The results underscore the potential of quantum-enhanced algorithms in advanc- ing machine learning models, paving the way for transformative applications in data analysis and classification tasks.

As we continue to explore the capabilities of quantum computing, future research will focus on scaling quantum kernels for larger datasets and fur- ther optimizing quantum algorithms. The potential for quantum-enhanced machine learning models to address complex real-world problems remains a fascinating avenue for future investigation.