Generalized Quantum Hadamard Test for Machine Learning

Quantum machine learning models are designed for performing learning tasks. Some quantum classifier models are proposed to assign classes of inputs based on fidelity measurements. Quantum Hadamard test is a well-known quantum algorithm for computing these fidelities. However, the basic requirement for deploying the quantum Hadamard test maps input space to L2-normalize vector space. Consequently, computed fidelities correspond to cosine similarities in mapped input space. We propose a quantum Hadamard test with the additional capability to compute the inner product in bounded input space, which refers to the Generalized Quantum Hadamard test. It incorporates not only L2-normalization of input space but also other standardization methods, such as Min-max normalization. This capability is raised due to different quantum feature mapping and unitary evolution of the mapped quantum state. We discuss the quantum circuital implementation of our algorithm and establish this circuit design through numerical simulation. Our circuital architecture is efficient in terms of computational complexities. We show the application of our algorithm by integrating it with two classical machine learning models: Logistic regression binary classifier and Centroid-based binary classifier and solve four classification problems over two public-benchmark datasets and two artificial datasets.