Approximate complex amplitude encoding algorithm and its application to data classification problems

Quantum computing has a potential to accelerate data processing efficiency, especially in machine learning, by exploiting special features such as the quantum interference. The major challenge in this application is that, in general, the task of loading a classical data vector into a quantum state requires an exponential number of quantum gates. The approximate amplitude encoding (AAE) method, which uses a variational means to approximately load a given real-valued data vector into the amplitude of a quantum state, was recently proposed as a general approach to this problem mainly for near-term devices. However, AAE cannot load a complex-valued data vector, which narrows its application range. In this work, we extend AAE so that it can handle a complex-valued data vector. The key idea is to employ the fidelity distance as a cost function for optimizing a parametrized quantum circuit, where the classical shadow technique is used to efficiently estimate the fidelity and its gradient. We apply this algorithm to realize the complex-valued-kernel binary classifier called the compact Hadamard classifier, and then we present a numerical experiment showing that it enables classification of the Iris dataset and credit card fraud detection.