Quantum embedding is a fundamental prerequisite for applying quantum machine learning techniques to classical data, and has substantial impacts on performance outcomes. In this study, we present Neural Quantum Embedding (NQE), a method that efficiently optimizes quantum embedding beyond the limitations of positive and trace-preserving maps by leveraging classical deep learning techniques. NQE enhances the lower bound of the empirical risk, leading to substantial improvements in classification performance. Moreover, NQE improves robustness against noise. To validate the effectiveness of NQE, we conduct experiments on IBM quantum devices for image data classification, resulting in a remarkable accuracy enhancement from 0.52 to 0.96. In addition, numerical analyses highlight that NQE simultaneously improves the trainability and generalization performance of quantum neural networks, as well as of the quantum kernel method.