  Quantum machine learning promises to efficiently solve important problems.
There are two persistent challenges in classical machine learning: the lack of
labeled data, and the limit of computational power. We propose a novel
framework that resolves both issues: quantum semi-supervised learning.
Moreover, we provide a protocol in systematically designing quantum machine
learning algorithms with quantum supremacy, which can be extended beyond
quantum semi-supervised learning. In the meantime, we show that naive quantum
matrix product estimation algorithm outperforms the best known classical matrix
multiplication algorithm. We showcase two concrete quantum semi-supervised
learning algorithms: a quantum self-training algorithm named the propagating
nearest-neighbor classifier, and the quantum semi-supervised K-means clustering
algorithm. By doing time complexity analysis, we conclude that they indeed
possess quantum supremacy.
