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Low resolution of positron emission tomography (PET) limits its diagnostic performance. Deep learning has been successfully applied to achieve super-resolution PET. However, commonly used supervised learning methods in this context require many pairs of low- and high-resolution (LR and HR) PET images. Although unsupervised learning utilizes unpaired images, the results are not as good as that obtained with supervised deep learning. In this paper, we propose a quasi-supervised learning method, which is a new type of weakly-supervised learning methods, to recover HR PET images from LR counterparts by leveraging similarity between unpaired LR and HR image patches. Specifically, LR image patches are taken from a patient as inputs, while the most similar HR patches from other patients are found as labels. The similarity between the matched HR and LR patches serves as a prior for network construction. Our proposed method can be implemented by designing a new network or modifying an existing network. As an example in this study, we have modified the cycle-consistent generative adversarial network (CycleGAN) for super-resolution PET. Our numerical and experimental results qualitatively and quantitatively show the merits of our method relative to the state-ofthe-art methods. The code is publicly available at https://github.com/PigYang-ops/CycleGAN-QSDL.