Positron emission tomography (PET) with fluorodeoxyglucose (FDG) or florbetapir (AV45) has been proved effective in the diagnosis of Alzheimer's disease. However, the expensive and radioactive nature of PET has limited its application. Here, employing multi-layer perceptron mixer architecture, we present a deep learning model, namely 3-dimensional multi-task multi-layer perceptron mixer, for simultaneously predicting the standardized uptake value ratios (SUVRs) for FDG-PET and AV45-PET from the cheap and widely used structural magnetic resonance imaging data, and the model can be further used for Alzheimer's disease diagnosis based on embedding features derived from SUVR prediction. Experiment results demonstrate the high prediction accuracy of the proposed method for FDG/AV45-PET SUVRs, where we achieved Pearson's correlation coefficients of 0.66 and 0.61 respectively between the estimated and actual SUVR and the estimated SUVRs also show high sensitivity and distinct longitudinal patterns for different disease status. By taking into account PET embedding features, the proposed method outperforms other competing methods on five independent datasets in the diagnosis of Alzheimer's disease and discriminating between stable and progressive mild cognitive impairments, achieving the area under receiver operating characteristic curves of 0.968 and 0.776 respectively on ADNI dataset, and generalizes better to other external datasets. Moreover, the top-weighted patches extracted from the trained model involve important brain regions related to Alzheimer's disease, suggesting good biological interpretability of our proposed method."