Methods for quantization-aware training have been used to compress deep neural network models and have demonstrated good performance at lower accuracies. However, due to the inherent nature of QAT, we have observed that most of these methods necessitate end-to-end training, which inevitably leads to increased training time and computational cost for the quantized model. Drawing inspiration from this, we propose a highly sensitive adaptive heterogeneous data sensing method that can accurately identify and preserve critical heterogeneous data, thereby reducing the training time of the quantized model through appropriate updates to the core data. To better adapt to both the quantized model and the full-precision model, we introduce an evaluation method based on first-order gradient information scores and Hessian matrix scores to identify data heterogeneity. Numerous experiments demonstrate the effectiveness of the GradHessian method, which enhances the performance of QAT while improving training efficiency by 50% compared to existing approaches.