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Heatmap regression has become the mainstream methodology for deep learning-based semantic landmark localization, including in facial landmark localization and human pose estimation. Though heatmap regression is robust to large variations in pose, illumination, and occlusion in unconstrained settings, it usually suffers from a sub-pixel localization problem. Specifically, considering that the activation point indices in heatmaps are always integers, quantization error thus appears when using heatmaps as the representation of numerical coordinates. Previous methods to overcome the sub-pixel localization problem usually rely on high-resolution heatmaps. As a result, there is always a trade-off between achieving localization accuracy and computational cost, where the computational complexity of heatmap regression depends on the heatmap resolution in a quadratic manner. In this paper, we formally analyze the quantization error of vanilla heatmap regression and propose a simple yet effective quantization system to address the sub-pixel localization problem. The proposed quantization system induced by the randomized rounding operation 1) encodes the fractional part of numerical coordinates into the ground truth heatmap using a probabilistic approach during training; and 2) decodes the predicted numerical coordinates from a set of activation points during testing. We prove that the proposed quantization system for heatmap regression is unbiased and lossless. Experimental results on popular facial landmark localization datasets (WFLW, 300W, COFW, and AFLW) and human pose estimation datasets (MPII and COCO) demonstrate the effectiveness of the proposed method for efficient and accurate semantic landmark localization. Code is available at http://github.com/baoshengyu/H3R.