Face presentation attack detection (PAD) aims to protect the security of face recognition systems. The existing depth-supervised method using stacked vanilla convolutions cannot explicitly extract efficient fine-grained information (e.g., spatial gradient magnitude) for the distinction between bona fide and attack presentations. To address this issue, the Sobel operator has been demonstrated effective to acquire gradient magnitude due to the fast calculation capacity for high-frequency information. However, the Sobel operator is hand-crafted so cannot deal with complex textures. Differently, we develop a learnable gradient operator (LGO) to adaptively learn gradient information in a data-driven way, which is a generalization of existing gradient operators and effectively captures detailed discriminative clues from raw pixels. In parallel, we propose an adaptive gradient loss for better optimization. Extensive experimental comparisons with the state-of-the-art methods on the widely used Replay-Attack, CASIA-FASD, OULU-NPU, and SiW datasets demonstrate the superior performance of the proposed approach.