With applications ranging from metabolomics to histopathology, quantitative phase microscopy (QPM) is a powerful label-free imaging modality. Despite significant advances in fast multiplexed imaging sensors and deep-learning-based inverse solvers, the throughput of QPM is currently limited by the pixel-rate of the image sensors. Complementarily, to improve throughput further, here we propose to acquire images in a compressed form so that more information can be transferred beyond the existing hardware bottleneck of the image sensor. To this end, we present a numerical simulation of a learnable optical compression-decompression framework that learns content-specific features. The proposed differentiable quantitative phase microscopy (?-QPM) first uses learnable optical processors as image compressors. The intensity representations produced by these optical processors are then captured by the imaging sensor. Finally, a reconstruction network running on a computer decompresses the QPM images post aquisition. In numerical experiments, the proposed system achieves compression of × 64 while maintaining the SSIM of ?0.90 and PSNR of ?30 dB on cells. The results demonstrated by our experiments open up a new pathway to QPM systems that may provide unprecedented throughput improvements.