Beyond Error Detection: Toward a Unified Picture of Magic State Distillation

Magic state distillation is often understood as a quantum error detection process and the distillation efficiency is usually associated with the distance of the stabilizer code used to characterize the protocol. However, many well-known counterexamples indicate that code distance alone does not capture their true distillation performance. For example, the $[[5, 1, 3]]$ protocol distills with only quadratic efficiency, and the $[[7, 1, 3]]$ Steane code distills with only linear efficiency. It still remains unclear how distillation efficiency is related with their stabilizer code. To address this gap, we establish a unified theory of magic state distillation that goes beyond the conventional error-detection picture. We provide a structural theorem that relating distillation efficiency to stabilizer code properties and intuitive picture behind the theorem. Our results provide a conceptual unification of MSD efficiency across protocols and noise models, and offer practical criteria for evaluating and designing distillation routines.