Fragility of Magic State Distillation under Imperfect Measurements

Magic state distillation (MSD) is the leading approach for producing the non-Clifford resources required for universal fault-tolerant quantum computation. Most prior analyses of MSD assume ideal projective stabilizer measurements, but this assumption becomes questionable on near-term hardware, where measurement fidelity is limited and large quantum error-correcting codes are unavailable. We consider imperfect measurements that are modelled as noisy projector controlled by \\textit{measurement strength}, which can be regarded as inverse of measurement noise, and develop a general framework for analyzing MSD under imperfect stabilizer measurements. We focus on MSD protocols that are based on CSS codes with transversal non-Clifford gates, and show that MSD exhibits a measurement-strength threshold that is distinct from the previously known threshold on input-state error. When measurement strength is below a critical threshold, MSD loses its distillation power entirely; When measurement strength is above this threshold, distillation under imperfect measurements remains possible, but the asymptotic target states may deviate from the ideal magic states. In particular, this deviation is at most first-order biased. Moreover, we show that imperfect measurements with finite measurement strength reduces the distillation efficiency to linear, implying exponentially larger distillation overheads than in the ideal-measurement case. We further show that this fragility can be mitigated generally up to the code capacity of MSD protocols by choosing the stabilizer generators to measure in the \\textit{standard form}. Our results reveal fundamental constraints imposed by imperfect measurements on MSD and provide guidance for designing more robust distillation protocols in realistic quantum hardware.