Magic State Distillation under Imperfect Measurements

Fault-tolerant quantum computation relies on magic state distillation (MSD) to provide high-fidelity non-Clifford resource states for universality. Although logical Clifford operations used in magic state distillation are considered noiseless due to their transversal nature, the logical measurement noise originating from imperfect physical measurements can still affect the output of distilled states. In this work, we investigate the effect of imperfect (weak) measurement noise on the magic state distillation process within the framework of stabilizer reduction, which characterizes MSD protocols using stabilizer codes. By mapping the distillation protocols to dynamic systems, we analyze these protocols under measurement noise using techniques from dynamic systems theory. We prove that for MSD protocols based on CSS codes with transversal non-Clifford gates, the distilled output states suffer from biased logical Pauli noise in the asymptotic limit under imperfect measurement. Additionally, we show that imperfect measurement noise degrades the convergence order to linear, regardless of the original order in the noiseless case. We simulate several common distillation protocols and numerically demonstrate our theoretical findings. Furthermore, we identify a threshold for measurement strength in MSD protocols, below which the asymptotic output states become fully mixed.