We examine the impact of imperfect measurement on magic state distillation (MSD) process by employing the framework of stabilizer reduction, which characterizes MSD protocols using stabilizer codes. We show the existence of thresholds for measurement strength in MSD protocols, below which there doesn't exist non-trivial target states and no input states can be distilled into better states. We prove that for MSD protocols based on CSS codes with transversal non-Clifford gates, the first-order effect of imperfect measurement will at most cause biased Pauli noise on the target states. Furthermore, we prove that we can minimize the effect of imperfect measurement noise on the asymptotically distilled states by measuring stabilizer generators in the standard form. We numerically demonstrate our theoretical results by simulating the [15, 1, 3] and [14, 2, 2] MSD protocols using the mapping from MSD protocols to dynamical systems. Our numerical results imply that the existence of imperfect measurement degrades the order of convergence rate to linear, regardless of the original order in the noiseless case. Our work will therefore contribute to understanding fault-tolerant quantum computing under imperfect measurement noise.
Zheng et al. (Sun,) studied this question.