Fault-Tolerant Postselection for Low-Overhead Magic State Preparation

We introduce a framework for fault-tolerant postselection (FTPS) of fault-tolerant codes and channels-such as those based on surface codes-using soft-information metrics based on visible syndrome and erasure information. We introduce several metrics for ranking configurations of syndromes and erasures. In particular, we introduce the logical gap (and variants thereof) as a powerful soft-information metric for predicting logical error rates of fault-tolerant channels based on topological error-correcting codes. The logical gap is roughly the unsigned weight difference between inequivalent logical corrections and is adaptable to any tailored noise model or decoder. We deploy this framework to prepare high-quality surface-code magic states with low overhead under a model of independent and identically distributed (IID) Pauli and erasure errors. Postselection strategies based on the logical gap can suppress the encoding error rate (EER) of a magic state preparation channel to the level of the physical error rate with low overhead. For example, when operating at 60% of the bulk threshold of the corresponding surface code, an overall reduction of the EER by a factor of 15 is achievable with a relative overhead factor of < 2 (approximately 23 times less than that of simple syndrome-counting rules). We analyze a schematic buffer architecture for implementing postselection rules on magic state factories in the context of magic state distillation. The FTPS framework can be utilized for mitigating errors in more general fault-tolerant logical channels.