Sample importance for data-driven decoding

Data-driven decoding (DDD) -- learning to decode syndromes of (quantum) error-correcting codes using training examples -- can be a difficult problem due to several atypical and poorly understood properties of the training data. We introduce a theory of example importance that clarifies these unusual aspects of DDD: For instance, we show that DDD of a simple error-correcting code is equivalent to a noisy, imbalanced binary classification problem. This suggests that an existing data augmentation technique -- turning the knob to increase error rates in training data -- actually introduces a tradeoff between class imbalance and label noise. We apply this technique in experiments showing robust improvements to decoder accuracy while explaining the failures of this technique in terms of example importance. We show similar improvements for decoding quantum codes involving multiple rounds of syndrome measurements and we characterize example importance in random stabilizer codes, suggesting broad applicability of both example importance and turning the knob for improving experimentally relevant data-driven decoders.