Importance sampling for data-driven decoding of quantum error-correcting codes

Data-driven decoding (DDD) - learning to decode syndromes of (quantum) error-correcting codes by learning from data - 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. We show that an existing importance sampling technique of training neural decoders on data generated with higher error rates introduces a tradeoff between class imbalance and label noise. We apply this technique to show robust improvements in the accuracy of neural network decoders trained on syndromes sampled at higher error rates, and provide heuristic arguments for finding an optimal error rate for training data. We extend these analyses to decoding quantum codes involving multiple rounds of syndrome measurements, suggesting broad applicability of both example importance and turning the knob for improving experimentally relevant data-driven decoders.

💡 Research Summary

The paper tackles a fundamental challenge in data‑driven decoding (DDD) of quantum error‑correcting codes: the training data are heavily skewed toward easy examples that baseline decoders already handle, while the rare “important” examples—those that a baseline decoder gets wrong but an optimal maximum‑likelihood decoder (MLD) would get right—are precisely the ones that a neural decoder must learn from. To formalize this, the authors introduce a notion of example importance. They categorize each training pair (syndrome σ, label y) as:

• Good – correctly decoded by the MLD;
• Important – good but mis‑decoded by a chosen baseline decoder f₀;
• Bad – mis‑decoded even by the MLD (i.e., the label is effectively noisy).

They define an importance weight J((σ,y);f₀) that counts the probability mass of important examples. Equation (3) shows that the total importance of a dataset upper‑bounds the possible improvement a neural decoder can achieve over the baseline. Computing J directly requires access to the MLD, which is generally intractable, underscoring the difficulty of assessing dataset usefulness a priori.

The authors then focus on the simplest non‑trivial DDD task: decoding an n‑bit classical repetition code under a biased bit‑flip channel. Proposition 1 proves that, for any fixed baseline decoder f₀, DDD on this code is mathematically equivalent to a noisy, imbalanced binary classification problem. The “noise” corresponds exactly to the fraction of bad examples, while the class‑imbalance is driven by the scarcity of important examples. In the low‑error regime (np ≪ 1) the class priors differ by at most ξₙ = exp