Learning Logical Error Classes Under Correlated Noise in a Seven Qubit Decoding Model

Abstract

A learned quantum decoder can fail because of sparse training, a misspecified noise representation or an observation that cannot distinguish logical classes. We separate these mechanisms in an X-only seven-qubit CSS model with a perfect syndrome. All 128 physical error strings are enumerated under six synthetic channels. Four training budgets and 60 repetitions compare minimum weight, independently fitted physical marginals, joint syndrome-class counts and model-assisted counts against privileged exact references. At 1024 labels, pair-correlated class learning reduces failure from 0.044696 to 0.013620, while the independent fit retains the larger risk. A pair01-trained class table transferred to pair24 instead reaches 0.055457. Adding an independent undetectable logical flip raises optimal failure to 0.052531, despite preserving the best decision table. The 12240 risk records use exact test integration; variation comes from sampled training sets. Physical and logical true-channel MAP coincide for these selected channels. The study establishes transparent statistical mechanisms without claiming circuit-level thresholds, quantum hardware measurements or large-code scaling.