Scale-Transferable Neural Belief Propagation for Quantum Protograph LDPC Codes and the Limits of Threshold-Guided Co-Design

Abstract

Quantum low-density parity-check (QLDPC) codes need decoders reusable across a code family and design tools that predict finite-length performance cheaply. We address both for protograph CSS codes. First, we propose a protograph-tied neural belief-propagation decoder in which the learned message weights are shared per protograph-edge group, so the parameter count is independent of the quasi-cyclic lift factor. A single pair of axis decoders, with 132 active weights per axis, is trained once on the smallest member of a code family and deployed without retraining across a four-fold range of blocklengths, staying within a factor of 1.42–1.54 of the BP+OSD-0 reference for the lift factors of the single base matrix studied here. Second, we develop a semi-quantum protograph-EXIT (P-EXIT) screening framework that tracks dual X/Z scalar-information trajectories under depolarizing noise. Its hashing-bound-calibrated thresholds provide an inexpensive screening score, and we test whether that score is a useful co-design objective. In the tested short-blocklength setting it is not: at matched rate and blocklength, candidates with an eight-fold threshold advantage lose to a low-threshold baseline, and gating the objective by Tanner girth and an estimated distance does not repair it; the sole strict Pareto-improver still fails to beat it. The cause is that neither the threshold nor the base-graph gate sees lift-specific short cycles and distance losses. A screen of 240 circulant lifts per lift factor finds 15–31% operationally failing lifts, indistinguishable from working lifts in blocklength, rate, rank and degree. For the separate post-lift structural measurements available on 40 seeds per lift factor, the observed failures and working cases are separated by girth and an estimated distance. Together they give a scale-transferable quantum decoder and an account of why threshold-guided co-design fails at short blocklength.