PRIME: A Neural-Prior BP-OSD Decoder for Quantum LDPC Codes

Quantum error correction protects logical information by encoding it redundantly across physical qubits, and a decoder infers a correction from the measured syndrome without introducing a logical error. For bivariate bicycle (BB) quantum low-density parity-check (qLDPC) codes, belief propagation with ordered-statistics decoding (BP-OSD) is widely used, and its channel prior, the assumed per-qubit error probability, comes from a predefined noise model. We introduce PRIME, which augments BP-OSD with a graph neural network (GNN) that maps each measured syndrome to a syndrome-conditioned channel prior over physical-qubit errors. The underlying BP-OSD procedure is used unchanged, which preserves syndrome consistency while letting the decoder exploit noise structure learned from data, and because the GNN operates directly on the Tanner graph, one model trained on small BB codes generalizes without retraining to larger BB codes and toric codes. We further introduce Jitter-PRIME, a perturbed-prior ensemble whose candidates are aggregated into logical equivalence classes, together with a learned confidence gate that sends only the riskiest instances to it. At physical error rate \(p=0.10\), Jitter-PRIME reduces logical error rate (LER) over BP-OSD-CS-6 by \(6.6\times\), \(9.0\times\), and \(57.8\times\) on BB codes with 288, 360, and 756 qubits. On the two held-out codes, the gate keeps the full ensemble gain with about 15% of the always-on ensemble's solver calls. Under circuit-level noise, Jitter-PRIME reduces LER over BP-OSD-CS-6 by \(1.5\)–\(15\times\) on the length-72 and length-144 BB codes.