QECFM: Quantum Error Correcting Code Discovery via Training Free Guidance

Designing quantum error-correcting codes is difficult: the search space is discrete and highly constrained, while evaluating a candidate often requires an expensive Monte Carlo estimate of its logical error rate. We introduce QECFM, a two-stage conditional flow matching model with a deterministic group-algebra readout layer, which maps each continuous sample to discrete parity-check matrices and enforces CSS commutativity. We then use D-Flow with a differentiable noise-aware critic to optimise the source variables toward codes with lower logical error rates. We evaluate the resulting method across six code families, code-capacity and circuit-level noise, and block lengths up to $n=288$. Our method finds codes with a lower median logical error rate than both learned and classical search baselines within the Bivariate Bicycle Code family. Under code-capacity depolarising noise, our best code also outperforms the Gross code at different physical error rates.