Generalize strict-resolution probabilities to multiple logical classes

Extend the two-class strict-resolution probability for approximate maximum-likelihood decoding to settings with an arbitrary number of logical classes.

Background

Approximate maximum-likelihood decoding estimates each logical class’s free energy from a truncated candidate pool and can resolve minimum-weight ties by comparing the sampled multiplicities of minimum-weight configurations. The theoretical strict-resolution probability derived in Eq. (PN_exact) applies only to a two-class model, in which one class has a unique minimum-weight configuration and the competing class has multiple degenerate ground states.

The paper establishes bounds on the truncated free-energy estimator for arbitrary logical-class counts, but does not derive the corresponding probability that AMLD strictly resolves degeneracies when more than two logical classes compete. Such a generalization would extend the theoretical analysis to codes and noise models with larger logical-class spaces, including the bivariate-bicycle-code setting considered numerically.