Universality of a lower bound for syndrome-based LEM protocols

Establish whether the lower bound on the shot overhead for syndrome-aware logical error mitigation, given by the harmonic expectation of per-subset bounds, Γ_SALEM ≥ H[f_bound(Λ_{L|k})], holds for any logical error mitigation protocol that utilizes error-correction syndrome information.

Background

The authors note that established lower bounds on the shot overhead of physical error mitigation and ExtLEM do not directly apply to adaptive, error-corrected circuits, but do apply to ExtLEM when syndrome data is ignored. They derive a weaker bound applicable to SALEM that depends on the harmonic mean over subset-conditioned logical channels.

It is unknown whether this harmonic-mean lower bound is universal for all syndrome-based logical error mitigation strategies. Proving (or disproving) such universality would clarify the fundamental limits of leveraging syndrome data and inform whether SALEM is close to optimal or if stronger constraints (or better protocols) exist.