Exponential reduction of the effective logical error rate for general even-distance codes

Prove that for any [[n,k,d]] stabilizer code with even distance d, in the low-error regime under local Pauli noise, quantum syndrome-aware estimation of a logical Pauli expectation value achieves an effective logical error rate that decreases exponentially with the number of logical qubits k. Concretely, establish that, beyond the restricted assumptions of Theorem 2 (which require a specific form of the conditional noise on dominant syndromes), the ratio between the effective logical error rate defined via the quantum Fisher information of the classical–quantum state Σ_s p_s |s⟩⟨s|⊗ ρ̄_{N_s}(ρ̄(θ)) and the logical error rate under decoding decays as O(2^{-k}) in the limit of vanishing physical error rate.

Background

The paper introduces an effective logical error rate to quantify how syndrome information improves logical observable estimation. For classical syndrome-aware protocols (syndrome used only in post-processing), Theorem 1 proves a universal constant-factor limitation. In contrast, for quantum syndrome-aware protocols (syndrome-conditioned logical measurements), Theorem 2 shows that the effective logical error rate can be exponentially smaller than the logical error rate, but only under a restricted assumption on the form of the conditional logical noise for the dominant ambiguous syndromes in even-distance codes.

Numerical evidence in Section 6.4 indicates that this exponential reduction persists more broadly for even-distance stabilizer codes beyond the assumptions of Theorem 2. The authors therefore pose proving this exponential decay for general even-distance codes as an open problem, aiming to extend the analytical result to wider code families and noise models.