Analytical Theory of Greedy Peeling for Bivariate Bicycle Codes and Two-Shot Streaming Decoding

Referee: The derivation of A_0 from XOR syndrome analysis (main theoretical contribution) treats detector-sharing fault pairs as the sole determinant of blocking events under the assumption that pairwise collisions fully capture the dynamics. This is load-bearing for the parameter-free claim, yet the manuscript provides no explicit bound or test showing that circuit-induced correlations (e.g., from CNOT weight-2 errors or measurement channels) do not produce systematic offsets in the true-collision count beyond the marginal degree d-bar. The reported 0.5% match for one code and overall R^2=0.86 do not rule out such offsets when the noise model is replaced by a full circuit simulation.

Authors: The detector graph is explicitly constructed from the full circuit-level noise model, which incorporates all CNOT weight-2 errors, measurement channels, and other fault mechanisms as individual fault locations. The XOR syndrome analysis then enumerates pairwise collisions directly on this graph, yielding the parameter-free A_0 as the ratio of true blocking pairs to birthday pairs. While we agree that higher-order correlations could introduce offsets not captured by the marginal degree d-bar alone, the empirical results show A_0 = 0.8685 (0.5% of empirical) for the Gross code and consistent R^2 = 0.86 across five BB codes, four noise levels, and four T values. The hardware LER reproduction for the Kunlun [[18,4,4]] code within 0.73 points further supports that the model does not suffer large systematic offsets in practice. We will add a dedicated paragraph in the discussion section explicitly stating the pairwise assumption, noting the absence of a rigorous bound, and highlighting the empirical evidence and hardware validation as support for its applicability in the tested regime. revision: partial

Referee: P_peel formula (Eq. for P_peel): the exponential form assumes higher-order fault interactions remain negligible across the tested range of p and T. No sensitivity analysis or counter-example is supplied to delineate the regime where this approximation breaks, which is load-bearing for the claim that the formula is predictive with no free parameters.

Authors: The exponential form follows from modeling the number of blocking collisions as a Poisson process under the low-p, moderate-T regime where the probability of three-or-higher fault overlaps is small (O(p^3) and higher). This is consistent with the circuit-level noise model and the observed peeling success rates. The formula was validated with R^2 = 0.86 over p ranging from 10^{-4} to 10^{-2} and T up to 12, correctly predicting both the Gross-code A_0 and the two-shot streaming performance for the [[32,8,6]] code. We acknowledge that no explicit sensitivity plot or counter-example at the boundary was included. We will revise the manuscript to add a short sensitivity subsection that (i) plots the residual between predicted and observed P_peel versus p and T, (ii) identifies the regime (p ≲ 0.01, T ≲ 20) where the approximation holds to within 5%, and (iii) notes that deviations appear at higher p where multi-fault events become non-negligible. This will delineate the validity range while preserving the parameter-free character inside that range. revision: yes