Fair Decoder Baselines and Rigorous Finite-Size Scaling for Bivariate Bicycle Codes on the Quantum Erasure Channel

Abstract

Fair threshold estimation for bivariate bicycle (BB) codes on the quantum erasure channel runs into two recurring problems: decoder-baseline unfairness and the conflation of finite-size pseudo-thresholds with true asymptotic thresholds. We run both uninformed and \emph{erasure-aware} minimum-weight perfect matching (MWPM) surface code baselines alongside BP-OSD decoding of BB codes. With standard depolarizing-weight MWPM and no erasure information, performance matches random guessing on the erasure channel in our tested regime -- so prior work that compares against this baseline is really comparing decoders, not codes. Using 200{,}000 shots per point and bootstrap confidence intervals, we sweep five BB code sizes from $N=144$ to $N=1296$. Pseudo-thresholds (WER = 0.10) run from $p^* = 0.370$ to $0.471$; finite-size scaling (FSS) gives an asymptotic threshold $p^*_\infty \approx 0.488$, within 2.4\% of the zero-rate limit and without maximum-likelihood decoding. On the fair baseline, BB at $N=1296$ has a modest edge in threshold over the surface code at twice the qubit count, and a 12$\times$ lower normalized overhead -- the latter is where the practical advantage sits. All runs are reproducible from recorded seeds and package versions.

Figures (7)

• Figure 1: FSS data collapse for bivariate bicycle codes on the erasure channel. Rescaling the erasure rate by $N^{1/\nu}$ with $\nu = 1.18$ collapses WER curves from all five code sizes onto a single master curve, confirming the scaling ansatz. The extrapolated asymptotic threshold is $p^*_\infty = 0.488 \pm 0.001$.
• Figure 2: Linearised FSS: pseudo-threshold $p^*$ vs $N^{-1/\nu}$ ($\nu = 1.18$). Under leading-order FSS, $p^*(N) \approx p^*_\infty + c\, N^{-1/\nu}$, so this plot should be linear with y-intercept $p^*_\infty$. The linear fit (dotted) extrapolates to $0.490$, consistent with the nonlinear FSS result $p^*_\infty = 0.488 \pm 0.001$ (dashed red, with 95% CI band at $N^{-1/\nu}=0$). This alternative fit does not assume a polynomial scaling function and serves as an independent consistency check.
• Figure 3: Encoding rate $K/N$ vs pseudo-threshold $p^*$ for BB codes (circles) and erasure-aware MWPM surface codes (squares). The dashed line shows the Shannon limit $K/N = 1-p$; points above are impossible. BB codes trace a Pareto frontier with higher rates at lower $p^*$ or lower rates at higher $p^*$.
• Figure 4: WER vs. erasure rate for bivariate bicycle codes (solid lines, $N=144$--$1296$) and surface code baselines. Dashed lines: erasure-aware MWPM surface codes ($N=288$--$2592$, $K=2$). Dotted lines: uninformed MWPM surface codes ($N=288$--$1152$, $K=2$, three sizes). Error bars show 95% Wilson confidence intervals. All BB code curves cross WER = 0.10 at $p^* = 0.37$--$0.47$; erasure-aware surface codes cross at $p^* = 0.41$--$0.46$.
• Figure 5: Pseudo-threshold $p^*$ (WER = 0.10) vs. code size $N$ for BB codes. Error bars show 95% bootstrap CIs (smaller than markers for large $N$). The dashed red line and shaded band show the FSS-extrapolated asymptotic threshold $p^*_\infty = 0.488 \pm 0.001$.