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

TL;DR

This paper evaluates decoding strategies for bivariate bicycle codes on the quantum erasure channel, addressing fairness issues and providing rigorous finite-size scaling to estimate true thresholds.

Contribution

It introduces a comprehensive analysis of decoder baselines and finite-size scaling for BB codes, revealing practical advantages over toric codes without maximum-likelihood decoding.

Findings

Pseudo-thresholds range from 0.370 to 0.471 for different code sizes.

Asymptotic threshold estimated at approximately 0.488.

BB codes show lower normalized overhead compared to toric codes at similar thresholds.

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) toric 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 toric 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.