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arXiv:2603.19062

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

Tushar Pandey

quant-phSubmitted Mar 19, 2026• Announced Mar 20, 2026• 0 citations• Revised Apr 21, 2026 (v2)
v1 · Mar 19, 2026v2 · Apr 21, 2026v3 · Apr 26, 2026
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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.

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Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-666 checks, exemplified by [[144,12,12]][[144,12,12]][[144,12,12]] with kd2/n=12kd^2/n=12kd2/n=12. We develop the algebraic structure theory of BB-type codes with weight-888 checks (weight-444 generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula k=2dim⁡R/(A,B)k=2\dim R/(A,B)k=2dimR/(A,B) (forcing even kkk), a 4ℓm4\ell m4ℓm-element symmetry group on generator pairs, an X/ZX/ZX/Z distance equality dX=dZd_X=d_ZdX​=dZ​, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At n=144n=144n=144 the pipeline returns a census of 535353 codes whose strongest members surpass the BB benchmark: [[144,6,d≥15]][[144,6,d\ge 15]][[144,6,d≥15]] exceeds the benchmark distance 121212 (certified d≥15d\ge 15d≥15), [[144,10,12]][[144,10,12]][[144,10,12]] reaches it with weight-888 checks, and [[144,16,10]][[144,16,10]][[144,16,10]] encodes a third more logical qubits at kd2/n=11.11kd^2/n=11.11kd2/n=11.11 (7.4%7.4\%7.4% below benchmark) while decoding no worse. At n=72n=72n=72, [[72,14,8]][[72,14,8]][[72,14,8]] attains kd2/n=12.44kd^2/n=12.44kd2/n=12.44---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-888 codes at ≈0.1%\approx 0.1\%≈0.1% pseudo-threshold versus ≈0.4%\approx 0.4\%≈0.4% for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.
Two-dimensional topological translationally-invariant (TTI) quantum codes, such as the toric code (TC) and bivariate bicycle (BB) codes, are promising candidates for fault-tolerant quantum computation. For such codes to be practically relevant, their decoders must successfully correct the most likely errors while remaining computationally efficient. For the TC, graph-matching decoders satisfy both requirements and, additionally, admit provable performance guarantees. Given the equivalence between TTI codes and (multiple copies of) the TC, one may then ask whether TTI codes also admit analogous graph-matching decoders. In this work, we develop a graph-matching approach to decoding general TTI codes. Intuitively, our approach coarse-grains the TTI code to obtain an effective description of the syndrome in terms of TC excitations, which can then be removed using graph-matching techniques. We prove that our decoders correct errors of weight up to a constant fraction of the code distance and achieve nonzero code-capacity and circuit-level thresholds in the specified noise models. We further numerically study a variant optimized for practically relevant BB codes and observe performance comparable to that of belief propagation with ordered statistics decoding (BP-OSD). We also study the code-capacity threshold for the hexagonal color code as a proof of concept. Our results indicate that graph-matching decoders are a viable approach to decoding BB codes and other TTI codes.
The discovery of new quantum error-correcting codes that encode several logical qubits into relatively few physical qubits motivates the development of efficient and accurate methods of decoding these systems. Here, we adopt the minimum-weight perfect matching algorithm, a subroutine invaluable to decoding topological codes, to decode bivariate bicycle codes. Using the equivalence of bivariate bicycle codes to copies of the toric code, we propose a method we call the `cylinder trick' to rapidly find a correction using matching on code symmetries. We benchmark our decoder on the gross code family, cyclic hypergraph-product codes, generalized toric codes, and recently proposed directional codes under code capacity and phenomenological noise models, demonstrating the general applicability of our protocol. For a subset of these codes, we find that our decoder can be significantly improved by augmenting matching with strategies including belief propagation and `over-matching', thus achieving performance competitive with state-of-the-art approaches.
Abstract Quantum low-density parity-check (qLDPC) memories promise lower-overhead fault tolerance, but reproducible finite-instance benchmarking has been hin- dered by the absence of explicit decoder representations and data provenance. We introduce adversarial channel equilibria (ACE), a bounded-Pauli channel game that organises fidelity guarantees, certificates and decoder benchmarks for bivariate-bicycle (BB) memories. Applying Sion’s minimax theorem to compact convex sets of completely positive trace-preserving maps, we prove existence of a saddle-point equilibrium and derive a closed-form fidelity floor. We provide the first SHA-256-anchored three-code benchmark comparing BP+OSD against two graphified baselines—DEM-chain MWPM and UF-BFS—on BB72, BB144 and BB288, using 10,000 trials per point and deterministic seeds. BP+OSD achieves risk ratios of 595× to 67.8× over DEM-chain MWPM on BB72 across tested error rates; at larger code sizes BP+OSD failure counts are zero while the graphi- fied baselines fail at rates above 50%. Repeated-cycle BP+OSD remains near zero through the tested range. A BB72 certificate verifies commutation, dimension, distance lower bound and radius-two correctability. Hardware-noise simulations are reported as model sensitivities only. All claims are mapped to verifiable data, code and SHA-256 hash artefacts archived at doi:10.5281/zenodo.20588303.