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REVIEW 4 major objections 7 minor 48 references

Erasure Decoding for Quantum LDPC Codes via Belief Propagation with Guided Decimation

T0 review · 4 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read BPGD, with two small tuning adjustments, is a general-purpose decoder for quantum erasure channels.

desk verdict Useful incremental results on BPGD for erasures, but the 'general-purpose' claim outruns evidence that is tuned per code and rate on the same data. read the letter →

arxiv 2411.08177 v2 pith:CXPR64LH submitted 2024-11-12 cs.IT math.ITquant-ph

Mert Gökduman , Hanwen Yao , Henry D. Pfister This is my paper · ORCID
classification cs.ITmath.ITquant-ph MSC 81P7094B35
keywords quantumLDPCcodeserasuredecodingbeliefpropagationguideddecimationdampinglog-likelihoodratioinitializationhypergraphproductlifted
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that belief propagation with guided decimation (BPGD), a decoder previously applied to bit-flip and depolarizing noise, also works well on quantum erasure channels. The natural adaptation is to assign erased qubits a near-zero log-likelihood ratio and known qubits a large positive ratio, so that belief propagation starts from the erasure locations. The paper then shows that scaling down that large ratio by a factor $c_{\mathrm{opt}}$ and damping the message updates by a factor $\gamma$ improves convergence, so that BPGD beats peeling and pruned peeling on two hypergraph-product codes and a lifted-product code, and approaches the vertical-horizontal decoder with lower computational cost. The conclusion is that BPGD is an effective general-purpose erasure decoder across the quantum LDPC landscape.

What carries the argument

The load-bearing mechanism is guided decimation: each round runs belief propagation for a fixed number of iterations; if the hard decision does not match the syndrome, the variable node with the largest belief magnitude is fixed to its most likely value, and belief propagation is rerun. On erasure channels this is initialized by setting the log-likelihood ratio of each erased bit to $\mathrm{llr}_{\min}\approx 0$ and each non-erased bit to $\mathrm{llr}_{\max}=25$. Two tuning parameters carry the performance gain: $c_{\mathrm{opt}}\in[0,1]$ scales down the confident priors, and $\gamma\in[0,1]$ damps each variable-to-check update as a weighted average of old and new messages.

What would settle it

Fix the tuning parameters found for the [[1600,64]] hypergraph-product code at erasure rate 0.1, run BPGD on a different quantum LDPC code of a different rate without retuning, and compare with the vertical-horizontal decoder; a failure rate that jumps well above the tuned-curve gap would show the parameters do not transfer.

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Extended reading notes

Core claim

The paper's central claim is that BPGD's guided decimation breaks the stopping-set deadlock that limits peeling decoders on quantum LDPC erasures: when belief propagation stalls, fixing the most reliable undecimated bit and rerunning belief propagation produces guesses that are often harmless, because an X-stabilizer fully covered by erasures can absorb the difference. The paper shows this mechanism works on erasures once the channel priors are set correctly, and that two small modifications—scaling the confident priors with $c_{\mathrm{opt}}$ and damping variable-to-check messages with $\gamma$—turn a decoder that is merely competitive into one that approaches the vertical-horizontal decoder on hypergraph-product codes and maximum likelihood on a lifted-product code. The supporting evidence is simulation-based, on codes of length 882, 1600, and 2025.

Load-bearing premise

The load-bearing premise is that the two tuning knobs, chosen by grid search on the same simulations used to plot the results, will keep working on other codes and erasure rates; if those choices are fragile, the claim that BPGD is a general-purpose erasure decoder is not established.

Editorial extensions

If this is right

  • On the tested hypergraph-product codes, BPGD with erasure-adapted priors outperforms both peeling and pruned peeling, and with damping or adjusted LLRs it approaches the failure rate of the vertical-horizontal decoder.
  • On the lifted-product B1 code, adjusted BPGD is much closer to maximum-likelihood decoding than pruned peeling, and displays a sharp threshold-like transition rather than the slow decay typical of pruned peeling.
  • Because BPGD already handles bit-flip and depolarizing noise, the same decoder is a candidate for channels that combine erasures and ordinary errors, a regime the paper notes is likely when erasure conversion is used.
  • BPGD offers a lower-complexity alternative to the vertical-horizontal decoder for hypergraph-product codes, trading a small performance gap for substantially lower computational cost.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper tunes $c_{\mathrm{opt}}$ and $\gamma$ by grid search on the same curves it plots; a transferable protocol for setting these parameters automatically (for example, from the code's degree distribution or the observed syndrome weight) would be needed to make the 'general-purpose solution' claim robust.
  • The threshold-like behavior reported for the lifted-product code suggests erasure decoding of quantum LDPC codes may admit a density-evolution or finite-length-scaling analysis analogous to classical LDPC erasure thresholds; the paper does not attempt this.
  • The random-guessing interpretation of BPGD on stopping sets could be quantified: if each guessed bit is correct with probability near $1/2$ and stabilizers absorb a constant fraction of mistakes, one might predict the observed failure-rate floor from the stopping-set statistics alone.
  • A natural test bed is a mixed erasure-plus-Pauli channel, where BPGD's prior LLR could be set from the erasure probability and the residual Pauli error rate; the paper mentions this as motivation but does not simulate it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The manuscript applies belief propagation with guided decimation (BPGD) to the quantum erasure channel, modifying the decoder by adjusting the prior LLRs of non-erased qubits and by damping variable-to-check messages. It reports Monte Carlo simulation results for two hypergraph product codes and one lifted-product code, comparing BPGD against peeling, pruned peeling, vertical-horizontal (VH), and maximum-likelihood decoders. The main claims are that BPGD outperforms peeling and pruned peeling, that adding damping and adjusted LLRs brings BPGD close to VH performance for HGP codes while retaining lower complexity, and that BPGD is an effective general-purpose erasure decoder across the quantum LDPC landscape.

Significance. The paper's strength is its direct empirical comparison of a simple, natural adaptation of BPGD against several established erasure decoders on nontrivial quantum LDPC codes, including an honest account of the cases where unadjusted BPGD does not outperform pruned peeling. If the reported performance were obtained with a principled parameter-setting rule, the finding that tuned BPGD approaches VH performance would be a useful contribution. However, the quantitative claims are not yet established: the optimal values of copt and γ are selected by grid search on the same failure-rate data used for the plots, no holdout or fixed-parameter evaluation is reported, no error bars are shown, and the complexity advantage is asserted without complexity analysis. As presented, the results demonstrate that tuned BPGD can perform well on three specific codes, but they do not support the 'general-purpose solution' conclusion.

major comments (4)
  1. [§IV, Tables 1–3 and §V, Figs. 1–3] The optimal values of copt and γ are obtained by grid search over the same simulation data that is later plotted as the failure-rate curves, with different parameter values reported for each code and each erasure-rate interval. This makes the plotted curves post-selection optima, and the paper provides no test on held-out codes or rates, nor any parameter-selection rule that avoids using the test data. Since the claimed advantage of adjusted BPGD over pruned peeling depends entirely on these tuned values, the central 'general-purpose solution' conclusion is not supported as stated. A concrete remedy would be to select (copt, γ) on one code or one rate and evaluate on others, or to show that the optimal parameters vary smoothly enough that a simple fixed or cross-validated rule recovers the same curves.
  2. [§V, Figs. 2–3 and §VI, Conclusion] The paper itself concedes that unadjusted BPGD performs worse than pruned peeling for the [[2025,81]] HGP code and 'does not uniformly outperform pruned peeling' for the B1 code. Thus the reported advantage of BPGD over pruned peeling is entirely due to the per-code, per-rate tuning described in Section IV. This is a load-bearing distinction: the abstract and conclusion claim that BPGD is an effective general-purpose solution, but the empirical content only shows that carefully tuned BPGD can outperform the baselines on the three tested instances. I would ask the authors to present a comparison under a single fixed parameter-setting protocol, or to clearly limit the claim to 'tuned BPGD' with a stated selection method.
  3. [Introduction and §VI, Conclusion] The manuscript repeatedly claims a 'significant computational advantage' and 'substantially lower' complexity compared to the VH decoder, but no complexity analysis or runtime measurement is provided. Algorithm 1 runs up to n rounds of BP, each with T iterations and message updates that are at least linear in the number of edges; a naive implementation could scale as O(T n^2) or worse, depending on code degree. To make the computational-advantage claim credible, the authors should state the per-round and total complexity of BPGD in terms of n and code parameters, and compare it explicitly with the O(n^2) complexity quoted for VH.
  4. [§III-D, Algorithm 1] The heuristic justification in Section III-D states that after peeling stops, BPGD 'simply guesses bits in the stopping set randomly and continues decoding,' adding that 'some of the guesses will be correct and the others can hopefully be absorbed by the stabilizer.' This is presented with no guarantees and no analysis of when such guess-and-continue succeeds or fails. Given that the paper's main claim is that BPGD is a general-purpose decoder, a failure-mode analysis or at least a detailed empirical characterization of when the guided-decimation guesses fail would be needed to substantiate that claim.
minor comments (7)
  1. [§III-D, Algorithm 1, line 12] In Algorithm 1, the decimation chooses vi = arg max γ(vi), but γ is never defined in the algorithm as a reliability measure; in Section IV-B the symbol γ denotes the damping factor. The pseudocode should instead write arg max |m_v| or introduce a separate symbol for the LLR magnitude.
  2. [§IV-A, line after Eq. (15)] The phrase 'In the content of BPGD decoding' appears to be a typo; it should read 'In the context of BPGD decoding.'
  3. [§VI, Conclusion] The sentence 'Due to its good performance for channels with erasures and channels with erros' contains a typo: 'erros' should be 'errors.'
  4. [§III-D, Algorithm 1, lines 3–5] The erased bits are assigned µ_vi = llr_min and non-erased bits µ_vi = llr_max, but the text says erased bits should have LLR 0 while llr_min is described as 'a very small value, llr_min≈0.' If llr_min is not exactly zero, the statement in Section III-D is slightly imprecise; please clarify the exact values and the rationale for using a nonzero value.
  5. [§V, Figs. 1–3] The figure captions state that the number of simulations per data point was chosen to ensure short error bars, but no error bars are shown in any figure. Please either show confidence intervals or explicitly state the estimated failure rates and their statistical uncertainty.
  6. [Tables 1–3] The tables label the values as 'optimized,' but the text does not specify the grid resolution or the exact criterion used to select them (e.g., minimum mean failure rate over some range). Stating the grid and the selection criterion would improve reproducibility.
  7. [References] Reference [47] is listed as 'M. T. S.' which is an incomplete author name; the full author list should be provided.

Circularity Check

1 steps flagged · score 4.0 of 10

The tuned BPGD variants are selected on the same failure-rate data used for evaluation, so the headline improvements over pruned peeling are post-selection optima rather than independent predictions; the core BPGD algorithm and external comparisons are otherwise self-contained.

  1. fitted input called prediction [Section IV-C (Tables 1-3) and Section V (Figures 1-3)]
    "To select the optimal value of copt, we performed experiments over different ranges of values for different erasure rates, fine-tuning to narrower intervals when necessary. ... Thus, we found suitable parameters by fixing γ for each erasure rate and searching over possible values of copt. The chosen values minimized the failure rates in our experiments."

    The reported failure-rate curves for 'Adjusted LLR BPGD', 'Damped BPGD', and 'Adjusted & damped BPGD' are generated at (copt, gamma) values selected to minimize the failure rate at the same erasure-rate points (Tables 1-3). Hence the plotted adjusted-BPGD performance is, by construction, the lower envelope of the grid search rather than the performance of a fixed, pre-specified decoder. The conclusion that tuning 'enables BPGD to approach the performance of VH' is therefore a statement about the best configuration found on the evaluation data, not an independent prediction; the general-purpose claim in the abstract extrapolates from per-code, per-rate tuned parameters without a holdout or fixed-parameter rule.

full rationale

The paper's only substantial circularity concern is the hyperparameter optimization loop: copt and gamma are tuned by grid search that minimizes the very failure-rate curves later plotted as evidence of decoder quality. This makes the headline 'BPGD with adjustments outperforms pruned peeling' a fitted claim rather than an independent prediction for the codes where unadjusted BPGD was worse. However, the BPGD algorithm itself is fully specified in Algorithm 1, the comparison decoders from [27] are external and reproduced from public code, and the self-citation to [28] is not load-bearing because the erasure-channel adaptation and algorithm details are given in the present paper. No uniqueness theorem is imported, no known result is merely renamed, and the base BPGD results are honestly reported alongside the tuned versions. The central claim retains independent content, but the magnitude of the reported improvement is inflated by in-sample parameter selection, so a moderate score is appropriate.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central performance claims rest on simulated decoding, with the BP equations and CSS reduction as standard assumptions. The key non-standard assumptions are the empirical effectiveness of random decimation in stopping sets and the transferability of the grid-search-optimized parameters. No new physical or algorithmic entities are introduced.

free parameters (5)
  • copt = varies by code and erasure rate (Tables 1, 3; e.g., 0.1 to 0.5 for [[1600,64]])
    Scaling factor for the initial LLR of non-erased bits (Eq. 15). Chosen to minimize failure rate on the same simulation data used for the performance plots.
  • gamma (damping factor) = varies by code and erasure rate (Table 2; e.g., 0.86 to 0.97)
    Weight in the damped BP update (Eq. 17). Chosen via grid search to minimize decoder failure rate.
  • llr_max = 25
    Large LLR clamp for non-erased bits (Section III-D). Set ad hoc for numerical stability; performance may depend on this value.
  • llr_min = approximately 0
    Small LLR for erased bits; used to avoid numerical issues while approximating the uniform prior.
  • BP iterations per round T = unstated
    The number of BP iterations in each decimation round is not specified in the paper, yet it affects convergence and runtime.
assumptions (5)
  • standard math BP update equations (12)-(14) correctly implement the sum-product algorithm on the Tanner graph.
    Standard message-passing updates; assumed to hold.
  • domain assumption The quantum erasure channel model: each erased qubit undergoes a uniform random Pauli error, and the decoder knows the erasure locations.
    Standard model from [12], [19]; used in all simulations.
  • domain assumption For CSS codes, correcting X and Z errors can be done independently using HZ and HX respectively (Eq. 5).
    Standard CSS reduction; relies on independent X/Z error probabilities for erased qubits.
  • ad hoc to paper In Section III-D, fixing bits in the stopping set essentially at random and continuing decoding leads to improved performance over pruned peeling; no proof is given.
    The paper provides an intuitive argument but not a rigorous guarantee or analysis of the probability of success.
  • ad hoc to paper The grid-search-selected values of copt and gamma (Tables 1-3) remain near-optimal for the code instances and erasure rates tested.
    The performance curves in Section V use these optimized values; the paper does not provide a rule for setting them for new codes or rates.

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Cite this review

Pith. "Pith review of Erasure Decoding for Quantum LDPC Codes via Belief Propagation with Guided Decimation." pith.science (2026). https://pith.science/paper/CXPR64LH

@misc{pith2026241108177,
  author       = {Pith},
  title        = {Pith review of: Erasure Decoding for Quantum LDPC Codes via Belief Propagation with Guided Decimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CXPR64LH}},
  note         = {Machine review of arXiv:2411.08177}
}
read the original abstract

Quantum low-density parity-check (LDPC) codes are a promising family of quantum error-correcting codes for fault tolerant quantum computing with low overhead. Decoding quantum LDPC codes on quantum erasure channels has received more attention recently due to advances in erasure conversion for various types of qubits including neutral atoms, trapped ions, and superconducting qubits. Belief propagation with guided decimation (BPGD) decoding of quantum LDPC codes has demonstrated good performance in bit-flip and depolarizing noise. In this work, we apply BPGD decoding to quantum erasure channels. Using a natural modification, we show that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes. Furthermore, we show that the performance of BPGD decoding on erasure channels can sometimes be improved significantly by either adding damping or adjusting the initial channel log-likelihood ratio for bits that are not erased. More generally, our results demonstrate BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape.

Figures

Figures reproduced from arXiv: 2411.08177 by the authors.

Figure 1
Figure 1. Comparison the quantum erasure channel of BPGD decod t [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Comparison on the quantum erasure channel of BPGD dec [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Comparison of BPGD decoding over the quantum erasure [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.