Sharp Error-Rate Transitions in Quantum QC-LDPC Codes under Joint BP Decoding

Daiki Komoto; Kenta Kasai

REVIEW 3 major objections 7 minor 8 references

Sharp Error-Rate Transitions in Quantum QC-LDPC Codes under Joint BP Decoding

T0 review · 3 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper reports that quantum QC-LDPC codes of non-vanishing rate, decoded by binary joint belief propagation, exhibit steep error-rate transitions and tie the residual error floor to small trapping-set subgraphs.

desk verdict A plausible new observation—steep error-rate transitions for finite-rate quantum QC-LDPC under binary joint BP—but the paper underspecifies its codes and decoder, so it is a good candidate for peer review rather than a finished result. read the letter →

arxiv 2507.11534 v2 pith:WPW3QIV7 submitted 2025-07-15 quant-ph cs.ITmath.IT

Daiki Komoto , Kenta Kasai This is my paper · ORCID

classification quant-phcs.ITmath.IT PACS 03.67.Pp

keywords quantumLDPCcodesquasi-cyclicjointbeliefpropagationerrorfloortrappingsetsframeratedepolarizingchannelcorrection

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 reports that quantum quasi-cyclic low-density parity-check (QC-LDPC) codes with non-vanishing rate, decoded by a binary joint belief-propagation decoder, show frame and bit error rates that drop steeply as the physical error rate $p_D$ decreases, while still exhibiting an error floor. To the authors' knowledge this is the first observation of such threshold-like behavior for quantum LDPC codes with non-vanishing coding rate, apart from a prior code decoded with a non-binary decoder. The paper also finds that residual error patterns in the floor are tiny: for the $(4,12,1452)$ code, $97\%$ of floor-region errors involve at most $3L$ bits, and for the $(4,12,4500)$ code, $98\%$ involve at most $2L$ bits. These small patterns point to trapping sets, specific short subgraphs of the Tanner graph, as the cause of the floor, and suggest that steering code design away from such subgraphs could lower the floor. A sympathetic reader would take the paper as establishing that steep finite-rate quantum error correction is accessible even to a comparatively simple binary decoder, and that the remaining floor is a local, graph-structural problem.

What carries the argument

The machinery is a quantum QC-LDPC construction whose two parity-check matrices $H_X$ and $H_Z$ are orthogonal, each composed of $J\times L$ circulant permutation blocks of size $P$, so the code length is $n=PL$; the paper specifies one example by exponent matrices $E_X$ and $E_Z$ with $J=3,L=8$ and says other lengths follow from the same construction with girth 6. On the decoding side, joint belief propagation estimates both noise vectors $\hat{x}$ and $\hat{z}$ from the syndromes $(s,t)$, approximating the posterior $p(x,z|s,t)$, and stops when $(H_Z\hat{x}, H_X\hat{z})=(s,t)$ or when the iteration limit is reached. This decoder is what makes the error-rate curve steep, and the floor analysis then counts, in failed events, how many bit positions differ between the true and estimated noise, connecting the dominant few-bit patterns to trapping sets in the Tanner graph.

What would settle it

Compute $H_X$ and $H_Z$ for each of the six simulated $(J,L,n)$ settings from the stated construction, verify $H_X H_Z^T = 0$ and that the Tanner graph has no cycles of length 4, and rerun the joint BP decoder on the depolarizing channel for at least one setting; failure on any check means the reported steep curves or floor statistics cannot be attributed to the claimed QC-LDPC code family.

Watch

Extended reading notes

Core claim

The central claim is that quantum QC-LDPC codes built from an orthogonal pair of parity-check matrices $H_X$ and $H_Z$, each a $J\times L$ array of circulant permutation matrices, with column weight $J$, row weight $L$, length $n=PL$, and rate $R=1-2J/L$, exhibit steep FER and BER curves under joint BP decoding on the depolarizing channel. The steepness increases with code length, and the error floor becomes more prominent as $n$ grows. In the floor region, failed decoding events leave residuals $\hat{x}+x$ and $\hat{z}+z$ concentrated in very few bit positions, quantified as $97\%$ of errors with at most $3L$ bits for the $(4,12,1452)$ code and $98\%$ with at most $2L$ bits for the $(4,12,4500)$ code. The paper interprets this concentration as evidence that trapping sets, small subgraphs of the Tanner graph, dominate the error floor, and therefore that identifying and avoiding such structures is a viable path to reducing it.

Load-bearing premise

The load-bearing assumption is that all six simulated code families are valid quantum codes with commuting X-checks and Z-checks and Tanner graphs free of short cycles, even though the text displays exact exponent matrices for only one example and asserts the remaining cases behave the same.

Editorial extensions

If this is right

If the central claim holds, steep finite-rate quantum error correction is accessible to binary joint BP, a comparatively simple decoder rather than a costly non-binary one.
Steepness grows with code length, so the observed transition behaves like a scaling phenomenon rather than a one-off curve.
Dominant floor events concentrate on a few bit positions, so avoiding the corresponding small Tanner-graph subgraphs should reduce the error floor.
The threshold-like transition appears for larger row weights and not for small ones, linking the effect to structural parameters rather than code length alone.
Lower-rate codes sit farther from the hashing bound, matching the classical binary LDPC pattern and giving a concrete target for future constructions.

Reading between the lines

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

The small-residual diagnosis implies that a post-processing pass which locates the few differing positions in a failed block and re-decodes only that subgraph could suppress the error floor without changing either the code or the main decoder.
The dependence of the transition on row weight suggests a testable design rule: for girth-6 exponent matrices with fixed column weight, there should be a minimum row weight below which the steep cliff disappears.
Spatially coupled versions of these codes, which the paper names as future work, would be a natural stress test: if coupling sharpens the transition and suppresses trapping sets, the finite-rate waterfall should coexist with a much lower floor.
The paper's bit-count metric is coding-theoretic, not physical; mapping residual bit positions onto logical Pauli errors would be required before using these floor counts as a guide for experimental error rates.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing worth knowing: this is the first report I've seen of finite-rate quantum QC-LDPC codes showing waterfall-like FER and BER transitions under a binary joint BP decoder, and the authors are appropriately modest about it. The figures make the central observation plausible: longer codes give steeper curves, and the error-floor residuals are concentrated on few-bit events. The comparison against the hashing bound is a reasonable sanity check. They also openly flag that their BER metric lacks a direct physical interpretation—that kind of honesty helps.

The real soft spot is reproducibility. For a short simulation report whose whole claim is an empirical first, the paper needs to give the exact codes and decoder settings. It provides the exponent matrices for only one (J=3, L=8) example, then says the other parameter sets are a “natural extension” with girth 6 “by appropriately choosing the sub-blocks.” That is not enough to confirm that the six simulated (J,L,n) codes actually satisfy H_X H_Z^T = 0 and have girth 6. We also get no maximum iteration count, message schedule, prior assignments, sample counts, or confidence intervals. The 97% and 98% error-floor percentages are point estimates with no error bars. These omissions are not fatal to the qualitative conclusion, but they make the paper a demonstration, not yet a verifiable result.

I checked the stress-test note about the missing exponent matrices. I think it lands. The decoder is standard joint BP, so the algorithm is not the weak point; the construction details are. That said, the text does not hide this—it says the girth can be made 6 by choosing sub-blocks, which is an assertion, not a specification. A referee can reasonably ask for the matrices or code artifacts.

The novelty claim is stated as “best of our knowledge” and is plausible given the cited prior work on non-binary BP, erasure channels, and turbo codes. It is an extension of the authors' own earlier construction [5], but that self-citation is not a problem in itself; the problem is that [5] is also an unverified preprint.

My overall take: this is a solid candidate for peer review. A serious referee should ask for the code, the full set of exponent matrices, and decoder configuration before the claim becomes fully load-bearing. I would not cite it in the next twelve months until those artifacts appear, but the observation deserves to be checked properly. Send it out; just make sure the reviewers know what to demand.

Referee Report

3 major / 7 minor

Summary. The paper reports simulation results for quantum quasi-cyclic LDPC (QC-LDPC) codes decoded with a binary joint belief-propagation (BP) decoder. The authors observe steep transitions in the frame error rate (FER) and bit error rate (BER) as the physical depolarizing error rate decreases, despite the presence of error floors. They further report that the dominant residual errors in the error-floor region involve only a small number of bits, which they interpret as evidence of trapping sets. The paper claims that this is the first observation of threshold-like behavior for quantum LDPC codes with non-vanishing coding rate under binary BP, excluding earlier non-binary BP results.

Significance. If the reported observation is reproducible, the paper makes a useful empirical contribution: it suggests that finite-rate quantum QC-LDPC codes can exhibit waterfall-like behavior under a relatively simple binary decoder, and that error floors are tied to small subgraph structures that could potentially be avoided by code design. The authors are honest about the non-physical nature of their BER metric and clearly state that the primary contribution is the observed transition, not a new decoder or code construction. The significance is currently limited by the lack of full construction details for the simulated codes and the absence of statistical reporting, both of which prevent independent verification.

major comments (3)

[II] The code construction is only fully specified for the J=3, L=8 example in Eq. (1). All six simulation settings use L=12 and J=3 or J=4, and the text states only that the construction is a 'natural extension' of [5] with girth 6 'by appropriately choosing the sub-blocks.' No exponent matrices, block sizes P, or sub-block choices are given for any of the simulated codes. Because orthogonality (H_X H_Z^T = 0), length n = P L, and girth are load-bearing for both the observed decoding curves and the trapping-set interpretation, the central observation is not reproducible from the manuscript as written. Please provide the full exponent matrices or a deterministic construction algorithm for all simulated (J, L, P), along with explicit verification of orthogonality and girth.
[III] Figure 1 reports FER and BER values down to about 1e-6, but the paper gives no sample counts, confidence intervals, or decoder settings (maximum number of iterations, message schedule, initial priors, or stopping criteria beyond the two described). The apparent steepness of the curves and the 97% and 98% error-floor percentages are point estimates with no stated uncertainty. Please report the number of decoded frames or failing frames per data point and, where feasible, confidence intervals so that the steepness and the error-floor composition can be assessed statistically.
[II] The BER metric counts residual bit mismatches only when decoding fails and is explicitly admitted to 'not have a direct physical interpretation.' Since the paper's central claim includes a steep BER transition, this non-standard metric could produce an artifact: the BER is conditioned on failure and may vary with the decoder's failure modes in a way that is not physically meaningful for degenerate quantum codes. Please either justify the BER metric as a meaningful figure of merit or restrict the central claim to the FER transition, which is physically well defined.

minor comments (7)

[II] The example exponent matrices in Eq. (1) do not specify the block size P or the resulting code length n. Please state P for the example or clarify that the entries are interpreted modulo a chosen P.
[II] The symbols C_X and C_Z in the success condition are not defined. They presumably denote the code spaces (kernels) of H_X and H_Z; please define them explicitly.
[II] The construction in [5] is not summarized, making the paper not self-contained. Please include a brief description of the base construction and how the J=3 and J=4 cases extend it.
[III] The hashing-bound reference lines in Fig. 1 are not defined. Please state the formula used and explain why it depends only on J and L.
[III] The percentages '97%' and '98%' refer to the 'error floor region,' but the physical error-rate range defining this region is not specified. Please define the range used for these statistics.
[References] Reference [3] lists '2015 IEEE International Symposium on Information Theory Proceedings, 2025,' which appears to contain an inconsistent year, and Reference [4] is missing the journal name (IEEE Transactions on Information Theory). Please correct the bibliographic details.
[I] The sentence 'Rather, the primary contribution of this study is to show...' begins with an orphaned 'Rather' that appears to be a leftover from an earlier draft; please rephrase.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported FER/BER transitions and error-floor bit counts come directly from simulations, with no fitted parameter or self-referential definition supplying the predicted curves.

full rationale

This is an empirical observation paper, not a derivation. The central curves in Fig. 1 are outputs of a joint BP decoder applied to quantum QC-LDPC codes and are benchmarked against the hashing bound; no parameter appearing in the plots is fitted from the plotted data and then relabeled as a prediction. The self-citations to [1], [5], and [6] serve to position the novelty claim, to motivate the code construction, and to interpret the error floor in terms of trapping sets, but none of these citations is the source of the simulated FER/BER values or the small-residual-error statistics. The paper's stated orthogonality and girth-6 property for the simulated codes are asserted rather than fully specified (exponent matrices are given only for one L=8 example while the figures use L=12), but that is a reproducibility or verification gap, not circularity. Under the requirement that circularity be exhibited as an equation or construction that reduces the output to an input, no such step exists, so the appropriate score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No numerical constants are fitted to data. The hand-chosen inputs are the QC-LDPC parameter sets and unspecified decoder settings. The central claim rests on the depolarizing channel model, the validity of the CSS codes, and the joint BP termination rules stated in Section II.

free parameters (2)

QC-LDPC construction parameters (J, L, P or n) = J=3 or 4, L=12, n in {516, 1428, 4524, 660, 1452, 4500}
Hand-selected from the authors' construction [5]; the steepness claim is demonstrated only for these parameter choices.
Joint BP decoder settings = not reported
Maximum iteration count, schedule, and damping are unspecified; these settings can determine whether a waterfall or an error floor appears.

assumptions (4)

domain assumption H_X and H_Z are orthogonal and define a valid CSS code; sub-block choices give girth 6 for the simulated lengths.
Invoked in Section II; the existence of these codes is taken from the authors' construction [5] and not independently verified in this paper.
domain assumption Channel noise is a depolarizing channel with physical error rate p_D.
All simulation results in Section III are conditional on this model; no other noise models are tested.
domain assumption The joint BP decoder's update rules and termination criterion adequately approximate the posterior p(x,z|s,t).
Decoder parameters such as maximum iteration count, schedule, and damping are unspecified, so part of the observed behavior may depend on these choices.
domain assumption The nonstandard bit-error metric, defined as the number of indices where x_i differs from x_hat_i or z_i differs from z_hat_i, is a meaningful proxy for error-floor size.
The paper itself states this metric has no direct physical interpretation, yet uses it to derive the small-error-count percentages in Section III.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Sharp Error-Rate Transitions in Quantum QC-LDPC Codes under Joint BP Decoding." pith.science (2026). https://pith.science/paper/WPW3QIV7

@misc{pith2026250711534,
  author       = {Pith},
  title        = {Pith review of: Sharp Error-Rate Transitions in Quantum QC-LDPC Codes under Joint BP Decoding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WPW3QIV7}},
  note         = {Machine review of arXiv:2507.11534}
}

read the original abstract

In this study, we report that quantum quasi-cyclic low-density parity-check codes decoded via joint belief propagation (BP) exhibit steep error-rate curves, despite the presence of error floors. To the best of our knowledge, this is the first observation of such threshold-like behavior for quantum LDPC codes with non-vanishing coding rate, excluding those decoded with non-binary BP decoders. Moreover, we find that dominant error events contributing to the error floor typically involve only a small number of bits. These findings suggest that the error floor is caused by trapping sets--specific subgraph structures in the Tanner graph--and indicate that identifying and avoiding such structures may lead to further reduction of the error floor.

Figures

Figures reproduced from arXiv: 2507.11534 by the authors.

**Figure 1.** BER and FER vs. physical error rate pD for quantum QC-LDPC codes with (J, L, n). corresponding to HX and HZ, are given as follows: EX =   −2 0 −2 1 −2 2 −2 3 −2 4 −2 5 −2 6 −2 7 −2 3 −2 0 −2 1 −2 2 −2 7 −2 4 −2 5 −2 6 −2 2 −2 3 −2 0 −2 1 −2 6 −2 7 −2 4 −2 5   EZ =   −2 4 −2 7 −2 6 −2 5 −2 0 −2 3 −2 2 −2 1 −2 5 −2 4 −2 7 −2 6 −2 1 −2 0 −2 3 −2 2 −2 6 −2 5 −2 4 −2 7 −2 2 −2 1 −2 0 −2 3   This code is a natural… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 4 canonical work pages

[5]

G \"o kduman, H

M. G \"o kduman, H. Yao, and H. D. Pfister, `` The Performance of Long Quantum LDPC Codes Based on the Hypergraph Product ,'' in 2015 IEEE International Symposium on Information Theory Proceedings, 2025

work page 2015
[1]

write newline

" write newline "" initialize.prev.this.status FUNCTION begin.bib preamble empty 'skip preamble write newline if " thebibliography " longest.label * " " * write newline " [1] #1 " write newline " url@rmstyle " write newline " " write newline " [2] #2 " write newline " =0pt " write newline " " ALTinterwordstretchfactor * " " * write newline " = 2 plus " wr...
[2]

11em plus .33em minus .07em 4000 4000 100 4000 4000 500 `\.=1000 = #1 \@IEEEnotcompsoconly \@IEEEcompsoconly #1 * [1] 0pt [0pt][0pt] #1 * [1] 0pt [0pt][0pt] #1 * \| ** #1 \@IEEEauthorblockNstyle \@IEEEcompsocnotconfonly \@IEEEcompsocconfonly \@IEEEauthorblockAstyle \@IEEEcompsocnotconfonly \@IEEEcompsocconfonly \@IEEEcompsocconfonly \@IEEEauthordefaulttex...

work page
[3]

Komoto and K

D. Komoto and K. Kasai, `` Quantum Error Correction near the Coding Theoretical Bound ,'' arXiv preprint arXiv:2412.21171, 2024

arXiv 2024
[4]

deMarti iOlius, P

A. deMarti iOlius, P. Fuentes, R. Or \' u s, P. M. Crespo, and J. Etxezarreta Martinez, ``Decoding algorithms for surface codes,'' Quantum , vol. 8, p. 1498, Oct. 2024

work page 2024
[6]

Poulin, J.-P

D. Poulin, J.-P. Tillich, and H. Ollivier, ``Quantum serial turbo codes,'' vol. 55, no. 6, pp. 2776 --2798, June 2009

work page 2009
[7]

Komoto and K

D. Komoto and K. Kasai, `` Explicit Construction of Classical and Quantum Quasi-Cyclic Low-Density Parity-Check Codes with Column Weight 2 and Girth 12 ,'' arXiv preprint arXiv:2501.13444, 2025

arXiv 2025
[8]

Kasai, ``Quantum error correction exploiting degeneracy to approach the hashing bound,'' 2025, arXiv:2506.15636

K. Kasai, ``Quantum error correction exploiting degeneracy to approach the hashing bound,'' 2025, arXiv:2506.15636

arXiv 2025

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [5]

G \"o kduman, H

M. G \"o kduman, H. Yao, and H. D. Pfister, `` The Performance of Long Quantum LDPC Codes Based on the Hypergraph Product ,'' in 2015 IEEE International Symposium on Information Theory Proceedings, 2025

work page 2015

[2] [1]

write newline

" write newline "" initialize.prev.this.status FUNCTION begin.bib preamble empty 'skip preamble write newline if " thebibliography " longest.label * " " * write newline " [1] #1 " write newline " url@rmstyle " write newline " " write newline " [2] #2 " write newline " =0pt " write newline " " ALTinterwordstretchfactor * " " * write newline " = 2 plus " wr...

[3] [2]

11em plus .33em minus .07em 4000 4000 100 4000 4000 500 `\.=1000 = #1 \@IEEEnotcompsoconly \@IEEEcompsoconly #1 * [1] 0pt [0pt][0pt] #1 * [1] 0pt [0pt][0pt] #1 * \| ** #1 \@IEEEauthorblockNstyle \@IEEEcompsocnotconfonly \@IEEEcompsocconfonly \@IEEEauthorblockAstyle \@IEEEcompsocnotconfonly \@IEEEcompsocconfonly \@IEEEcompsocconfonly \@IEEEauthordefaulttex...

work page

[4] [3]

Komoto and K

D. Komoto and K. Kasai, `` Quantum Error Correction near the Coding Theoretical Bound ,'' arXiv preprint arXiv:2412.21171, 2024

arXiv 2024

[5] [4]

deMarti iOlius, P

A. deMarti iOlius, P. Fuentes, R. Or \' u s, P. M. Crespo, and J. Etxezarreta Martinez, ``Decoding algorithms for surface codes,'' Quantum , vol. 8, p. 1498, Oct. 2024

work page 2024

[6] [6]

Poulin, J.-P

D. Poulin, J.-P. Tillich, and H. Ollivier, ``Quantum serial turbo codes,'' vol. 55, no. 6, pp. 2776 --2798, June 2009

work page 2009

[7] [7]

Komoto and K

D. Komoto and K. Kasai, `` Explicit Construction of Classical and Quantum Quasi-Cyclic Low-Density Parity-Check Codes with Column Weight 2 and Girth 12 ,'' arXiv preprint arXiv:2501.13444, 2025

arXiv 2025

[8] [8]

Kasai, ``Quantum error correction exploiting degeneracy to approach the hashing bound,'' 2025, arXiv:2506.15636

K. Kasai, ``Quantum error correction exploiting degeneracy to approach the hashing bound,'' 2025, arXiv:2506.15636

arXiv 2025