{"id":"3af91350-e895-433e-8096-a6d05f3f1d3e","arxiv_id":"2608.09583","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Noise-perturbed ensembles of Tanner forests with exact tree inference match or beat BP+OSD0 accuracy while cutting per-round decoding time by orders of magnitude.","lead":"A new decoder builds many simplified tree-like views of a quantum error-correcting code, adds small random perturbations to each view, and exactly solves each one to pick the most likely error. It matches or beats a standard high-accuracy decoder on surface and bivariate bicycle codes while running much faster per error-correction round.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sign inconsistency in the inference cost: Algorithm 2 uses C_q = -lambda_noise_q, while Section IIIB says lambda_q > 0 should discourage x_q = 1, so the claimed exact inference may be minimizing the opposite objective.","rationale":"The reader identified Eq. (7) as the weakest assumption, about which columns survive forest construction. Reading the full pipeline, I found a more immediate and distinct issue: the pseudocode for the exact inference stage appears to use the opposite sign of the cost derived in Section IIIB. The central claim is that NAED performs exact least-cost inference on a Tanner forest, and all LER and speed benchmarks depend on that inference stage. If Algorithm 2 does not minimize the stated objective, the simulation results cannot be traced to the described algorithm, regardless of how good the Eq. (7) ranking is. The proposed toy check is small and decisive: one check, two columns, and known lambdas. If the sign is wrong, the paper requires a substantive correction plus code/data verification; if the sign is a typo, the manuscript still needs to specify the intended convention precisely. This strengthens the reader's conditional verdict but does not change its category, because the reader already required code/data and sensitivity analysis before acceptance. A reject verdict would be premature since the simulations may well have used the correct sign despite the pseudocode error, but the inconsistency must be resolved before the central claim can be accepted.","tokens_in":25797,"tokens_out":13894,"duration_ms":141266,"concrete_test":"Implement Algorithm 2 exactly as written on the 2-column, 1-check tree with syndrome s = [1] and lambdas [1, 10] (root column q1 connected to check c, which is also connected to leaf column q2). The stated Eq. (16) objective requires the syndrome-valid assignment {q1=1, q2=0} with cost 1. If the pseudocode returns {q1=0, q2=1} with stated cost 10, the sign in Algorithm 2 line 7 is confirmed wrong. As a second check, replace line 7 with +lambda_noise_u * x_u; if and only if this corrected version returns the cost-1 assignment, the manuscript needs a sign correction and the released simulations must be rerun or verified to confirm which objective was actually optimized.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section IIIB defines w_q = f(x=0) - f(x') and sets lambda_q = -w_q, explicitly stating that lambda_q > 0 means flipping q increases the objective and q should stay 0, while lambda_q < 0 means q should flip. Therefore the per-column cost in Eq. (16) must be C_q = lambda_q: a positive lambda must penalize x_q = 1. Algorithm 2 line 7, however, sets the upward column message to -lambda_noise_u * x_u, i.e., C_q = -lambda_noise_q, the opposite sign. A column that the text designates as unlikely to be flipped therefore receives a negative flip cost and is preferentially selected by the dynamic program. Concretely, on a tree consisting of one root column q1, one check c, and one leaf column q2, with syndrome s_c = 1, lambda_q1 = 1, lambda_q2 = 10, the stated objective chooses {q1=1, q2=0} with cost 1, while Algorithm 2 as written returns {q1=0, q2=1} with stated cost 10. The claimed exactness of the Tanner-forest inference is thus not a property of the algorithm as specified; either the pseudocode has a sign typo or the simulations minimized a different objective than the one derived in Eqs. (7)-(9). Since all LER and timing claims depend on this DP, the central claim rests on an unresolved sign convention.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Noise Assisted Ensemble Decoding (NAED), a decoder for QLDPC codes that builds an ensemble of Tanner forests from the detector error model and performs exact inference on each forest by a linear-time dynamic program. The forest construction is driven by a syndrome- and channel-derived column score (Eq. 7), perturbed by ensemble-specific Gaussian noise (Eq. 10), with a residual-syndrome-aware variant (Eq. 15) for non-graph-like codes such as bivariate bicycle codes. Circuit-level simulation results are reported for rotated surface codes and BB codes, with claims of logical error rates comparable to or better than BP+OSD0 and of large per-round decoding speedups.","tokens_in":26168,"tokens_out":9973,"duration_ms":102409,"significance":"If correct, NAED would be a genuinely interesting decoder: exact inference on trees is linear-time, the ensemble is naturally parallel, and the Appendix B spanning argument for graph-like surface-code DEMs is standard and sound. The dynamic-programming derivation in Section III.D follows known tree-inference principles and is largely plausible. The paper is also honest about the main limitation, namely that syndrome-spanning forests cannot be guaranteed for general QLDPC codes. However, the central algorithm as written has a sign inconsistency that changes the optimized objective, and the empirical claims cannot be independently checked because no code or data are provided. These issues block acceptance in the current form.","major_comments":[{"comment":"Section III.B defines λ_q = -w_q and explicitly states that λ_q > 0 means flipping q increases the objective, so x_q=0 should be preferred, while λ_q < 0 means x_q=1 should be preferred. Consequently the per-column cost in Eq. (16) must be C_q = λ_q. Algorithm 2, lines 7 and 14, instead set the upward and root column costs to -λ_noise_q x_q, i.e. C_q = -λ_noise_q. Concretely, on a tree consisting of one check c with syndrome s_c=1 and two columns q1,q2 with λ_q1=1 and λ_q2=10, the objective derived from Section III.B chooses {q1=1,q2=0} at cost 1, whereas Algorithm 2 as written returns {q1=0,q2=1}, whose cost under the stated convention is 10. If the intended convention is that the variable λ_noise in Algorithm 2 is actually the perturbed weight n_b rather than the LLR -n_b, that contradicts the sentence after Eq. (10) and the call ETFI(F_b, s, n_b) in Algorithm 3. In either reading, the pseudocode does not implement the objective derived in Eqs. (7)-(9). Since every LER and timing claim is produced by this exact-inference routine, the numerical results must be re-derived under a consistent sign convention.","section":"Algorithm 2; Sections III.B and III.D"},{"comment":"The central reproducibility concern: Section VI states that the NAED code will be released only 'in the near future' and that numerical data are available only on request. The main claims—LER competitiveness in Figs. 3, 5, 7, 8 and the orders-of-magnitude timing improvement in Figs. 4 and 6—are therefore not independently verifiable from the manuscript. The timing comparison also omits baseline implementation details, such as whether BP+OSD0 is single-threaded, which OSD library or variant is used, and whether the NAED time in Fig. 4 includes construction of all ten forests or stops at the first valid result. These details are essential because the advertised speed advantage is a central contribution.","section":"Section IV and Section VI"},{"comment":"Algorithm 1 (CheckUpdate) does not define a result when a detector node c has no child column nodes but the required parity w = s_c ⊕ x_p equals 1: the loop over ch(c) is empty, the condition on line 8 is true, and line 9 takes the arg min over an empty set. This case is reachable for non-syndrome-spanning forests, which the paper explicitly allows for BB codes and which Algorithm 3 processes before the span check of line 7. The routine should propagate an infinite-cost message for infeasible parity; as written, the behavior is undefined for those inputs.","section":"Algorithm 1 and Algorithm 3"}],"minor_comments":[{"comment":"The 'Success percentage' in Tables I-IV does not state whether the denominator is all 10,000 sampled syndromes or only those not resolved by the previous decoding stage; Table V is clearer because it gives counts. Please state the denominator explicitly for each table.","section":"Tables I-IV"},{"comment":"The text around Fig. 1 refers to a 'target detector pair' {s139, s185} without explaining how this pair is selected. The ensemble-member examples would be easier to follow if the selection criterion were given.","section":"Section III.B, Fig. 1"},{"comment":"The condition 'If DEM matrix H is sparse' is ambiguous because the algorithm has just been run on both full and sparse DEMs. Please clarify what 'sparse' refers to and what the lifting operation does when the current DEM is the sparse one.","section":"Algorithm 3, line 11"},{"comment":"The scalar parameters κ=0.5, τ=0.75, β=2.0 are stated for the BB-code experiments, but no sensitivity analysis or selection criterion is provided. Given the heuristic nature of Eq. (15), a brief sensitivity study would strengthen the empirical claims.","section":"Section III.C and Fig. 5"}],"recommendation":"major_revision","confidential_remarks":"The sign inconsistency raised in the stress-test note is real and is the main technical blocker. The paper cannot be accepted while the pseudocode and the derived objective disagree, because all reported numbers depend on Algorithm 2. I also recommend asking for code or at least detailed reproducibility artifacts before any further round, since the timing and LER claims are entirely empirical. The paper is within scope, and the underlying tree-inference idea is worth pursuing if the sign issue is resolved and the experiments are rerun."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the framework is worth taking seriously—noise-perturbed syndrome-plus-LLR weights, an ensemble of Tanner forests, exact DP inference per tree, and an honest post-BP stage for BB codes. But Algorithm 2 has a sign error that contradicts the derivation in Section IIIB, and with no code or data I can't tell which objective the simulations minimized. That's the thing to fix first.\n\nWhat's actually new: the synthetic soft weight w_q = -llr_q + alpha * sum(2s_c-1) is a clean way to combine channel and syndrome information without BP; the noise-perturbed ensemble construction is a sensible diversity mechanism; and the residual-syndrome-aware dynamic reweighting (Eq. 15) is a useful contribution for non-graph-like codes. The DP itself is standard—two-state parity with cheapest flip—and Appendix B gives a sound argument that for weight-2 columns any ordered forest preserves the syndrome span. I also give the authors credit for stating plainly that syndrome-spanning forests are not guaranteed for BB codes and for adding a low-cost post-BP stage, and for benchmarking against BP+OTF in an appendix.\n\nThe soft spots, in order. First, the sign issue. Section IIIB defines lambda_q = -w_q and explicitly says positive lambda_q should keep x_q=0, so the flip cost in Eq. (16) must be lambda_q. Algorithm 2 line 7 uses -lambda_noise_q instead. On a two-column tree with lambda=(1,10) and syndrome 1, the derived objective picks the lambda=1 column; the pseudocode picks the lambda=10 column. The exactness claim is therefore not true of the algorithm as written. It could be a typo, but then the simulations may have minimized the opposite objective, and the reported LERs don't tell us which. Second, no code or data are released, which compounds the first issue—reproducibility is impossible as-is. Third, the main scalars (alpha, tau_b, kappa, tau, beta) are hand-set with no sensitivity analysis, and the timing benchmark against BP+OSD0 uses a 1000-iteration cap and time-to-first-valid from parallel workers; that's acceptable but should be documented more carefully.\n\nWho this is for: people working on real-time QLDPC decoding. The core idea is novel and the surface-code argument is solid; with the sign fixed and code/data released, this could be a useful decoder. Recommendation: send to peer review. The flaw is fixable and the framework deserves referee time, but the referee should ask for the corrected pseudocode and the simulation code before accepting the performance claims.","headline":"A promising QLDPC decoder framework with a real sign inconsistency in the inference algorithm that has to be resolved before the exactness and performance claims can be trusted.","tokens_in":26662,"tokens_out":6650,"would_cite":false,"duration_ms":62850,"reading_group":"yes","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Pp"],"model":"deepseek-v4-flash","headline":"Noise-perturbed Tanner forests make QLDPC decoding fast and accurate.","keywords":["quantum error correction","QLDPC codes","Tanner forest","dynamic programming decoding","ensemble decoding","circuit-level noise","belief propagation","ordered statistics decoding"],"falsifier":"The paper's Table V shows that for BB [[108,8,10]] and [[144,12,12]] codes at p=0.002, between 0.16% and 1.55% of sampled syndromes are not resolved by the full-DEM forest stage. A decisive test: take those unresolved syndromes, run NAED with the residual-aware dynamic construction while increasing the ensemble size B well beyond 100, and check whether any member ever produces a syndrome-spanning forest. If the failure set never shrinks, the noise-perturbation diversity claim is falsified for those codes.","tokens_in":25611,"feed_emoji":"🌲","tokens_out":5751,"duration_ms":53432,"temperature":0.7,"pith_summary":"This paper introduces a decoding framework, NAED, that replaces iterative message passing with exact inference over an ensemble of tree-like subgraphs (Tanner forests) of a quantum code's detector error model. The forest construction is guided by a syndrome- and channel-derived column score, and controlled noise perturbations generate diverse forests so that at least one forest spans the observed syndrome. On circuit-level simulations, NAED matches or beats the accuracy of BP+OSD0 for surface codes with only 11 ensemble instances, and for bivariate bicycle codes reaches OSD-level logical error rates when followed by a cheap post-BP stage. Per-round decoding times are reported to be orders of magnitude lower than BP+OSD0, because each forest is decoded by a single linear-time dynamic programming pass.","feed_headline":"Forest ensembles decode QLDPC codes orders of magnitude faster","feed_subtitle":"Exact inference on noise-perturbed Tanner forests matches BP+OSD0 accuracy in a fraction of the per-round time","key_machinery":"The load-bearing object is the noise-perturbed column score $n_b^q = w_q + \\tau_b \\varepsilon_q$ (Eq. (10)), with base score $w_q = -\\mathrm{llr}_q + \\alpha \\sum_{c \\in N(q)} (2s_c - 1)$, which equals the single-flip objective change $f(x=0)-f(x')$ in Eq. (9). This score orders the columns processed by a Kruskal-style forest constructor; for BB codes the score is upgraded to $n_b^q = \\kappa w_q + \\tau \\varepsilon_q + \\beta G_q(r)$ (Eq. (15)), where $G_q(r) = 2a_q(r) - |N(q)|$ is the dynamic gain in residual syndrome weight. Inference on each forest is performed by a two-stage dynamic program: an upward pass of two-entry messages with a parity-constrained cheapest-flip rule (Algorithm 1, CheckUpdate) and a downward traceback (Algorithm 2, ETFI), costing $\\Theta(|E_F|)$ time per forest. The ensemble pools member results by most-likely-error cost or first-valid output.","core_discovery":"The central claim is that a decoder does not need BP's iterative soft information: a single-column flip-cost heuristic computed from the syndrome and channel LLRs, numerically equal to the objective change $f(x=0)-f(x')$ in Eq. (9), is enough to prioritize decoding-matrix columns for building a Tanner forest, and adding Gaussian noise to these scores creates an ensemble of forests whose exact inference recovers the error. For graph-like codes such as surface codes, any forest built from the sparse detector error model spans the syndrome, so the ensemble's best solution is guaranteed to resolve it; for BB codes, a residual-syndrome-aware dynamic gain reorders columns online and, together with a low-cost post-BP stage, closes the gap to OSD accuracy. The paper thus targets the degenerate maximum-likelihood decoding problem rather than the standard QMLD formulation, arguing that the ensemble of forest inferences approximates DQMLD better than BP-based approximations.","pith_inferences":["The same synthetic-score mechanism could be tuned per code family: the residual-aware dynamic gain $G_q(r)$ is a drop-in replacement for the static score, so codes with higher column weights may inherit the BB-code treatment.","If the speedup survives hardware implementation, the per-round latency budget freed by NAED could be spent on denser measurement schedules or higher-distance codes, shifting the real-time bottleneck from decoding to readout.","A natural test is to combine NAED with decoder switching: use NAED as the always-on fast decoder and invoke a heavier decoder only on the few percent of syndromes that no forest spans.","The framework's dependence on syndrome-spanning forests suggests that code design criteria should include forest-spanning probability, not just distance and rate; codes engineered to maximize that probability would make the post-BP stage unnecessary."],"forward_implications":["Circuit-level decoding of surface codes can run without BP iterations, replacing the heuristic iteration cap with a single linear-time exact pass per forest.","Ensemble parallelism means wall-clock latency is set by the fastest syndrome-valid member, not by the slowest, making the decoder a candidate for FPGA/GPU real-time pipelines.","For BB codes, NAED's forest stage resolves over 98% of syndromes; only a cheap post-BP round is needed to reach OSD-level logical error rates.","The reported orders-of-magnitude speedup over BP+OSD0 grows with code distance, since the comparison decoder's per-round cost rises faster than the forest construction and inference."],"supporting_citations":[{"why":"It supplies the ordered Tanner forest construction, the sparse DEM sparsification routine, and the argument that graph-like surface-code mechanisms always allow syndrome-spanning forests.","marker":"[28]"},{"why":"It defines the BP+OSD0 decoder used as the accuracy and latency baseline throughout the paper.","marker":"[23]"},{"why":"It contributes the most-likely-error ensemble pooling rule that NAED uses to combine forest predictions.","marker":"[57]"},{"why":"It provides the Stim circuit-level simulator that generates the detector error models and syndrome samples for the memory experiments.","marker":"[58]"},{"why":"It supplies the bivariate bicycle code stabilizer circuits used in the BB-code benchmarks.","marker":"[46]"},{"why":"It introduces the bivariate bicycle code family that the paper targets as a general QLDPC test case.","marker":"[7]"},{"why":"It establishes trellis-based dynamic programming decoding for stabilizer codes, the conceptual ancestor of the forest inference DP.","marker":"[48]"}],"fun_headline_variants":["Noise-perturbed forest ensembles match BP+OSD accuracy at speed","Exact inference on Tanner forests delivers fast QLDPC decoding","NAED: fast ensemble decoding for QLDPC codes","Forest ensembles accelerate QLDPC decoding without accuracy loss","Forest ensembles achieve orders-of-magnitude speedup for QLDPC decoding"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the heuristic score of Eq. (7) ranks decoding-matrix columns so well that at least one noise-perturbed Tanner forest in the ensemble spans the observed syndrome or a logically equivalent correction; if no forest spans the syndrome, exact inference on the forests cannot recover the error.","fun_headline_variants_meta":{"raw":{"variants":["Noise-perturbed forest ensembles match BP+OSD accuracy at speed","Exact inference on Tanner forests delivers fast QLDPC decoding","NAED: fast ensemble decoding for QLDPC codes","Forest ensembles accelerate QLDPC decoding without accuracy loss","Forest ensembles achieve orders-of-magnitude speedup for QLDPC decoding"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001648,"raw_usage":{"total_tokens":6539,"prompt_tokens":932,"completion_tokens":5607,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":5514}},"tokens_in":548,"tokens_out":5607,"duration_ms":38688,"temperature":1.0,"reasoning_tokens":5514,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:37:28.652643+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The paper's Table V shows that for BB [[108,8,10]] and [[144,12,12]] codes at p=0.002, between 0.16% and 1.55% of sampled syndromes are not resolved by the full-DEM forest stage. A decisive test: take those unresolved syndromes, run NAED with the residual-aware dynamic construction while increasing the ensemble size B well beyond 100, and check whether any member ever produces a syndrome-spanning forest. If the failure set never shrinks, the noise-perturbation diversity claim is falsified for those codes.","supporting_citations":[{"cited_title":"We use the same sparsification procedures from Ref","cited_arxiv_id":null,"evidence_quote":"It supplies the ordered Tanner forest construction, the sparse DEM sparsification routine, and the argument that graph-like surface-code mechanisms always allow syndrome-spanning forests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It contributes the most-likely-error ensemble pooling rule that NAED uses to combine forest predictions."},{"cited_title":"Impulse Decoding of Quantum LDPC Codes: Equivalence of Degeneracy and Code-Shortening","cited_arxiv_id":"2606.18240","evidence_quote":"It supplies the bivariate bicycle code stabilizer circuits used in the BB-code benchmarks."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It introduces the bivariate bicycle code family that the paper targets as a general QLDPC test case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It establishes trellis-based dynamic programming decoding for stabilizer codes, the conceptual ancestor of the forest inference DP."}],"review_version":1}