{"id":"82f2ce12-f740-4538-be45-1145a737ef1c","arxiv_id":"2505.18592","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Concatenating (3,4)-random hypergraph product codes with distance-5 rotated surface codes yields a lower logical error per qubit than a distance-25 rotated surface code when the top-code size parameter is at least 4.","lead":"This paper studies a two-layer quantum error correction design: many small surface codes serve as the bottom layer, and a compact LDPC code links their logical qubits on top. The authors report that with 16 or more logical qubits and a target code distance of at least 25, this stacked design can match or beat a standard surface code at physical error rates near or below 1 percent.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central s>=4, d>=25 crossover rests on an unvalidated power-law extrapolation; no direct Monte Carlo comparison at the claimed operating point.","rationale":"The reader's weakest assumption is also the most load-bearing concern I find: the s>=4, d>=25 outperformance claim is derived by comparing fitted power laws for the concatenated code against a surface-code power law, with no direct simulation at the claimed operating point. The condition is mathematically sound given the fitted b and c, but it is numerically fragile. Small changes in c shift the integer crossover, and the BP-OS failure mode on high-weight errors, which the paper itself acknowledges, could corrupt the power-law behavior precisely where the comparison is made. The absence of prefactors in Eqs. (39)-(40) further weakens the comparison unless the pseudo-thresholds are interpreted as fully absorbing all constants, which is not stated. I also flag the typo in Eqs. (20)-(21) as a secondary but real reproducibility problem: the printed probability formula is dimensionally wrong and, if implemented as written, would change the soft information fed to the upper-layer decoder. Since the reader already made the power-law extrapolation the basis of a CONDITIONAL verdict, my assessment is unchanged. The proposed direct Monte Carlo test at p=10^-2 would settle the concern decisively, though it may be expensive for the d=25 surface code; the fallback at higher p with a crossover check preserves the test's practicality.","tokens_in":13215,"tokens_out":11347,"duration_ms":94400,"concrete_test":"Run a direct Monte Carlo comparison at p=10^-2: simulate the full s=4 concatenated code using the same BP-OS decoder with lookup-table syndrome-conditioned probabilities, and simulate a distance-25 rotated surface code with the improved BP decoder of Ref. [33], collecting at least 10^8 samples per code and recording the average logical error rate. If the concatenated rate is not below the surface-code rate at this physical error rate, the s>=4, d>=25 claim fails. If statistical resolution is insufficient at p=10^-2, repeat at p=2×10^-2 and p=3×10^-2 and check that the crossover predicted by Eq. (43) actually falls near or below 10^-2 when both measured curves are extrapolated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline condition s>=4, d>=25 at p≈1e-2 is not a directly measured fact but a comparison of two power-law models. Equation (39) uses the fit pL=(p/pth_c)^(b*s^c) with b=5.481 and c=0.667, while Eq. (40) uses pL=(p/pth_s)^((d+1)/2) with pth_s=0.1776 from Ref. [33]. Neither formula includes a constant prefactor, and Figure 10 plots these approximate forms rather than a direct simulation at the claimed crossover. Although s=4 is within the fitted range, the crossover condition is set by solving Eq. (43), and the result is sensitive to the fitted parameters: for c=0.6, srs(0.01,25) exceeds 4, so the integer condition would shift. The authors themselves note in Section III B that BP-OS may return incorrect corrections on high-weight errors, which is exactly the regime where a clean power law could break down. A secondary reproducibility issue is that Eqs. (20)-(21) write the configuration probability as p^{wt(e)}(1-p)^{wt(e)} rather than p^{wt(e)}(1-p)^{n-wt(e)}; if the implemented lookup table follows this text, the syndrome-conditioned probabilities are incorrectly normalized. These uncertainties mean the central claim is not established without a direct end-to-end check.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a hierarchical quantum error correction scheme in which a (3,4)-random hypergraph product (HGP) code is concatenated with a distance-5 rotated surface code, so that the non-local HGP code can be implemented with nearest-neighbor interactions via lattice surgery. The lower layer is decoded with a lookup table that supplies syndrome-conditioned logical error probabilities, and the upper layer is decoded with belief propagation plus ordered statistics (BP-OS). Under a code-capacity depolarizing noise model, Monte Carlo simulations are used to extract a power-law scaling of the logical error rate, p_L = (p/p_th)^alpha with alpha = b s^c, and the manuscript derives the condition s >= 4 and d >= 25 for the concatenated code to outperform a rotated surface code of distance d in both qubit efficiency and logical error rate. The central claimed operating point is s = 4, giving parameters [[10^4, 16, 2 alpha]] with alpha approximately 14.38, compared with a distance-25 rotated surface code.","tokens_in":13567,"tokens_out":7704,"duration_ms":62547,"significance":"The problem addressed is important: realizing constant-rate qLDPC codes on planar, nearest-neighbor hardware is a key bottleneck for low-overhead fault-tolerant quantum computing, and a concrete numerical study of a concatenated HGP-surface architecture with a soft-decision decoder is a useful contribution. The paper has clear strengths: a substantial Monte Carlo campaign with up to 10^9 samples, an explicit lookup-table decoder with precomputed syndrome-conditioned probabilities, a public data/code repository (DOI 10.5281/zenodo.15660987), and a transparent fitting procedure for the scaling parameters. If the central crossover claim were validated by direct simulation, the result would be practically relevant as a possible resource-efficient alternative to large surface codes. However, as discussed below, the headline condition s >= 4 and d >= 25 rests on an extrapolation of fitted power laws rather than on a direct end-to-end Monte Carlo comparison at the claimed operating point, and the printed probability formulas in Eqs. (20)-(21) contain a normalization error that affects the soft information used by the decoder unless silently corrected in the implementation.","major_comments":[{"comment":"Equations (20) and (21) define Pe(p,s) and Pc(p,s) with factors p^{wt(e)}(1-p)^{wt(e)}. For the L = 5 rotated surface code there are n = 25 data qubits per error type, so the probability of a configuration of weight w is p^w(1-p)^{n-w}. As printed, the two sums are not correctly normalized and systematically mis-weight configurations; if the implemented lookup table follows the text, every syndrome-conditioned logical error probability PL(p|s) fed into the BP decoder is biased. Please correct the second exponent to n - wt(e) and confirm that the posted code uses the corrected expression.","section":"III B, Eqs. (20)-(21)"},{"comment":"The central condition s >= 4 and d >= 25 is derived by substituting the fitted parameters b = 5.481, c = 0.667, and pbar_c^th = 0.157 into Eq. (43) and comparing the resulting power laws with Eq. (40). However, Eqs. (39)-(40) are scale-free power laws without constant prefactors, and Figure 10 plots these fitted formulas rather than Monte Carlo data at the claimed operating point. The crossover is therefore an extrapolation of the same fits used to define the scaling, and it is sensitive to the fit parameters: for example, with c = 0.6 the value of srs(0.01, 25) exceeds 4, which would invalidate the integer condition. The authors' own caveat in Section III B that BP-OS may return incorrect corrections for high-weight errors is relevant precisely in the extrapolated regime. A direct end-to-end Monte Carlo comparison at s = 4, d = 25, or an equivalent validation with uncertainty quantification on all fitted parameters, is needed before the central claim can be accepted.","section":"III C, Eqs. (39)-(43) and Figure 10"},{"comment":"Equation (50) gives [[10^4, 16, 2 alpha]] with alpha approximately 14.38 for s = 4, but Equation (8) gives [[625 s^2, s^2, 5 d_s]], and Equation (11) gives d_s approximately 2.76 * 4^0.660, which is about 7.0, implying a concatenated distance of roughly 35, not 2 alpha approximately 28.8. Moreover, alpha is already used for the scaling exponent in Eqs. (22)-(23), so the reuse of alpha in Eq. (50) is confusing. Please state the actual measured distance of the selected HGP instance for s = 4 and reconcile Eq. (8), Eq. (11), and Eq. (50).","section":"III C, Eq. (50)"}],"minor_comments":[{"comment":"The sentence before Eq. (50) says 'In Figure 7, we plot logical error rate...', but Figure 7 compares error scaling factors; the logical-error-rate comparison appears in Figure 10. Please correct the cross-reference.","section":"III C / Figure 10"},{"comment":"The function introduced as srs(p,d) in Eq. (43) is later written as s(p,d) in Eqs. (45)-(48); please unify the notation.","section":"III C, Eqs. (43)-(48)"},{"comment":"The phrase 'rigorously compute syndrome-conditioned logical error probabilities rigorously' contains a duplicated adverb; please remove one occurrence.","section":"III B"},{"comment":"The caption of Figure 3 refers to a 'blue line' and an 'orange line', while the surrounding text describes data points and a fitting line; please check that the colors and line styles are described consistently with the figure.","section":"III A / Figure 3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript addresses a relevant problem and contains a serious numerical effort with reproducible data, but the central quantitative claim is not yet established. The printed probability formulas in Eqs. (20)-(21) need correction, and the s >= 4, d >= 25 crossover should be verified by direct simulation at the claimed operating point rather than by extrapolating the same fitted power laws. These are fixable within the scope of the paper, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look if you care about 2D-compatible qLDPC: they concatenate (3,4)-random HGP codes with an L=5 rotated surface code and use a precomputed lookup table to feed syndrome-conditioned logical error probabilities into BP-OS for the upper layer. The lookup-table trick for L=5 is a genuinely practical contribution, and the Monte Carlo campaign (1e7 to 1e9 samples) is substantial. The data is openly available, and the references are appropriate. They claim the hierarchical code beats a distance-25 rotated surface code in both qubit efficiency and logical error rate when s>=4 (k>=16) at p around 1e-2.\n\nThe derivation of that condition is the soft spot. They fit pL ~ (p/pth)^(b*s^c) on s=2..6, then compare that power law against the surface-code scaling (p/pth)^((d+1)/2) without constant prefactors. The crossover condition s>=4 is an extrapolation of those fits; Figure 10 plots the formulas, not a direct MC comparison at the claimed operating point. The integer condition is sensitive to the fitted exponent: if c=0.6 instead of 0.667, the required s shifts to 5. So the headline is plausible but not established.\n\nThere are two more concrete issues. Eqs. (20)-(21) write the configuration probability as p^{wt(e)}(1-p)^{wt(e)}; for an n=25 code it should be p^{wt(e)}(1-p)^{n-wt(e)}. As printed, that biases the syndrome-conditioned probabilities, though the authors may well have implemented the correct formula in code. Also, for depolarizing noise the marginal flip rate for each CSS sector is 2p/3, not p; that may be a notational choice but deserves clarification. Eq. (50) gives [[10^4, 16, 2α]] with α≈14.38, which is inconsistent with the stated concatenated distance 5d_s and looks like a typo. The authors do acknowledge BP-OS can fail on high-weight errors; that limitation is real but does not undermine the low-p regime they target.\n\nWho it is for: anyone working on hierarchical qLDPC-surface architectures or soft-decision decoding for concatenated codes. The lookup-table idea is worth reusing. The paper deserves a serious referee, not a desk reject, but the referee should ask for a direct MC check at the crossover (at least s=4 vs d=25) and a cleanup of the probability formulas before the central claim is taken as established.","headline":"A practical hierarchical qLDPC-surface scheme with a clever lookup-table soft decoder, but the headline s>=4, d>=25 crossover rests on an extrapolated power-law comparison that needs a direct Monte Carlo check.","tokens_in":14073,"tokens_out":7462,"would_cite":true,"duration_ms":60772,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Pp"],"model":"deepseek-v4-flash","headline":"The paper claims a concatenated HGP-surface code can outperform the rotated surface code for size parameter s≥4 and distance d≥25 at physical error rates around or below 10^-2.","keywords":["quantum error correction","hypergraph product code","rotated surface code","code concatenation","belief propagation ordered statistics decoding","quantum LDPC codes","fault-tolerant quantum computation","nearest-neighbor architecture"],"falsifier":"Run direct Monte Carlo decoding at p=$10^{-2}$ and p=$10^{-3}$ for the s=4, d=25 concatenated code and compare the measured logical error rate per logical qubit with that of a distance-25 rotated surface code under the same depolarizing noise model; if the concatenated code's error rate is not lower, or if the power-law exponent flattens at s=4, the central claim fails.","tokens_in":13014,"feed_emoji":"🛡️","tokens_out":5495,"duration_ms":44303,"temperature":0.7,"pith_summary":"This paper proposes a hierarchical quantum error correction code that stacks a random hypergraph product (HGP) code on top of a distance-5 rotated surface code, so the non-local HGP code can be implemented with nearest-neighbor hardware via lattice surgery. The authors claim that with soft-decision decoding using belief propagation with ordered statistics plus syndrome-conditioned logical error probabilities from a lookup table, the concatenated code suppresses logical errors below threshold and removes an error floor. Under a code capacity depolarizing noise model, they derive explicit conditions for the hierarchical code to beat the rotated surface code on both qubit efficiency and logical error rate. The headline result is that at physical error rates around or below $10^{-2}$, the concatenated code wins for size parameter s≥4 (16 logical qubits) and surface distance d≥25, with parameters [[$10^{4}$,16,2α]] where α≈14.38. If right, this gives a resource-efficient, planar-compatible route to fault-tolerant quantum computing.","feed_headline":"A concatenated code can beat the surface code at distance 25","feed_subtitle":"At error rates near 10^-2, 16 logical qubits in one HGP-surface block match a distance-25 surface code.","key_machinery":"The load-bearing object is the concatenated pair: a distance-5 rotated surface code as the lower layer and a (3,4)-random HGP code as the upper layer, whose combined parameters are [[$625s^{2}$, $s^{2}$, 5d_s]] with d_s the largest distance found among 1000 sampled HGP instances. What makes the decoder work is a precomputed lookup table for the L=5 surface code: for every one of the ~3.4×$10^{7}$ X/Z syndromes it stores the most probable error, the set of errors that would cause a logical failure, and the syndrome-conditioned logical error probability P_L(p|s), which is fed as soft information into a belief propagation-plus-ordered-statistics (BP-OS) decoder of depth λ=10 for the upper HGP layer. This soft-decision chain yields the measured scaling law p_L=(p/̄p_th)^{b s^c}, which is the formula that carries the whole crossover argument against the surface code.","core_discovery":"The central claim is that concatenating a (3,4)-random HGP code with a rotated surface code of distance 5 yields a code that can beat the planar surface code once the lower-layer equivalent has distance d≥25 and the HGP size parameter s≥4. Qubit efficiency requires d≥25 because the concatenated code uses 625 physical qubits per logical qubit, while a d-by-d surface code uses $d^{2}$ per logical qubit. For logical error rate, the paper fits p_L = (p/̄p_th)^{b s^c} with b=5.481 and c=0.667, compares it with the surface-code scaling (p/̄p_s_th)^{(d+1)/2}, and solves for the crossover. The resulting threshold size parameter s_rs(p,d) is nearly flat below pseudo-threshold because the two pseudo-thresholds are close, giving s≥3.65 and hence s≥4. At s=4 the code has parameters [[$625s^{2}$, $s^{2}$, 5d_s]] = [[$10^{4}$,16,2α]] with α≈14.38, versus a single distance-25 surface block [[625,1,25]]; the concatenated code is claimed to match or exceed it in logical error suppression while carrying 16 logical qubits in the same per-logical-qubit footprint.","pith_inferences":["If the claimed crossover holds, 10^4 physical qubits for 16 logical qubits puts a planar-compatible quantum memory in a range relevant to near-term demonstrations, though the paper's code capacity estimate omits measurement noise and leakage that would raise this overhead.","The lookup-table soft-information construction suggests a template for other topological lower layers: any code whose syndrome-conditioned logical error probability can be computed offline can feed a qLDPC upper layer, potentially improving threshold estimates.","A direct Monte Carlo comparison at s=4 and d=25, rather than extrapolated scaling curves, would be the natural next check; the authors do not report one, so the concrete crossover point is an extrapolation.","Future work could replace the HGP upper layer with asymptotically good qLDPC codes; the paper notes this would lower the distance at which concatenation beats the surface code, implying an entire family of better hierarchical codes."],"forward_implications":["At p around or below 10^-2, a single [[10^4,16,2α]] block with s=4 replaces 16 distance-25 surface-code patches, giving the same 625 physical qubits per logical qubit while matching or improving the logical error rate.","For d>25, the hierarchical code becomes strictly more qubit-efficient than the rotated surface code, and the advantage grows with d.","Increasing s within one block suppresses logical errors more than preparing s^2 independent copies of the same code, because distance grows with s.","The general condition is s ≥ ((d+1)/(2b))^{1/c} and d ≥ 25, so any upper-layer qLDPC code with a higher encoding rate or better distance scaling lowers the required d and s.","Changing the lower-layer surface distance L1 to 3 or larger shifts the required HGP distance d_s to 8 or 10 in the hard-decision analysis, so the architecture is tunable."],"supporting_citations":[{"why":"Supplies the hypergraph product code construction and its distance and rate scaling, which defines the upper-layer code.","marker":"[12]"},{"why":"Provides the random (3,4)-classical-LDPC generation approach used to sample the HGP instances.","marker":"[17]"},{"why":"Provides the BP-OS decoder used for the upper-layer HGP code.","marker":"[24]"},{"why":"Reports the error floor of BP-only decoding on hierarchical codes, which this paper claims to remove with BP-OS and soft information.","marker":"[19]"},{"why":"Establishes the soft-decision decoding framework for concatenated codes that the lookup-table probabilities feed into.","marker":"[23]"},{"why":"Supplies the surface-code pseudo-threshold value used in the comparison with the rotated surface code.","marker":"[33]"},{"why":"Defines the prior hierarchical qLDPC-surface architecture that this work extends with concrete decoding and numerical comparison.","marker":"[18]"}],"fun_headline_variants":["Hierarchical QEC beats surface code at distance 25","HGP-surface code outperforms surface code at d=25","Concatenated qLDPC code beats surface code at distance 25","16 logical qubits beat a distance-25 surface code","Beat the surface code with a hierarchical qLDPC code"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusion hinges on the assumption that the fitted scaling formula for logical error rates, measured for small sizes, keeps holding unchanged for larger sizes; the paper does not test that point directly at the claimed crossover.","fun_headline_variants_meta":{"raw":{"variants":["Hierarchical QEC beats surface code at distance 25","HGP-surface code outperforms surface code at d=25","Concatenated qLDPC code beats surface code at distance 25","16 logical qubits beat a distance-25 surface code","Beat the surface code with a hierarchical qLDPC code"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000934,"raw_usage":{"total_tokens":4060,"prompt_tokens":1075,"completion_tokens":2985,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":2898}},"tokens_in":691,"tokens_out":2985,"duration_ms":18294,"temperature":1.0,"reasoning_tokens":2898,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:28:55.201390+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run direct Monte Carlo decoding at p=$10^{-2}$ and p=$10^{-3}$ for the s=4, d=25 concatenated code and compare the measured logical error rate per logical qubit with that of a distance-25 rotated surface code under the same depolarizing noise model; if the concatenated code's error rate is not lower, or if the power-law exponent flattens at s=4, the central claim fails.","supporting_citations":[{"cited_title":"Good quantum ldpc codes with linear time de- coders, 2022","cited_arxiv_id":null,"evidence_quote":"Provides the random (3,4)-classical-LDPC generation approach used to sample the HGP instances."},{"cited_title":"Concatenate codes, save qubits, 2024","cited_arxiv_id":null,"evidence_quote":"Provides the BP-OS decoder used for the upper-layer HGP code."},{"cited_title":"A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes","cited_arxiv_id":null,"evidence_quote":"Reports the error floor of BP-only decoding on hierarchical codes, which this paper claims to remove with BP-OS and soft information."},{"cited_title":"Pattison, and John Preskill","cited_arxiv_id":null,"evidence_quote":"Establishes the soft-decision decoding framework for concatenated codes that the lookup-table probabilities feed into."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the surface-code pseudo-threshold value used in the comparison with the rotated surface code."},{"cited_title":"Asymptotically good quantum and locally testable classical ldpc codes, 2022","cited_arxiv_id":null,"evidence_quote":"Defines the prior hierarchical qLDPC-surface architecture that this work extends with concrete decoding and numerical comparison."}],"review_version":1}