{"id":"6bb93882-17cb-4e6b-af6f-6be08f590c88","arxiv_id":"2608.02773","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Cornucopia codes are a family of quantum LDPC codes with encoding rate above 1/2, circuit-level pseudo-threshold above 0.4%, and a constant-depth 12-layer syndrome extraction schedule for neutral-atom arrays.","lead":"This paper introduces Cornucopia codes, quantum error-correcting patterns that pack many protected logical qubits into relatively few physical qubits while keeping the error threshold in a practical range. If they work as claimed, a single atom-array block could hold about 1,426 protected qubits using around 4,266 atoms, a big step toward useful quantum computers on near-term neutral-atom hardware.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The headline logical-error rates and the >0.4% pseudo-threshold both rest on a p^{d/2} exp(c0+c1p+c2p^2) fit extrapolated beyond its data; the claimed pseudo-threshold appears inconsistent with Tab. S1's own coefficients.","rationale":"The reader's weakest_assumption correctly identifies the extrapolation ansatz as the most fragile premise. My own reading confirms that the headline numbers depend on a fit whose range is narrow and whose extrapolated values are extremely sensitive to the assumed distance exponent. The internal inconsistency of the pseudo-threshold claim (0.37% by the paper's own coefficients versus the stated >0.4%) is a concrete, checkable manifestation of this fragility. I also considered the neglected idle error during atom routing as an alternative concern; it is real and reinforces the need for caution about the practical hardware claim, but the central claim as framed under the standard circuit-level noise model depends first on the extrapolation. I do not see a mathematical defect in the code construction or the syndrome schedule; the concerns are about the statistical extrapolation and the precision of the headline threshold. The verdict CONDITIONAL remains appropriate because the issues are addressable with additional simulations and certified circuit-level distances, not fatal to the code family. No change to the reader's verdict is warranted.","tokens_in":25726,"tokens_out":16541,"duration_ms":138115,"concrete_test":"Recompute the pseudo-threshold for [[2844,1426,18]] from the Tab. S1 coefficients by solving p^9 exp(c0+c1p+c2p^2)=p; if the root is below 0.004, the abstract's 'exceeding 0.4%' claim is refuted. To test the extrapolation, run additional circuit-level Monte Carlo (or Bravyi-Vargo rare-event simulation) for the [[2844,1426,18]] code at p=1.25e-3 and p=1.5e-3, and add these points to the fit; if the new points deviate from the p^{d/2} fit by more than the statistical error, the ansatz is invalid. Also re-fit using only p<=2e-3 and compare the extrapolated p_L at p=1e-3; a change by more than an order of magnitude indicates the fit is unstable. Additionally, search for a weight-17 fault pattern with BP-OSD; finding one would bound d_circ<=17 and invalidate the p^9 exponent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claims -- the extrapolated logical error rates p_L=2.6e-16 at p=1e-3 and p_L=1.9e-31 at p=1e-4, and the pseudo-threshold exceeding 0.4% -- are all outputs of the fitting ansatz p_L = p^{d/2} exp(c0+c1p+c2p^2) (Methods, 'Numerical simulation details'). For the headline [[2844,1426,18]] code, the fit is calibrated on Monte Carlo data only for p in [1.75e-3, 2.5e-3] (Tab. S2). The p=1e-3 claim is a downward extrapolation by a factor of 1.75 in p, which corresponds to a factor of roughly 870 in p^9; the p=1e-4 claim is 17.5x below the lowest simulated p. The ansatz fixes the exponent at d/2, i.e., 9 for d=18. But the paper only certifies upper bounds on the circuit-level distance d_circ (Tab. 1), and for [[1044,526,12]] the bound is 11<12, showing the exponent can be smaller than d/2. If the true circuit-level distance for [[2844,1426,18]] were 16, the extrapolated p_L at p=1e-3 would increase by a factor of 1000; if 14, by 10^6. The fitting procedure also uses unweighted least squares in log space over points with very different statistical weights. Moreover, the claimed 'pseudo-threshold exceeding 0.4%' is internally inconsistent with the reported fit: substituting the Tab. S1 coefficients c0=10.36, c1=18370, c2=-2448603 into the break-even condition p_L=p gives a crossing at p approximately 0.0037 for [[2844,1426,18]], not above 0.004. Thus the headline threshold claim is not supported by the paper's own fit, and the ultra-low error rates inherit the full sensitivity of the extrapolation to the assumed exponent and to unmeasured higher-order terms.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Cornucopia codes, a family of high-rate CSS qLDPC codes with a block-convolutional structure tailored to reconfigurable neutral-atom arrays. The codes have rate above 1/2, weight-12 checks, and a depth-12 syndrome extraction schedule implemented by rigid atom motions. The authors certify exact code distances, prove structural lemmas about the syndrome schedule and the logical action of a column-shift automorphism, and present Monte Carlo logical-memory simulations under a circuit-level depolarizing model for instances up to [[2844,1426,18]]. The headline claims are an encoding rate exceeding 1/2, a pseudo-threshold exceeding 0.4%, and extrapolated logical error rates of 2.6e-16 per logical qubit per cycle at p=1e-3 and 1.9e-31 at p=1e-4, which would give a more than tenfold physical-qubit overhead reduction relative to bivariate bicycle codes.","tokens_in":26160,"tokens_out":3877,"duration_ms":34990,"significance":"If the quantitative claims are supported, this would be a notable step toward demonstrating that highly efficient qLDPC codes with rate above 1/2 can be compatible with the connectivity constraints of neutral-atom hardware. The explicit construction, certified distances, constant-depth syndrome extraction schedule, and atom-routing estimates are valuable and largely self-contained. The paper also contributes a clear co-design principle and a logical-routing primitive. The main significance, however, rests on the pseudo-threshold and extrapolated error-rate figures, which are the load-bearing claims that the paper must defend; those figures currently rest on a fitting ansatz that is extrapolated well beyond the simulated range and appears inconsistent with the reported fit coefficients.","major_comments":[{"comment":"The claim that all simulated Cornucopia codes have a pseudo-threshold exceeding 0.4% is inconsistent with the paper's own fitting coefficients. For [[2844,1426,18]], substituting c0=10.36, c1=18370, c2=-2448603 into p_L = p^9 exp(c0+c1 p+c2 p^2) and solving p_L = p gives a crossing at approximately p=0.0038, below 0.004. The main text and Fig. 2a should be corrected to report the pseudo-threshold as determined by the simulated data or by the fit, and the statement 'exceeding 0.4%' must be revised.","section":"Methods, 'Numerical simulation details'; Tab. S1"},{"comment":"The extrapolated logical error rates at p=1e-3 and p=1e-4 rely on the ansatz p_L = p^{d/2} exp(c0+c1 p+c2 p^2), whose exponent d/2 is not established for these codes. Tab. 1 reports only upper bounds on the circuit-level distance, and for [[1044,526,12]] this bound is 11 < 12, demonstrating that the effective sub-threshold exponent can be smaller than d/2. If the circuit-level distance of [[2844,1426,18]] were 16 or 14, the extrapolated p_L at p=1e-3 would increase by roughly 10^3 or 10^6, respectively. The quoted rates 2.6e-16 and 1.9e-31 are therefore model-dependent extrapolations, not established predictions; the manuscript should either obtain direct estimates at lower p, use the certified circuit-level distance upper bounds to give conservative bounds, or clearly label these numbers as upper-bound-free extrapolations with explicit sensitivity.","section":"Methods, 'Numerical simulation details'; Tab. 1; Tab. S1"},{"comment":"The truncated-cycle simulations use N_c = 8 for the d=18 code and N_c = 6 for d<=12, well below the conventional N_c = d. The quantity p_L is derived from P_fail over N_c cycles as a per-cycle rate, which assumes that the logical failure probability accumulates independently per cycle. With N_c much smaller than the code distance, the per-cycle rate may not have converged to the steady-state value, and the fit could be biased. Please provide data for at least two or three N_c values per code to demonstrate convergence, or justify the truncated-cycle model quantitatively.","section":"Methods, 'Numerical simulation details'; Tab. S2"},{"comment":"The logical-error simulations neglect idle errors during the 10.4-16.2 ms atom-routing time per syndrome cycle, arguing only that this time is much smaller than the ~10 s coherence time. However, at p=1e-3 the per-cycle idle dephasing error is approximately 1.6e-3, comparable to the physical error rate per operation and to the break-even value. The statement that routing time is 'orders of magnitude shorter than coherence times' is true but does not make the idle error negligible relative to the depolarizing error budget. Please provide an explicit estimate of the idle-error contribution per cycle or include a decoherence term in the noise model for at least one representative code.","section":"Methods, 'Implementation in neutral-atom arrays'; 'Circuit-level noise simulation and decoding'"}],"minor_comments":[{"comment":"The fitting procedure uses unweighted least squares in log space over data points with very different statistical weights (e.g., 5 failures out of 2e6 samples versus 26 failures out of 1000 samples for the largest code). Please report the fit residuals or use a weighted fit, and state the uncertainty of the extracted coefficients.","section":"Methods, 'Numerical simulation details'; Tab. S1"},{"comment":"The extrapolated lines in Fig. 2 are presented with the same visual weight as the simulated marker points; consider using a different line style or shaded region to mark the extrapolation region beyond the lowest simulated p.","section":"Main text, Fig. 2"},{"comment":"The table lists p_L values at p=1e-3 and p=1e-4 without a footnote that these are extrapolations from fits, not direct simulation results; such a footnote would prevent misinterpretation.","section":"Main text, Tab. 1"},{"comment":"The phrase 'ultra-high encoding rate exceeding 1/2' is accurate for the presented instances but 'ultra-high' is subjective; consider using a more neutral phrasing such as 'encoding rate above 1/2' or 'rate greater than 1/2'.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a promising code family and a credible co-design approach, but the flagship quantitative claims (pseudo-threshold >0.4%, extrapolated error rates at 1e-3 and 1e-4) are not supported by the reported data and fitting coefficients. The authors should be asked to correct the pseudo-threshold statement, redo or reframe the extrapolations with proper sensitivity to the circuit-level distance and the fit range, and either provide convergence data for the truncated-cycle simulations or tone down the quantitative conclusions. With those changes, the manuscript could be publishable, but as written it overstates its own results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The Cornucopia construction is a genuine step forward: an explicit, rate>1/2 qLDPC family with exact distance certification for the listed instances, a 12-layer constant-depth syndrome extraction schedule, and a hardware co-design that makes the nonlocal connectivity plausible in neutral-atom arrays. The paper is honest about inheriting the broad framework from Refs. [47,48], and the mathematical core—the CSS construction, the scheduling proposition, the module decomposition for logical routing—looks solid. I agree with the reader that this is a real contribution, not a repackaging.\n\nThe soft spots are all in the quantitative claims. First, the 'pseudo-threshold exceeding 0.4%' is not supported by the paper's own fit. Plugging the Tab. S1 coefficients for [[2844,1426,18]] into the break-even condition gives a crossing around 0.37%. That is a direct internal inconsistency, and it should be fixed before publication. Second, the headline logical error rates (2.6e-16 at p=1e-3, 1.9e-31 at p=1e-4) come from extrapolating a p^{d/2} exp(c0+c1p+c2p^2) ansatz fitted to data only down to p=1.75e-3. The exponent d/2 is critical: the paper only certifies upper bounds on circuit-level distance, and for [[1044,526,12]] the bound is 11<12, showing the true exponent can be smaller. If the exponent for the largest code were 8 rather than 9, the quoted p=1e-3 rate jumps by three orders of magnitude. The fit is also unweighted in log space over samples with wildly different statistical weights. Third, the neglect of idle errors during 10–16 ms of atom routing is not self-evidently safe at p=1e-3; a 10 s coherence time gives about 1.6e-3 idle error per cycle, comparable to the physical error rate itself. And the truncated cycle counts (N_c=6–8 instead of d) may make the per-cycle rates optimistic, even if the comparison to surface and bivariate codes is fair.\n\nNone of this undermines the construction itself. The code family, the schedule, and the routing analysis are worth careful peer review. The fix is to refit or reframe the extrapolations, correct the threshold claim, and ideally release the distance-certification and simulation code. I would send it to a serious referee, with the expectation of a solid revision.","headline":"A serious qLDPC co-design paper whose headline numbers are extrapolations that need a closer look.","tokens_in":26739,"tokens_out":4456,"would_cite":true,"duration_ms":38300,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70","81P73","94B60"],"pacs":["03.67.Pp","03.67.Lx"],"model":"deepseek-v4-flash","headline":"Cornucopia codes claim to break the rate–threshold–hardware trade-off: rate above 1/2, pseudo-threshold above 0.4%, and constant-depth syndrome extraction.","keywords":["quantum error correction","quantum LDPC codes","high encoding rate","pseudo-threshold","neutral-atom arrays","syndrome extraction","logical routing","code automorphism"],"falsifier":"Run a direct Monte Carlo simulation of one full syndrome cycle of the $[[2844,1426,18]]$ code at $p=0.001$ including the circuit-level noise model together with the complete routing sequence using the paper's own trap-transfer, acceleration, and idle-error parameters; if the observed logical error rate per logical qubit per cycle is not near $2.6\\times10^{-16}$, or even below $p$, the extrapolation is falsified.","tokens_in":25458,"feed_emoji":"🛡️","tokens_out":7846,"duration_ms":65147,"temperature":0.7,"pith_summary":"Cornucopia codes are a newly constructed family of quantum error-correcting codes that aim to break the usual trade-off between encoding rate, error threshold, and hardware feasibility. Their defining claim is that a code can store more than one logical qubit for every two physical data qubits and still tolerate circuit-level physical error rates above 0.4%, while measuring all stabilizers in just twelve entangling layers. The concrete proposal is a [[2844,1426,18]] block that would encode 1,426 distance-18 logical qubits with an extrapolated logical error rate of $2.6\\times10^{-16}$ per logical qubit per cycle at a physical error rate of $0.1\\%$. If the extrapolations hold, this would bring ultra-low-overhead fault-tolerant memory into near-term reach on reconfigurable neutral-atom arrays, reducing the physical-qubit cost per logical qubit to about three rather than hundreds.","feed_headline":"Quantum codes clear 0.4% threshold at rate above 1/2","feed_subtitle":"A 2,844-qubit block would host 1,426 logical qubits at projected error 2.6e-16 per cycle.","key_machinery":"The load-bearing construction is a block-convolutional CSS code on the grid $\\mathbb{Z}_3\\times\\mathbb{Z}_q$: twelve data blocks and six check blocks are coupled by twelve coordinate permutations $A_i,B_i$ that are pure cyclic shifts in the column direction, with only four of them performing row swaps or row-cyclic shifts. This restricted permutation class is what makes nonlocal checks executable as rigid atom translations, keeps the CNOT schedule to twelve layers independent of code length, and enables exact distance certification via row-pair enumeration. The algebraic identity carrying the logical-routing claim is the module decomposition $L_{X/Z}\\simeq 18\\cdot\\mathbb{F}_2[C_q]\\oplus 4\\cdot\\mathbf{1}$, which says that a global column shift acts on each logical space as eighteen independent length-$q$ cyclic registers together with four fixed modes.","core_discovery":"The central claim is that the Cornucopia family of CSS quantum LDPC codes simultaneously achieves an encoding rate $r=k/n>1/2$, a circuit-level pseudo-threshold above $0.4\\%$ under the standard depolarizing noise model, and a syndrome extraction cycle of exactly twelve parallel CNOT layers whose required atom rearrangements are only collective cyclic shifts and a few row swaps. The largest instance, $[[2844,1426,18]]$, uses 2,844 data qubits and 1,422 check qubits to encode 1,426 logical qubits, and Monte Carlo simulations with a relay-BP/BP-OSD decoder extrapolate to $p_L=2.6\\times10^{-16}$ per logical qubit per cycle at $p=10^{-3}$ and $1.9\\times10^{-31}$ at $p=10^{-4}$. The paper also proves equality of $X$- and $Z$-distances, certifies all listed distances by exhaustive symmetry-reduced search, and identifies a logical-routing automorphism in which one physical column shift acts as eighteen simultaneous cyclic logical registers plus four invariant modes.","pith_inferences":["The paper does not construct universal fault-tolerant logical gates: lattice surgery for Cornucopia codes is left open, so the ultra-low-overhead memory does not yet translate into a full fault-tolerant computation stack.","Because each block contains hundreds or thousands of logical qubits, the block-level failure probability grows with $k$; readers comparing thresholds should separate per-logical-qubit error rates from the probability that all logical qubits in one block survive together.","The $O(\\sqrt{q})$ growth of ancilla routing time means the constant-depth claim is about entangling layers, not wall-clock time; at very large block sizes the cycle time could approach coherence limits, which is a testable scaling question the paper's estimates can be pushed to answer."],"forward_implications":["The $[[2844,1426,18]]$ block is projected to reach $p_L=2.6\\times10^{-16}$ per logical qubit per cycle at $p=0.1\\%$, using 4,266 physical qubits total, i.e. roughly three physical qubits per logical qubit.","Because a full syndrome cycle uses only twelve entangling layers and an estimated 10–16 ms of atom routing, the protocol fits within coherence times already demonstrated in neutral-atom experiments.","If the pseudo-threshold holds, Cornucopia codes offer an order-of-magnitude reduction in physical-qubit overhead compared with bivariate bicycle codes for the same logical workload at comparable logical error rates.","The column-shift automorphism gives a native logical-routing primitive: a single global physical shift moves logical information across eighteen cyclic registers, without deep SWAP networks or code deformation."],"supporting_citations":[{"why":"Supplies the bivariate bicycle code family used as the main rate and logical-error comparison baseline, and the fitting model for sub-threshold scaling.","marker":"[31]"},{"why":"Provides the affine-permutation ultra-high-rate code construction on which Cornucopia codes are based.","marker":"[47]"},{"why":"Adapts ultra-high-rate constructions to reconfigurable atom arrays and supplies the crossed-AOD compatible motion model used for routing-time estimates.","marker":"[48]"},{"why":"Defines relay belief propagation, the primary decoder in the cascaded decoding strategy used for all reported logical error rates.","marker":"[49]"},{"why":"Provides the BP-OSD fallback decoder used for hard error samples and the method used to upper-bound circuit-level distances.","marker":"[51]"},{"why":"Introduces the rare-event simulation methodology and the fitting ansatz $p_L=p^{d/2}\\exp(c_0+c_1p+c_2p^2)$ used to extrapolate logical error rates.","marker":"[57]"},{"why":"Supplies the neutral-atom array parameters (lattice spacing, maximum acceleration, trap-transfer time) underlying the syndrome-cycle routing-time estimates.","marker":"[59]"},{"why":"Documents long coherence times in neutral-atom qubits, supporting the paper's assumption that 10–16 ms routing per cycle is safely below coherence limits.","marker":"[53]"}],"fun_headline_variants":["Cornucopia codes: >1/2 rate and 0.4% threshold in one block","High-rate quantum codes clear 0.4% threshold","12-layer syndrome extraction: QEC at rate above 1/2","1426 logical qubits from 2844 physical with low overhead"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The headline logical error rates are extrapolations from Monte Carlo data at physical error rates 0.15%–0.25% down to 0.01%–0.1% under a noise model that ignores errors from the 10–16 ms of atomic motion per syndrome cycle; if the fitted curve bends upward or the motion introduces comparable errors, the quoted rates are not established.","fun_headline_variants_meta":{"raw":{"variants":["Cornucopia codes: >1/2 rate and 0.4% threshold in one block","High-rate quantum codes clear 0.4% threshold","12-layer syndrome extraction: QEC at rate above 1/2","1426 logical qubits from 2844 physical with low overhead"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000909,"raw_usage":{"total_tokens":3989,"prompt_tokens":1111,"completion_tokens":2878,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":727,"completion_tokens_details":{"reasoning_tokens":2797}},"tokens_in":727,"tokens_out":2878,"duration_ms":19610,"temperature":1.0,"reasoning_tokens":2797,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:01:03.309650+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a direct Monte Carlo simulation of one full syndrome cycle of the $[[2844,1426,18]]$ code at $p=0.001$ including the circuit-level noise model together with the complete routing sequence using the paper's own trap-transfer, acceleration, and idle-error parameters; if the observed logical error rate per logical qubit per cycle is not near $2.6\\times10^{-16}$, or even below $p$, the extrapolation is falsified.","supporting_citations":[{"cited_title":"Roffe, D","cited_arxiv_id":null,"evidence_quote":"Provides the BP-OSD fallback decoder used for hard error samples and the method used to upper-bound circuit-level distances."},{"cited_title":"Bravyi and A","cited_arxiv_id":null,"evidence_quote":"Introduces the rare-event simulation methodology and the fitting ansatz $p_L=p^{d/2}\\exp(c_0+c_1p+c_2p^2)$ used to extrapolate logical error rates."},{"cited_title":"Bluvstein, H","cited_arxiv_id":null,"evidence_quote":"Supplies the neutral-atom array parameters (lattice spacing, maximum acceleration, trap-transfer time) underlying the syndrome-cycle routing-time estimates."},{"cited_title":"Jenkins, J","cited_arxiv_id":null,"evidence_quote":"Documents long coherence times in neutral-atom qubits, supporting the paper's assumption that 10–16 ms routing per cycle is safely below coherence limits."}],"review_version":1}