{"id":"23c025b7-da3e-4ad2-affe-747171f903a4","arxiv_id":"2506.00130","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Romanesco codes are Clifford-deformed bivariate bicycle codes made from two cellular automaton codes that achieve an effective distance scaling linearly with qubit count in the infinite-bias limit.","lead":"This paper proposes a new family of quantum error-correcting codes, called Romanesco codes, built from two classical cellular automaton codes combined with a Clifford twist. The authors report that these codes maintain a high encoding rate and can reach an effective distance that grows linearly with qubit number under strongly biased noise, with simulations showing strong logical error suppression.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Linear infinite-bias distance scalings in Table I are empirical extrapolations from <=1000 qubits; open-boundary data are irregular, so the advertised dc proportional-to-n and v_infinity > 1 are not established.","rationale":"I examined the construction and the reduction argument in Section III.A/Appendix A. The claim that the infinite-bias decoding graph splits into two classical CA codes is exact: the X parts of the stabilizers are supported on disjoint sectors and equal to A and A^T, so the minimum all-Z logical weight is exactly the minimum distance of the classical codes. Thus the remaining load-bearing step is the asserted linear scaling of that classical distance. The reader's weakest assumption identifies exactly this. I see no internal inconsistency; the paper gives honest caveats about finite-bias behavior and decoder dependence. The numerical simulations and the small-size distance checks are credible independent support, but they do not prove the infinite-family scalings. Table VII/Fig. 8 show that for open boundaries the classical distance is non-monotonic (dc=40 at d=12, 36 at d=16, 40 at d=20), which makes the smooth formulas in Table I even less secure. Therefore the verdict should remain CONDITIONAL, pending a direct check at larger sizes or a proof. My concrete test would settle the question for the flagship family.","tokens_in":25568,"tokens_out":14993,"duration_ms":158472,"concrete_test":"Compute exact classical distances dc for the [[N,12,D1,1]] family (rules R1/R2 in Fig. 6) on tori with H=L=16,20,24,28,32, using the exact distance routine from the LDPC package (as in Appendix E) with an integer-programming fallback; accept the linear law dc=3N/16 only if every instance matches and no logical operator of weight < 3N/16 exists. Any deviation, or any failure of the upper-bound algorithm to certify tightness, refutes the claimed family scaling and invalidates the v_infinity=2.25 headline.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim—that the infinite-bias effective distance dc grows linearly with n for the families in Table I, giving v_infinity = k*dc/n > 1—rests on empirical extrapolation, not proof. Section III.B states that 'the scalings for k, d and dc are found empirically by generating multiple lattice sizes,' and Appendix B restricts the search to rules of size m <= 4, weight w <= 4, and lattices below 1000 data qubits. The families then assert functional forms such as dc = 3N/16 for all allowed lattice dimensions. This is a strong infinite-family statement. The open-boundary data in Table VII and Fig. 8 show that dc is highly irregular (e.g., d=12 -> dc=40, d=16 -> dc=36, d=20 -> dc=40), so the smooth linear formulas in Table I are not self-evident and may not persist at larger sizes. Moreover, Algorithm 1 returns an upper bound on distance; if a lower-weight logical operator was missed at the largest searched sizes, the reported dc would be an overestimate, directly inflating v_infinity. If the linear scaling fails for any allowed dimension, the headline advantage over 2D topological codes (which have d_infinity = O(sqrt(n)) with open boundaries) collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Romanesco codes, a family of Clifford-deformed bivariate bicycle codes whose input classical codes are cellular automaton (fractal) codes. The construction guarantees that in the infinite-noise-bias limit the decoding graph decouples into two independent classical cellular automaton codes, so the effective code distance approaches the classical distance dc. The authors numerically search over cellular automaton rules and identify several families of codes on the torus, cylinder, and open-boundary 2D plane. They report code families with encoding rates up to k=16 and overhead factors v∞ = kdc/n up to 2.67 on the torus, and they present code-capacity Monte Carlo simulations showing improved logical error rates compared with bivariate bicycle codes and rotated surface codes under biased noise. The paper is well structured and includes detailed appendices describing the search, decoding algorithms, and boundary constructions.","tokens_in":25844,"tokens_out":11825,"duration_ms":127758,"significance":"If the claimed asymptotic distance scalings are correct, the Romanesco construction provides a concrete route to qLDPC codes on a 2D local lattice whose infinite-bias effective distance grows linearly with n, surpassing the O(√n) distance limit of 2D topological codes under strong bias. The reduction to two decoupled classical cellular automaton codes is elegant and directly connects the quantum code performance to the well-studied distance properties of classical LDPC codes. The numerical results show substantial practical improvements under biased noise, and the paper is careful to discuss the regime of validity of the effective-distance picture. The main weakness is that the central asymptotic claim rests on empirical extrapolations from lattices with fewer than 1000 qubits, with irregular open-boundary data and no proof of linear scaling.","major_comments":[{"comment":"The claimed linear scalings of dc with n are presented as facts for the families in Table I, but they are inferred from lattices with fewer than 1000 qubits, and no finite-size data are shown for the torus or cylinder families. The open-boundary data in Table VII and Fig. 8 are highly irregular: for the k=2 family, dc/n equals 0.20, 0.11, 0.12, and 0.27 at d = 17, 18, 19, and 23, respectively, and for the k=4 family it drops from 0.16 at d=12 to 0.08 at d=16 and 0.04 at d=20 before returning to 0.15 at d=24. The statement in Section III.D that \"the infinite bias distance dc scales linearly with the number of data qubits n\" is therefore not supported by the evidence provided. Since v∞ = kdc/n > 1 is the central quantitative claim, the authors should either prove the linear scaling for the identified CA-rule families or provide complete finite-size tables/plots for each family and explicitly label the asymptotic scaling as an empirical conjecture.","section":"Section III.B and Table I"},{"comment":"The distances dc for the torus and cylinder families are computed with Algorithm 1, which the authors state returns an upper bound on the code distance. The convergence check (generating 100 new low-weight logicals) is heuristic, and if the bound is not tight at the largest searched sizes, the reported linear scalings and the corresponding v∞ values in Table I would be overestimates. The authors should run an exact distance computation for at least the largest instances of each torus family, as was done for the open-boundary codes in Table VII, and state explicitly whether the bounds are tight. Without this, the headline advantage over 2D topological codes is not firmly established.","section":"Appendix B, Algorithm 1"}],"minor_comments":[{"comment":"The constraints such as H%3=0 and L%3=0 are used without defining the modulo notation in the caption; please add a brief explanation.","section":"Table I"},{"comment":"The sentence \"We use this geometric constraint to find the most promising stabilizer shapes\" refers to the 9q^2 square-lattice search constraint; clarify that this is a search heuristic and not a requirement of the final code families.","section":"Section III.B"},{"comment":"The caption states that shaded regions are 95% confidence intervals for a Poisson distribution, but does not report the number of Monte Carlo shots or the number of logical errors per point; please include this information.","section":"Fig. 2 caption"},{"comment":"The phrase \"the classical distance does grow with 2d^2 > n\" is confusing; it should be rephrased as \"dc grows as Θ(d^2), which is proportional to n\" or similar.","section":"Appendix E, Fig. 8"},{"comment":"The term \"beam climbing\" in the description of the tesseract decoder is not standard and is not defined; please clarify whether this is a specific feature of the tesseract decoder and cite the relevant description.","section":"Section IV"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the construction is original and timely. The main issue is that the asymptotic distance scaling, which underlies the headline claim of exceeding 2D topological codes in the infinite-bias limit, is empirically extrapolated rather than proven, and the open-boundary data are quite irregular. The authors should be asked to provide the finite-size data for the torus and cylinder families, verify the tightness of the distance bounds at the largest sizes, and either prove the linear scaling for the identified CA-rule families or clearly downgrade the claim to a conjecture. The numerical search appears thorough, and the code-capacity simulation results are encouraging, so I do not see the issues as irreparable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the Romanesco paper. Here's my take.\n\nThe construction is genuinely new and quite elegant: take two cellular automaton codes related by a 180-degree rotation, use them as the A/B polynomials of a bivariate bicycle code, then Clifford-deform one sector to get a non-CSS code. The self-duality is built in, and in the infinite-bias limit the decoding graph splits exactly into the two classical CA codes. That is a clean mechanism, and it is not just a repackaging of known pieces—I don't know of prior work forming self-dual bicycles from reflected CA codes. The numerical search over rules is careful, and the comparison against XZZX and thin surface codes at comparable qubit counts is fair. The observation that some families have v_infinity > 1 is the right headline.\n\nThat said, the headline quantitative claim—linear scaling of dc with n, hence v_infinity > 1—is not established. The scalings in Table I are empirical fits over lattices up to 1000 qubits. Algorithm 1 returns an upper bound on distance, so a missed low-weight logical would inflate dc and v_infinity. The open-boundary data in Table VII are genuinely irregular (dc for d=12,16,20 is 40,36,40), so the smooth formulas in Table I look like they may not survive at larger sizes. The paper is honest about this in the appendices, but the abstract and intro state the linear scaling as fact. No code or data files are provided, which makes independent verification harder.\n\nI don't think there's a load-bearing flaw in the construction itself. The decoupling argument is solid, and the code parameters for the examples are what they are. The issue is whether the infinite-family claim holds. That is a proof gap, or at least a data gap. A referee should ask for either a proof of the linear distance scaling for the listed families (or a subset) or a much more extensive numerical characterization with exact distance checks at larger sizes, and release the search code.\n\nVerdict: worth a serious referee. The construction deserves to be in the literature even if the asymptotic claim is softened. For readers working on biased-noise QEC or cat qubits, this is a relevant construction, but I'd cite it as an empirical proposal rather than an established trade-off.\n\nMy recommendation: send to peer review, but with the expectation that the distance scaling claim gets tightened or explicitly downgraded to a conjecture.","headline":"Romanesco codes: a genuinely new biased-noise qLDPC construction whose headline linear-distance claim is plausible but rests on empirical extrapolation from small tori.","tokens_in":26356,"tokens_out":2858,"would_cite":true,"duration_ms":33319,"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":"Quantum code breaks the 2D distance barrier under biased noise.","keywords":["quantum LDPC codes","biased noise","noise bias","cellular automaton codes","fractal codes","bivariate bicycle codes","Clifford deformation","quantum error correction"],"falsifier":"Compute the exact classical distance $d_c$ for a larger instance of the $[[N,12,D_{1,1}]]$ family, for example $H = L = 24$ with $N = 1152$ data qubits, using an exact integer-program decoder; if $d_c$ is not $3N/16$ or the ratio $d_c/n$ decays with size, the claimed linear scaling and the $v_\\infty > 1$ overhead advantage fail.","tokens_in":25352,"feed_emoji":"🥦","tokens_out":8013,"duration_ms":89091,"temperature":0.7,"pith_summary":"This paper introduces Romanesco codes, a family of quantum low-density parity-check (qLDPC) codes built from two classical cellular automaton codes and tailored to noise that is biased toward phase-flip errors. The central claim is that in the limit of infinitely strong bias the code splits into two independent classical codes, so the effective quantum distance approaches the classical distance of those inputs, which can grow linearly with the number of data qubits. That would beat the square-root distance scaling of two-dimensional topological codes while keeping local, low-weight stabilizers. Concretely, several reported families have an overhead factor $v_\\infty = k d_c / n$ greater than 1, up to 2.67, meaning one patch encodes as much protection as several repetition or XZZX patches of the same size. Numerical simulations under code-capacity noise show strong suppression of the logical error rate compared with standard bivariate bicycle, XZZX, and thin surface codes.","feed_headline":"Quantum code breaks the 2D distance barrier under biased noise","feed_subtitle":"Two classical cellular automaton codes fuse so effective distance grows linearly with qubits, beating repetition and XZZX codes.","key_machinery":"The central object is the Clifford-deformed bivariate bicycle code built from two cellular automaton input codes. A cellular automaton code is a classical LDPC code whose stabilizer generators are translation-invariant parity checks shaped by a binary rule matrix $R$; sweeping $R$ across the lattice generates the code. The paper reads $R_1$ and $R_2$ (a 180-degree rotation of $R_1$) as monomials in the cyclic shift variables $x$ and $y$ to form polynomials $A$ and $B = A^T$, producing a self-dual bivariate bicycle code. A local Hadamard (Clifford) rotation on one sublattice converts the CSS code into a non-CSS code whose stabilizers are half X and half Z. The load-bearing mechanism is the decoupling: because the X parts of the stabilizers live on disjoint sublattices, in the infinite-bias limit the decoding graph splits into two independent classical cellular automaton codes, so the lowest-weight all-Z logical operators have length $d_c$ and the effective quantum distance approaches $d_c$.","core_discovery":"A Romanesco code is obtained by taking two cellular automaton (fractal) codes whose parity-check rules $R_1$ and $R_2$ are related by a 180-degree rotation, lifting them into a self-dual bivariate bicycle code with polynomials $A$ and $B = A^T$, and then applying a local Hadamard rotation on one of the two sublattices. The resulting code is non-CSS: every stabilizer generator is half X-type and half Z-type. Under strongly biased noise, the Z-type checks can be neglected, and the decoding graph separates into two decoupled sublattices, each supporting one of the input classical codes; the effective distance therefore approaches the classical distance $d_c$. For the families reported in Table I, $d_c$ scales linearly with the number $n$ of data qubits, for instance $d_c = 3N/16$ for the $[[N,12,D_{1,1}]]$ family, giving $v_\\infty = k d_c / n > 1$ and up to 2.67, in contrast to repetition and twisted XZZX codes for which $v_\\infty = 1$ and to 2D topological codes whose distance is bounded by $\\sqrt{n}$. The paper further shows that on tori, cylinders, and open planes the classical distance remains linear, and that code-capacity simulations of the $[[288,12,12]]$ and $[[244,4,12]]$ codes outperform comparable bivariate bicycle and surface codes at moderate to large bias.","pith_inferences":["If the linear $d_c$ scaling holds asymptotically, biased-noise hardware such as cat qubits could obtain qLDPC-level overhead reductions without long-range connectivity, since the layout is two-dimensionally local; this is an extrapolation the paper does not fully prove.","The fractal-like structure of logical representatives suggests that cellular automaton decoders, which the paper mentions as future work, could provide efficient decoding at large bias.","Different boundary terminations might preserve more logical qubits; the paper's simple truncation reduces $k$ by half on cylinders and to 4 on the plane, so boundary engineering is a natural lever for improving $v_\\infty$.","A direct proof or disproof of $d_c \\sim n$ for the Table I families would settle whether the advantage over 2D topological codes persists asymptotically."],"forward_implications":["In the infinite bias limit, the effective distance equals the classical distance $d_c$, and for the Table I families $d_c$ grows linearly with $n$, giving $v_\\infty = k d_c / n > 1$ up to 2.67.","A single Romanesco patch can replace several repetition or twisted XZZX patches of the same size for biased-noise quantum memory.","The $[[288,12,12]]$ code with $d_c = 54$ outperforms the $[[288,12,18]]$ bivariate bicycle code (infinite-bias distance 18) at moderate-to-large bias despite having a lower nominal quantum distance.","With open boundary conditions on a two-dimensional plane, $d_c$ still grows with $n$ (for example, $d_c > n/8$ for distance multiples of 12), unlike rotated XZZX codes whose infinite-bias distance scales as $\\sqrt{n}$.","The codes retain self-duality and limited-range weight-8 stabilizers, giving transversal H, S, and CNOT gates and a bipartite honeycomb layout amenable to local hardware."],"supporting_citations":[{"why":"Supplies the classical cellular automaton codes proposed for biased cat qubits, which are the input codes of the Romanesco construction.","marker":"[13]"},{"why":"Defines bivariate bicycle codes, the base code family from which Romanesco codes are built.","marker":"[33]"},{"why":"Provides the Clifford-deformation scheme used to turn a CSS code into a non-CSS bias-tailored code.","marker":"[34]"},{"why":"Shows that an XZ3 color code reduces to two cellular automaton codes in the infinite-bias limit, the seed of this construction.","marker":"[21]"},{"why":"Supplies the effective-distance framework and the twisted XZZX comparison with $v_\\infty = 1$.","marker":"[20]"},{"why":"Provides the $[[18q^2,4,4q]]$ honeycomb color code that Romanesco codes generalize.","marker":"[35]"},{"why":"Defines the XZZX surface code used as a bias-tailored topological baseline for comparison.","marker":"[15]"}],"fun_headline_variants":["Romanesco codes beat 2D distance limit under biased noise","Bias-tailored qLDPC codes reach linear distance scaling","Fractal codes fuse into high-rate quantum codes for biased noise","Non-CSS quantum code outperforms surface and repetition codes","Self-dual qLDPC codes from cellular automata boost distance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central claim rests on an empirical extrapolation: the classical distance of the cellular automaton input codes is inferred to grow linearly with lattice size from simulations on lattices up to about 1000 qubits, and no proof guarantees that larger lattices or other cellular automaton rules keep that growth.","fun_headline_variants_meta":{"raw":{"variants":["Romanesco codes beat 2D distance limit under biased noise","Bias-tailored qLDPC codes reach linear distance scaling","Fractal codes fuse into high-rate quantum codes for biased noise","Non-CSS quantum code outperforms surface and repetition codes","Self-dual qLDPC codes from cellular automata boost distance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000366,"raw_usage":{"total_tokens":2054,"prompt_tokens":1116,"completion_tokens":938,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":732,"completion_tokens_details":{"reasoning_tokens":849}},"tokens_in":732,"tokens_out":938,"duration_ms":11038,"temperature":1.0,"reasoning_tokens":849,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:12:06.673936+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact classical distance $d_c$ for a larger instance of the $[[N,12,D_{1,1}]]$ family, for example $H = L = 24$ with $N = 1152$ data qubits, using an exact integer-program decoder; if $d_c$ is not $3N/16$ or the ratio $d_c/n$ decays with size, the claimed linear scaling and the $v_\\infty > 1$ overhead advantage fail.","supporting_citations":[{"cited_title":"Gouzien, D","cited_arxiv_id":null,"evidence_quote":"Supplies the effective-distance framework and the twisted XZZX comparison with $v_\\infty = 1$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the $[[18q^2,4,4q]]$ honeycomb color code that Romanesco codes generalize."},{"cited_title":"Aliferis, F","cited_arxiv_id":null,"evidence_quote":"Defines the XZZX surface code used as a bias-tailored topological baseline for comparison."}],"review_version":1}