{"id":"ac6e16e9-5f24-4b9b-80b2-1dd181d8b744","arxiv_id":"2608.12509","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A search over non-abelian groups yields new moderate-blocklength quantum Tanner code instances whose randomized distance bounds exceed 20, with decoder pseudo-thresholds comparable to shorter codes.","lead":"This paper searches for small quantum Tanner codes, a family of quantum error-correcting codes, and reports new instances at blocklengths 500 to 1000 with randomized distance estimates above 20. It also measures error thresholds and releases open-source Julia software so the search can be repeated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Headline distances are randomized upper bounds, so the claimed d>20 and beyond-√n results are not established as true code distances; a certified lower-bound check is needed.","rationale":"The reader's weakest_assumption correctly identifies the central load-bearing point: the reported distances are upper bounds from randomized estimators, and the paper's significance claims—competitive parameters, d>20, and especially Section IV.C's 'beyond the square-root barrier'—depend on these bounds being close to the true distances. The paper itself is candid about this in Table III's caption and in the '≤' notation throughout, which prevents the issue from being an internal inconsistency, but it does not resolve it. The construction and search are real contributions: the code instances are explicitly specified in the appendices, the software is open source, and the algebraic framework follows prior work. However, without lower-bound certificates, the headline claims are not established. A single exact-distance computation on one or two representative codes would settle whether the upper bounds are misleading. Because the reader already rendered a CONDITIONAL verdict and my analysis points to the same assumption, no change to the verdict is needed; the paper should remain conditional on either providing certified lower bounds or clearly reframing the square-root and competitive-parameter claims as upper-bound-based heuristics.","tokens_in":41850,"tokens_out":5612,"duration_ms":56570,"concrete_test":"For at least the flagship [[720,6,(≤30,≤30)]] and [[480,8,(≤21,≤21)]] codes, reproduce the parity-check matrices from Table IX/X and compute exact X- and Z-distances by solving the coset minimum-weight problems d_X = min{wt(u): u∈C_Z \\ C_X^⊥} and d_Z = min{wt(v): v∈C_X \\ C_Z^⊥} with an exact branch-and-bound or information-set-decoding solver (e.g., Magma's MinimumWeight or a dedicated CSS exact-distance routine). If the exact d_min is at least 75% of the reported sQetch upper bound for all tested codes, the practical claims survive; if any exact distance falls well below the upper bound, then the beyond-√n and pseudo-threshold conclusions must be reframed as upper-bound-only statements.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central practical claim—that explicit QT codes with competitive distances d>20 exist at n∈[500,1000]—rests on treating the sQetch and DistRandCSS upper bounds as close to the true minimum distances. Every table entry is explicitly tagged with '≤d_X,≤d_Z', and Table III's own caption states that 'the true minimum distances, and hence the true values of kd²_min/n, may be smaller.' Section IV.C nevertheless labels these entries as 'Quantum Tanner codes exceeding the √n barrier' and computes BPT figures of merit from the upper bounds alone. Randomized distance estimation only finds low-weight logical operators; it cannot rule out the existence of smaller ones. The same upper bounds are also used as the number of memory rounds r=d_z/d_x in the pseudo-threshold benchmarks of Section IV.B, so the decoder results are computed under an assumed distance. If the true distances are substantially below the reported upper bounds, the 'competitive parameters' and 'beyond √n' conclusions collapse, while the algebraic construction itself remains valid. This is a load-bearing assumption about estimator tightness, not about the LRCC/lifting construction, and it is the same weak point identified by the reader.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports an extensive computational search for explicit quantum Tanner (QT) codes in the moderate-blocklength regime n∈[500,1000], using both the left-right Cayley complex description and the lifting perspective of Leverrier, Rozendaal, and Zémor. The authors search a wide range of non-abelian groups from GAP's SmallGrp library, assemble a catalogue of QT code instances, and estimate their dimensions and distances with the randomized estimators DistRandCSS and sQetch. They highlight several instances whose distance upper bounds lie near or above √n, report pseudo-thresholds for four representative codes under phenomenological and circuit-level noise using the Tesseract decoder, and release QuantumExpanders.jl, an open-source Julia library. The paper positions these results as an affirmative answer to the open question of whether competitive QT codes exist at moderate blocklength.","tokens_in":42084,"tokens_out":3367,"duration_ms":33627,"significance":"If the reported distance estimates are close to the true minimum distances, the paper provides a valuable set of explicit QT code instances and benchmarks in a previously under-explored regime, together with reproducible open-source tooling. The absence of circularity is a genuine strength: the distance and threshold values are outputs of search and simulation, not parameters fitted to target values, and the lifting construction is prior work used as an input. The randomized cross-validation with two independent estimators and the detailed appendices of multisets and permutations are also commendable. The central quantitative claims, however, rest on randomized upper bounds rather than certified distances; this is a load-bearing assumption that affects the 'beyond the √n barrier' conclusions and the pseudo-threshold benchmarks. A revision that either certifies distances or reframes the claims as upper-bound-based would substantially strengthen the paper.","major_comments":[{"comment":"The claim of codes 'exceeding the √n barrier' is not supported by the evidence as presented. Every distance in Table III is an sQetch upper bound, and the table caption explicitly states that 'the true minimum distances, and hence the true values of kd²_min/n, may be smaller.' The BPT figure of merit kd²_min/n is computed from these upper bounds, so it is an upper-bound-based estimate rather than an established value. For the [[480,8,(≤21,≤21)]] row, the upper bound d≤21 actually lies below √n≈21.9, so this row does not even satisfy the table's own title. The section should either provide certified lower bounds on distance (or exact distances for these small codes) or be reframed as 'instances whose estimated distance upper bounds lie above √n' without claiming that the true distances exceed the barrier.","section":"§IV C, Table III"},{"comment":"The pseudo-threshold benchmarks are computed with r=d_z rounds for the X-basis experiment and r=d_x rounds for the Z-basis experiment, where d_x and d_z are the reported upper bounds from randomized estimation. If the true distances are smaller than the reported bounds, then the memory experiments are run for more rounds than the code can actually sustain, and the reported thresholds are conditional on the assumed distances. The manuscript does not report the exact round counts used for each code or provide uncertainty estimates for the pseudo-thresholds. Please state the round counts explicitly and, ideally, rerun the benchmarks with certified distances or discuss quantitatively how sensitive the threshold estimates are to the choice of r.","section":"§IV B, Table II"},{"comment":"The reproducibility appendices contain mismatches that undermine the stated goal of making every instance easy to verify. Appendix A.2 refers to 'Table E' and constructs a [[324,8,(17,14)]] code, but there is no Table E in the appendix and Table V lists a [[324,8,(≤17,≤14)]] code with different multisets A and B; the relationship between the code in the snippet and the table entry is unclear. In addition, Table V contains a malformed parameter string '[[896,16,(≤,16≤16)]]' for the C14×C2 row. These inconsistencies should be corrected so that each code instance can be reproduced from the tables alone.","section":"Appendix A.2 and Table V"}],"minor_comments":[{"comment":"The abstract consistently says 'distance upper bounds exceeding 20', but the Discussion states that the code instances 'achiev[e] distance d>20'. Since the reported values are upper bounds, the Discussion should use the same qualified language as the abstract.","section":"Abstract and §V"},{"comment":"The caption 'Quantum Tanner codes exceeding the √n distance barrier' is inaccurate for the first row, [[480,8,(≤21,≤21)]], whose upper bound is below √n. The text acknowledges that this row 'sits on the boundary', but the title should reflect that the table contains instances whose estimated upper bounds are near or above √n.","section":"§IV C, Table III caption"},{"comment":"The sentence introducing the generator matrices appears to contain a typographical error: 'ker G_i = C′_i^⊥' should presumably read 'ker G′_i = C′_i^⊥'. Please correct the notation.","section":"§III B, text near Eq. (34)"},{"comment":"In the SL2(F4) example, the output tuple is printed as '(10, 3)' but the variables are named (dx, hz); one of them should be dz. This is a small typo but it matters for a reproducibility appendix.","section":"Appendix C, code snippet"},{"comment":"The caption states that distances are estimated using '50 million random-ISD trials of sQetch' with '1,000 trials of DistRandCSS', whereas Table I and the main text report 50,000 trials for DistRandCSS and 50–350 million for sQetch. The appendix captions should be reconciled with the main-text trial counts.","section":"Table V caption"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its main caveat in the Table III caption, but the section title and discussion overstate the strength of the evidence. Because the upper-bound issue affects the headline claims, a major revision with certified distances or carefully reframed claims is appropriate. The open-source library and reproducible data are strong assets, and the construction itself appears sound. I would also ask the editor to ensure the appendices are internally consistent before any revised version is sent out."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — this paper is a competent, well-documented search over QT code instances in the n≈500–1000 range. What's actually new: a broad sweep over non-abelian groups from GAP's SmallGrp, a bunch of explicit [[n,k,(≤dX,≤dZ)]] instances with raw distance estimates above 20, and QuantumExpanders.jl, which looks like a genuine contribution — both constructions implemented, with reproducible code data and a worked example in the appendix. I could verify the [[756,10]] example from the appendix and it checks out, which is more than most papers offer.\n\nThe main caveat is exactly the one you flagged: every distance in Tables I and III is an upper bound from sQetch/DistRandCSS, not a certified minimum. The paper is honest about this in captions, but Section IV.C still labels the table 'exceeding the √n barrier' and computes kd²/n figures from the bounds. If the true d's are substantially lower, those claims collapse — the construction remains valid, but the 'competitive parameters' are not established. This is load-bearing, not cosmetic. The pseudo-thresholds in Section IV.B also use the assumed distances to set the number of syndrome-extraction rounds r, so they inherit the same assumption to a lesser degree.\n\nMinor issues: some appendix typos (missing brackets in the [[720,6]] table row, a malformed multiset entry for [[576,40]], a couple of 'o Table III' slips), and the distance estimates across estimators sometimes diverge wildly (e.g., [[1024,16]]: QDistRnd gives (186,174), sQetch (16,16)) — the paper takes the tighter bound, which is optimistic and should at least be discussed. I don't see circularity or fitting-to-target; the search is genuine.\n\nBottom line: worth a serious referee. The right revision would add a lower-bound certificate for at least the headline instances (e.g., via LP or exhaustive decoding for small d), or reframe all distance claims as 'estimated' and damp down the √n language. The code and catalogue will be used by the qLDPC near-term community regardless. I'd bring it to reading group and I'd cite the code if I needed explicit instances. Send it out.","headline":"A useful search catalogue and open-source tooling for moderate-blocklength quantum Tanner codes, but its headline distances are randomized upper bounds, so the 'beyond √n' claims are provisional.","tokens_in":42576,"tokens_out":2395,"would_cite":true,"duration_ms":21746,"reading_group":"yes","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":"Explicit quantum Tanner codes at 500–1000 physical qubits are reported with distance upper bounds above 20, several crossing the square-root barrier.","keywords":["quantum Tanner codes","quantum LDPC codes","moderate blocklength","left-right Cayley complex","code lifting","distance estimation","CSS codes","pseudo-threshold"],"falsifier":"Run an exact minimum-distance computation on the $[[720,6,(\\le 30,\\le 30)]]$ code (or, failing that, a much larger certified branch-and-bound search on the $[[480,8,(\\le 21,\\le 21)]]$ code). If the true $d_X$ or $d_Z$ falls below 20—or below $\\sqrt{n}$ for the codes claimed to cross the barrier—the central practical claim is refuted for that instance.","tokens_in":41703,"feed_emoji":"⚛️","tokens_out":14825,"duration_ms":117953,"temperature":0.7,"pith_summary":"This paper asks whether quantum Tanner codes—a family of sparse quantum error-correcting codes built by placing small local codes on a square complex—with competitive rate and distance exist at moderate blocklength ($n\\in[500,1000]$), a gap between short explicit instances and asymptotic constructions. It answers affirmatively by searching non-abelian groups from a small-group library and constructing explicit codes through both the left-right Cayley complex description and the lifting construction. The reported instances have rates from 0.004 to 0.057, randomized distance upper bounds above 20, check weights 9 to 20, and pseudo-thresholds (the noise levels at which encoding starts to help) of 3.6–4.6% under phenomenological noise and 0.14–0.27% under circuit-level noise. If those bounds are tight, the instances provide practical sparse quantum error-correcting codes at sizes reachable by near-term hardware, with seven of eight listed codes exceeding the square-root distance line.","feed_headline":"Distance estimates pass 20 for quantum Tanner codes at 500–1000 qubits","feed_subtitle":"New explicit codes at 480–1120 qubits pass the square-root distance barrier and decode at up to 4.6% noise.","key_machinery":"The load-bearing mechanism is the lifted parity-check pair of Eq. (34): a seed CSS code on an $n_A\\times n_B$ qubit grid is expanded by replacing each base qubit with a fiber of $|G|$ qubits, using commuting left and right regular actions of a finite group $G$ encoded in permutation matrices $L_A$ and $R_B$. The geometric picture is the left-right Cayley complex, whose defining property is that every vertex link is an $A\\times B$ grid and adjacent $X$- and $Z$-generators share at most one row or column; this makes the local tensor codes $C_A\\otimes C_B$ and $C_A^\\perp\\otimes C_B^\\perp$ orthogonal where they meet, so the CSS condition holds by construction. The search reduces multisets to automorphism orbits, screens candidates by the minimum weight of a classical Tanner-code kernel basis, and uses two independent randomized distance estimators to set upper bounds.","core_discovery":"On the paper's own terms, the discovery is an explicit, searchable catalogue of quantum Tanner codes in the moderate-blocklength regime, obtained by lifting a small seed CSS code through commuting left and right regular actions of a finite group (equivalently, by placing local product codes on a left-right Cayley complex). The headline instances include $[[480,8,(\\le 21,\\le 21)]]$, $[[504,4,(\\le 36,\\le 27)]]$, $[[672,4,(\\le 48,\\le 28)]]$, $[[720,6,(\\le 30,\\le 30)]]$, and $[[864,8,(\\le 39,\\le 31)]]$; all distances are upper bounds from randomized estimation with up to 350 million trials. Seven of eight selected instances satisfy $d_{\\min}>\\sqrt{n}$, with the figure of merit $k d_{\\min}^2/n$ exceeding the surface-code value of 1. This is presented as an affirmative answer to the open question of whether competitive QT codes exist in this regime, complementing earlier searches restricted to a limited set of groups.","pith_inferences":["A natural next test is to certify the minimum distance of $[[720,6,(\\le 30,\\le 30)]]$ or $[[480,8,(\\le 21,\\le 21)]]$ with an exact or branch-and-bound method; if the certified distances stay above $\\sqrt{n}$, the square-root crossing becomes a theorem about specific codes rather than an estimate.","Because the lifting construction permits repeated group elements in the multisets (as shown in Appendix D), the same search machinery could be run over other small non-abelian groups or with different seed local codes to push blocklength, rate, or check weight in a chosen direction.","The reported circuit-level pseudo-thresholds, if reproduced on hardware, would make these codes competitive candidates for early fault-tolerance demonstrations at a few hundred to a thousand qubits, where 2D-local codes are the usual default.","A canonical basis for the logical operators of QT codes, which the paper identifies as open, would let such instances be used for fault-tolerant gates; the search results here provide concrete codes on which to develop that basis."],"forward_implications":["Moderate blocklength ($n\\in[500,1000]$) is no longer a dead zone for quantum Tanner codes: explicit instances with estimated distances above 20 now exist.","If the distance upper bounds are near the true distances, seven of the eight table entries cross the $\\sqrt{n}$ line, meaning the BPT bound's figure of merit $k d^2/n$ can exceed 1 at practical sizes.","The pseudo-thresholds of 3.6–4.6% (phenomenological) and 0.14–0.27% (circuit-level) show that decoding performance of shorter QT codes survives at larger blocklength.","The classical $[8,4,4]$ code stands out as a repeatable building block for good quantum Tanner codes, giving future searches a concrete starting point.","The open-source library makes every reported instance reproducible and provides a platform for extending the search to larger groups or other local codes."],"supporting_citations":[{"why":"Supplies the lifting construction, the search heuristics, and the open question this paper answers.","marker":"[17]"},{"why":"Provides the randomized distance estimator whose upper bounds are the paper's main evidence for distances above 20.","marker":"[19]"},{"why":"Provides the independent randomized distance estimator used to cross-check the upper bounds.","marker":"[20]"},{"why":"Supplies the decoder used to estimate the reported pseudo-thresholds under both noise models.","marker":"[30]"},{"why":"Concurrent QT code search that motivates the broader group sweep and serves as the comparison for this work.","marker":"[16]"},{"why":"Original QT code construction and asymptotic distance guarantee that the moderate-blocklength search tries to instantiate.","marker":"[10]"},{"why":"Introduces the left-right Cayley complex whose local grid structure is the geometric backbone of the construction.","marker":"[14]"},{"why":"Supplies the small-group library from which the searched non-abelian groups are drawn.","marker":"[18]"},{"why":"States the BPT bound used to define the square-root barrier and the figure of merit computed in Table III.","marker":"[2]"}],"fun_headline_variants":["Quantum Tanner codes beat sqrt(n) at 500–1000 qubits","Explicit QT codes with distance >20 at 480–1120 qubits","Tanner codes reach 4.6% pseudo-threshold at moderate sizes","Lift-based search yields QT codes with distance up to 36","Distance bounds pass sqrt(n) in Tanner codes at 500–1000 qubits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the randomized distance upper bounds are close to the true minimum distances; if a smaller logical operator exists that the estimator missed, the reported distances, beyond-barrier claims, and competitive thresholds would be overstated.","fun_headline_variants_meta":{"raw":{"variants":["Quantum Tanner codes beat sqrt(n) at 500–1000 qubits","Explicit QT codes with distance >20 at 480–1120 qubits","Tanner codes reach 4.6% pseudo-threshold at moderate sizes","Lift-based search yields QT codes with distance up to 36","Distance bounds pass sqrt(n) in Tanner codes at 500–1000 qubits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001016,"raw_usage":{"total_tokens":4360,"prompt_tokens":1090,"completion_tokens":3270,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":706,"completion_tokens_details":{"reasoning_tokens":3168}},"tokens_in":706,"tokens_out":3270,"duration_ms":22929,"temperature":1.0,"reasoning_tokens":3168,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:07:12.828597+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an exact minimum-distance computation on the $[[720,6,(\\le 30,\\le 30)]]$ code (or, failing that, a much larger certified branch-and-bound search on the $[[480,8,(\\le 21,\\le 21)]]$ code). If the true $d_X$ or $d_Z$ falls below 20—or below $\\sqrt{n}$ for the codes claimed to cross the barrier—the central practical claim is refuted for that instance.","supporting_citations":[{"cited_title":"naturally encoded by the LRCC associated with (A,B,G)","cited_arxiv_id":null,"evidence_quote":"Supplies the lifting construction, the search heuristics, and the open question this paper answers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the independent randomized distance estimator used to cross-check the upper bounds."},{"cited_title":"Sipser and D","cited_arxiv_id":null,"evidence_quote":"Supplies the decoder used to estimate the reported pseudo-thresholds under both noise models."},{"cited_title":"Panteleev and G","cited_arxiv_id":null,"evidence_quote":"Original QT code construction and asymptotic distance guarantee that the moderate-blocklength search tries to instantiate."},{"cited_title":"Leverrier and G","cited_arxiv_id":null,"evidence_quote":"Introduces the left-right Cayley complex whose local grid structure is the geometric backbone of the construction."},{"cited_title":"The construction follows Section III A and requires only the group, its generators, and the two local codes","cited_arxiv_id":null,"evidence_quote":"States the BPT bound used to define the square-root barrier and the figure of merit computed in Table III."}],"review_version":1}