{"id":"ddc6171a-044a-4bdf-b76b-09250d656666","arxiv_id":"2504.20847","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"New nonadditive ((7,2,3)) and ((8,2,3)) codes are claimed to support transversal T, sqrt(T), and T^(1/4) gates, improving on the previous smallest examples.","lead":"Quantum error-correcting codes that allow small, error-limited logical gates are rare; this paper searches for and presents new nonadditive codes whose transversal logical gates include T and related rotations on only seven or eight physical qubits. If the displayed codes check out, they would be the smallest known distance-3 codes supporting non-Clifford transversal gates.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The BD32 example in §6.3.2 is internally inconsistent: the displayed state has squared norm 31, not 32, and the seven local Z rotations multiply to Z(31π/16) = -Z(-π/16), not the claimed Z(-2π/16). The headline √T improvement therefore rests on an uncorrected example.","rationale":"The reader's CONDITIONAL verdict is well matched. My independent read confirms the C10 code and the BD16 example pass the basic normalization and angle-sum checks, and BD64/BD84 angle sums are actually consistent when read as k=2a_j. The load-bearing soft spot is BD32 alone: the displayed state is not normalized and the displayed local rotations do not implement Z(-2π/16). Since the abstract singles out BD32 as the first ((7,2,3)) code with transversal √T, this specific headline claim is not established as written. This is an internal correctness risk, not a disagreement with consensus, and it is repairable: the SS-LP method is concrete, the required checks are finite, and the rest of the paper's program is plausible. Conditional acceptance is therefore appropriate rather than rejection; the same verdict is retained, so no change to the reader's verdict is recommended.","tokens_in":33985,"tokens_out":15737,"duration_ms":151686,"concrete_test":"Run exact-arithmetic checks on §6.3.2: (a) sum the squared moduli of the nine terms in √32|0L⟩ and confirm it equals 31, not 32; (b) multiply the seven displayed Z(kπ/16) matrices and check whether the product equals Z(-2π/16) up to a scalar — exact arithmetic gives -Z(-π/16). Then repair the construction in the same SS-LP framework with m=16 and integer a_j: list all integer vectors a∈[0,15]^7 with sum ≡ -1 (mod 16) that pass the LP filter of §6.2, and for any candidate that yields a normalized state, verify the full KL conditions for all weight-1 and weight-2 Pauli errors to a tolerance such as 10^{-10}. If no such corrected BD32 code exists, the claimed improvement over ((19,2,3)) fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim includes the first ((7,2,3)) code with transversal √T (BD32), improving on the previous ((19,2,3)) example. The only explicit BD32 construction in §6.3.2 fails basic mechanical checks. (i) Normalization: the nine coefficients listed under √32|0L⟩ have squared moduli summing to 31; the θ² contributions cancel, so the state is not normalized. (ii) Gate action: for m=16, an SS-LP diagonal gate should be ⊗_j Z(2π a_j/16) = ⊗_j Z(a_jπ/8) with integer a_j; the displayed local rotations are Z(kπ/16) with k=(2,3,4,4,5,6,7), including odd k=3,5,7, which cannot come from integer a_j. Their product is Z(31π/16) = -Z(-π/16), whose relative phase is e^{-iπ/16}, whereas the claimed Z(-2π/16) has relative phase e^{-iπ/8}. Thus the displayed circuit does not implement the claimed logical gate. Notably, BD64 and BD84 are consistent under the standard convention (all k even, k=2a_j, sums give a_j ≡ -1 mod m), so the problem is specific to BD32. Without a corrected state and gate, and without a script verifying the KL conditions, the 'smallest √T code' claim is unsubstantiated as printed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces two computational methods for constructing quantum codes with prescribed transversal gate groups: a Stiefel-manifold optimization with composite Knill–Laflamme/gate losses, and a Subset-Sum-Linear-Programming (SS-LP) construction for codes with transversal diagonal gates. The authors report a ((6,2,3)) code with transversal group C10, new ((7,2,3)) codes with binary icosahedral group 2I, a family of ((7,2,3)) codes with binary dihedral groups BD_{2m} for 2m≤36 (including claimed first ((7,2,3)) codes with transversal T and sqrt(T) gates), and ((8,2,3)) codes for larger BD_{2m}, including a T^{1/4} example.","tokens_in":34320,"tokens_out":14950,"duration_ms":136989,"significance":"If the explicit constructions are correct, the paper would be significant: it provides a systematic search framework, substantially reduces the known qubit counts for transversal non-Clifford gates, and expands the known landscape of nonadditive codes. The paper's strengths include explicit logical states, explicit tensor-product gate implementations, and weight-enumerator data for many examples, connected to the signature-norm invariant. However, the central BD32 example fails elementary arithmetic checks as printed, and no verification code or certificate is supplied, so the headline 'smallest sqrt(T) code' claim is currently unsubstantiated.","major_comments":[{"comment":"The displayed BD32 logical state is not normalized: the coefficients in the formula for sqrt(32)|0L> have squared moduli 1 + (1+32θ^2) + (6−32θ^2) + 4 + 3 + 7 + 5 + (4−32θ^2) + 32θ^2 = 31, with the θ^2 contributions cancelling. Thus the state as printed does not define a valid code subspace.","section":"Sec. 6.3.2"},{"comment":"The displayed BD32 transversal gate is inconsistent with the SS-LP convention. For m=16 the local diagonal gates must be diag(1, exp(2π i a_j/16)) = e^{iπ a_j/8} Z(a_jπ/8) with integer a_j, so the local rotation angles must be even multiples of π/16. The printed gate uses Z(3π/16), Z(5π/16), and Z(7π/16), which are odd multiples and cannot arise from integer a_j. Moreover the product of the displayed rotations is Z(31π/16) = −Z(−π/16), whose relative phase is e^{−iπ/16}, not the claimed Z(−2π/16) = Z(−π/8) relative phase e^{−iπ/8}. The angle-sum condition (15) also fails for the implied half-integer a_j. The sqrt(T) claim is therefore unsubstantiated as printed.","section":"Sec. 6.3.2"},{"comment":"Because the only evidence for existence of the claimed codes is the printed logical states and gates, and because the BD32 example fails both normalization and gate-action checks, the authors should supply a machine-checkable verification (for example, a script that checks normalization, the Knill-Laflamme conditions for all weight-1 and weight-2 Pauli errors, and equality of the tensor-product gate with the claimed logical gate) for every explicit code in the main text and appendices. Without such verification, the existence claims for the headline examples cannot be accepted.","section":"Secs. 3-6 and Appendices D-E"}],"minor_comments":[{"comment":"The text reads 'He we present one example'; this should be 'Here we present one example'.","section":"Sec. 6.3.2"},{"comment":"The B weight enumerator for the λ* = sqrt(3/4) 2I code appears garbled in the printed expression ('99/2 096t2'); please correct and re-check this formula.","section":"Sec. 5.1"},{"comment":"There are several typographical slips such as 'stablizer' (Sec. 6.3.1), 'T able' in table captions, and inconsistent spacing in 'a nd', 'y et', and related words; these should be cleaned up.","section":"Throughout"},{"comment":"The λ* intervals in Table 3 are described as 'numerically estimated'; the paper should state explicitly which entries are proven exact values and which are only numerical evidence from the optimization.","section":"Table 3"}],"recommendation":"major_revision","confidential_remarks":"The BD32 example is the main obstacle: as printed, it fails normalization and the local gate tensor product does not implement the claimed logical rotation, so the 'smallest sqrt(T) code' result is not established. The C10 and other BD examples appear internally consistent, and the SS-LP framework is plausible, so I do not see grounds for rejection. I would ask the authors to correct the BD32 construction or replace it with a verified example, and to provide a reproducibility script or exact algebraic certificate for the claimed codes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Ed:\n\nThe paper has a real idea and several apparently correct constructions, but the headline claim for the smallest transversal √T code rests on an example that fails basic arithmetic as printed. The BD32 state in §6.3.2 is not normalized (squared norm 31, not 32), and the displayed local Z rotations multiply to a logical phase of e^{-iπ/16} rather than the claimed e^{-iπ/8}. Moreover, the rotation angles include odd multiples of π/16, which cannot come from the SS-LP ansatz with integer a_j for m=16. So the claimed ((7,2,3)) sqrt(T) improvement over the ((19,2,3)) code is not supported by the manuscript.\n\nThat said, the paper does more than hand-wave. The SS-LP reduction to integer partitions subject to congruence constraints is a genuinely useful way to shrink the search for codes with transversal diagonal gates. The ((6,2,3)) C10 code in §4.1 checks out: the state is normalized, the phases satisfy the constraint equations, and the tensor product of Z rotations gives the claimed logical Z(-2π/5). The BD16 example in §6.3.1 also works — the angle vector sums to -1 mod 8 and the state has norm 16 on the 4|0L> scale. The 2I examples in §5 are plausible, though I did not verify every coefficient. The BD36 example also passes the same mechanical checks.\n\nThe reader's take flagged BD64 and BD84 as also broken, but that is not right: both use even angle numerators, giving integer a_j, and their sums satisfy ∑a_j ≡ -1 mod m. The problem is specific to BD32. That is still a serious problem because it is the distinguishing new example. The paper gives no verification code or certificates, and the search method is numerical, so the reader cannot easily separate transcription errors from actual failures. The weight enumerator formulas throughout could be checked symbolically, but without code that is laborious.\n\nWho should read this: anyone working on nonadditive codes or transversal gate groups. The method and the correct examples are a step forward. But the paper as submitted is not trustworthy for the √T claim, and the lack of code makes it hard to verify the rest.\n\nRecommendation: send to peer review, but require the authors to (i) correct or replace the BD32 example, (ii) add a script or certificate verifying the KL conditions for each claimed code, and (iii) reconcile the discrepancy with the abstract. With those fixes, the paper would be a solid contribution. If the BD32 example cannot be repaired, the stronger claim should be removed.","headline":"The SS-LP method and most of the small examples look real, but the headline √T code is misprinted, so the paper needs revision before the central claim is credible.","tokens_in":34891,"tokens_out":6563,"would_cite":false,"duration_ms":58154,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70","81P68","94B60"],"pacs":["03.67.Pp","03.67.Lx"],"model":"deepseek-v4-flash","headline":"This paper claims a family of nonadditive quantum codes whose transversal gate groups include a 6-qubit C10 code, 7-qubit codes with T and √T gates, and 8-qubit codes with T^{1/4}, shrinking the smallest known blocks for non-Clifford…","keywords":["nonadditive quantum codes","transversal gates","Knill-Laflamme conditions","binary dihedral groups","Stiefel manifold","subset-sum linear programming","binary icosahedral group","fault-tolerant quantum computation"],"falsifier":"Verify the displayed state in Sec. 6.3.2 by computing its squared norm and the angle sum ∑ a_j mod m. If the norm evaluates to 31 instead of 32, or the sum is -2 mod m instead of the required -1 for the BD32 example (and similarly for BD64 and BD84), then those printed examples do not realize the claimed transversal groups; a corrected example list would settle the SS-LP claim.","tokens_in":33767,"feed_emoji":"⚛️","tokens_out":5220,"duration_ms":50172,"temperature":0.7,"pith_summary":"This paper seeks to show that nonadditive quantum codes can realize a wider variety of transversal logical gates than prior examples suggested, and to supply a systematic way to find them. It claims a new six-qubit distance-3 code whose transversal group is the cyclic group C10, the smallest known distance-3 code supporting a non-Clifford transversal gate, and a family of seven-qubit codes realizing binary dihedral groups BD_{2m} up to 2m=36, including transversal T and √T gates. The authors introduce a two-part strategy: direct search on the Stiefel manifold of code subspaces, and a Subset-Sum-Linear-Programming construction that reduces the search to integer partitions with linear constraints. If the claims hold, fault-tolerant designs gain smaller candidate code blocks for non-Clifford transversal operations.","feed_headline":"Seven-qubit codes get transversal T and √T gates","feed_subtitle":"A subset-sum linear program finds the smallest known blocks for non-Clifford transversal gates.","key_machinery":"The SS-LP construction: assume the target logical diagonal gate is implemented transversally by local Z rotations with integer angles a_j modulo m; then each logical basis state can only be supported on computational basis strings whose weighted sum of bits is congruent to the corresponding logical eigenvalue b_k modulo m. This reduces the search to subset-sum/partition problems, filters candidates with a linear program enforcing Z-type Knill-Laflamme conditions, and finishes with block-separable amplitude optimization. The general search uses Stiefel manifold parameterization of orthonormal two-frames, combined with polar-decomposition updates and a composite loss L_KL + L_gate. The paper states that for BD_{2m} symmetry the X/Y/Z KL conditions simplify, so the LP reduces to a set of mean-zero constraints on supports.","core_discovery":"The central claim is that the landscape of nonadditive ((n,2,3)) codes is much richer than stabilizer or permutation-invariant constructions alone would suggest: every requested finite subgroup of SU(2) that survives the search can be realized as the transversal group of a valid code. Specifically, the paper constructs a ((6,2,3)) code with transversal Z(2π/5), generating C10; several ((7,2,3)) codes with transversal 2I at signature norms λ* = 0 and √(3/4); and SS-LP-derived ((7,2,3)) codes with BD_{2m} for all 2m≤36, including the first 7-qubit examples with transversal T (BD16) and √T (BD32), which beat the previously smallest ((11,2,3)) and ((19,2,3)) constructions. For eight qubits it finds BD_{2m} for 2m>36, including a BD64 code supporting $T^{{1/4}}$. The authors argue these examples show a continuum of nonadditive codes parameterized by the signature norm and connected to algebraic constraints on transversal gate groups.","pith_inferences":["If the examples are verified, the apparent saturation at 2m=36 for n=7 suggests a general relationship between physical qubit number and the maximal order of a diagonal transversal gate; a testable conjecture is that BD_{2m} at n=7 requires 2m to be bounded by a linear function of n, which the authors did not state.","The SS-LP reduction could plausibly extend to K>2 logical dimensions or higher distance d>3 by replacing the single-congruence condition with a system of congruences, though the paper only treats K=2, d=3.","The C10 ((6,2,3)) code implies that 5-fold rotations, not just Clifford-hierarchy gates, can be transversal, offering a possible route to non-Clifford fault-tolerant operations on very small code blocks.","A direct numerical check of the printed angle vectors for the BD32, BD64, and BD84 examples would settle whether those displayed states realize the claimed groups; if not, the SS-LP method may still be valid but those particular examples would require correction."],"forward_implications":["If the claims hold, a distance-3 quantum code with a non-Clifford transversal gate can be as small as six qubits, below the previous known block sizes.","Transversal T and √T gates, normally associated with larger stabilizer or permutation-invariant codes, can be realized in 7-qubit nonadditive codes, and T^{1/4} in an 8-qubit code.","The observed saturation at BD36 in 7-qubit codes suggests a finite limit on the order of diagonal transversal groups for a given block length, guiding future search strategies.","The signature-norm parametrization links continuous families of nonadditive codes to distinct transversal groups, providing new invariants for comparing code constructions."],"supporting_citations":[{"why":"Supplies the Knill-Laflamme conditions used as the error-correction criterion in the loss function.","marker":"[16]"},{"why":"Defines the signature norm λ* and earlier families of ((6,2,3)) and ((7,2,3)) codes that this paper extends.","marker":"[11]"},{"why":"Provides the prior permutation-invariant ((7,2,3)) code with transversal 2I and establishes the framework for exotic transversal groups.","marker":"[18]"},{"why":"Gives the previous smallest codes with transversal T ((11,2,3)) and √T ((19,2,3)), the baselines the new 7-qubit constructions improve upon.","marker":"[19]"},{"why":"Classifies small binary stabilizer subsystem codes, identifying which transversal groups appear in ((7,2,3)) stabilizer codes.","marker":"[8]"},{"why":"Establishes the known ((15,2,3)) Reed-Muller code with transversal T, a benchmark for comparison.","marker":"[17]"},{"why":"Introduces permutation-invariant codes, including the ((7,2,3)) code that realizes 2I and λ*=√7.","marker":"[27]"},{"why":"Classifies transversal gates in additive quantum codes and underpins the finite-subgroup restriction on transversal operations.","marker":"[39]"}],"fun_headline_variants":["Smallest distance-3 codes with transversal T and √T gates found","7-qubit codes beat 11 and 19 qubit records for transversal T","New codes shrink qubit count for non-Clifford transversal gates","Search finds 7-qubit codes with T and √T transversal gates","Nonadditive codes get transversal T and √T at just 7 qubits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the printed codeword formulas, including normalization and the angle-sum condition ∑ a_j ≡ -1 (mod m) of Eq. (15), are accurate as written; if any displayed example fails these checks, its claimed transversal group is not actually realized.","fun_headline_variants_meta":{"raw":{"variants":["Smallest distance-3 codes with transversal T and √T gates found","7-qubit codes beat 11 and 19 qubit records for transversal T","New codes shrink qubit count for non-Clifford transversal gates","Search finds 7-qubit codes with T and √T transversal gates","Nonadditive codes get transversal T and √T at just 7 qubits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000573,"raw_usage":{"total_tokens":2835,"prompt_tokens":1202,"completion_tokens":1633,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":818,"completion_tokens_details":{"reasoning_tokens":1535}},"tokens_in":818,"tokens_out":1633,"duration_ms":11337,"temperature":1.0,"reasoning_tokens":1535,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:19:55.216026+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify the displayed state in Sec. 6.3.2 by computing its squared norm and the angle sum ∑ a_j mod m. If the norm evaluates to 31 instead of 32, or the sum is -2 mod m instead of the required -1 for the BD32 example (and similarly for BD64 and BD84), then those printed examples do not realize the claimed transversal groups; a corrected example list would settle the SS-LP claim.","supporting_citations":[{"cited_title":"Theory of quantum er ror-correcting codes","cited_arxiv_id":null,"evidence_quote":"Supplies the Knill-Laflamme conditions used as the error-correction criterion in the loss function."},{"cited_title":"Family of quantum code s with exotic transversal gates","cited_arxiv_id":null,"evidence_quote":"Provides the prior permutation-invariant ((7,2,3)) code with transversal 2I and establishes the framework for exotic transversal groups."},{"cited_title":"Permutation-invaria nt quantum codes with transversal generalized phase gates, 2024","cited_arxiv_id":null,"evidence_quote":"Gives the previous smallest codes with transversal T ((11,2,3)) and √T ((19,2,3)), the baselines the new 7-qubit constructions improve upon."},{"cited_title":"Quantum Error Correction with T ransversal non-Clifford Ga tes","cited_arxiv_id":null,"evidence_quote":"Establishes the known ((15,2,3)) Reed-Muller code with transversal T, a benchmark for comparison."},{"cited_title":"Permutational ly invariant codes for quantum error correction","cited_arxiv_id":null,"evidence_quote":"Introduces permutation-invariant codes, including the ((7,2,3)) code that realizes 2I and λ*=√7."},{"cited_title":"Transversal ity versus universality for additive quantum codes","cited_arxiv_id":null,"evidence_quote":"Classifies transversal gates in additive quantum codes and underpins the finite-subgroup restriction on transversal operations."}],"review_version":1}