{"id":"65721b8b-df13-4a12-8cb6-fa361808a67a","arxiv_id":"2507.05392","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"An asymptotically good qubit CSS code family with constant rate and distance and transversally addressable logical CCZ gates is claimed, built from Stichtenoth's transitive iso-orthogonal algebraic geometry codes.","lead":"This paper constructs the first family of asymptotically good quantum error-correcting codes over qubits that support transversally addressable non-Clifford logical CCZ gates on any three logical qubits. If correct, it improves on the authors' earlier construction by removing a polylogarithmic overhead and achieving constant rate and distance for fault-tolerant quantum computation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claim 3.14's fourwise orthogonality is unsupported: the printed proof supplies only a triple-product inclusion, while the fourfold product requires C_{4a,4b}⊆C_{ai-a,bi-b}, i.e. 5a≤ai, which fails for a=⌊ai/4⌋. The CCZ addressability theorems, and hence Theorem 1.1, rest on this identity.","rationale":"I read the construction in good faith: using Stichtenoth's transitive, iso-orthogonal AG codes to obtain a constant-alphabet code with addressable CCZ gates is plausible, and the claimed asymptotic parameters are consistent with the rate and distance bounds in Lemma 3.11. However, the exact step flagged by the reader is indeed load-bearing. Claim 3.14 is invoked verbatim in both the intra-block proof (Theorem 3.12) and the inter-block proof (Theorem 3.15); without Eq. (64), the physical phase sum cannot be reduced to the logical phase w_A w_B w_C. The proof of Claim 3.14 as written contains only a triple-product containment, and the natural fourfold version requires 5a≤ai, which fails for a=⌊ai/4⌋. The defect is internal to the argument, not a matter of disagreeing with the community's consensus. I also checked whether the obvious parameter change a=⌊ai/5⌋ would restore the proof: the distance inequality (22) then requires r≥10, so the repair is not a trivial substitution for the stated r≥8. Thus the reader's REJECT verdict remains appropriate; I do not see grounds to change it, and I do not see a way to certify the central claim from the submitted text.","tokens_in":14705,"tokens_out":13833,"duration_ms":166162,"concrete_test":"Independently re-derive Claim 3.14 from Proposition 4.7 of [Sti06]. The only product-code route is: f1f2f3f4∈C_{4a,4b}, and the claim requires C_{4a,4b}⊆C_{ai-a,bi-b}. Substitute a=⌊ai/4⌋; for ai=4m, this gives 4a=4m and ai−a=3m, so the required containment fails coefficientwise. As a computational check, instantiate the smallest Stichtenoth tower level with these parameters, pick a nonconstant f∈L(aA+bB), and evaluate S=∑_j u_j f(α_j)^4 plus the corresponding beta sum; Claim 3.14 predicts the two weighted sums are equal, while the failed containment predicts they need not be. If the test is run and S is nonzero for any such f, the claim is false as stated; if the paper supplies a different proof route, the test should be redone along that route.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing defect is in Claim 3.14, the fourwise orthogonality identity. To prove Eq. (64), the authors need the vector u·(f1 f2 f3 f4) to be orthogonal to C, where each fi∈L(aA(i)+bB(i)). Since the product lies in C_{4a,4b}, the proof needs C_{4a,4b}⊆C_{ai-a,bi-b}, equivalently 4a≤ai−a and 4b≤bi−b. The printed proof instead states only the triple-product inclusion u·(C*C*C)⊆u·C_{3a,3b} and then asserts the fourwise conclusion. Replacing that line with the required fourfold containment gives 4a=4⌊ai/4⌋>ai−⌊ai/4⌋=ai−a for every ai≥4, so the needed coefficient inequality fails on the A-divisor. Theorems 3.12 and 3.15 use Claim 3.14 in the phase computations (Eqs. (67)–(71) and (88)–(93)) to collapse the physical CCZ phase sum to the logical w_A w_B w_C; without the identity, the logical gate action is not established. This is a missing key lemma in the submitted proof, not merely a disagreement about the plausibility of the construction. The reader's suggested a=⌊ai/5⌋ repair is not automatic either: checking Lemma 3.8's Eq. (22) asymptotically requires r≥10 for this substitution to preserve the distance bound, so the fix is nontrivial and absent from the text.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper constructs a family of qudit CSS codes over a fixed alphabet F_q with q = r^2 a power of two and r ≥ 8, using Stichtenoth's transitive, iso-orthogonal algebraic geometry codes. The code QAG has length n(i) = N(i) - k(i), dimension k(i) = N(i)/(r(r-1)), and relative distance at least 1/4 - 3/(2(r-1)) - 1/(r(r-1)), so it is asymptotically good over qudits of constant dimension. The main technical contribution is a proof that any logical CCZ gate on three logical qudits in one, two, or three codeblocks can be implemented by physical CCZ gates (depth 7 intra-block, depth 3 for two blocks, depth 1 for three blocks, and depth 1 after a constant-factor duplication). Using the qudit-to-qubit conversion from [HVWZ25], the authors infer the first asymptotically good qubit CSS code family with a transversally addressable non-Clifford gate, stated as Theorem 1.1.","tokens_in":15077,"tokens_out":26610,"duration_ms":284104,"significance":"If the construction is correct, Theorem 1.1 settles an open problem posed in [HVWZ25] and represents a genuine advance: previous qubit codes with transversal non-Clifford gates either had poor parameters or required qudit dimension growing with length. The paper's parameter estimates in Section 3.1 are explicit and checkable, and the key orthogonality claim underlying the gate theorems is in fact proved correctly; the apparent fourfold-containment gap identified in the stress-test does not exist. The main caveat is the paper's heavy reliance on the prior qudit-to-qubit conversion and notation from [HVWZ25], but this is a standard follow-up pattern and not a defect in the central derivation.","major_comments":[{"comment":"The apparent gap reported in the stress-test does not actually arise. To prove Eq. (64), one must show that for any f1,...,f4 in L(aA^(i)+bB^(i)), the weighted sum over all N(i) places vanishes. The proof does this by establishing u·(f1 f2 f3) ∈ C_{a,b}^{(i)⊥} via the containment chain in Eq. (65), using only the triple-product inclusion C_{3a,3b} ⊆ C_{ai-a,bi-b}, which holds because a = floor(ai/4) gives 3a ≤ ai - a (and similarly for b). Since the evaluation vector of f4 is a codeword of C_{a,b}, the vanishing of the dot product of u·(f1 f2 f3) with that vector is exactly the needed fourfold identity. The proof does not require the fourfold product containment C_{4a,4b} ⊆ C_{ai-a,bi-b}; it never forms the product f1 f2 f3 f4 as a single function in a divisor space. Thus Theorems 3.12 and 3.15 are not undermined by the concern about 5a ≤ ai.","section":"3.2, Claim 3.14 (Eqs. (64)-(66))"}],"minor_comments":[{"comment":"In the proof of Claim 3.14, after the containment chain in Eq. (65), the text should explicitly state that the evaluation vector of f4 lies in C_{a,b}^{(i)} and that the containment places u·(f1 f2 f3) in the dual, so the standard dot product of these two vectors vanishes; the current compressed wording invites the misreading that a fourfold product containment is being used.","section":"3.2, Claim 3.14 proof"},{"comment":"The notation QAG := CSS(X, G0; Z, G⊥) is ambiguous: Claim 3.10 establishes orthogonality with respect to the u-weighted inner product, not the standard dot product. The authors should state explicitly how G⊥ is defined (u-dual, or the monomial equivalence that reduces it to the standard CSS condition), since a reader unfamiliar with [HVWZ25] cannot verify the CSS condition from the text.","section":"3.1.2, Eq. (49) and Assumption 3.9"},{"comment":"The circuit-depth argument says each physical qudit appears in exactly three physical CCZ gates; because a Galois automorphism can fix an αk, the same gate may contain a repeated coordinate, so the correct statement is 'at most three'. The subsequent counting of at most six neighboring gates still goes through with this weaker bound.","section":"3.2, Circuit Depth"},{"comment":"Remark 3.3's reduction of depth to one by duplicating qudits is stated without argument; since Theorem 1.1's depth-one claim relies on it, a brief explanation or pointer to the standard construction would be helpful.","section":"3.3, Remark 3.3"},{"comment":"There are several typographical errors: 'Throuhgout' at the start of Section 3.1, 'transveral' in the Introduction, and 'We also that P′' in Definition 2.1 (should be 'We also say that P′').","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern about Claim 3.14 is a false positive: the proof is correct as written, and the fourfold identity follows from the triple-product containment together with the definition of the dual code. I recommend minor revision rather than reject. The only substantive request is to clarify the u-twisted CSS definition in Section 3.1.2 and to expand the one-line step in Claim 3.14 so that future readers do not repeat the misreading."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's the take on arXiv:2507.05392. If the main theorem were established, this would be a genuine advance: the first asymptotically good qubit codes with transversally addressable non-Clifford gates. The construction replaces the Reed-Solomon component codes from the prior paper with Stichtenoth's transitive, iso-orthogonal AG codes, keeping q = Theta(1), which is exactly what removes the polylog loss after qudit-to-qubit conversion. The parameter estimates in Section 3.1 are coherent, and the presentation is clear.\n\nBut the gate theorem has a load-bearing gap. Claim 3.14 asserts a fourwise orthogonality identity (Eq. 64) that the proof does not actually establish. The proof line (65) shows u·(C*C*C) ⊆ u·C_{3a,3b} ⊆ C^⊥, i.e., a triple-product statement. For the fourfold sum in Eq. (66) you would need C_{4a,4b} ⊆ C_{ai-a,bi-b}, which requires 5a ≤ ai. With a = floor(ai/4), that fails for large i (e.g., ai = 4s+2 gives 4a = 4s > ai - a = 3s+2 when s > 2). Theorems 3.12 and 3.15 use this identity to collapse the physical CCZ phase sum to the logical w_A w_B w_C, so without it the logical gate action is not proven. This is a missing key lemma, not a stylistic complaint.\n\nThe good news: the gap looks repairable. Taking a = floor(ai/5) with a larger fixed r would likely restore the containment, but that fix isn't in the text, and the parameter estimates in Lemma 3.8 would need rechecking for the distance bound.\n\nWho is this for? Researchers working on transversal gates and constant-overhead fault tolerance. The paper deserves a serious referee: the construction is natural, the claim is important, and the flaw is concrete and possibly fixable. I would not cite it as a proven result in the next 12 months, but I'd bring it to a reading group to work through the AG details.","headline":"A natural and promising construction whose central addressability claim rests on a fourwise orthogonality identity that the proof does not actually deliver.","tokens_in":15661,"tokens_out":2895,"would_cite":false,"duration_ms":31034,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70","94B27"],"pacs":["03.67.Pp","03.67.Lx"],"model":"deepseek-v4-flash","headline":"The paper claims the first asymptotically good qubit CSS codes whose logical CCZ gate on any three logical qubits is implemented by a depth-one physical CCZ circuit.","keywords":["quantum error correction","CSS codes","asymptotically good codes","transversal gates","non-Clifford gates","CCZ gate","algebraic geometry codes","transitive codes"],"falsifier":"Compute the fourfold weighted sum in Claim 3.14 for small $i$ with $a = \\lfloor a_i/4 \\rfloor$, $b = \\lfloor b_i/4 \\rfloor$: choose four functions in $L(aA^{(i)}+bB^{(i)})$, evaluate at the $\\alpha$-places and $\\beta$-places, and compare the weighted sums using the vector $u$ from [Sti06]. Any disagreement shows the identity false; in particular one should check whether $u \\cdot (C_{a,b}^{\\ast 4})$ lies in $u \\cdot C_{a_i-a,b_i-b}$, which would require $4a \\le a_i-a$, i.e. $5a \\le a_i$.","tokens_in":2155,"feed_emoji":"⚛️","tokens_out":3472,"duration_ms":135436,"temperature":0.7,"pith_summary":"This paper aims to close a long-standing gap: asymptotically good quantum codes, with constant rate and constant relative distance, have not before been constructed to support transversal non-Clifford gates. It claims the first such qubit code family, with parameters $[[n, \\Theta(n), \\Theta(n)]]_2$, where a logical CCZ gate on any three logical qubits, whether in one, two, or three blocks, is executed by a depth-one physical circuit of CCZ gates. The method replaces the growing qudit dimension of earlier Reed-Solomon constructions with algebraic geometry codes over a fixed field, so the conversion from qudits to qubits preserves the linear rate and distance.","feed_headline":"First qubit codes merge good parameters with transversal CCZ","feed_subtitle":"A constant field size makes the qudit-to-qubit conversion preserve linear rate and distance.","key_machinery":"The central object is the tower of function fields $E_0 \\subseteq E_1 \\subseteq \\cdots$ over $\\mathbb{F}_q$ with $q=r^2$ from [Sti06], in which each extension $E_i/E_0$ is Galois and the place $(z=1)$ splits completely. From this tower one defines the classical algebraic geometry codes $C^{(i)}_{a,b} = C_L(D^{(i)}, aA^{(i)}+bB^{(i)})$ with $a = \\lfloor a_i/4 \\rfloor$ and $b = \\lfloor b_i/4 \\rfloor$. The iso-orthogonality property gives a fixed nonzero vector $u$ with $(C^{(i)}_{a,b})^\\perp = u \\cdot C^{(i)}_{a_i-a,b_i-b}$. The load-bearing identity is the fourwise orthogonality relation in Claim 3.14: for functions $f_1,\\dots,f_4$ in $L(aA^{(i)}+bB^{(i)})$, the weighted sum of $f_1 f_2 f_3 f_4$ over the physical places equals the same weighted sum over the logical places. Together with the transitive action of $\\mathrm{Gal}(E_i/F_0)$ on the rational places, which sends logical place $\\beta_A$ to $\\beta_B$ and $\\beta_C$, this identity collapses the phases of the many physical CCZ gates into the single logical phase $\\mathrm{tr}(\\gamma w_A w_B w_C)$.","core_discovery":"The central claim is Theorem 1.1: there exists a family of quantum CSS codes over qubits with parameters $[[n, \\Theta(n), \\Theta(n)]]_2$ supporting a transversally addressable non-Clifford gate. Concretely, any three logical qubits labeled $A, B, C$ in one, two, or three blocks of the code can receive the logical $\\mathsf{CCZ}_\\gamma$ gate, which multiplies a computational basis state by $(-1)^{\\mathrm{tr}(\\gamma \\eta_1\\eta_2\\eta_3)}$, by applying a depth-one circuit of physical CCZ gates. The construction builds a CSS code from classical transitive, iso-orthogonal algebraic geometry codes over $\\mathbb{F}_q$ with $q=r^2$ a fixed power of two. Physical qudits correspond to a subset of rational places above $(z=1)$ in a tower of function fields, logical qudits to another subset, and Galois automorphisms move logical addresses around while preserving the places. The iso-orthogonal structure supplies a fixed nonzero weight vector that turns the sum of physical CCZ phases into exactly the logical phase $w_A w_B w_C$. Because $q$ is constant, the qudit-to-qubit conversion gives a qubit code that remains asymptotically good.","pith_inferences":["If the fourwise orthogonality identity holds for the full family, the transitive Galois action may give more than single-gate addressability: products of CCZ gates on disjoint logical triples could also be implemented in constant depth, a strong addressability property the paper does not claim.","The constant-field construction suggests a route to concatenating the asymptotically good code with itself or with small codes while keeping the field size fixed, which the paper does not explore.","Testing the fourwise identity numerically on the smallest tower levels would be a natural finite-size experiment; if it holds there, the mechanism is concrete enough to simulate, and if not, the gate derivation needs a different containment argument.","The same coordinate symmetry that makes logical qudits addressable could also permute physical coordinates under fault-tolerant scheduling, an operational benefit not discussed in the paper."],"forward_implications":["An asymptotically good qubit code can host transversal, addressable non-Clifford gates, so fault-tolerant schemes no longer have to choose between linear-rate/linear-distance parameters and transversal non-Clifford logic.","The intra-block logical CCZ gate has a depth-7 physical implementation, the inter-block case has depth 1, and duplicating qudits a constant number of times makes both cases depth 1 while preserving asymptotic goodness.","The construction generalizes to other diagonal gates with $\\pm 1$ diagonal entries acting on a constant number of qudits, and to qudit dimensions beyond $q$ a power of two.","Because the field size is constant, the qudit-to-qubit conversion preserves the $\\Theta(n)$ rate and $\\Theta(n)$ distance, removing the polylogarithmic loss of the earlier Reed-Solomon-based construction.","The construction resolves one open problem from the authors' previous paper; the other listed open problems, including strong addressability of arbitrary products of CCZ gates, remain open."],"supporting_citations":[{"why":"Supplies the transitive, iso-orthogonal algebraic geometry codes, the function-field tower with Galois extensions, and the weight vector $u$ via its Proposition 4.7.","marker":"[Sti06]"},{"why":"Defines the CSS construction, the addressable-CCZ framework, and the qudit-to-qubit conversion used to turn the constant-field qudit code into the qubit code.","marker":"[HVWZ25]"},{"why":"Provides methods for converting qudit transversal non-Clifford gates into qubit codes, used in the concatenation step.","marker":"[GG24]"},{"why":"Supplies another conversion method for binary quantum codes with transversal CCZ gates, used in the same step.","marker":"[Ngu24]"},{"why":"Provides standard facts on function fields, places, Galois extensions, and algebraic geometry code parameters used throughout the proof.","marker":"[Sti09]"}],"fun_headline_variants":["Good qubit codes now support transversal CCZ","First asymptotically good codes with transversal non-Clifford","Addressable CCZ on good quantum codes","Qubit codes merge good parameters with CCZ","Asymptotically good qubit codes with transversal CCZ"],"cache_read_input_tokens":17536,"weakest_assumption_plain":"The load-bearing assumption is that Claim 3.14's fourwise orthogonality identity holds for all code levels used: for any four functions in $L(aA^{(i)}+bB^{(i)})$, the weighted sum over physical places equals the weighted sum over logical places. If this identity fails for even one level, the physical CCZ phases do not collapse to the logical phase and the gate implementation is not established.","fun_headline_variants_meta":{"raw":{"variants":["Good qubit codes now support transversal CCZ","First asymptotically good codes with transversal non-Clifford","Addressable CCZ on good quantum codes","Qubit codes merge good parameters with CCZ","Asymptotically good qubit codes with transversal CCZ"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000372,"raw_usage":{"total_tokens":2005,"prompt_tokens":976,"completion_tokens":1029,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":954}},"tokens_in":592,"tokens_out":1029,"duration_ms":8909,"temperature":1.0,"reasoning_tokens":954,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:30:18.283069+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the fourfold weighted sum in Claim 3.14 for small $i$ with $a = \\lfloor a_i/4 \\rfloor$, $b = \\lfloor b_i/4 \\rfloor$: choose four functions in $L(aA^{(i)}+bB^{(i)})$, evaluate at the $\\alpha$-places and $\\beta$-places, and compare the weighted sums using the vector $u$ from [Sti06]. Any disagreement shows the identity false; in particular one should check whether $u \\cdot (C_{a,b}^{\\ast 4})$ lies in $u \\cdot C_{a_i-a,b_i-b}$, which would require $4a \\le a_i-a$, i.e. $5a \\le a_i$.","supporting_citations":[],"review_version":1}