{"id":"3870092d-3d97-461d-80d6-ec7aac9d8e37","arxiv_id":"2502.01864","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors construct qubit CSS codes with addressable transversal CCZ gates and introduce an addressable orthogonality framework that generalizes Bravyi-Haah triorthogonality.","lead":"This paper constructs the first quantum error-correcting codes whose non-Clifford gates can be applied to selected logical qubits rather than to all logical qubits at once. Its main code is near-asymptotically good and implements a logical CCZ gate on any chosen triple of logical qubits with a single layer of physical CCZ gates.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.8 is asserted by analogy to [Ngu24] without proof; if the ℓ-generalization fails, the asymptotically good addressable-CCZ construction has no instantiation.","rationale":"I read the paper in good faith. The central Reed–Solomon construction (Section 3) is self-contained: the phase calculations in Theorems 3.7 and 3.8 check out under the stated degree condition m < n/3+1, and the distance bound d ≥ min(n−m+1, m−k+1) follows from the puncturing/shortening identities. The qubit conversions in Section 6.1 are detailed and the multiplication-friendly embedding lemma, while imported from [Ngu24], is a standard construction with r=t^3 and appears adequate for the depth-one CCZ implementation; I do not see a fatal flaw there. The weakest point is indeed Theorem 5.8. The paper states it as a simple corollary of prior work without proof, and the parameter constraints for an AG code with strong ℓ-multiplication and linear dual distance require a nontrivial choice of field size and divisor degree. Since this theorem is the only instantiation behind the asymptotically good addressable-CCZ code (Theorem 1.2), the conditionality of the reader's verdict is justified. I found no other concern that outweighs this one. Therefore the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":106,"tokens_out":44471,"duration_ms":517178,"concrete_test":"Independently re-derive Theorem 5.8 for the required ℓ=8 case: write out the AG-code family obtained from Theorem 3.6 of [Ngu24] using the Garcia–Stichtenoth tower over a fixed field F_q with q=2^t chosen large enough (e.g., q=2^10), exhibit an effective divisor G with deg(G)=cN and c<1/8, verify that the code contains the all-ones vector, compute the resulting m,d,d^⊥ and m−k,d−k,d^⊥−k, and confirm they are Θ(N). If the computation cannot be carried out or the linearity fails, Theorem 1.2 has no valid instantiation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The asymptotically good qubit code of Theorem 1.2 rests entirely on Theorem 5.8, which asserts the existence of an explicit family of classical AG codes over a fixed field with the strong ℓ-multiplication property and m,d,d^⊥ all linear in the block length. The proof is a single sentence: it claims this follows as a simple generalization of Theorem 3.6 of [Ngu24] by replacing 37 with ℓ. This does not verify the crucial parameter balance. For an AG code C=C_L(D,G) with deg(G)=cN and genus g=εN, the ℓ-multiplication property requires c < 1/ℓ, while the dual-distance bound is d^⊥ ≥ (c−2ε)N; after puncturing k=Θ(N) columns, one needs d^⊥−k = Θ(N), i.e., c−2ε > c−ε (up to constants), which forces 2ε < c < 1/ℓ. This is feasible only if the field is chosen large enough relative to ℓ (e.g., 1/(√q−1) < 1/(2ℓ)), but the paper never states such a condition nor proves that the Garcia–Stichtenoth tower produces the required divisor. It also does not confirm that the code contains the all-ones vector, which Remark 5.2 relies on to upgrade from ordinary to strong ℓ-multiplication. Since Corollary 5.9 and the construction of Section 4.4 feed directly into Theorem 5.10 and then Theorem 1.2, a failure of Theorem 5.8 collapses the asymptotically good result. I find no internal error in the self-contained Reed–Solomon construction or the qubit conversion of Section 6.1; Theorem 1.1 additionally depends on the cited multiplication-friendly embedding lemma from [Ngu24], but that lemma is a stated external result and appears correct. The unproved AG instantiation is the most load-bearing gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a framework of \"addressable orthogonality\" for CSS codes, generalizing the triorthogonality framework of Bravyi and Haah, and uses it to construct quantum codes supporting transversal, addressable non-Clifford gates. The primary result (Theorem 1.1) is an explicit qubit code family with parameters [[n, Ω(n/polylog(n)), Ω(n/polylog(n))]] that supports a logical CCZ gate on any triple of logical qubits across one, two, or three code blocks via a depth-one physical circuit of CCZ gates. This is obtained by combining a self-contained Reed-Solomon qudit construction (Section 3) with qudit-to-qubit conversion and multiplication-friendly embeddings from [Ngu24] (Sections 2.3 and 6.1). The paper also develops the addressable orthogonality framework (Section 4) and proves Theorem 1.2: an asymptotically good qubit code with CCZ gates on pre-designed, pairwise disjoint intra-block triples, based on an algebraic-geometry instantiation (Theorem 5.8) and a further conversion (Section 6.2). Additional results include generalizations to higher C^{ℓ-1}Z and polynomial gates, and an appendix on addressable T gates up to Clifford corrections.","tokens_in":53343,"tokens_out":20132,"duration_ms":175920,"significance":"If the proofs are completed, this is a substantial advance: it gives the first quantum code families with genuinely addressable transversal non-Clifford gates, a flexible framework that extends triorthogonality, and near-asymptotically optimal parameters for the strongest form of addressability. The Reed-Solomon construction and the qubit conversion are presented explicitly and in detail; the near-asymptotically good Theorem 1.1 is internally consistent up to imported lemmas that are standard in the area. However, the asymptotically good theorem rests on an unproved algebraic-geometry existence statement (Theorem 5.8), and Theorem 1.1 depends on the multiplication-friendly embedding lemma from [Ngu24]. Once these dependencies are made precise, the results would be a clear and useful contribution to quantum coding theory.","major_comments":[{"comment":"Theorem 5.8 is load-bearing for Theorem 1.2, but its proof is a single sentence asserting that the result follows by replacing 37 with ℓ in Theorem 3.6 of [Ngu24]. This does not verify the required parameter balance. In particular, for an algebraic-geometry code C_L(D,G) with deg G = cN and genus g = εN, the ℓ-multiplication property requires roughly c < 1/ℓ, while the dual-distance condition d^⊥ - k = Θ(N) requires c to be sufficiently above 2ε plus the puncturing rate; the paper does not state the necessary field-size condition or prove that the Garcia-Stichtenoth tower supplies the required divisor. The statement also uses k in the conclusion m-k, d-k, d^⊥-k = Θ(N) without defining k. Finally, the proof does not establish that the code contains the all-ones vector, which Remark 5.2 needs to upgrade from regular to strong ℓ-multiplication. Because Corollary 5.9, Theorem 5.10, Section 6.2, and Theorem 1.2 all depend on this result, a complete proof or a precise statement of the imported theorem with all parameter conditions is needed.","section":"Section 5.3, Theorem 5.8"},{"comment":"The conversion of the W gate to a logical CCZ gate in Theorem 6.10 requires the existence of β̂ satisfying tr(β̂ Σ_{cyc} α_a^4 α_b^2 α_c) = 1. The paper relegates this to a footnote claiming that the needed elements are \"easily seen to be obtainable,\" without a proof. This is a small but load-bearing finite-field point: if no such β̂ exists, the logical action of Π^{β̂} would not reduce to CCZ after the gauge-fixing step. Please supply a short proof, or state explicitly the condition on q and the choice of elementary basis elements α_1, α_2, α_3.","section":"Section 6.2, Eq. (190) and footnote 22"}],"minor_comments":[{"comment":"The parameter k is used in the statement of Theorem 5.8 before it is defined; please introduce it explicitly, e.g., as the dimension of the punctured code used in Lemma 5.3.","section":"Section 5.3, Theorem 5.8"},{"comment":"The scheduling of the n physical CCZ gates into a depth-4 circuit is correct but terse. The four triples associated with a given α are the four 3-subsets of a coset of the two-dimensional subspace spanned by Δ_AB and Δ_AC, and the depth-4 coloring can be described directly on these cosets. The current wording, which says there is exactly one gate with a given qudit in a given position, is easy to misread and could be clarified.","section":"Section 3.1, depth-4 argument"},{"comment":"Theorem 1.1 depends crucially on the multiplication-friendly embedding lemma imported from [Ngu24] and on the qudit-to-qubit conversion lemma cited to [WHY24]. These dependencies should be listed explicitly in the proof of Theorem 6.1, so that the reader can see exactly which ingredients are proved here and which are imported.","section":"Section 6.1.2, Lemma 6.3"},{"comment":"Several small typos and notation inconsistencies appear: in Eq. (37) a parenthesis is missing in θ(U)(|v⟩) = ψUψ^{-1}(|v⟩); in Section 4.4 the proof of Theorem 4.11 writes U^{βΛ_a}_7 with a lower-case 'a'; in Section 2.2 the phrase \"for l > 3\" uses 'l' instead of 'ℓ'. These should be corrected.","section":"Throughout"},{"comment":"The proof uses α_A to denote an evaluation point, which can be confused with the self-dual basis elements α_i introduced later in Section 6.1.1. Please choose a different symbol for the evaluation points, or state the relation explicitly.","section":"Section 5.2, Theorem 5.4"}],"recommendation":"major_revision","confidential_remarks":"The main gap is Theorem 5.8, which is asserted by a one-sentence generalization of results in [Ngu24] and [WHY24]. Since one co-author of the present manuscript is also an author of [WHY24], the editor may wish to pay particular attention to correct attribution and to the completeness of the cited generalization. My recommendation is based on the technical gap itself, not on the author overlap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result is stronger than I expected: a qudit code where any triple of logical qudits across one, two, or three blocks can be hit with CCZ via depth-one physical CCZ. I checked the phase calculation in Section 3 against the dual-RS quadrature relation, and everything holds under the degree condition 3m-3 < n. The addressable orthogonality framework is a genuine generalization of Bravyi-Haah triorthogonality, and it does real work: with Reed-Solomon codes they get addressable CCZ with the same asymptotic parameters as the global, non-addressable constructions, which is new and worth having. The qudit-to-qubit conversion is mostly cited from Ngu24's MFE machinery, but they use it carefully, and the hardcoding argument with a self-dual basis is sound.\n\nThe soft spot is exactly where the stress-test lands: Theorem 5.8, the AG-code instantiation with strong ℓ-multiplication and m,d,d⊥ all linear, is proven in one sentence as a \"simple generalization\" of Theorem 3.6 of [Ngu24]. That theorem is load-bearing for the asymptotically good Theorem 1.2 and for Corollary 5.9. The parameter-balance concern from the stress-test—needing 2ε < c < 1/ℓ with a sufficiently large fixed field—is never addressed, and the all-ones-vector condition for upgrading regular to strong multiplication is not checked. I do not think this is fatal; the generalization likely goes through with the right function-field tower, but as written it is an assertion, not a proof. The paper should either carry out the construction or cite a precise statement with proof. The beta-hat existence in Section 6.2 is also waved at in a footnote; that one is minor, but worth tightening.\n\nThe paper is also honest about its limits: it explicitly leaves open the asymptotically good version of Theorem 1.1, and the addressable T-gate appendix has no instantiation. That honesty counts in its favor.\n\nMy take: this deserves a serious referee. The RS construction is a clean contribution, and the framework is publishable even if the AG instantiation needs work. I would ask the referee to verify Theorem 5.8 before accepting, but I would not desk-reject. Worth a read if you work on transversal gates or magic state distillation.","headline":"Strong Reed-Solomon construction with a genuinely new addressable-orthogonality framework; the asymptotically good AG instantiation is asserted by analogy and needs a real proof before acceptance.","tokens_in":53941,"tokens_out":2203,"would_cite":true,"duration_ms":22668,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70","94B05","94B27"],"pacs":["03.67.Pp","03.67.Lx"],"model":"deepseek-v4-flash","headline":"This paper constructs explicit qubit CSS codes in which any three logical qubits across one, two, or three code blocks can be addressed with a logical CCZ gate implemented by a depth-one circuit of physical CCZ gates, with…","keywords":["quantum error correction","CSS codes","transversal gates","addressable logical gates","CCZ gate","triorthogonality","Reed-Solomon codes","algebraic geometry codes"],"falsifier":"Search for the claimed family of algebraic geometry codes with the strong $\\ell$-multiplication property for the needed $\\ell$ (in particular $\\ell = 7$) over a fixed field: either construct it explicitly from the cited tower of function fields and verify that $m$, $d$, and $d^\\perp$ are all $\\Theta(N)$, or exhibit an $\\ell$ and a field where the simple generalization fails, which would disprove Theorem 5.8 and with it the asymptotically good qubit code of Theorem 1.2.","tokens_in":52738,"feed_emoji":"🎯","tokens_out":11027,"duration_ms":92586,"temperature":0.7,"pith_summary":"Fault-tolerant quantum computers need error-correcting codes whose logical gates do not spread errors. Transversal gates satisfy this, but nearly all known constructions apply a logical gate to every logical qubit at once, which is far too coarse for high-rate codes that encode many qubits. This paper shows that fine-grained, addressable transversal gates are possible: it constructs an explicit family of qubit CSS codes whose rate and distance are within polylogarithmic factors of optimal, in which any triple of logical qubits across one, two, or three code blocks can be acted on by a logical CCZ gate through a depth-one circuit of physical CCZ gates. It also proves an asymptotically good version in which pre-designed, pairwise disjoint triples inside one code block are addressable. The unifying tool is a new framework, addressable orthogonality, that extends the triorthogonality framework and reduces the code construction to classical codes with multiplication properties.","feed_headline":"New qubit codes run CCZ on any chosen triple","feed_subtitle":"Near-optimal rate and distance with a depth-one physical circuit: a first for addressable non-Clifford gates.","key_machinery":"The load-bearing object is the addressable orthogonality property of a generator matrix $G$: for each logical index $A$ there is a weight vector $\\Gamma^A$ such that weighted sums of products of rows vanish except when all rows coincide at index $A$, where the sum equals a nonzero value. When such a matrix is converted into a CSS code, this condition makes a transversal physical diagonal gate induce the intended logical gate on the addressed qubits. Reed-Solomon codes supply addressable strong $\\ell$-orthogonality directly through their evaluation structure, while algebraic geometry codes supply strong $\\ell$-multiplication, which yields addressable $(\\ell-1)$-orthogonality through a general lemma. Qudit-to-qubit conversion then uses self-dual bases and multiplication-friendly embeddings, with $r = t^3$ qubits per qudit (or a duplicated version for depth-one intra-block gates), to turn the qudit CCZ gate into a depth-one circuit of qubit CCZ gates.","core_discovery":"The central discovery is an explicit qubit CSS code family with parameters $[[n, \\Omega(n/\\mathrm{polylog}(n)), \\Omega(n/\\mathrm{polylog}(n))]]_2$ that supports a transversal, addressable CCZ gate: given any three logical qubits lying in one, two, or three code blocks, a depth-one circuit of physical CCZ gates applies a logical CCZ gate to exactly those qubits. The paper further proves an asymptotically good family with parameters $[[n, \\Theta(n), \\Theta(n)]]_2$ in which the logical qubits of one block are partitioned into triples and each pre-designed triple can be addressed with a logical CCZ gate via a depth-one circuit of physical Z, CZ, and CCZ gates. These qubit codes are obtained by first building qudit codes from Reed-Solomon codes (for full addressability) and from algebraic geometry codes (for pre-designed addressability), then converting them to qubit codes while preserving addressability. The authors claim these are the first quantum codes with transversal, addressable non-Clifford gates in a strong sense, going beyond the weak single-index addressability that follows automatically from a transversal higher-level Clifford gate.","pith_inferences":["A natural next step, not claimed by the paper, is to bring the Reed-Solomon internal-structure argument to algebraic geometry codes; if that succeeds, the near-asymptotically good full-addressability result would likely become asymptotically good, resolving the paper's Open Problem 1.","The framework is strictly stronger than the folklore that a transversal gate at level $\\ell+1$ gives an addressable gate at level $\\ell$: for univariate gates it avoids a factor-of-two parameter loss, and for polynomial gates the saving is much larger, so the addressable orthogonality conditions should become the default reduction for diagonal Clifford-hierarchy gates.","The depth-one physical circuits closely match neutral-atom hardware, where three-atom CCZ interactions are native; a small demonstration of the inter-block addressable CCZ on the qubit code could be a concrete experimental testbed.","Combining the qudit-to-qubit conversion (self-dual bases plus multiplication-friendly embeddings) with homological or balanced-product LDPC constructions is a plausible route to addressable non-Clifford gates on low-weight stabilizer codes, which the current non-LDPC codes do not yet provide."],"forward_implications":["Full addressability of the CCZ gate is achievable on qubit CSS codes with dimension and distance within polylogarithmic factors of optimal, so high-rate codes need not be limited to global-only transversal non-Clifford operations.","The same construction works for $C^{\\ell-1}Z$ gates for any $\\ell$ and for other diagonal gates with $\\pm 1$ diagonal entries, making the addressability result a family of gates rather than a single gate.","The addressable orthogonality framework subsumes triorthogonality and its generalizations, giving a unified reduction: build a classical code with a multiplication property, derive an orthogonal matrix, and read off an addressable transversal gate.","An asymptotically good qubit code supports addressable CCZ on pre-designed disjoint intra-block triples via a constant-depth circuit of Z, CZ, and CCZ gates, so the asymptotically good regime is compatible with addressability in a weaker sense.","Up to Clifford corrections, the framework also yields addressable T gates through addressable triorthogonality, although the paper does not provide an instantiation."],"supporting_citations":[{"why":"introduces the triorthogonal-matrix framework and matrix-to-quantum-code construction that the addressable orthogonality framework generalizes.","marker":"[BH12]"},{"why":"supplies the qudit-to-qubit conversion method, multiplication-friendly embeddings (including the degree-3 MFE with r=t^3), and the algebraic-geometry-code family that Theorem 5.8 claims to generalize.","marker":"[Ngu24]"},{"why":"provides algebraic geometry code preliminaries, the U_{q,7} gate's Clifford-hierarchy level, and prior qudit-to-qubit conversion results.","marker":"[WHY24]"},{"why":"gives asymptotically good quantum codes with transversal non-Clifford gates and a qudit-to-qubit conversion alternative; it is the baseline the present addressability results extend.","marker":"[GG24]"},{"why":"constructs a tower of Artin-Schreier function fields attaining the Drinfeld-Vladut bound, the resource on which the asserted AG-code family rests.","marker":"[GS95]"},{"why":"supplies the theory of algebraic function fields and codes used to derive AG codes with the multiplication property and linear parameters.","marker":"[Sti09]"},{"why":"provides the puncturing and shortening results used to turn classical codes into quantum CSS codes and to bound their distances.","marker":"[HP03]"},{"why":"extends triorthogonality to prime fields, a direct predecessor whose framework the addressable orthogonality definitions subsume.","marker":"[KT19]"}],"fun_headline_variants":["Quantum codes with CCZ on any triple, near-optimal parameters","Addressable CCZ gates: first quantum codes with depth-one circuits","Any triple of logical qubits: one transversal CCZ gate","Near-optimal quantum codes with addressable non-Clifford gates","Addressable orthogonality yields CCZ gates on chosen triples"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The asymptotically good construction rests on the asserted existence of an explicit family of classical algebraic geometry codes over a fixed finite field with the strong $\\ell$-multiplication property and with dimension, distance, and dual distance all linear in the block length; the paper cites a simple generalization of a known theorem but does not carry out that construction, so if such a family does not exist the asymptotically good claim collapses.","fun_headline_variants_meta":{"raw":{"variants":["Quantum codes with CCZ on any triple, near-optimal parameters","Addressable CCZ gates: first quantum codes with depth-one circuits","Any triple of logical qubits: one transversal CCZ gate","Near-optimal quantum codes with addressable non-Clifford gates","Addressable orthogonality yields CCZ gates on chosen triples"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000292,"raw_usage":{"total_tokens":1815,"prompt_tokens":1169,"completion_tokens":646,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":785,"completion_tokens_details":{"reasoning_tokens":556}},"tokens_in":785,"tokens_out":646,"duration_ms":6547,"temperature":1.0,"reasoning_tokens":556,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T14:11:45.602124+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for the claimed family of algebraic geometry codes with the strong $\\ell$-multiplication property for the needed $\\ell$ (in particular $\\ell = 7$) over a fixed field: either construct it explicitly from the cited tower of function fields and verify that $m$, $d$, and $d^\\perp$ are all $\\Theta(N)$, or exhibit an $\\ell$ and a field where the simple generalization fails, which would disprove Theorem 5.8 and with it the asymptotically good qubit code of Theorem 1.2.","supporting_citations":[],"review_version":1}