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Asymptotically Good Quantum Codes with Addressable and Transversal Non-Clifford Gates

T0 review · 1 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A natural and promising construction whose central addressability claim rests on a fourwise orthogonality identity that the proof does not actually deliver. read the letter →

arxiv 2507.05392 v1 pith:BYOCOFEH submitted 2025-07-07 quant-ph

Zhiyang He , Vinod Vaikuntanathan , Adam Wills , Rachel Yun Zhang This is my paper
classification quant-ph MSC 81P7094B27 PACS 03.67.Pp03.67.Lx
keywords quantumerrorcorrectionCSScodesasymptoticallygoodtransversalgatesnon-CliffordCCZgatealgebraicgeometrytransitive
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $\mathrm{tr}(\gamma w_A w_B w_C)$.

What would settle it

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. $5a \le a_i$.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

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.

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 (1)
  1. [3.2, Claim 3.14 (Eqs. (64)-(66))] 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.
minor comments (5)
  1. [3.2, Claim 3.14 proof] 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.
  2. [3.1.2, Eq. (49) and Assumption 3.9] 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.
  3. [3.2, Circuit Depth] 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.
  4. [3.3, Remark 3.3] 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.
  5. [Throughout] 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′').

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the CCZ construction is built on external Stichtenoth codes, and the disputed Claim 3.14 is an unsupported proof step, not a circular one.

full rationale

The paper's central chain is not circular. The quantum code is defined from the algebraic-geometry codes C(i)_{a,b} of Stichtenoth [Sti06], and the key structural inputs—transitivity of the Galois action, the iso-orthogonality relation C(i)^\perp_{a,b} = u·C(i)_{ai-a,bi-b}, and the divisor/parameter estimates—are cited from external sources (Propositions 2.2, 2.6, 3.7, 4.7 of [Sti06]; Lemma 3.8 uses [Sti09]). The logical CCZ action in Theorems 3.12 and 3.15 is then derived by direct phase computation from the physical CCZ gates, with no fitted parameter, no target quantity redefined as an input, and no assumption of the theorem being proved. Self-citations to [HVWZ25] are used for background, for the qudit-to-qubit conversion, and for comparison, but those are prior techniques that do not assume the present result; they adapt methods from [GG24] and [Ngu24]. The manuscript does contain a serious non-circular defect that must be flagged: Claim 3.14 is asserted with an incomplete proof. The printed proof gives only the triple-product containment u·(C*C*C) ⊆ u·C_{3a,3b} ⊆ u·C_{ai-a,bi-b}, while Eq. (64) requires a fourwise orthogonality statement for f1 f2 f3 f4. That would need the fourfold containment u·(C*C*C*C) ⊆ u·C_{ai-a,bi-b}, i.e. roughly C_{4a,4b} ⊆ C_{ai-a,bi-b}, which is not established and in fact fails for a = floor(ai/4) since 4a > ai - a for large ai. Theorems 3.12 and 3.15 rely on this identity in their phase computations (Eqs. (67)-(71) and (88)-(93)), so the logical gate action is not rigorously established as written. This is a soundness/completeness gap, not a circularity: the missing step is a mathematical containment, not the re-importation of the paper's own conclusion into its assumptions.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on cited AG-code properties, a prior qudit-to-qubit conversion result, and a fourwise orthogonality premise whose proof is incomplete. No new physical entities are introduced.

assumptions (3)
  • domain assumption Stichtenoth's tower [Sti06] supplies Galois extensions Ei/E0, complete splitting of (z=1), divisor invariance of A(i) and B(i), and the iso-orthogonality relation C⊥ = u*C_{ai-a,bi-b}.
    Invoked throughout Sections 2.1 and 3.1 as black boxes; the paper does not reprove these algebraic geometry facts.
  • domain assumption The qudit-to-qubit conversion of [HVWZ25] preserves transversally addressable CCZ gates and yields asymptotically good qubit codes when q = Theta(1).
    Remark 3.5 uses this conversion to lift Theorem 3.1 to Theorem 1.1, but the transformation is not restated or proved in this paper.
  • ad hoc to paper Fourfold products of functions in L(aA+bB) lie in L((ai-a)A + (bi-b)B), as needed for Claim 3.14.
    The proof of Claim 3.14 only demonstrates the triple-product inclusion, and the stated parameters do not imply the fourfold inclusion. This is an unsupported load-bearing premise.

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Cite this review

Pith. "Pith review of Asymptotically Good Quantum Codes with Addressable and Transversal Non-Clifford Gates." pith.science (2026). https://pith.science/paper/BYOCOFEH

@misc{pith2026250705392,
  author       = {Pith},
  title        = {Pith review of: Asymptotically Good Quantum Codes with Addressable and Transversal Non-Clifford Gates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BYOCOFEH}},
  note         = {Machine review of arXiv:2507.05392}
}
abstract

Constructing quantum codes with good parameters and useful transversal gates is a central problem in quantum error correction. In this paper, we continue our work in arXiv:2502.01864 and construct the first family of asymptotically good quantum codes (over qubits) supporting transversally addressable non-Clifford gates. More precisely, given any three logical qubits across one, two, or three codeblocks, the logical $\mathsf{CCZ}$ gate can be executed on those three logical qubits via a depth-one physical circuit of $\mathsf{CCZ}$ gates. This construction is based on the transitive, iso-orthogonal algebraic geometry codes constructed by Stichtenoth (IEEE Trans. Inf. Theory, 2006). This improves upon our construction from arXiv:2502.01864, which also supports transversally addressable $\mathsf{CCZ}$ gates and has inverse-polylogarithmic rate and relative distance.

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Forward citations

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Reference graph

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