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REVIEW 5 major objections 5 minor 35 references

Reinforcement Learning Enhanced Greedy Decoding for Quantum Stabilizer Codes over $\mathbb{F}_q$

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A Hermitian curve over F_3 yields a [[27,13,4]]_3 qutrit stabilizer code, and an RL-on-Greedy decoder cuts logical failures from about 8 percent to about 0.5 percent.

desk verdict The claimed [[27,13,4]]_3 code is unsupported: Section 3 builds on the wrong curve and an inconsistent divisor, so the decoder benchmark is unanchored. read the letter →

arxiv 2506.03397 v1 pith:LBBPWZI4 submitted 2025-06-03 quant-ph math.QA

Vahid Nourozi This is my paper
classification quant-phmath.QA MSC 94B2781P70
keywords GoppacodesquantumstabilizerHermitiancurvesseparatedpolynomialsreinforcementlearningdecodingdeepQ-networkqutritcodealgebraicgeometry
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 tries to establish two claims. The first is that plane curves given by separated polynomials, in particular $y^q + y = x^m$ with $q-m=1$, produce classical Goppa codes that are Hermitian self-orthogonal in a suitable range, and that for $q=3$ the Hermitian curve $y^3+y=x^4$ yields a ternary code of length 27, dimension 13, distance 4 and hence a $[[27,13,4]]_3$ qutrit stabilizer code. The second is that a two-stage decoder, first a greedy syndrome-reduction pass and then a deep Q-network trained on residual syndromes, dramatically lowers logical failure, from about 8 percent to about 0.5 percent at physical error rate $p=0.01$, and approaches the performance of an optimal decoder for weight-two errors. A sympathetic reader would care because this points to a practical way to protect quantum information in three-level systems with both a new code family and a learned decoder that fixes the greedy decoder's weaknesses.

What carries the argument

The machinery is the one-point algebraic-geometry code on a separated-polynomial curve, together with the monomial basis claimed in Lemma 3.1, $\{x^i y^j : i\ge 0,\ 0\le j\le q-1,\ iq+j(q-1)\le r\}$, and the Hermitian self-orthogonality condition $2r\le q^3+q^2-3q$ that lets the nonbinary stabilizer-code lemma convert $C_r$ into a quantum code. The decoding machinery is the two-stage RL-on-Greedy algorithm: greedily flip qudits to reduce syndrome weight, then let a deep Q-network act on whatever residual syndrome remains, with actions and syndrome values taken in $\mathbb{F}_q$.

What would settle it

For $q=3$ and $r=7$, count the monomials in Lemma 3.1 under its own inequality: $i\cdot3+j\cdot2\le 7$ has 7 solutions, while the Riemann-Roch dimension of $L(49P_\infty)$ on $y^3+y=x^4$ is 47; recomputing the code dimension directly from $C_L(D,G)$, or checking whether the function $t$ with divisor $G-r(q^2-q+1)P_\infty$ exists, would settle which side is right.

Watch

Extended reading notes

Core claim

The central discovery on the code side is a parameterized family: take $X:y^q+y=x^m$ with $q-m=1$, let $D$ be the sum of $\mathbb{F}_{q^2}$-rational points outside a chosen set $G$, and set $G=r(q^2-q+1)P_\infty$; the paper claims the code $C_r=C_L(D,G)$ is monomially equivalent to a one-point code, is Hermitian self-orthogonal for $r\le q^2+q-3$, and converts through the nonbinary stabilizer-code lemma into a $q$-ary stabilizer code with parameters $[[q^3,\,q^3+q^2-3q-2r,\,r+2q-q^2]]$. For $q=3$, $r=7$, this gives $[[27,13,4]]_3$; for $q=5$ it gives a family of $5$-ary codes. The decoder-side discovery is the RL-on-Greedy algorithm: a standard greedy syndrome decoder first reduces the syndrome, and a trained deep Q-network then cleans up the residual syndrome, with simulations showing near-optimal correction on the $[[27,13,4]]_3$ code.

Load-bearing premise

Everything hinges on Lemma 3.1: the dimension of the code is taken from a count of monomials $x^i y^j$ with $i q + j(q-1) \le r$, as if the divisor were $rP_\infty$ rather than $r(q^2-q+1)P_\infty$, and the argument also assumes an auxiliary rational function $t$ exists with divisor $G - r(q^2-q+1)P_\infty$; if either is wrong, the stated dimension, distance, and quantum code parameters collapse.

Editorial extensions

If this is right

  • If the construction is right, the $q=3$ case gives three qutrit codes: $[[27,13,4]]_3$, $[[27,11,5]]_3$, and $[[27,9,6]]_3$ for $r=7,8,9$.
  • For $q=5$, the same theorem produces 5-ary codes from $[[125,99,3]]_5$ up to $[[125,91,7]]_5$.
  • Every code in this family with $2r\le q^3+q^2-3q$ is Hermitian self-orthogonal, so the nonbinary stabilizer-code lemma applies uniformly to turn the whole family into quantum stabilizer codes.
  • The RL-on-Greedy decoder is generic: it only needs the parity-check matrix, so it can be trained for any $\mathbb{F}_q$ stabilizer code, including nonbinary codes.
  • On the $[[27,13,4]]_3$ code, the learned decoder corrects essentially every weight-$\le 2$ error pattern that is correctable in principle, so its logical failure rate is close to the code's true capability.

Reading between the lines

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

  • If the parameters survive a direct check, $[[27,13,4]]_3$ has a high encoding rate (13 logical qutrits in 27 physical qutrits) for distance 4; a fair judgment of that rate needs a systematic table of known qutrit codes, which the paper does not include.
  • The reported near-optimal performance is inferred from correcting weight-$\le 2$ patterns; a direct count of which weight-2 patterns still fail would quantify the gap to an exact maximum-likelihood decoder.
  • The decoder is trained per code and noise model, so the 8%-to-0.5% drop should not be extrapolated to other codes or noise channels without retraining; testing on $[[27,11,5]]_3$ and $[[27,9,6]]_3$ would show whether the advantage persists as distance increases.
  • The RL stage's one-or-two-step corrections suggest the residual syndromes after greedy are highly structured; examining their distribution could suggest cheaper deterministic rules that capture the same benefit without training.
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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

5 major / 5 minor

Summary. The paper proposes a construction of classical Goppa codes and quantum stabilizer codes from plane curves defined by separated polynomials, focusing on curves of the form y^q + y = x^m with |m-q|=1. It claims a ternary code of length 27, dimension 13, distance 4 from the Hermitian curve y^3 + y = x^4, yielding a [[27,13,4]]_3 quantum code. The authors also introduce an RL-on-Greedy decoder that applies a greedy syndrome decoder followed by a deep Q-network, and report a reduction in logical failure rate from roughly 8% to 0.5% at p=0.01 on this code. The paper argues that the decoder approaches optimal performance for the claimed code.

Significance. If the code parameters were valid, the [[27,13,4]]_3 code would be a useful addition to qutrit stabilizer codes, and the RL-on-Greedy decoder would be a practical and generic enhancement for nonbinary decoding. The public availability of the implementation code is a strength for reproducibility. However, the central mathematical construction contains several load-bearing errors that invalidate the claimed code parameters, and the decoder's near-optimality claim is not benchmarked against any independent optimal decoder. As a result, the significance as stated is not realized by the manuscript in its current form.

major comments (5)
  1. [Section 3.1, Lemma 3.1 and Proposition 3.3] Lemma 3.1 claims a basis for L(G) with G = r(q^2-q+1)P_∞, but the monomial counting condition iq+j(q-1) ≤ r is the condition for L(rP_∞), not for L(r(q^2-q+1)P_∞); the divisor degree has been dropped by a factor of q^2-q+1. Independently, the pole orders (x)_∞ = qP_∞ and (y)_∞ = (q-1)P_∞ correspond to the curve y^q + y = x^{q-1}, not to the Hermitian curve y^q + y = x^{q+1} used in the abstract and in Section 5.4. For q=3, the paper's general condition q-m=1 gives m=2, while the example uses m=4. These two issues invalidate the dimension count in Proposition 3.3(3) and the parameter formulas in Theorem 4.2 for the claimed Hermitian-based code.
  2. [Section 3, definition of G and deg(G)] The divisor G is defined as G = sum_{P in X(F_q)} rP, and the paper claims deg(G) = r(q^2-q+1). This requires |X(F_q)| = q^2-q+1, but no evidence is given for this count. For q=3, the Hermitian curve y^3+y=x^4 has at most four F_3-rational points (three affine plus possibly one at infinity), not seven, so the claimed degree 7r is false on that curve. Since deg(G) directly determines the designed distance d* = n - deg(G) and the Riemann-Roch dimension k = deg(G)-g+1, the dimension and distance calculations in Section 3 and Example 4.3 are unsupported.
  3. [Section 3, Proposition 3.4 and Lemma 3.2] Proposition 3.4 asserts that G - (q^2-q+1)P_∞ is the divisor of a rational function t, which would make C(D,G) monomially equivalent to a one-point code, and Lemma 3.2 then derives the dual code formula. No such rational function is exhibited, and the existence of a principal divisor supported on the finite point set X(F_q) with the specified degree is not a general fact; for q=3 it would require a degree-7 principal divisor supported on the claimed seven points, which cannot exist because X(F_3) does not have seven points on either candidate curve. The monomial equivalence and the dual-distance formula are therefore not justified.
  4. [Section 5.4, decoder near-optimality claim] The claim that RL-on-Greedy 'approaches the performance of an optimal decoder' is not supported by the presented evidence. The paper only compares against a pure greedy decoder; it does not simulate a maximum-likelihood decoder, compute a quantum Hamming bound or other optimality bound, or report statistical uncertainty (error bars, number of trials, confidence intervals) for the failure rates. The agent is trained and evaluated on the same code and noise model, so the reported improvement is self-referential and does not establish near-optimal performance.
  5. [Section 5.4, dependence on unsupported code] The experimental section evaluates the decoder on a [[27,13,4]]_3 code whose existence depends on the classical code C_r from Theorem 4.2. As argued above, the dimension and distance calculations for C_r are invalid due to the incorrect pole orders, divisor degree, and unproved monomial equivalence. The simulation therefore does not demonstrate the decoder's value on a code that the paper has actually constructed; the experimental results are uninterpretable as a validation of the proposed construction.
minor comments (5)
  1. [Section 5.3] The text contains a dangling reference 'Eq., ??' after 'state transition'; this should be fixed to a proper equation number or removed.
  2. [Acknowledgments] The acknowledgment refers to 'the first author' and to support from TWAS/CNPq, but the paper lists only one author; this should be clarified.
  3. [Introduction and References] References [5,6,7] concern power-systems optimization and are not connected to the coding-theoretic content of the paper; they appear out of place in the introduction.
  4. [Example 4.4] For q=5, r=21 the paper correctly computes [[125,93,6]]_5, but then states that '[[125, 93, 12]]5 appears in tables of best-known codes [32]', which is inconsistent with the just-derived distance value of 6; the quoted table entry or the distance value needs correction.
  5. [Section 2.2 and Section 3] The symbol G is used both for the point set X(F_q) and for the divisor sum over that set; this notation is confusing and should be clarified, e.g., by using different symbols for the set and the divisor.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the claimed quantum-code parameters rest on standard AG-code machinery, and the RL decoder result is an empirical evaluation rather than a prediction forced by construction.

full rationale

The derivation chain is not circular. The code construction applies standard algebraic-geometry code definitions and a standard nonbinary stabilizer-code conversion (Lemma 4.1, attributed to Ashikhmin–Knill). The dimension and self-orthogonality claims are meant to follow from Riemann–Roch and standard one-point-code formulas, with proofs citing external references ([28,29]) rather than the authors' own results. The self-references that appear are background citations to the authors' prior separated-polynomial papers and are not load-bearing for the new theorem. No equation is equivalent to its input by definition. The paper's serious defects are mathematical correctness gaps, not circularity: the text fixes q−m=1 while the abstract's Hermitian example y^3+y=x^4 has m=q+1; Lemma 3.1 counts monomials against iq+j(q−1)≤r while claiming to describe L(r(q^2−q+1)P∞); and the asserted degree deg(G)=r(q^2−q+1) is unsupported and inconsistent with the actual number of F_3-points on the stated curve. Similarly, the RL-on-Greedy 'near-optimal performance' claim is not a derived prediction: the agent is trained and evaluated on the same code and noise model, so the simulation is self-referential as validation, but no fitted parameter is renamed as a predicted quantity in the mathematical chain. Therefore no circular step is exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the Hermitian curve identification, the point-count and divisor equivalence assumptions, and an asserted dual-distance bound. Each of these is either unproved or internally inconsistent. The decoder claim adds a further set of unspecified training parameters. No new physical entities are introduced.

free parameters (2)
  • Integer divisor coefficient r = 7, 8, 9 for q=3; 18 to 22 for q=5
    The code dimension and distance are controlled by choosing r. The q=5 example uses r-values outside the range stated in Theorem 4.2, so the parameter selection is not disciplined by the stated theorem.
  • RL training hyperparameters = Not stated
    Network architecture, learning rate, replay buffer size, reward penalties, episode count, and trial count are not given. The reported failure rates depend on these choices.
assumptions (4)
  • ad hoc to paper The Weierstrass semigroup at P_infty for the curve y^q + y = x^m with |m-q|=1 is generated by q and q-1, so L(rP_infty) has dimension counted by #{i q + j(q-1) <= r}.
    Used in Lemma 3.1. For the Hermitian curve y^q + y = x^{q+1}, the pole orders are q and q+1, not q and q-1. No proof or reference is given for the claimed semigroup.
  • ad hoc to paper There exists a rational function t on X with divisor G - (q^2-q+1)P_infty, so C(D,G) is monomially equivalent to a one-point code.
    Proposition 3.4 asserts this equivalence without constructing t or proving that the chosen point set G has the required linear equivalence. The preceding definition G=X(F_q) has q+1 points, not q^2-q+1 points.
  • ad hoc to paper The dual minimum distance of C_r satisfies d_perp >= r + 2q - q^2 in the stated r-range.
    Theorem 4.2 invokes this bound without proof. It is not derived from the dual AG code formula and is not shown to hold for the actual divisor choices.
  • domain assumption The curve has q^3 affine F_{q^2}-rational points and the set G has q^2-q+1 points.
    These counts justify n=q^3 and deg(G)=r(q^2-q+1). They are not proved in the paper and are inconsistent with the earlier definition G=X(F_q), which would give q+1 points.

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Cite this review

Pith. "Pith review of Reinforcement Learning Enhanced Greedy Decoding for Quantum Stabilizer Codes over $\mathbb{F}_q$." pith.science (2026). https://pith.science/paper/LBBPWZI4

@misc{pith2026250603397,
  author       = {Pith},
  title        = {Pith review of: Reinforcement Learning Enhanced Greedy Decoding for Quantum Stabilizer Codes over $\mathbbF_q$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LBBPWZI4}},
  note         = {Machine review of arXiv:2506.03397}
}
abstract

We construct new classical Goppa codes and corresponding quantum stabilizer codes from plane curves defined by separated polynomials. In particular, over $\mathbb{F}_3$ with the Hermitian curve $y^3 + y = x^4$, we obtain a ternary code of length 27, dimension 13, distance 4, which yields a [[27, 13, 4]]$_3$ quantum code. To decode, we introduce an RL-on-Greedy algorithm: first apply a standard greedy syndrome decoder, then use a trained Deep Q-Network to correct any residual syndrome. Simulation under a depolarizing noise model shows that RL-on-Greedy dramatically reduces logical failure compared to greedy alone. Our work thus broadens the class of Goppa- and quantum-stabilizer codes from separated-polynomial curves and delivers a learned decoder with near-optimal performance.

Figures

Figures reproduced from arXiv: 2506.03397 by the authors.

Figure 1
Figure 1. Decoding failure rate vs. physical error probability for the [[27 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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Reference graph

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