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REVIEW 3 major objections 5 minor 49 references

Quantum Tanner Codes at Moderate Blocklength

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Explicit quantum Tanner codes at 500–1000 physical qubits are reported with distance upper bounds above 20, several crossing the square-root barrier.

desk verdict A useful search catalogue and open-source tooling for moderate-blocklength quantum Tanner codes, but its headline distances are randomized upper bounds, so the 'beyond √n' claims are provisional. read the letter →

arxiv 2608.12509 v1 pith:SE6GGJX6 submitted 2026-08-12 quant-ph

Feroz Ahmed Mian , Vaishnavi L. Addala , Arman Meraj , Adhiraj Chadha , Stefan Krastanov This is my paper · ORCID
classification quant-ph PACS 03.67.Pp
keywords quantumTannercodesLDPCmoderateblocklengthleft-rightCayleycomplexcodeliftingdistanceestimationCSSpseudo-threshold
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 asks whether quantum Tanner codes—a family of sparse quantum error-correcting codes built by placing small local codes on a square complex—with competitive rate and distance exist at moderate blocklength ($n\in[500,1000]$), a gap between short explicit instances and asymptotic constructions. It answers affirmatively by searching non-abelian groups from a small-group library and constructing explicit codes through both the left-right Cayley complex description and the lifting construction. The reported instances have rates from 0.004 to 0.057, randomized distance upper bounds above 20, check weights 9 to 20, and pseudo-thresholds (the noise levels at which encoding starts to help) of 3.6–4.6% under phenomenological noise and 0.14–0.27% under circuit-level noise. If those bounds are tight, the instances provide practical sparse quantum error-correcting codes at sizes reachable by near-term hardware, with seven of eight listed codes exceeding the square-root distance line.

What carries the argument

The load-bearing mechanism is the lifted parity-check pair of Eq. (34): a seed CSS code on an $n_A\times n_B$ qubit grid is expanded by replacing each base qubit with a fiber of $|G|$ qubits, using commuting left and right regular actions of a finite group $G$ encoded in permutation matrices $L_A$ and $R_B$. The geometric picture is the left-right Cayley complex, whose defining property is that every vertex link is an $A\times B$ grid and adjacent $X$- and $Z$-generators share at most one row or column; this makes the local tensor codes $C_A\otimes C_B$ and $C_A^\perp\otimes C_B^\perp$ orthogonal where they meet, so the CSS condition holds by construction. The search reduces multisets to automorphism orbits, screens candidates by the minimum weight of a classical Tanner-code kernel basis, and uses two independent randomized distance estimators to set upper bounds.

What would settle it

Run an exact minimum-distance computation on the $[[720,6,(\le 30,\le 30)]]$ code (or, failing that, a much larger certified branch-and-bound search on the $[[480,8,(\le 21,\le 21)]]$ code). If the true $d_X$ or $d_Z$ falls below 20—or below $\sqrt{n}$ for the codes claimed to cross the barrier—the central practical claim is refuted for that instance.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is an explicit, searchable catalogue of quantum Tanner codes in the moderate-blocklength regime, obtained by lifting a small seed CSS code through commuting left and right regular actions of a finite group (equivalently, by placing local product codes on a left-right Cayley complex). The headline instances include $[[480,8,(\le 21,\le 21)]]$, $[[504,4,(\le 36,\le 27)]]$, $[[672,4,(\le 48,\le 28)]]$, $[[720,6,(\le 30,\le 30)]]$, and $[[864,8,(\le 39,\le 31)]]$; all distances are upper bounds from randomized estimation with up to 350 million trials. Seven of eight selected instances satisfy $d_{\min}>\sqrt{n}$, with the figure of merit $k d_{\min}^2/n$ exceeding the surface-code value of 1. This is presented as an affirmative answer to the open question of whether competitive QT codes exist in this regime, complementing earlier searches restricted to a limited set of groups.

Load-bearing premise

The load-bearing premise is that the randomized distance upper bounds are close to the true minimum distances; if a smaller logical operator exists that the estimator missed, the reported distances, beyond-barrier claims, and competitive thresholds would be overstated.

Editorial extensions

If this is right

  • Moderate blocklength ($n\in[500,1000]$) is no longer a dead zone for quantum Tanner codes: explicit instances with estimated distances above 20 now exist.
  • If the distance upper bounds are near the true distances, seven of the eight table entries cross the $\sqrt{n}$ line, meaning the BPT bound's figure of merit $k d^2/n$ can exceed 1 at practical sizes.
  • The pseudo-thresholds of 3.6–4.6% (phenomenological) and 0.14–0.27% (circuit-level) show that decoding performance of shorter QT codes survives at larger blocklength.
  • The classical $[8,4,4]$ code stands out as a repeatable building block for good quantum Tanner codes, giving future searches a concrete starting point.
  • The open-source library makes every reported instance reproducible and provides a platform for extending the search to larger groups or other local codes.

Reading between the lines

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

  • A natural next test is to certify the minimum distance of $[[720,6,(\le 30,\le 30)]]$ or $[[480,8,(\le 21,\le 21)]]$ with an exact or branch-and-bound method; if the certified distances stay above $\sqrt{n}$, the square-root crossing becomes a theorem about specific codes rather than an estimate.
  • Because the lifting construction permits repeated group elements in the multisets (as shown in Appendix D), the same search machinery could be run over other small non-abelian groups or with different seed local codes to push blocklength, rate, or check weight in a chosen direction.
  • The reported circuit-level pseudo-thresholds, if reproduced on hardware, would make these codes competitive candidates for early fault-tolerance demonstrations at a few hundred to a thousand qubits, where 2D-local codes are the usual default.
  • A canonical basis for the logical operators of QT codes, which the paper identifies as open, would let such instances be used for fault-tolerant gates; the search results here provide concrete codes on which to develop that basis.
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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

3 major / 5 minor

Summary. The paper reports an extensive computational search for explicit quantum Tanner (QT) codes in the moderate-blocklength regime n∈[500,1000], using both the left-right Cayley complex description and the lifting perspective of Leverrier, Rozendaal, and Zémor. The authors search a wide range of non-abelian groups from GAP's SmallGrp library, assemble a catalogue of QT code instances, and estimate their dimensions and distances with the randomized estimators DistRandCSS and sQetch. They highlight several instances whose distance upper bounds lie near or above √n, report pseudo-thresholds for four representative codes under phenomenological and circuit-level noise using the Tesseract decoder, and release QuantumExpanders.jl, an open-source Julia library. The paper positions these results as an affirmative answer to the open question of whether competitive QT codes exist at moderate blocklength.

Significance. If the reported distance estimates are close to the true minimum distances, the paper provides a valuable set of explicit QT code instances and benchmarks in a previously under-explored regime, together with reproducible open-source tooling. The absence of circularity is a genuine strength: the distance and threshold values are outputs of search and simulation, not parameters fitted to target values, and the lifting construction is prior work used as an input. The randomized cross-validation with two independent estimators and the detailed appendices of multisets and permutations are also commendable. The central quantitative claims, however, rest on randomized upper bounds rather than certified distances; this is a load-bearing assumption that affects the 'beyond the √n barrier' conclusions and the pseudo-threshold benchmarks. A revision that either certifies distances or reframes the claims as upper-bound-based would substantially strengthen the paper.

major comments (3)
  1. [§IV C, Table III] The claim of codes 'exceeding the √n barrier' is not supported by the evidence as presented. Every distance in Table III is an sQetch upper bound, and the table caption explicitly states that 'the true minimum distances, and hence the true values of kd²_min/n, may be smaller.' The BPT figure of merit kd²_min/n is computed from these upper bounds, so it is an upper-bound-based estimate rather than an established value. For the [[480,8,(≤21,≤21)]] row, the upper bound d≤21 actually lies below √n≈21.9, so this row does not even satisfy the table's own title. The section should either provide certified lower bounds on distance (or exact distances for these small codes) or be reframed as 'instances whose estimated distance upper bounds lie above √n' without claiming that the true distances exceed the barrier.
  2. [§IV B, Table II] The pseudo-threshold benchmarks are computed with r=d_z rounds for the X-basis experiment and r=d_x rounds for the Z-basis experiment, where d_x and d_z are the reported upper bounds from randomized estimation. If the true distances are smaller than the reported bounds, then the memory experiments are run for more rounds than the code can actually sustain, and the reported thresholds are conditional on the assumed distances. The manuscript does not report the exact round counts used for each code or provide uncertainty estimates for the pseudo-thresholds. Please state the round counts explicitly and, ideally, rerun the benchmarks with certified distances or discuss quantitatively how sensitive the threshold estimates are to the choice of r.
  3. [Appendix A.2 and Table V] The reproducibility appendices contain mismatches that undermine the stated goal of making every instance easy to verify. Appendix A.2 refers to 'Table E' and constructs a [[324,8,(17,14)]] code, but there is no Table E in the appendix and Table V lists a [[324,8,(≤17,≤14)]] code with different multisets A and B; the relationship between the code in the snippet and the table entry is unclear. In addition, Table V contains a malformed parameter string '[[896,16,(≤,16≤16)]]' for the C14×C2 row. These inconsistencies should be corrected so that each code instance can be reproduced from the tables alone.
minor comments (5)
  1. [Abstract and §V] The abstract consistently says 'distance upper bounds exceeding 20', but the Discussion states that the code instances 'achiev[e] distance d>20'. Since the reported values are upper bounds, the Discussion should use the same qualified language as the abstract.
  2. [§IV C, Table III caption] The caption 'Quantum Tanner codes exceeding the √n distance barrier' is inaccurate for the first row, [[480,8,(≤21,≤21)]], whose upper bound is below √n. The text acknowledges that this row 'sits on the boundary', but the title should reflect that the table contains instances whose estimated upper bounds are near or above √n.
  3. [§III B, text near Eq. (34)] The sentence introducing the generator matrices appears to contain a typographical error: 'ker G_i = C′_i^⊥' should presumably read 'ker G′_i = C′_i^⊥'. Please correct the notation.
  4. [Appendix C, code snippet] In the SL2(F4) example, the output tuple is printed as '(10, 3)' but the variables are named (dx, hz); one of them should be dz. This is a small typo but it matters for a reproducibility appendix.
  5. [Table V caption] The caption states that distances are estimated using '50 million random-ISD trials of sQetch' with '1,000 trials of DistRandCSS', whereas Table I and the main text report 50,000 trials for DistRandCSS and 50–350 million for sQetch. The appendix captions should be reconciled with the main-text trial counts.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the code construction, group search, distance estimates, and decoder benchmarks are self-contained; reported distances are explicitly randomized upper bounds, not fitted targets.

full rationale

The paper's derivation chain is not circular. It applies two external prior constructions (Leverrier–Zémor quantum Tanner codes and the Leverrier–Rozendaal–Zémor lifting perspective) to a new combinatorial search over non-abelian GAP SmallGrp groups. The code parameters n and k are computed exactly from the constructed parity-check matrices; d_X and d_Z are not solved-for outputs of a fitting procedure but are reported as upper bounds from two independent randomized estimators (sQetch and DistRandCSS), explicitly tagged with '≤' in Tables I, III, V, VIII, and with the caption of Table III itself noting that 'the true minimum distances, and hence the true values of kd²_min/n, may be smaller.' The decoder pseudo-thresholds in Table II are simulation outputs under stated phenomenological and circuit-level noise models, and using the estimated distances to set the number of memory rounds is an explicitly stated experimental convention, not an input that manufactures the threshold. The screening heuristic of Section IV A ranks candidate multisets by a classical Tanner-code proxy, but the final reported quantum code distances and decoder results are evaluated independently after that screening; this is a search heuristic, not a fitted parameter renamed as a prediction. The load-bearing references are external: the lifting construction [17], the Tesseract decoder [30], sQetch [19], and DistRandCSS [20] are all prior works by authors disjoint from the present paper, so no self-citation chain forces the conclusions. The only substantive weakness—that randomized distance estimation gives upper bounds rather than certified lower bounds—affects the strength of the practical claims about exceeding √n, but it is an estimator-tightness assumption and a correctness risk, not a circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No fitted constants enter the code construction itself; the only free choice affecting reported distances is the trial budget of the randomized estimators. The code parameters are search outputs, not fitted predictions. No new physical or mathematical entities are postulated. The main assumptions are the validity of the cited lifting construction and the reliability of the estimators and decoder simulations.

free parameters (1)
  • Randomized distance-estimation trial count = 50,000 DistRandCSS trials; 50,000,000 to 350,000,000 sQetch trials
    Chosen by computational budget. The reported upper bounds are only as tight as the search effort; more trials could lower them, so the headline distance values are conditional on this choice.
assumptions (4)
  • domain assumption The lifting construction in Eq. (34), taken from [17], produces valid CSS codes with the stated blocklengths and dimensions for arbitrary group multisets A and B.
    Used throughout Section III B and IV A. The paper does not rederive the commutation condition H_X H_Z^T = 0 beyond citing [17], and the Appendix D example is only for small codes.
  • domain assumption Randomized distance estimators sQetch and DistRandCSS return correct upper bounds on d_X and d_Z after 50,000 to 350,000,000 trials.
    All reported distances are upper bounds and are never certified by exhaustive search or a lower-bound proof. The estimators' correctness and sufficient trial counts are assumed.
  • domain assumption The Tesseract decoder and the Stim-based noise models faithfully represent memory experiments under phenomenological and circuit-level noise.
    Pseudo-thresholds in Table II rest on this simulation pipeline; no independent verification or error bars are given in the main text.
  • standard math The GAP SmallGrp and Oscar.jl group-theory routines correctly enumerate groups and automorphism orbits for the search.
    The search relies on these libraries to prune isomorphic multisets via automorphisms; this is standard computational group theory but is not independently checked here.

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Cite this review

Pith. "Pith review of Quantum Tanner Codes at Moderate Blocklength." pith.science (2026). https://pith.science/paper/SE6GGJX6

@misc{pith2026260812509,
  author       = {Pith},
  title        = {Pith review of: Quantum Tanner Codes at Moderate Blocklength},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SE6GGJX6}},
  note         = {Machine review of arXiv:2608.12509}
}
abstract

We present explicit constructions of quantum Tanner (QT) codes with good rate and distance, obtained through two complementary approaches: the left-right Cayley complex (LRCC) description and the "lifting" perspective, in which a seed Calderbank-Shor-Steane (CSS) code is lifted by commuting left-right regular actions of a finite group $\mathcal{G}$. Through an extensive search over non-abelian groups from GAP's SmallGrp library, we investigate the moderate-blocklength regime ($n \in [500,1000]$) and identify several new code instances with distance upper bounds exceeding $20$. These include $[[480,8,(\leq 21,\leq 21)]]$, $[[504,4,(\leq 36,\leq 27)]]$, $[[672,4,(\leq 48,\leq 28)]]$, $[[720,6,(\leq 30,\leq 30)]]$, and $[[864,8,(\leq 39,\leq 31)]]$, with these bounds obtained using up to $350$ million trials of sQetch, a randomized distance estimator. The code instances presented have check weights ranging from $9$ to $20$. Using the Tesseract decoder, we estimate pseudo-thresholds of $3.6\%$-$4.6\%$ under phenomenological noise and $0.14\%$-$0.27\%$ under circuit-level noise, comparable to prior results at shorter code lengths. We also provide QuantumExpanders.jl, an open-source Julia library for constructing QT codes and explicit constructions of Ramanujan graphs.

Figures

Figures reproduced from arXiv: 2608.12509 by the authors.

Figure 1
Figure 1. FIG. 1. Expander graphs must be simultaneously sparse and highly connected; the three graphs on [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Location of the non-trivial eigenvalues of Γ relative [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Tanner code construction from [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. A bipartite graph [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Three Cayley graphs, with generators shown by edge color. (a) [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The neighborhood [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The notion of a fiber in three settings. In each case a total space projects onto a base space, and the fiber over [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Logical error rate per round [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Logical error rate per round [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]

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Reference graph

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    Morgenstern’s Graph Construction We use the explicit (q+ 1)-regular Ramanujan graphs of Morgenstern [25] for even prime powersq= 2 l. Let ε∈F q be chosen such thatt 2 +t+εis irreducible over Fq, and letg(x)∈F q[x] be irreducible of even degreed. WritingFqd∼= Fq[x]/(g(x)), let ...

  41. [49]

    Starting from the Morgenstern gen- erating setB={b 0,b 1,...,b q}, Dinur [14] provides the alternative symmetric generating set A={b 0bj, bjb0|j= 1,...,q},(C7) of size 2q

    Alternating Generating Sets The QT code construction requires two symmetric gen- erating setsAandBsatisfying the total non-conjugacy (TNC) condition. Starting from the Morgenstern gen- erating setB={b 0,b 1,...,b q}, Dinur [14] provides the alternative symmetric generating set...

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Reviewed August 16, 2026 · model on record in the stance chip above.