Construction and Decoding of Quantum Margulis Codes

Bane Vasic; Dimitris Chytas; Michele Pacenti

arxiv: 2503.03936 · v3 · submitted 2025-03-05 · 🪐 quant-ph · cs.IT· math.IT

Construction and Decoding of Quantum Margulis Codes

Michele Pacenti , Dimitris Chytas , Bane Vasic This is my paper

Pith reviewed 2026-05-23 00:43 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT

keywords quantum LDPC codesMargulis codesmin-sum decodingerror degeneracytwo-block group algebraTanner graphbivariate bicycle codesgirth control

0 comments

The pith

Quantum Margulis codes from classical Margulis constructions admit effective decoding by the standard min-sum algorithm with linear complexity under the code capacity model.

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

The paper introduces quantum Margulis codes as a new family of quantum LDPC codes obtained from Margulis' classical LDPC construction through the two-block group algebra framework. It shows that these codes decode successfully with the ordinary min-sum decoder, which runs in linear time, whereas bivariate bicycle codes typically need the heavier ordered-statistics decoder to reach useful performance. The authors link the difference to the Tanner graph structure, which lacks the group symmetry found in other constructions and thereby reduces the error-degeneracy problem that degrades many QLDPC decoders. They supply an explicit algorithm that builds 2BGA codes with girth at least 6 or 8 and use it to produce examples of length 240 and 642; simulations then confirm that the new codes exhibit lower error floors than BB codes when both are decoded with min-sum.

Core claim

Quantum Margulis codes, obtained from Margulis' classical LDPC codes via the two-block group algebra framework, can be decoded by the standard min-sum algorithm with linear complexity under the code-capacity noise model. Their Tanner graphs lack group symmetry, which mitigates the error-degeneracy obstacle that forces more elaborate decoders on other QLDPC families such as bivariate bicycle codes. An auxiliary construction algorithm enforces a minimum girth of 6 or 8, and the resulting length-240 and length-642 codes display improved error-floor behavior relative to BB codes under identical min-sum decoding.

What carries the argument

The Tanner graph of a quantum Margulis code, which lacks the group symmetry present in bivariate bicycle constructions and thereby reduces error degeneracy for min-sum decoding.

If this is right

Quantum Margulis codes of length 240 and 642 can be built with guaranteed girth 6 or 8 using the controlled-girth 2BGA algorithm.
Min-sum decoding alone produces lower error floors for these codes than for bivariate bicycle codes of comparable length.
The codes require only linear-complexity decoding and avoid the need for ordered-statistics post-processing under the code-capacity model.
The same construction framework yields multiple distinct quantum Margulis codes whose performance is validated by numerical simulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If symmetry removal is the decisive factor, other QLDPC families might be improved by deliberate symmetry-breaking modifications to their graphs.
Hardware implementations could become simpler because the decoder no longer needs the extra circuitry required by ordered-statistics methods.
Extending the same codes to circuit-level noise models would test whether the observed advantage survives beyond the code-capacity assumption.
The girth-control algorithm might be reusable for other two-block group algebra constructions beyond Margulis base graphs.

Load-bearing premise

The absence of group symmetry in the Tanner graph is what primarily reduces error degeneracy and enables effective min-sum decoding.

What would settle it

A direct comparison in which a quantum Margulis code is modified to add group symmetry and then shown to require ordered-statistics decoding for the same performance level as the unmodified code.

Figures

Figures reproduced from arXiv: 2503.03936 by Bane Vasic, Dimitris Chytas, Michele Pacenti.

**Figure 1.** Figure 1: (a) A square associated with the chain complex X. (b) A square associated with the co-chain complex X∗ . co-chain complex X∗ : F G00 2 ∂ 1 ←− F G01 2 ⊕ F G10 2 ∂ 2 ←− F G11 2 , (7) such that ∂ 1 = [AT , BT ] T and ∂ 2 = [BT , AT ]. Hence, the parity check matrices of the code will have the form ( HX = ∂2 = [A B] HZ = ∂ 2 = [BT AT ] (8) Note that AT , BT correspond to the biadjacency matrices of the double… view at source ↗

**Figure 2.** Figure 2: 4-cycle in the expanded neighborhood of a check with dc = 4. Red, square nodes indicate checks, while circles indicate variable nodes. Black and gray variable nodes indicate qubits belonging to the two blocks A and B [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: 6-cycle in the expanded neighborhood of a check with dc = 4. the graph are directed, and we highlight in red the edges that form a cycle. 6 Decoding of quantum Margulis code In this Section we analyze the performance of quantum Margulis codes under iterative decoding. In particular, we show that a simple nMS decoder with flooding schedule is able to approach the BPOSD performance. First, let us recall … view at source ↗

**Figure 4.** Figure 4: A symmetric stabilizer for a dv = 3 2BGA code. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Depth-3 neighborhood of a check node in the symmetric stabilizer in the J72, 12, 6K BB code. Due to the graph’s symmetry, every check node in the graph has this neighborhood. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Evolution of the weight of the estimated error in the MS decoder. node vi can be calculated as Ei = log    X 1 u=0 Y dv j=1 mj→i(u)    . (13) Conversely, the free-entropy Ej corresponding to each check node cj can be calculated as Ej = log    X i1+...+idc=0 Y dc j=1 min→j (in)    . (14) Finally, the free-entropy on an edge between vi and cj Eij can be calculated as Eij = log (X 1 u=0 mj→i(u)mi… view at source ↗

**Figure 7.** Figure 7: Depth-3 neighborhood of three different check node [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

**Figure 8.** Figure 8: Trajectory of the Bethe free-entropy function (16) of the nMS decoder while decoding a symmetric stabilizer on a BB code and a quantum Margulis code. entropy (which is correlated with W) before being able to converge. We validate our observation through simulations, illustrated in [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗

**Figure 9.** Figure 9: Performance of MS decoding compared with BPOSD0 for a length 240 quantum Margulis code, for different amounts of maximum decoder iterations. different dimension3 . For each code, we simulate nMS decoding under a simple depolarizing noise model on data qubits (indicated with the parameter ǫ), and perfect syndrome measurements. The nMS decoder utilizes a flooding (parallel) schedule, and a normalization … view at source ↗

**Figure 11.** Figure 11: Simulation results for several codes sampled from the output of Algorithm 1. formance, approaching a logical error rate of approximately 2 · 10−6 for ǫ = 0.02, while the best is the one with k = 2, able to approach a logical error rate of 3 · 10−8 for ǫ = 0.03. It is worth noticing the code with k = 10, that approaches a logical error rate of around 7 · 10−8 for ǫ = 0.03, which is remarkable given its hig… view at source ↗

**Figure 12.** Figure 12: Performance comparison of a BB code and a quantum Margulis code under circuit-level noise. Each measurement is repeated four times, and decoding is performed using MSOSD10 on the circuit-level Tanner graph. ldpc Python package [24]. Each value is obtained after simulating 106 error samples. From the Figure, it is evident how, in this setting, the quantum Margulis code has slightly worse performance tha… view at source ↗

read the original abstract

Quantum low-density parity-check codes are a promising approach to fault-tolerant quantum computation, offering potential advantages in rate and decoding efficiency. In this work, we introduce quantum Margulis codes, a new class of QLDPC codes derived from Margulis' classical LDPC construction via the two-block group algebra framework. We show that quantum Margulis codes, unlike bivariate bicycle codes which require ordered statistics decoding for effective error correction, can be efficiently decoded using a standard min-sum decoder with linear complexity, when decoded under the code capacity noise model. This is attributed to their Tanner graph structure, which does not exhibit group symmetry, thereby mitigating the well-known problem of error degeneracy in QLDPC decoding. To further enhance performance, we propose an algorithm for constructing 2BGA codes with controlled girth, ensuring a minimum girth of 6 or 8, and use it to generate several quantum Margulis codes of length 240 and 642. We validate our approach through numerical simulations, demonstrating that quantum Margulis codes behave significantly better than BB codes in the error floor region, under min-sum decoding.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

New Margulis-based QLDPC family with girth control that decodes under min-sum, but the symmetry explanation for why it works is not isolated from other factors.

read the letter

The paper defines quantum Margulis codes by lifting the classical Margulis construction into the two-block group algebra framework and adds an explicit algorithm to enforce girth 6 or 8. It reports that the resulting length-240 and length-642 codes decode with ordinary min-sum under code-capacity noise and avoid the error-floor problems seen in bivariate bicycle codes with the same decoder. That combination of construction plus decoder behavior is the concrete new piece relative to the BB literature cited in the abstract. The girth-control procedure itself is described clearly enough to be reproducible from the text. The central claim is that the absence of group symmetry in the Tanner graph is what reduces error degeneracy enough for min-sum to succeed. The abstract states this directly, but the provided material gives no controlled comparison that holds girth, degree, and cycle distribution fixed while toggling only the symmetry property. Without that isolation, the performance difference could come from the girth algorithm or other 2BGA choices rather than symmetry alone. Numerical results are invoked but no tables, error bars, or decoder parameters appear in the abstract, so the size of the claimed gain cannot be checked yet. This work is aimed at researchers building or decoding QLDPC codes who need concrete constructions that run with standard message-passing. It is worth sending to referees because the construction is new and the decoding claim is testable; the symmetry story can be tightened or replaced during review without discarding the codes themselves.

Referee Report

2 major / 1 minor

Summary. The paper introduces quantum Margulis codes as a new class of QLDPC codes obtained from Margulis' classical construction via the two-block group algebra (2BGA) framework. It claims these codes admit efficient decoding by a standard min-sum decoder of linear complexity under the code-capacity noise model (in contrast to bivariate bicycle codes, which require ordered-statistics decoding), attributes this to the absence of group symmetry in their Tanner graphs (thereby reducing error degeneracy), presents an algorithm for constructing 2BGA codes with controlled girth (minimum 6 or 8), generates explicit length-240 and length-642 examples, and reports numerical simulations showing improved error-floor behavior relative to BB codes under min-sum decoding.

Significance. If the performance claims and causal attribution are substantiated, the work would supply a concrete family of QLDPC codes that are practically decodable without OSD, together with an explicit girth-control construction; both would be useful additions to the QLDPC literature. The controlled-girth algorithm itself is a clear constructive contribution.

major comments (2)

[Abstract] Abstract (and the numerical-validation paragraph): the central claim that quantum Margulis codes 'can be efficiently decoded using a standard min-sum decoder' and that this is 'attributed to their Tanner graph structure, which does not exhibit group symmetry' is supported only by invocation of 'numerical simulations' with no methods section, no decoder parameters, no data tables or figures, and no error-bar statistics supplied. This renders the decoding-efficiency assertion unverifiable.
[Abstract] Abstract (performance comparison): the assertion that the performance difference versus BB codes stems from the absence of group symmetry (rather than from the explicit girth-control construction, the chosen degrees, or other 2BGA parameters) is not isolated by any controlled experiment that holds girth, variable-node degree, and cycle structure fixed while toggling only the symmetry property. Without such isolation the causal link remains an untested hypothesis.

minor comments (1)

[Abstract] The abstract states that the codes 'behave significantly better than BB codes in the error floor region' but supplies no quantitative thresholds or block-length scaling to make the comparison precise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive comments on our manuscript. We will revise the paper to address the concerns raised regarding the verifiability of our claims and the attribution of performance improvements.

read point-by-point responses

Referee: [Abstract] Abstract (and the numerical-validation paragraph): the central claim that quantum Margulis codes 'can be efficiently decoded using a standard min-sum decoder' and that this is 'attributed to their Tanner graph structure, which does not exhibit group symmetry' is supported only by invocation of 'numerical simulations' with no methods section, no decoder parameters, no data tables or figures, and no error-bar statistics supplied. This renders the decoding-efficiency assertion unverifiable.

Authors: The full manuscript includes Section 4 on the min-sum decoding algorithm and Section 5 on numerical simulations, which detail the decoder implementation (including iteration count, damping factor, and stopping criteria), the code-capacity depolarizing noise model, and present performance curves with statistical error bars in Figures 3-5. However, we agree that the abstract does not sufficiently reference these details. In the revised manuscript, we will update the abstract to include a concise mention of the simulation setup and explicitly direct readers to the relevant sections for methods and data. This revision will make the efficiency claim verifiable directly from the abstract. revision: yes
Referee: [Abstract] Abstract (performance comparison): the assertion that the performance difference versus BB codes stems from the absence of group symmetry (rather than from the explicit girth-control construction, the chosen degrees, or other 2BGA parameters) is not isolated by any controlled experiment that holds girth, variable-node degree, and cycle structure fixed while toggling only the symmetry property. Without such isolation the causal link remains an untested hypothesis.

Authors: We recognize that isolating the effect of group symmetry alone would require additional constructions that match all other parameters except symmetry, which is not straightforward given the distinct algebraic origins of the codes. Our attribution is based on the fundamental difference in Tanner graph symmetry arising from the Margulis construction versus the cyclic group structure in BB codes, and the observed reduction in error degeneracy aligns with theoretical expectations in the literature. In the revision, we will expand the discussion in Section 6 to clarify the role of symmetry and note that while girth control contributes to performance, the lack of symmetry is the key differentiator enabling effective min-sum decoding without OSD. We will also qualify the language in the abstract to present it as a plausible explanation supported by the structural analysis rather than a definitively isolated cause. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives quantum Margulis codes from the established Margulis classical LDPC construction via the two-block group algebra (2BGA) framework, then reports simulation results for min-sum decoding performance on explicitly constructed length-240/642 codes with controlled girth. The attribution of effective decoding to absent group symmetry is an interpretive claim supported by the generated Tanner graphs and numerical outcomes rather than any quantity defined by construction, fitted parameter, or self-citation chain. No equation or central result reduces to its own inputs; the performance comparison is external simulation output, not a renaming or self-referential prediction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only view yields minimal ledger entries; the two-block group algebra framework is invoked as the derivation route but no explicit free parameters or new entities are named.

axioms (1)

domain assumption The two-block group algebra framework can be applied to the classical Margulis construction to produce valid quantum codes with the stated Tanner-graph properties.
Invoked in the abstract as the method that yields quantum Margulis codes.

pith-pipeline@v0.9.0 · 5719 in / 1318 out tokens · 48248 ms · 2026-05-23T00:43:23.256458+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

quantum Margulis codes ... Tanner graph structure, which does not exhibit group symmetry, thereby mitigating the well-known problem of error degeneracy in QLDPC decoding
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

girth of at least six and eight ... nMS decoder

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Simplified circuit-level decoding using Knill error correction
quant-ph 2026-03 accept novelty 7.0

Knill error correction reduces circuit-level decoding for quantum LDPC codes to the simpler code-capacity decoder while remaining fault-tolerant under locally decaying noise.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · cited by 1 Pith paper

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[1] [1]

Asymptotically Good Quantum and Lo- cally Testable Classical LDPC Codes

Pavel Panteleev and Gleb Kalachev. Asymptotically Good Quantum and Lo- cally Testable Classical LDPC Codes. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Com- puting, page 375–388, 2022

work page 2022

[2] [2]

Quantum Tanner Codes

Anthony Leverrier and Gilles Zémor. Quantum Tanner Codes. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) , pages 872– 883, 2022

work page 2022

[3] [3]

Good Quan- tum LDPC Codes with Linear Time De- coders

Irit Dinur, Min-Hsiu Hsieh, Ting-Chun Lin, and Thomas Vidick. Good Quan- tum LDPC Codes with Linear Time De- coders. In Proceedings of the 55th Annual ACM Symposium on Theory of Comput- ing, pages 905–918, June 2023. 13

work page 2023

[4] [4]

High-Threshold and Low-Overhead Fault-Tolerant Quan- tum Memory

Sergey Bravyi, Andrew W Cross, Jay M Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J Yoder. High-Threshold and Low-Overhead Fault-Tolerant Quan- tum Memory. Nature, 627(8005):778–782, 2024

work page 2024

[5] [5]

Renyu Wang and Leonid P. Pryadko. Distance Bounds for Generalized Bicycle Codes. Symmetry, 14(7), 2022

work page 2022

[6] [6]

Hsiang-Ku Lin and Leonid P. Pryadko. Quantum Two-Block Group Algebra Codes. Physical Review A , 109:022407, Feb 2024

work page 2024

[7] [7]

Explicit Construc- tions of Graphs without Short Cycles and Low Density Codes

Grigorii A Margulis. Explicit Construc- tions of Graphs without Short Cycles and Low Density Codes. Combinatorica, 2(1):71–78, 1982

work page 1982

[8] [8]

Malcolm, Andrew N

Alexander J. Malcolm, Andrew N. Glaudell, Patricio Fuentes, Daryus Chan- dra, Alexis Schotte, Colby DeLisle, Rafael Haenel, Amir Ebrahimi, Joschka Roﬀe, Armanda O. Quintavalle, Ste- fanie J. Beale, Nicholas R. Lee-Hone, and Stephanie Simmons. Computing Eﬃciently in QLDPC Codes, 2025. URL: https://arxiv.org/abs/2502.07150

work page arXiv 2025

[9] [9]

Quantum Margulis Codes

Michele Pacenti and Bane Vasić. Quantum Margulis Codes. In 60th Annual Allerton Conference on Communication, Control, and Computing , Sept. 25-27 2024

work page 2024

[10] [10]

M. Pacenti. Quantum Margulis Codes, GitHub repository. A valiable here

work page

[11] [11]

A. R. Calderbank and Peter W. Shor. Good Quantum Error-Correcting Codes Exist. Physical Review A , 54(2):1098– 1105, August 1996

work page 1996

[12] [12]

Bohdanowicz, Aleksander Kubica, Steven T

Oscar Higgott, Thomas C. Bohdanowicz, Aleksander Kubica, Steven T. Flammia, and Earl T. Campbell. Improved Decoding of Circuit Noise and Fragile Boundaries of Tailored Surface Codes. Physical Review X, 13(3):031007, July 2023

work page 2023

[13] [13]

Anqi Gong, Sebastian Cammerer, and Joseph M. Renes. Toward Low-latency It- erative Decoding of QLDPC Codes Un- der Circuit-Level Noise, 2024. URL: https://arxiv.org/abs/2403.18901

work page arXiv 2024

[14] [14]

Enhanced Min- Sum Decoding of Quantum Codes Us- ing Previous Iteration Dynamics

Dimitris Chytas, Nithin Raveendran, and Bane Vasić. Enhanced Min- Sum Decoding of Quantum Codes Us- ing Previous Iteration Dynamics. URL: https://arxiv.org/abs/2501.05021

work page arXiv

[15] [15]

Flanagan, and Bane Vasić

Dimitris Chytas, Michele Pacenti, Nithin Raveendran, Mark F. Flanagan, and Bane Vasić. Enhanced Message-Passing Decod- ing of Degenerate Quantum Codes Utiliz- ing Trapping Set Dynamics. IEEE Com- munications Letters, pages 1–1, 2024

work page 2024

[16] [16]

Collective bit flipping-based decoding of quantum ldpc codes,

Dimitris Chytas, Nithin Raveendran, and Bane Vasić. Collective Bit Flipping-Based Decoding of Quan- tum LDPC Codes, 2024. URL: https://arxiv.org/abs/2406.17070

work page arXiv 2024

[17] [17]

Kschischang, B.J

F.R. Kschischang, B.J. Frey, and H.-A. Loeliger. Factor graphs and the sum- product algorithm. IEEE Transactions on Information Theory , 47(2):498–519, 2001

work page 2001

[18] [18]

Dholakia, E

Jinghu Chen, A. Dholakia, E. Elefthe- riou, M.P.C. Fossorier, and Xiao-Yu Hu. Reduced-Complexity Decoding of LDPC Codes. IEEE Transactions on Communi- cations, 53(8):1288–1299, 2005

work page 2005

[19] [19]

Rosenthal and P.O

J. Rosenthal and P.O. Vontobel. Con- structions of Regular and Irregular LDPC Codes Using Ramanujan Graphs and Ideas from Margulis. In IEEE Interna- tional Symposium on Information Theory , page 4, June 2001

work page 2001

[20] [20]

MacKay and Michael S

David J.C. MacKay and Michael S. Postol. Weaknesses of Margulis and Ramanujan-Margulis Low-Density Parity- Check codes. Electronic Notes in Theoret- ical Computer Science , 74:97–104, 2003

work page 2003

[21] [21]

Trap- ping Sets of Quantum LDPC Codes

Nithin Raveendran and Bane Vasić. Trap- ping Sets of Quantum LDPC Codes. Quantum, 5:562, October 2021

work page 2021

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Planjery

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