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arxiv: 2605.24177 · v1 · pith:QXAJW3ZYnew · submitted 2026-05-22 · 🪐 quant-ph · cs.IT· math.IT

Towards Scalable Quaternary Message-Passing Decoding for Quantum Error Correction

Boqing Zhang , Henry D. Pfister , Hanwen Yao , Siyuan Niu This is my paper

Pith reviewed 2026-06-30 15:53 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords quantum error correctionmessage passing decodingsurface codesmin-sum decoderdilution methoddepolarizing thresholdX noise
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The pith

A dilution method lets quaternary Min-Sum decoders reach apparent 16% depolarizing thresholds up to distance 20.

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

The paper develops a dilution method for quaternary message-passing decoders applied to quantum surface codes. This technique enables the Min-Sum decoder to display an apparent threshold of 16 percent under depolarizing noise for codes up to distance 20, surpassing Minimum-Weight Perfect Matching at these sizes. Under X-noise the diluted decoder runs in O(N log squared d) time and beats BP-OSD at distance 65. A graph-dilution argument is provided to explain the method's effectiveness and to clarify when such algorithms can scale. The results aim to improve the reliability and theoretical grounding of message-passing methods for larger quantum error-correcting codes.

Core claim

We introduce a dilution method, which allows a quaternary Min-Sum (MS) decoder to exhibit an apparent depolarizing threshold of 16% up to distance 20, outperforming Minimum-Weight Perfect Matching in finite-length regimes. Notably, for X-noise, the standard MS decoder under dilution has worst-case complexity O(N log² d) and outperforms BP-OSD at d=65. The observed ∼9% threshold may correspond to a true asymptotic threshold. Finally, we give a graph-dilution argument that interprets the success of the dilution method and offers insight into when MP algorithms can genuinely scale.

What carries the argument

The dilution method applied to the decoding graph for quaternary Min-Sum decoding, together with the accompanying graph-dilution argument.

Load-bearing premise

The dilution procedure and graph-dilution argument remain effective at distances much larger than 65 and the reported thresholds are not artifacts of the specific simulation parameters or code instances used.

What would settle it

Running the diluted Min-Sum decoder on surface codes of distance 100 or higher under depolarizing noise to determine if the threshold stays near 16% or falls.

Figures

Figures reproduced from arXiv: 2605.24177 by Boqing Zhang, Hanwen Yao, Henry D. Pfister, Siyuan Niu.

Figure 1
Figure 1. Figure 1: Illustration of the canonical Tanner graph T of the distance-5 surface code (left) and the corresponding component lattices GX (blue) and GZ (red) (right). In T , black variable nodes represent physical qubits, blue and red factor nodes correspond to X- and Z-type stabilizers, respectively. This Tanner graph decomposes into two canonical lattices in which qubits lie on edges and blue (resp. red) vertices r… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of quaternary message passing for CSS codes. The left panel shows the Z- (resp. X-) factor￾to-variable update νˆ (t) aZ→i (xi) (resp. µˆ (t) aX→i (zi) ) from a Z (resp. a X) to variable node i, which proceeds independently and is not coupled. The right panel shows the variable-to-factor (Z- and X-factor) update ν (t+1) i→aZ (xi) and µ (t+1) i→aX (zi), where the messages become coupled through … view at source ↗
Figure 3
Figure 3. Figure 3: Examples of diluted graphs obtained by diagonal s-sparsification. From left to right: the D 1 V -diluted lattice G 1 DV , the D 1 H-diluted lattice G 1 DH , and the D 3 V -diluted lattice G 3 DV . Definition 2 (Cartesian s-Sparsification). Let {Ck}k∈Z denote the ordered family of Cartesian grid lines, either all horizontal or all vertical grid lines, in a lattice G. The Cartesian s-sparsification retains o… view at source ↗
Figure 4
Figure 4. Figure 4: Examples of diluted graphs obtained by Cartesian s-sparsification. From left to right: the C 1 V -diluted lattice G 1 CV , the C 1 H-diluted lattice G 1 CH , and the C 3 V -diluted lattice G 3 CV . 6 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of effective errors on diluted lattices. The middle panels show physical errors (red strings) on the original lattice, and their corresponding effective errors on the diluted lattices are shown in purple. The effective errors on the right are stabilizer-equivalent to the physical error, while the one on the left differs by a logical operator. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Examples of diluted Tanner graphs T s induced by the D s V -diluted lattice G s DV via diagonal sparsification pattern D s V : a D 1 V -diluted Tanner graph T 1 DV (left panel), and a D 3 V -diluted Tanner graph T 3 DV (right panel). By making explicit the correspondence between the diluted Tanner graph and its component lattices, the error-correcting radius of the diluted Tanner graph admits a natural cha… view at source ↗
Figure 7
Figure 7. Figure 7: An example of the graph-dilution sequence {T (0) , T (1) , T (2)} for a distance-5 surface code. The sparsifi￾cation pattern is the diagonal pattern D s V . In this section, given a distance-d surface code and a sparsification pattern (DV , DH, CV or CH), we show how the dilution method constructs the corresponding graph-dilution sequence {T (k)} K k=0. Furthermore, Proposition 1 and Theorem 1 also suggest… view at source ↗
Figure 8
Figure 8. Figure 8: Performance of quaternary Min-Sum decoding with dilution under depolarizing noise (left), together with a finite-length comparison against the standard global decoder MWPM at distances 9 and 15 (right). For MS under dilution, the damping factor is ϵ = 0.15 for all distances. Under X-noise, the situation is different, as shown in [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Performance of Min-Sum decoding with dilution under X-noise (left), together with a finite-length comparison against the standard global decoders MS-OSD under X-noise at distances 33 and 65 (right). For MS￾OSD, the number of Min-Sum iterations is set to 30. For MS under dilution, the damping factor is ϵ = 0.1 for d = 33, ϵ = 0.05 for d = 65, and ϵ = 0.15 for all other distances. 4.2 Complexity and Converge… view at source ↗
Figure 10
Figure 10. Figure 10: Left: Average decoding complexity, measured by the median number of iterations until convergence over 104 trials. The inset shows the maximum iteration budget Imax as a function of code distance. Right: Non￾convergence rate with and without dilution and damping, where ϵ denotes the damping factor. Next, we examine numerically how dilution affects the convergence behavior of the Min-Sum decoder. As shown i… view at source ↗
Figure 11
Figure 11. Figure 11: Histograms of the number of iterations until convergence for the d = 9 (left) and d = 65 (right) surface code at threshold 9%. The black line shows the baseline without dilution. 5 Interpretation of Dilution Method While many practical modifications have improved MP decoders for quantum LDPC codes, their behavior remains poorly understood. In this section, we use the dilution method as a lens to first rev… view at source ↗
Figure 12
Figure 12. Figure 12: After two iterations, the incoming messages [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 12
Figure 12. Figure 12: Left: Short 4-cycles and 8-cycles in the diluted graph formed by different stabilizer types. Right: The cavity picture of the mixed-type 4-cycle. For this cycle, we are interested in the correlation between the incoming messages ν (2) i→aZ (xi) and ν (2) j→aZ (xj ), shown by the red arrows. After removing the Z-factor node a Z , these incoming messages correspond to the cavity marginals µ \a (xi | σX) and… view at source ↗
Figure 13
Figure 13. Figure 13: Schematic illustration of the dilution process on a one-dimensional strip. Left: The graph-dilution sequence G (0) , G (1) , G (2) , . . .. Right: One step in the evolution of the effective error from stage k to stage k + 1. BP is first applied on G (k) to estimate the errors on the edges that will be removed. After decimation, diluting the corresponding qubits induces a renormalization of the residual er… view at source ↗
Figure 14
Figure 14. Figure 14: Numerical experiment for testing the evolution of effective error weight under dilution. The experiment is performed locally on one block B (k) at different stages k. Left: Illustration of cells and blocks, green regions denote cells, while red regions denote blocks. Each block consists of two neighboring cells. Middle: A stage-k block B (k) j , formed by two neighboring cells, is mapped to a stage-(k + 1… view at source ↗
Figure 15
Figure 15. Figure 15: Local regions and worst-case configurations used in Lemma 1. The first two panels define the local decompositions. For the diagonal pattern, Di is the green local region and ∂Di consists of the red retained boundary edges. For the Cartesian pattern, Ci is the green horizontal band and ∂Ci consists of the two red retained boundary lines. The last two panels illustrate worst-case local configurations. In th… view at source ↗
read the original abstract

The scalability and interpretability of message-passing (MP) decoding, such as (quaternary) Belief Propagation, remain open challenges in quantum error correction. Even for surface codes, arguably the first testbed for decoding methods, studies of improved MP decoders have mostly been restricted to small distances ($d \lesssim 19$). Moreover, the mismatch with established message-passing theory limits the decoder's interpretability, making it unclear whether MP decoding can sustain its effectiveness at large system sizes. This work takes a step toward a more principled and interpretable MP decoding framework, with the goal of making MP-based decoding more reliable and bridging theory and practice. We introduce a dilution method, which allows a quaternary Min-Sum (MS) decoder to exhibit an apparent depolarizing threshold of $16\%$ up to distance $20$, outperforming Minimum-Weight Perfect Matching in finite-length regimes. Notably, for $X$-noise, the standard MS decoder under dilution has worst-case complexity $O(N \log^2 d)$ and outperforms BP-OSD at $d=65$. The observed $\sim 9\%$ threshold may correspond to a true asymptotic threshold. Finally, we give a graph-dilution argument that interprets the success of the dilution method and offers insight into when MP algorithms can genuinely scale. Taken together, these results provide encouraging progress toward scalable and interpretable MP decoding in quantum error correction.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a dilution method for quaternary Min-Sum (MS) message-passing decoding of quantum surface codes. It reports an apparent depolarizing threshold of 16% up to distance 20 (outperforming MWPM in finite-length regimes), an O(N log² d) complexity for X-noise that outperforms BP-OSD at d=65, and a possible ~9% asymptotic threshold, interpreted via a graph-dilution argument that aims to explain when MP decoders can scale.

Significance. If the reported thresholds and complexity advantages hold asymptotically and generalize beyond the tested distances and code instances, the work would provide a concrete step toward scalable, interpretable MP decoding in QEC, addressing the current restriction of such studies to small distances and offering a principled way to suppress trapping sets via dilution.

major comments (3)
  1. [Simulation results] Results on depolarizing noise (up to d=20): the central 16% threshold claim and its comparison to MWPM rest on Monte-Carlo data whose statistical reliability cannot be assessed without reported trial counts, error bars, or the precise dilution implementation (e.g., how edges or messages are diluted).
  2. [X-noise performance] X-noise results at d=65: the O(N log² d) MS decoder outperforming BP-OSD is load-bearing for the scalability claim, yet the manuscript provides no details on the surface-code instances, cycle structure after dilution, or whether the observed advantage persists when the underlying Tanner graph contains longer cycles.
  3. [Graph-dilution argument] Graph-dilution argument: presented as interpretive rather than a closed-form bound, it does not yet demonstrate that short-cycle suppression remains effective at distances ≫65; without such analysis the extrapolation to a possible 9% asymptotic threshold remains an open risk for the central scaling claim.
minor comments (2)
  1. [Preliminaries] Notation for the quaternary alphabet and message alphabet should be defined once at first use and used consistently.
  2. [Figures] Figure captions for threshold plots should explicitly state the number of logical errors observed per data point.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Simulation results] Results on depolarizing noise (up to d=20): the central 16% threshold claim and its comparison to MWPM rest on Monte-Carlo data whose statistical reliability cannot be assessed without reported trial counts, error bars, or the precise dilution implementation (e.g., how edges or messages are diluted).

    Authors: We agree these statistical and implementation details are necessary to assess reliability. In the revised manuscript we will report the number of Monte-Carlo trials per data point, include error bars on all logical-error-rate plots, and give a precise description of the dilution procedure including the selection of edges and messages. revision: yes

  2. Referee: [X-noise performance] X-noise results at d=65: the O(N log² d) MS decoder outperforming BP-OSD is load-bearing for the scalability claim, yet the manuscript provides no details on the surface-code instances, cycle structure after dilution, or whether the observed advantage persists when the underlying Tanner graph contains longer cycles.

    Authors: We will add the requested details: the specific surface-code instances used at d=65, the cycle structure of the diluted Tanner graph, and a discussion of decoder performance when longer cycles are present. These additions will be placed in the relevant results section. revision: yes

  3. Referee: [Graph-dilution argument] Graph-dilution argument: presented as interpretive rather than a closed-form bound, it does not yet demonstrate that short-cycle suppression remains effective at distances ≫65; without such analysis the extrapolation to a possible 9% asymptotic threshold remains an open risk for the central scaling claim.

    Authors: The graph-dilution argument is offered as an interpretive lens rather than a closed-form proof. We will revise the text to state its limitations explicitly, clarify that it does not constitute a demonstration for distances ≫65, and note that the ~9% figure remains a conjecture pending further numerical or analytic work at larger distances. revision: partial

Circularity Check

0 steps flagged

No circularity; thresholds from finite-distance Monte Carlo simulations and graph-dilution argument offered as interpretation only.

full rationale

The paper's central results consist of empirical performance curves obtained via Monte Carlo decoding simulations on surface codes up to d=65 (X-noise) and d=20 (depolarizing), together with an interpretive graph-dilution argument that explains observed behavior without claiming a closed-form derivation or asymptotic proof. No equations reduce any reported threshold to a fitted parameter by construction, no self-citation chain bears the load of the main claim, and the dilution method is presented as an algorithmic modification whose effectiveness is measured directly rather than assumed. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only view limits visibility into parameters or axioms. The dilution method is the primary new element introduced. Relies on standard surface-code and noise-model assumptions common to the field.

axioms (1)
  • domain assumption Surface codes under depolarizing or X noise behave according to established quantum error correction models.
    Invoked implicitly when reporting thresholds for these codes and noise types.
invented entities (1)
  • Dilution method no independent evidence
    purpose: To improve scalability and interpretability of quaternary message-passing decoding.
    New technique introduced without external validation beyond the reported simulations.

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Reference graph

Works this paper leans on

35 extracted references · 8 canonical work pages · 4 internal anchors

  1. [1]

    On the iterative decoding of sparse quantum codes

    D. Poulin and Y. Chung, “On the iterative decoding of sparse quantum codes,”arXiv preprint arXiv:0801.1241, 2008

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  2. [2]

    M ¨uller, T

    T. Müller, T. Alexander, M. E. Beverland, M. Bühler, B. R. Johnson, T. Maurer, and D. Vandeth, “Improved belief propagation is sufficient for real-time decoding of quantum memory,”arXiv preprint arXiv:2506.01779, 2025

    work page arXiv 2025

  3. [3]

    Fifteen years of quantum ldpc coding and improved decoding strategies,

    Z. Babar, P. Botsinis, D. Alanis, S. X. Ng, and L. Hanzo, “Fifteen years of quantum ldpc coding and improved decoding strategies,”iEEE Access, vol. 3, pp. 2492–2519, 2015

    2015

  4. [4]

    Trapping sets of quantum ldpc codes,

    N. Raveendran and B. Vasić, “Trapping sets of quantum ldpc codes,”Quantum, vol. 5, p. 562, 2021

    2021

  5. [5]

    Degeneracy and its impact on the decoding of sparse quantum codes,

    P. Fuentes, J. E. Martinez, P. M. Crespo, and J. Garcia-Frías, “Degeneracy and its impact on the decoding of sparse quantum codes,”IEEE Access, vol. 9, pp. 89093–89119, 2021

    2021

  6. [6]

    Richardson and R

    T. Richardson and R. Urbanke,Modern coding theory. Cambridge university press, 2008

    2008

  7. [7]

    Mezard and A

    M. Mezard and A. Montanari,Information, physics, and computation. Oxford University Press, 2009

    2009

  8. [8]

    Quantum ldpc codes with positive rate and minimum distance proportional to the square root of the blocklength,

    J.-P. Tillich and G. Zémor, “Quantum ldpc codes with positive rate and minimum distance proportional to the square root of the blocklength,”IEEE Transactions on Information Theory, vol. 60, no. 2, pp. 1193–1202, 2013

    2013

  9. [9]

    Quantum ldpc codes with almost linear minimum distance,

    P. Panteleev and G. Kalachev, “Quantum ldpc codes with almost linear minimum distance,”IEEE Transactions on Information Theory, vol. 68, no. 1, pp. 213–229, 2021

    2021

  10. [10]

    Balanced product quantum codes,

    N. P. Breuckmann and J. N. Eberhardt, “Balanced product quantum codes,”IEEE Transactions on Information Theory, vol. 67, no. 10, pp. 6653–6674, 2021

    2021

  11. [11]

    Belief propagation decoding of quantum ldpc codes with guided decimation,

    H. Yao, W. A. Laban, C. Häger, A. G. i Amat, and H. D. Pfister, “Belief propagation decoding of quantum ldpc codes with guided decimation,” in2024 IEEE International Symposium on Information Theory (ISIT). IEEE, 2024, pp. 2478–2483

    2024

  12. [12]

    Exploiting degeneracy in belief propagation decoding of quantum codes,

    K.-Y. Kuo and C.-Y. Lai, “Exploiting degeneracy in belief propagation decoding of quantum codes,” npj Quantum Information, vol. 8, no. 1, p. 111, 2022

    2022

  13. [13]

    Generalized belief propagation algorithms for decoding of surface codes,

    J. Old and M. Rispler, “Generalized belief propagation algorithms for decoding of surface codes,”Quan- tum, vol. 7, p. 1037, 2023

    2023

  14. [14]

    Enhanced feedback iterative decoding of sparse quantum codes,

    Y.-J. Wang, B. C. Sanders, B.-M. Bai, and X.-M. Wang, “Enhanced feedback iterative decoding of sparse quantum codes,”IEEE transactions on information theory, vol. 58, no. 2, pp. 1231–1241, 2012

    2012

  15. [15]

    Stabilizer inactivation for message-passing decoding of quantum ldpc codes,

    J. Du Crest, M. Mhalla, and V. Savin, “Stabilizer inactivation for message-passing decoding of quantum ldpc codes,” in2022 IEEE Information Theory Workshop (ITW). IEEE, 2022, pp. 488–493

    2022

  16. [16]

    Topological quantum memory,

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,”Journal of Mathe- matical Physics, vol. 43, no. 9, pp. 4452–4505, 2002

    2002

  17. [17]

    Almost-linear time decoding algorithm for topological codes,

    N. Delfosse and N. H. Nickerson, “Almost-linear time decoding algorithm for topological codes,”Quan- tum, vol. 5, p. 595, 2021

    2021

  18. [18]

    Proof of finite surface code threshold for matching,

    A. G. Fowler, “Proof of finite surface code threshold for matching,”Physical review letters, vol. 109, no. 18, p. 180502, 2012

    2012

  19. [19]

    Proof of a finite threshold for the union-find decoder

    S. Yoshida, E. Lake, and H. Yamasaki, “Proof of a finite threshold for the union-find decoder,”arXiv preprint arXiv:2602.20238, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  20. [20]

    Degenerate quantum ldpc codes with good finite length performance,

    P. Panteleev and G. Kalachev, “Degenerate quantum ldpc codes with good finite length performance,” Quantum, vol. 5, p. 585, 2021. 17

    2021

  21. [21]

    Localized statistics decoding 13 for quantum low-density parity-check codes

    T. Hillmann, L. Berent, A. O. Quintavalle, J. Eisert, R. Wille, and J. Roffe, “Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes,”arXiv preprint arXiv:2406.18655, 2024

    work page arXiv 2024

  22. [22]

    Improved single-shot decoding of higher-dimensional hypergraph- product codes,

    O. Higgott and N. P. Breuckmann, “Improved single-shot decoding of higher-dimensional hypergraph- product codes,”PRX Quantum, vol. 4, no. 2, p. 020332, 2023

    2023

  23. [23]

    Statistical physics of inference: Thresholds and algorithms,

    L. Zdeborová and F. Krzakala, “Statistical physics of inference: Thresholds and algorithms,”Advances in Physics, vol. 65, no. 5, pp. 453–552, 2016

    2016

  24. [24]

    Good quantum error-correcting codes exist,

    A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,”Phys. Rev. A, vol. 54, pp. 1098–1105, Aug 1996. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.54.1098

    work page doi:10.1103/physreva.54.1098 1996

  25. [25]

    Multiple-particleinterferenceandquantumerrorcorrection,

    A.Steane, “Multiple-particleinterferenceandquantumerrorcorrection,”ProceedingsoftheRoyalSociety ofLondon.SeriesA:Mathematical, PhysicalandEngineeringSciences, vol.452, no.1954, pp.2551–2577,

    1954

  26. [26]

    Available: https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1996.0136

    [Online]. Available: https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1996.0136

    work page doi:10.1098/rspa.1996.0136 1996

  27. [27]

    Refined belief propagation decoding of sparse-graph quantum codes,

    K.-Y. Kuo and C.-Y. Lai, “Refined belief propagation decoding of sparse-graph quantum codes,”IEEE Journal on Selected Areas in Information Theory, vol. 1, no. 2, pp. 487–498, 2020

    2020

  28. [28]

    Log-domain decoding of quantum ldpc codes over binary finite fields,

    C.-Y. Lai and K.-Y. Kuo, “Log-domain decoding of quantum ldpc codes over binary finite fields,”IEEE Transactions on Quantum Engineering, vol. 2, pp. 1–15, 2021

    2021

  29. [29]

    Quaternary neural belief propagation decoding of quantum ldpc codes with overcomplete check matrices,

    S. Miao, A. Schnerring, H. Li, and L. Schmalen, “Quaternary neural belief propagation decoding of quantum ldpc codes with overcomplete check matrices,”IEEE Access, 2025

    2025

  30. [30]

    Stabilizer Codes and Quantum Error Correction

    D.Gottesman, “Stabilizercodesandquantumerrorcorrection.caltechph.d,” Ph.D.dissertation, Thesis, eprint: quant-ph/9705052, 1997

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  31. [31]

    Fault-tolerant quantum computation by anyons,

    A. Y. Kitaev, “Fault-tolerant quantum computation by anyons,”Annals of physics, vol. 303, no. 1, pp. 2–30, 2003

    2003

  32. [32]

    A message-passing algorithm with damping,

    M. Pretti, “A message-passing algorithm with damping,”Journal of Statistical Mechanics: Theory and Experiment, vol. 2005, no. 11, pp. P11008–P11008, 2005

    2005

  33. [33]

    Sufficient conditions for convergence of Loopy Belief Propagation

    J. Mooij and H. Kappen, “Sufficient conditions for convergence of loopy belief propagation,”arXiv preprint arXiv:1207.1405, 2012

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  34. [34]

    Beyond trees: Analysis and convergence of belief propagation in graphs with multiple cycles,

    R. Zivan, O. Lev, and R. Galiki, “Beyond trees: Analysis and convergence of belief propagation in graphs with multiple cycles,” inProceedings of the AAAI Conference on Artificial Intelligence, vol. 34, no. 05, 2020, pp. 7333–7340

    2020

  35. [35]

    Belief propagation decoding on a sparsified graph ensemble of the surface code,

    B. Zhang, H. Yao, and H. D. Pfister, “Belief propagation decoding on a sparsified graph ensemble of the surface code,” in2025 IEEE International Symposium on Information Theory (ISIT). IEEE, 2025, pp. 1–6. 18 A Proof of Theorem 1 We use two different local decompositions for the two sparsification patterns, as shown in the Fig. 15. For the diagonal patter...

    2025