pith. machine review for the scientific record. sign in

arxiv: 2605.00038 · v2 · submitted 2026-04-28 · 💻 cs.AR · quant-ph

Recognition: unknown

Lottery BP: Unlocking Quantum Error Decoding at Scale

Yanzhang Zhu , Chen-Yu Peng , Yun Hao Chen , Yeong-Luh Ueng , Di Wu
Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:42 UTC · model grok-4.3

classification 💻 cs.AR quant-ph
keywords quantum error correctionbelief propagationdecodingtopological codessurface codetoric codescalable architecture
0
0 comments X

The pith

Introducing randomness into belief propagation decoding raises accuracy for topological quantum codes by 2 to 8 orders of magnitude.

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

This paper seeks to bridge the gap between existing quantum error decoders that fall short in accuracy, speed, generality, or scalability and an ideal decoder suitable for fault-tolerant quantum computing with millions of qubits. Lottery BP adds randomness to the belief propagation process to achieve much higher accuracy on topological codes like surface and toric codes. Syndrome vote pre-processing compresses multi-round syndromes to ease the backlog issue. The PolyQec architecture uses Lottery BP locally to invoke the global ordered statistics decoder far less often. A new GPU-based simulator called Syndrilla enables fair and fast evaluation of such decoders.

Core claim

Lottery BP introduces randomness during belief propagation decoding, improving accuracy over standard BP by 2 to 8 orders of magnitude for topological codes. Syndrome vote serves as a pre-processing step to handle multi-round measurement errors by compressing syndromes. The PolyQec architecture implements Lottery BP as a local decoder paired with ordered statistics decoding as global, reducing the need for the costly global step by 3 to 5 orders of magnitude while remaining configurable for different codes and check types.

What carries the argument

Lottery BP, the decoder that introduces randomness into belief propagation iterations to enhance convergence and accuracy on quantum error correction codes.

If this is right

  • Real-time decoding becomes possible for large-scale quantum systems with millions of qubits.
  • The backlog problem from accumulating syndromes is mitigated, increasing the time available for decoding.
  • Hybrid decoding architectures can scale better by relying on fast local decoding most of the time.
  • The modular simulator allows rapid development and testing of new decoding methods on GPUs.
  • Configurable designs support both surface/toric codes and X/Z checks in a unified way.

Where Pith is reading between the lines

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

  • If randomness reliably helps BP, similar stochastic elements might improve other iterative algorithms in quantum decoding.
  • Reduced reliance on global decoding could simplify hardware requirements for quantum processors.
  • Further tests on bivariate bicycle codes and other code families would confirm the broad applicability.
  • Integration with neural network decoders might yield even higher performance combinations.

Load-bearing premise

That introducing randomness during BP decoding will reliably boost accuracy across a broad set of codes without introducing new failure modes or excessive computational cost.

What would settle it

A benchmark on a large surface code instance or under a different error model where Lottery BP shows no accuracy gain or slower performance than plain BP.

Figures

Figures reproduced from arXiv: 2605.00038 by Chen-Yu Peng, Di Wu, Yanzhang Zhu, Yeong-Luh Ueng, Yun Hao Chen.

Figure 1
Figure 1. Figure 1: Design space of decoder architecture. The chip size [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: 3D decoding graph for measurement error. Red and [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example of randomness resolving a degenerate deadlock in a [[13,1,3]] surface code. We consider error events occurring [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Lottery policies on surface code at 𝑑 = 9. 10 −3 10 −2 10 0 BP d=3 d=5 d=7 d=9 d=11 d=13 10 −3 10 −2 10 0 Probability Relay BP 10 1 10 3 Iteration count 10 −3 10 −2 10 0 Lottery BP (a) Surface code. 10 −3 10 −2 10 0 BP [[72,12,6]] [[90,8,10]] [[108,8,10]] [[144,12,12]] [[288,12,18]] 10 −3 10 −2 10 0 Probability Relay BP 10 1 10 3 10 5 Iteration count 10 −3 10 −2 10 0 Lottery BP (b) BB code [PITH_FULL_IMAG… view at source ↗
Figure 5
Figure 5. Figure 5: Distribution of decoding iterations at 𝑝 = 0.05. where the middle three variants are adopted in prior works [39]. The resulting accuracy is given in Figure 4a 1 . (1) Global optimal: Among all VNs with the most unsatisfied CNs, flip the sign of the one with the minimum absolute LLR. This policy builds an upper bound but is impractical for hardware implementation due to global search. (2) Global connectivit… view at source ↗
Figure 6
Figure 6. Figure 6: Syndrome vote to handle measurement errors on [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Overview of PolyQec. On the right plane, vertical dashed lines separate each fully pipelined stages [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: CN layout in the memory. whose cost increases quadratically with the code distance. Fortu￾nately, the parity-check matrices of surface and toric codes are regular, which facilitates converting the message layout on the fly instead of storing the parity-check matrix. 2 C2V converter converts the C2V message from CN layout to VN layout, which groups the messages needed by each VN together. On the other hand,… view at source ↗
Figure 9
Figure 9. Figure 9: Resolving read bank conflict of LLR memory. [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: outlines Syndrilla. The workflow of Syndrilla is similar to other QEC simulators [14, 15, 53, 57, 64, 71, 76], including error and syndrome generation, followed by decoder and logical check. What distinguishes Syndrilla is integrated metrics to fair evaluation and cross-platform execution for accelerated simulation. Syndrilla also interacts with Stim [30] to support broader error models. Integrated Metric… view at source ↗
Figure 11
Figure 11. Figure 11: Simulating surface code at 𝑑 = 11. CPU and GPU de￾note Syndrilla running on Intel Core i5-13450HX and NVIDIA GeForce RTX 4080. CPU_cpp marks C++ BP+OSD [70] run￾ning on the Intel CPU with no batch support. The runtime represents the total simulation time for each configuration. (a) and (c) have a batch size of 100,000 for BP+OSD and 200,000 for Lottery BP+OSD. (c) and (d) use FP64 data. (d) uses 𝑝 = 0.05.… view at source ↗
Figure 12
Figure 12. Figure 12: Logical error rate without measurement error. [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 15
Figure 15. Figure 15: Quantization on surface code at 𝑑 = 9. 7.1.3 Quantization Analy￾sis. Unlike using floating￾point in simulation, we need to quantize the soft mes￾sages to fixed-point format for efficient hardware imple￾mentation [PITH_FULL_IMAGE:figures/full_fig_p010_15.png] view at source ↗
Figure 14
Figure 14. Figure 14: Invoke rate of OSD [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗
Figure 18
Figure 18. Figure 18: Scalability study. 7.2.4 Scalability. To support millions of qubits, decoding has to be massively paral￾lel within a tight latency margin due to syndrome generation [29]. An optimal decoder is expected to de￾code as many logical qubits as possible, with minimum area and time. We quantify this advantage using a novel metric, decoding efficiency, defined in Equation 1. Decoding efficiency = Num of decoding … view at source ↗
Figure 17
Figure 17. Figure 17: Area and power at 𝑝 = 10−3 . All BP support 𝑑 ≤ 32. have an SRAM area, since its memory is implemented with register file. All BP variants occupy similar area, with Lottery BP having 0.57𝑚𝑚2 . Among all decoders, Micro Blossom always has the largest area, since it implements the most accurate and costly MWPM. AFS has the smallest area among all, since UF is cheaper to implement and logic area is not repor… view at source ↗
read the original abstract

To enable fault tolerance on millions of qubits in real time, scalable decoding is necessary, which motivates this paper. Existing decoding algorithms (decoders), such as clustering, matching, belief propagation (BP), and neural networks, suffer from one or more of inaccuracy, costliness, and incompatibility, upon a broad set of quantum error correction codes, such as surface code, toric code, and bivariate bicycle code. Therefore, there exists a gap between existing decoders and an ideal decoder that is accurate, fast, general, and scalable simultaneously. This paper contributes in three aspects, including decoder, decoder architecture, and decoding simulator. First, we propose Lottery BP, a decoder that introduces randomness during decoding. Lottery BP improves the decoding accuracy over BP by 2~8 orders of magnitude for topological codes. To efficiently decode multi-round measurement errors, we propose syndrome vote as a pre-processing step before Lottery BP, which compresses multiple rounds of syndromes into one. Syndrome vote increases the latency margin of decoding and mitigates the backlog problem. Second, we design a PolyQec architecture that implements Lottery BP as a local decoder and ordered statistics decoding (OSD) as a global decoder, and it is configurable for surface/toric code and X/Z check. Since Lottery BP boosts the local decoding accuracy, PolyQec invokes the costly global OSD decoder less frequently over BP+OSD to enhance the scalability, e.g., 3~5 orders of magnitude less for topological codes. Third, to evaluate decoders fairly, we develop a PyTorch-based decoding simulator, Syndrilla, that modularizes the simulation pipeline and allows to extend new decoders flexibly. We formulate multiple metrics to quantify the performance of decoders and integrate them in Syndrilla. Running on GPUs, Syndrilla is 1~2 orders of magnitude faster than CPUs.

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

2 major / 2 minor

Summary. The manuscript proposes Lottery BP, a belief-propagation decoder augmented with randomness injection, claiming 2-8 orders of magnitude higher decoding accuracy than standard BP on topological codes (surface, toric, bivariate bicycle). It introduces syndrome vote as a pre-processor to compress multi-round syndromes, the PolyQec architecture that pairs local Lottery BP with global ordered-statistics decoding (OSD) to reduce OSD invocations, and the Syndrilla PyTorch-based simulator for modular, GPU-accelerated decoder evaluation with multiple performance metrics.

Significance. If the accuracy and scalability claims hold under standard noise models and code distances, the work could meaningfully advance real-time decoding for large-scale quantum error correction by lowering the frequency of expensive global post-processing while providing a reusable simulation framework. The modular Syndrilla simulator is a clear strength for enabling reproducible and extensible decoder comparisons in the field.

major comments (2)
  1. [Abstract] Abstract: The central claim that Lottery BP improves decoding accuracy over BP by 2-8 orders of magnitude is stated without any supporting numerical results, code distances, noise models, logical error rates, or baseline details (e.g., whether comparisons include OSD or not). This is load-bearing for the primary contribution and cannot be assessed from the available information.
  2. [Abstract] Abstract: No ablation is described that isolates the effect of the randomness mechanism in Lottery BP from the syndrome-vote pre-processor or from OSD post-processing. Given well-known BP failure modes on girth-4 quantum LDPC graphs, this omission leaves open whether the reported gains are robust or introduce new variance/cost issues.
minor comments (2)
  1. [Abstract] Abstract: The phrasing 'upon a broad set of quantum error correction codes' is slightly awkward; 'on' would be clearer.
  2. [Abstract] Abstract: The claim that PolyQec invokes OSD '3~5 orders of magnitude less' should specify the exact metric (e.g., invocation rate per logical qubit or per syndrome) and the reference BP+OSD configuration.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the abstract and the importance of clearly isolating contributions. We address each major comment below and outline revisions to improve assessability and robustness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that Lottery BP improves decoding accuracy over BP by 2-8 orders of magnitude is stated without any supporting numerical results, code distances, noise models, logical error rates, or baseline details (e.g., whether comparisons include OSD or not). This is load-bearing for the primary contribution and cannot be assessed from the available information.

    Authors: We agree that the abstract presents the improvement claim at a high level. The full manuscript details the supporting results in Section 4, including logical error rates for surface, toric, and bivariate bicycle codes at distances d=5 to d=13 under depolarizing and circuit-level noise models. These show 2-8 orders of magnitude gains over standard BP (without OSD post-processing). We will revise the abstract to include a concise example of these metrics, such as the improvement for a representative distance and noise model, to enable immediate assessment. revision: yes

  2. Referee: [Abstract] Abstract: No ablation is described that isolates the effect of the randomness mechanism in Lottery BP from the syndrome-vote pre-processor or from OSD post-processing. Given well-known BP failure modes on girth-4 quantum LDPC graphs, this omission leaves open whether the reported gains are robust or introduce new variance/cost issues.

    Authors: The manuscript includes direct comparisons of Lottery BP to standard BP and evaluates the full PolyQec architecture incorporating syndrome vote and OSD. However, we acknowledge that an explicit ablation isolating the randomness injection would better address concerns about robustness and variance on girth-4 graphs. We will add a dedicated ablation subsection in the revised version, presenting results for Lottery BP variants with and without the randomness mechanism while holding syndrome vote and OSD fixed. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on empirical proposal without self-referential derivations

full rationale

The paper introduces Lottery BP as a new decoder variant that injects randomness into standard belief propagation, then reports measured accuracy gains (2-8 orders) and architectural benefits on specific codes. No equations, parameter fits, uniqueness theorems, or derivation steps appear in the provided text that would allow any claimed result to reduce to its own inputs by construction. The contributions are framed as algorithmic proposals, pre-processing steps, and a simulator implementation, all evaluated externally via simulation rather than derived tautologically. Self-citations are absent from the load-bearing claims, and no 'prediction' is obtained by fitting then renaming. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are specified in the abstract.

pith-pipeline@v0.9.0 · 5648 in / 996 out tokens · 45690 ms · 2026-05-09T20:42:21.498378+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

87 extracted references · 49 canonical work pages

  1. [1]

    Panos Aliferis, Daniel Gottesman, and John Preskill. 2005. Quantum accuracy threshold for concatenated distance-3 codes.arXiv preprint quant-ph/0504218 (2005)

    work page arXiv 2005

  2. [2]

    Thomas Ayral, Pauline Besserve, Denis Lacroix, and Edgar Andres Ruiz Guzman

  3. [3]

    Quantum computing with and for many-body physics.The European Physical Journal A59, 10 (2023), 227

    2023

  4. [4]

    Zunaira Babar, Panagiotis Botsinis, Dimitrios Alanis, Soon Xin Ng, and Lajos Hanzo. 2015. Fifteen years of quantum LDPC coding and improved decoding strategies.IEEE Access3 (2015), 2492–2519. https://doi.org/10.1109/ACCESS. 2015.2503267

    work page doi:10.1109/access 2015

  5. [5]

    Kahng, Naveen Muralimanohar, Ali Shafiee, and Vaishnav Srinivas

    Rajeev Balasubramonian, Andrew B. Kahng, Naveen Muralimanohar, Ali Shafiee, and Vaishnav Srinivas. 2017. CACTI 7: New Tools for Interconnect Exploration in Innovative Off-Chip Memories.Transactions on Architecture and Code Opti- mization(2017)

    2017

  6. [6]

    Large language models encode clinical knowledge

    Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J. Yoder. 2024. High-threshold and low-overhead fault-tolerant quantum memory.Nat.627, 8005 (2024), 778–782. https://doi.org/10.1038/S41586- 024-07107-7

    work page doi:10.1038/s41586- 2024

  7. [7]

    Harry Buhrman, Noah Linden, Laura Mančinska, Ashley Montanaro, and Maris Ozols. 2022. Quantum majority vote.arXiv preprint arXiv:2211.11729(2022)

    work page arXiv 2022

  8. [8]

    A. R. Calderbank and P. W. Shor. 1996. Good quantum error-correcting codes exist.Physical Review A54 (1996), 1098–1105. https://doi.org/10.1103/PhysRevA. 54.1098

    work page doi:10.1103/physreva 1996

  9. [9]

    Yudong Cao, Jonathan Romero, Jonathan P Olson, Matthias Degroote, Peter D Johnson, Mária Kieferová, Ian D Kivlichan, Tim Menke, Borja Peropadre, Nico- las PD Sawaya, et al. 2019. Quantum chemistry in the age of quantum computing. Chemical reviews119, 19 (2019), 10856–10915

    2019

  10. [10]

    Jinghu Chen and Marc P. C. Fossorier. 2002. Density evolution for two improved BP-Based decoding algorithms of LDPC codes.IEEE Commun. Lett.6, 5 (2002), 208–210. https://doi.org/10.1109/4234.1001666

    work page doi:10.1109/4234.1001666 2002

  11. [11]

    R. Chen, S. Siriyal, and V. Prasanna. 2015. Energy and Memory Efficient Mapping of Bitonic Sorting on FPGA. InProceedings of the 2015 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays. ACM, 240–249

    2015

  12. [12]

    James CL Chow. 2024. Quantum computing in medicine.Medical Sciences12, 4 (2024), 67

    2024

  13. [13]

    David G Cory, Mark D Price, Wojciech Maas, Emanuel Knill, Raymond Laflamme, Wojciech H Zurek, Timothy F Havel, and Shyamal S Somaroo. 1998. Experimental quantum error correction.Physical Review Letters81, 10 (1998), 2152

    1998

  14. [14]

    Ben Criger and Imran Ashraf. 2018. Multi-path Summation for Decoding 2D Topological Codes.Quantum2 (2018), 102. https://doi.org/10.22331/Q-2018-10- 19-102

    work page doi:10.22331/q-2018-10- 2018

  15. [15]

    Poulami Das, Aditya Locharla, and Cody Jones. 2022. Lilliput: a lightweight low-latency lookup-table decoder for near-term quantum error correction. In Proceedings of the 27th ACM International Conference on Architectural Support for Programming Languages and Operating Systems. 541–553

    2022

  16. [16]

    Poulami Das, Christopher A Pattison, Srilatha Manne, Douglas M Carmean, Krysta M Svore, Moinuddin Qureshi, and Nicolas Delfosse. 2022. Afs: Accu- rate, fast, and scalable error-decoding for fault-tolerant quantum computers. In 2022 IEEE International Symposium on High-Performance Computer Architecture (HPCA). IEEE, 259–273. https://doi.org/10.1109/HPCA539...

    work page doi:10.1109/hpca53966.2022.00027 2022

  17. [17]

    Delfosse and N

    N. Delfosse and N. H. Nickerson. 2017. Almost-linear time decoding algorithm for topological codes. arXiv:1709.06218 [quant-ph] arXiv preprint arXiv:1709.06218

    work page arXiv 2017

  18. [18]

    Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. 2002. Topological quantum memory.J. Math. Phys.43, 9 (2002), 4452–4505. https://doi.org/10. 1063/1.1499754

    2002

  19. [19]

    Simon J Devitt, William J Munro, and Kae Nemoto. 2013. Quantum error correc- tion for beginners.Reports on Progress in Physics76, 7 (2013), 076001

    2013

  20. [20]

    Alberto Di Meglio, Karl Jansen, Ivano Tavernelli, Constantia Alexandrou, Srini- vasan Arunachalam, Christian W Bauer, Kerstin Borras, Stefano Carrazza, Ar- ianna Crippa, Vincent Croft, et al. 2024. Quantum computing for high-energy physics: State of the art and challenges.Prx quantum5, 3 (2024), 037001

    2024

  21. [21]

    Thomas JS Durant, Elizabeth Knight, Brent Nelson, Sarah Dudgeon, Seung J Lee, Dominic Walliman, Hobart P Young, Lucila Ohno-Machado, and Wade L Schulz

  22. [22]

    A primer for quantum computing and its applications to healthcare and biomedical research.Journal of the American Medical Informatics Association31, 8 (2024), 1774–1784

    2024

  23. [23]

    A. A. Emran and M. Elsabrouty. 2014. Simplified variable-scaled min-sum LDPC decoder for irregular LDPC codes. InProceedings of the IEEE Consumer Commu- nications and Networking Conference (CCNC). 518–523. https://doi.org/10.1109/ CCNC.2014.6940497

    work page arXiv 2014

  24. [24]

    Tiago M Fernandez-Carames and Paula Fraga-Lamas. 2020. Towards post- quantum blockchain: A review on blockchain cryptography resistant to quantum computing attacks.IEEE access8 (2020), 21091–21116

    2020

  25. [25]

    Ferris and David Poulin

    Andrew J. Ferris and David Poulin. 2014. Tensor Networks and Quantum Error Correction.Phys. Rev. Lett.113 (Jul 2014), 030501. Issue 3. https://doi.org/10. 1103/PhysRevLett.113.030501

    2014

  26. [26]

    M. P. C. Fossorier and S. Lin. 1995. Soft-decision decoding of linear block codes based on ordered statistics.IEEE Transactions on Information Theory41 (Sept. 1995), 1379–1396. https://doi.org/10.1109/18.412683

    work page doi:10.1109/18.412683 1995

  27. [27]

    Austin G. Fowler. 2013. Minimum weight perfect matching of fault-tolerant topological quantum error correction in average 𝑂( 1) parallel time.arXiv preprint arXiv:1307.1740(2013). https://arxiv.org/abs/1307.1740

    work page arXiv 2013

  28. [28]

    Austin G Fowler, Matteo Mariantoni, John M Martinis, and Andrew N Cleland

  29. [29]

    Surface codes: Towards practical large-scale quantum computation.Physical Review A—Atomic, Molecular, and Optical Physics86, 3 (2012), 032324

    2012

  30. [30]

    Fowler, Adam C

    Austin G. Fowler, Adam C. Whiteside, and Lloyd CL Hollenberg. 2012. Towards practical classical processing for the surface code.Physical Review Letters108, 18 (2012), 180501. https://doi.org/10.1103/PhysRevLett.108.180501

    work page doi:10.1103/physrevlett.108.180501 2012

  31. [31]

    Crespo, and Javier Garcia- Frías

    Patricio Fuentes, Josu Etxezarreta Martinez, Pedro M. Crespo, and Javier Garcia- Frías. 2021. Degeneracy and Its Impact on the Decoding of Sparse Quantum Codes. IEEE Access9 (2021), 89093–89119. https://doi.org/10.1109/ACCESS.2021.3089829

    work page doi:10.1109/access.2021.3089829 2021

  32. [32]

    Fowler, and Michael R

    Joydip Ghosh, Austin G. Fowler, and Michael R. Geller. 2012. Surface code with decoherence: An analysis of three superconducting architectures.Physical Review A86, 6 (2012), 062318. https://doi.org/10.1103/PhysRevA.86.062318

    work page doi:10.1103/physreva.86.062318 2012

  33. [33]

    Craig Gidney. 2021. Stim: a fast stabilizer circuit simulator.Quantum5 (July 2021), 497. https://doi.org/10.22331/q-2021-07-06-497

    work page doi:10.22331/q-2021-07-06-497 2021

  34. [34]

    Anqi Gong, Sebastian Cammerer, and Joseph M. Renes. 2024. Graph Neural Networks for Enhanced Decoding of Quantum LDPC Codes. InIEEE International Symposium on Information Theory, ISIT 2024, Athens, Greece, July 7-12, 2024. IEEE, 2700–2705. https://doi.org/10.1109/ISIT57864.2024.10619589

    work page doi:10.1109/isit57864.2024.10619589 2024

  35. [35]

    PhD thesis, California Institute of Technology, 1997.doi:10.7907/rzr7-dt72

    Daniel Gottesman. 1997.Stabilizer Codes and Quantum Error Correction. Ph. D. Dissertation. California Institute of Technology. https://doi.org/10.7907/rzr7-dt72

    work page doi:10.7907/rzr7-dt72 1997

  36. [36]

    Antoine Grospellier, Lucien Grouès, Anirudh Krishna, and Anthony Leverrier

  37. [37]

    https://doi.org/10.22331/Q-2021-04-15-432

    Combining hard and soft decoders for hypergraph product codes.Quantum 5 (2021), 432. https://doi.org/10.22331/Q-2021-04-15-432

    work page doi:10.22331/q-2021-04-15-432 2021

  38. [38]

    Nils Herrmann, Daanish Arya, Marcus W Doherty, Angus Mingare, Jason C Pillay, Florian Preis, and Stefan Prestel. 2023. Quantum utility–definition and as- sessment of a practical quantum advantage. In2023 IEEE International Conference on Quantum Software (QSW). IEEE, 162–174

    2023

  39. [39]

    Breuckmann

    Oscar Higgott and Nikolas P. Breuckmann. 2023. Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes.PRX Quantum4 (May 2023), 020332. Issue 2. https://doi.org/10.1103/PRXQuantum.4.020332

    work page doi:10.1103/prxquantum.4.020332 2023

  40. [40]

    Oscar Higgott and Craig Gidney. 2025. Sparse Blossom: correcting a million errors per core second with minimum-weight matching.Quantum9 (2025), 1600. https://doi.org/10.22331/Q-2025-01-20-1600 12 Lottery BP : Unlocking Quantum Error Decoding at Scale Conference’17, July 2017, Washington, DC, USA

    work page doi:10.22331/q-2025-01-20-1600 2025

  41. [41]

    Adam Holmes, Mohammad Reza Jokar, Ghasem Pasandi, Yongshan Ding, Mas- soud Pedram, and Frederic T Chong. 2020. NISQ+: Boosting quantum computing power by approximating quantum error correction. In2020 ACM/IEEE 47th annual international symposium on computer architecture (ISCA). IEEE, 556–569

    2020

  42. [42]

    Min-Hsiu Hsieh and François Le Gall. 2011. NP-hardness of decoding quantum error-correction codes.Physical Review A83, 5 (May 2011), 052331. https: //doi.org/10.1103/PhysRevA.83.052331

    work page doi:10.1103/physreva.83.052331 2011

  43. [43]

    Tzu-Hsuan Huang, Ting-An Hu, and Yeong-Luh Ueng. 2023. Branch-assisted sign-flipping belief propagation decoding for topological quantum codes based on hypergraph product structure.IEEE Transactions on Quantum Engineering4 (2023), 1–15. https://doi.org/10.1109/TQE.2023.3279379

    work page doi:10.1109/tqe.2023.3279379 2023

  44. [44]

    Richard Jozsa and Noah Linden. 2003. On the role of entanglement in quantum- computational speed-up.Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences459, 2036 (2003), 2011–2032

    2003

  45. [45]

    Emanuel Knill. 2005. Quantum computing with realistically noisy devices.Nature 434, 7029 (2005), 39–44

    2005

  46. [46]

    K. Y. Kuo and C. Y. Lai. 2020. Refined belief propagation decoding of sparsegraph quantum codes.IEEE J. Sel. Area. Inf. Theory1, 2 (Aug 2020), 487–498. https: //doi.org/10.1109/JSAIT.2020.3011758

    work page doi:10.1109/jsait.2020.3011758 2020

  47. [47]

    Kao-Yueh Kuo and Ching-Yi Lai. 2022. Exploiting degeneracy in belief propa- gation decoding of quantum codes.npj Quantum Information8, 1 (2022), 111. https://doi.org/10.1038/s41534-022-00623-2

    work page doi:10.1038/s41534-022-00623-2 2022

  48. [48]

    Ching-Yi Lai and Kao-Yueh Kuo. 2021. Log-domain decoding of quantum LDPC codes over binary finite fields.IEEE Transactions on Quantum Engineering2 (2021), 1–15. https://doi.org/10.1109/TQE.2021.3113936

    work page doi:10.1109/tqe.2021.3113936 2021

  49. [49]

    Andrew J Landahl, Jonas T Anderson, and Patrick R Rice. 2011. Fault-tolerant quantum computing with color codes.arXiv preprint arXiv:1108.5738(2011)

    work page arXiv 2011

  50. [50]

    Moritz Lange, Pontus Havström, Basudha Srivastava, Isak Bengtsson, Valde- mar Bergentall, Karl Hammar, Olivia Heuts, Evert van Nieuwenburg, and Mats Granath. 2025. Data-driven decoding of quantum error correcting codes using graph neural networks.Physical Review Research7, 2 (2025), 023181

    2025

  51. [51]

    Benjamin P Lanyon, James D Whitfield, Geoff G Gillett, Michael E Goggin, Marcelo P Almeida, Ivan Kassal, Jacob D Biamonte, Masoud Mohseni, Ben J Powell, Marco Barbieri, et al. 2010. Towards quantum chemistry on a quantum computer.Nature chemistry2, 2 (2010), 106–111

    2010

  52. [52]

    Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. 2015. Deep learning.Nature 521, 7553 (2015), 436–444. https://doi.org/10.1038/nature14539

    work page doi:10.1038/nature14539 2015

  53. [53]

    Ye-Hua Liu and David Poulin. 2019. Neural belief-propagation decoders for quantum error-correcting codes.Physical review letters122, 20 (2019), 200501

    2019

  54. [54]

    MacKay and Radford M

    David J.C. MacKay and Radford M. Neal. 1997. Near Shannon Limit Performance of Low-Density Parity-Check Codes.Electronics Letters33, 6 (March 1997), 457–

    1997

  55. [55]

    https://doi.org/10.1049/el:19961141

    work page doi:10.1049/el:19961141

  56. [56]

    David J. C. MacKay, Graeme Mitchison, and Peter L. McFadden. 2004. Sparse- graph codes for quantum error correction.IEEE Trans. Inf. Theory50 (2004), 2315–2330. https://doi.org/10.1109/TIT.2004.834737

    work page doi:10.1109/tit.2004.834737 2004

  57. [57]

    Ryan Mandelbaum. 2025. Scaling for quantum advantage and beyond. IBM Quantum Blog. https://www.ibm.com/quantum/blog/qdc-2025 Accessed: 2026- 02-22

    2025

  58. [58]

    Satvik Maurya, Joshua Viszlai, Nithin Raveendran, Poulami Das, and Swamit Tannu. 2025. decoder-bench: Benchmarking Decoders for Quantum Error Correc- tion. In2025 IEEE International Symposium on Workload Characterization (IISWC). IEEE, 286–295. https://doi.org/10.1109/iiswc66894.2025.00032

    work page doi:10.1109/iiswc66894.2025.00032 2025

  59. [59]

    Vasileios Mavroeidis, Kamer Vishi, Mateusz D Zych, and Audun Jøsang. 2018. The impact of quantum computing on present cryptography.arXiv preprint arXiv:1804.00200(2018)

    work page arXiv 2018

  60. [60]

    Tristan Müller, Thomas Alexander, Michael E Beverland, Markus Bühler, Blake R Johnson, Thilo Maurer, and Drew Vandeth. 2025. Improved belief propaga- tion is sufficient for real-time decoding of quantum memory.arXiv preprint arXiv:2506.01779(2025)

    work page arXiv 2025

  61. [61]

    NVIDIA. 2023. CUDA-Q: The Platform for Integrated Quantum-Classical Comput- ing.arXiv preprint(2023). https://www.researchgate.net/publication/373970091

    work page arXiv 2023

  62. [62]

    NVIDIA. 2026. CUDA-Q QEC - Quantum Error Correction Library. https: //nvidia.github.io/cudaqx/components/qec/introduction.html. Accessed: 2026- 03-28

    2026

  63. [63]

    Pavel Panteleev and Gleb Kalachev. 2021. Degenerate quantum LDPC codes with good finite length performance.Quantum5 (2021), 585

    2021

  64. [64]

    Pavel Panteleev and Gleb Kalachev. 2021. Degenerate quantum LDPC codes with good finite length performance.Quantum5 (2021), 585. https://doi.org/10.22331/ q-2021-11-22-585

    2021

  65. [65]

    Michael A. Perlin. 2023. qLDPC. https://github.com/qLDPCOrg/qLDPC

    2023

  66. [66]

    M. S. Postol. 2001. A proposed quantum low density parity check code. https: //doi.org/10.48550/arXiv.quant-ph/0108131

    work page doi:10.48550/arxiv.quant-ph/0108131 2001

  67. [67]

    David Poulin. 2005. Stabilizer formalism for operator quantum error correction. Physical Review Letters95, 23 (2005), 230504. https://doi.org/10.1103/PhysRevLett. 95.230504

    work page doi:10.1103/physrevlett 2005

  68. [68]

    Poulin and Y

    D. Poulin and Y. Chung. 2008. On the iterative decoding of sparse quantum codes. Quantum Inf. Comput.8 (2008), 987–1000. https://doi.org/10.48550/arXiv.0801. 1241

    work page doi:10.48550/arxiv.0801 2008

  69. [69]

    2025.Deltakit

    Guen Prawiroatmodjo, Angela Burton, Adrien Suau, Chidi Nnadi, Abbas Bracken Ziad, Adam Melvin, Adam Richardson, Adnaan Walayat, Alex Moylett, Alise Virbule, AmirReza Safehian, Andrew Patterson, Anton Buyskikh, Archi Ruben, Ben Barber, Brendan Reid, Cai Rees Manuel, Dan Seremet, David Byfield, Elisha Matekole, Gabriel Gallardo, Gyorgy Geher, Jack Turner, J...

    work page doi:10.5281/zenodo.17145113 2025

  70. [70]

    John Preskill. 2018. Quantum computing in the NISQ era and beyond.Quantum 2 (2018), 79

    2018

  71. [71]

    John Preskill. 2025. Beyond nisq: The megaquop machine. , 7 pages

    2025

  72. [72]

    Daniel Price, Prabhu Vellaisamy, Patricia Gonzalez, George Michelogiannakis, John P Shen, and Di Wu. 2026. A-Graph: A Unified Graph Representation for At-Will Simulation across System Stacks.arXiv preprint arXiv:2602.04847(2026). https://doi.org/10.48550/arXiv.2602.04847

    work page doi:10.48550/arxiv.2602.04847 2026

  73. [73]

    Nithin Raveendran and Bane Vasić. 2021. Trapping sets of quantum LDPC codes. Quantum5 (2021), 562. https://doi.org/10.22331/q-2021-10-14-562

    work page doi:10.22331/q-2021-10-14-562 2021

  74. [74]

    Baker, Arash Fayyazi, Sophia Fuhui Lin, Ali Javadi-Abhari, Massoud Pedram, and Frederic T

    Gokul Subramanian Ravi, Jonathan M. Baker, Arash Fayyazi, Sophia Fuhui Lin, Ali Javadi-Abhari, Massoud Pedram, and Frederic T. Chong. 2023. Better Than Worst-Case Decoding for Quantum Error Correction. InInternational Conference on Architectural Support for Programming Languages and Operating Systems

    2023

  75. [76]

    Decoding across the quan- tum low-density parity-check code landscape

    Joschka Roffe, David R. White, Simon Burton, and Earl Campbell. 2020. Decoding across the quantum low-density parity-check code landscape.Physical Review Research2, 4 (Dec 2020). https://doi.org/10.1103/physrevresearch.2.043423

    work page doi:10.1103/physrevresearch.2.043423 2020

  76. [77]

    Salamat, A

    S. Salamat, A. Haj Aboutalebi, B. Khaleghi, J. H. Lee, Y. S. Ki, and T. Rosing. 2021. NASCENT: Near-Storage Acceleration of Database Sort on SmartSSD. InPro- ceedings of the 2021 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays. ACM, 262–272

    2021

  77. [78]

    Ashley M Stephens. 2014. Fault-tolerant thresholds for quantum error correction with the surface code.Physical Review A89, 2 (2014), 022321

    2014

  78. [79]

    Barbara M Terhal. 2015. Quantum error correction for quantum memories. Reviews of Modern Physics87, 2 (2015), 307–346

    2015

  79. [80]

    Tillich and G

    J.-P. Tillich and G. Zemor. 2014. Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength.IEEE Transactions on Information Theory60, 2 (Feb. 2014), 1193–1202. https://doi.org/ 10.1109/TIT.2013.2292061

    work page doi:10.1109/tit.2013.2292061 2014

  80. [81]

    Suhas Vittal, Poulami Das, and Moinuddin Qureshi. 2023. Astrea: Accurate quantum error-decoding via practical minimum-weight perfect-matching. In Proceedings of the 50th Annual International Symposium on Computer Architecture. 1–16. https://doi.org/10.1145/3579371.3589037

    work page doi:10.1145/3579371.3589037 2023

Showing first 80 references.