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REVIEW 3 major objections 6 minor 51 references

Union-Intersection Union-Find for Decoding Depolarizing Errors in Topological Codes

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read UIUF provably corrects depolarizing errors on toric and surface codes up to half the code distance in linear time, and outperforms MWPM on rotated surface codes in simulations.

desk verdict A genuinely new UF variant with strong simulations, but the half-distance proof has a load-bearing gap: diameter is substituted for erasure count, and the diameter bound itself is asserted, not proved. read the letter →

arxiv 2506.14745 v1 pith:5V3NOJUW submitted 2025-06-17 quant-ph cs.ITmath.IT

Tzu-Hao Lin , Ching-Yi Lai This is my paper · ORCID
classification quant-phcs.ITmath.IT MSC 81P7081P68 PACS 03.67.Pp03.67.Lx
keywords quantumerrorcorrectionUnion-FinddecoderdepolarizingerrorstopologicalcodessurfaceX/Zcorrelationerasuredecodinglinear-time
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

The paper introduces the Union-Intersection Union-Find (UIUF) decoder for depolarizing errors in toric and surface codes. Its aim is to combine the standard Union-Find decoder's proven half-distance guarantee with the better practical performance of iterative Union-Find decoders. UIUF first runs syndrome validation separately on the X and Z decoding graphs, then marks any qubit whose edge is covered by a cluster in both graphs as an erasure, and finally reruns Union-Find with these extra erasures. The paper proves that the decoder corrects r erasure errors plus an additional Pauli error of weight t whenever $r+2t

What carries the argument

The load-bearing object is the cluster produced by syndrome validation: a connected component of check nodes in a decoding graph, grown until it is consistent with some error matching the observed syndrome. The new mechanism is the intersection step, in which a qubit is added to the erasure set whenever the edge it represents in $G_X$ lies inside a cluster of $L_X$ and the corresponding edge in $G_Z$ lies inside a cluster of $L_Z$. This converts likely Y-error locations into known erasures before the final Union-Find calls, which is how the decoder uses X/Z correlations without iterating. The proof then uses cluster-diameter bounds, at most $2t+r$ for a Pauli error of weight $t$ with $r$ erasures, to show the enlarged erasure set still satisfies the erasure-plus-error condition of the standard Union-Find guarantee, so Lemma 6 applies.

What would settle it

Enumerate all Pauli errors of weight $\lfloor (d-1)/2\rfloor$ on a small code where the bound matters, such as the [[25,1,4]] surface code or the [[72,2,6]] toric code, and run UIUF on each syndrome; a single logical failure would disprove Theorem 8. To isolate the proof's weak step, construct two valid clusters, one in $G_X$ and one in $G_Z$, whose intersection as edge sets has diameter strictly larger than the smaller of the two cluster diameters; that would contradict the diameter claim used to derive the inequalities in the proof.

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Extended reading notes

Core claim

The central discovery, stated on the paper's own terms, is that a non-iterative decoder can exploit X/Z correlations and still keep a distance guarantee. The UIUF procedure is: syndrome-validation on each decoding graph produces a collection of valid clusters; the intersection step marks a qubit as an erasure exactly when the edges it represents in $G_X$ and $G_Z$ are both covered by their respective clusters; two ordinary Union-Find calls then produce the corrections using the enlarged erasure set. Because a Y error contributes to both decoding graphs, its position tends to be covered by both clusters and is thereby turned into a known erasure, which makes the remaining X and Z subproblems easier. Theorem 8 states that on a distance-$d$ toric or surface code, UIUF corrects up to r erasure errors plus a Pauli error of weight t on the remaining qubits whenever $r+2t<d$; in particular it corrects all errors of weight at most $\lfloor (d-1)/2\rfloor$. The complexity remains $O(n)$, and the proof works by bounding the diameter of the intersection erasure set by the smaller of the two cluster diameters plus $r$, which reduces the required condition to the same $r+2t<d$ as the standard Union-Find guarantee.

Load-bearing premise

The proof of Theorem 8 rests on the asserted, not separately proved, bound that the erasure set formed by intersecting an X-cluster and a Z-cluster has diameter no larger than the smaller of the two cluster diameters; if that diameter bound gives way, the half-distance guarantee does not follow.

Editorial extensions

If this is right

  • UIUF corrects every error of weight at most $\lfloor (d-1)/2\rfloor$ on a distance-$d$ toric or surface code, giving it the same half-distance guarantee as the standard Union-Find decoder.
  • The guarantee is stated for the mixed setting of $r$ erasures plus $t$ Pauli errors with $r+2t<d$, so the decoder also applies directly to erasure-assisted decoding problems.
  • The decoder keeps $O(n)$ worst-case complexity, with measured average runtime about 1.53 times that of Union-Find, so the performance gain does not sacrifice linear-time decoding.
  • On rotated surface codes, UIUF with weighted growth achieves lower logical error rates than MWPM for distance at least 7 under both code-capacity and phenomenological noise, even where its phenomenological threshold is lower than MWPM's.
  • Under biased noise with $p_Y=p_X$, UIUF's advantage over Union-Find persists for bias ratios below 100 and disappears above about 1000.

Reading between the lines

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

  • The diameter-based structure of the proof suggests the guarantee likely transfers to other bounded-degree CSS topological codes whose decoding graphs obey the same cluster-diameter bound, so UIUF is a natural candidate for other lattice geometries.
  • The intersection erasure set could be exported as a preprocessing step for other decoders such as reweighted MWPM or belief propagation; one could test whether marking the same qubits as erasures improves their low-error-rate performance without iteration.
  • Because UIUF's advantage over MWPM concentrates at low error rates and larger distances, a practical decoder might use UIUF's fast clustering for most shots and invoke a more expensive decoder only when the intersection step produces an unusually large or ambiguous erasure set.
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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 / 6 minor

Summary. The paper proposes the Union-Intersection Union-Find (UIUF) decoder for CSS topological codes, which uses the intersection of syndrome-validation clusters in the X and Z decoding graphs as an erasure set before running two UF decoding steps. The authors claim that UIUF corrects all errors of weight up to half the code distance while retaining O(n) time complexity, and they support this with extensive simulations under code-capacity, phenomenological, and biased noise models, reporting logical error rates below UF and, on rotated surface codes, below MWPM, as well as threshold estimates.

Significance. If the distance guarantee of Theorem 8 is valid, UIUF is a valuable practical decoder: it offers an order-of-magnitude improvement in logical error rate over UF in the low-error regime, a provable worst-case linear-time complexity, and a parameter-free algorithm with no fitted constants. The simulation study is broad (four code families, three noise models, runtime benchmarks) and the comparisons with standard UF and PyMatching-based MWPM are informative. However, the central theoretical contribution is Theorem 8, and the correctness of its proof is essential to the paper's claims. The manuscript's proof of Theorem 8 has significant gaps, detailed below; the numerical results alone cannot establish the claimed distance guarantee. With a complete proof, this would be a strong contribution; in its current form, the main theoretical claim is not rigorously supported.

major comments (3)
  1. [§IV-A, proof of Theorem 8] The proof asserts without proof that the set I of qubits covered by both an LX cluster and an LZ cluster has connected components of diameter at most min{2s+2q, 2t−2s}+r. A connected component of I may be formed by qubits whose GX-edges belong to one cluster while their GZ-edges belong to several different clusters (and vice versa); it need not be contained in any single GX or GZ cluster. The manuscript does not provide a bound for such alternating components, so the inequalities in Eqs. (2) and (3) are not justified.
  2. [§IV-A, invocation of Lemma 6] Lemma 6 is stated for r erasure qubits, i.e., an erasure count, not for an erasure set of bounded diameter. The proof of Theorem 8 substitutes a claimed diameter bound for the erasure count and then invokes Lemma 6 without deriving a separate statement that a UF decoder corrects t errors when the erasure set has diameter D and D+2t<d. Such a lemma is plausible and may follow from the methods of [13], but it is not stated or proved, so the logical step from diameter to correction is missing.
  3. [§IV-A, coverage of Y errors] The sentence 'Note that the Y errors are now covered by erasures' is central to the proof, but no argument is given that every Y error's edges are always contained in the final LX and LZ clusters. If a Y error lies outside either cluster, the remaining error counts in the subsequent UF subroutines exceed s or t−s−q, and the diameter bounds (2) and (3) no longer apply. The proof needs an explicit argument that the syndrome-validation clusters cover the support of every Y error.
minor comments (6)
  1. [§II-B] The word 'deomposed' should be 'decomposed'.
  2. [§I] The phrase 'Then we introduced the UIUF decoder' should be 'Then we introduce the UIUF decoder'.
  3. [Abstract and §V-C] The abstract claims that UIUF achieves lower logical error rates than MWPM on rotated surface codes under both noise models without qualification; in Figures 13 and 14, MWPM performs better for small distances (d=3,5) and, in the phenomenological model, for d=13 above ϵ=0.03. The abstract should state the regime where UIUF is superior.
  4. [Table II] The caption says 'total errors' but it is unclear what set of errors is counted; please clarify whether these are all weight-(t+1) Pauli errors and specify the counting convention.
  5. [§IV-A] The term 'diameter' is used throughout the proof of Theorem 8 but never formally defined; please specify the graph metric (decoding graph or lattice) and whether it applies to vertices or edges of the clusters and erasure sets.
  6. [Algorithm 1 and Figure 8] The notation for the iteration limit is inconsistent: 'iter max' in Algorithm 1, 'itermax' in Figures 8 and the text. Please unify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: UIUF's distance guarantee rests on external UF/peeling lemmas and simulation thresholds are fitted outputs, not fitted inputs.

full rationale

The derivation chain is self-contained relative to external prior work. Theorem 8's guarantee is built on Lemma 3 (the Delfosse–Zemor peeler) and Lemma 6 (the Delfosse–Nickerson Union-Find decoder), both cited from outside the authors' own prior results, so the central claim is not justified by a self-citation chain. Algorithm 2 contains no fitted constants: the intersection erasure set is defined directly from the valid clusters produced by syndrome validation, and the error-correction claim is argued from cluster-diameter bounds plus the external Lemma 6. All reported thresholds are outputs of Monte Carlo data fitting to the finite-size scaling ansatz pL = f(d^{1/ν}(ε−τ)), so the thresholds are not inputs disguised as predictions or renamed outputs. The self-citations that do appear ([26], [27], [45], [46], [47]) are used only for background on belief propagation, for the phenomenological-noise model definition, and for the logical-error-rate formula; none carries the load-bearing decoding guarantee. One genuine weakness should be flagged, but it is not circularity: in Section IV-A the proof asserts, without proof, that the intersection erasure set has diameter at most min{2s+2q, 2t−2s}+r, and it applies Lemma 6 using this diameter bound rather than Lemma 6's stated erasure-count condition r+2t<d. That is a possible correctness gap in the proof of Theorem 8, but it does not reduce the theorem to its own assumptions, to a fitted parameter, or to a self-citation. Because the central claim does not reduce by construction to its inputs, the circularity score is 0.

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

The central theoretical claim rests on the standard UF and peeling decoder guarantees from prior work, plus one unproved geometric assertion about the intersection of clusters. No fitted parameters or new physical entities are introduced; the simulation thresholds are outputs of data fitting, not inputs to the decoder.

assumptions (4)
  • domain assumption Union-Find decoder guarantee (Lemma 6 of [13])
    Invoked in the proof of Theorem 8 to conclude that UIUF corrects the error once the clusters are shown not to cover a logical operator.
  • domain assumption Peeling decoder erasure-correction capability (Lemma 3 of [12])
    Supports the erasure-decoding step used inside the UF subroutine.
  • domain assumption Cluster diameter bound after syndrome validation is at most 2t+r for t errors and r erasures
    Taken from the analysis of the standard UF decoder [13] and used to bound the sizes of LX and LZ clusters in Theorem 8.
  • ad hoc to paper Connected components of the intersection of LX and LZ clusters have diameter at most the minimum of the two cluster diameters
    Asserted in the proof of Theorem 8 without a separate proof; the authors do not explain why any two qubits in the same intersection component must lie in the same pair of clusters in both graphs.

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Pith. "Pith review of Union-Intersection Union-Find for Decoding Depolarizing Errors in Topological Codes." pith.science (2026). https://pith.science/paper/5V3NOJUW

@misc{pith2026250614745,
  author       = {Pith},
  title        = {Pith review of: Union-Intersection Union-Find for Decoding Depolarizing Errors in Topological Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5V3NOJUW}},
  note         = {Machine review of arXiv:2506.14745}
}
abstract

In this paper, we introduce the Union-Intersection Union-Find (UIUF) algorithm for decoding depolarizing errors in topological codes, combining the strengths of iterative and standard Union-Find (UF) decoding. While iterative UF improves performance at moderate error rates, it lacks an error correction guarantee. To address this, we develop UIUF, which maintains the enhanced performance of iterative UF while ensuring error correction up to half the code distance. Through simulations under code capacity, phenomenological, and biased noise models, we show that UIUF significantly outperforms UF, reducing the logical error rate by over an order of magnitude (at around $10^{-5}$). Moreover, UIUF achieves lower logical error rates than the Minimum Weight Perfect Matching (MWPM) decoder on rotated surface codes under both the code capacity and phenomenological noise models, while preserving efficient linear-time complexity.

Figures

Figures reproduced from arXiv: 2506.14745 by the authors.

Figure 1
Figure 1. (a) Layout graph of the surface code for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The decoding graph GX consists of the red check nodes and solid lines, whereas the decoding graph GZ consists of the blue check nodes and dashed lines. C. Peeling decoder Delfosse and Zemor proposed a Peeling decoder for surface ´ and toric codes that corrects erasure errors [12]. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. UF decoding for a toric code. (a) The two thick blue [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: illustrates the joint decoding graph for the [[9, 1, 3]] rotated surface code under the phenomenological noise model. At each layer, a check node is nontrivial if its syndrome difference with the previous round is nontrivial. Since a measurement error affects the syndr…
Figure 5
Figure 5. Figure 5: An example of IRUF decoding for a weight-3 error [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Applying syndrome validation and the intersection step [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: The number of undecodable errors of weights [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Comparison of the decoding performance of UF, UIUF, [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 10
Figure 10. Figure 10: Comparison of the decoding performance of the UF [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 9
Figure 9. Figure 9: Comparison of the decoding performance of the UF, [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 12
Figure 12. Figure 12: Estimated threshold in the phenomenological noise [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]
Figure 11
Figure 11. Figure 11: Estimated thresholds in the code capacity noise model [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 14
Figure 14. Figure 14: Comparison of the decoding performance between [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]
Figure 13
Figure 13. Figure 13: Comparison of the decoding performance between [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]
Figure 15
Figure 15. Figure 15: Comparison of the UF and UIUF decoders for the [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]
Figure 16
Figure 16. Figure 16: Decoding time comparison for the three decoders on [PITH_FULL_IMAGE:figures/full_fig_p012_16.png]

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Works this paper leans on

51 extracted references · 39 canonical work pages

  1. [13]

    Almost-linear time decoding algo- rithm for topological codes,

    N. Delfosse and N. H. Nickerson, “Almost-linear time decoding algo- rithm for topological codes,” Quantum, vol. 5, p. 595, Dec. 2021

    work page 2021

  2. [1]

    Algorithms for quantum computation: Discrete logarithms and factoring,

    P. W. Shor, “Algorithms for quantum computation: Discrete logarithms and factoring,” in Proc. Annu. Symp. Found. Comput. Sci. (FOCS), 1994, pp. 124–134

    work page 1994

  3. [2]

    A fast quantum mechanical algorithm for database search,

    L. K. Grover, “A fast quantum mechanical algorithm for database search,” in Proc. Annu. ACM Symp. Theory Comput. (STOC) , 1996, pp. 212–219

    work page 1996

  4. [3]

    Scheme for reducing decoherence in quantum computer memory,

    P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A, vol. 52, pp. 2493–2496, 1995

    work page 1995

  5. [4]

    Good quantum error-correcting codes exist,

    A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,” Phys. Rev. A, vol. 54, p. 1098, 1996

    work page 1996

  6. [5]

    Error correcting codes in quantum theory,

    A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev. Lett., vol. 77, p. 793, 1996

    work page 1996

  7. [6]

    Stabilizer codes and quantum error correction,

    D. Gottesman, “Stabilizer codes and quantum error correction,” Ph.D. dissertation, California Institute of Technology, Pasadena, CA, USA,

    work page

  8. [7]

    Fault-tolerant quantum computation by anyons,

    A. Y . Kitaev, “Fault-tolerant quantum computation by anyons,” Ann. Phys., vol. 303, no. 1, pp. 2–30, 2003, arXiv preprint quant-ph/9707021

    arXiv 2003

Show all 51 references
  1. [8]

    Quantum codes on a lattice with boundary,

    S. B. Bravyi and A. Y . Kitaev, “Quantum codes on a lattice with boundary,” arXiv preprint quant-ph/9811052 , 1998

    1998 arXiv

  2. [9]

    Optimal resources for topolog- ical two-dimensional stabilizer codes: Comparative study,

    H. Bombin and M. A. Martin-Delgado, “Optimal resources for topolog- ical two-dimensional stabilizer codes: Comparative study,” Phys. Rev. A, vol. 76, p. 012305, 2007

    2007

  3. [10]

    Surface codes: Towards practical large-scale quantum computation,

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,” Phys. Rev. A, vol. 86, p. 032324, 2012

    2012

  4. [11]

    Topological quantum memory,

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,” J. Math. Phys. , vol. 43, no. 9, pp. 4452–4505, 2002

    2002

  5. [12]

    Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel,

    N. Delfosse and G. Z ´emor, “Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel,” Phys. Rev. Res. , vol. 2, p. 033042, Jul 2020

    2020

  6. [14]

    Paths, trees, and flowers,

    J. Edmonds, “Paths, trees, and flowers,” Can. J. Math. , vol. 17, pp. 449–467, 1965. 12

    1965

  7. [15]

    An interpretation of union-find decoder on weighted graphs,

    Y . Wu, N. Liyanage, and L. Zhong, “An interpretation of union-find decoder on weighted graphs,” 11 2022

    2022

  8. [16]

    Worst-case analysis of set union algorithms,

    R. E. Tarjan and J. van Leeuwen, “Worst-case analysis of set union algorithms,” J. ACM, vol. 31, no. 2, p. 245–281, Mar. 1984. [Online]. Available: https://doi.org/10.1145/62.2160

    1984 doi

  9. [17]

    Union-find quantum decoding without union-find,

    S. J. Griffiths and D. E. Browne, “Union-find quantum decoding without union-find,” Physical Review Research, vol. 6, p. 013154, Feb. 2024

    2024

  10. [18]

    Fault-tolerant weighted union-find decoding on the toric code,

    S. Huang, M. Newman, and K. R. Brown, “Fault-tolerant weighted union-find decoding on the toric code,” Physical Review A , vol. 102, no. 1, p. 012419, 2020

    2020

  11. [19]

    Scalable neural decoder for topological surface codes,

    K. Meinerz, C.-Y . Park, and S. Trebst, “Scalable neural decoder for topological surface codes,” Phys. Rev. Lett. , vol. 128, p. 080505, Feb

  12. [20]

    Fusion blossom: Fast mwpm decoders for qec,

    Y . Wu and L. Zhong, “Fusion blossom: Fast mwpm decoders for qec,” in 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 1. IEEE, 2023, pp. 928–938

    2023

  13. [21]

    Sparse blossom: correcting a million errors per core second with minimum-weight matching,

    O. Higgott and C. Gidney, “Sparse blossom: correcting a million errors per core second with minimum-weight matching,” Quantum, vol. 9, p. 1600, 2025

    2025

  14. [22]

    Bubble clustering decoder for quantum topological codes,

    D. Forlivesi, L. Valentini, and M. Chiani, “Bubble clustering decoder for quantum topological codes,” IEEE Transactions on Communications, pp. 1–1, 2025

    2025

  15. [23]

    A decoding algorithm for CSS codes using the X/Z correlations,

    N. Delfosse and J.-P. Tillich, “A decoding algorithm for CSS codes using the X/Z correlations,” in 2014 IEEE International Symposium on Information Theory, 2014, pp. 1071–1075

    2014

  16. [24]

    Low-density parity-check codes,

    R. G. Gallager, “Low-density parity-check codes,” IRE Trans. Inf. Theory, vol. 8, no. 1, pp. 21–28, 1962

    1962

  17. [25]

    Pearl, Probabilistic reasoning in intelligent systems: networks of plausible inference

    J. Pearl, Probabilistic reasoning in intelligent systems: networks of plausible inference. Kaufmann, 1988

    1988

  18. [26]

    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 J. Sel. Areas Inf. Theory , vol. 1, no. 2, pp. 487–498, 2020

    2020

  19. [27]

    Exploiting degeneracy in belief propagation decoding of quantum codes,

    ——, “Exploiting degeneracy in belief propagation decoding of quantum codes,” npj Quantum Inf., vol. 8, 2022, article no. 111. (See a complete version at https://arxiv.org/abs/2104.13659)

    2022 arXiv

  20. [28]

    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

    2021

  21. [29]

    Decoding across the quantum low-density parity-check code landscape,

    J. Roffe, D. R. White, S. Burton, and E. T. Campbell, “Decoding across the quantum low-density parity-check code landscape,” Phys. Rev. Res., vol. 2, p. 043423, 2020

    2020

  22. [30]

    On belief propagation decoding of quantum codes with quaternary reliability statistics,

    C.-F. Kung, K.-Y . Kuo, and C.-Y . Lai, “On belief propagation decoding of quantum codes with quaternary reliability statistics,” arXiv preprint arXiv:2305.03321, 2023

    2023 arXiv

  23. [31]

    Optimal complexity correction of correlated errors in the surface code,

    A. G. Fowler, “Optimal complexity correction of correlated errors in the surface code,” arXiv preprint arXiv:1310.0863 , 2013

    2013 arXiv

  24. [32]

    Pipelined correlated minimum weight perfect matching of the surface code,

    A. Paler and A. G. Fowler, “Pipelined correlated minimum weight perfect matching of the surface code,” Quantum, vol. 7, p. 1205, 2023

    2023

  25. [33]

    A modified MWPM decoding algorithm for quantum surface codes over depolarizing channels,

    Y . Yuan and C.-C. Lu, “A modified MWPM decoding algorithm for quantum surface codes over depolarizing channels,” arXiv preprint arXiv:2202.11239, 2022

    2022 arXiv

  26. [34]

    Performance enhancement of surface codes via recursive minimum-weight perfect- match decoding,

    A. d. iOlius, J. E. Martinez, P. Fuentes, and P. M. Crespo, “Performance enhancement of surface codes via recursive minimum-weight perfect- match decoding,” Physical Review A , vol. 108, no. 2, p. 022401, 2023

    2023

  27. [35]

    A real-time, scalable, fast and highly resource efficient decoder for a quantum computer,

    B. Barber, K. M. Barnes, T. Bialas, O. Bu ˘gdaycı, E. T. Campbell, N. I. Gillespie, K. Johar, R. Rajan, A. W. Richardson, L. Skoric, C. Topal, M. L. Turner, and A. B. Ziad, “A real-time, scalable, fast and highly resource efficient decoder for a quantum computer,” 2023. [Onlin...

    2023 arXiv

  28. [36]

    Hierarchical decoding to reduce hardware requirements for quantum computing,

    N. Delfosse, “Hierarchical decoding to reduce hardware requirements for quantum computing,” ArXiv, vol. abs/2001.11427, 2020. [Online]. Available: https://api.semanticscholar.org/CorpusID:210966495

    2001 arXiv

  29. [37]

    Scalable quantum error correction for surface codes using fpga,

    N. Liyanage, Y . Wu, A. Deters, and L. Zhong, “Scalable quantum error correction for surface codes using fpga,” in 2023 IEEE 31st Annual International Symposium on Field-Programmable Custom Computing Machines (FCCM), 2023, pp. 217–217

    2023

  30. [38]

    Actis: A Strictly Local Union–Find Decoder,

    T. Chan and S. C. Benjamin, “Actis: A Strictly Local Union–Find Decoder,” Quantum, vol. 7, p. 1183, Nov. 2023. [Online]. Available: https://doi.org/10.22331/q-2023-11-14-1183

    2023 doi

  31. [39]

    Threshold error rates for the toric and planar codes,

    D. S. Wang, A. G. Fowler, A. M. Stephens, and L. C. L. Hollenberg, “Threshold error rates for the toric and planar codes,” Quantum Inf. Comput., vol. 10, no. 5, pp. 456–469, 2010

    2010

  32. [40]

    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

  33. [41]

    Projective plane and planar quantum codes,

    M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes,” Foundations of Computational Mathematics, vol. 1, pp. 325–332, 2001

    2001

  34. [42]

    Fault-tolerant quantum computation with high threshold in two dimensions,

    R. Raussendorf and J. Harrington, “Fault-tolerant quantum computation with high threshold in two dimensions,” Phys. Rev. Lett. , vol. 98, p. 190504, 2007

    2007

  35. [43]

    Codes for the quantum erasure channel,

    M. Grassl, T. Beth, and T. Pellizzari, “Codes for the quantum erasure channel,” Physical Review A , vol. 56, no. 1, p. 33, 1997

    1997

  36. [44]

    Efficiency of a good but not linear set union algorithm,

    R. E. Tarjan, “Efficiency of a good but not linear set union algorithm,” J. ACM, vol. 22, no. 2, p. 215–225, apr 1975

    1975

  37. [45]

    Quantum data-syndrome codes,

    A. Ashikhmin, C.-Y . Lai, and T. A. Brun, “Quantum data-syndrome codes,” IEEE J. Sel. Areas Commun. , vol. 38, no. 3, pp. 449–462, 2020

    2020

  38. [46]

    Generalized quantum data-syndrome codes and belief propagation decoding for phenomenological noise,

    K.-Y . Kuo and C.-Y . Lai, “Generalized quantum data-syndrome codes and belief propagation decoding for phenomenological noise,” IEEE Transactions on Information Theory , vol. 71, no. 3, pp. 1824–1840, 2025

    2025

  39. [47]

    Entanglement-assisted quantum error- correcting codes with imperfect ebits,

    C.-Y . Lai and T. A. Brun, “Entanglement-assisted quantum error- correcting codes with imperfect ebits,” Phys. Rev. A, vol. 86, p. 032319, Sep 2012

    2012

  40. [48]

    T. M. Cover and J. A. Thomas, Elements of information theory . USA: Wiley-Interscience, 1991

    1991

  41. [49]

    Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory,

    C. Wang, J. Harrington, and J. Preskill, “Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory,” Ann. Phys., vol. 303, no. 1, pp. 31–58, 2003. 13

    2003

  42. [1997]

    Available: https://doi.org/10.7907/rzr7-dt72

    [Online]. Available: https://doi.org/10.7907/rzr7-dt72

    doi

  43. [2022]

    Available: https://link.aps.org/doi/10.1103/PhysRevLett

    [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett. 128.080505

    doi

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Reviewed August 15, 2026 · model on record in the stance chip above.